ECE Board Exam cheat sheet

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); discriminant > 0 gives two distinct real roots, = 0 a repeated root, < 0 a complex-conjugate pair.

Standard derivatives: d/dx(e^x) = e^x, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(tan x) = sec²x.

Integration by parts: ∫u dv = uv − ∫v du; pick u by LIATE order (logarithmic, inverse trig, algebraic, trig, exponential).

Triangle rules: a/sin A = b/sin B = c/sin C = 2R (circumradius) and c² = a² + b² − 2ab cos C.

Series sums: arithmetic Sₙ = n(a₁ + aₙ)/2; geometric Sₙ = a₁(1 − rⁿ)/(1 − r), with infinite sum a₁/(1 − r) only when |r| < 1.

Special limits: lim(x→0) sin x / x = 1 (x in radians) and lim(x→∞)(1 + 1/x)^x = e ≈ 2.71828.

Laplace pairs: L{1} = 1/s, L{t} = 1/s², L{e^(at)} = 1/(s − a), L{sin ωt} = ω/(s² + ω²).

Counting: nPr = n!/(n − r)! for ordered arrangements; nCr = n!/[r!(n − r)!] for unordered selections.

Straight lines: m = (y₂ − y₁)/(x₂ − x₁); parallel lines share the same slope, perpendicular lines satisfy m₁m₂ = −1.

Ideal op-amp: infinite open-loop gain and input impedance, zero output impedance; with negative feedback V₊ = V₋ and no current enters the inputs.

Op-amp closed-loop gains: inverting = −Rf/Rin, non-inverting = 1 + Rf/R1, voltage follower = 1.

BJT relations: Ic = βIb, Ie = (β + 1)Ib, α = β/(β + 1); a conducting silicon junction shows VBE ≈ 0.7 V.

MOSFET saturation: Id = ½k(Vgs − Vt)², valid when Vgs > Vt and Vds > Vgs − Vt, so the device acts as a current source.

Diode drops: silicon ≈ 0.7 V, germanium ≈ 0.3 V, Schottky ≈ 0.3 V; reverse saturation current roughly doubles per 10 °C rise.

Series RLC resonance: f₀ = 1/(2π√(LC)), Q = (1/R)√(L/C), bandwidth = f₀/Q.

Ideal transformer: Vp/Vs = Np/Ns = Is/Ip; impedance reflected to the primary scales as (Np/Ns)².

Rectifier ripple frequency equals the source frequency for half-wave and twice the source frequency for full-wave; larger C and RL reduce ripple.

Thevenin/Norton: Vth = open-circuit voltage, Rth = Voc/Isc; maximum power transfers to a load when RL = Rth.

Constant-acceleration kinematics: v = u + at, s = ut + ½at², v² = u² + 2as.

Newton's laws: F = ma; without a net force a body keeps its velocity; action and reaction are equal, opposite, and on different bodies.

Work and energy: KE = ½mv², PE = mgh, W = Fd cos θ, power P = W/t = Fv.

Ideal gas law: PV = nRT with R = 8.314 J/(mol·K); one mole occupies about 22.4 L at 0 °C and 101.3 kPa.

Thermodynamics: ΔU = Q − W (Q in, W out); Carnot efficiency η = 1 − Tc/Th using absolute temperatures.

Materials: stress σ = F/A, strain ε = ΔL/L, Young's modulus E = σ/ε, valid in the linear elastic region.

Fluids: continuity A₁v₁ = A₂v₂; Bernoulli P + ½ρv² + ρgh = constant; buoyant force equals the weight of displaced fluid.

Heat transfer: conduction rate Q/t = kAΔT/L; sensible heat Q = mcΔT; latent heat Q = mL.

Electrostatics: Coulomb force F = kq₁q₂/r² with k ≈ 8.99 × 10⁹ N·m²/C²; field E = kq/r².

Nyquist criterion: sample at fs ≥ 2fmax; components above fs/2 fold back into the baseband as aliasing.

Shannon–Hartley: C = B log₂(1 + S/N) bits per second, the error-free capacity limit for a band-limited channel.

AM: modulation index μ = Vm/Vc must stay ≤ 1; single-tone total power Pt = Pc(1 + μ²/2) with sidebands at fc ± fm.

FM: Carson's rule bandwidth ≈ 2(Δf + fm); deviation ratio D = Δf/fm; FM trades bandwidth for noise immunity.

PCM: SQNR ≈ 6.02n + 1.76 dB for a full-scale sinusoid, and bit rate = fs × n for uniform quantization.

Satellite: geostationary altitude ≈ 35,786 km with a 24-hour period; free-space loss increases 6 dB per doubling of range or frequency.

Networking: IPv4 addresses are 32 bits and IPv6 are 128 bits; TCP is connection-oriented and reliable, UDP is connectionless.

Cellular evolution: 1G analog FM, 2G digital (GSM), 3G WCDMA, 4G LTE with OFDMA, 5G NR on sub-6 GHz and mmWave bands.

Microwave bands: L 1–2 GHz, S 2–4 GHz, C 4–8 GHz, X 8–12 GHz, Ku 12–18 GHz, K 18–27 GHz, Ka 27–40 GHz.