topicwise_answers_explanation/T-15_IM_Equivalent_Circuit.md

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← T-14: IM as Rotating Transformer | 🏠 Index | T-16: Torque Equations & Curves →

T-15: IM Equivalent Circuit

ECE 2207 — Explanation Answers (Sorted by Topic)

Topic Overview: This document compiles all semester final questions and explanation answers on IM Equivalent Circuit from 7 years of exams (2017–2024). Repeated questions appear once with all exam appearances noted.

T-15: Step-by-Step Derivation of the Induction Motor Equivalent Circuit

Appears in: 2017 Q3(a), 2020 Q5(c)

The physical challenge: different frequencies on stator and rotor

In a transformer, both primary and secondary windings operate at the identical electrical frequency $f$ . In an induction motor, however, the stator operates at supply frequency $f$ , but the rotor conductors rotate at mechanical speed $N$ . The relative speed between the stator rotating magnetic field ( $N_{s}$ ) and the rotor ( $N$ ) is the slip speed $s N_{s}$ .

Consequently, the induced rotor EMF alternates at slip frequency: $f_{r} = s f$

Because electrical circuits operating at different frequencies cannot be directly joined into a single equivalent mesh, we must convert the variable-frequency rotating rotor into a mathematically equivalent stationary circuit operating at the supply frequency $f$ .

Step 1: Stator Model and Standstill Rotor (The Transformer Analogy at $s = 1$ )

At standstill (locked rotor, slip $s = 1$ ), the rotor is stationary, and both stator and rotor operate at the supply frequency $f$ .

The stator primary consists of winding resistance $R_{1}$ , leakage reactance $j X_{1}$ , and a parallel magnetizing branch ( $R_{c} ∥ j X_{m}$ ) carrying no-load excitation current $I_{0}$ ( $I_{c}$ for core loss and $I_{m}$ for air-gap magnetization).
The stator induced EMF $E_{1}$ couples magnetically to the rotor standstill EMF $E_{2}$ across the air gap via an ideal transformer of effective turns ratio $a = N_{1} / N_{2}$ .
The rotor winding has resistance $R_{2}$ and standstill leakage reactance $j X_{2}$ , short-circuited on itself.

Step 1: Stator and rotor transformer model at standstill

Step 2: Rotor Circuit at Any Running Slip $s$ (Actual Rotor Frequency $f_{r} = s f$ )

When the rotor rotates at mechanical speed $N$ , the relative speed between the stator rotating magnetic field ( $N_{s}$ ) and the rotor is $(N_{s} - N) = s N_{s}$ . Consequently:

Rotor induced EMF: $E_{r} = s E_{2}$
Rotor frequency: $f_{r} = s f$
Rotor leakage reactance: $X_{r} = 2 π f_{r} L_{2} = 2 π (s f) L_{2} = s X_{2}$
Rotor resistance: $R_{2}$ remains constant (independent of slip).

The rotor current per phase at running slip $s$ is: $I_{2} = \frac{E _{r}}{Z _{r}} = \frac{s E _{2}}{R _{2}^{2} + ( s X _{2} ) ^{2}} = \frac{s E _{2}}{R _{2} + j s X _{2}}$

Step 2: Rotor circuit at operating slip s

Step 3: Frequency Transformation to Stator Line Frequency $f$ (Variable Resistance Model)

Because the stator operates at frequency $f$ while the rotor operates at slip frequency $s f$ , they cannot be coupled directly into a unified conductive circuit.

Dividing both the numerator and denominator of the rotor current expression by slip $s$ : $I_{2} = \frac{\frac{s E _{2}}{s}}{\frac{R _{2} + j s X _{2}}{s}} = \frac{E _{2}}{\frac{R _{2}}{s} + j X _{2}}$

Physical Significance:

The induced EMF is now fixed at $E_{2}$ at constant line frequency $f$ .
The rotor leakage reactance is fixed at its standstill value $j X_{2}$ at frequency $f$ .
The rotor resistance becomes a variable fictitious resistance $\frac{R _{2}}{s}$ , which reflects the effect of mechanical rotation.
Both stator and rotor models now operate at the same supply frequency $f$ .

Step 3: Electrically equivalent rotor circuit at line frequency f

Step 4: Separation of Rotor Power (Copper Loss vs. Mechanical Power Output)

The total electrical power transferred across the air gap into the rotor per phase ( $P_{g}$ , air-gap power) is: $P_{g} = I_{2}^{2} (\frac{R _{2}}{s})$

We split the variable resistance $\frac{R _{2}}{s}$ into two components: $\frac{R _{2}}{s} = R_{2} + R_{2} (\frac{1 - s}{s}) = R_{2} + R_{L}$

$R_{2}$ (Constant): Actual rotor winding resistance, which accounts for the internal rotor copper loss: $P_{c u} = I_{2}^{2} R_{2}$
$R_{L} = R_{2} (\frac{1 - s}{s})$ (Variable): Fictitious electrical load resistance representing the gross mechanical power developed by the rotor ( $P_{m}$ ): $P_{m} = I_{2}^{2} R_{L} = I_{2}^{2} R_{2} (\frac{1 - s}{s})$

Step 4: Rotor circuit separating copper loss and mechanical load

Step 5: Referring Rotor to Stator (Complete Exact Per-Phase Equivalent Circuit)

By referring all rotor quantities across the ideal transformer to the stator side using the effective transformation ratio $a = N_{1} / N_{2}$ (where impedances scale by $a^{2}$ and currents by $1/ a$ ):

$E_{2}^{'} = a E_{2} = E_{1}$
$I_{2}^{'} = \frac{I _{2}}{a}$
$R_{2}^{'} = a^{2} R_{2}$
$X_{2}^{'} = a^{2} X_{2}$
$R_{L}^{'} = a^{2} R_{L} = R_{2}^{'} (\frac{1 - s}{s})$
Total rotor branch resistance: $\frac{R _{2}^{'}}{s} = R_{2}^{'} + R_{L}^{'}$

The ideal transformer is eliminated, resulting in the Complete Exact Per-Phase Equivalent Circuit:

Step 5: Complete exact per-phase equivalent circuit of 3-phase induction motor

Step 6: Approximate Per-Phase Equivalent Circuit (Engineering Model)

Under normal load conditions, the stator impedance drop $I_{1} (R_{1} + j X_{1})$ is negligible ( $\approx 2 - 5%$ ). Shifting the shunt magnetizing branch ( $R_{c} ∥ j X_{m}$ ) directly across the stator supply terminals $V_{1}$ simplifies calculation without substantial loss of accuracy:

Total equivalent series resistance: $R_{01} = R_{1} + R_{2}^{'}$
Total equivalent series leakage reactance: $X_{01} = X_{1} + X_{2}^{'}$
Load resistance: $R_{L}^{'} = R_{2}^{'} (\frac{1 - s}{s})$

Step 6: Approximate per-phase equivalent circuit

This complete per-phase model enables exact calculation of stator current, power factor, electromagnetic torque, developed power, and efficiency across the entire speed range.

← T-14: IM as Rotating Transformer | 🏠 Index | T-16: Torque Equations & Curves →

topicwise_answers_explanation/T-15_IM_Equivalent_Circuit.md

6.4 KB · commit d2f3b6e

← T-14: IM as Rotating Transformer | 🏠 Index | T-16: Torque Equations & Curves →

T-15: IM Equivalent Circuit

ECE 2207 — Explanation Answers (Sorted by Topic)

Topic Overview: This document compiles all semester final questions and explanation answers on IM Equivalent Circuit from 7 years of exams (2017–2024). Repeated questions appear once with all exam appearances noted.

T-15: Step-by-Step Derivation of the Induction Motor Equivalent Circuit

Appears in: 2017 Q3(a), 2020 Q5(c)

The physical challenge: different frequencies on stator and rotor

Consequently, the induced rotor EMF alternates at slip frequency: $f_{r} = s f$

Step 1: Stator Model and Standstill Rotor (The Transformer Analogy at $s = 1$ )

At standstill (locked rotor, slip $s = 1$ ), the rotor is stationary, and both stator and rotor operate at the supply frequency $f$ .

The stator primary consists of winding resistance $R_{1}$ , leakage reactance $j X_{1}$ , and a parallel magnetizing branch ( $R_{c} ∥ j X_{m}$ ) carrying no-load excitation current $I_{0}$ ( $I_{c}$ for core loss and $I_{m}$ for air-gap magnetization).
The stator induced EMF $E_{1}$ couples magnetically to the rotor standstill EMF $E_{2}$ across the air gap via an ideal transformer of effective turns ratio $a = N_{1} / N_{2}$ .
The rotor winding has resistance $R_{2}$ and standstill leakage reactance $j X_{2}$ , short-circuited on itself.

Step 1: Stator and rotor transformer model at standstill

Step 2: Rotor Circuit at Any Running Slip $s$ (Actual Rotor Frequency $f_{r} = s f$ )

When the rotor rotates at mechanical speed $N$ , the relative speed between the stator rotating magnetic field ( $N_{s}$ ) and the rotor is $(N_{s} - N) = s N_{s}$ . Consequently:

Rotor induced EMF: $E_{r} = s E_{2}$
Rotor frequency: $f_{r} = s f$
Rotor leakage reactance: $X_{r} = 2 π f_{r} L_{2} = 2 π (s f) L_{2} = s X_{2}$
Rotor resistance: $R_{2}$ remains constant (independent of slip).

The rotor current per phase at running slip $s$ is: $I_{2} = \frac{E _{r}}{Z _{r}} = \frac{s E _{2}}{R _{2}^{2} + ( s X _{2} ) ^{2}} = \frac{s E _{2}}{R _{2} + j s X _{2}}$

Step 2: Rotor circuit at operating slip s

Step 3: Frequency Transformation to Stator Line Frequency $f$ (Variable Resistance Model)

Because the stator operates at frequency $f$ while the rotor operates at slip frequency $s f$ , they cannot be coupled directly into a unified conductive circuit.

Physical Significance:

The induced EMF is now fixed at $E_{2}$ at constant line frequency $f$ .
The rotor leakage reactance is fixed at its standstill value $j X_{2}$ at frequency $f$ .
The rotor resistance becomes a variable fictitious resistance $\frac{R _{2}}{s}$ , which reflects the effect of mechanical rotation.
Both stator and rotor models now operate at the same supply frequency $f$ .

Step 3: Electrically equivalent rotor circuit at line frequency f

Step 4: Separation of Rotor Power (Copper Loss vs. Mechanical Power Output)

The total electrical power transferred across the air gap into the rotor per phase ( $P_{g}$ , air-gap power) is: $P_{g} = I_{2}^{2} (\frac{R _{2}}{s})$

We split the variable resistance $\frac{R _{2}}{s}$ into two components: $\frac{R _{2}}{s} = R_{2} + R_{2} (\frac{1 - s}{s}) = R_{2} + R_{L}$

$R_{2}$ (Constant): Actual rotor winding resistance, which accounts for the internal rotor copper loss: $P_{c u} = I_{2}^{2} R_{2}$
$R_{L} = R_{2} (\frac{1 - s}{s})$ (Variable): Fictitious electrical load resistance representing the gross mechanical power developed by the rotor ( $P_{m}$ ): $P_{m} = I_{2}^{2} R_{L} = I_{2}^{2} R_{2} (\frac{1 - s}{s})$

Step 4: Rotor circuit separating copper loss and mechanical load

Step 5: Referring Rotor to Stator (Complete Exact Per-Phase Equivalent Circuit)

$E_{2}^{'} = a E_{2} = E_{1}$
$I_{2}^{'} = \frac{I _{2}}{a}$
$R_{2}^{'} = a^{2} R_{2}$
$X_{2}^{'} = a^{2} X_{2}$
$R_{L}^{'} = a^{2} R_{L} = R_{2}^{'} (\frac{1 - s}{s})$
Total rotor branch resistance: $\frac{R _{2}^{'}}{s} = R_{2}^{'} + R_{L}^{'}$

The ideal transformer is eliminated, resulting in the Complete Exact Per-Phase Equivalent Circuit:

Step 5: Complete exact per-phase equivalent circuit of 3-phase induction motor

Step 6: Approximate Per-Phase Equivalent Circuit (Engineering Model)

Total equivalent series resistance: $R_{01} = R_{1} + R_{2}^{'}$
Total equivalent series leakage reactance: $X_{01} = X_{1} + X_{2}^{'}$
Load resistance: $R_{L}^{'} = R_{2}^{'} (\frac{1 - s}{s})$

Step 6: Approximate per-phase equivalent circuit

This complete per-phase model enables exact calculation of stator current, power factor, electromagnetic torque, developed power, and efficiency across the entire speed range.

← T-14: IM as Rotating Transformer | 🏠 Index | T-16: Torque Equations & Curves →

azazarfin60/Electrical_Machine_I_Semester_Final

topicwise_answers_explanation/T-15_IM_Equivalent_Circuit.md

T-15: IM Equivalent Circuit

T-15: Step-by-Step Derivation of the Induction Motor Equivalent Circuit

The physical challenge: different frequencies on stator and rotor

Step 1: Stator Model and Standstill Rotor (The Transformer Analogy at $s = 1$ )

Step 2: Rotor Circuit at Any Running Slip $s$ (Actual Rotor Frequency $f_{r} = s f$ )

Step 3: Frequency Transformation to Stator Line Frequency $f$ (Variable Resistance Model)

Step 4: Separation of Rotor Power (Copper Loss vs. Mechanical Power Output)

Step 5: Referring Rotor to Stator (Complete Exact Per-Phase Equivalent Circuit)

Step 6: Approximate Per-Phase Equivalent Circuit (Engineering Model)

Report a problem

azazarfin60/Electrical_Machine_I_Semester_Final

topicwise_answers_explanation/T-15_IM_Equivalent_Circuit.md

T-15: IM Equivalent Circuit

T-15: Step-by-Step Derivation of the Induction Motor Equivalent Circuit

The physical challenge: different frequencies on stator and rotor

Step 1: Stator Model and Standstill Rotor (The Transformer Analogy at $s = 1$ )

Step 2: Rotor Circuit at Any Running Slip $s$ (Actual Rotor Frequency $f_{r} = s f$ )

Step 3: Frequency Transformation to Stator Line Frequency $f$ (Variable Resistance Model)

Step 4: Separation of Rotor Power (Copper Loss vs. Mechanical Power Output)

Step 5: Referring Rotor to Stator (Complete Exact Per-Phase Equivalent Circuit)

Step 6: Approximate Per-Phase Equivalent Circuit (Engineering Model)

Report a problem

T-15: IM Equivalent Circuit

T-15: Step-by-Step Derivation of the Induction Motor Equivalent Circuit

The physical challenge: different frequencies on stator and rotor

Step 1: Stator Model and Standstill Rotor (The Transformer Analogy at s=1)

Step 2: Rotor Circuit at Any Running Slip s (Actual Rotor Frequency fr​=sf)

Step 3: Frequency Transformation to Stator Line Frequency f (Variable Resistance Model)

Step 4: Separation of Rotor Power (Copper Loss vs. Mechanical Power Output)

Step 5: Referring Rotor to Stator (Complete Exact Per-Phase Equivalent Circuit)

Step 6: Approximate Per-Phase Equivalent Circuit (Engineering Model)

T-15: IM Equivalent Circuit

T-15: Step-by-Step Derivation of the Induction Motor Equivalent Circuit

The physical challenge: different frequencies on stator and rotor

Step 1: Stator Model and Standstill Rotor (The Transformer Analogy at s=1)

Step 2: Rotor Circuit at Any Running Slip s (Actual Rotor Frequency fr​=sf)

Step 3: Frequency Transformation to Stator Line Frequency f (Variable Resistance Model)

Step 4: Separation of Rotor Power (Copper Loss vs. Mechanical Power Output)

Step 5: Referring Rotor to Stator (Complete Exact Per-Phase Equivalent Circuit)

Step 6: Approximate Per-Phase Equivalent Circuit (Engineering Model)

Step 1: Stator Model and Standstill Rotor (The Transformer Analogy at $s = 1$ )

Step 2: Rotor Circuit at Any Running Slip $s$ (Actual Rotor Frequency $f_{r} = s f$ )

Step 3: Frequency Transformation to Stator Line Frequency $f$ (Variable Resistance Model)

Step 1: Stator Model and Standstill Rotor (The Transformer Analogy at $s = 1$ )

Step 2: Rotor Circuit at Any Running Slip $s$ (Actual Rotor Frequency $f_{r} = s f$ )

Step 3: Frequency Transformation to Stator Line Frequency $f$ (Variable Resistance Model)