Circuit Analysis 2: AC circuits: cram pack

Polar form: z = r∠θ = r(cos θ + j sin θ) = r·e^jθ, with r = |z| and θ = arg z. Back to rectangular form: a = r cos θ, b = r sin θ. The argument needs the quadrant: θ = atan(b/a) only when a > 0; for a < 0 add or subtract 180°. In polar form multiplication is easy: multiply the moduli and add the arguments; for division divide the moduli and subtract the arguments.

De Moivre and roots: (r∠θ)ⁿ = rⁿ∠nθ. The equation zⁿ = R∠φ has exactly n roots, all with modulus R^1/n, at the arguments (φ + 360°·k)/n for k = 0, 1, …, n − 1: they lie evenly spaced on a circle, 360°/n apart. A quadratic z² + pz + q = 0 with p² < 4q has the complex roots z = −p/2 ± j√(q − p²/4); a polynomial with real coefficients always has its complex roots in conjugate pairs a ± jb.

AC circuits with phasors: resistors, inductors and capacitors have impedances R, jωL and 1/(jωC), with ω = 2πf. In series, impedances add like resistances, but as complex numbers. Rms current and voltage follow Ohm’s law with |Z|.

AC circuits with phasors: resistors, inductors and capacitors have impedances R, jωL and 1/(jωC), with ω = 2πf. In series, impedances add like resistances, but as complex numbers. Rms current and voltage follow Ohm’s law with |Z|.

AC power: apparent power S = V·I (VA) splits into real power P = S·cos φ (W), which does useful work, and reactive power Q = S·sin φ (var), which moves back and forth between source and load. The power factor is cos φ. A capacitor in parallel supplies reactive power locally, so the line current drops for the same real power.

AC power: apparent power S = V·I (VA) splits into real power P = S·cos φ (W), which does useful work, and reactive power Q = S·sin φ (var), which moves back and forth between source and load. The power factor is cos φ. A capacitor in parallel supplies reactive power locally, so the line current drops for the same real power.

Q factor and bandwidth: in a resonant circuit Q = 2π·(energy stored)/(energy lost per cycle). For series RLC Q = ω₀L/R = (1/R)√(L/C); for parallel RLC Q = R/(ω₀L) = R√(C/L). The bandwidth between the half-power frequencies f₁ and f₂ (where the power is half and the current or voltage 1/√2 of its peak) is BW = f₂ − f₁ = f₀/Q, which is R/(2πL) in series and 1/(2πRC) in parallel. The resonance frequency is the geometric mean f₀ = √(f₁f₂), and f_1,2 = f₀(√(1 + 1/(4Q²)) ∓ 1/(2Q)). Losses also make the oscillation decay: in series RLC the amplitude falls as e^−Rt/(2L).

Q factor and bandwidth: in a resonant circuit Q = 2π·(energy stored)/(energy lost per cycle). For series RLC Q = ω₀L/R = (1/R)√(L/C); for parallel RLC Q = R/(ω₀L) = R√(C/L). The bandwidth between the half-power frequencies f₁ and f₂ (where the power is half and the current or voltage 1/√2 of its peak) is BW = f₂ − f₁ = f₀/Q, which is R/(2πL) in series and 1/(2πRC) in parallel. The resonance frequency is the geometric mean f₀ = √(f₁f₂), and f_1,2 = f₀(√(1 + 1/(4Q²)) ∓ 1/(2Q)). Losses also make the oscillation decay: in series RLC the amplitude falls as e^−Rt/(2L).

Balanced three-phase systems: in a Y (star) connection the line-to-line voltage is √3 times the phase voltage, V_L = √3·V_ph, and the line current equals the phase current. In a Δ (delta) connection the line-to-line voltage equals the phase voltage and the line current is √3 times the phase current, I_L = √3·I_ph. The total power is three times the power of one phase: P = 3·V_ph·I_ph·cos φ = √3·V_L·I_L·cos φ.

Balanced three-phase systems: in a Y (star) connection the line-to-line voltage is √3 times the phase voltage, V_L = √3·V_ph, and the line current equals the phase current. In a Δ (delta) connection the line-to-line voltage equals the phase voltage and the line current is √3 times the phase current, I_L = √3·I_ph. The total power is three times the power of one phase: P = 3·V_ph·I_ph·cos φ = √3·V_L·I_L·cos φ. Power factor correction: capacitors supply reactive power locally, Q_C = P·(tan φ₁ − tan φ₂). Each of the three capacitors supplies Q_C/3 = V²·ω·C, where V is the voltage across that capacitor: V_L in a Δ bank, V_L/√3 in a Y bank.

First-order transients: in a circuit with one capacitor or one inductor, every voltage and current moves exponentially from its initial value to its final value: x(t) = x(∞) + [x(0) − x(∞)]·e^−t/τ, with τ = RC or τ = L/R. After about 5τ the circuit has settled.

First-order transients: in a circuit with one capacitor or one inductor, every voltage and current moves exponentially from its initial value to its final value: x(t) = x(∞) + [x(0) − x(∞)]·e^−t/τ, with τ = RC or τ = L/R. After about 5τ the circuit has settled.

Thévenin’s theorem: any linear two-terminal network behaves like a single voltage source V_th in series with a resistance R_th. Norton’s theorem gives the same network as a current source I_N = V_th/R_th in parallel with R_th.

Thévenin’s theorem: any linear two-terminal network behaves like a single voltage source V_th in series with a resistance R_th. Norton’s theorem gives the same network as a current source I_N = V_th/R_th in parallel with R_th.

Frequency response: substitute s = jω in the transfer function. The magnitude in decibels is 20·log₁₀|H(jω)| and the phase is the sum of the angles of the numerator factors minus those of the denominator factors. A first-order RC or RL filter has its corner (−3 dB) frequency at ω_c = 1/(RC) or R/L, and f_c = ω_c/(2π); a low-pass filter gives |H| = 1/√(1 + (ω/ω_c)²) and φ = −atan(ω/ω_c). In the asymptotic (straight-line) Bode plot each pole bends the magnitude down by 20 dB/decade at its corner frequency, each zero bends it up by 20 dB/decade, and an integrator 1/s gives −20 dB/decade at all frequencies. Gain crossover ω_c: |L(jω)| = 1 (0 dB); phase margin PM = 180° + ∠L(jω_c). Phase crossover ω₁₈₀: ∠L(jω) = −180°; gain margin GM = −20·log₁₀|L(jω₁₈₀)| dB. The closed loop is stable when both margins are positive.

Frequency response: substitute s = jω in the transfer function. The magnitude in decibels is 20·log₁₀|H(jω)| and the phase is the sum of the angles of the numerator factors minus those of the denominator factors. A first-order RC or RL filter has its corner (−3 dB) frequency at ω_c = 1/(RC) or R/L, and f_c = ω_c/(2π); a low-pass filter gives |H| = 1/√(1 + (ω/ω_c)²) and φ = −atan(ω/ω_c). In the asymptotic (straight-line) Bode plot each pole bends the magnitude down by 20 dB/decade at its corner frequency, each zero bends it up by 20 dB/decade, and an integrator 1/s gives −20 dB/decade at all frequencies. Gain crossover ω_c: |L(jω)| = 1 (0 dB); phase margin PM = 180° + ∠L(jω_c). Phase crossover ω₁₈₀: ∠L(jω) = −180°; gain margin GM = −20·log₁₀|L(jω₁₈₀)| dB. The closed loop is stable when both margins are positive.