Electric recruit response (transient analysis)

When a switch is closed or opened in a circuit containing inductors or capacitors, currents and voltages do not jump instantly to their final values. Energy stored in magnetic and electric fields must change gradually, so there is a temporary "transient" period before the circuit settles into its "steady state". This topic (listed by NEC as electric circuit response) covers the transient and steady-state behaviour of R-L, R-C and R-L-C circuits, the time constant, and the damping cases of second-order circuits.

Key concepts

• Steady state: the condition reached after a long time when voltages/currents are constant (DC) or periodic (AC). For DC steady state, an inductor acts as a short circuit and a capacitor as an open circuit.
• Transient: the response between one steady state and the next. It decays to zero in stable circuits.
• Total response = natural (transient, homogeneous solution) + forced (steady-state, particular solution).
• Continuity rules: current through an inductor cannot change instantly, $i_L(0^+) = i_L(0^-)$; voltage across a capacitor cannot change instantly, $v_C(0^+) = v_C(0^-)$.
• At the instant of switching ($t = 0^+$), an uncharged capacitor acts as a short circuit; an inductor with zero initial current acts as an open circuit.

R-L circuit (DC source V switched on)

• $i(t) = \frac{V}{R}(1 - e^{-t/\tau})$, time constant $\tau = L/R$ (s).
• At $t = \tau$ current reaches 63.2% of final value; after about $5\tau$ it is >99%.
• Decay (source removed): $i(t) = I_0e^{-t/\tau}$, falls to 36.8% in one τ.

R-C circuit

• Charging: $v_C(t) = V(1 - e^{-t/\tau})$, $i(t) = \frac{V}{R}e^{-t/\tau}$, with $\tau = RC$.
• Discharging: $v_C(t) = V_0e^{-t/\tau}$.
• Initial current at switching is $V/R$ (capacitor acts as short).

R-L-C series circuit (second order) Characteristic equation: $s^2 + \frac{R}{L}s + \frac{1}{LC} = 0$.

Condition	Damping	Response
$R > 2\sqrt{L/C}$ ($\zeta > 1$)	Overdamped	two real roots, slow, no oscillation
$R = 2\sqrt{L/C}$ ($\zeta = 1$)	Critically damped	repeated real root, fastest without overshoot
$R < 2\sqrt{L/C}$ ($\zeta < 1$)	Underdamped	complex roots, decaying oscillation
$R = 0$	Undamped	sustained oscillation at $\omega_n$

• Undamped natural frequency $\omega_n = 1/\sqrt{LC}$; damping ratio $\zeta = \frac{R}{2}\sqrt{\frac{C}{L}}$; damped frequency $\omega_d = \omega_n\sqrt{1-\zeta^2}$.

AC transients: when a sinusoidal source is switched onto an RL circuit, the current has a steady sinusoidal part plus a decaying DC offset whose size depends on the switching instant. If the switch closes at the instant where the steady-state current would naturally be zero, no transient occurs. This explains the DC offset in fault currents.

Laplace method: elements become impedances $R$, $sL$, $1/(sC)$; initial conditions appear as sources ($Li(0)$ in series with inductor, $v(0)/s$ in series with capacitor). Solve algebraically, then inverse transform.

High-yield points

• $\tau = L/R$ for RL and $\tau = RC$ for RC; 63.2% of the change occurs in one τ, ~5τ is steady state.
• Inductor current and capacitor voltage cannot change instantaneously.
• DC steady state: L acts as a short, C as an open circuit.
• Series RLC critical resistance $R = 2\sqrt{L/C}$; $\zeta = \frac{R}{2}\sqrt{C/L}$.
• ζ < 1 gives decaying oscillation; ζ > 1 gives a slow non-oscillatory response.

Formulas

$\tau_{RL} = \frac{L}{R},\qquad \tau_{RC} = RC\quad\text{(seconds)}$

$x(t) = x(\infty) + [x(0^+) - x(\infty)]e^{-t/\tau}$

This general first-order formula works for any current/voltage in a single-time-constant circuit.

$s_{1,2} = -\frac{R}{2L} \pm \sqrt{\left(\frac{R}{2L}\right)^2 - \frac{1}{LC}}$

Energy stored: $W_L = \frac12 LI^2$, $W_C = \frac12 CV^2$ (J).

Worked example

A 10 μF capacitor is charged through a 100 kΩ resistor from a 12 V DC source. Find τ and the voltage after 2 s.

1. $\tau = RC = 100\times10^3 \times 10\times10^{-6} = 1$ s.
2. $v_C(2) = 12(1 - e^{-2}) = 12(1 - 0.1353) = 12 \times 0.8647 = 10.38$ V.
3. Initial current $= 12/100\text{ k} = 0.12$ mA; at 2 s, $i = 0.12e^{-2} = 0.0162$ mA.

Exam tips

• One time constant → 63.2% of rise or 36.8% remaining on decay; 5τ is taken as "steady state".
• Inductor resists change of current; capacitor resists change of voltage.
• At t = 0⁺: uncharged C = short, de-energised L = open. At t = ∞ (DC): C = open, L = short.
• Critical resistance for series RLC is $2\sqrt{L/C}$.
• Increasing R in an RL circuit reduces τ (faster); increasing R in RC increases τ (slower).

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. An initially uncharged capacitor is charged from a dc source through a resistor. After one time constant, its voltage reaches about what percentage of the supply voltage?

• A. 86.5 %
• B. 63.2 %
• C. 50 %
• D. 36.8 %

Answer: B. $v_C = V(1 - e^{-t/\tau})$, and at $t = \tau$ this is $V(1 - e^{-1}) = 0.632V$.

2. A series RLC circuit has $R = 2\ \Omega$, $L = 1$ H and $C = 0.2$ F. What is the damped natural frequency of its transient response?

• A. 5 rad/s
• B. 1 rad/s
• C. 2 rad/s
• D. 2.24 rad/s

Answer: C. Characteristic equation $s^2 + (R/L)s + 1/(LC) = s^2 + 2s + 5 = 0$, roots $s = -1 \pm j2$, so $\omega_d = 2$ rad/s.

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