Chapter 5: Scalar control methods

This chapter introduces two-inductance, T and T'', per-phase equivalent circuits of the induction motor for explanation of the scalar control meth-ods. The open-loop, Constant Volts Hertz, and closed-loop speed control methods are presented, and field weakening and compensation of sup and stator voltage drop are explained. Finally, scalar torque control, based on decomposition of the stator current into the flux-producing and torque-producing components, is described.

5 SCALAR CONTROL METHODS This chapter introduces two-inductance, T and T', per-phase equivalent circuits of the induction motor for explanation of the scalar control methods The open-loop, Constant Volts Hertz, and closed-loop speed control methods are presented, and field weakening and compensation of sUp and stator voltage drop are explained Finally, scalar torque control, based on decomposition of the stator current into the flux-producing and torqueproducing components, is described 5.1 TWO-INDUCTANCE EQUIVALENT CIRCUITS OF THE INDUCTION MOTOR As a background for scalar control methods, it is convenient to use a pair of two-inductance per-phase equivalent circuits of the induction motor They differ from the three-inductance circuit introduced in Section 2.3, which can be called a T-model because of the configuration of inductances (see Figure 2.14) Introducing the transformation coefficient given by = ^, (5.1) 93 94 CONTROL OF INDUCTION MOTORS the T-model of the induction motor can be transformed into the so-called T-model, shown in Figure 5.1 Components and variables of this equivalent circuit are related to those of the T-model as follows: Rotor resistance (referred to stator), RR (5.2) = y% Magnetizing reactance, (5.3) ^M "^ 7^m ~ ^s- Total leakage reactance XL (5.4) = 7^is + % Rotor current referred to stator (5.5) Rotor flux AR (5.6) = -yAr The actual radian frequency, (Op of currents in the rotor the induction motor is given by Wj = SO) (5.7) This frequency, subsequently called rotor frequency, is proportional to the slip velocity, (a^i, as (5.8) Wr = PpWsl ;x Is Rs •AMr-S-, o-.—^AAr jcoAs FIGURE 5.1 UM )JXH JUAB The T equivalent circuit of the induction motor CHAPTER / SCALAR CONTROL METHODS Taking into account that R^/s = be expressed as /JRW/WJ, current /R 4=141=1^-=%=, 95 in the F-model can (5.9) where Tr = ^^L^R- Symbol LL denotes the total leakage inductance = XL/CD) The electrical power, Pgiec consumed by the motor is ^elec = / ? R ^ (LL (5.10) (Or and the mechanical power, Pmech* ^^^ be obtained from P^i^^ by subtracting the resistive losses, S/JR/R Finally, the torque, T^, developed in the motor can be calculated as the ratio of Pmech to the rotor angular velocity, 0)^, which is given by a>M = '• (5.11) Pp The resultant formula for the developed torque is EXAMPLE 5.1 For the example motor, find parameters of the F-model and the developed torque The F-model parameters are: = 1.0339, R^ = 0.1668 fl/ph, LM = 0.0424 H/ph, andLL = 0.00223 H/ph This yields Tr = 0.00223/ 0.1668 = 0.0134 s The rms value, A^, of the stator flux under rated operating conditions of the motor, calculated from the T-model in Figure 2.14, is 0.5827 Wb/ph Under the same conditions, the slip of 0.027 results in the rotor frequency, Wp of 0.027 X 377 = 10.18 rad/s These data allow calculation of the rotor current using Eq (5.9) as/R = 0.5827/0.1668 X 10.18/[(0.0134 X 10.18)^ + 1]^^^ = 35.24 A/ph, which corresponds to /^ = 1.0339 X 35.24 = 36.44 A/ph The developed torque, T^, can be found from Eq (5.12) as T^ = X X 0.5827^/0.1668 X 10.18/[(0.0134 X 10.18)^ + 1] = 183.1 Nm These values can be verified using the program used in Example 2.1 • Another two-inductance per-phase equivalent circuit of the induction motor, called an inverse-T or T'-model, is shown in Figure 5.2 The 96 CONTROL OF INDUCTION MOTORS FIGURE 5.2 The T' equivalent circuit of the induction motor coefficient, y\ for transformation of the T-model into the F'-model is given by ,f _ ^ m Xr' (5.13) and the rotor resistance, magnetizing reactance, total leakage reactance, rotor current, and rotor flux in the latter model are y'% (5.14) •^M ~" -^n (5.15) Xi = X„ + y'X^ (5.16) RR I' = =^ (5.17) and K = 7'A (5.18) respectively The electrical power is given by an equation similar to Eq (5.10), that is ^elec ~ 3/?R / R , (Or (5.19) CHAPTER / SCALAR CONTROL METHODS 97 and the developed torque can again be calculated by subtracting the resistive losses and dividing the resultant mechanical power by the rotor velocity This yields r/2 ^M = ^PpRk^^ (Or (5.20) which, based on the T' equivalent circuit, can be rearranged to TM = ^P^MIKIM^ (5.21) where L^ = ^ V ^ denotes the magnetizing inductance in the F'-model EXAMPLE 5.2 Repeat Example 5.1 for the F'-model of the example motor The r'-model parameters are: 7' = 0.9823, R!^ = 0.1505 ft/ph, L;^ = 0.0403 H/ph, and Li = 0.0021 H/ph Phasors of the rated stator and rotor current calculated from the T-model (see Example 2.1) are = 35.35 -7*17.66 A/ph and/^ = -36.21 + ;4.06 A/ph, respectively Hence, = (-36.21 +74.06)70.9823 = -36.86 +7*4.13 A/ph and /M = "•" ~ —1.51 — 7*13.6 A/ph The magnitudes, /R and 7^, of these currents are 37.09 A and 13.68 A, respectively, which yields TM = X X 0.0403 X 37.09 X 13.68 = 184.03 Nm, a result practically the same as that obtained in Example 5.1 • 5.2 OPEN-LOOP SCALAR SPEED CONTROL (CONSTANT VOLTS/HERTZ) Analysis of Eq (5.12) leads to the following conclusions: If (Of = 1/Tp then the maximum (pull-out) torque, T^ij^^^, is developed in the motor It is given by Tu^max = l-5Pp—, (5.22) and the corresponding critical slip, s^^ is TrO) (5.23) Typically, induction motors operate well below the critical slip, so that (Oj < l/Tp Then, (Trw^)^ + ^ , and the torque is practically proportional to coj For a stiff mechanical characteristic 98 CONTROL OF INDUCTION MOTORS of the motor, possibly high flux and low rotor resistance are required When the stator flux is kept constant, the developed torque is independent of the supply frequency, / On the other hand, the speed of the motor strongly depends o n / [see Eq (3.3)] It must be stressed that Eq (5.12) is only valid when the stator flux is kept constant, independently of the slip In practice, it is usually the stator voltage that is constant, at least when the supply frequency does not change Then, the stator flux does depend on slip, and the critical slip is different from that given by Eq (5.23) Generally, for a given supply frequency, the mechanical characteristic of an induction motor strongly depends on which motor variable is kept constant EXAMPLE 5.3 Find the pull-out torque and critical slip of the example motor if the stator flux is maintained at the rated level of 0.5827 Wb (see Example 5.1) The pull-out torque, rM,niax' is 1.5 X X 0.5827^/0.00223 = 685.2 Nm, and the corresponding critical slip, s^j, at the supply frequency of 60 Hz is 1/(377 X 0.0134) = 0.198 Note that these values differ from those in Table 2.2, which were computed for a constant stator voltage • Assuming that the voltage drop across the stator resistance is small in comparison with the stator voltage, the stator flux can be expressed as V o) 1V 2TT f Thus, to maintain the flux at a constant, typically rated level, the stator voltage should be adjusted in proportion to the supply frequency This is the simplest approach to the speed control of induction motors, referred to as Constant Volts/Hertz (CVH) method It can be seen that no feedback is inherently required, although in most practical systems the stator current is measured, and provisions are made to avoid overloads For the low-speed operation, the voltage drop across the stator resistance must be taken into account in maintaining constant flux, and the stator voltage must be appropriately boosted Conversely, at speeds exceeding that corresponding to the rated frequency,/^at, the CVH condition cannot be satisfied because it would mean an overvoltage Therefore, the stator voltage is adjusted in accordance to the following rule: f Vs (K,rat - VS,0)T- ^s,rat + ^s.O M for /