HIGH PERFORMANCE DRIVES  E Levi, 2001 63 Summarising the procedure: speed loop is held open, d-axis current command is set to rated and q-axis current command of the alternating square waveform is imposed. If the rotor time constant value in the controller is correctly set, speed response has to be triangular function. Figure 3.21a is valid for the correct (rated) value of the rotor time constant and speed response is a triangular function in this case. However, if the rotor time constant significantly deviates from the correct value (1.7 times rated value in Fig. 3.21b) the speed response is unsatisfactory and far from required triangular waveform. 3.6. DESIGN OF THE CONTROL SYSTEM FOR AN INDIRECT FEED-FORWARD CURRENT-FED ROTOR FLUX ORIENTED INDUCTION MACHINE Calculation of all the necessary values, required for indirect rotor flux oriented control, will be illustrated using an example. Consider a three-phase four-pole star connected squirrel-cage induction machine, whose parameters at 50 Hz are RRXXX srsrm =====10 6 3 135 12.6 132ΩΩΩΩΩ γγ . Rated current and voltage equal 2.1 A and 380 V. The machine is to be operated as an indirect rotor flux oriented current-fed induction machine. Current control is performed with phase current controllers so that actual and reference phase currents can be assumed to be the same for the purpose of calculation. The speed is to be controlled from zero up to its rated value, using a constant, rated value of rotor flux. The rated torque and inertia of the machine are 5.07 Nm and 0.1 kgm 2 , respectively. It is required to determine all the values needed for realisation of indirect rotor flux oriented control. These are the rated rotor flux, the rated stator d-axis and q-axis currents (all in terms of rms and peak values) and the value of the slip speed for rated torque operation. Next, since the scheme under consideration contains only the speed controller, it is necessary to determine the parameters of the speed PI controller. This is done using the so-called symmetrical optimum method. The time delay due to the inverter and signal processing may be approximated with a first order delay block whose time constant is 0.05 ms (the reasons and the need for this will be explained later). The controller under consideration is the one shown in Fig. 3.10 for the general case of variable rotor flux reference. Since in this example rotor flux reference is kept constant at all times, the overall drive configuration is as shown in Fig. 3.11. The complete indirect vector controller (for variable rotor flux reference case) and its implementation as a part of the indirect feed-forward rotor flux orientation scheme (for the case of constant rotor flux reference) are shown for convenience once more in the upper and the lower part of Fig. 3.22. Constants in the implementation in the lower part of Fig. 3.22 follow directly from (3.30)-(3.31), taking into account that, due to constant rotor flux reference, ψ rm d s Li ** = : iKT KiT P L LP L Li Ki K i L TTi qs e qs e r mr r mds sl qs sl qs m rr r d s ** ** ** ** ** ** = == = = === 11 2 22 2 3 12 3 1 1 ψ ωω ψ (3.32) On the basis of Fig. 3.22 and (3.32) one concludes that the required values for an indirect vector controller are the rated rotor flux value and constants KK 1 2 , . The inductances of the machine and rotor time constant are needed in (3.32). Note that the stator parameters are not required (although they are given, for the sake of completeness). From given reactances one easily finds the inductances and the rotor time constant: () () LX LX LLL TLR mm rr rmr rrr === === =+= == = 2 50 132 100 0 42 2 50 12.6 100 0 04 046 046 63 73ms π π ππ γγ γ /. /. . ./. H H H HIGH PERFORMANCE DRIVES  E Levi, 2001 64 ψ r * i ds * T r s 1/ L m T e * 2 L r i qs * 3 PL m ω sl * ω r φ r e L r 1/ s ω ψ r * i ds * i α s * i a * i a 1/ L m 2C R j φ r e i b * P i b e W ω * T e * i qs * M PI K 1 3 _ i β s * i c * i c K 2 φ r e ω sl * ω r I 1/ s M ωω mech P Fig. 3.22 - Full indirect vector controller and its implementation for the case of constant rotor flux reference operation (base speed region only). When the machine operates in steady-state under rated operating conditions (index n), then it follows from (3.9) that TP L L i en m r rn qsn = 3 2 ψ () ψ ωψ rn m dsn sl n m qsn r rn Li Li T = = (3.33) since ψψ ωω rrnL e nsl sln TT * ,,===. Furthermore, the stator current has to be equal to its rated value as well. Note however that the given rated current is rms value, while the one obtained from d-q axis currents is the peak value. Hence iii iI sn dsn qsn sn sn =+ == = 22 222.12.97xA (3.34) As the next step, it is necessary to calculate the rated stator d-q axis current components. Torque equation of (3.33) and stator current equation (3.34) are used for this purpose, since the rated torque value and the rated stator current are known quantities: HIGH PERFORMANCE DRIVES  E Levi, 2001 65 TP L L iP L L ii ii ii i i en m r rn qsn m r dsn qsn dsn qsn dsn qsn qsn dsn == = = = 3 2 3 2 507 3 042 046 507 115 4.407 2 2 ψ . . . Substitution into stator current equation (3.34) yields iii iii ii ii ii xi xx xi sn dsn qsn sn dsn qsn dsn dsn dsn dsn dsn dsn dsn dsn =+= ==+ =+ =+ −+= = −+= = = 22 2 222 22 24 42 2 2 12 12 2.97 88209 88209 19.42 8821 19.42 8821 19.42 0 8821 19.42 0 4.59 4.23 2.1426 A 2.057 A / . . . . . ,, The correlation between d-q axis currents yields values of stator q-axis current: ii i qsn dsn qsn = = 4.407 2.0568 12, A 2.1424 A One notes that the same two values appear in inverse order as solutions for rated stator d-q axis currents. Normally the correlation ii dsn qsn < holds true, so that finally ii dsn qsn ==2.057 2.1424AA Once when the stator rated d-q axis currents are known, it becomes possible to determine the rated rotor flux and the constants of (3.32): K P L Li K Ti Li r mds rds rn m dsn 1 22 2 2 3 1 1 3 0 46 042 1 2.057 0 4226 11 0 073x2.057 533rad 0 42 2.057 0864 == = == = == = * * . . . . . A/Nm /As xWb ψ The calculated values of the stator d-q axis current components and the rotor flux are the peak values. Corresponding rms values are Ψ rn dsn qsn dsn qsn sn I I II I == == == += + = ≡ 0 864 2 0 61 2.057 2 14545 A 2.1424 2 1515 A 14545 1515 2.1 22 2 2 . . Wb A It is now possible to determine the rated angular slip frequency as well. Note however that the slip frequency is a variable, determined by the instantaneous value of the stator q-axis current command. Hence the rated value of the angular slip frequency will exist only when the stator q-axis current command is exactly equal to its rated value. From (3.33) HIGH PERFORMANCE DRIVES  E Levi, 2001 66 () ωψ ωω sl n m qsn r rn qsn r dsn sl n mech sl n Li T i Ti P rad / s rad / s === = == = 2.1424 0 073x2.057 14.267 14.267 2 7.1337 /( . ) // () Since rated synchronous (mechanical) speed is known (1500 rpm for 50 Hz for a four-pole machine), one may calculate the rated speed of rotation of the machine as well, n n sl n mech =− =− =1500 60 2 1500 30 7.1337 14319 π ωπ xrpm () /. This completes the necessary calculations related to the machine parameters. What now remains to be done is to design a PI speed controller. In order to perform this task, it is necessary to somehow represent the whole control system and the machine with a simple transfer function diagram. Consider the structure of Fig. 3.22. In this scheme the induction machine can be represented with the block diagram of Fig. 3.2. The resulting complete block diagram is shown in Fig. 3.23. ψ r * i ds * i α s * i a * i a 1/ L m 2C R j φ r e i b * P i b e W ω * T e * i qs * M PI K 1 3 _ i β s * i c * i c K 2 φ r e ω sl * ω r 1/ s ω ω i a i αs i ds 1 ψ r L m i b 3 − j φ r 1+sT r e i c 2i βs i qs L m ω sl T r φ r 3P L m T e P ω 2L r Js T L ω r 1 s Fig. 3.23 - Block diagram of the indirect vector controller and the induction motor. According to (3.1), reference and actual phase currents can be regarded as being mutually equal. Hence the current controlled PWM inverter can be omitted from the scheme in Fig. 3.23. Furthermore, in the absence of detuning, the estimated and the actual rotor flux position angles are the same (i.e. axes d* and d of Fig. 3.20 coincide). Hence the two co-ordinate transformation blocks exp(jφ r ) and exp(-jφ r ) cancel each other. The same applies to transformation blocks 2/3 and 3/2 in Fig. 3.23. Since stator d- axis current reference is constant, rotor flux in the machine is, after initial transient (initial excitation) constant as well. This enables Fig. 3.23 to be simplified to the form shown in Fig. 3.24. Note that the flux channel does not contain a controller. Furthermore, after initial transient (initial excitation) rotor flux is constant and equal to its reference. Hence the upper channel of Fig. 3.24 can be omitted from further consideration. Only the lower channel, in which the speed PI controller is, has to be considered further on. Note that in this channel the constant K 1 and the block 32PL L mr cancel each other. Thus the transfer function block diagram, required for the design of the speed controller, reduces to the one shown in Fig. 3.25. Figure 3.25 is the final structure of the transfer function block diagram. Only one minor modification is still necessary. Speed PI controller is designed for the response to the input speed command - hence disturbance term (load torque) can be ignored. The resulting structure is the one of Fig. 3.26. HIGH PERFORMANCE DRIVES  E Levi, 2001 67 Controller Machine ψ r * i ds * i ds = i ds * ψ r = ψ r * 1/L m L m ω * T e * i qs * i qs = i qs *3PL m PI K 1 _2L r T e − T L ω mech ω P 1/(Js) Fig. 3.24 - Simplified block diagram of an indirect rotor flux oriented induction machine. T L ω* T e *= T e − PI − ω P Js Fig. 3.25 - Speed control loop. ω* T e *=T e P PI − Js ω Fig. 3.26 - Speed control loop with omitted disturbance term. Recall that the inverter was taken as ideal in the very beginning. However, some delay always occurs in the system due to signal processing and delay in inverter response to the command. These delays are conveniently represented in a somewhat superficial way, by inserting a first order delay block in between the torque command and the actual torque. Hence the transfer function block diagram that is used for the speed controller design takes the final form of Fig. 3.27. Time constant of the first order delay block, T delay , is in this example equal to 50 micro-seconds. Transfer function of a PI controller is given with HIGH PERFORMANCE DRIVES  E Levi, 2001 68 () Gs K sT KsT sT p i pi i ()=+= + 1 1 1 (3.35) ω* T e *1 T e P PI − 1+sT delay Js ω Fig. 3.27 - Final structure of the block diagram for the speed controller design. Parameters that need to be determined are the proportional gain and the integral time constant, K p and T i . The transfer function of the process is essentially represented with a first order delay block (representing all the small delays in the system) and a pure integrator (representing mechanical subsystem of the motor). Note that, without the first order delay block (that was artificially added at the end of the transfer function block diagram derivation) the complete indirect rotor flux oriented induction machine is represented with a single, pure integrator block (Fig. 3.26). This means that the complete electro-magnetic part of the motor behaves as an ideal plant that instantaneously responds to the given command (recall that, by definition, torque response is instantaneous in a rotor flux oriented induction machine and is achieved by an appropriate, again instantaneous, change in the stator q-axis current; instantaneous change in the current is possible since ideal current feeding was assumed). Since the complete drive transfer function consists of a first order delay block with very small time constant and a pure integrator block, so-called symmetrical optimum method is used to calculate the parameters of the PI controller (this method is in general used when there is a pure integrator in the plant transfer function; when there is no pure integrator and the plant is represented with first order delay blocks only, another method called modulus optimum is used). Calculation of the parameters of the PI controller depends on the ratio of time constants in the plant transfer function. Let the sum of all the small delays be denoted with σ (in this case sum of all the small delays equals T delay ) and let the dominant time constant in the plant transfer function be T dom .Dominant time constant is here the time constant of the integrator block: T dom = J/P = 0.1/2 = 0.05 sec. One then looks at the ratio of the dominant time constant to four times the sum of small delays in the system. For the data of this drive, shown in Fig. 3.28, one finds that the ratio of dominant time constant to four times the sum of small time delays is ω* K p (1+sT i ) T e *1 T e 1 − sT i 1+0.00005s 0.05s ω PI First order delay Integrator block block Fig. 3.28 - Transfer block diagram for the speed PI controller design, using data of the drive under consideration. HIGH PERFORMANCE DRIVES  E Levi, 2001 69 () T dom 4 005 450 10 250 1 6 σ ==>> − . x When this ratio is much larger than one (this is in general always the case in vector controlled drives), the proportional gain and the integral time constant of the speed PI controller are determined with KT T pdom i = = 2 4 σ σ (3.36) Therefore, for the numerical values of this drive, one gets () () K T T p dom i == = == = = − − 2 005 250 10 500 4 4 50 10 0 0002 0 2 6 6 σ σ . x xsms When the speed controller is designed using symmetrical optimum, an overshoot of 43% will results in the speed response to application of a step speed command. This in general cannot be tolerated and the overshoot is reduced by inserting a smoothing element in the channel of the speed command. The smoothing block is a first order delay block, whose time constant equals four times the sum of small delays in the system. Hence the transfer function of the smoothing element is Gs s smooth () () = + 1 14 σ (3.37) and its time constant equals the PI controller integrator time constant, 0.2 ms here. The overshoot in the speed response is reduced, by inserting the smoothing element, to 8.1%. The final outlay of the speed controller part of the drive is shown in Fig. 3.29, both in general form and with the calculated values for the drive under consideration. Limiter ω *1 K p (1+ sT i ) T e * 1+(4σ)s − sT i ω Reference PI smoothing controller ω Limiter ω* 1 500(1+0.0002 s ) T e * 1+0.0002s − 0.0002 s ω Reference PI smoothing controller ω Fig. 3.29 - Speed control loop designed using symmetrical optimum: general outlay and specific values for the drive under consideration. Fig. 3.29 includes, apart from the reference smoothing element and the speed PI controller, one specific control block that has not been discussed so far in any considerable depth. The block is called ‘limiter’ and it serves the purpose of limiting the output of the speed controller to the maximum permissible value. For the purpose of explaining its role, suppose that the step speed reference, equal to rated (electrical) angular speed of the machine under consideration, is applied from standstill at time instant zero. Rated speed has already been calculated and it corresponds to 299.9 rad/s ≈ 300 rad/s. The smoothing element has very small time constant (meaning that its output becomes equal to the input in very short time interval of approximately five times 4σ, that is 1 ms) and therefore its existence can be neglected for the sake of this discussion. Since the reference speed at the input of the summator equals 300 rad/s, while the actual speed is zero, the speed error equals 300 rad/s. Passing of this speed error through the speed controller whose proportional gain is 500 means that the torque reference at the output of the speed controller initially equals 300 x 500 = 150,000 Nm! Recall that the rated torque is HIGH PERFORMANCE DRIVES  E Levi, 2001 70 5.07 Nm. This means that stator q-axis current would initially be required to be equal to approximately 30,000 times its rated value. This obviously must not be allowed since the inverter would be destroyed immediately. The output of the speed controlled must therefore be limited to the maximum permitted torque value, that corresponds to the maximum permitted stator q-axis current value. The maximum permitted stator q-axis current value is determined by the rating of the inverter. Typically in high performance drives, continuous current rating of the inverter would be 100% to 150% of the machine’s current rating, while the inverter short term (transient rating) may be up to 300% (and rarely higher) of the continuous current rating. Short term means that such a high current can persist during a transient only. Hence the torque would be typically limited to 200% to 300% of the machine’s rated torque. When the output of the speed controller is smaller than the limiter adjustment, the limiter passes through the torque command (i.e. speed controller output) without affecting its value. If the speed controller output exceeds the limiter adjustment, the limiter gives at the output the adjusted maximum value. Hence the limiter is often represented with the block illustrated in Fig. 3.30. T e(out) T e(lim) T e(in) *T e(in) T e(out) * −T e(lim) Fig. 3.30 - Limiter. Adjustment of the limiter for negative and positive inputs is always the same in electric drives. The action of the limiter can be described with If T e(in) * < T e(lim) then T e(out) * = T e(in) * (3.38a) If T e(in) * >T e(lim) then T e(out) * = T e(lim) (3.38b) where T e(lim) is the limiter adjustment. 3.7. PROBLEMS 1. An induction machine has the following parameters (all rotor quantities are referred to the stator): RR LL L sr sr m == == =10 63 004 04HHΩΩ γγ The machine has four poles and rated power and rated speed of 0.75 kW and 1400 rpm, respectively, for the rated 380 V, 50 Hz three-phase supply with a star connected stator winding. (a) Give the complete time domain model of the machine in an arbitrary rotating reference frame in terms of d-q axis variables. (b) Derive the space vector model of the induction machine in an arbitrary reference frame, using the model given in (a) as the starting point. Sketch the dynamic space vector equivalent circuit of an induction machine in the arbitrary reference frame. (c) Derive and calculate the stator phase voltage d-q axis components and the stator phase voltage space vector, if the machine is fed with the sinusoidal phase voltages vVtvVt vVt ab c ==−=−2223243cos cos( / ) cos( / ) ωωπ ωπ of rated rms value and frequency, and the speed of the reference frame is selected as synchronous (d- axis of the reference frame aligned with the stator phase voltage space vector). (d) Calculate the stator current space vector and the stator current d-q axis components in the synchronously rotating frame if the machine operates with rated load torque and is fed with the voltages given in (c). . HIGH PERFORMANCE DRIVES  E Levi, 2001 63 Summarising the procedure: speed loop. 100 0 42 2 50 12.6 100 0 04 046 046 63 73ms π π ππ γγ γ /. /. . ./. H H H HIGH PERFORMANCE DRIVES  E Levi, 2001 64 ψ r * i ds * T r s 1/ L m T e * 2 L r i qs * 3 PL m ω sl * ω r φ r e L r 1/ s ω ψ r * i ds * i α s * i a * i a 1/ L m 2C R j φ r e i b * P i b e W ω *