GATE ECE Formula Sheet 2027: Key Formulas by Subject

Q: Is the reciprocity condition for h-parameters the same as for z-parameters?

No. A reciprocal two-port has z₁₂ = z₂₁, y₁₂ = y₂₁ and AD - BC = 1, but for h-parameters the condition is h₁₂ = -h₂₁. The minus sign is the trap. A symmetric network adds z₁₁ = z₂₂, y₁₁ = y₂₂ or A = D.

By MD ANISH AHAMADUpdated 4 Oct 20268 min read

GATE ECE Formula Sheet 2027: Key Formulas by Subject

This GATE ECE formula sheet collects the formulas that the five counted GATE EC papers (2010, 2019, 2020, 2025 and 2026) used most, grouped by syllabus section, with the trap beside each one. It is a free selection from the last-minute revision sheet of the GATE ECE 2027 book. Use it to test your recall, then practise each formula on past questions.

In this guide

Key takeaways
How to read this sheet
Networks, Signals and Systems
Engineering Mathematics
Analog Circuits
Communications
Digital Circuits
Electronic Devices
Control Systems
Electromagnetics
General Aptitude
Subject-wise formula sheets
Quick revision

Key takeaways

Most GATE ECE numerical questions reuse a small set of standard formulas with new numbers.
A formula is only safe with its conventions: $\omega$ in rad/s, $f$ in Hz, rms unless "peak" is written.
Many traps are a factor of 2 or $2\pi$ : rad/s against Hz, ring against Johnson counters, BPSK against orthogonal signalling.
Several traps are a sign: $h_{12} = -h_{21}$ , $1 + GH$ against $1 - GH$ , and the minus in an inverting gain.
Sections below follow counted marks, from Networks at a mean of 17.0 marks a paper to Electromagnetics at 7.4.
You cannot carry notes into the hall, so these formulas must be in memory. Confirm exam rules at gate2027.iitm.ac.in.

How to read this sheet

Some terms first. A phasor is the complex amplitude of a sinusoid at one frequency. The region of convergence (ROC) is the set of $z$ for which a z-transform sum converges. A trap is the specific slip that a wrong option is built to catch.

Notation: $\omega = 2\pi f$ ; $V_m$ is a peak value and $V_{\text{rms}} = V_m/\sqrt{2}$ ; $\log$ is base 10, $\ln$ natural, and $\log_2$ wherever the unit is bits. $V_T = kT/q \approx 26\text{ mV}$ at 300 K. Op-amps and sources are ideal unless a question says otherwise.

Networks, Signals and Systems

The largest section, at a mean of 17.0 marks across the counted papers.

Formula	Watch out for
$Z_L = j\omega L$ , $\displaystyle Z_C = \dfrac{1}{j\omega C}$ ; $P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\theta$ , $\theta = \angle Z$	A 50 Hz source is $\omega = 314$ rad/s; meters read rms, $v(t) = V_m \cos\omega t$ is peak
Reciprocal: $z_{12} = z_{21}$ , $y_{12} = y_{21}$ , $AD - BC = 1$ , $h_{12} = -h_{21}$ ; symmetric: $z_{11} = z_{22}$ , $A = D$	Only the h-parameter condition carries a minus sign
$R_{\text{th}}$ : independent $V$ short, $I$ open; with dependent sources use a test source	Dependent sources are never deactivated; superposition never applies to power
$x(t) = x(\infty) + \big[x(0^+) - x(\infty)\big]e^{-t/\tau}$ , $\tau = RC$ or $L/R$	$v_C$ and $i_L$ jump if switching makes a capacitor loop or inductor cut-set
$\displaystyle \omega_0 = \dfrac{1}{\sqrt{LC}}$ , $\text{BW} = \omega_0/Q$ ; series $Q = \omega_0 L/R$ , parallel $Q = R/(\omega_0 L)$	Parallel $Q$ is the reciprocal form
Right-sided: ROC $\vert z \vert > \vert p \vert_{\max}$ ; causal and stable iff all poles inside $\vert z \vert = 1$	The ROC never contains a pole; stable iff ROC contains the unit circle
$f_s > 2f_{\max}$ ; alias at $\vert f - kf_s \vert$	Nyquist rate is $2f_{\max}$ , Nyquist frequency is $f_s/2$
$\lim_{t\to\infty} f(t) = \lim_{s\to 0} sF(s)$	Valid only if every pole of $sF(s)$ is in the open left half-plane
Lengths $N$ and $M$ convolve to $N + M - 1$	Linear and circular convolution differ
$\displaystyle E = \int \vert x \vert^2 dt = \dfrac{1}{2\pi}\int \vert X(j\omega) \vert^2 d\omega$	The $1/2\pi$ is present in $\omega$ and absent in $f$

Fourier series symmetry: even symmetry removes sine terms ( $b_n = 0$ ), odd symmetry removes cosine terms and $a_0$ , and half-wave symmetry, $x(t) = -x(t \pm T/2)$ , removes the even harmonics. A system is LTI only if every block in a cascade or parallel connection is.

FIR and IIR filter design is newly named in the 2027 syllabus, though 2019 Q39 and Q54 were already filter-design questions. The GATE ECE 2027 syllabus changes guide sets out all four changes.

Trap: Half-wave symmetry removes harmonics; even and odd symmetry remove terms. Swapping those two rules lands on a printed distractor.

Engineering Mathematics

Formula	Watch out for
Polar $dA = r\,dr\,d\theta$ ; spherical $dV = \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta$	The dropped $r$ is this section's most expensive slip
$Ax = b$ consistent iff $\text{rank}(A) = \text{rank}([A \mid b])$ ; unique iff that rank $= n$	$\det A = 0$ means no solution or infinitely many; read which is asked
$\oint_C f\,dz = 2\pi j \sum \text{Res}$ ; simple pole: $\lim_{z \to z_0}(z - z_0)f(z)$	Count only poles strictly inside; clockwise flips the sign
$\sum \lambda_i = \text{trace}$ , $\prod \lambda_i = \det A$	A triangular matrix shows its eigenvalues on the diagonal
Roots $m_1 \ne m_2$ : $C_1 e^{m_1 x} + C_2 e^{m_2 x}$ ; repeated: $(C_1 + C_2 x)e^{mx}$ ; $\alpha \pm j\beta$ : $e^{\alpha x}(C_1 \cos\beta x + C_2 \sin\beta x)$	The repeated root needs the factor $x$
$E[aX + bY] = aE[X] + bE[Y]$ ; $\text{Var}(X + Y) = \text{Var}\,X + \text{Var}\,Y$ if uncorrelated	$E[X^2] \ne (E[X])^2$ ; the gap is the variance
$\displaystyle \rho = \dfrac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}$ ; $\displaystyle b_{yx} = \dfrac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$ , $a = \bar{y} - b_{yx}\bar{x}$	The line of $y$ on $x$ differs from $x$ on $y$ ; $r^2 = b_{yx}\,b_{xy}$

Correlation and regression analysis is new in the 2027 syllabus. 2026 Q38 already asked correlation; regression has no counted precedent.

Analog Circuits

Formula	Watch out for
Half-wave: $V_{\text{avg}} = V_m/\pi$ , $V_{\text{rms}} = V_m/2$ ; full-wave: $2V_m/\pi$ , $V_m/\sqrt{2}$	$V_m$ is the peak after the stated diode drops
Inverting $-R_f/R_1$ ; non-inverting $1 + R_f/R_1$	Virtual short needs negative feedback; not in a Schmitt trigger
$g_m = I_C/V_T$ , $r_\pi = \beta/g_m$ , $A_v = -g_m R_C$	Unbypassed $R_E$ : $\displaystyle A_v = \dfrac{-g_m R_C}{1 + g_m R_E}$ , $R_{\text{in}} = r_\pi + (\beta + 1)R_E$
MOSFET $\displaystyle g_m = \dfrac{2I_D}{V_{ov}} = \sqrt{2\mu C_{ox}(W/L)I_D}$	$g_m \propto \sqrt{I_D}$ for a MOSFET, $\propto I_C$ for a BJT
$\displaystyle R_s = \dfrac{V_{\text{in,min}} - V_z}{I_{z,\min} + I_{L,\max}}$	Check Zener power at the opposite corner
$A_d = g_m R_C$ differential out, $g_m R_C/2$ single-ended; $\text{CMRR} = \vert A_d / A_{cm} \vert$	The factor of 2 on single-ended output
$\displaystyle f_c = \dfrac{1}{2\pi RC}$ Hz, $\displaystyle \omega_c = \dfrac{1}{RC}$ rad/s	The answer unit decides the $2\pi$

Remember: Removing the emitter bypass capacitor lowers the gain and raises the input resistance. Both directions, or the mark is gone.

This page is a selection. The book's last-minute sheet lists every ranked concept section by section, with the counted question numbers behind each formula, a trap on every entry and the MCQ guessing arithmetic. It is part of the GATE ECE 2027 book, with 925 questions with worked solutions and 10 full mock tests.

Communications

Formula	Watch out for
Antipodal (BPSK, QPSK): $P_b = Q\big(\sqrt{2E_b/N_0}\big)$ ; orthogonal: $Q\big(\sqrt{E_b/N_0}\big)$	$N_0/2$ is the two-sided PSD; MAP equals ML only with equal priors
FM $\beta = \Delta f / f_m$ ; Carson $\text{BW} = 2(\Delta f + f_m)$ ; power $A_c^2/2$	$k_f$ in Hz/V or rad/s/V differs by $2\pi$
$\displaystyle \mu = \dfrac{V_{\max} - V_{\min}}{V_{\max} + V_{\min}}$ ; $P_t = P_c(1 + \mu^2/2)$ ; $\displaystyle \eta = \dfrac{\mu^2}{2 + \mu^2}$	Maximum efficiency is $1/3$ at $\mu = 1$
$S_y(f) = \vert H(f) \vert^2 S_x(f)$	$\vert H \vert^2$ , not $H$
$H = -\sum p_i \log_2 p_i$ ; $C = B\log_2(1 + \text{SNR})$ ; BSC $C = 1 - H(p)$	Use base 2 for bits
Hamming: $2^{n-k} \ge n + 1$ , $d_{\min} = 3$ ; corrects $t$ if $d_{\min} \ge 2t + 1$	Two errors decode to the wrong codeword
$R_b = n f_s$ ; $\text{SQNR} = 6.02n + 1.76$ dB	Symbol rate times $\log_2 M$ is bit rate

Digital Circuits

Formula	Watch out for
Mod- $N$ counter: $\lceil \log_2 N \rceil$ flip-flops; $n$ toggle stages divide by $2^n$	Ring counter $n$ states, Johnson $2n$
Maximal LFSR length $2^n - 1$	XOR feedback never enters all zeros
A $2^n$ -to-1 mux realises any function of $n + 1$ variables	A 4-to-1 mux does any three-variable function
Two's complement: $-2^{n-1}$ to $2^{n-1} - 1$	Overflow is two same-sign operands giving the opposite sign
$T_{\min} = t_{cq} + t_{\text{comb,max}} + t_{su}$ ; hold: $t_{cq} + t_{\text{comb,min}} \ge t_{\text{hold}}$	Slowing the clock never fixes hold
DAC $V_{\text{out}} = V_{\text{ref}} \cdot D/2^n$	Full scale is $V_{\text{ref}}(2^n - 1)/2^n$

Electronic Devices

Formula	Watch out for
$I = I_S\big(e^{V/\eta V_T} - 1\big)$	One decade of current costs about 60 mV at $\eta = 1$
$D/\mu = kT/q$ ; $\sigma = q(n\mu_n + p\mu_p)$	cm²/s against m²/s is a factor $10^4$
$\displaystyle V_{bi} = \dfrac{kT}{q}\ln\dfrac{N_A N_D}{n_i^2}$	Natural logarithm
$\displaystyle I_D = \tfrac{1}{2}\mu C_{ox}\dfrac{W}{L}(V_{GS} - V_{th})^2$ , for $V_{DS} \ge V_{GS} - V_{th}$	Check whether the given $k$ already holds the $\displaystyle \tfrac{1}{2}$
Threshold at surface potential $2\phi_F$ , $\phi_F = V_T \ln(N_A/n_i)$	$2\phi_F$ , not $\phi_F$
$\lambda(\mu\text{m}) = 1.24/E_g(\text{eV})$ ; $\alpha = \beta/(1 + \beta)$	Zener has a negative temperature coefficient, avalanche positive

Scaling in MOSFETs is new for 2027. Under constant-field scaling by $\kappa$ , delay falls by $\kappa$ , power by $\kappa^2$ , and power density stays constant.

Control Systems

Formula	Watch out for
$G(s) = C(sI - A)^{-1}B + D$ ; poles are eigenvalues of $A$	A cancellation can hide an unstable state
Closed loop $\displaystyle \dfrac{G}{1 + GH}$ (negative feedback)	Positive feedback gives $1 - GH$
Routh: sign changes in column one $=$ right-half-plane poles	A row of zeros: differentiate the auxiliary polynomial from the row above
Overshoot fraction $M_p = e^{-\pi\zeta/\sqrt{1 - \zeta^2}}$ ; $\displaystyle t_s \approx \dfrac{4}{\zeta\omega_n}$ (2 per cent)	Use $3/(\zeta\omega_n)$ only when 5 per cent is stated
Step $\displaystyle e = \dfrac{1}{1 + K_p}$ ; ramp $1/K_v$ ; parabola $1/K_a$	Unity feedback only; the "1 +" is for the step
Asymptotes $\displaystyle \dfrac{(2k + 1)180^\circ}{n - m}$ , centroid $\displaystyle \dfrac{\sum p - \sum z}{n - m}$	Real-axis locus: odd count to the right
PID: $K_p + K_i/s + K_d s$	The integral term raises the system type by one

Gain margin is read at the phase crossover and phase margin at the gain crossover. Swapping them is a reachable wrong answer.

Electromagnetics

Formula	Watch out for
$\eta_0 = 120\pi \approx 377\ \Omega$ ; $\displaystyle \Gamma = \dfrac{\eta_2 - \eta_1}{\eta_2 + \eta_1}$ , $\tau = 1 + \Gamma$	Power fractions are $\vert \Gamma \vert^2$ and $1 - \vert \Gamma \vert^2$ ; $\tau$ can exceed 1
$\displaystyle P_{\text{avg}} = \dfrac{\vert E_0 \vert^2}{2\eta}$ for peak $E_0$	No $\displaystyle \tfrac{1}{2}$ for rms
$\displaystyle Z_{\text{in}} = Z_0 \dfrac{Z_L + jZ_0 \tan\beta l}{Z_0 + jZ_L \tan\beta l}$ ; quarter-wave $Z_0^2/Z_L$	$\beta$ uses the line wavelength $\lambda_0/\sqrt{\varepsilon_r}$
$\displaystyle \text{VSWR} = \dfrac{1 + \vert \Gamma \vert}{1 - \vert \Gamma \vert}$	One Smith chart turn is $\lambda/2$
$\displaystyle f_c = \dfrac{c}{2}\sqrt{(m/a)^2 + (n/b)^2}$ ; $\text{TE}_{10}$ at $c/2a$	No $\text{TM}_{10}$ mode
Friis $\displaystyle \dfrac{P_r}{P_t} = G_t G_r \left(\dfrac{\lambda}{4\pi R}\right)^2$	Gains as ratios, not dB
$\delta = \sqrt{2/(\omega\mu\sigma)}$	Syllabus-named; no counted question computes it

For how much each section is worth, see the GATE ECE subject-wise weightage, and for the concepts behind these formulas, the GATE ECE important topics.

General Aptitude

Average speed over equal distances is $\displaystyle \dfrac{2v_1 v_2}{v_1 + v_2}$ ; over equal times it is the arithmetic mean.
A rise of $x$ per cent then a fall of $x$ per cent is a net loss of $x^2/100$ per cent.
Two workers at rates $1/a$ and $1/b$ finish together in $\displaystyle \dfrac{ab}{a + b}$ .

The full method for these 15 marks is in how to prepare for GATE General Aptitude.

In one line: A blind four-way MCQ guess has expected value exactly zero, so eliminate one option before you guess.

Subject-wise formula sheets

Each of these goes deeper on one part of the GATE ECE paper, with a worked example and the trap for every formula:

Quick revision

Reciprocity is $h_{12} = -h_{21}$ for h-parameters; every other set has no minus sign.
A stable, causal discrete system has every pole strictly inside the unit circle.
Antipodal is $Q(\sqrt{2E_b/N_0})$ ; orthogonal drops the 2.
FM power is $A_c^2/2$ whatever the modulation index.
Johnson counters have $2n$ states; ring counters have $n$ .
$g_m$ grows with $I_C$ for a BJT and with $\sqrt{I_D}$ for a MOSFET.
Gain margin at the phase crossover, phase margin at the gain crossover.
Write units beside every number, and round only the final answer.

Frequently asked questions

Can I take a formula sheet into the GATE ECE exam?

No. You cannot carry paper, notes or a physical calculator into the exam hall. The exam provides a virtual calculator and rough sheets. A useful habit is to write the few formulas you forget under pressure on the rough sheet in the first five minutes. Confirm the current exam-day rules at gate2027.iitm.ac.in.

Which GATE ECE formulas are asked most often?

In the five GATE EC papers the book counts, 2010, 2019, 2020, 2025 and 2026, phasor analysis, the z-transform region of convergence, the two-port reciprocity conditions, detection error probability and angle modulation appeared in all five. Learn those formulas together with their traps first.

What is the difference between BPSK and orthogonal signalling error probability?

Antipodal signalling such as BPSK has bit error probability $Q(\sqrt{2E_b/N_0})$ , while orthogonal signalling has $Q(\sqrt{E_b/N_0})$ . The factor of 2 inside the square root is the whole difference, and it is a common distractor. Also keep $N_0/2$ as the two-sided noise density and $N_0$ as the one-sided one.

Is the reciprocity condition for h-parameters the same as for z-parameters?

No. A reciprocal two-port has $z_{12} = z_{21}$ , $y_{12} = y_{21}$ and $AD - BC = 1$ , but for h-parameters the condition is $h_{12} = -h_{21}$ . The minus sign is the trap. A symmetric network adds $z_{11} = z_{22}$ , $y_{11} = y_{22}$ or $A = D$ .

How should I revise formulas before GATE ECE?

Read a compact formula sheet once a day in the last two weeks and once more on the exam morning. For each formula, know when it applies and the trap built around it. Write the unit beside every number, keep rad/s and Hz apart, carry full precision and round only the final numerical answer.

Sources

Dates, fees and the syllabus are set by the GATE 2027 organising institute and can change. Always confirm at gate2027.iitm.ac.in.