ELEC3565 Electric Machines
Appendix to Part 2 of Course Assignment - Induction Motors
The objective of this lab is to derive the equivalent circuit parameters of an induction motor from two simple tests, namely, locked-rotor test and no-load tests; neither in- volves supplying large amounts of power to the motor.
Section 1: Overview on the induction motor
The motor investigated in this lab is a 3-phase, 50 Hz, 4-pole, cage induction machine rated for output power 2.2kW at 1420rpm.
The per-phase equivalent circuit for the induction motor is shown in Figure 1. Note that it is very similar to that of a transformer, with primary and secondary becoming stator and rotor respectively. There is one vital difference: the mechanical output power of the motor is represented by the power dissipated in the resistance Note that this resistance changes with slip, reflecting the fact that the mechanical power is a function of slip.
When the slip is zero (at synchronous speed), the load resistance becomes infinite, so there is no rotor current, and hence no output power. And when the slip is 1 (at standstill) the load resistance becomes zero, and again there is no mechanical out- put power. Note that in the later condition, however, the rotor current will be large, because the load resistance is zero (i.e. a short-circuit). The corresponding torque is of course the starting torque of the motor.
Figure 1 – The ‘per phase’ equivalent circuit of 3-phase induction motor
R1 = Stator winding resistance per phase
R2' = Referred rotor resistance per phase
X1 + X'2 = Total referred leakage reactance per phase
Xm = Magnetising reactance per phase
Rc = 'Core loss' resistance per phase
The equivalent circuit parameters (R1 , R'2, X1 , X'2, Rc and Xm ) are derived from two tests - the locked-rotor test and the no-load test. They are explained in the following section.
Section 2: Overview on the locked-rotor and no-load tests
Under locked-rotor condition (at S = 1)
The part of the circuit to the right of the dotted line in Figure 1 becomes short-cir-cuited. The impedance of the remaining elements R'2 and X'2 is much lower than the impedance of parallel-connected element Xm , so the latter can be ignored and the per-phase equivalent circuit for locked-rotor test reduces to that shown in Figure 2. The locked-rotor test of induction motor is equivalent to the test of a transformer with short-circuited secondary.
To perform. this test safely in practice, the voltage must be increased only up to the point when Iph corresponds to that of the rated value while the rotor is kept to be standstill (w = 0). The supply voltage in this test therefore will be significantly lower than that of the rated value.
Figure 2 - Reduced equivalent circuit under locked-rotor condition
The per-phase voltage, current and power related to (R1 + R'2) and (X1 + X'2) are de-fined by following equations:
We normally will be able to measure R1 directly across the windings using multime- ter. Using the measured quantities of per-phase voltage, current and power through locked-rotor test, R'2 can be calculated from Equation (1) and with known
R1 and R2', X1 + X'2 can be calculated from Equation (2). X1 and X'2 are taken to have equal value, i.e. X1 = X'2.
Under no-load condition
The slip (s) is very small and the impedance to the right of the dotted line in Figure 1 is much higher than that of the parallel-connected Rc and Xm . So the per-phase equivalent circuit for no-load test reduces to that shown in Figure 3a).
Figure 3 - Reduced equivalent circuit under no-load condition
Since the currents of parallel elements Rc and Xm differ from the phase current Iph , the values of parameters Rc and Xm cannot be directly obtained from measured quantities in no-load test (voltage, current and power). Nonetheless, by applying the complex numbers notation, the parallel-connected elements Rc and Xm in the equiva- lent circuit for no-load test can be transformed into an equivalent series-connected elements R and X shown in Figure 3b) where Rc and Xm are respectively related to R and X by the following:
This transformation is derived at the end of this lab sheet.
The no-load test of induction motor is equivalent to the test of a transformer with open secondary. Normally this test is done under rated voltage level while the load torque is set to zero.
Series-connected elements (R1 + R2(I)) and (X1 + X2(I)) can now be easily calculated from vph , Iph and pph in similar manner as done for the locked-rotor test, i.e. by ap-plying Equations (1) and (2) in which the parameters R2(I) and X2(I) are replaced by R and X , respectively. Knowing the reactance X1 obtained from the locked-rotor test and the resistance R1 , the values of R and X can be determined. With the calculated and X , Rc and Xm can be calculated from Equations (3) and (4).
Section 4: Derivation of the Equivalent Circuit Parameters
Based on the equations given in Section 3 and the following measurements obtained from your lab:
No-load test:
R1 = Ω,
Vph = V,
Iph = A,
Pph = W.
Locked-rotor test:
Vph = V,
Iph = A,
Pph = W,
you can now derive the parameters R2(I) , X1 , X2(I) , Rc and Xm .
Transformation of parallel elements Xm and Rc into equiva- lent series elements R and X
Applying complex numbers notation, the impedance of parallel elements Xm and Rc is expressed as
The real and imaginary terms on the right-hand side of this expression represent re- spectively the resistive and reactive elements of the equivalent (series) impedance R + jX , i.e.,
and
These can be re-arranged so that we have and