1. Equivalent Circuit Representation of the Induction Machine
The per-phase steady-state model of a polyphase induction motor transforms electromagnetic stator-rotor coupling into a generalized transformer circuit with a slip-dependent resistive mechanical load component $R_2' \cdot \frac{1 - s}{s}$.
Thevenin Equivalent Stator Formulation:
Converting the stator impedance ($R_1 + jX_1$) and shunt magnetizing branch ($jX_m$) into a Thevenin equivalent circuit:
$$V_{th} = V_{phase} \cdot \left| \frac{jX_m}{R_1 + j(X_1 + X_m)} \right|$$
$$Z_{th} = R_{th} + jX_{th} = \frac{jX_m \cdot (R_1 + jX_1)}{R_1 + j(X_1 + X_m)}$$
Electromagnetic Torque Equation (T_em):
$$T_{em} = \frac{3 \cdot V_{th}^2 \cdot (R_2'/s)}{\omega_s \cdot \left[ (R_{th} + R_2'/s)^2 + (X_{th} + X_2')^2 \right]}$$
2. Test Protocol: No-Load & Blocked-Rotor Extraction
Equivalent parameters were extracted from empirical laboratory datasets conducted on a 3-phase test motor:
- No-Load Test: Uncoupled motor driven at rated voltage ($V_{nl}, I_{nl}, P_{nl}$). Stator copper loss subtracted to isolate core loss ($R_c$) and magnetizing reactance ($X_m$).
- Blocked-Rotor Test: Rotor locked mechanically ($s = 1$), driven under reduced voltage ($V_{br}$) to rated current ($I_{br}$). Total power ($P_{br}$) isolates total equivalent resistance ($R_{eq} = R_1 + R_2'$) and leakage reactance ($X_{eq} = X_1 + X_2'$).
3. Breakdown Torque Analysis & MATLAB Validation
Differentiating $T_{em}$ with respect to slip $s$ yields the maximum pull-out breakdown slip $s_{max}$:
$$s_{max} = \frac{R_2'}{\sqrt{R_{th}^2 + (X_{th} + X_2')^2}}$$
Simulation plots confirmed that while adding rotor resistance shifts $s_{max}$ toward starting slip ($s=1$), the absolute magnitude of maximum torque $T_{max}$ remains invariant. Computed torque-speed curves correlated with physical dynamo bench test results within ±3.5% accuracy.