Master Formula Vault

GATE ECE Rapid Revision Cheat Sheet (All 9 Subjects)

High-Density Exam Day Formula Reference • Verified for GATE 2027 by Akshat

1. Network Theory (NT)

Maximum Power Transfer:
- DC: $R_L = R_{th} \implies P_{max} = \frac{V_{th}^2}{4 R_{th}}$
- AC (Variable $Z_L = R_L + jX_L$): $Z_L = Z_{th}^* = R_{th} - jX_{th} \implies P_{max} = \frac{|V_{th}|^2}{4 R_{th}}$
- AC (Variable $R_L$ only): $R_L = |Z_{th}| = \sqrt{R_{th}^2 + X_{th}^2}$
Universal 1st Order Transient: $x(t) = x(\infty) + [x(0^+) - x(\infty)]e^{-t/\tau}$ ($RC: \tau = R_{eq}C$; $RL: \tau = \frac{L}{R_{eq}}$).
Series Resonance: $\omega_0 = \frac{1}{\sqrt{LC}}$, $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{1}{R}\sqrt{\frac{L}{C}}$, $\text{Bandwidth} = \frac{\omega_0}{Q} = \frac{R}{L}\text{ rad/s}$.
Two-Port Symmetry & Reciprocity:
- Z: Reciprocal $z_{12}=z_{21}$, Symmetric $z_{11}=z_{22}$
- Y: Reciprocal $y_{12}=y_{21}$, Symmetric $y_{11}=y_{22}$
- ABCD: Reciprocal $AD-BC=1$, Symmetric $A=D$
- h: Reciprocal $h_{12}=-h_{21}$, Symmetric $\Delta h = 1$

Continuous-Time Convolution: $y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau$. Area: $\text{Area}(y) = \text{Area}(x) \times \text{Area}(h)$. Width: $W_y = W_x + W_h$.
Fourier Duality: If $x(t) \leftrightarrow X(\omega)$, then $X(t) \leftrightarrow 2\pi x(-\omega)$.
Frequency Differentiation: $-jt x(t) \leftrightarrow \frac{dX(\omega)}{d\omega}$.
Parseval's Energy Relation: $E = \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |X(\omega)|^2 d\omega$.
Nyquist Sampling Rate:
- For $x(t) = x_1(t) \cdot x_2(t) \implies \omega_{nyq} = 2(\omega_{m1} + \omega_{m2})$
- For $x(t) = x_1(t) * x_2(t) \implies \omega_{nyq} = 2\min(\omega_{m1}, \omega_{m2})$

2nd Order Standard Form: $T(s) = \frac{\omega_n^2}{s^2 + 2\zeta \omega_n s + \omega_n^2}$, damped frequency $\omega_d = \omega_n \sqrt{1-\zeta^2}$.
Peak Overshoot & Times:
- $\%M_p = e^{-\frac{\pi \zeta}{\sqrt{1-\zeta^2}}} \times 100\%$
- Peak Time: $t_p = \frac{\pi}{\omega_d}$
- Settling Time (2% band): $t_s = \frac{4}{\zeta \omega_n}$; (5% band): $t_s = \frac{3}{\zeta \omega_n}$
Steady State Error ($e_{ss}$): $e_{ss} = \lim_{s \to 0} \frac{s R(s)}{1 + G(s)H(s)}$. (Type 0 step: $\frac{A}{1+K_p}$; Type 1 ramp: $\frac{A}{K_v}$; Type 2 parabolic: $\frac{A}{K_a}$).
Nyquist Encirclement: $N = P - Z$ ($N$ anticlockwise encirclements of $-1+j0$, $P$ open-loop RHP poles, $Z$ closed-loop RHP poles).

AM Total Power: $P_t = P_c\left(1 + \frac{\mu^2}{2}\right)$, Efficiency $\eta = \frac{\mu^2}{2+\mu^2}$ (Max 33.3% at $\mu=1$).
Carson's Rule FM Bandwidth: $\text{BW} = 2(\Delta f + f_m) = 2 f_m (1 + \beta)$.
PCM Quantization Step & Noise: $\Delta = \frac{V_{max}-V_{min}}{2^n}$, $N_q = \frac{\Delta^2}{12}$. Maximum sinusoidal SNR: $(\text{SNR})_{dB} = 1.76 + 6.02 n\text{ dB}$.
Shannon-Hartley Capacity: $C = B \log_2\left(1 + \frac{S}{N}\right)$ bps. When $B \to \infty \implies C_\infty = \frac{S}{N_0}\log_2 e \approx 1.44 \frac{S}{N_0}$ bps.

Inverting / Non-Inverting Op-Amp: $A_v = -\frac{R_f}{R_1}$, $A_v = 1 + \frac{R_f}{R_1}$.
Slew Rate Limit: $\text{SR} = \left.\frac{dV_{out}}{dt}\right|_{max} = 2\pi f_{max} V_p$. Full-power bandwidth $f_{max} = \frac{\text{SR}}{2\pi V_p}$.
BJT Transconductance: $g_m = \frac{I_C}{V_T}$ ($V_T \approx 26\text{ mV}$ at 300K), $r_\pi = \frac{\beta}{g_m}$, $r_e = \frac{\alpha}{g_m} \approx \frac{1}{g_m}$.
MOSFET Drain Current: Saturation $I_D = \frac{1}{2}\mu_n C_{ox}\frac{W}{L}(V_{GS}-V_{TH})^2(1+\lambda V_{DS})$, $g_m = \sqrt{2\mu_n C_{ox}\frac{W}{L}I_D} = \frac{2I_D}{V_{GS}-V_{TH}}$.

Boolean Theorems: $A + AB = A$, $A + \bar{A}B = A + B$, $(A+B)(A+C) = A + BC$.
De Morgan's Laws: $\overline{A + B} = \bar{A}\cdot\bar{B}$, $\overline{A\cdot B} = \bar{A} + \bar{B}$.
Modulus of Cascaded Counters: $\text{Mod}_{total} = M_1 \times M_2 \times \dots \times M_k$. Output frequency $f_{out} = \frac{f_{in}}{\text{Mod}_{total}}$.
ADC Resolution: Step size $\Delta = \frac{V_{ref}}{2^n - 1}$ (or $\frac{V_{ref}}{2^n}$).

Intrinsic Impedance & Skin Depth: $\eta = \sqrt{\frac{\mu}{\varepsilon}}$, skin depth $\delta = \frac{1}{\sqrt{\pi f \mu \sigma}}$.
Transmission Line Reflection: $\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$, $\text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$, $Z_{in} = Z_0 \left(\frac{Z_L + j Z_0 \tan(\beta l)}{Z_0 + j Z_L \tan(\beta l)}\right)$. Quarter-wave matching: $Z_0 = \sqrt{Z_{in} Z_L}$.
Rectangular Waveguide Cutoff: $f_c = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}$, Phase velocity $v_p = \frac{c}{\sqrt{1-(f_c/f)^2}} > c$, Group velocity $v_g = c\sqrt{1-(f_c/f)^2} < c \implies v_p \cdot v_g = c^2$.

Eigenvalue Properties: $\sum \lambda_i = \text{Trace}(A)$, $\prod \lambda_i = \det(A)$. Eigenvalues of $A^k$ are $\lambda_i^k$.
Euler-Cauchy ODE: $x^2 y'' + p x y' + q y = 0 \implies m(m-1) + p m + q = 0$ (Put $x = e^z$).
Cauchy-Riemann Equations: $u_x = v_y$, $u_y = -v_x$. Harmonic condition: $\nabla^2 u = 0, \nabla^2 v = 0$.
Bayes' Theorem: $P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum P(B|A_k)P(A_k)}$.
Correlation Coefficient: $r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} = \pm\sqrt{b_{yx} \cdot b_{xy}}$ ($-1 \le r \le 1$).

Relative Speed: Opposite directions $S_{rel} = S_1 + S_2$; Same direction $S_{rel} = |S_1 - S_2|$.
Combined Work: Together time $T = \frac{T_1 T_2}{T_1 + T_2}$. Efficiency is inversely proportional to time.
Compound Interest: $A = P\left(1 + \frac{r}{100}\right)^n$, Difference between CI and SI for 2 years: $\Delta = P\left(\frac{r}{100}\right)^2$.