Scalar Control
description
Transcript of Scalar Control
1
SCALAR CONTROL OF INDUCTION MOTOR DRIVES We have seen that applying balanced, sinusoidal 3-‐phase supply to a 3-‐phase sinusoidal distributed winding produces a rotating MMF wave and hence rotating magnetic flux in the airgap at the synchronous frequency. The rotating magnetic flux will induce
(i) EMF on the stator, 𝐸!" (ii) EMF on the rotor, 𝐸!
The induced EMF on stator is known as the back EMF (or also known as the airgap voltage), is proportional to the airgap flux and the frequency of the applied voltage, thus
𝐸!" = 𝑘!𝑓𝜙!" (1) The stator voltage equation can be written as
𝑉! = 𝑅!𝐼! + 𝑗2𝜋𝑓𝐿!"𝐼! + 𝐸!", (2) where 𝑅!𝐼! and 𝑗2𝜋𝑓𝐿!"𝐼! represent the voltage drop due to stator resistance and stator flux leakage respectively. The induced EMF on the rotor circuit, on the other hand, will be at slip frequency since the rotor rotates at slip frequency with respect to the rotating magnetic flux. The induced EMF on rotor can be written as
𝐸! = 𝑘!𝑓!"𝜙!" (3) Since the rotor bars are shorted by end rings, rotor current will flow, hence the rotor voltage equation can be written as
𝐸! = 𝑅!𝐼! + 𝑗2𝜋𝑓!"𝐿!"𝐼! (4)
where Llr is the rotor leakage inductance to represent for the leakage rotor flux, and Rr is the rotor resistance. The rotor current, in turn, induced the rotor flux, which rotates at a slip frequency with respect to the rotor and therefore at synchronous frequency with respect to the stationary stator. The torque production can be considered as a result of the interaction between the rotor flux and the airgap flux. The per-‐phase equivalent circuit, which we have derived before, is shown in Figure 1(a). Figure 1(b) shows the corresponding phasor diagram.
Figure 1: (a) Per-‐phase equivalent circuit, (b) Phasor diagram It can be shown that the steady state torque of the induction motor is given by
𝑇! = 𝐾𝐼!𝜙!"𝑠𝑖𝑛𝛿, (5)
𝛿 𝐸!" 𝐼!
𝐼! 𝐼!
𝑉!
𝑉!
𝐼! 𝐼!
𝐼! 𝐸!"
+
_
+
_
𝐿!" 𝐿!"
𝐿! Rr/s 𝜃 𝑗𝜔𝐿!"𝐼!
(a)
(b)
2
where K is a constant, and 𝛿 is the angle between 𝐼! and 𝐼!. For small slip operation, 𝛿 is close to 90o (i.e. 𝜃 ≈ 0) and hence (5) can be written as
𝑇! = 𝐾𝐼!𝜙!" (6) When the rotor is running at small slip, 𝑅! ≫ 2𝜋𝑓!"𝐿!", hence combining (3), (4) and (6), we can write
𝑇! = 𝐾′𝜙!"! 𝑓!" (7) To ensure maximum torque capability at any frequency, it is therefore necessary to maintain the airgap flux, 𝜙!" at its rated value. With constant 𝜙!", the torque is proportional to the slip frequency, fsl. We have seen that in the per-‐phase equivalent circuit, the current that flows through Lm (i.e. magnetizing current Im) is responsible for the airgap flux production. Thus from the per-‐phase steady-‐state equivalent circuit point of view, in order to maintain the rated airgap flux at any frequency, it is therefore necessary to ensure that Im is at its rated value at any frequency. From the equivalent circuit, the magnetizing current Im can be written as (8).
𝐼! = !!!!"!!
! 𝐼!,!"#$% =!!,!"#$%
!!!!"#$%!! (8)
According to (8), the magnitude of the magnetizing current can be maintained constant at its rated by maintaining the ratio Eg/f equals to Eg,rated/frated. If the frequency is reduced, Eg has to be reduced proportionally to maintain constant Im. If operation at small slip is considered, and the ratio of Eg/f is maintained constant, the motor characteristics at different synchronous frequencies are as shown in Figure 2.
Figure 2: Constant airgap characteristics at different frequencies At high speed, where the induced back EMF, Eg, is large and the voltage drop across the stator leakage and resistance are relatively small; under this condition, Eg/f is maintained constant by maintaining Vs/f constant. In other words, we can assume
!!!≃ !!
! (9)
However at low speed, Eg is small and thus the voltage drop across the stator impedances is significant and approximation (9) cannot be used. If (9) is assumed, then the rated flux cannot be maintained hence torque capability will be reduced. In order to improve the torque capability at low speed, the following method can be used:
𝜔!"#$,!"#$% 𝜔!"#!
𝜔!"#!
𝑇!"#$
T
𝜔! (rad/s) 𝜔!"#$,!"#$% 𝜔!"#$! 𝜔! ,!"#$% 𝜔! ,!
𝑇!"#$%
𝑇!
𝑇!"#$%𝜔!"#$,!"#$%
=𝑇!
𝜔!"#$!=
𝑇!𝜔!"#$!
𝑇!
𝜔!"#$! 𝜔! ,!
3
(i) Boosting the voltage at low frequency:
To accurately boost the voltage, stator current needs to be measured. The voltage drop across the stator impedance is then calculated and added to the stator voltage. Alternatively, one can approximate the amount of voltage boost needed at low speed, which depends on the stator current and hence on the load. Low frequency voltage boost can be either a linear boost of a non-‐liner boost (Figure 2)
Figure 3: Voltage boost at low frequency
ii) Stator current control
It also possible to control the magnetizing current, Im, in order to ensure rated magnetizing current at all times. The relationship between the stator current and the magnetizing current can be obtained from the per-‐phase equivalent circuit. Thus the magnetizing current can be indirectly controlled via the stator current. This can be accomplished, for example, using a current-‐controlled voltage source inverter.
From the per-‐phase equivalent circuit,
𝐼!!!"!!"!
!!!
!" !!!!! !!!!
𝐼! (10)
which gives
𝐼!!!"!!!
!!!
!"!!"!!!!
𝐼! (11)
Let 𝐿!" = 𝜎!𝐿! , where 𝜎! is the rotor leakage factor, then we can write (11) as
𝐼!!!"!!!
!!!
!" !!!!!!
!!!!!!
𝐼! (12)
Recognizing that 𝜔!"#$ = 𝑠𝜔 and 𝜏! =
!!!! , (12) can be written as
𝐼!!!!!"#$!!!!
!!!"#$!!
!!!!!!!!
𝐼! (13)
Constant magnetizing current Im can be obtained by controlling Is according to (13). With Im set to its rated value and motor parameters assumed constant, (13) indicated that Is is a function of slip frequency. One possible scheme is shown in Figure 4. The speed controller generates the slip frequency,
4
which is fed to the function generator to produce the stator current magnitude according to (13). The stator current reference generator generates the 3-‐phase current references based on this magnitude and the synchronous speed, which is obtained by adding the slip speed with the rotor speed. Three-‐phase stator currents are synthesized using current controlled scheme as discussed in earlier module. As can be seen from (13), the generation of the stator current reference is highly dependent on the motor parameters (Rr, Lr and Lm), which will change with operating temperature. If motor parameters varies from their nominal values, Im will not be at its rate value.
Figure 4: Constant magnetizing current with stator current control
Open-‐loop V/f control For low cost, low performance drive, open-‐loop constant V/f control is normally employed. With open-‐loop speed control, the rotor speed will be less than the synchronous speed by slip speed. In other words, the desired speed, 𝜔!!∗ , will differ from the actual speed, 𝜔!,!, by slip speed 𝜔!"#$!, as shown in Figure 5. To improve on the speed regulation, slip speed has to be estimated and added to the reference speed – this is known as the slip compensation technique. According to Figure 5, the new reference speed, 𝜔!!∗ , is obtained by adding 𝜔!!∗ with the estimated 𝜔!"#$!. With the new reference speed, the new rotor speed 𝜔!! will be approximately equal to 𝜔!!∗ . In actual, 𝜔!"#$! will be slightly higher than 𝜔!"#$!; if the load torque is constant, then, 𝜔!"#$! = 𝜔!"#$!. A typical open-‐loop constant V/f control scheme is as shown in Figure 6.
Figure 5: Slip compensation
𝜔!!∗
𝑇!"#$
T
𝜔! (rad/s) 𝜔!"#$! 𝜔!!
𝑇!
𝜔!!∗ = 𝜔!!
∗ + 𝜔!"#$!
𝜔!! ≈ 𝜔!!∗
𝑇!
Motor characteristic AFTER slip compensation
Motor characteristic BEFORE slip compensation
𝜔!"#$!
5
Figure 6: Constant V/f drive with slip compensation How is the slip speed estimated? The slip frequency is proportional to the torque, hence it can be estimated by estimating the torque. The torque is estimated from the air-‐gap power, which is obtained by subtracting the input power. Thus,
𝑇! =𝑃!"#!!"#𝜔!
Input power, on the other hand, is calculated by subtracting the input DC power with the inverter losses, as shown in Figure 7.
Figure 7: Airgap power estimation Closed-‐loop speed control by slip compensation Speed regulation can be improved by employing closed-‐loop speed control system with tachometer feedback, as shown in Figure 8 [2]. The reference and actual speed are compared and the error is fed to the speed controller,
6
which generates the slip frequency. The slip frequency is limited to its maximum value and added to the rotor frequency that gives the synchronous frequency; the slip frequency is limited in order to avoid the synchronous frequency from reaching the breakdown frequency. Using the synchronous frequency, constant V/f is implemented.
Figure 8: Closed-‐loop speed control by slip compensation Further readings: [1] Power Electronic Control of AC Motors – J.M.D. Murphy and F.G. Turnbull, Pergamon Press, 1988 [2] Modern Power Electronics and AC Drives – BK Bose, Prentice Hall, 2001 [3] Power Electronics: Converters, applications and design – Ned Mohan, TM Undeland, WP Robbins, John Wiley, 2003