GATE EC Networks Formula Sheet

Organised by the official GATE 2027 EC syllabus, Section 2: Networks, Signals and Systems (the network topics). Each worked example was checked in code. Practise these: GATE EC Networks, Signals and Systems previous year questions.

Circuit analysis

Syllabus: Node and mesh analysis, superposition, Thevenin's theorem, Norton's theorem, reciprocity.

Voltage divider

V_2 = V_s\,\frac{R_2}{R_1 + R_2}

Symbols

$V_s$ : voltage across the series pair (V)
$R_1, R_2$ : series resistors (Ω)
$V_2$ : voltage across $R_{2}$ (V)

Applies when

The same current flows through both resistors: nothing else draws current from the middle node.

Common mistake

Using it after a load is connected to the middle node. The load is in parallel with

R_{2}

, so use

R_{2}

in parallel with the load instead.

Worked example

V_{s}

= 12 V,

R_{1}

= 2 kΩ,

R_{2}

= 1 kΩ: 4 V

Current divider

I_1 = I\,\frac{R_2}{R_1 + R_2}

Symbols

$I$ : total current into the parallel pair (A)
$R_1, R_2$ : parallel resistors (Ω)
$I_1$ : current through $R_{1}$ (A)

Applies when

Two branches in parallel. With more branches, use conductances:

I_{k}

= I

G_{k}

/ ΣG.

Common mistake

Putting

R_{1}

on top. The current through

R_{1}

takes the other resistor on top: the larger share goes through the smaller resistor.

Worked example

I = 6 mA,

R_{1}

= 1 kΩ,

R_{2}

= 2 kΩ: 4 mA

Node analysis (KCL at a node)

\sum_k \frac{V_n - V_k}{R_k} = I_{\text{in}}

Symbols

$V_n$ : the unknown node voltage (V)
$V_k$ : voltages of the nodes joined to it through $R_{k}$ (V)
$I_{\text{in}}$ : source current into the node (A)

Applies when

Write one equation per unknown node; a voltage source between two unknown nodes needs a supernode.

Common mistake

Mixing sign conventions within one equation. Pick 'current leaving is positive' and keep it for every branch.

Worked example

A node joined to 10 V through 2 Ω and to ground through 3 Ω, with 1 A injected: 7.2 V

Thevenin and Norton equivalents

V_{th} = V_{oc}, \qquad R_{th} = \frac{V_{oc}}{I_{sc}}, \qquad I_N = I_{sc} = \frac{V_{th}}{R_{th}}

\begin{aligned} V_{th} &= V_{oc}, \\ R_{th} &= \frac{V_{oc}}{I_{sc}}, \\ I_N &= I_{sc} \\ &= \frac{V_{th}}{R_{th}} \end{aligned}

Symbols

$V_{oc}$ : open-circuit voltage at the terminals (V)
$I_{sc}$ : short-circuit current at the terminals (A)
$R_{th}$ : Thevenin resistance (Ω)

Applies when

Any linear network. With only dependent sources inside,

V_{oc}

= 0, so find

R_{th}

by applying a test source.

Common mistake

Turning off dependent sources when finding

R_{th}

. Only independent sources are turned off; dependent ones stay.

Worked example

10 V behind 2 kΩ, with 3 kΩ across the terminals: 1.2 kΩ (

V_{oc}

= 6 V,

I_{sc}

= 5 mA)

Superposition

x = \sum_k x_k \quad (x_k\text{: response to source }k\text{ alone})

\begin{gathered} x = \sum_k x_k \\ x_k\text{: response to source }k \\ \text{acting alone} \end{gathered}

Symbols

$x$ : any voltage or current in the circuit
$x_k$ : its value with only independent source k on

Applies when

Linear circuits. It adds voltages and currents, not powers.

Common mistake

Adding the powers from each source. Add the currents first, then find the power.

Worked example

10 V through 4 Ω into a node with 4 Ω to ground and 2 A injected; voltage of the node: 9 V

Reciprocity

z_{12} = z_{21}, \qquad y_{12} = y_{21}

Symbols

$z_{12}, z_{21}$ : transfer impedances of a two-port (Ω)
$y_{12}, y_{21}$ : transfer admittances (S)

Applies when

Networks of R, L, C and transformers only: no dependent sources.

Common mistake

Assuming it holds with a dependent source inside. A controlled source usually makes

z_{12}

≠

z_{21}

Worked example

A T network of 2 Ω, 4 Ω (shunt) and 2 Ω:

z_{12}

with port 1 open: 4 Ω (

z_{21}

= 4 Ω too)

Sinusoidal steady state

Syllabus: Phasors, complex power, maximum power transfer.

Impedance of R, L and C

Z_R = R, \qquad Z_L = j\omega L, \qquad Z_C = \frac{1}{j\omega C}

\begin{aligned} Z_R &= R, \\ Z_L &= j\omega L, \\ Z_C &= \frac{1}{j\omega C} \end{aligned}

Symbols

$\omega$ : angular frequency, 2πf (rad/s)
$L$ : inductance (H)
$C$ : capacitance (F)

Applies when

Sinusoidal steady state at a single frequency.

Common mistake

Using f instead of ω. At 50 Hz, ω is 314.16 rad/s, not 50.

Worked example

L = 10 mH at f = 1 kHz: the reactance

X_{L}

= ωL: 62.83 Ω

Complex power

S = V_{rms} I_{rms}^{*} = P + jQ, \qquad P = V_{rms} I_{rms}\cos\theta

\begin{aligned} S &= V_{rms} I_{rms}^{*} \\ &= P + jQ, \\ P &= V_{rms} I_{rms}\cos\theta \end{aligned}

Symbols

$S$ : complex power (VA)
$P$ : real power (W)
$Q$ : reactive power (var)
$\theta$ : angle of V minus angle of I

Applies when

RMS phasors. With peak phasors,

S = VI^{*}/2

Common mistake

Forgetting the conjugate, which flips the sign of Q: an inductive load must give Q > 0.

Worked example

V = 230∠0° V rms, I = 10∠−30° A rms: real power: 1991.86 W (Q = 1150 var)

Maximum power transfer

Z_L = Z_{th}^{*}, \qquad P_{max} = \frac{V_{th,rms}^{2}}{4R_{th}}

\begin{aligned} Z_L &= Z_{th}^{*}, \\ P_{max} &= \frac{V_{th,rms}^{2}}{4R_{th}} \end{aligned}

Symbols

$Z_{th} = R_{th} + jX_{th}$ : Thevenin impedance (Ω)
$V_{th,rms}$ : RMS Thevenin voltage (V)

Applies when

Load free to take any impedance. If the load must be a pure resistance, the best is

R_{L}

= |

Z_{th}

Common mistake

Using a peak voltage in

V^2/(4R_{th})

. With a peak value the formula is

V_{peak}^2/(8R_{th})

Worked example

V_{th}

= 10 V rms,

Z_{th}

= 50 Ω: 0.5 W

Time and frequency domain analysis

Syllabus: RL, RC and RLC circuits, solution of network equations using the Laplace transform. Linear 2-port network parameters, wye-delta transformation.

First-order step response

x(t) = x(\infty) + \big[x(0^+) - x(\infty)\big]\,e^{-t/\tau}, \qquad \tau = RC \text{ or } \frac{L}{R}

\begin{aligned} x(t) &= x(\infty) \\ &\quad + \big[x(0^+) - x(\infty)\big]\,e^{-t/\tau}, \\ \tau &= RC \text{ or } \frac{L}{R} \end{aligned}

Symbols

$x$ : any capacitor voltage, inductor current, or other variable
$\tau$ : time constant (s)
$R$ : resistance seen by C or L with sources off (Ω)

Applies when

One energy-storage element (or ones that combine into one) and DC sources for t > 0.

Common mistake

Taking R as the resistor next to the capacitor. It is the Thevenin resistance across the capacitor's terminals.

Worked example

A capacitor charging from 0 to 5 V with τ = 1 ms: value at t = 1 ms: 3.161 V

Series RLC

\omega_0 = \frac{1}{\sqrt{LC}}, \qquad \zeta = \frac{R}{2}\sqrt{\frac{C}{L}}, \qquad Q = \frac{1}{R}\sqrt{\frac{L}{C}}

\begin{aligned} \omega_0 &= \frac{1}{\sqrt{LC}}, \\ \zeta &= \frac{R}{2}\sqrt{\frac{C}{L}}, \\ Q &= \frac{1}{R}\sqrt{\frac{L}{C}} \end{aligned}

Symbols

$\omega_0$ : undamped natural (resonant) frequency (rad/s)
$\zeta$ : damping ratio
$Q$ : quality factor, 1/(2ζ)

Applies when

R, L and C in series. ζ < 1 is underdamped, ζ = 1 critical, ζ > 1 overdamped.

Common mistake

Using the parallel formula. In series, more R lowers Q; in parallel, more R raises it.

Worked example

R = 10 Ω, L = 1 mH, C = 1 µF: quality factor: 3.162 (

ω_{0}

= 31 623 rad/s, ζ = 0.158)

Parallel RLC

\omega_0 = \frac{1}{\sqrt{LC}}, \qquad Q = R\sqrt{\frac{C}{L}}

Symbols

$R$ : resistance in parallel with L and C (Ω)

Applies when

R, L and C all in parallel.

Common mistake

Expecting the series result. Here Q grows with R.

Worked example

R = 1 kΩ, L = 1 mH, C = 1 µF: quality factor: 31.62

Initial and final value theorems

f(0^+) = \lim_{s\to\infty} sF(s), \qquad f(\infty) = \lim_{s\to 0} sF(s)

\begin{aligned} f(0^+) &= \lim_{s\to\infty} sF(s), \\ f(\infty) &= \lim_{s\to 0} sF(s) \end{aligned}

Symbols

$F(s)$ : Laplace transform of f(t)

Applies when

Final value: every pole of sF(s) in the left half-plane. Initial value: F(s) strictly proper.

Common mistake

Applying the final value theorem to an unstable or oscillating response, where f(∞) doesn't exist.

Worked example

F(s) = 5 / (s(s + 2)): final value: 2.5

Laplace models of L and C with initial conditions

V_L(s) = sL\,I(s) - L\,i(0^-), \qquad V_C(s) = \frac{I(s)}{sC} + \frac{v(0^-)}{s}

\begin{aligned} V_L(s) &= sL\,I(s) \\ &\quad - L\,i(0^-), \\ V_C(s) &= \frac{I(s)}{sC} + \frac{v(0^-)}{s} \end{aligned}

Symbols

$i(0^-)$ : inductor current just before t = 0 (A)
$v(0^-)$ : capacitor voltage just before t = 0 (V)

Applies when

Use the values at 0⁻, which inductor current and capacitor voltage keep across a switching instant.

Common mistake

Leaving out the initial-condition sources, which gives the zero-state response only.

Worked example

1 H inductor carrying 2 A at 0⁻, discharging into 4 Ω: i(t = 0.5 s): 0.271 A

Z-parameters and Y-parameters

\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix}\begin{bmatrix} I_1 \\ I_2 \end{bmatrix}, \qquad [y] = [z]^{-1}

\begin{aligned} &\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} \\ &\quad = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix}\begin{bmatrix} I_1 \\ I_2 \end{bmatrix}, \\ [y] &= [z]^{-1} \end{aligned}

Symbols

$z_{11}$ : $V_{1}$ / $I_{1}$ with port 2 open (Ω)
$z_{21}$ : $V_{2}$ / $I_{1}$ with port 2 open (Ω)
$y_{11}$ : $I_{1}$ / $V_{1}$ with port 2 shorted (S)

Applies when

Each parameter is measured with the other port open (z) or shorted (y).

Common mistake

Measuring z with the other port shorted, or y with it open.

Worked example

T network, series 2 Ω, shunt 4 Ω, series 2 Ω:

z_{11}

: 6 Ω (

z_{12}

= 4 Ω, so

y_{11}

= 6/20 = 0.3 S)

Wye-delta transformation

R_a = \frac{R_{ab}R_{ca}}{R_{ab} + R_{bc} + R_{ca}}, \qquad R_{ab} = \frac{R_aR_b + R_bR_c + R_cR_a}{R_c}

\begin{aligned} R_a &= \frac{R_{ab}R_{ca}}{R_{ab} + R_{bc} + R_{ca}}, \\ R_{ab} &= \frac{R_aR_b + R_bR_c + R_cR_a}{R_c} \end{aligned}

Symbols

$R_a, R_b, R_c$ : wye resistors at nodes a, b, c (Ω)
$R_{ab}, R_{bc}, R_{ca}$ : delta resistors between the node pairs (Ω)

Applies when

Three-terminal networks; the same formulas work with impedances.

Common mistake

Pairing the wrong resistors. A wye resistor at node a uses the two delta resistors that touch a.

Worked example

Three equal 30 Ω delta resistors: each wye resistor: 10 Ω

Practise these: GATE EC Networks, Signals and Systems previous year questions

Source for the section headings: the official GATE 2027 EC syllabus, IIT Madras. The formulas, examples and notes are Studyloaf's own.

GATE EC Networks Formula Sheet

Circuit analysis

Voltage divider

Current divider

Node analysis (KCL at a node)

Thevenin and Norton equivalents

Superposition

Reciprocity

Sinusoidal steady state

Impedance of R, L and C

Complex power

Maximum power transfer

Time and frequency domain analysis

First-order step response

Series RLC

Parallel RLC

Initial and final value theorems

Laplace models of L and C with initial conditions

Z-parameters and Y-parameters

Wye-delta transformation

Other GATE EC formula sheets