Circuit Analysis (RC, RL & AC)

Beyond DC resistor networks: capacitors and inductors store energy and make circuits time- and frequency-dependent. Learn RC and RL transients and time constants, reactance and impedance, phasors, AC analysis of RLC circuits, resonance and the first-order RC filter.

Almost every electronic circuit contains capacitors and inductors: power-supply filters, timing circuits, reset delays, audio filters and radio tuners. Analysing them needs two tools — the exponential transient response and phasor (complex-number) AC analysis.

An analogy

A capacitor is like a bucket filling through a narrow tap — quick at first, then slower as it nears full. The time constant tells you how big the bucket and how narrow the tap are.

Transients

When a switch closes, a capacitor voltage cannot jump — it rises exponentially; an inductor current cannot jump — it rises exponentially. After 1τ the change is 63.2 % complete; after 5τ, 99.3 % (considered settled).

Phasors and impedance

For sinusoidal AC, each element becomes a complex impedance: R, jωL, 1/(jωC). Ohm's and Kirchhoff's laws then work with complex numbers. The magnitude gives amplitude; the angle gives phase shift.

Frequency response

An RC low-pass passes low frequencies and attenuates highs by 20 dB/decade above f_c (−3 dB at f_c). Swapping R and C gives a high-pass. Bode plots show gain and phase versus log frequency.

Resonance and Q

In series RLC, X_L = X_C at f₀ — impedance is minimum (just R) and current peaks. Quality factor Q = X_L/R = f₀/bandwidth sets how sharp the peak is — the basis of radio tuning and LC filters.

Formulas

• RC charging: v_C(t) = V (1 − e^(−t/τ)), τ = RC
• RL current rise: i(t) = (V/R)(1 − e^(−t/τ)), τ = L/R
• Reactances: X_L = 2πfL, X_C = 1/(2πfC)
• Series RLC impedance: Z = R + j(X_L − X_C), |Z| = √(R² + (X_L − X_C)²)
• RC low-pass cut-off: f_c = 1/(2πRC)
• Resonance: f₀ = 1/(2π√(LC))

Worked example: Microcontroller reset delay

Given: R = 10 kΩ, C = 1 µF from 3.3 V; Reset releases at 70 % of V_CC

Find: Delay.

• τ = 10 × 10³ × 1 × 10⁻⁶ = 10 ms
• 0.7 = 1 − e^(−t/τ) → t = −τ ln 0.3 = 1.204 τ
• t = 12.0 ms

Answer: ≈ 12 ms

Worked example: Series RL impedance

Given: R = 100 Ω, L = 0.1 H at 50 Hz

Find: |Z| and phase.

• X_L = 2π × 50 × 0.1 = 31.4 Ω
• |Z| = √(100² + 31.4²) = 104.8 Ω
• φ = atan(31.4/100) = 17.4°

Answer: 104.8 Ω, current lags by 17.4°

Common mistakes

• Wrong: Assuming a capacitor voltage can change instantly. — Right: It follows the exponential; only current can jump.
• Wrong: Adding impedance magnitudes directly. — Right: Add as complex numbers (phasors).

Applications

• Timing & reset circuits: RC delays for power-on reset and debouncing.
• Filtering: RC anti-aliasing and decoupling.
• Tuned circuits: LC resonance in radios and oscillators.

Key takeaways

• τ = RC or L/R; settle in ≈ 5τ.
• X_L = 2πfL, X_C = 1/(2πfC).
• Use complex impedance for AC.
• f_c = 1/(2πRC); f₀ = 1/(2π√LC).

Going further

• Laplace analysis: s-domain impedances for arbitrary inputs.
• Second-order systems: Damping ratio and overshoot in RLC.
• SPICE simulation: Transient and AC sweeps in LTspice.