GATE Network Theory formula sheet

Networks (Electric Circuits), also known as Network Theory. 166 formulas in 7 topics, each with when to use it. Each is explained in the topic notes.

RESISTOR (OHM'S LAW)

v = R i, p = v i = i^{2} R = \frac{v ^{2}}{R} \geq 0

Linear resistor, passive sign convention. Conductance $G = 1/ R$ (siemens), so $i = G v$ .

INDUCTOR

v i (t) w_{L} = L \frac{d i}{d t} = i (t_{0}) + \frac{1}{L} \int_{t_{0}}^{t} v d τ = \frac{1}{2} L i^{2}

Passive sign convention. The stored energy depends only on the present current. In DC steady state $d i / d t = 0$ , so an inductor behaves as a short circuit.

Network theorems

All 17 formulas

POWER AND RMS VALUE WITH COMPONENTS AT DIFFERENT FREQUENCIES

P = I_{0}^{2} R + k \sum I_{k}^{2} R, I_{rms} = I_{0}^{2} + k \sum I_{k}^{2}

$I_{0}$ = DC component; $I_{k}$ = rms value of the component at frequency $ω_{k}$ , with all the $ω_{k}$ non-zero and distinct. Here average powers do add, because the cross terms average to zero over time. Two components at the same frequency do not: combine them as phasors first. Example: $i = 3 + 42 sin ω t$ A gives $I_{rms} = 9 + 16 = 5$ A.

THEVENIN–NORTON RELATIONS

V_{t h} = V_{oc}, I_{N} = I_{sc} Z_{t h} = Z_{N} = \frac{V _{oc}}{I _{sc}}, V_{t h} = I_{N} Z_{t h}

$V_{oc} = V_{a} - V_{b}$ with a–b open; $I_{sc}$ flows from a to b through a short across a–b. The ratio cannot be used when both are zero. That is always the case for a network with no independent source, and it can also happen when the sources inside produce nothing at a–b (a balanced bridge, for example). Then use the test-source method.

Transients in RL, RC and RLC circuits

All 31 formulas

CONTINUITY OF THE STATE VARIABLES

v_{C} (0^{+}) = v_{C} (0^{-}), i_{L} (0^{+}) = i_{L} (0^{-})

Holds whenever $i_{C}$ and $v_{L}$ contain no impulse. It can fail in two ways: a source that itself contains an impulse (such as $δ (t)$ ), or a switching that creates a loop of capacitors and ideal voltage sources whose voltages do not already satisfy KVL, or a cut-set of inductors and ideal current sources whose currents do not already satisfy KCL (see the tip below). Put in terms of charge and flux linkage: $q_{C} = C v_{C}$ and $λ_{L} = L i_{L}$ are continuous.

FIRST DERIVATIVES AT 0+

\frac{d v _{C}}{d t}_{0^{+}} = \frac{i _{C} ( 0 ^{+} )}{C}, \frac{d i _{L}}{d t}_{0^{+}} = \frac{v _{L} ( 0 ^{+} )}{L}

Find $i_{C} (0^{+})$ and $v_{L} (0^{+})$ from the $t = 0^{+}$ circuit, using KCL and KVL with capacitors replaced by voltage sources and inductors by current sources. A second-order circuit needs this as its second initial condition. For the derivative of a resistor quantity, differentiate its KVL or KCL equation and substitute these values.

Sinusoidal steady state, AC power and resonance

All 30 formulas

AVERAGE VALUE

V_{avg} = \frac{1}{T} \int_{t_{0}}^{t_{0} + T} v (t) d t

Any periodic waveform; integrate over any one full period.

RMS VALUE

V_{rms} = \frac{1}{T} \int_{t_{0}}^{t_{0} + T} v^{2} (t) d t

Any periodic waveform. Square first, then average, then take the root.

Magnetically coupled circuits

All 15 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.

Two-port networks

All 23 formulas

Z (OPEN-CIRCUIT IMPEDANCE) PARAMETERS

[V_{1} V_{2}] = [z_{11} z_{21} z_{12} z_{22}] [I_{1} I_{2}], z_{11} z_{21} = \frac{V _{1}}{I _{1}}_{I_{2} = 0}, = \frac{V _{2}}{I _{1}}_{I_{2} = 0}, z_{12} z_{22} = \frac{V _{1}}{I _{2}}_{I_{1} = 0} = \frac{V _{2}}{I _{2}}_{I_{1} = 0}

All four are in ohms, and each is measured with the other port open. $z_{11}$ and $z_{22}$ are driving-point (input and output) impedances. $z_{12}$ and $z_{21}$ are transfer impedances.

Y (SHORT-CIRCUIT ADMITTANCE) PARAMETERS

[I_{1} I_{2}] = [y_{11} y_{21} y_{12} y_{22}] [V_{1} V_{2}], y_{11} y_{21} = \frac{I _{1}}{V _{1}}_{V_{2} = 0}, = \frac{I _{2}}{V _{1}}_{V_{2} = 0}, y_{12} y_{22} = \frac{I _{1}}{V _{2}}_{V_{1} = 0} = \frac{I _{2}}{V _{2}}_{V_{1} = 0}

All four are in siemens, and each is measured with the other port shorted. $[y] = [z]^{- 1}$ whenever both exist. In general $y_{11} \neq = 1/ z_{11}$ , because the two are measured under different terminations.

Balanced three-phase circuits

All 18 formulas

BALANCED POSITIVE-SEQUENCE (ABC) SET

V_{an} = V_{p} ∠ 0^{\circ}, V_{bn} = V_{p} ∠ - 12 0^{\circ}, V_{c n} = V_{p} ∠ + 12 0^{\circ}

Phase b lags phase a by $12 0^{\circ}$ and phase c lags phase b by $12 0^{\circ}$ . For negative sequence (acb) the angles of $V_{bn}$ and $V_{c n}$ are interchanged: $V_{bn} = V_{p} ∠ + 12 0^{\circ}$ , $V_{c n} = V_{p} ∠ - 12 0^{\circ}$ .

OPERATOR A AND THE ZERO SUM OF A BALANCED SET

a = 1∠12 0^{\circ} = - \frac{1}{2} + j \frac{3}{2}, 1 + a + a^{2} = 0

For abc: $V_{bn} = a^{2} V_{an}$ and $V_{c n} = a V_{an}$ . Hence any balanced set sums to zero, $V_{an} + V_{bn} + V_{c n} = 0$ , and the same holds for a balanced set of currents.