Induction machines formulas

Three-phase and single-phase induction machines. 18 formulas for GATE EE, IN Electrical Machines, each with when to use it. Each is explained in the topic notes.

All 18 Induction machines formulas

SYNCHRONOUS SPEED (RPM)

n_{s} = \frac{120 f}{P}

$f$ = supply frequency (Hz), $P$ = number of poles. This is the speed of the rotating stator field. In rad/s: $ω_{s} = \frac{2 π n _{s}}{60} = \frac{4 π f}{P}$ .

AMPLITUDE OF THE RESULTANT ROTATING MMF (THREE-PHASE)

F_{R} = \frac{3}{2} F_{ma x}

$F_{ma x}$ is the peak MMF of one phase winding. Holds for a balanced three-phase winding carrying balanced three-phase currents. The resultant has constant amplitude and rotates at $n_{s}$ .

SLIP

s = \frac{n _{s} - n}{n _{s}}

Dimensionless. $s = 1$ at standstill and $s \to 0$ near synchronous speed. $0 < s < 1$ : motoring. $s < 0$ : the rotor is driven above $n_{s}$ (induction generator). $s > 1$ : the rotor turns against the field (braking, as in plugging).

ROTOR (INDUCED) FREQUENCY

f_{r} = s f

Frequency of the currents induced in the rotor at slip $s$ , where $f$ is the stator supply frequency. At standstill ( $s = 1$ ), $f_{r} = f$ .

SPLIT OF THE ROTOR BRANCH RESISTANCE

\frac{R _{2}^{'}}{s} = R_{2}^{'} + R_{2}^{'} (\frac{1 - s}{s})

The fixed part $R_{2}^{'}$ dissipates rotor copper loss. The slip-dependent part $R_{2}^{'} \frac{1 - s}{s}$ is the equivalent load resistance, and the power it absorbs is the gross mechanical power developed, per phase.

ROTOR CURRENT REFERRED TO THE STATOR (APPROXIMATE CIRCUIT)

I_{2}^{'} = \frac{V _{1}}{( R _{1} + \frac{R _{2}^{'}}{s} ) ^{2} + ( X _{1} + X _{2}^{'} ) ^{2}}

Per-phase rms rotor current, with $V_{1}$ the per-phase voltage. The stator current is the phasor sum of $I_{2}^{'}$ and the no-load (magnetizing) current.

AIR-GAP POWER SPLIT

P_{g} : P_{c u 2} : P_{m} = 1 : s : (1 - s)

$P_{g} = 3 (I_{2}^{'})^{2} R_{2}^{'} / s$ is the total air-gap power, $P_{c u 2} = s P_{g}$ the total rotor copper loss and $P_{m} = (1 - s) P_{g}$ the gross mechanical power. The split gives rotor power quantities without computing currents. Since $P_{m} / P_{g} = 1 - s$ , the motor's overall efficiency cannot exceed $1 - s$ .

DEVELOPED (ELECTROMAGNETIC) TORQUE

T = \frac{P _{g}}{ω _{s}} = \frac{P _{m}}{ω _{r}} = \frac{3 ( I _{2}^{'} ) ^{2} R _{2}^{'}}{s ω _{s}}

$P_{g}$ is the total three-phase air-gap power and $ω_{s} = 2 π n_{s} /60$ the synchronous speed in rad/s. $ω_{r} = (1 - s) ω_{s}$ is the rotor speed in rad/s. Divide air-gap power by synchronous speed, or mechanical power by rotor speed, and never mix the two. Shaft torque is lower by the rotational-loss torque.

TORQUE IN TERMS OF APPLIED VOLTAGE

T = \frac{3 V _{1}^{2} ( R _{2}^{'} / s )}{ω _{s} [ ( R _{1} + \frac{R _{2}^{'}}{s} ) ^{2} + ( X _{1} + X _{2}^{'} ) ^{2} ]}

Approximate equivalent circuit. $V_{1}$ is the per-phase stator voltage. For the exact circuit use the Thevenin values $V_{t h}, R_{t h}, X_{t h}$ in place of $V_{1}, R_{1}, X_{1}$ .

SLIP AT MAXIMUM TORQUE

s_{m T} = \frac{R _{2}^{'}}{R _{1}^{2} + ( X _{1} + X _{2}^{'} ) ^{2}}

Approximate equivalent circuit. Independent of the applied voltage and proportional to $R_{2}^{'}$ . Adding rotor resistance moves the maximum-torque point towards standstill.

MAXIMUM (PULL-OUT OR BREAKDOWN) TORQUE

T_{ma x} = \frac{3 V _{1}^{2}}{2 ω _{s} [ R _{1} + R _{1}^{2} + ( X _{1} + X _{2}^{'} ) ^{2} ]}

Approximate equivalent circuit. Independent of $R_{2}^{'}$ : changing the rotor resistance moves where the maximum torque occurs, not its value. Proportional to $V_{1}^{2}$ .

STARTING TORQUE ($S=1$)

T_{s t} = \frac{3 V _{1}^{2} R _{2}^{'}}{ω _{s} [ ( R _{1} + R _{2}^{'} ) ^{2} + ( X _{1} + X _{2}^{'} ) ^{2} ]}

The general torque expression with $s = 1$ .

TORQUE RELATIVE TO MAXIMUM TORQUE

\frac{T}{T _{ma x}} = \frac{2}{\frac{s}{s _{m T}} + \frac{s _{m T}}{s}}, \frac{T _{s t}}{T _{ma x}} = \frac{2 s _{m T}}{1 + s _{m T}^{2}}

Valid when the stator resistance is neglected ( $R_{1} \approx 0$ ), so that $s_{m T} = R_{2}^{'} / (X_{1} + X_{2}^{'})$ . The same supply voltage and frequency apply to $T$ and $T_{ma x}$ .

QUANTITIES FROM THE NO-LOAD TEST

P_{N L} - 3 I_{N L}^{2} R_{1} = P_{cor e} + P_{r o t}, \frac{V _{N L}}{3 I _{N L}} \approx X_{1} + X_{m}

$V_{N L}$ and $I_{N L}$ are line values and $P_{N L}$ is the total three-phase input power. Per-phase values are taken for an equivalent star connection. The second relation uses the low power factor (impedance almost entirely reactive).

PARAMETERS FROM THE BLOCKED-ROTOR TEST

R_{1} + R_{2}^{'} = \frac{P _{B R}}{3 I _{B R}^{2}}, Z_{B R} = \frac{V _{B R}}{3 I _{B R}}, X_{1} + X_{2}^{'} = Z_{B R}^{2} - (R_{1} + R_{2}^{'})^{2}

$V_{B R}$ and $I_{B R}$ are line values and $P_{B R}$ is the total three-phase input power. Results are per phase for an equivalent star connection. If the test is run at a reduced frequency $f_{B R}$ , multiply the reactance by $f / f_{B R}$ . The test cannot separate $X_{1}$ from $X_{2}^{'}$ , so a split (often equal) must be assumed or given.

STARTING TORQUE RELATIVE TO FULL-LOAD TORQUE

\frac{T _{s t}}{T _{f l}} = x^{2} (\frac{I _{sc}}{I _{f l}})^{2} s_{f l}

From $T \propto (I_{2}^{'})^{2} R_{2}^{'} / s$ with $R_{2}^{'}$ constant and the magnetizing current neglected. $I_{sc}$ is the DOL (full-voltage) starting current, $I_{f l}$ the full-load current, $s_{f l}$ the full-load slip. $x = 1$ for DOL, $x = 1/ 3$ for star-delta, and $x$ = tap ratio for an autotransformer.

ROTOR RESISTANCE FOR MAXIMUM STARTING TORQUE

R_{2}^{'} + R_{e x t}^{'} = R_{1}^{2} + (X_{1} + X_{2}^{'})^{2} \Rightarrow T_{s t} = T_{ma x}

Wound-rotor motor, approximate equivalent circuit. $R_{e x t}^{'}$ is the added rotor resistance referred to the stator. This choice makes $s_{m T} = 1$ .

SLIPS RELATIVE TO THE FORWARD AND BACKWARD FIELDS

s_{f} = s = \frac{n _{s} - n}{n _{s}}, s_{b} = \frac{n _{s} + n}{n _{s}} = 2 - s, T = T_{f} - T_{b}

$n$ is the rotor speed in the forward direction. At standstill $s_{f} = s_{b} = 1$ and $T_{f} = T_{b}$ , so the net torque $T = 0$ .

Where this was asked: 2018 (1), 2020 (1), 2022 (5), 2023 (3), 2024 (2), 2025 (3), 2026 (2). See the 17 previous-year questions on Induction machines. This shows what was asked, not what will be asked.

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Induction machines: the notes and 17 solved previous-year questions