ESE ECE Prelims cheat sheet

SI base units: length metre (m), mass kilogram (kg), time second (s), current ampere (A), temperature kelvin (K), amount mole (mol), luminous intensity candela (cd).

Dimensional analysis: every valid physical equation is dimensionally homogeneous; dimensionless groups (Reynolds, Mach, Froude) govern similitude in scale models.

Standards and ethics: professional codes require competence-limited practice, disclosure of conflicts, and public-safety precedence over employer interest.

Project management: critical path is the longest-duration path through the network; total float = LS − ES = LF − EF, and zero float marks critical activities.

Engineering economics: simple payback ignores time value of money; net present value discounts all cash flows at the minimum acceptable rate of return.

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

Materials: Young's modulus E = stress/strain in the linear elastic region; Poisson's ratio ν = −lateral strain/axial strain, typically 0.25–0.35 for metals.

Energy and environment: efficiency = useful output/input; a thermal plant's overall efficiency is the product of cycle, boiler, and generator efficiencies.

Information and society: number systems convert by repeated division (to base b) or positional weighting (from base b); binary-coded decimal encodes each decimal digit in four bits.

KCL: algebraic sum of currents at a node is zero; KVL: algebraic sum of voltages around any closed loop is zero - both hold for linear and nonlinear elements.

Thevenin equivalent: V_th = open-circuit voltage, R_th = V_oc/I_sc, or the resistance seen with independent sources deactivated (voltage sources shorted, current sources opened).

Norton is the dual: I_N = I_sc and R_N = R_th; source transformation swaps V_th in series with R_th for I_N in parallel with R_N.

Superposition applies only to linear circuits: response equals the sum of responses to each independent source acting alone; dependent sources stay active.

First-order transient: v(t) = v(∞) + [v(0⁺) − v(∞)]e^(−t/τ), with τ = R_thC for capacitive and τ = L/R_th for inductive circuits.

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

Maximum power transfer: load receives maximum power when R_L = R_th (DC), giving P_max = V_th²/(4R_th); efficiency at that point is 50%.

Two-port parameters: h-parameters are hybrid (h₁₁ = input impedance with output shorted); z-parameters are open-circuit, y-parameters short-circuit parameters.

Reciprocity theorem: in a linear bilateral network, a source in branch 1 producing current in branch 2 yields the same current ratio when the source and response positions are exchanged.

PN junction diode: Shockley equation I = I_s(e^(V/nV_T) − 1), with V_T ≈ 26 mV at 300 K; forward drop ≈ 0.7 V (Si) and 0.3 V (Ge).

Depletion width under reverse bias grows as W ∝ √(V_bi + V_R); junction capacitance C_j = εA/W therefore falls with increasing reverse bias - the basis of varactor diodes.

BJT regions: cutoff (both junctions off), active (BE forward, BC reverse), saturation (both forward); in active mode I_C = βI_B and I_E = (β+1)I_B.

MOSFET: threshold voltage V_T separates cutoff from conduction; in saturation I_D = ½k′(W/L)(V_GS − V_T)²(1 + λV_DS), and the body effect raises V_T when V_SB > 0.

Zener diode used as a shunt regulator operates in reverse breakdown; a series resistor must drop the difference between V_in and V_Z at the rated current.

Bias stability: the stability factor S = ∂I_C/∂I_CO should be small; emitter degeneration (R_E) and voltage-divider bias reduce S toward 1.

Small-signal hybrid-π model: g_m = I_C/V_T, r_π = β/g_m; voltage gain of a common-emitter stage ≈ −g_m(R_C ∥ R_L).

Thermal runaway occurs when I_C rises with temperature faster than the bias network can compensate; the condition is V_CE·I_C exceeding the rated power dissipation without adequate heat sinking.

Op-amp ideal assumptions: infinite open-loop gain, infinite input impedance, zero output impedance; with negative feedback the summing-point constraint forces V₊ = V₋.

Inverting amplifier gain = −R_f/R_in; non-inverting gain = 1 + R_f/R_1; integrator uses a capacitor in feedback (output ∝ −∫v dt), differentiator swaps R and C.

Slew rate limits large-signal frequency response: maximum sine frequency without distortion is f_max = SR/(2πV_peak).

Boolean identities: De Morgan's laws (AB)' = A' + B' and (A+B)' = A'B'; consensus and absorption rules simplify multi-level expressions.

Karnaugh map grouping uses powers of two (1, 2, 4, 8); don't-care terms may be included to enlarge groups and reduce literal count.

A multiplexer implements any n-variable function with 2^(n−1) select lines by tying inputs to constants or variables; a decoder with NAND gates realizes minterm sums.

Flip-flops: SR is level-sensitive and has a forbidden state; JK removes it (toggle when J = K = 1); D is transparent data; T toggles on T = 1.

Counters: an n-bit ripple counter has modulus 2ⁿ but accumulates propagation delay; synchronous counters clock all stages together and can be truncated with reset logic to any modulus.

CMOS logic: dynamic power P = αCV²f (α = activity factor), plus short-circuit and leakage power; noise margin = V_OH(min) − V_IH(min).

LTI systems are fully characterized by their impulse response h(t); output is the convolution y(t) = x(t)*h(t), and in the transform domain Y = XH.

Stability of a causal LTI system requires all poles of H(s) in the open left half-plane; causality requires the ROC to be to the right of the rightmost pole.

Fourier series coefficients of a periodic signal: a_k = (1/T)∫x(t)e^(−jkω₀t)dt; a real even signal has purely real, even coefficients.

Parseval's theorem: total energy equals the integral of |X(f)|² over all frequencies; energy spectral density is |X(f)|².

Sampling theorem: a bandlimited signal with maximum frequency f_m is recoverable from samples taken at f_s > 2f_m; below that, aliasing folds high frequencies into the baseband.

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

Z-transform: a causal sequence has ROC |z| > r; the unit circle in the z-plane corresponds to the DTFT, so poles inside the unit circle give a stable causal system.

Convolution properties: commutative, associative, distributive; convolution with δ(t − t₀) shifts a signal to t₀.

A system is BIBO stable if and only if ∫|h(t)|dt < ∞ (continuous) or Σ|h[n]| < ∞ (discrete).

AM (DSB-FC): modulation index μ = A_m/A_c; total power P_t = P_c(1 + μ²/2); bandwidth = 2f_m, and sidebands each carry μ²P_c/4.

FM: Carson's rule gives bandwidth ≈ 2(Δf + f_m); modulation index β = Δf/f_m, and FM trades bandwidth for output SNR improvement over AM.

Superheterodyne receiver: image frequency f_image = f_LO + f_IF (high-side injection); the RF filter must reject it before mixing.

Quantization: uniform PCM with n bits gives SNR ≈ 6.02n + 1.76 dB for a full-scale sinusoid; step size Δ = full-scale range/2ⁿ.

Nyquist criterion for zero ISI: minimum baseband bandwidth = R_s/2 for symbol rate R_s; raised-cosine roll-off α expands it to (R_s/2)(1 + α).

Shannon capacity: C = B log₂(1 + S/N) bits/s; it is an upper bound independent of modulation and coding scheme.

Digital modulation: BPSK and QPSK have the same E_b/N₀ performance for a given error rate (QPSK carries 2 bits/symbol); M-QAM requires higher E_b/N₀ as M grows.

Matched filter maximizes output SNR at the sampling instant; its impulse response is the time-reversed signal, and output peak SNR = 2E/N₀.

Error control: ARQ uses retransmission with error detection (CRC), while FEC adds redundancy (Hamming, convolutional, LDPC) to correct without feedback.

Maxwell's equations in differential form: ∇·D = ρ, ∇·B = 0, ∇×E = −∂B/∂t, ∇×H = J + ∂D/∂t; the displacement-current term enables wave propagation.

Intrinsic impedance of free space η₀ = √(μ₀/ε₀) ≈ 377 Ω; in a general medium η = √(μ/ε), and wave velocity v = 1/√(με).

Poynting vector S = E × H gives power flow density; time-average for a sinusoid is ½Re(E × H*).

Boundary conditions: tangential E and normal B are continuous; tangential H differs by surface current density J_s, and normal D differs by surface charge density ρ_s.

Transmission line: characteristic impedance Z₀ = √(L/C); input impedance of a line of length l terminated in Z_L is Z_in = Z₀(Z_L + jZ₀tanβl)/(Z₀ + jZ_Ltanβl).

Reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀); VSWR = (1 + |Γ|)/(1 − |Γ|); a quarter-wave transformer matches when Z₀' = √(Z_in Z_L).

Antenna fundamentals: radiation resistance, directivity D, gain G = ηD; effective aperture A_e = λ²D/(4π), and Friis free-space path loss = (4πd/λ)².

Half-wave dipole has D ≈ 1.64 (2.15 dBi) and length ≈ 0.47λ; a short monopole over a ground plane has the same pattern as a dipole but half the input resistance.

Waveguides: cutoff frequency f_c = c/(2a) for the dominant TE₁₀ mode in a rectangular guide of broad dimension a; below cutoff the wave is evanescent.

Transfer function = ratio of Laplace transforms of output to input with zero initial conditions; valid only for linear time-invariant systems.

Second-order step response: ω_n natural frequency, ζ damping ratio; peak overshoot M_p = e^(−πζ/√(1−ζ²)), and 2% settling time ≈ 4/(ζω_n).

Routh–Hurwitz array gives the number of right-half-plane roots as the number of sign changes in the first column; a zero row uses the auxiliary polynomial.

Steady-state error: for a type-0 system a step gives error 1/(1+K_p); for type-1 a ramp gives 1/K_v; a parabolic input needs type 2 for finite error.

Bode plot: a pole contributes −20 dB/decade and −90° of phase; gain margin and phase margin are read at the phase-crossover and gain-crossover frequencies respectively.

Nyquist criterion: N = Z − P, where N is encirclements of −1, P open-loop RHP poles, Z closed-loop RHP poles; stability requires Z = 0.

Root locus rules: branches start at open-loop poles and end at zeros (or infinity); a point on the real axis is on the locus if the number of poles-plus-zeros to its right is odd.

PID control: proportional acts on present error, integral eliminates steady-state error but adds lag, derivative anticipates error and amplifies noise; integral windup requires anti-windup limiting.

State-space: controllability requires the controllability matrix [B AB A²B …] to have full rank; observability uses [C; CA; CA²; …].

PMMC movement: deflection θ ∝ I (uniform scale), used for DC only; adding a series multiplier extends voltage range, and a shunt extends current range.

Ammeter insertion error is minimized by low resistance; voltmeter loading error is minimized by high resistance - a voltmeter of R_v across R loads the circuit by R ∥ R_v.

Wheatstone bridge balance: R₁/R₂ = R₃/R₄ with galvanometer current zero; Kelvin double bridge is used for low resistances because it cancels lead and contact resistance.

AC bridges: Maxwell bridge measures inductance, Schering bridge measures capacitance and dissipation factor, Wien bridge measures frequency.

Instrument transformers: a current transformer secondary must never be open-circuited while energized (dangerous voltage); a potential transformer secondary must never be shorted.

CRO: deflection sensitivity in V/div, time base in s/div; bandwidth is the frequency at which displayed amplitude falls to 0.707 of the low-frequency value.

Digital meters: resolution is the smallest change displayed; accuracy is closeness to true value, precision is repeatability; a 3½-digit meter resolves 2000 counts.

Transducers: strain gauge sensitivity is the gauge factor GF = (ΔR/R)/ε; thermocouple output follows the Seebeck effect and needs cold-junction compensation.

Error analysis: for y = f(x₁, x₂), relative uncertainty combines as √(Σ(∂y/∂xᵢ · Δxᵢ)²) for independent random errors.

8085 basics: 8-bit data bus, 16-bit address bus (64 KB address space), accumulator-based ALU with registers B, C, D, E, H, L; ALE latches the low address byte.

8085 flags: S (sign), Z (zero), AC (auxiliary carry), P (parity), CY (carry); conditional jumps test these flags after an instruction that affects them.

8086: 20-bit address bus (1 MB), 16-bit data bus; physical address = segment register × 16 + offset, giving four overlapping segments.

Interrupts on 8085: TRAP is non-maskable and highest priority, followed by RST 7.5, 6.5, 5.5, then INTR; INTA acknowledges INTR.

Memory-mapped I/O uses normal memory instructions and shares the address space; port-mapped I/O uses separate instructions (IN, OUT) and control signals.

8051 microcontroller: 8-bit CPU, 4 KB internal ROM, 128 bytes RAM, two 16-bit timers, one serial port, four 8-bit ports; bit-addressable RAM from 20H to 2FH.

Addressing modes: immediate, direct, register, register-indirect, and implied; only register-indirect (via HL or BC/DE) provides full 16-bit addressing in 8085.

Computer organization: cache reduces average memory access time (AMAT = hit time + miss rate × miss penalty); pipelining increases throughput, not single-instruction latency.

Number representation: two's complement range for n bits is −2^(n−1) to 2^(n−1) − 1; subtraction is addition of the two's complement.