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RC/RL Transients & Impedance Tricks (IPhO E&M)

IPhO E&M lesson on first-order RC/RL switching, finding time constants via Thevenin resistance, and fast impedance shortcuts for AC.

  • International Physics Olympiad preparation

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Most IPhO “transient” circuits are secretly one first-order system after a Thevenin reduction. The core skill is to get the correct initial condition and the correct time constant, then write the exponential in one line. For AC, impedance turns the differential equations into algebra, and high/low frequency limits kill most of the complexity.

Prerequisites (quick refresh)
First-order + impedance template (write this every time)
  1. Identify the state variable: v_C(t) or i_L(t).
  2. Initial condition: v_C is continuous, i_L is continuous.
  3. Final value: replace C by open circuit and L by short circuit for steady DC.
  4. Time constant: compute Rₜₕ seen by the reactive element with independent sources turned off. Use τ = RₜₕC (RC) and τ = L/Rₜₕ (RL).
  5. Write the one-line solution: x(t) = x_∞ + [x(0⁺)-x_∞]e^(-t/τ).
  6. For AC: replace R,L,C by impedances and use dividers: Z_R = R, Z_L = iω L, Z_C = 1/(iω C).

1. Definitions (Must Know)

  • Capacitor relation:
    i_C = Cdv_C/dt.
  • Inductor relation:
    v_L = Ldi_L/dt.
  • Time constant for RC: τ = RₜₕC.
  • Time constant for RL: τ = L/Rₜₕ.
  • Capacitor voltage cannot jump: v_C(0⁺) = v_C(0⁻).
  • Inductor current cannot jump: i_L(0⁺) = i_L(0⁻).
  • Sinusoidal steady state (phasors):
    Z_R = R, Z_L = iω L, Z_C = 1/(iω C).

2. Key Ideas (What Earns Marks)

  • Reduce first, solve later. Replace everything seen by the reactive element by its Thevenin equivalent (Vₜₕ,Rₜₕ) and the ODE becomes trivial.
  • Final and initial values are often diagram-only. At t → ∞ for DC: C is open, L is short. At t = 0⁺: enforce continuity on v_C or i_L.
  • One exponential is the default. If you see one capacitor or one inductor, expect one time constant unless there is an explicit second energy storage element.
  • Impedance limits (low frequency). Z_C is large (open-like) and Z_L is small (short-like).
  • Impedance limits (high frequency). Z_C is small (short-like) and Z_L is large (open-like).
  • Energy sanity checks:
    U_C = (1/2)Cv_C², U_L = (1/2)Li_L².

3. Detailed Explanations

A. The one-line solution for any first-order circuit.
Once you know x(0⁺), x_∞, and τ, you can write

x(t) = x_∞ + [x(0⁺)-x_∞]e^(-t/τ).

This works for v_C(t) and i_L(t) after the switching event.

B. Finding Rₜₕ seen by the reactive element.
Remove the capacitor (open) or the inductor (open) and look into its terminals with independent sources turned off:

  • Voltage sources turned off become shorts.
  • Current sources turned off become opens. Then reduce the remaining resistors by series/parallel to get Rₜₕ.

C. Fixed “instant” pictures that save time.
For DC switching:

  • At t = 0⁺: treat v_C as fixed (voltage source equal to v_C(0⁻)), and treat i_L as fixed (current source equal to i_L(0⁻)).
  • At t → ∞: capacitor open, inductor short.

D. Impedance tricks that appear in olympiad circuits.
Once you are in sinusoidal steady state, any linear circuit becomes a resistor network with complex numbers. Two high-yield moves:

  • Use a complex voltage divider: Vₒᵤₜ = Vᵢₙ Z₂/(Z₁ + Z₂).
  • Do limit checks without algebra: replace Z_C and Z_L by open/short in the low and high frequency limits to get the correct asymptotes.

4. Common Mistakes

  • Using τ = RC with the wrong R (it must be the resistance seen by the element, not a random series resistor).
  • Forgetting continuity: setting v_C(0⁺) = 0 because “the switch just closed”, or setting i_L(0⁺) = 0 because “the switch just opened”.
  • In AC, mixing peak and RMS values without stating which one you use.
  • Treating Z_C as 1/(ω C) (that drops the phase and changes the algebra).
  • Forgetting that inductors are shorts in steady DC and capacitors are opens in steady DC.

5. Exam Tips

  • Before any algebra, write three lines: x(0⁺) = ? , x_∞ = ? , τ = ?.
  • Draw the t = 0⁺ and t → ∞ equivalent circuits explicitly (two quick sketches).
  • In phasors, compute the transfer function symbolically and then do the low and high frequency limits as a check.
  • If a network looks messy, Thevenin it. That is usually what the question is testing.

6. Worked Examples

Example 1 (RC with a divider: Thevenin seen by the capacitor).
A battery of emf V feeds two resistors R₁ and R₂ in series. A capacitor C is connected from the midpoint node to the bottom node at t = 0. Initially the capacitor is uncharged. Find v_C(t) for t after the connection.

Solution sketch

The capacitor sees the rest of the circuit as a Thevenin source.

Open-circuit midpoint voltage:

Vₜₕ = VR₂/(R₁ + R₂).

Turn off the battery (short it) and look into the midpoint node: R₁ and R₂ are in parallel:

Rₜₕ = R₁∥ R₂ = R₁R₂/(R₁ + R₂).

Initial condition: v_C(0⁺) = 0. Final value: v_C(∞) = Vₜₕ.

Therefore

v_C(t) = Vₜₕ(1-e^(-t/(RₜₕC))).

Example 2 (RL step: current growth and inductor voltage).
A battery of emf V is connected at t = 0 to a series combination of resistor R and inductor L. Initially i_L(0⁻) = 0. Find i_L(t) and v_L(t) for t after the switch closes.

Solution sketch

For a series RL driven by a step, the time constant is

τ = L/R.

Final current is i_∞ = V/R and initial current is 0, so

i_L(t) = (V/R)(1-e^(-tR/L)).

Inductor voltage is v_L = L di/dt:

v_L(t) = V e^(-tR/L).

Check: at t = 0⁺, v_L = V (inductor initially resists current change). At long times, v_L → 0.

Example 3 (impedance: RC low-pass transfer function).
A resistor R is in series with a capacitor C to ground. The input sinusoidal voltage is across the series pair, and the output is the capacitor voltage. Find H(ω) = Vₒᵤₜ/Vᵢₙ.

Solution sketch

Use impedances: Z_C = 1/(iω C). Voltage divider gives

H(ω) = Z_C/(R + Z_C) = 1/(1 + iω RC).

Magnitude and phase:

|H| = 1/(square root of (1 + (ω RC)²)), arg(H) = - arctan(ω RC).

Checks:

  • Low frequency: |H| ≈ 1 (capacitor sees the full signal).
  • High frequency: |H| ≈ 1/(ω RC) (output is suppressed).

7. Mind Stretchers

Mind stretcher: infinite RC ladder input impedance

Consider an infinite ladder network: a series resistor R, then a capacitor C to ground, then the same pattern repeats forever. Find the input impedance Z(ω) in sinusoidal steady state.

Solution idea: use self-similarity. After the first resistor, the capacitor is in parallel with the rest of the infinite ladder, which has the same impedance Z.

Let Z_C = 1/(iω C). Then

Z = R + (Z_C∥ Z) = R + (Z_C Z)/(Z_C + Z).

Rearranging:

(Z-R)(Z_C + Z) = Z_C Z ⇒ Z²-RZ-RZ_C = 0.

Solve the quadratic:

Z = (R + square root of (R² + 4RZ_C))/2,

choosing the branch with positive real part.

Limit checks:

  • High frequency: Z_C is small, so Z → R (first capacitor shorts the node).
  • Low frequency: Z_C is large, so |Z| grows (ladder looks increasingly open).

8. Practice

Practice loop (fast and correct)
  1. For any switching problem, force yourself to write x(0⁺), x_∞, and τ before solving.
  2. For each RC/RL problem, compute Rₜₕ seen by C or L by source-killing and looking into the terminals.
  3. For each impedance problem, do low and high frequency limits as a built-in check.
Syllabus and review details

No official syllabus alignment is listed for this lesson.