GATE EC cheat sheet

GATE EC: Core Reference

• BJT active-region relations: IC = βIB, IE = (β+1)IB, and α = β/(1+β); transconductance gm = IC/VT with VT ≈ 26 mV at 300 K.

• MOSFET saturation condition: VDS ≥ VGS − VTH; then ID = ½μnCox(W/L)(VGS − VTH)²(1 + λVDS), and gm = 2ID/(VGS − VTH) = √(2μnCox(W/L)ID).

• Op-amp ideal assumptions: infinite open-loop gain, infinite input impedance, zero output impedance, and virtual short (V+ = V−) only when negative feedback is present.

• Inverting amplifier gain = −Rf/Rin; non-inverting gain = 1 + Rf/R1; integrator output = −(1/RC)∫vin dt; differentiator output = −RC dvin/dt.

• Stability rule: closed-loop system is stable iff all poles of 1 + G(s)H(s) lie in the left half-plane; Routh array sign changes in the first column give the number of right-half-plane poles.

• Nyquist criterion: N = Z − P, where N is clockwise encirclements of −1, Z closed-loop RHP poles, P open-loop RHP poles; stable closed loop needs N = −P.

• Second-order step response: ωn = √(ωn²), damping ζ from denominator s² + 2ζωn s + ωn²; peak overshoot Mp = exp(−πζ/√(1−ζ²)) and peak time tp = π/(ωd), ωd = ωn√(1−ζ²).

• Sampling: a bandlimited signal with maximum frequency fm must be sampled at fs > 2fm to avoid aliasing; aliased frequency folds to |f − kfs| within [0, fs/2].

• Fourier pairs to memorize: rect(t/τ) ↔ τ sinc(fτ); tri(t/τ) ↔ τ sinc²(fτ); e^(−at)u(t) ↔ 1/(a + j2πf); δ(t) ↔ 1; u(t) ↔ 1/(j2πf) + ½δ(f).