The authors acknowledge the review work and advices of Prof. Mátyás Hunyár, the typing of Mrs. Écsi Antalné, the drawings of Miss Ilona Wibling.
Tartalom
The electronic lecture notes entitled Drive Control is prepared for the MSc. students of the Budapest University of Technology and Economics, Faculty of Electrical Engineering and Informatics, Specialization of Electrical Machines and Drives. The lecture notes deals with the theoretical and practical investigation of the modern drivespecific and taskspecific control methods of the semiconductor electrical drives, containing power electronics and electric motors.
The topic assumes the basic knowledge of the electric machines, power electronics and control theory. Besides, for the investigation of the threephase semiconductor controlled drives, the knowledge of the Parkvector (spacevector) representation methods is definitely necessary. Those students, graduating at our BSc. courses in Electric Power Engineering specialization have these knowledges.
The figures of this lecture notes have been available for the students for many years as paper copy study material.
Besides the study purpose, this lecture notes can be well utilized by drive designers or specialists applying electric drives.
The content can be extended by the help of references at the end.
Budapest, 2011.
Dr.Schmidt István professor emeritus 
Dr.Veszprémi Károly professor 
In the Drive Control subject motion controls implemented by electric drives are investigated. Oneshaft equivalent (referred) model of the system can be derived in mechanically rigid and knockingfree drive chain. The referring can be done to the shaft of the electric motor or the mechanical load. Fig.1.1. shows the oneshaft model of a rotating system referred to the motor shaft.
In the figure: m is the motor torque, m_{t} is the load torque, θ_{m} is the motor inertia, θ_{t} is the inertia of the load, w is the angular speed of the motor shaft, α is its rotation angle. Assuming constant resultant inertia (θ=θ_{m}+θ_{t}), the motion equation is:
(1.1.a,b)
Where m_{d} is dynamic (accelerating) torque. Eq. (1.1.a) is the equivalent expression of Newton’s II. law for rotation. The motion parameters: the dw/dt angular acceleration, the w angular speed and the α rotation angle can be controlled by the motor torque m. The m_{t} load torque can be considered as disturbance.
A modern controlled drive contains an electric motor, a power electronic circuit and a drive controller (see Fig. 1.2).
The drive controller usually operates in subordinated structure. Assuming position control, Fig.1.3. shows the blockscheme of the drive control. The aim is tracking the α_{a} position reference.
According to the subordinated structure, the position controller provides the w_{a} speed reference, the speed controller provides the m_{a} torque reference, the torque controller provides the i_{a} current reference. The current controller controls the power electronics by the v control signal. α, w, m and i are the feedback signals of the angle, speed, torque and current respectively. The current control is drivespecific, the position control is driveindependent (taskspecific). The speed control is usually driveindependent, but there are drivespecific versions (sensorless). There is not explicit torque control, since it can be deduced to current control. There is always a current control. Its aim is to control the torque and perhaps the flux, furthermore to perform overload (overcurrent) protection.
The controlled electrical drives can be found in every modern equipment, in the developed country they consume approx. the half of the produced electric power.
First the drivespecific current and torque controls are investigated for DC, synchronous, induction and reluctance machines. Then the taskspecific speed and position controls are discussed and some practical applications are presented.
Among the synchronous and induction motor drives only the control of the modern, widely applied DC link frequency converterfed drives are investigated.
Tartalom
First the DC machines, then the supplying power electronics and finally the current controllers are discussed.
The separately excited, permanent magnet and series excited DC machines are applied in practice. Separately excited, compensated DC machine is assumed in the following. Its equivalent circuit in oneshaft model is shown in Fig.2.1.
The input signals of the motor are the u terminal voltage and the ϕ=f(i_{g}) flux, its output signals are the I armature current, the m toque, the w speed and the α angle, disturbance signal is the m_{t} load troque. Assuming ϕ=const. and θ=const., the drive equations are summarized in (2.12.5):
u(t)=Ri(t)+L·di(t)/dt+u_{b}(t), 
u(s)=Ri(s)+Lsi(s)+u_{b}(s). 
(2.1.a,b) 
u_{b}(t)=kϕw(t), 
u_{b}(s)=kϕw(s). 
(2.2.a,b) 
m(t)m_{t}(t)=θ·dw(t)/dt, 
m(s)m_{t}(s)=θsw(s). 
(2.3.a,b) 
m(t)=kϕi(t), 
m(s)=kϕi(s). 
(2.4.a,b) 
w(t)=dα(t)/dt, 
w(s)=sα(s). 
(2.5.a,b) 
Column “a” is a time dependent linear differential equation system, column “b” is an algebraic equation system with Laplace transforms. Considering eqs. (2.2.a és 2.4.a) the p_{m}=mw mechanical power can be expressed as p_{m}=u_{b}i, since the brush, core, friction and ventilation losses are neglected at the rotor. In steadystate: di/dt=0 and dw/dt=0. Considering these, from the expressions of column “a” the W(M) mechanical characteristic of the drive can be expressed:
(2.6.a,b)
Accordingly, for a given M torque the w speed can be modified by U, ϕ and R (the last one is lossy). In controlled drives mainly the U terminal voltage (the W_{0}=U/kϕ noload speed) is modified. By the ϕ flux (fieldweakening) the speed range can be extended. Assuming 4/4 quadrant operation Fig.1.2. presents the linear W(M) mechanical characteristics (U=const. and ϕ=const.) for the normal (ϕ=ϕ_{n}, U_{n}≤U≤U_{n}) and for the fieldweakening (U=U_{n}, or U=U_{n}, ϕ<ϕ_{n}, ϕ=ϕ_{n}│W_{0n}/W_{0}│≈ϕ_{n}│W_{n}/W│) ranges. The longtime loadability limit corresponding to I=±I_{n} nominal current is also shown. Eg. in the W>0 and M>0 quadrant at the normal range maximum M=M_{n}=kϕ_{n}I_{n} nominal torque is allowed, while in fieldweakening maximum P_{m}=MW=U_{b}I=M_{n}W_{n}=P_{n} nominal power is allowed.
Using the column “b” in (2.12.5):
(2.7.a,b)
the blockdiagram of the constant flux DC machine can be drawn (Fig.2.3., T_{v}=L/R is the electric time constant of the armature circuit).
For linear system the superposition can be applied, so e.g. the w speed can be calculated as
(2.8)
The Y_{wu} and Y_{wmt} transfer functions can be derived from the blockdiagram:
(2.9)
(2.10)
Here: T_{m}=θR/(kϕ)^{2}=θ·dW/dM is the electromechanical time constant for m_{t}=const, is the equivalent time constant, is the damping factor. If ξ>1 (i.e. T_{m}>4T_{v}), then aperiodic, if ξ<1 (i.e. T_{m}<4T_{v}), then oscillating speed tracking is got for a step change in terminal voltage or load torque. With s=0 (t=∞) the transfer factors are the same as the factors got from (2.6.a) for steadystate. For the inner part of Fig.2.3. the voltagedimension block diagram (Fig.2.4) or the perunit dimensionless block diagram (Fig.2.5) are frequently used. In the latest, the quantities are dimensionless, they are related to the nominal values: u’=u/U_{n}, i’=i/I_{n}, ϕ’=ϕ/ϕ_{n}, m=m/M_{n}, w’=w/W_{0n}, R’=RI_{n}/U_{n}, T_{in}=θW_{0n}/M_{n}=θW_{0n}W_{n}/P_{n }is the nominal starting time, T_{m}/T_{in}=(W_{0n}W_{n})/W_{0n}=R’. For the motors T_{v} and T_{m }are few tens ms,_{ } T_{in} is few hundreds ms.
The DC machine is fed by the converter ÁI (Fig.2.6). The converter can be linecommutated AC/DC converter or DC/DC chopper. In both cases the u_{k} mean value of the pulsating u terminal voltage of the converter (and so the w=w_{k} speed of the drive) can be controlled continuously. The expression (2.6) is valid here also, but the U_{k},_{ }I_{k},_{ }M_{k} mean values must be substituted. The speed pulsation can be neglected (w≈w_{k}) because of the integration (1.1.a). A filter choke with L_{F} and R_{F} parameters is inserted frequently between the converter and the motor to reduce the pulsation of the I current and m torque. So in (2.6.a) R+R_{F} must be used instead of R.
For AC/DC line commutated converter in an industrial drive most commonly threephase, sixpulse symmetrically controlled thyristor bridge is applied. In this case the pulsation frequency of the u voltage, i current and m torque is f_{0}=6f_{h}=6·50 Hz=300 Hz, the pulse time is . The equivalent circuit is in Fig.2.7. and the output characteristic U_{k}(I_{k}) is in Fig.2.8.
The transformer (or commutation coil) is represented by its equivalent circuit in Fig.2.7. The I_{k} _{r}=f(α) critical current curve is the border of the continuous and discontinuous conduction modes in Fig.2.8. At I_{k}>I_{kr} current (e.g. in point 1) the conduction is continuous, at I_{k}<I_{kr} (eg. in point 2) it is discontinuous. In quadrant I. the converter is in rectifying mode, in quadrant IV. it is in inverter mode. Using R=R_{F}=0 approximation, U_{k}=U_{b} and so Fig.2.8. shows the W(M) mechanical characteristic (at ϕ=const.) approximately. As it can be seen, the thyristor bridge drive is capable of 2/4 quadrant operation (quadrants I. and IV. in Fig.1.1.c.).
At continuous conduction the U_{k}(I_{k}) characteristic is linear:
(2.11.a,b,c)
Where: is the mean value of the maximal continuous operation voltage, R_{f}=(3/π)ω_{h}L_{t} is a fictive resistance caused by the overlap (U_{tm} is the peak value of the phase voltage, ω_{h}=2πf_{h} is the angular frequency of the lines).
For transients in continuous conduction mode (Fig.2.9.a) the approximating equivalent circuit in Fig.2.9.b. can be drawn for the important mean values.
Accordingly, a similar armature voltage equation with mean values can be used as in (2.1.a) for the transient process:
(2.12)
(2.13.a,b)
For α input value Eq. (2.12) is nonlinear caused by cosα, so small deviations should be considered and the equation must be linearized. Assuming constant phase voltage amplitude (so U_{km}=const.) u_{ko} depends only on α:
(2.14)
Using Laplace transformation and rearranging:
(2.15.a,b)
Here: T_{ve}=L_{e}/R_{e}. For the w speed and u_{b} induced voltage the k index marking mean value can be omitted, since these quantities can not follow the torque pulsation with f_{o}=6f_{h}=300Hz frequency (w=w_{k}, u_{b}=u_{bk}). Using (2.15) and Fig. 2.3. the block diagram valid for continuous conduction mode can be drawn (Fig.2.10.).
The dead time should be considered by the factor , since Δu_{k0} can be modified only after the firing. The average dead time is . As can be seen from the block diagram, the transfer function of the GV firing controller is also necessary. u_{v} is the input signal of the firing control, at analogue implementation it is a control voltage. It is usual to make nonlinear firing control, to get cosα proportional to u_{v}. In this way the nonlinearities of GV and ÁIF compensate each other, i.e. the relation between u_{v} and u_{ko} becomes linear.
For discontinuous conduction the U_{k}(I_{k}) characteristic is nonlinear (see Fig.2.8.). In this case i_{k} can stepchange during transients, since the current i starts from zero value at the beginning of every pulse, and becomes zero at the end of the pulse. This is demonstrated in Fig.2.11. for Δα decrease.
Using mean values averaged to T_{0} pulse period, according to Fig.2.6. and Fig.2.8. the following voltage equation can be written:
(2.16)
There are no inductances in the expression, since i_{k }can step change. It must be linearized, since according to Fig.2.8. u_{k} is nonlinearly depends on α and i_{k}:
(2.17)
Linearizing (2.16), using Laplace transform and rearranging:
(2.18.a,b,c)
Using this expression, the block diagram for discontinuous conduction can be drawn.
R_{esz} _{ }(2.18.b) is much larger than R_{e} (2.13.a). It comes from the fact that both R_{ÁI}, and R_{ÁISZ} are proportional to the gradient of the static (steadystate) U_{k}(I_{k}) characteristic at the given point (e.g. points 1 and 2 in Fig.2.8.) and R_{ÁISZ}>>R_{ÁI}.
Comparing Fig.2.10. and Fig.2.12. it is clear, that there is a significant difference between the block diagrams of the continuous and discontinuous conduction. Consequently, different type current controllers are necessary in the two operation modes. The T_{h} dead time will be neglected during the investigation of the current control.
The threephase thyristor bridge drive (Fig.2.72.8.) is capable of two quadrant (2/4) operation (quadrant I. and IV. in Fig. 2.2.). The block diagram of the current control loop is given in Fig. 2.13. According to (2.4.a) the current control corresponds to indirect torque control.
Both the i_{a} current reference and the i current feedback signal can be only positive. The current control loop implements the torque control (m_{a}≥0 is possible), the current limitation (I_{korl}) and the firing angle limitation (0≤α≤α_{max}=150160^{0}, U_{vkorl}).
For fourquadrant (4/4) operation two thyristor bridges are necessary, which are practically connected antiparallely (Fig.2.14.). The ÁI1 converter can conduct i>0, while the converter ÁI2: i<0.
The current control of the two sets converters can be implemented with circulatingcurrent or circulatingcurrentless control.
Fig.2.15. shows the block diagram of the circulatingcurrentless control. The circulatingcurrent logic ensures by the K1 and K2 electronic switches, that always only one converter is fired. In this way circulatingcurrent can not develop, so there is no need for any L_{k} circulatingcurrent limiting chokes. The GV1 firing controller is controlled by u_{v1}=u_{v}, while GV2 by u_{v2}=u_{v} signal. The later negative sign is necessary to get the same operation for the converters in the reverse rotation but the same operation mode quadrants (IIII. driving, IIIV. braking, Fig. 1.1.c).
Fig. 2.16.demostrates a transient process, where the m_{a} torque reference is provided by an external speed controller (according to Fig.1.3). A step decrease in point 1 instant is assumed in the n_{a} speed reference. For M_{t}=const. load torque the operation of the drive is transferred from the driving point 1 to the driving point 3, meanwhile the drive is braking. In this case a T_{0}=12ms currentless period must be ensured between the reversal of i_{k} armature current for the recovery of the insulation capability of the previously conducting thyristors. The avoidless discontinuous condunctions slow down the reversal of i_{k} current. The overshoot of the i_{k} current in point 2 (thin line) can be avoided, if α_{2} is set correspondingly to u_{b}=kϕw. In circulatingcurrentless mode the reversal of the armature current is executed slowly, in more ms.
Fig.2.17. shows the block diagram of the control with circulatingcurrent. In this case both converters are fired always simultaneously. The basic aim is to provide the same output voltage mean value by both converters (U_{1}=U_{2}). By u_{k}=u_{ko} approximation (Fig.2.9.b.):
(2.19.a,b)
Because of the rule for the firing angles of the two converters e.g. from α_{max}=150^{o}, α_{min}=30^{o} is resulted in. As a result, the utilization of the converters is decreased, since this operation is possible in the cosα_{max}=0,87≤u_{k0}/U_{km}≤0,87=cosα_{min} range. In spite of the equality of U_{1k} and U_{2k} mean values, the u_{1} and u_{2} instantaneous values are different, consequently circulating current will flow. This is limited by the L_{k} chokes. The circulating current results in a faster motor current reversal comparing with the circulatingcurrentless operation. There are other possible control methods: circulatingcurrentweak control with α_{1}+α_{2}>180^{o} and circulatingcurrent regulation. The later needs two current controller: one of them controls to i_{ka}, the other to i_{a}+i_{ka} (i_{a} is the reference of the motor current, i_{ka} is the reference of the circulating current).
All discussed current control (Fig.2.13., 2.15., 2.17_{.}) regulate the mean value of the motor current i_{k} (Fig.2.9., 2.11.) and accordingly the mean value of the motor torque m_{k}=kϕi_{k}.
The DC/DC converter can be 1/4, 2/4 and 4/4 quadrant chopper (Fig.2.18.b.) on the U_{k}(I_{k}) plane. In this chapter only the 4/4 quadrant version is investigated, since this is applied widely with permanent magnet DC motor in servo and robot drives. The circuit diagram is given in Fig.2.18.a.
The circuit composed by two legs is a threestate converter. For controlling the legs, care must be taken to switch on only one transistor (IGBT) in a leg. If both of them are switched on simultaneously (e.g. T1 and T2), then a shortcircuit between the P and N bars is formed. Assuming ideal T1T4 transistors and D1D4 diodes the u output voltage can be +U_{e}, U_{e} and 0 value. Applying high frequency switchings between these voltage levels, the mean value of the output voltage can be controlled continuously in the range U_{e}≤u_{k}≤U_{e}. This method is the Pulse Width Modulation (PWM). Unipolar and bipolar operation can be distinguished, the bipolar does not use the u=0 voltage value (Fig.2.19.). The mean value of the voltage in bipolar operation is: u_{k}=(2b1)U_{e}, In unipolar operation is u_{k}=±bU_{e} (0≤b=t_{b}/T_{u}≤1).
The pulsation frequency of the u voltage is f_{u}=1/T_{u} _{, }which is more kHz in practice. That is why, to smooth the i armature current no series filter choke is necessary, the L inductance of the armature circuit is enough. With 4/4 quadrant chopper there is no discontinuous conduction, the current with i_{k}=0 mean value is continuous also. Assuming lossless energy conversion chain, the mean value of the powers are: P_{m} _{k}=M_{k}W=P_{k}=U_{k}I_{k}=P_{e} _{k}=U_{e}I_{ek}. In motor (driving) mode: I_{ek}>0, in generator (braking) mode: I_{ek}<0.
The aim of the current control is to track the i_{a}=m_{a}/kϕ current reference (determined by the torque demand) with zero error: Δi=i_{a}i=0. It is not possible with the discretestates chopper. From the (2.1.a) voltage equation using i=i_{a}Δi, the derivative of the current error can be expressed:
(2.20.a,b)
The e fictive voltage (using the RΔi=0 approximation) means the continuous terminal voltage corresponding to the errorless reference tracking i=i_{a}. The current controller can select from three voltage levels (+U_{e}, U_{e}, 0) in every instant If the selection is optimal, then i current tracks the i_{a} reference with small error (with small switching frequency), i oscillates around i_{a}.
There are two current control methods applied widely in practice: with PWM modulator and with hysteresis control.
The block diagram of the current control with PWM modulator is presented in Fig.2.20.a. Here SZI is a traditional e.g. PI type current controller, the block PWM is a PWM modulator. The PWM modulator generates the v1v4 twolevel control signals from the u_{v} control signal. The PWM modulator in the chopper plays similar role as the GV firing controller in the line commutated converters. The PWM modulator can operate in pushpull or alternate control mode. The pushpull control results in bipolar operation, the alternate control results in unipolar operation. Commonly true for both of them, that the mean value of the chopper output u voltage (u_{k}) is proportional to the u_{v} control signal. If analogue PWM modulator is used, the A_{u} coefficient in the
(2.21)
expression is the voltage amplification factor of the PWM chopper. The current control with PWM modulator regulates the mean value of the armature current: i_{k}.
The block diagram of the hysteresis current control is presented in Fig.2.20.b. Here SZI is a hysteresis current controller, which provides directly the v1v4 twolevel control signals. There is no need for an additional element between the SZI current controller and the Chopper (no PWM modulator), since both are discretestates unit. The hysteresis current control regulates the instantaneous value of the I armature current.
The voltage dimension block diagram of the PWM modulator based current control of the 4/4 quadrant DC chopperfed DC drive is given in Fig.2.21. The MotorLoad part corresponds to Fig.2.4., R^{*} is the Ω dimension transfer factor of the current sensor, A_{i}=R*/R, Δi_{k}=i_{a}i_{k} is the current error. Let’s assume first the current transients is so fast that the changing of the speed and the induced voltage can be neglected. The second part of the induced voltage in U_{b}+u_{b}=kϕW+kϕw is zero. So the feedback from u_{b} can be neglected in the small deviation block diagram of the current control loop (Fig.2.21.).
The practically applied PI type SZI current controller has the following transfer function:
(2.22)
Selecting properly the K_{sz}, T_{sz} parameters, the T_{v} electrical time constant can be eliminated from the current control loop. So the transfer function of the open current control loop is:
(2.23.a,b,c)
The transfer function of the closed current control loop is a firstorder lag element:
(2.24)
Consequently the controlled i_{k} current tracks the i_{a} reference by T_{i} delay:
(2.25.a)
(2.25.b)
The last equation for the Δi_{k} current error (2.25.b) is true if i_{a}=const. Then Δi_{k} is changing exponentially:
(2.26)
In practice T_{i} is given by the user (in servo drives it is around 1ms). Knowing it, the PI type SZI controller can be set in the following way (see (2.23.b,c)):
(2.27.a,b)
As an example let’s examine the effect of the current reference step:
(2.28)
The characteristic time functions are presented in Fig. 2.22. In Fig.2.22.a. the current controller operates in linear mode, in Fig.2.22.b. it is limited (saturated) at the beginning.
At linear operation for t>0 the following expressions are valid:
(2.29.a)
(2.29.b)
(2.29.c)
The condition of the linear operation is that the demanded u_{k} (2.29.c) has to fall into the range U_{e}≤u_{k}≤+U_{e}. If ΔI_{o}=I_{v}I_{o} greater than ΔI_{0max}=T_{i}·(U_{e}U_{b}RI_{0})/L, then u_{k}(t=+0)>U_{e} is required for the linear operation. In this case for a while saturated (limited) operation occurs.
In the 0<t<t ^{*;} saturated range U_{k}=U_{e} (Fig.2.22.b.). The i_{k} current tends to the I_{p}=(U_{e}U_{b})/R steadystate value by exponential function with T_{v} time constant:
(2.30)
At t* time instant . In the time interval next to saturation t>t* linear operation occurs, and similarly to (2.29.a) the current time function is:
(2.31)
if at t=t* time instant the integrator of the PI controller sets voltage. This can be ensured by setting the integral part of the u_{v}=u_{vp}+u_{vI} control voltage to u_{vI}=(U_{b}+Ri_{k})/A_{u} during the saturated operation. As an approximate solution the output of the integrator can be kept on the value which was at the beginning of the saturation (in our example it is u_{vI}=(U_{b}+Ri_{0})/A_{u}).
In servo motors because of , neglecting the variation of the u_{b} induced voltage in Fig.2.21. is not allowed. In this case the effect of the u_{b} on the current control loop can be compensated by a feedback (Fig. 2.23.) If the u_{b} part of the terminal voltage u_{k} is set by the compensating feedback, the current controller sees a passive RL circuit:
(2.32)
In a line commutated converterfed DC drive e.g. in Fig.2.13. the role of the PWM modulator is played by the GV firing control. Neglecting the T_{h} deadtime, in continuous conducting mode the PI type current controller must be set in the same way as in the 4/4 quadrant chopper (2.27). According to the block diagram in Fig.2.10. the role of A_{u} is played by , the role of T_{v} is played by T_{ve}=L_{e}/R_{e}. Since here the frequency of the subsequent firings is f_{o}=300 Hz and the pulse period is , T_{i}≈10ms can be selected according to practical experiences (it is larger by approx. one order than at the 4/4 quadrant chopper). If in discontinuous conduction mode the transfer function of the current control loop should be the same as in (2.24) then considering the block diagram in Fig.2.12. the current controller must be I type:
(2.33.a,b)
In (2.33.b) is assumed (see Fig.2.8.). If the drive operates in continuous and in discontinuous mode too (such case is e.g. the 4/4 quadrant circulatingcurrentless drive in Fig.2.15.), then adaptive SZI current controller is necessary, in which the structure (PI→I) and the integrator parameter ( ) can be modified depending on the mode of operation.
In the hysteresis current control of the 4/4 quadrant chopperfed DC drive a ±ΔI width tolerance band is allowed around the i_{a} reference signal. The hysteresis current controller observes the instant when the Δi=i_{a}i current error reaches the border of the ±ΔI band (in sampled system when it is first out of the tolerance band). Then a following evaluation process selects the best from the 3 possible voltages (+U_{e}, U_{e}, 0). This new voltage (u) moves the Δi error back into the tolerance band (Fig.2.24.). This method regulates the instantaneous value of the i current. The current bangbang control in Fig.2.25. have been spread widely in practice, where the applied u voltage depends on the Δi current error only.
The block containing the SZI current controller and the Chopper can be a twostage (Fig.2.26.a., b, c.) or threestage unit (Fig.2.26.d.). The control in Fig.2.26.a. reults in bipolar, while in Fig.2.2.b., c, d. unipolar operation. The versions a. and d. can operate in all 4 quadrants (Fig.2.18.b.), the version b. only in the I. and II. quadrant (U_{k}≥0), the version c. only in III. and IV. quadrant (U_{k}≤0)
The versions a. and d. capable of 4/4 quadrant operation result in different i_{a} reference current tracking. It is demonstrated in Fig.2.27. where a transition from A state to B state in Fig.2.18.b. is displayed with i_{a}=const. current reference.
With twostage current controller the I current is in a ±ΔI width band around the reference i_{a}, while with threestage depending on the sign of the u_{b}=kϕw induced voltage it is either in +ΔI, or in ΔI width band. Its reason is the fact that (according to Fig. 2.26.d.) u=0 can increase or also decrease the current i or current error Δi:
(2.34.a,b)
Consequently, the i_{k} mean value of the current is equal to the current reference with twostage controller (i_{k}≌i_{a}), while they are different with threestage controller ( ).
All bangbang current control are robust, only the width of the tolerance band (ΔI) can be modified, it provides reference tracking without overshoot with analogue implementation. The ΔI has a minimal value, limited by the switching frequency of the transistors (Fig.2.18.a.) The pulsation frequency of the voltage (current, torque) is f_{u}=1/T_{u}=1/(t_{b}+t_{k}) according to Fig.2.19. From the (2.1.a) voltage equation used for t_{b} and t_{k} time periods (assuming R≈0) the pulsation frequency can be expressed for bipolar (f_{ub}) and unipolar (f_{uu}) operation:
(2.35.a,b)
For versions a., b. and c. in Fig.2.26. ΔI^{*;}=2ΔI, for version d. ΔI^{*;}=ΔI. The maximum of the pulsation frequency is at b=t_{b}/T_{u}=1/2 dutycycle in both cases:
(2.36.a,b)
The pulsation frequency as a function of the voltage mean value is given in Fig.2.28. The voltage mean value is u_{k}=(2b1)U_{e} in bipolar and u_{k}=±bU_{e} (0≤b≤1) in unipolar operation. The f_{k} switching frequency of the T1T4 transistors (Fig.2.18.a.) in bipolar mode equals to the pulsation frequency, while in unipolar mode to its half:
(2.37.a,b)
Considering Fig.2.28. and the allowed maximal switching frequency of the transistors (f_{kmax}) the minimal tolerance band width (ΔI_{min}) can be determined. Selecting a larger ΔI, f_{k}<f_{kmax} is got.
A 2/4 quadrant linecommutated thyristor bridge converterfed drive (Fig.2.7., Fig.2.8.) with speed control capable of fieldweakening also is investigated as an example. Its block diagram is given in Fig.2.29.a.
Here the ÁIG excitation circuit converter is also a thyristor bridge. SZW is the speed controller, SZI is the armature current controller, SZU is the armature voltage controller and SZIG is the excitation current controller. All controllers are PI type in practice. The beginning of the fieldweakening is determined by the armature voltage. In the range u_{k}<U_{n} (approximately w<W_{n}): i_{ga}=I_{gkorl}=I_{gn} and consequently ϕ=ϕ_{n}. In the range w>W_{n}: u_{k}=U_{n}, approximately u_{b}=kϕw=U_{bn}=kϕW_{n}, i.e ϕ≈(W_{n}/w)·ϕ_{n}, ϕ_{min}≈(W_{n}/W_{max})·ϕ_{n}. In the fieldweakening range also the converter in the armature circuit (ÁI) reacts first for any change (w_{a}, or m_{t} modification). Accordingly here U_{km}>U_{n} is necessary (Fig.2.8.). Instead of SZU voltage controller a nonlinear setpoint element for the excitation current reference is also can be applied (Fig.2.29.b.). Neglecting the saturation of the core, the excitation current is proportional to the flux, so in the w>W_{n} range: i_{ga}=(W_{n}/w)I_{gn}. The normal and the fieldweakening range on the i_{k}i_{gk} plane are demonstrated in Fig.2.30. In the figure: I_{gmin}=I_{gn}/2, neglecting the saturation: ϕ_{min}=ϕ_{n}/2, W_{max}=2W_{n}. The range I_{n}≤i_{k}≤I_{n} can be allowed for long time, the range I_{n}<i_{k}<I_{meg} only for short time (the commutation limits are not considered). For ordinary motor: I_{meg}≈1,5 I_{n}, but for servo motor: I_{meg}≈5 I_{n} can be.
The threephase drive controls are described with Parkvectors (Spacevectors, shortly: vectors). For the sake of simplicity, the rotor of the machine is assumed to be cylindrical, wounded and symmetrical. Both the stator and the rotor are Y (star) connected and the starpoint is isolated (not connected) (Fig.3.1).
The a, b, c. notations are for the phases, the stator quantities are without indices, the rotor quantities are with index r. The machine vector equation s valid for transient processes also can be written simply in the natural coordinate systems ( own coordinate system, where the quantities exist ):
stator:
(3.1.a,b)
rotor:
(3.1.c,d)
Here the stator vectors are in a coordinate system fixed to the stator, the rotor vectors are in a coordinate system fixed to the rotor. R is the stator resistance, R_{r} is the rotor resistance, L is the stator inductance, L_{r} is the rotor inductance, L_{m} is the mutual (main) inductance. In the equations of the flux linkage (shortly: flux) the e^{j} ^{α} factor can be eliminated, if a common coordinate system is used. The relations of the quantities (e.g. the currents) in the own and the common coordinate system (marked by *) are, Fig.3.1.b:
stator:
(3.2.a,b)
rotor:
(3.2.c,d)
Using these expressions, the machine equations in the common coordinate system can be written:
stator:
(3.3.a,b)
rotor:
(3.3.c,d)
Where w=dα/dt is the angular speed of the rotor, w_{k}=dα_{k}/dt is the angular speed of the common coordinate system. The flux equations become more simple, the voltage equations become more complicated. The equivalent circuits corresponding to equations (3.3) are in Fig. 3.2.
The equivalent circuit for the fluxes is the same as for a transformer. L_{s} is the stator, L_{rs} is the rotor leakage (stray) inductance, L=L_{m}+L_{s}, L_{r}=L_{m}+L_{rs}. The main flux is linked both to the stator and the rotor. In Fig.3.2.a,b. reduction to 1:1 effective number of turns is assumed. In the drive control practice the rotor quantities have a further reduction in the following way:
(3.4.ad)
If the fictive ‘a’ ratio is selected to a=L_{m}/L_{r}<1, then the leakage inductance of the rotor is eliminated: L’_{rs}=0 (Fig.3.3.a.), if it is selected to a=L/L_{m}>1, then L’_{s}=0 (Fig.3.3.b.).
L’ is the stator transient inductance, σ is the resultant stray factor:
(3.5.a,b)
Common coordinate system and the modified equivalent circuits are used in the following, but the * and ’ notations are not used (except in L’, u’ and ψ’). E.g. the equivalent circuit got by the elimination of the rotor leakage inductance is given in Fig.3.4.
In the induction machine it is usual to call the reduced rotor flux to flux behind the transient inductance (shortly transient flux), the voltage to transient voltage. The machine equations corresponding to Fig.3.4:
stator:
(3.6.a,b)
rotor:
(3.6.c,d)
These equations are valid for squirrelcage and slipring induction machines and cylindrical, symmetrical rotor synchronous machines. The last means that the d and q axis synchronous inductances and subtransient inductances are equal: L_{d}=L_{q} and . The usual flux equivalent circuits for synchronous machines are presented in Fig.3.5. Fig.3.5.a. corresponds to Fig.3.4.a, while Fig.3.5.b. corresponds to the following equation got from (3.6.b,d):
(3.7)
Here L_{d}=L”+L_{m} is the synchronous inductance, is the subtransient flux vector, is the pole flux vector proportional to the rotor current vector.
Assuming sinusoidal flux density and excitation spatial distribution, the torque can be expressed by the stator flux ( ) and current (ī) vectors:
(3.8.a,b)
The symbol × means vector product, the ‘·’ means scalar product. The torque is provided as vector and signed scalar by (3.8.a) and (3.8.b) respectively.
The above Parkvector voltage, flux and torque equations together with the motion equations (1.3.a and 1.5.a) form the differential equation system of the drive. In the next chapters for the theoretical calculations always 2p=2 (twopole) machine is considered (p=1, and not written).
Tartalom
The sinusoidal field generated by the permanent magnet is represented by a Ψ_{p}=const. amplitude pole flux vector in the d direction (Fig.4.1.), so in a stationary coordinate system (w_{k}=0):
(4.1)
In a wounded rotor it would be provided by a current source supply. The pole flux rotating with the rotor induces the ū_{p} pole voltage in the stator coils:
(4.2)
Using Fig.3.4. and Fig.3.5. quite simple flux and voltage equivalent circuits can be derived (Fig.4.2.). As can be seen in Fig.3.4.a. the stator flux depends on the ī stator current too:
(4.3)
Using it in (3.8.a,b) the torque can also be calculated with the pole flux vector:
(4.4.a,b)
Neglecting the friction and the windage losses, using (4.2) and (4.4.b) the p_{m} mechanical power can be calculated in the following way:
(4.5)
In the dq coordinate system fixed to the pole flux vector (w_{k}=w):
(4.6.a,b)
(4.7.a,b,c)
(4.8)
In (4.8) i_{q} is the torque producing current component, ϑ_{p} is the torque angle.
In the case of inverter supply normal and fieldweakening operations are usual.
In normal operation mode : i_{d}=0, at m>0 i_{q}>0, ϑ_{p}=90^{o}, sinϑ_{p}=1, at m<0 i_{q}<0, ϑ_{p}=90^{o}, sinϑ_{p}=1. As can be seen in (4.8), in this way for a given torque the required current is the smallest. In the vector diagram for normal mode (Fig.4.3.a.) besides the currents and fluxes the fundamental voltages are also drawn. Steadystate operation and ω_{1}=2πf_{1}=w fundamental angular frequency are assumed, furthermore the harmonics in the voltages, currents and fluxes (caused by the inverter supply) are neglected. Accordingly e.g. the voltage induced by the flux can be calculated similarly as (4.2): (the index 1 denotes fundamental harmonic).
The amplitude of the induced voltage vector using the approximations above:
(4.9.a,b)
The index 0 denotes normal operation (i_{d}=0). If R≈0 the approximation is used, then the induced voltage is equal to the terminal voltage: u_{i10}≈u_{10}. At a given torque (at i_{q}=(2/3)m/Ψ_{p} current): ψ_{0}=const, while u_{i10} is proportional to w. The equality u_{i10}=U_{n} (U_{n} is the nominal voltage) determines the limit of the normal operation on the wm plane and the maximal speed which can be reached in normal mode:
(4.10)
Fieldweakening operation mode: Increasing w further, because u_{i10} would be greater than U_{n} (u_{i10}>U_{n}) the amplitude of the stator flux must be reduced by i_{d}<0, by the L_{d}i_{d} component of the armature reaction (Fig.4.3.b.). In this way the induced voltage can be reduced:
(4.11.a,b)
The necessary i_{d} fieldweakening current component (by R≈0 approximation) is determined by the u_{i1}=U_{n} equality:
(4.12)
At the largest fieldweakening: Ψ_{p}+L_{d}I_{dmeg}=0, i.e. I_{dmeg}=Ψ_{p}/L_{d}. In this case the value under the square root in (4.12) is zero. That is why (by R≈0 approximation) with i_{d}=I_{dmeg} the torque is hyperbolically decreases with the increasing speed: m=(3/2)Ψ_{p}U_{n}/(wL_{d}). The ranges of the operation modes and borders considering also the limits are given in Fig.4.4.
The following ranges and limits can be identified in Fig.4.4.a.:
0M1 section: 
normal m>0, ϑ_{p}=+90°, i_{d}=0. 
in the 0M1M20’ „square”: 
fieldweakening m>0, 90^{o}<ϑ_{p}<180^{o}, i_{d}<0. 
0G1 section: 
normal m<0, ϑ_{p}=90^{o}, i_{d}=0. 
in the 0G1G20’ „square”: 
fieldweakening m<0, 90^{o}>ϑ_{p}>180^{o}, i_{d}<0. 
M1M2, G1G2 border: 
current limit, i=I_{max}. 
M2G2 border: 
d current limit, i_{d}=I_{dmeg}. 
00’ section: 
i_{q}=0, m=0 mechanical noload, ϑ_{p}=180^{o}. 
These ranges can be seen in Fig.4.4.b. also (M_{max}=(3/2)Ψ_{p}I_{max}). The given wm range is valid for abc phasesequence, at acb phasesequence its reflection to the m axis must be considered. If the demanded operation point is given in the wm plane, then using (4.8) and (4.12) the necessary i_{q} torque producing and i_{d} filedweakening components of the ī current vector can be determined. The block diagram of the torque controlled drive is presented in Fig. 4.5.a. Here the m_{a} torque reference according to (4.8) determines the reference i_{qa}, m_{a} and w according to (4.12) determine the reference i_{da}. The current vector controller ensures the tracking of the current references: i_{q}=i_{qa}, i_{d}=i_{da} by the power electronic circuit (VSI).
Similarly to Fig.2.29.a., also a SZU voltage controller can set the i_{da} reference (Fig.4.5.b.). In this case the amplitude of the ū_{1} fundamental voltage vector (u_{1}=│ū_{1}│) must be controlled to U_{n} in the fieldweakening range. SZU must be limited in such a way to get i_{da}=0 in the normal range.
A current vector control oriented to the pole flux vector (to the polefield generated by the permanent magnet) is necessary, since the i_{qa} and i_{da} current references are given directly. The contradiction as the references are available in d,q and the feedback signals are in a,b,c components must be absolved. Same type reference and feedback signals (in the same coordinate system) are necessary for the current vector control. The possibilities are demonstrated in Fig.4.6.b. by a coordinate transformation chain.
In the crosssections a,b,c,d,e, in the possible two coordinate systems, five different sametype reference and feedback signal combinations can be considered. Accordingly the following current vector controls can be implemented theoretically:
a section: coordinate system rotating with the polefield, Cartesian coordinates,
b section: coordinate system rotating with the polefield, polar coordinates,
c section: stationary coordinate system, polar coordinates,
d section: stationary coordinate system, Cartesian coordinates,
e section: stationary coordinate system, phase quantities.
In the versions a,b,c the reference and feedback signal of the i_{d}, i_{q} and │ī│ current controllers and the ϑ_{p }and α_{i} angle controllers are DC type quantities. In the versions d, e the reference and feedback signal of the i_{x}, i_{y,} i_{a}, i_{b}, i_{c} current controllers are AC type quantities (with f_{1} fundamental frequency in steadystate). It can be established, that the coordinate transformation cannot be avoided, and for the stationary→polefield and the polefield→stationary coordinate transformations the α angle of the pole flux vector must be known. The number of the computational demanding coordinate transformations is determined by the fact as the sensing is possible in stationary coordinate system (i_{a}, i_{b}, i_{c}, α) and the intervention is possible also in stationary coordinate system (the inverter is connected to the stator), while the references are available in polefield coordinate system directly (i_{da}, i_{qa}).
In practice, the a, or the e versions are used for current vector control (Fig.4.7.). In version a the references, in version e the feedback signals can be used directly. In version a two, in version e one coordinate transformation is necessary.
As can be seen in Fig.4.5.a. and Fig.4.7.a.,b., the motor is fed by PWM voltage source inverter (VSI) in all cases. In electrical drive practice two or threelevel voltage source inverters are applied. These generate the threephase voltages of variable f_{1} frequency and variable u_{1} amplitude from the U_{e}=const. DC voltage by Pulse Width Modulation (PWM).
In industrial drives, the U_{e} DC voltage is generated from the threephase f_{h}=50Hz AC lines by a converter (see chapter 7.). In the twolevel inverter the 0 point is fictive, in the threelevel version it is real, it can be loaded. Accordingly the u_{a0}, u_{b0}, u_{c0} voltages can be set to two values in the twolevel inverter (+U_{e}/2, U_{e}/2), and three values in the threelevel inverter (+U_{e},/2, 0, U_{e}/2). The number of states which can be provided by the switches are: 2^{3}=8 in the twolevel inverter, 3^{3}=27 in the threelevel inverter (generally: levelnumber^{phasenumber}). As the twolevel version is spread widely, only it is investigated in the following. Most frequently the twolevel IGBT voltage source inverters are applied (Fig.4.9.a.).
The legs of the phases are the same as in Fig. 2.18.a. Assuming ideal transistors T1T6 and diodes D1D6 the a,b,c phases can be connected either to the P positive bar or to the N negative bar. In one leg either the upper or the lower transistor can be ON, conducting together would result in PN shortcircuit. The switched on transistor or the antiparallel diode conducts depending on the direction of the phase current. It is true, if the voltage condition
(4.13)
is valid (i.e. the DC voltage is larger than the maximum of the lineto line voltages between a,b,c points), ensuring the controllability of the inverter. If it is not true, the freewheeling diodes occasionally conduct (when there is a positive voltage on them) even the parallel transistor is off.
It is assumed in the following, that one transistor is switched on in every phase leg by the twolevel va, vb, vc control signals and the (4.13) condition is fulfilled. Table 4.1. shows to which bar the phases are connected in the possible 8 states.
Table 4.1. The 8 switching states.
k 
1 
2 
3 
4 
5 
6 
7P 
7N 
a 
P 
P 
N 
N 
N 
P 
P 
N 
b 
N 
P 
P 
P 
N 
N 
P 
N 
c 
N 
N 
N 
P 
P 
P 
P 
N 
There can be only 7 different voltage vectors on the output of the inverter (ū=0 can be provided in two ways: 7P and 7N):
(4.14)
In this way the twolevel VSI with U_{e}=const. is a 7state vector actuator unit. The demanded fundamental voltage vector with u_{1} amplitude and ω_{1}=2πf_{1} angular frequency is generated by PWM control switching between these 7 possible ū(k) vectors:
(4.15)
Fig.4.10. shows a characteristic inverter voltage time function.
The energy flow is possible in both direction, if the DC circuit is capable of it. In the case of the intermediate DC link versions it depends on the way how the U_{e}=const. DC voltage is generated (chapter 7). In the most modern version (Fig.4.9.b.) either the machineside converter ÁG or the lineside converter ÁH are VSI type. In this way the power can flow in both directions. In the simplest case ÁH is a diode bridge, when only motor mode operation is possible. In motor mode (driving mode) the mean value of the DC current is positive I_{ek}>0, while in generator (brake) mode it is negative I_{ek}<0. Assuming lossless energy conversion chain the power mean values are (with the notation in Fig. 4.9.b.):
The aim is to track the ī_{a} current reference vector (determined by the driving task) without error . By a noncontinuous state VSI it is not possible. The derivative of the current error vector corresponding to the ū=ū(k) voltage vector can be expressed using the ū=Rī+L_{d}·dī/dt+ū_{p} voltage equation (derived from Fig.3.2.b.), and considering expression:
(4.16.a,b)
The ē fictive voltage vector (using approximation), means the necessary continuous voltage vector for the errorless tracking: ī=ī_{a}. In every instant the current controller can select from 7 kinds of ū(k) voltage vectors (4.14). If the selection is optimal, then the ī current vector tracks the ī_{a} reference with small error (ī oscillates around ī_{a}).
Similarly to the chopperfed DC drive (Fig.2.20.) two kinds of current vector control spread widely in practice: the PWM modulator based and the hysteresis control. In the PWM modulator based current vector control (Fig.4.11.a.) the PWM VSI has a PWM modulator, and the current controller acts through this modulator indirectly. The hysteresis current vector controllers (Fig. 4.11.b.) control the PWM VSI directly. In Fig.4.7.a.,b. PWM modulator based version is assumed.
The PWM modulator based current vector control (Fig.4.11.a.) has more versions, depending on in which coordinate system the components of the ī current vector are controlled, and which are the input signals of the PWM modulator. If the SZI controllers control the dq components, then the two versions in Fig.4.12.a.,b., if the abc components (phase currents), then the two versions in Fig. 4.12.d.,e. are possible. The SZI current controllers are PI type in the practice. Fig.4.12.c. presents how the components dq are produced. The control signals (with index v) control spacevector PWM modulator in the a,e versions and threephase PWM modulator in the b,d versions. The necessity of the coordinate transformations is obvious in all cases.
Le’s examine the versions in Fig.4.12.a. and Fig. 4.12.d. in a little bit more detail.
Current vector control with dq components, by spacevector PWM (SPWM) (Fig.4.12.a.). The detailed block diagram of this version is in Fig.4.13.
The blocks SZID and SZIQ are usually PI type current controllers, their outputs (u_{vd} and u_{vq}) form a control vector , which is proportional to the ū_{1} fundamental voltage vector of the PWM inverter (the motor), if the f_{ISZM} switching frequency is large enough. According to the experience, if f_{ISZM}>20f_{1} then ū_{1}=K_{u}ū_{v}. As a synchronous machine is investigated, the maximal value of f_{1} is determined by the maximal speed (n=n_{1}=f_{1}/p). In practice: f_{1max}≤100Hz, so with f_{ISZM}≥2kHz the above proportionality is well correct. The input signals of the SPWM are the u_{v} _{ }amplitude and α_{v} angle of the ū_{v} control vector, its output signals are the twolevel va, vb, vc inverter control signals. In practice, the SPWM is operating in sampled mode, the sample frequency is equal to the f_{ISZM} frequency.
Using sampled SPWM in the nth sample period with the control vector
(4.17.a,b,c)
fundamental voltage vector is prescribed. K_{u} is the voltage gain factor of the SPWM controlled VSI. The ū_{1}(n) vector can be produced by switching on the neighbour three ū(k) voltage vectors (Fig. 4.14.b) for proper time interval. In the time instant presented in Fig.4.14.a. the ū_{1}(n) is in the sector 1 of 60° degree.
Here ū(1), ū(2) and ū(7) are the three neighbour vectors. The ū_{1}(n) vector is developed as the time weighted mean value of these vectors:
(4.18)
Where τ_{1n}+τ_{2n}+τ_{7n}=τ=const. is the sampling period, b_{1n}+b_{2n}+b_{7n}=1 is the sum of the duty cycles. The b_{1n}, b_{2n} and b_{7n} duty cycles can be derived from the geometric considerations based on Fig.4.14.a.:
(4.19.a,b,c)
Where U_{1max} is the possible maximum fundamental peak value, which is according to Fig.4.14.a. is:
(4.20)
The function of the duty cycles in sector 1 is presented in Fig.4.15. for 0,8U_{1max} amplitude ū_{1} fundamental voltage. b_{1n} and b_{2n} are proportional to the prescribed u_{1}(n)=0,8U_{1max} amplitude, the b_{1n}/b_{2n} ratio depends on the α_{1}(n) angle. The switching between the 3 possible vectors can be done in two ways (Table 4.2.).
Table 4.2. Switching methods in sector 1.
Method I. 
Method II. 

k 
1 
2 
7 
1 
2 
7 
1 
2 
7 
1 
2 
7P 
2 
1 
7N 
1 
2 
7P 

sample 
n 
n+1 
n+2 
n 
n+1 
n+2 
There is one double switch at method I. in every sampling period, even by 7P or by 7N the ū(7)=0 voltage vector is produced. It is eliminated using method II. by the periodical changing of the switching order of ū(1), ū(2), 7P and 7N. Considering Table 4.2. and Fig.4.14.b. it can be established, that at method I. 4, at method II. 3 switchings correspond to one sampling period. I.e. using method II. the switching number can be reduced to ratio ¾ and also the switching loss proportional to it, comparing with method I.
The operation of the SPWM is investigated in sector 1, but it operates in the other sectors similarly.
Current vector control with abc phase quantities, by threephase PWM modulator (3Ph PWM) (Fig.4.12.d.).
The SZIA, SZIB and SZIC are usually PI type current controllers, their output signals are the u_{va}, u_{vb} and u_{vc} phase control signals (modulating signals). Processing them the 3Ph PWM generates the twolevel control signals va, vb, vc for the inverter. The 3Ph PWM consists of 3 onephase modulator, but the carrier wave of the modulators (u_{Δ}) is common (Fig.4.17.).
Operation of the analogue PWM modulator is demonstrated in Fig.4.18. for phase a. (Nowadays digital modulators implemented by counters are applied.) While u_{va}>u_{Δ}, then va=H (high level), phase a is on the P bar: u_{a0}=+U_{e}/2. When u_{va}<u_{Δ}, then va=L (low level), phase a is on the N bar: u_{a0}=U_{e}/2. There exist f_{Δ}/f_{1}=const. synchronous modulation and, f_{Δ}=const., f_{Δ}/f_{1}=var. asynchronous modulation. It can be proved, that in steadystate in the output voltage of the inverter besides the fundamental component with frequency f_{1} upperharmonics with frequencies f_{Δ}±2f_{1}, f_{Δ}±4f_{1},…, 2f_{Δ}±f_{1}, 2f_{Δ}±3f_{1},… also appear.
The synchronous machine thanks to its L_{d} synchronous inductance is very good filter for the current and the torque. It is demonstrated in Fig.4.19. drawn according to Fig. 4.2. Here Δū and Δī are the resultants of the upperharmonics:
(4.21.a,b,c)
Where ν is the order number of the harmonics. It is assumed, that the ū_{p} pole voltage (4.2) is purely fundamental (ω_{1}=2πf_{1}=w_{1}=w). If f_{Δ}≥2kHz, then well approximately the current pulsation ( ) and the torque pulsation (Δm) caused by can be neglected (it is also true for modulation method in Fig.4.13., if f_{ISZM}≥2kHz). The cagerotor induction machine (chapter 5.) is also a good filter if PWM VSI is the supply.
It is true with good approximation either for synchronous or asynchronous modulation that the VSI controlled by 3phase PWM modulator can be considered as a proportional element if f_{Δ}/f_{1}>20. E.g. for phase a:
(4.22.a,b)
According to Fig.4.18. maximum of u_{va} may be U_{Δ} _{m}/2, consequently the maximal fundamental peak value is:
(4.23)
Comparing with (4.20) it is clear that the maximal fundamental voltage with 3phase PWM modulator is 15% less than with SPWM. The utilization of the inverter can be improved if the 3phase PWM is controlled by modified (u_{v}*) control signals (Fig.4.20).
If the control signals are modified in the way presented in Fig.4.20., then the voltage amplification factor is increased to and so as with SPWM.
The hysteresis current vector controls operate the VSI directly (without PWM modulator). As the control is fulfilled with vectors, instead of tolerance band tolerance area should be given. It can operate in stationary or polefield coordinate system. The tolerance area in stationary coordinate system can be a circle or a regular hexagon, in polefield coordinate system can be circle or square. The operation can be analogue or digital.
Analogue control in stationary coordinate system is investigated in the following. The analogue hysteresis current vector controller senses that instant (comparing instant), when the current error vector ( ) reaches the border of the tolerance area. It means the condition at ΔI radius tolerance circle, at hexagon with 2ΔI side distance: │Δi_{a}│=ΔI or │Δi_{b}│=ΔI or │Δi_{c}│=ΔI conditions (Δi_{a}=i_{aa}i_{a}, Δi_{b}=i_{ba}i_{b}, Δi_{c}=i_{ca}i_{c} are the phase current errors). The tolerance area is circle in Fig.4.21.a. and hexagon in Fig.4.21.b. After sensing the reaching of the tolerance band the adaptive version a twostep procedure selects the optimal voltage vector ū(k) from the possible seven switchable ones by the VSI.
The block diagram of the adaptive hysteresis current vector control is given in Fig.4.22. The comparing and convergence conditions are valid for circle shape tolerance area (Fig.4.23.). is the derivative of the current error vector corresponding to the ū(k) voltage vector at the t_{0} comparing instant (4.16.a), ‘·’ means scalar product. According to Fig.4.23., is the condition to move current error vector back to the tolerance area by the ū(k) voltage vector. According this vector convergence condition the algorithm selects among the available seven ū(k) voltage vectors (4.14) the possible N vectors. Usually N>1, so a criterion is necessary to select the optimal. E.g. the criterion given in Fig.4.22. (max(T_{k}/S_{k})) has the aim to get less switching (S_{k}) and more T_{k} for staying inside the tolerance area.
Let’s assume that Fig.4.24. corresponds to the t_{0} comparing instant. Fig.4.24.a. corresponds to (4.16.a), in Fig.4.24.b. the dotted lines are the derivatives (speeds) of the current error vector at point, the numbers are the k values in ū(k). According to Fig.4.24.a,b. the following can be established: ū_{old}=ū(2), N=3: ū(1), ū(6), ū(5), the switching numbers are: with selecting ū(1): S_{1}=1, with selecting ū(6): S_{6}=2, with selecting ū(5): S_{5}=3, the relation of the expected times to the next comparing is T_{1}>T_{6}>T_{5}. Considering these, using the max(T_{k}/S_{k})) criterion: ū_{new}=ū(1) should be selected. The adaptive hysteresis current vector control described in Fig.4.22. is quite complicated, consequently it is not applied in practice.
In the practically widely applied simple hysteresis current vector control the ū_{new} voltage vector depends on the current error vector only. In the simplest hexagon shape tolerance case selection of ū(k) depends on the Δi_{ao}, Δi_{bo} és Δi_{co} phase current errors. This is the 3phase bangbang current control (Fig.4.25.).
Only phase a is detailed, since circuits of the phase b and c are similar. It can be proved, that because of the interaction of the phases coming from i_{a}+i_{b}+i_{c}=0 the error per phase can be larger (max. ±2ΔI) then the tolerance band ±ΔI. It means, that the vector convergence condition for hexagon is not always satisfied. Consequently the ī current vector can move to the shaded triangles around the tolerance area (Fig.4.21.b.). This control is simple and robust, only the ΔI phase tolerance band must be set, taking into consideration the allowed switching frequency of the inverter. Note: if the 0 and 0’ point would have been connected (Fig.4.25.), then the current bangbang controls in the phases would be independent, so the phase current errors would stay in the ±ΔI band.
Tartalom
The rotor ‘winding’ is a shortcircuited squirrelcage (shortly: cage). The cage rotor induction machine can be substituted by a wounded rotor machine, which has shortcircuited rotor coils with terminals ra, rb, rc (Fig.3.1.a.) so the rotor terminal voltage is zero: ū_{r}=0. Considering this and (3.6.c), the rotor voltage equation in rotor coordinate system (w_{k}=w) shows, that the ī_{r} rotor current vector can modify the rotor flux vector only:
(5.1.a,b)
The flux linked with the shortcircuited rotor coil can be modified only slowly because of the small R_{r} rotor resistance. The rotor flux vector must be developed by the ī stator current. The constant rotor flux fieldoriented controls are examined in the following.
The operation of the cage rotor induction machine depends on the flux linked with the shortcircuited rotor principally. Accordingly the flux equivalent circuits (Fig.3.3.a. and Fig.3.4.a.) should be used and the coordinate system should be fixed to the rotor flux vector (Fig.5.1).
The equations (3.6.ad) must be actualized, by using w_{k}=w_{ψr}=dα_{ψr}/dt and ū_{r}=0:
stator:
(5.2.a,b)
rotor:
(5.2.c,d)
Where w_{r}=w_{ψ} _{r}w is the angular speed of the rotor flux vector relatively to the rotor. In the coordinate system fixed to the rotor flux vector (so called field coordinate system):
(5.3.a,b,c)
Decomposing the rotor voltage equation (5.2.c) to d real and q imaginary parts:
(5.4.a,b)
(5.5.a,b)
Decomposing the rotor flux equation (5.2.d) to real and imaginary parts and considering (5.4.b) and (5.5.b):
(5.6.a,b)
(5.7.a,b)
(5.6.b) shows that the rotor flux vector amplitude (ψ_{r}) can be modified by the i_{d} flux producing component only, i_{q} has no effect on it. Modifying i_{d}, ψ_{r} tracks the L_{m}i_{d} value like a first order lag elemet with T_{ro} time constant, caused by the flux modification damping effect (5.4.b) of the shortcircuited rotor (Fig.5.2.). So the ψ_{r} amplitude can only be modified slowly, as the T_{ro}=L_{m}/R_{r} noload rotor time constant is more tenth of sec. Consequently for a high dynamic drive the rotor flux amplitude (ψ_{r}) should be kept constant. Then as dψ_{r}/dt=0 and ψ_{r}=L_{m}i_{d}:
(5.8.a,b,c,d)
The torque with Parkvectors (3.8.a), considering (5.2.b) is:
(5.9)
(5.10)
(5.10) shows, that the m torque can be set by the i_{q} torque producing current component. For m>0, i_{q}>0 (Fig.5.1.), for m<0, i_{q}<0. From (5.7.b) the angular speed of the rotor flux vector relatively to the stator can be expressed:
(5.11)
As can be seen from (5.6.a) and (5.10), the supply of the cage rotor induction machine should be oriented to the rotor flux vector (shortly to the field).
The block diagram of the current source supplied cage rotor induction machine in field coordinate system is drawn using Fig.5.3 (1.1.a, 5.6.b, 5.10 and 5.11). The figure is extended by the i_{a},i_{b},i_{c}→i_{x},i_{y}→i_{d},i_{q} transformation boxes. According to this block diagram the fieldoriented current source supply must feed the induction machine with such i_{a}, i_{b}, i_{c} currents (resulting in ī=(2/3)(i_{a}+āi_{b}+ā^{2}i_{c}) current vector) to get i_{d}=ψ_{r}/L_{m}=const. d current component and the q current component (i_{q}) must be proportional to the demanded torque.
Consequently the current vector control of the induction machine in field coordinate system is decoupled to two independent i _{d} and i _{q} (rotor flux and torque ) control loop. The induction machine supplied in this way behaves similarly to the compensated, separately excited DC machine. The i_{d} flux producing component corresponds to the excitation current (or the permanent magnet), the i_{q} torque producing component corresponds to the armature current, and i_{q} modifies only the torque, in the same way as the armature current in the DC machine. It should be emphasized, that the decoupling is true only in the dq rotor flux coordinate system. The critical point of the fieldoriented control is the determination of the position of this coordinate system. The switchingon of the fieldoriented drive must be started with the development of the ψ_{r} rotor flux (as in the DC machine with the switching on of the excitation), and this flux must be kept until the switchingoff the drive.
The motor voltage can be modified directly by the PWM VSI in practice. The fieldoriented control can be implemented by voltage source supply also, if the ū voltage vector necessary to develop the previously defined i_{d}, i_{q} currents is connected to the motor. For the investigation of the voltage source supply, let’s substitute the (5.2.b) expression of the stator flux vector into (5.2.a):
(5.12)
The real and imaginary parts of the voltage equation are:
(5.13.a)
(5.13.b)
As can be seen, the decoupling is not exact for the voltages in field coordinate system, since the d axis equation contains q quantity, the q axis equation contains d quantity also (there is a crosscoupling). Dividing the u_{d} and u_{q} voltages by R and arranging, the following equations are got:
(5.14.a)
(5.14.b)
The i_{d} and i_{q} currents track the left side quantities with T’=L’/R stator transient time constant (it is few 10 ms, i.e. less than the T_{R0} by one order). The block diagram in Fig.5.4. is drawn using these two equations and Fig.5.3. (the dotted box here corresponds to the part of Fig.5.3. surrounded by dotted line). As can be seen in the block diagram, e.g. the modification of i_{q} component (the m torque) requires not only the modification of u_{q}, but the modification of u_{d} also, if the i_{d} component (the ψ_{r} flux) should be kept constant. As the inverter acts in stationary coordinate system by u_{a}, u_{b}, u_{c} voltages, the block diagram in dq coordinate system is extended by the abc/xy and the xy/dq transformations.
Also the steadystate symmetrical sinusoidal operation can be got from the equations derived in the previous chapter. In the case of inverter supply, the statements are valid for the fundamental quantities with f_{1} frequency. Capitals denote steadystate values, index 1 denotes fundamental quantities in the following. The summarised equations of the Ψ_{r1}=L_{m}I_{1d}=const. operation are the following:
(5.15.a,b)
(5.16.a,b)
(5.17.a,b,c)
(5.18.a,b)
Using these equations and assuming Ψ_{r1}=Ψ_{rn}=const. nominal rotor flux, the current vector diagram (Fig.5.5.a.) and the mechanical characteristics (Fig.5.6.a.) are drawn. It comes from (5.18.a) and (2.6.a) that the W(M) mechanical characteristics (Fig.2.2. and Fig.5.6.a.) are similar to that of the DC machine, but the role of the U terminal voltage is played by f_{1} frequency, the role of the ϕ flux is played by the Ψ_{r1} rotor flux, and the R armature resistance must be substituted by the R_{r} rotor resistance. The Fig.5.5.b. and Fig.5.6.b. are for Ψ_{1}=Ψ_{n}=const. nominal stator flux operation (without deduction).
The W(M) curves are for abc positive phase sequence supply, the acb phase sequence case can be got by reflecting the curves to the origin. Modifying the f_{1} frequency, the W(M) curves are shifted parallel. The Ψ_{r1}=const. rotor flux operation is more advantageous, since then the W(M) mechanical characteristics have not a breakdown point (at Ψ_{1}=const.: and at Ψ_{1}=Ψ_{n}: M_{b}=(22,5)M_{n}). Above the nominal f_{1n} frequency neither the rotor flux (Ψ_{r1}) nor the stator flux (Ψ_{1}) amplitude can be kept at the nominal value. The reasons are:
The inverter cannot provide significantly larger voltage than U_{1n}=U_{n} nominal voltage, and the motor could not withstand it too.
The stator core losses (P_{coreH} hysteresis and P_{coreE} eddycurrent losses) can reach not allowed value:
(5.19)
Accordingly, in the range f_{1}>f_{1n} (W_{1}>W_{1n}) the flux must be reduced, the field must be weakened. E.g. the Ψ_{r1} rotor flux must be modified approximately in the following way in the fieldweakening W_{1}>W_{1n} (approx. W>W_{n}) range:
(5.20)
Fig.5.5.a. and Fig.5.6.a. correspond to Ψ_{r1}=Ψ_{rn} normal operation. Fig.5.7. shows the fieldweakening ranges also (assuming 4/4 quadrant operation) on the W(M) plane (Fig.5.7.a.) and the Ī_{1} current vector ranges in the dq coordinate system (Fig.5.7.b.).
The aim is to keep the ψ_{r} amplitude constant by closedloop regulation or by openloop control. There are two widely spread methods in practice:
Direct rotor flux control. In this case the ψ_{r} amplitude and α_{ψ} _{r} angle of the rotor flux vector (Fig.5.1)
(5.21)
is created (usually by a machine model). The ψ_{r} rotor flux amplitude is kept through i_{d} by control, and the m torque is controlled by i_{q}. This method is implemented by current vector control oriented to the rotor flux vector (shortly fieldoriented current vector control). Only this practically widely applied closedloop regulated version is investigated in the following.
Indirect rotor flux control . In this case the ψ_{r} and α_{ψ} _{r} are not created directly, and ψ_{r} is not regulated in closedloop, only it is kept by openloop control. There will be an example for this method at the current source inverterfed drives (Chapter 8).
The block diagram of the drive controlling the torque by fieldoriented control in given in Fig.5.8. From the m_{a} torque reference the reference value of the torque producing current component can be derived using (5.10):
(5.22)
According to (5.20) the reference value of the rotor flux in the simplest case depends on the w_{ψ} _{r}=w_{1}=2πf_{1} fundamental angular frequency (approximately on the w speed) (Fig.5.9.a.):
(5.23.a,b)
The SZΨ flux controller provides the reference value of the flux producing current component: i_{da}. If there is only normal operation (Fig.5.7.a.), then the SZΨ can be omitted, and i_{da}=I_{dn}=Ψ_{rn}/L_{m}=const. flux producing current reference can be set. Similarly to Fig.4.5., a SZU voltage controller controlling the fundamental voltage vector amplitude (u_{1}) also can provide the i_{da} reference value (Fig.5.8.b.). It controls to u_{1}=U_{n} in the fieldweakening range, its upper limit must be set to I_{dn}, its lower limit must be set to I_{dmin} (Fig.5.7).
In energysaving operation, the ψ_{ra} flux reference can depend on the load (on the m_{a} torque reference). The torque expression (5.10) with ψ_{r}=const. operation by substitutions ψ_{r}=L_{m}i_{d}, i_{d}=icosϑ and i_{q}=isinϑ can be written in the following form:
(5.24.a,b)
As coming from (5.24.a), for m=const. torque the i_{d}i_{q} product is constant, i.e. it is a hyperbolic function on the i_{d}i_{q} plane. It is demonstrated in Fig.5.9.b. for current references (at the permanent magnet synchronous machine for m=const. torque the i_{q} is constant according to (4.8)). As can be seen in (5.24.b), a m>0 torque can be developed with the minimal current with ϑ=45° torque angle. Since besides the dependent copper losses there are core losses also depending on , the maximal efficiency energysaving operation for m>0 is at ϑ_{opt}>45°. ϑ_{opt} depends on the f_{1} frequency, since the core losses (5.19) are frequency dependent also. Near f_{1}≈0 frequency: ϑ_{opt}≈45°, since the core losses are zero, near f_{1}≈f_{1n} frequency: ϑ_{opt}≈60°. If independently of the frequency the torque angle is kept at ϑ_{n}=arctg(I_{qn}/I_{dn})=arctg(W_{rn}T_{ro}) corresponding to the N nominal point (f_{1n}, Ψ_{rn}=L_{m}I_{dn}, M_{n}=(3/2)Ψ_{rn}I_{qn}) (Fig.5.9.c.), then a suboptimal energysaving operation is got (it means ϑ=ϑ_{n}, W_{r}=W_{rn} rotor frequency). In this case the current references at m>0, w<W_{n} depend on the torque reference in the following way:
(5.25.a,b)
The operation is similar to the series excited DC machine.
If good dynamics is the goal, then i_{da}=I_{dn}=const. is necessary in the a normal range.
By the twolevel VSI (Fig.4.9) it is again not possible to track the reference value without error ( ). Fig.5.10. is redrawn from Fig.3.4.b. From the corresponding voltage equation (ū=Rī+L’dī/dt+ū’), substituting , the derivative of the current error vector for the ū=ū(k) voltage vector can be expressed (w_{k}=0):
(5.26)
The ū’ transient voltage can be calculated from (5.21):
(5.27)
The first term is zero in the ψ_{r}=Ψ_{rn}=const. normal range. A well operating current vector controller selects from the available seven ū(k) voltage vectors (4.14) that one, which results in small current error and small switching frequency.
Similarly to the permanent magnet synchronous machine the current references are available in d,q and the feedback signals are in a,b,c components and “same type” reference and feedback signals are necessary for the current vector control. The possibilities are demonstrated in Fig.5.11. which is very similar to Fig.4.6.
At the crosssections a,b,c,d,e, in the possible two coordinate systems, five different “sametype” reference and feedback signal combinations can be considered. Accordingly the following current vector controls can be implemented theoretically:
a section: coordinate system rotating with the rotor field, Cartesian coordinates,
b section: coordinate system rotating with the rotor field, polar coordinates,
c section: stationary coordinate system, polar coordinates,
d section: stationary coordinate system, Cartesian coordinates,
e section: stationary coordinate system, phase quantities.
It can be established, that the coordinate transformation cannot be avoided, and for the stationary→rotor field and the rotor field→stationary coordinate transformations the α_{ψ} _{r} angle of the rotor flux vector must be known. In practice, the a, or the e versions are used for current vector control (Fig.5.12.). In version a the references (i_{da}, i_{qa}), in version e the feedback signals (i_{a}, i_{b}, i_{c}) can be used directly. In version a two, in version e one coordinate transformation is necessary. Comparing Fig.5.12.a,b. with Fig.4.7.a,b. the high similarity between the current vector control of the cage rotor induction machine and the permanent magnet synchronous machine can be seen. The only one but significant difference comes from the fact, that different flux vector is used for the field orientation. At the PM synchronous machine it is the pole flux vector rotating with the rotor. Its angle (α) can be measured by a position encoder (P), its amplitude in ideal case is constant (Ψ_{p}=const.). At the cage rotor induction machine it is the rotor flux vector, neither its α_{ψ} _{r} angle nor its ψ_{r} amplitude can be measured directly. These can be produced by a machine model.
At the cage rotor induction machine the PWM modulator based and hysteresis current vector control methods (Fig.4.11.a,b.) also can be applied. Also the PWM modulator based methods in Fig.4.13. and Fig.4.16. are used widely in the practice. Thanks to the high similarity, universal drives are developed (UNIDRIVE), capable of current vector control of either PM synchronous machine or cage rotor induction machine.
A new component has appeared in Fig. 5.12.a,b: the machine model. The machine model manipulates measured signals and machine equations (Fig.5.13.) The machine equations need machine parameters. The parameters can be determined offline (before operation, offline identification) or online (during operation, online identification). These two methods are frequently applied together.
There are two models applied widely in the practice: the stator model and the rotor model. It should be considered in both of them, that the measurement can be done only in stationary coordinate system (w_{k}=0).
Stator model. Using the stator voltage equation (3.6.a) with w_{k}=0 and Fig.3.4.a., the x and y components of the rotor flux vector ( ) can be calculated:
(5.28.a,b)
(5.29.a,b)
The machine model in Fig.5.14. uses these equations. Besides the α_{ψ} _{r} angle of the rotor flux vector it provides the angular speed (w_{ψ} _{r}=dα_{ψ}/dt), the ψ_{r} amplitude and the m electromagnetic torque. The machine model uses the measured voltages and currents, and machine parameters R and L’. It is enough to measure two linetoline voltages and two line currents in practice. At low f_{1} frequency the term Rī is significant relatively to ū, so the inaccuracy of R stator resistance (caused by the temperature) can result in large error. The openloop analogue integrators calculating ψ_{x} and ψ_{y} flux components have error caused by the offset and drift. Consequently this model has a lower frequency limit in the practice (approx. 0.05f_{1n}=0.05⋅50=2.5 Hz). Because of these problems this model is not applied in servo and electric vehicle drives.
Rotor model. The rotor voltage equation (3.6.c) is the starting point. w_{k}=0 and ū_{r}=0 are substituted, and the not measurable rotor current is eliminated by (3.6.d):
(5.30)
The x and y components of can be expressed:
(5.31.a)
(5.31.b)
The machine model in Fig.5.15. uses these equations. It uses the measured the currents and speed, calculates the ψ_{r} _{, }α_{ψ} _{r}, w_{ψ} _{r} and m signals using the L_{m}, R_{r} and T_{ro}=L_{m}/R_{r} machine parameters. It has a great advantage against to the previous one: it does not contain openloop integration. The negative feedbacked integrators result in first order lag elements (the time constant is T_{r0}), so the offset and drift problems are avoided. This model can operate even until zero frequency. There are problems associated with the variation of the R_{r} rotor resistance (caused by the temperature, it is a slow process) and the variation of the L_{m} magnetizing inductance (caused by the variation of saturation in the fieldweakening range, it is a much faster process). In a sophisticated drive online identification is necessary to get the actual value of R_{r} and L_{m}. If the R_{r} and L_{m} parameters used in the model are inaccurate, then the calculated ψ_{r} and α_{ψ} _{r} values do not correspond to the actual values in the motor. Consequently the fieldoriented control uses these inaccurate values, so the control is done not exactly in rotor flux coordinate system, the decoupling in the current components i_{d} and i_{q} is deteriorated.
There is a combined model also, which produces not only the rotor flux, but the w speed also. If the rotor voltage equation (5.30) is multiplied by the conjugate of the rotor flux vector ( ), the folloving vector equation is got:
(5.32)
The x real and y imaginary components are:
(5.33.a)
(5.33.b)
The w speed can be expressed from the imaginary component:
(5.34)
This expression makes possible to calculate the speed without mechanical sensor (sensorless). The ψ_{rx} and ψ_{ry} flux components are calculated by (5.29.a,b), the w speed is calculated by (5.34) in Fig.5.16. The stator voltages and currents must be sensed. (5.33.a) can be used for parameter identification: if the parameters R, L’, R_{r}, L_{m} are not accurate, then v_{x}≠0. At constant flux operation (ψ_{r}=const., L_{m}=const.) the v_{x}=0 equality makes possible a simple online identification of one parameter, e.g. R_{r}.
Considering (5.2.b) (at PM synchronous machine (4.3) ) the ī current vector control can be implemented by the control of the stator flux. This principle is the background of the direct flux and torque control (shortly direct torque control DTC). It is described in the following for cage rotor induction machine, since DTC is applied widely for this one.
In the most commonly used application the m_{a} torque reference is set by the SZW speed controller (Fig.5.17.), the ψ_{a} flux amplitude reference is set by the FΨA setpoint element. The ψ_{a} flux reference is practically speed dependent, in the w≤W_{n} range it is: ψ_{a}=Ψ_{n}, in the w>W_{n} range it is: ψ_{a}=(W_{n}/w)Ψ_{n}.
Let’s substitute (5.2.b) to the torque expression of the induction machine (5.9):
(5.35.a)
(5.35.b)
So the torque can be calculated by the fluxes also (δ is the small angle between and ). In steadystate in the stationary reference frame the rotor flux vector rotates with w_{ψr}=dα_{ψr}/dt≈w_{1}=2πf_{1} fundamental angular frequency, while the stator flux vector can be modified by the applied ū(k) terminal voltage vector (3.6.a):
(5.36)
The twolevel VSI can switch 7 kinds of ū(k) voltage vector (4.14) to the machine terminals, so in every instant 7 different kinds of flux speed vector is possible. The amplitude and angle of the stator flux vector can be changed much faster than that of the rotor flux vector (caused by the L’ī term). The fastest torque modification can be done by changing the angle between them (d). The fastest d modification can be done by ū voltage vectors, nearly perpendicular to , since the d angle is small. E.g. at the instant demonstrated by Fig.5.18. the m>0 torque and the corresponding d>0 angle can be increased the fastest by switching the ū(1) voltage vector to the induction machine terminals. The fastest torque and d decrease can be reached by the ū(4) voltage vector. The ū(7)=0 voltage vector stops the vector, so the d angle and the torque decreases.
Identifying the flux vector angular position by six 60° sectors, general rules depending on sector number (N=1,...6) can be given for the selection of the voltage vectors. The flux vector sectors must be defined relatively to the ū(1), … ū(6) voltage vectors according to Fig.5.18. Neglecting the R resistance, the possible flux speed vectors are identical with the ū(k) voltage vectors. For examining the ith sector, the ū(1), … ū(6) voltage vectors are identified as in Fig.5.19. (the indices overflow at 6). Let’s assume w_{y} _{r}>0 and m>0 motor operation. By eq. (5.36) geometrically can be proved, that if the flux vector is inside the sector, then the absolute value of the flux vector is increased by the ū(i), ū(i+1) and ū(i+5), while decreased by the ū(i+3), ū(i+2) and ū(i+4) voltage vectors. In the same time, the torque (the d angle) is increased by ū(i+1) and ū(i+2), while decreased by ū(i+4) and ū(i+5) voltage vectors. The ū(7) zero voltage vector does not change the flux, but decreases the torque.
This direct flux and torque control keeps both the stator flux vector amplitude and the m torque in a prescribed band by bangbang control. The voltage vector to be switched to the induction machine is determined by three signals: the Δψ=ψ_{a}ψ flux amplitude error, the Δm=m_{a}m torque error, and the angular position of the flux vector given by the sector number N. A possible block diagram of the control scheme is given in Fig.5.20.
The generation of the y_{a} flux amplitude reference and the m_{a} torque reference is not given in the figure. The SZY flux controller is a twolevel hysteresis comparator, the SZM torque controller is a threelevel hysteresis comparator. So the possible values of KY are 1 and 0, the possible values of a KM are 1, 0 and 1. The block ARC determines the actual sector of from y_{x} and y_{y} components. The machine model is a simplified version of the stator model in Fig.5.14., since the y_{r} and a_{y} _{r} are not needed now. The torque is calculated with the m=(3/2)(ψ_{x}i_{y}–;ψ_{y}i_{x})expression.
By the rules determined for the i^{th} sector, in the function of KY, KM and N, the identifying number of the necessary ū(k) voltage vector can be given (Table.5.1.a.). The ū(7)=0 vector can be generated in two ways: all phases are connected to the P bar (7P) or to the N bar (7N), see Fig.4.9.a. This table is the Switching Table in Fig.5.20. The digitally stored table is addressed by a 6bit binary number composed from KY (1 bit), KM (2 bits) and N (3 bits).
Table 5.1.a. The identifying numbers of the ū(k) voltage vector. 
Table 5.1.b. The identifying numbers of the ū(k) voltage vector, if the ū(7)=0 vector is not used. 
KY 
KM 
N 
KY 
KM 
N 

1 
2 
3 
4 
5 
6 
1 
2 
3 
4 
5 
6 

1 
1 
2 
3 
4 
5 
6 
1 
1 
1 
2 
3 
4 
5 
6 
1 

0 
7P 
7N 
7P 
7N 
7P 
7N 
0 
6 
1 
2 
3 
4 
5 

1 
6 
1 
2 
3 
4 
5 
0 
1 
3 
4 
5 
6 
1 
2 

0 
1 
3 
4 
5 
6 
1 
2 
0 
5 
6 
1 
2 
3 
4 

0 
7N 
7P 
7N 
7P 
7N 
7P 

1 
5 
6 
1 
2 
3 
4 
The flux vector path of the direct flux and torque control using Table 5.1.a. as switching table is given in Fig.5.21. for the sector N=6, with w>0 and m>0 operating point qualitatively. The switchings are initiated by the flux controller in points A,B,C, while in points by the torque controller. In points A and C: Dy=DY, so KY changes from 1 to 0, in point B: Dy=+DY, so KY changes from 0 to 1. Accordingly between points A and B: KY=0, between points B and C: KY=1. In the points on the flux vector path: KM=0, else: KM=+1. The changing of the flux vector sector alone does not cause switching. From the ū(7P)=ū(7N)=0 voltage vectors that one is selected, which causes less switching number (Table 5.1.a.). Using the direct flux and torque control in Fig.5.20., only the references and the tolerance bends can be set, consequently the control is robust. In a practical implementation the tolerance bands (±ΔM and ±ΔΨ) are ±(0.01005) in perunit. Generally the torque has larger band than the flux (DM>DY). The minimal value of the DM and DY bands is determined by the allowed switching frequency of the inverter.
The switchon of the drive must be started by the development of the flux for the DTC also. KM must be set to 1, N to any value (1, 2,…6), and the derivative of the flux reference (dψ_{a}/dt) should be limited to limit the flux producing current also. The m_{a} torque reference may be enabled only after the development of the flux.
By the ū(7) voltage vector the positive m>0 torque is decreased at w_{ψ} _{r}>0, while it is increased at w_{ψ} _{r}<0. It can be seen clearly in Fig.5.22.a. ū(7)=0 stops the flux vector (5.36). In this case, if w_{ψ} _{r}>0, then angle δ and the m torque decreases (5.35.b), if w_{ψ} _{r}<0, then angle δ and the torque m increases. Consequently, at w_{ψ} _{r}>0 the rows KM=+1 and 0 act in Table 5.1.a., at w_{ψ} _{r}<0 the rows KM=1 and 0 act. As a result at w_{ψ} _{r}>0 the torque error is not negative: Δm=m_{a}m≥0, at w_{ψ} _{r}<0 it is not positive: Δm≤0 (Fig.5.22.b.). Accordingly the mean value of the torque (m_{k}) at w_{ψ} _{r}>0 is smaller by approx. ΔM/2, at w_{ψ} _{r}<0 is greater by approx. ΔM/2 than the m_{a}>0 torque reference.
It is an advantage of the described version (capable of 4/4 quadrant operation), that it controls the torque fast, and the controllers are robust.
It can be proved, that in a one rotation direction (w_{ψ} _{r}>0) 2/4 or 1/4 quadrant drive (e.g. in a wind turbine generator) the KM=1 rows of Table 5.1.a. are never used. So in this case the SZM hysteresis torque controller also can be twolevel comparator.
There is such a Switching Table (Table 5.1.b.), where the ū(7)=0 voltage vector is not used. In this case the SZM controller is ab ovo twolevel comparator. This strategy should be used in that case, when the torque (the d angle) must be controlled fast. Easily can be proved, that this strategy significantly increases the switching number.
Besides the cage rotor induction machines, the DTC is also applied for VSIfed PM synchronous and doublefed induction machine drives in the practice.
The 3phase wounded rotor slipring induction machine (Fig.6.1.a.) can be supplied from two side (stator and rotor sides). In sinusoidal symmetrical steadystate operation its speed can be modified by the stator and rotor frequency (f_{1} and f_{r}=f_{2}):
(6.1)
The sign of f_{2} is positive, if the phase sequences in the stator and the rotor are the same, and negative, if they are opposite. The powers can be expressed in the following way:
(6.2.a,b,c)
P_{1} is the stator terminal (input) power, P_{t} is the stator cupper loss, P_{core} is the stator core loss, P_{l} is the airgap power, P_{r} is the rotor power, P_{m} is the mechanical power, P_{tr} is the rotor cupper loss, P_{core} _{r} is the rotor core loss, P_{2} is the rotor terminal power. Neglecting the losses:
(6.3.a,b,c)
The powers can be expressed by the torque and angular speeds:
(6.4.a,b,c)
Where W_{1}=ω_{1}/p is the angular speed of the rotating field, (ω_{1}=2πf_{1} is its angular frequency), W is the rotor angular speed, W_{r}=W_{1}W=ω_{2}/p is the angular speed of the rotating filed relative to the rotor (ω_{2}=2πf_{2}), s=W_{r}/W_{1} is the slip. 2p=2 pole number is assumed in the following, so the angular speeds and angular frequencies are identical.
In the modern version of the doublefed induction machine (Fig.6.1.b.) the stator is connected directly to the lines (f_{1}=f_{h}=50Hz, W_{1}=2πf_{1}≌314/s), while to the rotor a VSI is connected. Both the machineside (ÁG) and lineside (ÁH) converters are twolevel VSIs. Neglecting the losses and using the notations in Fig.6.1.b.:
(6.5.a,b)
The power flow is presented in Fig.6.2. for lossless case. As can be seen, in under synchronous speed (subsynchronous) drive and above synchronous speed (oversynchronous) brake operation power is drawn from the rotor, (P_{2}=P_{r}>0), while in oversynchronous drive and subsynchronous brake operation power is supplied to the rotor (P_{2}=P_{r}<0). It can be established, that the power directions are W_{r} and M dependent. The power circuit in Fig.6.1.b. is capable of bidirectional power flow (P_{2}>0 and P_{2}<0), since U_{e}=const.>0 but I_{ek} can be bidirectional (I_{ek}>0 and I_{ek}<0). If ÁG would be a diode bridge, then only P_{2}>0 is possible, (this is the case of the subsynchronous cascade drive). Only the rotor power (P_{2}=P_{r}=MW_{r}) flows through ÁG and ÁH converters. Consequently they must be designed to the power (designed rating):
(6.6)
│M│_{max} and │W_{r}│_{max} are not surely developed in the same time. │M│_{max} determines the rotor current, │W_{r}│_{max} determines the rotor voltage. A usual operation range is given in Fig.6.3. Here: W_{max}/W_{min}=2, P_{ÁItip}=M_{n}W_{1}/3≌P_{n}/3. In this case the ÁG and ÁH converters should be designed to onethird of the nominal power of the induction machine (P_{n}=M_{n}W_{n}≌M_{n}W_{1}) only, but below W_{min}=(2/3)W_{1} speed the converter ÁG must be disconnected from the rotor, since large rotor induced voltage is developed in it.
As a result of the current vector control of ÁG (Fig.6.1.b.) the rotor is supplied with constrained current (currentfed). Assuming ideal lines, the stator is supplied with constrained voltage (voltagefed) (approximately constrained flux). Consequently fieldweakening is not possible in this case.
Because of the constrained stator flux, the rotor current vector control of the ÁG converter should be oriented to the stator flux. For the same reason, the equivalent circuit for fluxes in Fig.6.4. should be used (it corresponds to Fig.3.3.b, L’_{r} is the rotor transient inductance).
The constrained ū=ū_{h} voltage and f_{1}=f_{h}=50Hz frequency means approximately flux constrain also (consider (3.6.a) with R=0 and w_{k}=0: ):
(6.7)
(6.8)
Because of the flux constraint the field coordinate system is fixed to the stator flux (Fig.6.5.). In this field coordinate system:
(6.9.a,b,c)
By the current Kirchhoff’s law (see Fig.6.4.) , two component equations can be given:
(6.10.a,b)
The rotor current components (i_{rd} and i_{rq}) can be controlled directly (Fig.6.1.b.), but it means indirect stator current components (i_{d} and i_{q}) control also. According to (6.10.a), the ψ=L_{m}(i_{d}+i_{rd}) flux development task can be shared between the stator and the rotor flux producing current components (i_{d} and i_{rd}). The torque expression with Parkvectors (spacevectors), considering is:
(6.12)
The torque is determined by the torque producing current components (i_{q}=i_{rq}). With R=0 approximation, according to (6.8): . That is why approximately the i_{q} component is proportional to the stator active power (p), while the i_{d} component is proportional to the stator reactive power (q):
(6.14.a,b)
As can be seen, active power is demanded for torque production, and reactive power for flux production.
Coming from (6.13), the demanded torque determines the i_{q}=i_{rq} components only. The d current components can be chosen freely (keeping the rule (6.10.a)). If i_{rd}=Ki_{m},_{ }then i_{d}=(1K)i_{m} is required. Fig.6.6. shows the current vectors in field coordinate system for m=const.>0 and different K sharing constant. At K=0 the stator, at K=1 the rotor, at K=0.5 fiftyfifty the stator and the rotor develop the ψ flux. At K>1 the doublefed induction machine is overexcited, at K<1 underexcited. If R=R_{r}, then the minimum of the P_{t}+P_{tr} resultant cupper loss is at K=0.5.
The block diagram of the drive, controlling the torque by stator flux fieldoriented control is given in Fig.6.7.a. The reference values of the rotor current components can be derived from the torque reference (m_{a}) and the flux amplitude (ψ):
(6.15.a,b)
To determine i_{rda} the K sharing coefficient and the L_{m} magnetizing inductance must be given. Considering (6.14.b), i_{rda} can be provided by a reactive power controller also (Fig.6.7.b.).
Fig.6.7. Fieldoriented torque controlled drive. a. With i_{rda} setpoint element, b. with SZQ reactive power controller.
The rotor current references are available in d,q and the feedback signals are in r _{a}, r_{b}, r_{c} components. Sametype reference and feedback signals are necessary for the rotor current vector control. The possibilities are demonstrated in Fig.6.8.b. which is very similar to Fig.4.6. and Fig. 5.11.
In the crosssections a,b,c,d,e, in the possible two coordinate systems, five different “sametype” reference and feedback signal combinations can be considered. Accordingly the following current vector controls can be implemented theoretically:
a section: coordinate system rotating with the stator field, Cartesian coordinates,
b section: coordinate system rotating with the stator field, polar coordinates,
c section: coordinate system rotating with the rotor, polar coordinates,
d section: coordinate system rotating with the rotor, Cartesian coordinates,
e section: coordinate system rotating with the rotor, phase quantities.
It can be established, that the coordinate transformation cannot be avoided, and for the rotor →stator field and the stator field→rotor coordinate transformations the α_{ψ} angle of the stator flux vector must be known. In practice, the a, or the e versions are used for current vector control (Fig.6.9.). In version a the references (i_{rd} _{a}, i_{rqa}), in version e the feedback signals (i_{ra}, i_{rb}, i_{rc}) can be used directly. In version a two, in version e one coordinate transformation is necessary.
Comparing Fig. 6.9.a,b. with Fig.4.7.a.,b. and Fig.5.12.a.,b. the high similarity between the current vector control of the different type machines can be found.
The PWM modulator based and the hysteresis current vector control methods (Fig.3.11.a.,b.) here also can be applied. The PWM modulator based current vector control methods similar to Fig.4.13. and Fig.4.16. are applied widely in the practice.
The machine model in Fig.6.7. and Fig.6.9.a.,b. must be made in the following way. The task of the machine model is to provide the ψ amplitude and α_{ψ} angle (relative to the ra rotor axis) of the stator flux. Using the stator voltage equation (3.6.a) with w_{k}=0 the xy components, the amplitude and the angle of the stator flux vector can be calculated:
(6.16.a,b)
(6.17.a,b)
The angle of the vector in the coordinate system rotating with the rotor according to Fig.6.5. is:
(6.18)
As can be seen, this calculation needs the angle of the rotor (α). In a machine model using the equations above, with the notation of Fig.5.13., the measured signals are: u_{a}, u_{b}, u_{c}, i_{a}, i_{b}, i_{c} and α, the used machine parameter is: R, the calculated values are: ψ and α_{ψ}.
The stator model used in chapter 5.3.2. has the same equations (5.28.a,b) as here (6.16.a,b). There the error caused by the inaccuracy of resistance R at low f_{1} frequency was described. It is not a problem here, since the frequency is high: f_{1}=f_{h}=50Hz.
The direct torque and flux control can be applied for the doublefed induction machine also, (chapter 5.4.), but here the ψ_{r} amplitude of the rotor flux vector is controlled by bangbang control. It can be proved, that the ψ_{ra} reference value is well proportional to the ψ amplitude of the stator flux vector.
Tartalom
The U_{e} DC voltage of the PWM VSI converters (chapter 4.2.2.) is available directly only in few cases, e.g. in batteryfed, solarcellfed, fuelcellfed or DC overheadcontact linefed vehicle. In industrial drives it must be produced from the 3phase lines (f_{h}=50Hz frequency) by an AC/DC converter (ÁH). That is why there is a DC link circuit between the ÁH and ÁG converters. The simplest AC/DC converter is a diode bridge, connected to a C smoothing capacitance (Fig.7.1. without voltage limiter). After the initial charging of C, the R_{t} charging resistance is shortcircuited. In this version the mean value of the i_{e} DC current can be only positive steadily (i_{ek}≥0) so DC power (p_{e}) and its mean value can be only positive: p_{ek}=U_{e}i_{ek}≥0. That is why for PM synchronous machine and for cage rotor induction machine only driving operation (motor mode) is possible: p_{m}=mw>0.
In servo drives the generator mode braking (p_{m}<0, p_{e}<0, i_{e}<0) exists only for short time transients. During these operations the voltage limiting brake circuit (Fig.7.1.) plays role: the braking energy is dissipated on the R_{f} brake resistance. Assuming bangbang voltage limiting, Fig.7.2. shows i_{e} and u_{e} during the braking process qualitatively.
The diode bridge in Fig.7.1. operates as a peakvalue rectifier, consequently its currents supplied from the lines have quasi pulse shape.
In modern VSI drives the ÁH converter is capable of bidirectional power flow and networkfriend operation. In a networkfriend operation the phase currents are symmetrical, sinusoidal and their phase angle (φ_{h1}) relative to the corresponding voltage can be set. These tasks can provided by VSI type ÁH lineside converter (Fig.4.9.a.)
The power circuit diagram of the VSI type ÁH lineside converter (more and more spread in practice) is given in Fig.7.3. The machineside ÁG is not detailed now. The ÁH VSI connected to the lines via filter circuits. The simplest filter is a 3phase choke or a transformer. It is assumed in the following. Accordingly L_{h} and R_{h} contain the inductance and resistance of the filter also. The R_{t} charging resistance is shortcircuited during operation.
Assuming lossless energy flow, in steadystate the line fundamental power (P_{h1}) is equal to the DC mean power (P_{ek}) and the motor (PM synchronous or shortcircuited induction) mechanical power (P_{m}).
(7.1)
Where U_{h} is the peak value of the sinusoidal phase voltage of the lines, I_{h1} is the peak value of the fundamental line current, U_{e} is the smooth DC voltage, I_{ek}=I_{ehk}=I_{egk} is the mean value of the DC current, M_{k} is the mean value of the torque, W is the constant speed. In motor drive operation (P_{m}>0) the mean value of the DC current is: I_{ek}>0 and the active component of the line current is: I_{h1p}=I_{h1}cosφ_{h1}>0. At generator brake (P_{m}<0): I_{ek}<0 and I_{h1p}<0. A given power can be provided with the smallest line current (I_{h1}) with cosφ_{h1}=±1 power factor.
The fundamental controlling task of ÁH is the DC voltage (u_{e}) control. From the current Kirchhoff’s law for the DC link in Fig.7.3. (i_{c}=i_{eh}i_{eg}) and multiplying it by u_{e}, the DC power equation is got:
(7.2)
The aim is u_{e}=U_{e}=const., du_{e}/dt=0, i_{c}=0, p_{c}=0, which can be provided by p_{eh}=p_{eg} (p_{eg} is approx. the same as the mechanical power p_{m}=mw). Since both the lineside (p_{eh}) and machineside (p_{eg}) power are pulsating, the balance of the DC power can be ensured only for mean values: p_{ehk}=p_{egk}, p_{ck}=0. So the aim can be implemented by line power (p_{h}) control subordinated to a DC voltage control. Assuming ideal lines, the line power control can be reduced to ī_{h} line current vector control.
The line (incl. the filter) is modelled by an ideal voltage source and series L_{h}R_{h} elements in Fig.7.3. In this ideal case the control of the ÁH converter is oriented to the
(7.3)
line voltage vector, or rather to its integral:
(7.4)
which is a fictive flux vector (ω_{h}=2πf_{h}, ).
Fig.7.4. shows the Parkvector equivalent circuit and the vector diagram of the ī_{h} current vector in stationary coordinate system and in coordinate system fixed to the line voltage vector. In Fig.7.4.a. the ū_{H} is the voltage vector provided by the VSI type ÁH, which has, according to (4.14) 7 discrete values, if the controllability condition ( ) is satisfied. Comparing the equivalent circuit in Fig.7.4.a. with the equivalent circuits in Fig.4.2.b. and Fig.5.10., it can be established, that here the line plays the role of the machines (PM synchronous or shortcircuited induction motor). In the coordinate system fixed to the voltage vector (Fig.7.4.c.):
(7.5.a,b)
( ). Using the i_{hp} active and the i_{hq} reactive current components, the line active and reactive powers can be calculated:
(7.6.a,b)
The DC voltage can be controlled by the p_{h}≌p_{eh} active power and by the i_{hp} current component. With i_{hq}=0 only active, with i_{hp}=0 only reactive power flows on the lineside.
From voltage equation (considering Fig.7.4.a.) ū_{H}=R_{h}ī_{h}+L_{h}dī_{h}/dt+ū_{h} substituting ī_{h}=ī_{ha}Δī_{h} the derivative of the line current error vector can be expressed:
(7.7.a,b)
These equations are very similar to (4.16 and 5.26). A well operating current vector controller selects always the optimal from the available seven ū_{H}(k) voltage vectors (4.14).
The current references are available in pq and the feedback signals are in abc components. “Sametype” reference and feedback signals are necessary for the line current vector control. The possibilities are demonstrated in Fig.7.5. which is very similar to Fig.4.6.b. and Fig. 5.11.b. The reference values of the active and reactive current components can be derived from the active and reactive power references (7.6.a,b):
(7.8.a,b)
In the crosssections a,b,c,d,e, in the possible two coordinate systems, five different “sametype” reference and feedback signal combinations can be considered. In practice, the a, or the e versions are used for current vector control:
a section: coordinate system rotating with the line voltage vector, Cartesian coordinates,
e section: stationary coordinate system, phase quantities.
These two versions are demonstrated in Fig.7.6. The Line model is a new element, which provides the U_{h} amplitude and α_{uh} angle of the ū_{h} ideal line voltage vector (Fig.7.4.a.). The L_{sz} in Fig.7.6.a.,b is the inductance of the filter (choke or transformer) (it is the larger part of L_{h} in Fig.7.3. and Fig.7.4.a).
Fig.7.6. The block diagram of the current vector control. a. In ccordinate system rotating with the line voltage, Cartesian coordinates (version a), b. In stationary coordinate system, phase quantities (version e).
The current vector control methods similar to Fig.4.13. and Fig.4.16. are applied widely in the practice in this case also.
An example for the application of VSI type converter on line and machineside is given with a cage rotor induction machine drive (Fig.7.7). The torque control subordinated to the SZW speed control is the same as in Fig. 5.8.a. On the lineside the SZU DC voltage controller provides the reference value of the active current component (i_{hpa}), i_{hqa} is determined by the demanded q_{ha} reactive power (7.8.b). SZIG and SZIH are the PWM based current vector controllers of the machine and lineside respectively (see Fig.4.13. and Fig.4.16.)
During the charging of C the converter ÁG is disabled. In the initial period converter ÁH is also disabled. In this period the diode bridge in ÁH charges C via the R_{t} charging resistance, assuming i_{eg}=0 up to linetoline peak voltage. At the end of this charging period R_{t} is shortcircuited. Enabling the SZIH line current vector controller subordinated to the SZU DC voltage controller (it is generally PI) the u_{e} DC voltage increases up to the u_{ea}>U_{hvcsúcs} reference value. Meanwhile i_{hpa}>0 because of charging C. E.g. at 3x400V+10% line voltage, . Accordingly in this case the DC voltage is near U_{e}=U_{ea}=700V. After the charging of C the machineside controllers are also enabled. During control, the same 7 kinds of voltage vectors ū(k) (4.14) can be switched by ÁH to the lineside terminals ha, hb, hc as by ÁG to the machine terminals a,b,c. The conditions of the inverter’s controllability are: U_{e}>U_{hvcsúcs} and U_{e}>U_{gvcsúcs} (U_{gvcsúcs} is the peak value of the linetoline induced voltage in the induction machine)
Similarly to the machineside direct torque and flux control (chapter 5.4.), hysteresis direct active and reactive power control (DPC) on the lineside also can be applied. The role of the stator flux vector ( ) is played by , the role of the rotor flux vector ( ) is played by the fictive flux vector (7.4), the role of L’ is played by L_{h}. In this case instead of the m torque and the ψ flux amplitude the active power (p_{h}) and the reactive power (q_{h}) are controlled respectively, by bangbang control. Both hysteresis controllers are twolevel. The switching tables (5.1.a. and b.) can be used, but in Table 5.1.a. the row KM=1 is omitted, since the rotates in one direction only. The advantage of DPC is its robust behaviour and the lack of the coordinate transformation.
Tartalom
The current source inverter (CSI)fed drives similarly to the industrial VSIfed drives are in the DC link inverter category. There are two types in practice: the CSI with thyristors and the pulse width modulated (PWM) CSI.
The power circuit of the thyristor CSIfed induction machine (IM or AM in Fig.8.1.) drive is given in Fig.8.1.a. The ÁH is a linecommutated thyristor bridge converter, the ÁG is the CSI with thyristors. GVÁH and GVÁG are the firing controllers of ÁH and ÁG respectively.
There is a choke L_{e} directly on the terminals of ÁG in the DC link, which provides current constraint for short time. The current source type of the DC current (i_{e}=I_{e}) is supported by the current control with ÁH too. There are no dedicated turnoff circuits to the thyristors in ÁG, they have so called phase sequence commutation. The firing of the subsequent thyristor starts the turnoff process of the conducting thyristor, and the current is transferred to the new phase gradually in the given bridge side. There are no antiparallel diodes on the thyristors, since the DC current may be only positive: i_{e}≥0. The series diodes (DPA,…DNC) separate the properly charged capacitors C from the motor, preventing their discharge between the commutations.
In practice squirrelcage type IM is supplied by CSI. Attention must be payed on the fact, that the commutation process (not detailed here) is determined by the C capacitors and the L’ transient inductance of the IM together (Fig.5.10. and Fig.8.1.b.). So the C capacitors of the CSI must be fitted to the motor parameters (approximately to the motor power).
The 2/4 quadrant operation of ÁH lineside converter is enough for the 4/4 quadrant (regenerating) operation of the drive, too. Considering the P_{ek}=U_{ek}I_{e}≌P_{m}=M_{k}W power equation and (2.11) in motor mode U_{ek}≌U_{ekm}cosα_{h}>0 (α_{h}<90^{o}) the operation mode of ÁH is rectifying, in generator mode it is inverter mode: U_{ek}<0 (α_{h}>90^{o}).
The firing control of ÁG is done with variable f_{1} fundamental frequency. Neglecting the commutation process (i.e. assuming instantaneous commutation) the motor phase currents (i_{a}, i_{b}, i_{c}) vs. ω_{1}t=2πf_{1}t are shown in Fig.8.2.a., the current vector ī is shown in Fig.8.2.b. in steadystate. There are two conducting phases in every instant, one on the P positive side, one on the N negative side. According to the six possible twophase conduction modes 6 different current vector can be developed:
(8.1)
This expression is similar to the expression of VSI: (4.14). The fundamental current vector is:
(8.2.a,b)
Assuming lossless CSI and IM (R≈0), the power mean values in Fig.8.1. are identical:
(8.3)
Index k denotes mean value, index 1 denotes fundamental quantity. Considering (8.2.b):
(8.4)
Consequently the φ’_{1} phase angle of the fundamental current vector ( ) relative to the transient voltage vector ( ) in the ÁG current source inverter is similar to the firing angle (α) of a linecommutated converter.
There are two means for intervention in a thyristor CSI:
In ÁH by α_{h} through the U_{ek} DC voltage the i_{e} DC current, and so the i_{1} fundamental current amplitude can be controlled.
In ÁG the α_{i1} angle of the ī_{1} current vector, and so the dα_{i1}/dt=ω_{1}=2πf_{1} fundamental angular frequency can be controlled.
In this case, considering the two intervention possibilities, the fieldoriented current vector control is implemented by method c in the coordinate transformation chain (Fig.5.11.b.). The current references are produced directly in dq components here also. According to (5.22) the i_{1qa} torque producing fundamental current component is determined by the m_{a} torque reference. The i_{1da} flux producing fundamental current component is set by the rotor flux controller. The fundamental current reference vector and its components are demonstrated in Fig.8.3.
Fig.8.4. shows the block scheme of the fieldoriented CSIfed cage rotor IM drive for direct rotor flux control.
By Descartes(Cartesian)/Polar transformation from the i_{1da} and i_{1qa} components the fundamental current amplitude (i_{1a}=│ī_{1a}│) and the torque angle (ϑ_{1a}) reference values can be got. According to Fig.8.3. the angle of the fundamental current reference vector (ī_{1a}) in stationary coordinate system is:
(8.5)
The ψ_{r} amplitude and α_{ψ} _{r} angle of the rotor flux is calculated by the motor model (Fig.5.15.). The ψ_{r} rotor flux amplitude is controlled by the SZΨ flux controller. The rotor flux amplitude refernce (ψ_{ra}) depends on the w angular speed only in the simplest case (5.23). The SZI current controller directly controls the i_{e} DC current, indirectly the i_{1}=│ī_{1}│ amplitude of the ī_{1} fundamental current vector. The α_{i1}=α_{i1a }angle of the ī_{1} current vector for ω_{1}>0 positive sequence operation can be ensured by firings given in Fig.8.5. E.g. when the ī_{1a }vector at α_{i1a}=0^{o} enters to the bold 60^{o}sector, the NC thyristor should be fired to move the current vector from ī(1) to ī(2). Next at α_{i1a}=60^{o} PB must be fired.
Because of the noninstantaneous commutation, at high speed with the previously described firings the α_{i1} angle would be inaccurate. The compensation of the effect of the practically constant commutation time on the firing instant can be solved.
Fig.8.6. shows the block scheme of the fieldoriented CSIfed drive for indirect rotor flux control. In this case there is no machine model, ψ_{r} and α_{ψ} _{r} are not available.
Using (5.6.b) and (5.10) for references, the fundamental current component references can be derived:
(8.6.a,b)
The angular speed and angle of the rotor flux vector (5.11) relative to the stator are calculated from references also:
(8.7)
α_{ψ} _{ro} is the initial angle of the rotor flux vector, which is determined by the firstly fired two thyristors in ÁG. The angle of the ī_{1a} current vector reference can be calculated similarly to (8.5), but α_{ψ} _{ra} is used:
(8.8)
The bold part of Fig.8.6. is drawn using (8.6, 8.7, 8.8). It can be seen from the expressions, that the R_{r} and L_{m} machine parameters must be known here also. The thin line drawn part is the same as that of in Fig.8.4.
Formerly the thyristor CSI–fed drives are widely applied thanks to its robustness in medium power 4/4 quadrant drives.
In a PWM CSI fully controllable semiconductors are used. Fig.8.7. shows the IGBT version, while Fig.8.8. shows the GTO version. Only the ÁG converter is drawn, since basically ÁH converter is the same.
The IGBTs can not withstand more than 1015V blocking voltage, that is why the series diodes (DPA,…DNC) are necessary in the IGBT version. The diodes parallel to the IGBTs are not necessary principally, but they are used for the sake of safety. In the GTO version the motor is represented by the equivalent circuit in Fig.8.1.b. and a Space Vector PWM (SPWM) is also indicated. The DC current can not be interrupted because of L_{e}. It can be avoided by overlapping the conduction of the switches in one bridge side, i.e. the switchon precedes the switchoff. The motor currents also can not be interrupted because of the L’ transient inductance, that is why the C capacitors are necessary.
The current vector of a PWM CSI (ī) (Fig.8.9.) similarly to the PWM VSI (4.14) can have 7 different states:
(8.9)
The ī(7)=0 zero current vector can be developed by controlling ON both switches in a leg simultaneously (e.g. PA and NA) while the others are off. Then in spite of i_{e}=I_{e}>0, ī=0 is developed (ī_{m}=ī_{c}). The motor current vector (ī_{m}) is the difference of the PWM CSI current (ī) and the capacitors’ current (ī_{c}):
(8.10)
The C capacitance is fitted to the L’ inductance of the motor to get: ī_{m1}≈ī_{1}, ī_{c1}≈0 for the fundamental components, and ī_{mv}≈0, ī_{v}≈ī_{cv} for the upper harmonics. So in steadystate the motor current is approximately sinusoidal.
From the PWM methods described with the VSI (Chapter 4.2.3.1.) the space vector PWM method can be applied without any changes for the CSI. In the nth sampling period the ī_{1}(n) vector prescribed by the controllers can be produced as an average of the 3 neighbour vectors ī(k) switching them for the proper time interval. In Fig.8.9. ī_{1}(n) is in sector 1 (the sector is 60^{o} wide), now ī(1), ī(2) and ī(7) are the 3 neighbour vectors. ī_{1}(n)similarly to (4.18) is provided as a time average of these 3 vectors. As PA is on for ī(1) and ī(2) also, in this sector to reduce the switching number ī(7)=0 current vector should be produced by switching on PA and NA. Similarly in sector 2: PC and NC, in sector 3: PB and NB, in sector 4: PA and NA, in sector 5: PC and NC, in sector 6: PB and NB are the proper selection for ī(7)=0. By the space vector PWM the maximal fundamental harmonic current amplitude is I_{1max}=I_{e}. Using the scheme of the fieldoriented control in Fig.8.4., the inputs of the PWM controller are the amplitude (i_{1a}) and the angle (α_{i1a}) of the current reference vector. Very fast current control can be implemented, since the fundamental current can be controlled in spite of the i_{e}=I_{e}=const. DC current. There are modern, networkfriend versions, where the lineside converter (ÁH) is also a PWM CSI circuit.
Nowadays the CSIfed drives with thyristors are used rarely, since the VSI with f_{ISZM}≥2000Hz can provide better current, flux and torque behaviour. The wide spread application of the PWM CSI is limited by the associated resonance problem. Using the ÁH converter in Fig.7.3., no problem to make 4/4 quadrant networkfriend operation with VSI.
The circuit diagram of the converterfed synchronous motor (CFSM) drive is given in Fig.9.1. Here all of the converters: the lineside ÁH, the motorside ÁM and the excitationside ÁG converter operate with line commutation. The line commutation of the thyristors in ÁM is possible, while the overexcited synchronous machine can provide the reactive power necessary for the commutation. In ÁM the commutations are done by the subtransient voltages of the SZ synchronous machine, that is why this commutation is called machine (load) commutation also. The supply is currentsourcetype, caused by the DC filter choke L_{e}.
The converter ÁM can be controlled to rectifying and to inverter mode, so in spite of the unidirectional DC current mean value (I_{ek}>0) the CFSM is capable of motor and generator mode operation. In motor mode ÁH is a rectifier, ÁM is an inverter, the mean value of the DC voltage is negative: U_{ek}<0. In generator mode the converter modes are exchanged, consequently: U_{ek}>0. Reversing the phase sequence of firing the thyristors of ÁM bidirectional rotation in driving and braking mode is possible (4/4 quadrant operation).
Fig.9.2. shows the block diagram of the flux and speed controlled CFSM. α_{h} firing angle is the acting signal of the speed controller, α_{g} firing angle is that of the flux (excitation) controller. Usually both controllers have subordinated current control loop. The α firing angle of ÁM is set by a selfcontrolled firing controller operated from signals of the synchronous machine SZ. By the selfcontroller the torque development can be optimized in motor (M) and generator (G) mode.
From the DC sides of converters ÁH and ÁG the selfcontrolled ÁM CFSM (the dottedline surrounded part of Fig.9.2.) looks like a DC machine. In a real DC machine only u_{e} and u_{g} can be modified, the modification of the brush rocker position corresponding to the firing angle of the machineside converter (α) is not used for this purpose. In a CFSM the excitation must be controlled always, because of the large armature reaction of the synchronous machine.
Assuming ideal, zero resistance (R_{r}=0) rotor winding, for a given excitation current i_{g} zero rotor voltage is necessary: ū_{r}=0. In this way in w_{k}=w coordinate system the (3.6.c) rotor voltage equation is the (3.6.d) rotor flux equation is . This is the principle of the so called flux constancy: the resistanceless shortcircuited coil does not allow the variation of the flux linked with it. So in every operating point the subtransient flux vector linking with the rotor winding is constant. In stationary coordinate system (w_{k}=0) the subtransient flux vector and the corresponding induced voltage vector are (assuming constant speed operating point: w=dα_{r}/dt=const., α_{r} is the angle of the rotor):
(9.1.a,b)
It means, that in steadystate both and ū” rotate with W=W_{1}=2πf_{1} rotor/fundamental angular speed and their amplitudes (Ψ” and U”) are constant. Selecting t=0 instant to the positive maximum of u_{a}”:
(9.2.a,b,c)
The stator voltage equation in stationary reference frame (3.6.a) considering (3.7) is:
(9.3)
Using (9.3) the equivalent circuit of CFSM can be drawn (Fig.9.3). Comparing with Fig.2.7. high similarity can be found with R→R_{t}, L”→L_{t}, u_{a}”→u_{ta} substitution.
In the ÁM motorside converter according to the 6 thyristors the commutation frequency is variable: 6f_{1} since the fundamental frequency is variable. Considering ideal thyristors, smooth DC current (i_{e}=I_{e}) and R=0 stator resistance the classical linecommutated converter theory with overlap for steadystate can be applied (the overlap must be considered, since L” is much greater with one order than L_{t}). This theory gives the following expressions for the DC voltage and current mean values:
(9.4.a,b)
Where . The α firing angle, the κ extinction angle (δ=κα is the overlap angle) and the μ=180^{o}κ commutationreserve angle are related to the subtransient voltage. Fig.9.4.shows the vectors of the terminal voltage (ū), the subtransient voltage (ū_{”}) (9.2.b) and the current (ī) in inverter mode operation. The 60° sector started with the firing of the NC thyristor is drawn in bold. Using (9.3) (and approximation R=0) the derivative of the current vector (ī) is:
(9.5)
E.g. this is the speed of the current vector movement during the commutation NB→NC from point 1 to point 2. Considering the L”dī/dt vector movement speed, the control limits of the thyristor NC (B: firing ON limit (α=0º); K: extinction limit (µ=0º)) are marked on ū_{”}. In generator/rectifier mode the drive can operate at the firing ON limit: α=α_{min}=0^{o} also. In motor/inverter mode for the sake of safety the extinction limit (κ=κ_{max}=180^{o}) must not be reached, only maximum κ_{meg}=160^{o} extinction angle is allowed approximately.
In steadystate, neglecting the losses the P_{mk} mechanical power is equal to the mean values of the P_{ℓk} airgap power and the P_{ek} DC link power (in motor/inverter mode: P_{mk}>0, P_{ek}<0):
(9.6)
Using (9.4) and (9.6) the mean values of the speed and the torque can be expressed:
(9.7.a,b)
The maximal torque is developed by the CFSM at κ_{max}=180^{o} extinction limit in motor mode, and at α_{min}=0^{o} firing ON limit in generator mode. Using (5.9, 5.10) the expression of the torque is:
(9.8.a,b,c)
According to (8.2.b) the amplitude of the fundamental current (I_{1}) is proportional to the DC current (i_{e}=I_{e}) with good approximation:
(9.9)
With given Ψ” and I_{1} the maximal M_{k} is at ϑ_{1}=±90^{o} torque angle. Fig.9.5. shows M_{k}/M_{n} relative torque (referred to M_{n}=(3/2)Ψ_{n}I_{n} nominal torque) vs. I_{1}/I_{n}≌I_{e}/I_{en} (where ). Besides the machine (load) commutation operation limits of ÁM (α_{min}=0^{o}os and κ_{max}=180^{o}) the safe motor/inverter mode limit curve is also drawn (κ_{meg}=160^{o} extinction angle). It can be established, that similarly to the separately excited DC machine the torque is proportional to the DC current. In motor mode: M_{k}=K_{M}I_{e}, in generator mode: M_{k}=K_{G}I_{e}, K_{M}>0, K_{G}<0. In the shaded areas the drive can operate only with forced commutation (VSI or CSI supply). A given M_{k} torque should be developed with the possibly minimal I_{1} current. For the CFSM it requires a twostage self controlled firing controller, which provides κ=κ_{meg} operation in motor/inverter mode, and α=α_{min} operation in generator/rectifier mode. In practice the controls from the position of the shaft (α) or from the position of the subtransient flux vector (α_{ψ}”) are used. The former is called: firing control from the shaft, the later: firing control from the flux.
The firing controller from the subtransient flux vector results in fieldoriented firing control. Fig.9.6.a. presents the block diagram, Fig.9.6.b. shows the firing levels (for w>0 and a,b,c phase sequence). The flux vector is provided by stator machine model (detailed in Fig.5.14), the α_{ψ}” angle and the ψ” amplitude are provided by the blocks ARC and AMPL respectively. The M/G motor/generator twostage signal changes the comparing levels according to the κ_{meg} and α_{min} operation modes. In motor mode to the κ=κ_{meg}=const. extinction angle a operation point dependent firing angle is corresponding (α=κ_{meg}δ). Therefore in this case the Δ signal of the FG function generator corrects the firing comparing levels using the load dependant input value I_{e} (in fieldweakening more accurately I_{e}/ψ”). The firing control from the subtransient flux vector is identical with the firing control from the subtransient voltage vector, since e.g. at w>0: α_{u}”=α_{ψ}”+90°.
More practical to fire from , since it moves on a more smooth path than the voltage vector, and its amplitude in the normal range is constant, while the amplitude of ū_{”} is proportional to w. The firing from the shaft position (α) has larger load dependency in motor/inverter mode, and Δ correction is necessary in generator/rectifier mode also.
Fig.9.7. shows the current vector loci in coordinate system fixed to the flux vector (field coordinate system) with fieldoriented firing control and machine commutation, assuming smooth DC current (i_{e}=I_{e}). The marked amplitudes and angles come from Fig.9.4.; Fig.9.4.a. and Fig.9.7.b. correspond to operation point with κ=κ_{meg}=160° and approx. nominal motor mode I_{1} current. In this case approx. ϑ_{1}=120° can be reached as best torque angle.
It can be proved, that in the range R>WL” the safe machine commutation is not possible. The border angular speed is:
(9.10)
With the usual machine parameter it is reached at f_{1h}≌5Hz. In the W<W_{h}, f_{1}<f_{1h} range step commutation is used. Since in this case f_{h}/f_{1}>50 Hz/5 Hz=10, so there are at least 10 firings in ÁH between two firings of ÁG. Consequently in this case the commutation can be done by ÁH, by controlling the current vector to zero in every 1/6^{th} machine period (Fig.9.8.a.). The step commutation using the ī=0 current vector can be made faster, if during the commutation the TE thyristor connected parallel with L_{e} (Fig.9.1.) is fired ON. Therefore the current flowing in L_{e} can remain unchanged (i_{e}=I_{e}) during the commutation, only the motor current must be reduced to zero, and then must be increased back to I_{e}. There is not a limit for the phase angle of Ī_{1} in step mode, so the torque angle in motor mode can be ϑ_{1}=90°, in generator mode ϑ_{1}=270°(90°) also (Fig.9.8.b.,c).
In the range W>W_{n}, f_{1}>f_{1n} field weakening must be applied. In this case the amplitude of the subtransient flux vector must be controlled in the following way approximately:
(9.11)
Fig.9.9. presents the possibly covered operation range, assuming 4/4 quadrant and field weakening operation on the W(M_{k}) plane. In the step commutation range there is not continuous operation usually.
Let’s mention, that the largest variable speed drive in the word is a CFSM. This 101MW drive is used for a fan of a wind tunnel of the NASA. This huge wind tunnel is used for aerodynamic investigations of supersonic aircrafts.
If a cage rotor induction machine is made resultantly capacitive by parallel capacitors (Fig.8.7) it is also capable of operating from line commutated converter (Fig.9.1.). This so called capacitively compensated converterfed induction machine is capable of operating in narrow frequency range only, that is why it is rarely used only.
In the switched reluctance motor (SRM) both the stator and the rotor are cogged. The number of cogs on the wounded stator is Z=2pm^{*}, the number of cogs on the nonwounded rotor is usually Z_{r}=Z±2p (2p is the number of poles, m^{*} is the number of the phases). In the machine in Fig.10.1. these values are: m*=3, 2p=2, Z=6 and Z_{r}=4. Only the coils of phase a are drawn in the figure. The most commonly applied SRMs have m^{*}=3 and 4 phases.
Using the energy theorem (calculating the energy modification for Δt time interval) the torque can be expressed. The result is simple, if at a given time current flows only in one phase, the saturation is neglected and only the cupper looses are considered. In this case the torque developed by the ith phase is:
(10.1)
Where i_{i} is the current of the ith phase, L_{i} is its selfinductance, α is the angle of rotation of the rotor. According to this expression, the torque is independent of the current direction (that is why the power circuit capable of one current direction in Fig.10.1. is enough in each phase), and the torque exists only if dL_{i}/dα≠0. If the mutual inductances can be neglected comparing with the L_{i} selfinductances of the phases (it is usually a good approximation), then more phases can conduct simultaneously. In this case the resultant torque is
(10.2)
The factor dL_{i}/dα is determined by the motor, the value is determined by the supply. The selfinductance of the phases (L_{i}) depends on the α and changing periodically with Z_{r}α=2π periodicity, repeated Z_{r} times in one revolution. Assuming trapezoidal inductance change, Fig.10.2. shows the inductance of the ith phase L_{i}(α) and its dL_{i}/dα factor.
The larger the L_{max}L_{min} difference, the larger the dL_{i}/dα factor and the torque which can be developed with a given current. Positive torque (m_{i}>0) can be developed by current flowing during dL_{i}/dα>0 section, negative torque (m_{i}<0) can be developed by current flowing during dL_{i}/dα<0 section. Both sections have β length. At dL_{i}/dα=0 sections current flow is useless, since it does not develop torque, only losses would be generated. Accordingly, the phase currents must be synchronised to the rotor position (α).
The phase currents must be fitted to the motor according to the drive demand. That is why the SRM drives are designed and manufactured in complex way. The smooth, pulsation free torque is frequently a demand (e.g. in servo and vehicle drives). Fig.10.3. presents the fitted supply to develop smooth torque for threephase machine with trapezoidal phase selfinductances L_{a}(α), L_{b}(α) and L_{c}(α). In this case the condition of the fitted supply is the given length of the torque development sections: β>360°/3=120°. According to Fig.10.3.b.,c.,d. the rectangular shape β_{i}=120° wide current pulses in these sections provide the ideal fitted supply. The solid line current curves develop positive torque (m>0), the dotted ones negative torque (m<0). The currents to develop positive and negative torque have the same direction, but shifted by δ≈180° from each other. In the real case the phase currents can not be increased and decreased instantaneously because of the selfinductances. Consequently the shape of the phase currents is significantly modified at high speed.
The best utilization of the trapezoidal selfinductance SRM is got if the width of the current flow is equal to β (β_{i}=β). However in this case (except at β_{i}=120°) the torque is pulsating with 6nZ_{r} frequency (n is the rotation speed).
If the threephase machine is starconnected, applying the connection in Fig.10.1. for threephase, the simple power electronic circuit in Fig.10.4 can be got. Assuming ideal semiconductors, it can switch +U_{e}, U_{e} and 0 voltages to the phases. Switching between these three values with high frequency (PWM) the phase currents can be controlled. Since only positive phase currents are necessary, in unipolar mode +U_{e} and 0, in bipolar mode +U_{e} and U_{e} are switched. The mean value of the i_{e} DC current is positive in motor mode (I_{ek}>0) and negative in generator mode (I_{ek}<0). Assuming lossless power electronics and motor the power mean values are: P_{mk}=M_{k}W=P_{ek}=U_{e}I_{ek}. The constant DC voltage (U_{e}≈const.) is provided by an AC/DC converter presented in Fig.7.1. if generator (brake) mode occurs only during transients.
Fig.10.5. presents the blockscheme of a speed controlled threephase SRM drive. The SZW speed controller provides the torque reference m_{a}. The squareroot of its absolute value │m_{a}│ (by NG) is proportional to the phase current amplitude:
(10.3)
According to (10.2) the NG squareroot block linearizes the torque control loop (it is reduced to current control). The phase current references (i_{aa}, i_{ba}, i_{ca}) correspond to the fitted supply (to Fig.10.3).
The synchronisation to the rotor position is done by the FGA, FGB, FGC function generators, using Z_{r}α’. If w>0 and m_{a}>0, then the operation is in motor mode: Z_{r}α’=Z_{r}α. If w>0 and m_{a}<0, then the operation is in generator mode: Z_{r}α’=Z_{r}αδ. The phase current amplitudes are set by the × multiplication, using I_{a}. The PWM current controllers per phase can be implemented by PWM controllers or by hysteresis controllers. The control signals va, vb, vc switch the transistors TA, TB, TC (Fig.10.4.), the control signal vo switches transistor T0. In practice, the PWM is implemented by T0.
In a real case the step change of the phase currents is not possible because of the inductances. It can be compensated by a speed dependent prefiring. The PWM operation is possible until such speed and torque, where the onsection of the +U_{e} voltage is long enough to develop the phase current with amplitude I_{a}. Beyond this a so called single pulse operation is possible. In this range the torque is pulsating and the torque loadability decreases.
Obviously the smooth, pulsation free torque operation can be implemented not only with trapezoidal phase selfinductances. The shape of the fitted phase currents i_{i}(α) can be determined by the expression (10.2) always. To do it, the machine characteristic selfinductance angle dependency perphase L_{i}(α) must be known.
Tartalom
Among the task specific controls the speed and the position controls are discussed.
The speed control can be drive specific also. The speed signal of a DC machine can be provided by a machine model using (1.1.a and 1.2.a):
(11.1)
The combined machine model of the cage rotor IM (Fig.5.15.) can provide the w speed signal also (5.34). If these calculated w speed signals are used as feedback signal of the speed controller, then the speed control is drive specific, so called sensorless type.
In the next coming investigations the speed feedback signal is provided by a speed sensor. According to Fig.1.3. there is a subordinated torque loop to the speed controller. However the torque control in the investigated drives can be reduced to current/currentcomponent control. Accordingly, current/currentcomponent control is subordinated to the speed control in practice.
In this chapter the speed control of a 4/4 quadrant PWM chopperfed DC drive with subordinated current control is investigated as an example. Its blockscheme is presented in Fig.11.1. According to the subordinated structure the reference value of the SZI current controller (i’_{a}) is set by the SZW speed controller. In the dottedline surrounded control loop the signals with prime have [V] dimension in analogue implementation, and dimensionless in digital implementation. A_{vw}, A_{vi} and A_{u} are the transfer factors of the speed sensor, the current sensor and the PWM DC chopper (Fig.1.21.) respectively.
The controller SZW can operate in the saturated and in the linear range.
The saturated range is realized, if the speed has such a value, which results in SZW output i’_{a} reaching the limit (saturated) value (+I’_{korl} or I’_{korl}). (SZI is in saturation, if its output is at ±U’_{vm} value.) These correspond to the allowed current limit for the motor and the chopper (±I’_{korl}). The current limitation provides protection functions. In the investigation of the saturated operation of the speed controller it is assumed, that i’_{a}=±I’_{korl} and the time function of i’_{k}(t) from I_{0} approaches I’_{korl} with T_{i} time constant:
(11.2)
I.e. the inner current control loop is in linear range described by (2.24).
At start up: I_{0}=0. Current i’_{k}(t) reaches I’_{korl} value in approx. 3T_{i}, then while i’_{a}=I’_{korl} the motor accelerates with maximal current (i’_{k}=±I’_{korl}) and maximal torque (m_{k}=M_{korl}=kϕI_{korl}). The time function of the speed can be calculated from the following differential equation:
(11.3)
Meanwhile the speed controller has not any effect. The acceleration rate depend on m_{t} and θ. The current limitation periods can be avoided by limiting the gradient of the speed reference (ramping, Fig.11.2.). The drive can track such a speed reference (w_{a}), for which the following condition is true:
(11.4)
The speed and current tracking properties of the controller operating in linear range are determined by both controllers. First the current controller, then the speed controller should be tuned.
The linear and saturated operation of the SZW speed controller is presented in Fig.11.3 by its typical transient response to w_{a} step change in the reference.
Section I (saturated): the speed curve corresponds to (11.3), i_{k}=I_{korl}. Controller SZW comes out the saturation at speed error Δw^{*}.
Section II (linear): depending on the structure and setting of the controller, the speed reaches the reference, and the current its stationer value (i_{ks}) with or without oscillation, with or without error. i_{ks} can be calculated from the necessary torque m_{ks}=m_{ts} to maintain the steadysate value of the speed (w=w_{a}): i_{ks}=m_{ts}/(kϕ).
Section III: it is again saturated: i’_{a}=I’_{korl}.
Section IV: it is linear, stationary state, with m_{t}=0 load torque.
To design the speed control in the linear range, the symmetrical optimum method is widely used. Let’s assume, that the transfer function of the subordinated current control loop has been adjusted to (2.24). Since i’_{a}=A_{vi}i_{a} so . To determine the setting of the speed controller, the simplified block scheme in Fig.11.4. is used.
Fig.11.4.a. can be derived from Fig.11.1. by neglecting the effect of the speed change (kϕw) to the current control. The surrounded part is the drive specific part. In Fig.11.4.b. voltage dimension signals are used using Fig. 1.4. The transfer function of the open speed control loop using Fig.11.4.b. is:
(11.5)
Where T=CT_{m} is a resultant time constant, C=(kϕ/R)(A_{vi}/A_{vw})is a dimensionless value. The transfer function of the PI type SZW speed controller is:
(11.6.a,b)
Substituting s→jω, the Bode diagram of the Y(jω) frequencyfunction is given in Fig.11.5. According to the practice: T_{i}<<T. The crossover angular frequency (ω_{cw}) and the cutoff angular frequency (1/T_{w}) can be modified by the speed controller (by Y_{F}(jω)). The design of the controller based on selecting T_{i}<T_{w}=BT_{i}<T, so inserting a 20 dB/decade slope section between the 40 dB/decade slope sections.
The cutoff frequency is set by K_{F} to get ω_{cw} at the middle of the 20 dB/decade slope section. It can be proved, that in this case at ω_{cw} the phase lag φ=arc(Y_{w,} _{Δ} _{w}) is minimal (the phase margin φ_{t} is maximal), i.e. the system is the best considering the stability. For this case using Fig.11.5. the following expression can be written: . Using│Y(jω_{cw})│=1, the setting rules for the parameters of the speed controller are:
(11.7.a,b)
The coefficient B is selected using simulations, depending on the desired tracking property. Its optimal value for reference step is B≈10, for load step is B≈5 (fast, small overshoot tracking). Proper behaviour for both cases can be got at B≈7.5. It can be established, that the faster the current control loop (the smaller the time constant T_{i}), the larger parameter K_{F} and the smaller parameter T_{F} can be selected, i.e. the faster the speed control will be.
It is used in servo drives most commonly. Its basic types are:
PTP point to point control, (e.g. spot welder robot),
CP continuous path, path tracking control (e.g. arc welder robot).
The scheme of the subordinated structure position control is presented in Fig.11.6. (the position is represented by the α angle of the motor shaft). The inner speed control loop coincides with Fig.11.1. The w’_{a} speed reference is set by the SZP position controller. A_{vp} is the transfer factor of the position sensor.
The behaviour of the position control is investigated in detail for the PTP control, in that case when the speed reference is limited to ±W’_{poz} (Fig.11.6) (W_{poz} is usually less than W_{max} allowed for the drive). The transient response functions of the system for an α_{a} magnitude step change in the position reference are given in Fig.11.7. Such a large reference step is considered, which results in limited (saturated) operation of the SZP, SZW and SZI controllers. The saturated operation of the SZP, SZW and SZI controllers mean, that their outputs are limited to±W’_{poz}, ±I’_{korl} and ±U’_{vm} values respectively. The reason of the saturated operation is the high and/or longlasting error on the input of the controllers.
In sections I, II (acceleration and constant speed operation): SZP is saturated, w_{a}=W_{poz}. SZW is saturated in section I, and linear in section II. SZP comes out the saturation at Δα_{2} position error.
In sections III, IV (deceleration and positioning): SZP is in linear mode, w_{a}<W_{poz}. In section III (current limited section) SZW is saturated. In section IV every controller is in linear mode. The optimal design of SZP can be determined in this section.
For small α_{a} reference step the sections II and III can be missing.
The simplified block scheme of the position control loop for the linear section IV is shown in Fig.11.8. Here represents the speed controlled drive (Fig.11.1.). is the transfer function of the closed speed control loop.
The transfer function of the SZP position controller is usually PID type:
(11.8)
The transfer function of the open position control loop is:
(11.9)
(11.10.a,b,c)
It can be established, that the natural α=∫wdt integration effect (1/s) results in modified effects of the position controller’s PID parameters. E.g. the D parameter (T_{pd}) affects the loop gain (K_{1}), the P parameter (K_{p}) affects the integration time (T_{1}). In the practice, for PTP control P and PD position controllers are applied. By these controllers “type 1” position control loop can be implemented, i.e. for position reference step the tracking is errorless in steadystate.
Position control with proportional (P type) controller.
In this case Y_{p}=K_{p}, the transfer function of the open position control loop is:
(11.11.a,b)
Assuming PI type speed controller, Fig.11.9. presents the Bode diagram of the open position control loop (s→jω). As can be seen, in the ω<ω_{cw} frequency range: , so in this range│Y_{α,Δα}│=│1/(jωT)│. The phase shift of the open position control loop ( ) is independent of K_{p}. Modifying K_{p}, │Y_{α,Δα}│ can be shifted up and down. Increasing K_{p}, the crossover frequency of the position control loop (ω_{cp}=1/T=A_{p}) getting closer to the crossover frequency of the speed control loop (ω_{cw}). The maximal value of K_{p} is determined by the phase margin (φ_{t}):
(11.12)
where are the angles of the frequency functions. To get acceptable control behaviour for the position control loop, φ_{t}≥45^{°} phase margin is required. This requirement determines the maximal value of K_{p} and A_{p}, and the minimal value of T. It can be established, that the faster the speed control loop (the larger ω_{cw}), the larger crossover frequency/loopgain (ω_{cp}=A_{p}), the smaller the time constant T=1/ω_{cp} and the faster the positioning can be.
This rule can be generalised: fast speed control needs fast current control, fast position control needs fast speed control. I.e. in a subordinated control structure the speed of the inner control loop limits the speed of the outer control loop.
If ω_{cw}>>ω_{cp} (if φ_{t}≥60^{°}), then approximately. Considering it in (11.11.a), the transfer function of the closed position control loop is:
(11.13)
Consequently the controlled α position tracks the α_{a} reference with lag T:
(11.14.a)
(11.14.b)
The later expression of the Δα position error (11.14.b) is valid if α_{a}=const. Then Δα and w change exponentially:
(11.15.a,b)
(11.16)
With the Δα_{0}/T=A_{p}Δα_{0}=W_{poz} expression it is assumed, that the section III (current limited deceleration) misses. The torque by (1.a) motion equation and m_{t}=0 assumption is:
(11.17)
Because of the missing section III the inequality θW_{poz}/T<M_{korl}=kϕI_{korl} must be fulfilled. It determines a minimum value for T and a maximum value for A_{p}.
The exponential tracking (solid curves in Fig.11.10.) is overshoot free, but slow.
A finite time (T_{f}) linear tracking can be reached by constant deceleration (dw/dt=W_{poz}/T_{f}) braking. Here the time functions for 0≤t≤T_{f}, with m_{t}=0 are the following:
(11.18.a)
(11.18.b)
(11.18.c)
Now the θW_{poz}/T_{f}<M_{korl} condition must be satisfied. At the exponential tracking: Δα_{o}=W_{poz}T, at the linear tracking: Δα_{o}=W_{poz}T_{f}/2. Consequently: T_{f}=2T. The time functions of the linear tracking are given in dotted lines in Fig.11.10. From (11.15, 11.16) and (11.18.a,b) it is clear, that the positioning speed at the exponential tracking is proportional to Δα, at the linear tracking to :
(11.19.a,b)
(11.20.a,b)
At the linear tracking for ± signed Δα the next expression is valid:
(11.21)
At the linear tracking the position controller is P type also, but K_{p} and A_{p} (11.11.b) are not constant, they depend on Δα, i.e. SZP is a nonlinear variable gain controller.
Fig.11.11. shows the w(Δα) (in section II and IV the w_{a}(Δα)) function for such a positioning process, where the section III is missing and m_{t}=0. Difference between the exponential and linear tracking can be identified only in section IV. The initial position error for the exponential tracking is Δα_{o}>Δ, not to get current limit. (Δ is the rotation angle in section I, in the section with current limit). At linear tracking the gradient of the w(Δα) parabola is:
(11.22)
Its value at the origin would be ∞. Because of stability problems, near the origin the exponential tracking should be followed in this case also, with a A_{pmax}>>A_{p} parameter setting.
Only the basic principles of the speed and position control are described in this subject. In both cases other control methods can be used, too: slidingmode control, model reference control, fuzzy control, neural network based control, etc.
Tartalom
The modern practical applications of the VSIfed drives and the CFSM drive are described in this chapter.
One possible way of electric energy storage is the flywheel electrical drive, which stores the energy in kinetic form.
The flywheel storage uses the E_{L} kinetic energy of a mass with q_{L} inertia rotating with w_{L} angular speed. The maximal kinetic energy corresponds to the maximal speed:
(12.1)
If the k^{th} part of the E_{Lmax} energy should be utilised, then:
(12.2)
(12.3)
Usual practical values are: k=0.75, w_{Lmin}=0.5w_{Lmax}.
The kinetic energy can be modified by the m_{L} torque of the flywheel’s drive, i.e. by its p_{L} power:
(12.4)
During deceleration (decreasing w_{L}, discharging) energy is withdrawn, during acceleration (increasing w_{L}, charging) energy is supplied to the flywheel. In a modern flywheel drive (Fig.12.1.a.) L is the flywheel, Á is the gearbox, VG is the electrical driving machine, TE is the power electronic circuit, H is the electric grid, q_{L} is the resultant inertia referred to the shaft of the flywheel. The VG, TE and H units must be capable of bidirectional power flow. The VG electric machine is in motor mode at p_{L}>0 (charging), and in generator mode at p_{L}<0 (discharging). The modern, lowloss applications are gearless, they use direct drive.
The usual operation range of the TEVG electric drive is given in Fig.12.1.b. on the w_{L}m_{L} plane. In the w_{Lmin}£ω_{L}£w_{Lmax} operation range the maximal power is +P_{Lmax} at charging and –;P_{Lmax} at discharging. The maximal driving torque of the drive is M_{Lmax}=P_{Lmax}/ω_{Lmin}, the maximal brake torque is –M_{Lmax}. It can be established, that the flywheel drive is a monodirectional two quadrant drive, and its normal operation range is the field weakening. The nominal point of the drive should be selected to point 2: M_{Ln}=M_{Lmax}, w_{Ln}=w_{Lmin} and P_{Ln}=M_{Ln}w_{Ln}=P_{Lmax}. Fig.2.1.c. shows that case, when the limits are fully used, when the p_{L} power pulsates in the range ±P_{Lmax} with 2DT periodicity and the energy is changing between E_{Lmin} and E_{Lmax} linearly. It can be derived for DT:
(12.5.a,b)
Where T_{Lin} is the nominal stating time of the drive.
The principal task of the flywheel drive is to compensate (smooth) the pulsating electric power. The control of the cage rotor induction machine (Fig.7.7) driven flywheel drive is described as an example. One possible block scheme of the control loop to compensate the power pulsation is presented in Fig.12.2.
The pulsating power of the G consumer or generator should be compensated. From the measured p_{G} instantaneous value the SZ filter provides the mean value (p_{Gk}) and the difference of these two powers sets the electric power reference of the flywheel drive (p_{LGa}):
(12.6)
From p_{LGa} and w_{L} unit MA provides a torque reference:
(12.7.a)
(12.7.b)
Where p_{LGa}p_{Lv} is the mechanical power of the drive, p_{Lv} is the w_{L} dependent loss of the drive, m_{Lv}=p_{Lv}/ω_{L} is the corresponding torque. The m_{Lv} motor mode torque is necessary to keep the w_{L} angular speed constant. Instead of the MA torque set point element power controller also can be used, but the power of the flywheel drive (p_{L}) must be measured too in this case. From the w_{L} and m_{La} signals the FA block provides the rotor flux reference of AL machine. It mainly depends on the speed:
(12.8)
Where Y_{rn} is the nominal rotor flux. The machineside SZÁLG current vector controller controls the torque and the flux of the AL induction machine by the ÁLG converter. The control can be implemented by fieldorineted control (see chapter 5.3.1.). From the flux reference (y_{La}) and the torque reference (m_{La}) the current components references can be calculated in the field reference frame:
(12.9)
These are constrained by the SZÁLG current vector controller.
The gridside voltage controller (SZULE) controls the DC voltage (u_{Le}) by its active power reference (p_{LHa}). The reference of the reactive power (q_{LHa}) is determined by external grid demands. The SZÁLH current vector controller controls the active and reactive power of the flywheel drive by the ÁLH converter. The lineoriented current vector control can be implemented according to chapter 7.1.1. From the active and reactive power references (p_{LHa} and q_{LHa}) the current components references can be calculated:
(12.10)
These are constrained by the SZÁLH current vector controller.
The structure of a permanent magnet synchronous machine driven flywheel drive is similar. The block scheme of the doublefed induction machine driven flywheel drive is different because of the missing field weakening possibility.
The block scheme of the control (Fig.12.2.) does not contain the initial charging part (the starting and acceleration form zero speed).
As an example, the compensation of a sinusoidally pulsating p_{G} power is demonstrated in Fig.12.3. in perunit system. The amplitude of the pulsation is set to such a value for k=0.75, which results in reaching the speed (w_{Lmin} and w_{Lmax}) and power (±P_{Lmax}) limits. At the beginning of the power pulsation compensation the speed of the flywheel is set to such a value, which results in symmetrical compensation reserve. The corresponding values are (w_{Ln}=w_{Lmin}):
(12.11)
For k=0.75: _{,} _{ }E_{Lk}=2,5E_{Lmin}, E_{Lmax}=4E_{Lmin}.
The example drive can not compensate perfectly larger amplitude or larger period power pulsation.
Among the practical implementations, the product of Beacon Power can be mentioned (Smart Energy 25). In this product the flywheel is driven by permanent magnet synchronous machine, it rotates in vacuum with magnetically levitated bearing, with 800016000rpm (k=0,75). It can provide P_{Lmax}=100kW power for 15min, i.e. E_{Lmax}E_{Lmin}=25kWh.
As examples, among the railway traction drives a modern locomotive, among the urban transportation drives a modern VSIIM trolleybus drive are described.
It is singlephase (50Hz, 25kV) DClink VSIfed vehicle (Taurus locomotive). Five power components can be distinguished: line transformer, lineside converters, the DC link, the motorside converters and the induction machines (Fig.12.4.).
The figure presents in detail the circuit of one doublemachine driven bogie. Every machine has own inverter. Consequently the inverters and motors can be controlled independently, so e.g. the adhering force can be utilised better. Every bogie has a power electronic unit. It contains three parallel connected lineside 4QS (Four Quadrant System) twolevel converters (ÁH1, ÁH2, ÁH3) and two twolevel VSIs (INV1, INV2). This configuration makes possible to use exactly the same type GTO legs in the lineside converters (ÁHx) and in the inverters (INVx). Such configuration is used for high power locomotives (e.g. 4·1600kW=6400kW).
The 4QS fourquadrant lineside converters make possible the regenerative electrical brake operation, and the currents in the input contact wire are sinusoidal with cosj=±1 power factor. The 4QS converters are controlled by active power control subordinated to DC voltage control. Since the singlephase power pulsates with 2f_{h}=100Hz frequency, there is a filter (L1,C1) in the DC link tuned to 100Hz.
The principle of the control of the VSIfed vehicle drive is the fieldoriented current vector (chapter 5.3.). Until the load makes possible, constant torque angle (ϑ_{1}) control (constant f_{r} rotor frequency control) is used, resulting in energy saving operation. The regions of the control are presented in Fig.12.5. for motor operation. In Fig.12.5.a. the Ī_{1} current vector in dq reference frame is given, in Fig.12.5.b. the torque is given in the Mw_{1} plane with the regions and the limits.
Region I.: Energy saving operation, the torque angle is: ϑ_{1}=ϑ_{1opt}=ϑ_{1n}, the torque is (M≤M_{n}):
(12.12)
Region II.: Nominal rotor flux operation (Y_{r1}=Y_{r1n}), w_{1}£w_{1n}=2pf_{1n}, M³M_{n}, the torque is:
(12.13)
The maximal torque (M_{max}) is determined by the current limit (I_{1max}).
Region III.: Field weakening operation, w_{1}>w_{1n}, the flux and the torque are:
(12.14.a,b)
(12.15)
For regenerative brake operation the Fig.12.5. should be reflected to the horizontal axis.
The locomotives have torque (traction force) control, subordinated to speed control (Fig.12.6.). In forward and reverse running the torques have opposite sign, the sign inverting is done by block E/H. During starting the vehicle accelerates till the v_{a} speed set by the driver, with traction force settable by the limit torque m_{korl} (in the w_{1}>w_{1n} speed range the KORL block can decrease the m_{korl} value set by the driver). Reaching the v_{a} speed the SZV speed controller sets the torque reference necessary to keep the required speed. Instead of torque limitation, acceleration control is also an option.
The operation of the urban transportation vehicles between two stops contains acceleration, coasting and deceleration (braking). The motor develops tracking/braking force only during the acceleration and braking. Therefore speed control is not applied in these vehicles, only the acceleration and deceleration process are controlled usually by the torque.
Because of the frequent starting and braking processes, with lossless starting and regenerative braking significant energy can be saved. By regenerative braking, according to the measurements in normal traffic conditions the 3035% of the supply energy can be supplied back.
The main power circuit of a VSIIM trolleybus drive is given in Fig.12.7.
The AM induction motor is connected to the U_{T} DC supply through an IGBT twolevel voltage source inverter (INV). The U_{T} supply should be provided by the circuit given in the figure, since the trolleys of the vehicle can connect shortly opposite polarity voltage to the vehicle in the cross roads. It is rectified by the D1D4 diode bridge. At normal polarity the regenerative braking is possible through IGBTs T1, T2. Smoothing of U_{T} voltage is done by filter L_{sz}C_{sz}, the initial charging of C_{sz} is done by the KT, RT charging circuit. There is a TL surge absorber, KF1, KF2 main contactor and a noise filter on the supply side.
The controlled motor operation and the controlled regenerative braking can be implemented by the control of the inverter. The condition of the regenerative braking is that the supply voltage should stay bellow the allowed U_{T }£U_{Tm}. If during regenerative braking the opposite energy flow causes reaching U_{Tm} value, then the TF transistor can connect resistance RF parallel to C_{sz}. With ONOFF switching the resistance RF the U_{T } voltage can be controlled.
The basic principle of the control of the VSIfed trolleybus drive is the fieldoriented current vector control, but here only the acceleration and the braking is controlled. The different control regions for motor/acceleration mode are shown in Fig. 12.8.: Fig.12.8.a. presents the Ī_{1} current vector in dq reference frame, Fig.12.8.b. shows the torque on the Mw_{1} plane with the regions and the limit curves. Opposite to the railway VSI drive (Fig.12.5.) there are only two regions here, since the constant speed energy saving operation is not necessary in the urban transportation.
Region I.: Nominal flux operation: Y_{r1}=Y_{r1n}, w_{1}£w_{1n}. The torque can be calculated by (12.13), M_{max} is determined by the I_{1max} current limit.
Region II.: Field weakening operation, w_{1}>w_{1n}, the flux can be calculated by (12.14), the torque by (12.15).
For regenerative brake operation the Fig.12.8. should be reflected to the horizontal axis. The generator mode current limit I_{1max} is usually less, than in motor mode.
Basically the trolleybus has torque (traction force) control (Fig.12.9).
The driver sets the m_{a} torque reference by the GY acceleration pedal for starting/acceleration and by the F brake pedal for stop/braking. The torque reference is positive for acceleration and negative for braking in forward running.
The wind power plant contains the generator, the wind turbine, the mechanical gearbox, the power electronic circuit, the control system and the auxiliary equipments. The modern wind turbine generators are VSIfed induction or synchronous machine drives operating in generator mode.
The characteristic P_{T}(v) powerwind speed diagram and the M(W) torqueangular speed (on the shaft of the turbine) diagram are given in Fig.12.10a. and b. respectively. At the region AB where the wind speed is v_{0}£v£v_{N} (this is the optimal pitch angle region) the power factor (aerodynamic efficiency) of the wind turbine (12.16) is optimal (maximal):
(12.16)
P_{T}=M_{T}W_{T}=MW is the power of the wind turbine, P_{SZ} is the power of the wind rotating the wind turbine. In this region the angular speed of the wind turbine W_{T} (the angular speed of the generator W) must be controlled approximately proportionally to the v wind speed to get maximal power factor C_{Pmax} between point A and B. At the nominal wind speed (v_{N}) in point B the power of the wind turbine is (neglecting the losses):
(12.17)
In the region AB the power P_{T} and torque M are approximately:
(12.18.a,b)
In region BC (v_{N}£v£v_{max}) constant P_{TN} nominal power should be provided by the limitation/control of the power at the wind turbine and at the generator.
The basic aim of the control in both regions to utilise the wind turbine power (limited to P_{TN}) as much as possible. At a given v wind speed the P_{T} power can be controlled at the wind turbine by turning the nacelle relatively to the direction of the wind and by turning the blade around its longitudinal axis (b angle pitch control), at the generator by the control of the angular speed. The limitation of the power is done at the wind turbine.
Nowadays the direct driven synchronous machine and doublefed induction machine are applied as generators most frequently. The block schemes of the permanent magnet synchronous machine (PMSM) and the doublefed induction machine (DFIM) wind turbine generators are given in Fig.12.11. and Fig.12.12. respectively.
The central controller (KSZV) determines the speed reference (w_{a}) of the generator and the pitch angle reference (b_{a}) using the wind speed (v) and the power (P_{h}»P_{T}). The generatorside controller (SZG) is a fieldoriented current vector control (chapters 4.2. and 6.1) subordinated to speed control. The grid(line)side controller (SZH) is a lineoriented current vector control (chapter 7.1.1.) subordinated to DC voltage control.
There are direct driven synchronous machine wind turbine generators with excited rotor. In this case Fig.12.11. must be extended with the excitation control. Rarely cage rotor induction machine is used with gearbox. Its block scheme is similar to Fig.12.11.
At the synchronous machine wind turbine generator the current vector control in SZH, at the DFIM wind turbine generator the current vector control in SZG and SZH can ensure symmetrical, sinusoidal line currents with power factor cosj=1 at terminals A,B,C.
Let’s assume the practical case: W_{max}=2W_{0}, W_{max}=1,2W_{N}, W_{0}=0,6W_{N} (Fig.12.10.b.). In this case the synchronous speed of the DFIM wind turbine generator should be selected to W_{1}=(W_{0}+W_{max})/2=0,9W_{N} (Fig.6.3.). With these data the nominal power of the generator (P_{Gn}) and the design rating power of the power electronic circuit (P_{TEtip}) are given in Table 12.1. referred to the nominal wind turbine power (P_{TN}), the losses are neglected.
Table 12.1. The power conditions for the different wind turbine generators.
Generator type 
P _{Gn} /P _{TN} 
P _{TEtip} /P _{TN} 
Field weakening 
Permanent magnet synchronous machine PMSM 
1.2 
1,2 
No 
1 
1 
Yes 

Cage rotor induction machine CRIM 
1 
1 
Yes 
Doublefed induction machine DFIM 
0.9 
0.3 
Not possible 
The nominal power of the generator is P_{Gn}=M_{n}W_{n}. The nominal torque of the generator (M_{n}) is always the same as the nominal torque of the wind turbine reduced to the generator shaft (M_{N}), M_{n}=M_{N}. The nominal speed of the generator at the PMSM without field weakening is: W_{n}=W_{max}=1,2W_{N} (in this case the generator can operate in point C’ also), at PMSM and CRIM with wield weakening: W_{n}=W_{N}. At DFIM: W_{n}»W_{1}=0,9W_{N}. The design rating of the power electronic circuit at the PMSM and CRIM is the same as the nominal power of the generator: P_{TEtip}=P_{Gn}. At DFIM the power electronic circuit must be designed according to (6.6), in the investigated example it is: P_{TEtip}=M_{N}(W_{max}W_{n})=0,3M_{N}W_{N}=0,3P_{TN}. This is the reason, why the DFIM generator is used mainly for the high power wind turbine generators. E.g. for a P_{TN}=3MW power DFIM wind turbine generator a power circuit with design rating power P_{TEtip}=900kW is enough. However it means, that the DFIM wind turbine generators must not be connected to the grid bellow W_{0} speed, since in this case the voltage on the ÁG rotor side converter would be too high.
It should be mentioned, that the controlled electrical drives with VSI type lineside converter (the wind turbine generators in Fig.12.11. and Fig.12.12. are among them too) can provide auxiliary services not requiring active power besides the principal service. These are the reactive power compensation, the asymmetry compensation and the harmonics compensation. Furthermore these additional services can be provided at nowind also in the case of modern wind turbine generators. The additional services affect the design rating of the lineside converter (ÁH) and the DC link capacitance (C).
The compensation of the asymmetry and the reactive power is demonstrated in Fig.12.13. as an example in perunit. The examined system is similar to Fig.12.2. G represents the consumers, instead of the flywheel drive a wind turbine generator is assumed (Fig.12.11). In period 1 the asymmetrical inductive currents pollute the lines (the negative sequence current is 20%). In period 2 the ÁH converter of the wind turbine generator makes symmetrical the line currents (cosj_{1}@0,7). In period 3 ÁH compensates the reactive power, so the line is loaded by much less currents at cosj_{1}@1. The mean value of the threephase power (p=p_{la}+p_{lb}+p_{lc}) is constant always. From period 4 also the active power of the consumers is provided by the wind power generator, so the line is not loaded.
By compensating the asymmetry, the ellipse becomes circle in xy reference frame, the small circle becomes point in pq reference frame. By compensating the reactive power the current vector jumps to the p axis in p‑q reference frame.
The gas turbine – synchronous generator systems are widely used for electric energy generation, because of their economic operation, reliability and low maintenance demand. The high power units (100200 MW) are widely applied in peakload power plants, since they can be started quickly. The gas turbine systems are frequently applied with steam turbine in combined cycle, when the high temperature exhaust gas of the gas turbine is used even for steam generation. The applied synchronous generator is usually turbogenerator type with 2p=2 pole, so its nominal synchronous speed at f_{1n}=50Hz is n_{n}=n_{1n}=60f_{1n}rpm=3000rpm.
The gas turbine as an internalcombustion engine is not capable of self starting, a starting equipment is necessary. It is practical to use the synchronous generator of the gas turbine system as starting motor, since in this case no need for a separated electrical starting motor. During starting the synchronous generator is usually operated as converterfed synchronous motor (CFSM, see Fig.9.1. and Fig.9.2). The block scheme of the CFSM starter of a gas turbine is presented in Fig.12.14. During operation the G generator driven by the GT gas turbine is connected to the grid through the KF main switch and the TF main transformer. During starting it is connected to the static starter (containing the converters ÁM and ÁH) through the K1 and K2 switches. The excitation coil is supplied by the static exciter (containing the ÁG converter).
The starting is initiated by the plant manager. The main characteristic signals of the starting process lasting T_{i}=1520min are given in Fig.12.15.: speed (n), the amplitude of the subtransient voltage (U”), the DC current (i_{e}), the excitation current (i_{g}).The starting has the following main parts (Fig.12.15):
Startup and acceleration until approx. (1/3)n_{n}=1000rpm speed.
Few minutes constant speed operation (n=const.) to ventilate the gas turbine, then at the end of this period decreasing the current (i_{e}) and the torque to zero.
After decreasing the current to zero next is the deceleration, meanwhile the ignition of the gas turbine is prepared.
After the deceleration the next interval is the development of the current again (i_{e}), acceleration and keeping the speed during the ignition of the gas turbine.
After the ignition of the gas turbine further acceleration with constant flux by the CFSM and by the gas turbine.
Acceleration above approx. (1/3)n_{n} speed in field weakening, when U_{1}»U”»const. The CFSM connected generator plays role in the acceleration of the machine set until approx. (2/3)n_{n}=2000 rpm.
After decreasing the current of the CFSM (i_{e}) and the excitation current (i_{g}) to zero the turbinegenerator machine set is accelerated further by the gas turbine until n_{n}=3000 rpm speed.
In the 1^{st} and 4^{th} phase the CFSM, in the 5^{th} and 6^{th} the CFSM and the gas turbine together, in the 7^{th} only the gas turbine accelerates.
The static starter is usually designed to few percent of the nominal power of the synchronous generator. The typical values are in Table 12.2. (100% is the nominal data of the generator).
Table 12.2. The main data of the CFSM starter.
CFSM starter 

Power, P_{m} 
2% 
Torque, M 
5% (10%) 
Voltage, U_{1} 
10% 
Current, I_{1} 
20% 
The value in brackets for the torque is the maximal starting torque, which is necessary for the friction of rest.
The 10% voltage means, that in the CFSM operation the nominal flux can be kept maximum until 0.1n_{n}. Consequently practically the starter accelerates with approx. nominal flux only in the step motor operation after the startup (n<0.1n_{n}), and then the flux is much less than the nominal during the whole next starting process. The 10% torque can be developed with the assumed 20% current with 50% flux.
Large advantage of the CFSM starter, that with one single static starter through the starting bar any of the other identical units of the gas turbine power plant can be started.
The problem is similar in the pumped storage hydroelectric plants, when the electrical machine is used in motor mode in pumping operation, it also must be started. Practically the CFSM starter is used most frequently in this case also.
Calculate the settings of the controller of a chopperfed speed controlled DC drive (chapter 11 and 2.2).
The parameters of the system are the following:
The nominal data of the motor:
P_{n}=2800W n_{n}=960rpm
U_{n}=220V I_{n}=14A
U_{gn}=220V I_{gn}=0.7A
R=1.4Ω L=46mH
Θ_{motor}=0,1kgm^{2} Θ_{load}=0,15kgm^{2}
The data of the sensors:
Current sensor: 2A→1V→A_{vi}
Speed sensor: 100/min→1V→A_{vw}
The transfer factor of the chopper:
A_{u}=220V/10V
Notes:
PI type controllers should be used.
The time constant of the closed current loop should be 10ms.
Tasks:
Optimise the settings to reference jump and load jump too.
Look for a setting, which is close to the optimum for both jumps.
Check the calculation results by simulations.
Used 
Meaning 

t index 
load quantity 

P_{t} 
cupper loss 

Θ 
inertia 

M, m 
torque 

a index 
reference value 

v, v index 
control signal 

v index 
electrical 

v index 
lineto line quantity 

v index 
feedback value 

u, U 
voltage 

g index 
excitation 

U_{b} 
induced voltage 

T_{in} 
nominal starting time 

ÁI, Á block 
converter circuit 

T_{h} 
deadtime 

h index 
lines, grid quantity 

kr index 
critical value 

korl index 
limit value 

GV block 
firing controller 

e index 
DC quantity 

SZ block 
controller 

k index 
mean value 

meg index 
allowed quantity 

SZM unit 
synchronous machine 

ISZM 
PWM 

D (coordinate system) 
Cartesian (coordinate system) 

l l index 
airgap quantity 

L index 
flywheel quantity 

P_{tip} 
power design rating 

csúcs index 
peak value 

AM unit 
asynchronous machine (induction machine IM) 
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