Coupled circuits formulas

Magnetically coupled circuits. 15 formulas for GATE EC, EE, IN Networks (Electric Circuits), each with when to use it. Each is explained in the topic notes.

All 15 Coupled circuits formulas

TERMINAL EQUATIONS OF TWO COUPLED COILS (TIME DOMAIN)

v_{1} = L_{1} \frac{d i _{1}}{d t} \pm M \frac{d i _{2}}{d t}, v_{2} = L_{2} \frac{d i _{2}}{d t} \pm M \frac{d i _{1}}{d t}

Each coil's voltage follows the passive sign convention for that coil (+ at the terminal where its own current enters). Use $+ M$ when both currents enter (or both leave) their dotted terminals and $- M$ when one enters and the other leaves.

PHASOR (SINUSOIDAL STEADY-STATE) FORM

V_{1} = j ω L_{1} I_{1} \pm j ω M I_{2}, V_{2} = j ω L_{2} I_{2} \pm j ω M I_{1}

Same sign rule as the time-domain equations. $ω M$ is the mutual reactance in ohms.

COUPLING COEFFICIENT

k = \frac{M}{L _{1} L _{2}}, 0 \leq k \leq 1 \Rightarrow M \leq L_{1} L_{2}

$k = 0$ : no coupling; $k = 1$ : perfect coupling (all the flux of each coil links the other). Coils wound on a common iron core are typically tightly coupled ( $k$ close to 1); air-cored coils are usually more loosely coupled.

SERIES CONNECTION

L_{eq} = L_{1} + L_{2} \pm 2 M

$+ 2 M$ for series-aiding (the junction joins an undotted terminal to a dotted one); $- 2 M$ for series-opposing (dotted to dotted, or undotted to undotted). Both values are non-negative because $k \leq 1$ .

THREE COUPLED COILS IN SERIES

L_{eq} = L_{1} + L_{2} + L_{3} \pm 2 M_{12} \pm 2 M_{23} \pm 2 M_{13}

The same current flows through all three coils. Decide each pair separately with that pair's own marks: $+ 2 M_{ij}$ if the current enters both of the pair's marked terminals (or leaves both), $- 2 M_{ij}$ otherwise.

MUTUAL INDUCTANCE FROM TWO SERIES MEASUREMENTS

M = \frac{L _{aid} - L _{opp}}{4}, L_{1} + L_{2} = \frac{L _{aid} + L _{opp}}{2}

Measure the series pair, reverse one coil and measure again. The larger reading is always the aiding one.

PARALLEL CONNECTION

L_{eq} = \frac{L _{1} L _{2} - M ^{2}}{L _{1} + L _{2} \mp 2 M}

$- 2 M$ in the denominator when like terminals are joined (both dots on the same node: parallel-aiding); $+ 2 M$ when unlike terminals are joined (parallel-opposing). Coil resistances neglected. With $k = 1$ the numerator is zero, so $L_{eq} = 0$ , except for two equal coils ( $L_{1} = L_{2}$ ) in parallel-aiding, where the formula reads $0/0$ and $L_{eq} = L_{1}$ .

ENERGY STORED IN TWO COUPLED COILS

w = \frac{1}{2} L_{1} i_{1}^{2} + \frac{1}{2} L_{2} i_{2}^{2} \pm M i_{1} i_{2}

Instantaneous currents. $+ M$ when both currents enter (or both leave) their dotted terminals, $- M$ otherwise: the same rule as for the voltages.

T-EQUIVALENT OF COUPLED COILS

dots on the same side: dots on opposite sides: L_{a} = L_{1} - M, L_{b} = L_{2} - M, L_{c} = + M L_{a} = L_{1} + M, L_{b} = L_{2} + M, L_{c} = - M

$L_{a}$ and $L_{b}$ are the series arms on the coil-1 and coil-2 sides, $L_{c}$ the shunt arm to the common terminal. Valid only when the two coils share a common terminal (or their lower terminals are joined in the circuit), because the T adds that connection. "Same side" means both dots away from the common terminal or both on it. A negative inductance in the model is acceptable for analysis, although it is not a physical element.

INPUT IMPEDANCE OF A LINEAR TRANSFORMER (REFLECTED IMPEDANCE)

Z_{in} = Z_{11} + Z_{r}, Z_{r} = \frac{ω ^{2} M ^{2}}{Z _{22}}

$Z_{22}$ is the whole secondary loop impedance, load included. $Z_{r}$ does not depend on the dot positions ( $M$ enters squared), and its reactance has the opposite sign to that of $Z_{22}$ : an inductive secondary reflects as a capacitive reactance.

INPUT INDUCTANCE WITH THE SECONDARY SHORT-CIRCUITED

L_{sc} = L_{1} - \frac{M ^{2}}{L _{2}} = L_{1} (1 - k^{2})

Lossless coils, from $Z_{r}$ with $Z_{22} = j ω L_{2}$ . With the secondary open, the input is simply $j ω L_{1}$ ; comparing the two measurements gives $k$ .

IDEAL TRANSFORMER: VOLTAGE AND CURRENT RATIOS

\frac{V _{1}}{V _{2}} = \frac{N _{1}}{N _{2}} = n, \frac{I _{1}}{I _{2}} = \frac{N _{2}}{N _{1}} = \frac{1}{n}

With the convention above. If one dot is moved to the other end of its winding while the voltage and current references stay on the same physical terminals, both ratios change sign: $V_{1} / V_{2} = - n$ and $I_{1} / I_{2} = - 1/ n$ . In the two-port convention ( $I_{2}$ entering the secondary dot) the current relation reads $I_{1} = - I_{2} / n$ .

IMPEDANCE REFLECTION THROUGH AN IDEAL TRANSFORMER

Z_{in} = \frac{V _{1}}{I _{1}} = n^{2} Z_{L}

$Z_{L}$ connected across the secondary; independent of the dot positions. In the other direction, an impedance on the primary side appears on the secondary side as $Z / n^{2}$ .

POWER CONSERVATION IN AN IDEAL TRANSFORMER

V_{1} I_{1}^{*} = V_{2} I_{2}^{*}

It neither stores nor dissipates energy: complex power in equals complex power out (convention above).

MAXIMUM POWER TRANSFER THROUGH AN IDEAL TRANSFORMER

n^{2} R_{L} = R_{s} \Rightarrow n = \frac{R _{s}}{R _{L}}

Resistive source $R_{s}$ on the primary, resistive load $R_{L}$ on the secondary, $n = N_{1} / N_{2}$ . The load then receives $∣ V_{s} ∣^{2} / (4 R_{s})$ , with $V_{s}$ the rms source voltage.

Where this was asked: 2023 (1), 2024 (2), 2025 (3). See the 6 previous-year questions on Coupled circuits. This shows what was asked, not what will be asked.

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Coupled circuits: the notes and 6 solved previous-year questions