ESE Prelims Electrical cheat sheet

Ohm's law V = IR; power P = VI = I²R = V²/R; energy W = ∫P dt.

KCL: algebraic sum of currents at a node is zero; KVL: algebraic sum of voltages around any closed loop is zero.

Series resistances add; parallel conductances add. Two in parallel: R = R₁R₂/(R₁+R₂).

Star-delta: R_A = (R_AB·R_AC)/(R_AB+R_BC+R_CA); delta-star: R_AB = (R_A R_B + R_B R_C + R_C R_A)/R_C.

Thevenin: V_th = open-circuit voltage, R_th = V_oc/I_sc; Norton: I_N = short-circuit current, R_N = R_th.

Maximum power transfer when R_L = R_th; efficiency then 50%. For AC, Z_L = Z_th* (conjugate match).

First-order RC/RL: τ = RC or L/R; step response reaches 63.2% at t = τ, 99.3% at 5τ.

Series RLC: resonant ω₀ = 1/√(LC), Q = (1/R)√(L/C), bandwidth = ω₀/Q; at resonance Z is purely resistive.

Reciprocity and superposition hold only for linear bilateral networks; superposition does not apply to power.

DC machine EMF: E = PφNZ/60A; torque T = 0.159·PφIZ/A (SI, N·m) with Z = total conductors, A = parallel paths.

Shunt DC motor: N ∝ (V − I_aR_a)/φ; series motor must never be started unloaded because speed rises as flux falls.

Transformer EMF per turn: E = 4.44 f Φ_max; induced EMF E = 4.44 f N Φ_max.

Transformer: V₁/V₂ = N₁/N₂ = I₂/I₁; equivalent impedance referred to primary = a²Z₂, a = N₁/N₂.

Three-phase induction motor: synchronous speed N_s = 120f/P; slip s = (N_s − N)/N_s; rotor frequency = sf.

Induction motor torque T ∝ sE₂²R₂/(R₂² + (sX₂)²); maximum torque occurs at R₂ = sX₂, independent of R₂ magnitude.

Synchronous machine: EMF E = 4.44 f N Φ k_w; voltage regulation by synchronous impedance method uses E₀ = √((V cos φ + IR)² + (V sin φ ± IX_s)²).

Single-phase induction motors are not self-starting; double-revolving-field theory gives zero starting torque.

Efficiency is maximum when variable losses (copper) equal constant losses (core plus friction and windage).

Per-unit: Z_pu = Z_actual/Z_base, Z_base = V_base²/S_base; on base change Z_pu,new = Z_pu,old(S_new/S_old)(V_old/V_new)².

Short-circuit MVA = √3 × V_L × I_sc; fault current from a source of reactance X is I = V/(√3 X).

Transmission line: surge impedance Z_c = √(L/C); SIL = V_L²/Z_c; surge impedance loading is the natural load.

ABCD constants: V_S = AV_R + BI_R, I_S = CV_R + DI_R; for a symmetrical network AD − BC = 1 and A = D.

Voltage regulation = (|V_R,noload| − |V_R,full load|)/|V_R,full load| × 100%.

Ferranti effect: receiving-end voltage exceeds sending-end voltage on lightly loaded or open long lines due to charging current.

Load flow: slack bus specifies |V| and δ, PV bus specifies P and |V|, PQ bus specifies P and Q; Newton-Raphson converges quadratically.

Equal-area criterion: stability limit is when accelerating area equals decelerating area on the P-δ curve.

Corona: critical disruptive voltage depends on conductor radius, spacing, air density factor and surface irregularity; corona loss rises with frequency.

Transfer function is defined only for linear time-invariant systems with zero initial conditions.

Second-order standard form: ω_n²/(s² + 2ζω_n s + ω_n²); ζ < 1 underdamped, ζ = 1 critically damped, ζ > 1 overdamped.

Peak overshoot M_p = e^(−πζ/√(1−ζ²)); peak time t_p = π/(ω_n√(1−ζ²)); settling time (2%) ≈ 4/(ζω_n).

Routh-Hurwitz: number of sign changes in the first column equals the number of right-half-plane roots.

Steady-state error: type 0 gives finite error to step, type 1 zero error to step and finite to ramp; e_ss = lim s→0 sR(s)/(1+G(s)H(s)).

Bode: a pole contributes −20 dB/decade and −90°; a zero contributes +20 dB/decade and +90°; gain margin is measured at the phase crossover.

Nyquist criterion: Z = N + P, where N is encirclements of −1 and P is open-loop right-half-plane poles.

PID: proportional acts on present error, integral removes offset, derivative adds damping and anticipates error.

State space: ẋ = Ax + Bu, y = Cx + Du; stability requires all eigenvalues of A in the left half-plane.

PMMC movement: deflection θ ∝ I; sensitivity = NAB/k (rad/A); it responds to average value and suits DC only.

Ammeter: low resistance in series; voltmeter: high resistance in parallel; multiplying factor of a voltmeter with series R_m is 1 + R_m/R_v.

Wheatstone bridge balance: R₁/R₂ = R₃/R₄; Maxwell bridge measures inductance, Schering bridge capacitance and loss angle.

Two-wattmeter method: total power = W₁ + W₂; power factor angle tan φ = √3(W₁ − W₂)/(W₁ + W₂).

Energy meter constant K = revolutions per kWh; error % = (measured − true)/true × 100%.

Instrument transformer: CT burden expressed in VA; ratio correction factor = true ratio/rated ratio.

CRO: deflection sensitivity in V/cm; time base sets sweep; Lissajous figures give frequency ratio from tangent counts.

Digital meters: resolution is the smallest change detectable; accuracy is closeness to true value, precision is repeatability.

Errors: systematic (instrumental, environmental) can be corrected; random errors are reduced by averaging repeated readings.

BJT: I_E = I_B + I_C, β = I_C/I_B, α = β/(1+β); in active region V_BE ≈ 0.7 V (Si).

MOSFET saturation current I_D = ½μ_nC_ox(W/L)(V_GS − V_th)²; triode region behaves as a voltage-controlled resistor.

Op-amp ideal: infinite gain and input impedance, zero output impedance; inverting gain = −R_f/R_in, non-inverting gain = 1 + R_f/R_in.

Op-amp integrator output = −(1/RC)∫v_in dt; differentiator output = −RC dv_in/dt.

Boolean identities: A + A'B = A + B; De Morgan: (A + B)' = A'B', (AB)' = A' + B'.

Karnaugh map: group 1, 2, 4, 8 adjacent cells; a group of 2ⁿ cells eliminates n variables.

Flip-flops: JK toggles when J = K = 1; D stores input; a mod-N counter needs ⌈log₂N⌉ flip-flops.

ADC: n-bit resolution = V_FS/2ⁿ; successive approximation takes n clock cycles; flash ADC is fastest with 2ⁿ − 1 comparators.

CMOS static power dissipation is negligible; dynamic power = CV²f.

SCR: turns on when anode-cathode voltage exceeds forward breakover with gate pulse; turns off only when current falls below holding current.

Half-wave controlled rectifier with resistive load: V_dc = (V_m/2π)(1 + cos α).

Full-wave controlled rectifier: V_dc = (2V_m/π)cos α; for a single-phase fully controlled bridge the same average applies.

Single-phase AC voltage controller with resistive load: V_rms = V_s√[(π − α + sin 2α/2)/π].

Step-down chopper: V_o = D·V_s; step-up chopper: V_o = V_s/(1 − D).

Buck-boost: V_o = −V_s·D/(1 − D); in continuous conduction these are ideal, lossless relations.

Three-phase inverter 180° mode: line voltage has six steps; 120° mode gives two switches conducting at a time.

DC motor speed control by armature voltage: N ∝ V_a − I_aR_a; chopper control varies V_a while torque capability stays constant.

V/f control of induction motor keeps flux constant; below base speed constant torque, above base speed constant power (field weakening).

LTI system output = convolution of input with impulse response: y(t) = x(t) * h(t); in frequency domain Y = XH.

Convolution with δ(t − t₀) shifts a signal to t₀; convolution of two rectangular pulses gives a trapezoid.

Fourier series coefficients of a real signal satisfy c_{−n} = c_n*; a periodic signal has a line spectrum.

Fourier transform pair: rect(t/T) ↔ T sinc(fT); a time shift adds a linear phase −2πft₀.

Sampling theorem: sample at f_s > 2f_max; otherwise aliasing folds high frequencies to f_s − f.

Laplace transform: L{e^{−at}u(t)} = 1/(s+a); final value theorem valid only if poles of sX(s) lie in the left half-plane.

z-transform: Z{aⁿu[n]} = z/(z − a), ROC |z| > |a|; a causal stable system has all poles inside the unit circle.

Parseval's theorem: energy in time domain equals energy in frequency domain, ∫|x(t)|²dt = ∫|X(f)|²df.

A system is causal if h(t) = 0 for t < 0; BIBO stable if ∫|h(t)|dt is finite.

Maxwell's equations: ∇·D = ρ, ∇·B = 0, ∇×E = −∂B/∂t, ∇×H = J + ∂D/∂t.

Coulomb's law: F = q₁q₂/(4πε₀r²); ε₀ = 8.854 × 10⁻¹² F/m, μ₀ = 4π × 10⁻⁷ H/m.

Gauss's law for a sphere of charge Q gives E = Q/(4πε₀r²); inside a uniformly charged sphere E ∝ r.

Boundary conditions: tangential E and normal B are continuous; normal D jumps by surface charge, tangential H by surface current.

Poynting vector S = E × H gives power flow density; average power = ½Re(E × H*).

Plane wave: intrinsic impedance η = √(μ/ε) = 377 Ω in free space; phase velocity v = 1/√(με).

Skin depth δ = √(2/(ωμσ)); attenuation constant α = 1/δ for a good conductor.

Transmission line reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀); VSWR = (1 + |Γ|)/(1 − |Γ|).

Capacitance of a parallel-plate capacitor C = εA/d; energy stored W = ½CV² = ½∫εE²dv.

SI base units: metre, kilogram, second, ampere, kelvin, mole, candela; derived units are products of these.

Dimensional analysis: both sides of a valid equation have the same dimensions; check exponents, not constants.

Standard prefixes: k = 10³, M = 10⁶, G = 10⁹, m = 10⁻³, μ = 10⁻⁶, n = 10⁻⁹, p = 10⁻¹².

Engineering ethics: public safety and welfare take priority over employer or client interests; disclose conflicts of interest.

Indian electricity regulation: CEA advises on technical standards; CERC regulates inter-state, SERC intra-state transmission and tariffs.

Environmental impact: EIA screens projects before clearance; the precautionary principle applies where evidence is uncertain.

Project management: critical path is the longest path through the network; slack is zero on critical activities.

Basic economics: break-even output = fixed cost/(unit price − unit variable cost).

Probability: P(A∪B) = P(A) + P(B) − P(A∩B); for independent events P(A∩B) = P(A)P(B).