**Tests to Determine Circuit Model Parameters**

Circuit Model Parameters – As the Circuit Model Parameters of an induction motor is similar to that of a transformer, the parameters of the model can be obtained by means of nonloading tests as in the case of the transformer—no-load test (corresponding to the OC test on the transformer) and the blocked-rotor test (corresponding to the SC test on transformer).

**The No****–****Load (NL) Test**

In this test the motor is run on no-load at rated voltage and frequency. The applied voltage and current and power input to motor are measured by the metering as per Fig. 9.17.

Let the meter readings be

Power input = P_{0} (3-phase)

Current =I_{0} (average of the three meter readings)

Voltage = V_{0} (line-to-line rated voltage)

Power input at no-load (I_{0}) provides losses only as the shaft output is zero. These losses comprise,

P_{o} (no-load loss) = P_{c1} (stator copper loss) + P_{c1} (iron/core loss) + P_{wf} (windage and friction loss) = Rotational loss]

wherein core loss occurs only in the stator as the slip is extremely low (of the order of 0.001) and so the frequency of rotor current is as low as 0.05 Hz.

The magnitude of no-load current in an induction motor is about 30-40% of full-load current because of the air-gap. So the stator copper loss at no-load needs to be accounted for. This can be estimated by measuring dc stator resistance and correcting to ac value (50 Hz) and corrected for temperature (°C).

The mechanical power developed corresponds to P_{wf }only and so, as already mentioned above the slip is very low and the output resistance

Also R_{2}‘/s_{0} >> X2′ and so X2′ can be ignored. The corresponding no-load Circuit Model Parameters is drawn in Fig. 9.18(a) wherein R_{2}‘/s_{0} appears in parallel to R_{i}. By combining the parallel shunt resistances, the final circuit at no-load is as given in Fig. 9.18 (b). Here R_{iwf} accounts for rotational loss, i.e., core loss and windage and friction loss. Magnitude-wise R_{iwf} >>

R_{1}, the stator resistance, is found by dc testing of the stator winding and correcting the value to ac operation (at 50 Hz). X_{1}, the stator leakage reactance, will be found from the blocked-rotor test which follows. We can then find X_{m} and R_{iwf}. from the no-load (NL) test data. By simplification of the circuit of Fig. 9.18(b), we get

The equivalent circuit is drawn in Fig. 9.18(c).

It can be justifiably assumed that (X_{m}/R_{iwf})^{2} = 0, so we get from the above equations

From the NL test data (V_{0}, I_{0}, P_{o}) we can find from the circuit of Fig. 9.18(c).

By substituting the values of R_{o} and X_{o} in Eqs. (9.37) and (9.38) respectively, we obtain X_{m} and R_{iwf}.

R_{i}, the stator core loss resistance can be found out if the additional test of separating core loss from the windage and friction loss, as described below, is carried out.

**Separating Out Core-Loss from Windage and ****Friction Loss**

The separation of these two losses can be carried out by the no-load test conducted from the variable-voltage, rated-frequency supply. As the voltage is reduced below the rated value, the core-loss decreases almost as the square of voltage. Since the slip does not increase significantly, the windage and friction loss remains almost constant. The voltage is continuously reduced till the machine slip suddenly begins to increase and the motor tends to stall. At no-load this happens at a sufficiently reduced voltage. The plot of P_{o} versus Vas shown in Fig. 9.19 is extrapolated to V= 0 which gives P_{wf} as P_{i}= 0 at zero voltage.

**Voltage-Ratio Test**

This test can only be conducted on a slip-ring motor by exciting the stator at rated voltage and frequency while keeping the rotor open-circuited; the rotor will not rotate. The ratio of rotor to stator voltage can then be measured by means of a voltmeter; it may be noted that the rotor-induced emf which appears at the slip-rings is of a supply frequency as the rotor is at a standstill.

**Blocked****–****Rotor (BR) Test**

This test is used to determine the series parameters of the Circuit Model Parameters of an induction motor. The circuit is similar to that of a transformer short-circuit test. Short circuiting the load resistance in the circuit model of Fig. 9.8 corresponds to making s = 1 so that R_{2}‘ (1/s — 1) = 0. This means that the rotor must be stationary during this test, which requires that it be blocked mechanically from rotating while the stator is excited with appropriate reduced voltage. The Circuit Model Parameters seen under these conditions is given in Fig. 9.20(a).

The current drawn by the motor in the BR test should be close to its rated value as the motor reactances are sensitive to saturation effects in the magnetic core. Rated current value is obtained by applying reduced voltage to the stator as blocked rotor presents short-circuited condition at the stator terminals (low impedance Z_{BR}). The core loss at this reduced voltage can be ignored but as the magnetizing reactance (X_{m}) is much lower in an induction motor compared to a transformer, its effect cannot be ignored. This justifies the BR circuit model of Fig. 9.20 presented above.

In normal operating range of an induction motor the slip is low (2-8%). This means low rotor frequency and negligible rotor core loss. However, in BR test the rotor frequency is the same as the stator frequency which is much higher than rotor frequency in normal operation (it is almost negligible). Though with reduced voltage applied to stator the rotor core loss is small. The higher rotor frequency would affect the value of R_{BR} and the rotor resistance as determined from the test, it will be smaller. (see the last para of this section). So for obtaining accurate results for rotor resistance, the BR test needs to be conducted at reduced frequency (25% of rated frequency). The reactances thus obtained are then scaled up to the rated frequency (50 Hz). However, for motors rated less than 25 kW, reduced frequency test is not warranted.

Metering and connection diagrams for the BR test are the same as in the connection diagram of Fig. 9.17. Of course the motor must be fed from appropriate low voltage (variable) source of frequency as discussed above. The following readings are recorded during this test:

V_{BR} = stator voltage (line-to-line)

I_{BR} = stator current (average of three ammeter readings)

P_{BR }= power fed into stator; this mainly constitutes the copper loss in stator and rotor. At reduced voltage core loss (even in stator) is negligible.

From these test readings we can compute

__These values constitute the series equivalent of the BR test (Fig. 9.20(b)).__

We, however, need to determine the Circuit Model Parameters R_{2}‘, X_{1}, X_{2}‘, while R_{1} is known from the dc test. From the motor circuit in BR test as given in Fig. 9.20(b) we can write

By making certain assumptions certain simplifications are carried out below:

Equation (9.46) then takes the form

The following results can then be written down with the knowledge that

If (X_{m} + X_{2}‘) > 10 R2’, which is usually the case, the approximations made in the Eq. (9.50) for R_{2}‘ cause an error of less than 1%.

At this stage we need to separate out X_{1} and X_{2}‘, which is not possible by the data of this (BR) test. Generally it is fairly accurate to assume that

If the BR test is conducted at rated frequency, two factors affect the value of R2′ as observed above. Firstly the rotor (winding) resistance increases as the frequency of rotor currents is the same as the rated frequency, while under normal operating conditions it is very small; few hertz or so. Secondly the frequency of rotor flux alterations is also at rated frequency. The rotor core then presents an effective resistance in parallel to R2′ thereby reducing effective R2′ as measured. These two effects tend to cancel each other out. So no reduced frequency testing is needed for small size motors (less than 25 kW).

**Circuit Model as Obtained from NL and BR Tests**

With the circuit parameters obtained as above, circuit model in two alternative forms are drawn in Figs. 9.21(a) and (b).

- Model of Fig. 9.21(a)

Here R_{iwf} represents core loss and windage and friction loss. Approximation lies in the fact that actual windage and friction is taken off the shaft in mechanical form. A more exact circuit would be possible if these two losses are separated out by the test described above. Usually much accuracy is not needed.

It may be noted here that power absorbed by the output resistance R2′ (1/s — 1) is the net mechanical output.

- Model of Fig. 9.21(b) – IEE circuit Model

Here R_{iwf }is removed so that only shunt reactance is the magnetizing reactance X_{m} Core loss and winding and friction loss must now be subtracted from the mechanical output (power in R_{2}‘(1/s — 1). This is a different type of approximation and is found more accurate than the approximation made in model of Fig. 9.21(a). This circuit is, however, simpler to analyze and therefore, as mentioned already, is commonly adopted.

**Approximate Circuit Model**

The approximate Circuit Model Parameters has been given in Fig. 9.9. With reference to this circuit, the circuit model seen during NL test is as given in Fig. 9.22(a) where the rotor is regarded as open circuit. The Circuit Model Parameters during BR test is as given in Fig. 9.22(b) wherein the shunt branch is ignored.

R_{1} is obtained from dc test and is connected to operating temperature (75 °C). No ac resistance correction is normally needed for induction motors.

The approximate Circuit Model Parameters as obtained from NL and BR tests is drawn in two versions — Fig. 9.23(a) and Fig. 9.23(b). In the circuit of Fig. 9.23(b) the rotational output has to be subtracted from the gross output. This gives better results than the circuit model of Fig. 9.23(a) and so would be adopted wherever such approximation (shifting shunt branch to motor input terminals) is warranted. The approximate circuit of Fig. 9.23(a) is mainly employed in circle diagram method of determining induction motor performance.