Determining single phase IM equivalent circuit parameters

The parameters of the single-phase induction motor equivalent circuit ($R_1$, $X_1$, $R_2'$, $X_2'$, $X_m$ and rotational loss) are found from three simple tests on the main winding (auxiliary winding left open): a DC resistance test, a blocked-rotor test and a no-load test.

     A        W
 o--(A)--+--[W]--+-------+
         |       |       |
 1-ph    (V)   Main     rotor
 variac  |    winding  (free or
 supply  |       |      blocked)
 o-------+-------+-------+

1. DC resistance test

Pass DC through the main winding and measure $V_{dc}$ and $I_{dc}$:

$$R_1 = \frac{V_{dc}}{I_{dc}} \quad(\text{multiply by about } 1.1\text{–}1.2 \text{ for AC/skin effect})$$

2. Blocked-rotor (short-circuit) test

Procedure: hold the rotor stationary, apply a reduced voltage through a variac until rated current flows. Record $V_{sc}$, $I_{sc}$, $P_{sc}$.

At $s = 1$ the forward and backward halves are equal, and since $X_m \gg$ rotor impedance, the magnetising branch is neglected. The circuit becomes $R_1 + jX_1 + R_2' + jX_2'$.

$$\begin{aligned} Z_{sc} &= \frac{V_{sc}}{I_{sc}}, \qquad R_{sc} = \frac{P_{sc}}{I_{sc}^2} \\ X_{sc} &= \sqrt{Z_{sc}^2 - R_{sc}^2} \\ R_2' &= R_{sc} - R_1 \\ X_1 &= X_2' = \frac{X_{sc}}{2} \end{aligned}$$

3. No-load test

Procedure: run the motor at rated voltage with no load (start it with the auxiliary winding, then open it). Record $V_0$, $I_0$, $P_0$.

At no load $s \approx 0$:

• Forward branch: $0.5R_2'/s \to \infty$, so only $j0.5X_m$ remains.
• Backward branch: $0.5R_2'/(2-s) \approx R_2'/4$; since $0.5X_m \gg$ this, only $R_2'/4 + j0.5X_2'$ remains.

 R1   X1   0.5Xm     R2'/4   0.5X2'
o-/\/-mmm---mmm---+--/\/\---mmm---o

$$\begin{aligned} Z_0 &= \frac{V_0}{I_0}, \qquad \cos\phi_0 = \frac{P_0}{V_0 I_0} \\ X_0 &= Z_0 \sin\phi_0 = X_1 + 0.5X_m + 0.5X_2' \\ X_m &= 2\left(X_0 - X_1 - 0.5X_2'\right) = 2(X_0 - 1.5X_1) \end{aligned}$$

Rotational (core + friction + windage) loss

$$P_{rot} = P_0 - I_0^2\left(R_1 + \frac{R_2'}{4}\right)$$

This loss is assumed constant and is subtracted from the mechanical power when finding output.

Summary

With these values, the full forward/backward equivalent circuit is drawn, and current, power factor, torque and efficiency at any slip can be calculated.