Questions: First-Order Transient Circuit Response

Question 1

An RC circuit has a capacitor C, a resistor R1 in series between the source and the capacitor, and a resistor R2 in parallel with the capacitor. After zeroing all independent sources, what resistance should be used to compute the time constant τ?

• A. R1 alone — only the series resistor limits charging current into the capacitor
• B. R2 alone — only the parallel resistor directly discharges the capacitor
• C. R1 + R2 — add all resistors in the circuit
• D. R1 in parallel with R2 — the Thevenin resistance seen from the capacitor's terminals, with sources zeroed

Answer: (D)

The time constant τ = R_th × C, where R_th is the Thevenin resistance seen from the capacitor's terminals with all independent sources zeroed (voltage sources replaced with short circuits, current sources with open circuits). Looking into the capacitor's terminals with the source zeroed: R1 connects from one terminal through the source (now a short) to ground, while R2 connects from one terminal to ground — so R1 and R2 are in parallel from the capacitor's perspective. R_th = R1 ‖ R2. Using only R1 or R2 misidentifies which resistors actually form the discharge/charge path seen by the storage element — the most common mistake in computing τ.

Question 2

A capacitor in a circuit holds 8V just before a switch opens at t = 0. The switch opening disconnects the voltage source. What is the capacitor voltage v(0⁺) immediately after the switch opens?

• A. 0V — the switch opening removes the source, so the capacitor immediately starts from zero
• B. 8V — capacitor voltage cannot change instantaneously, so it remains at its pre-switching value
• C. The supply voltage — the capacitor immediately charges to match the new circuit configuration
• D. Indeterminate — you need to solve the differential equation to find the initial value

Answer: (B)

Capacitor voltage cannot change instantaneously — this is the continuity condition. The capacitor stores energy in its electric field (E = ½CV²), and instantaneous change would require infinite power (P = C dV/dt → ∞ if dV is finite and dt → 0). Therefore v(0⁺) = v(0⁻) = 8V, the value just before switching. The initial condition for the transient response is always taken from the circuit state just before the switching event (t = 0⁻), not after. This is the second most common mistake after computing τ from the wrong resistance.

Question 3

The natural response of a first-order RC circuit (with no external source, only initial stored energy) is a special case of the general shortcut formula v(t) = v(∞) + [v(0⁺) − v(∞)]·e^(−t/τ), obtained by setting v(∞) = 0.

• T. True
• F. False

Answer: True

When there is no source, the circuit eventually reaches v(∞) = 0 (all stored energy dissipated into the resistor). Substituting into the shortcut formula: v(t) = 0 + [v(0⁺) − 0]·e^(−t/τ) = v(0⁺)·e^(−t/τ). This is exactly the natural response — a pure exponential decay from the initial stored energy. Similarly, the step response (capacitor initially uncharged, source applied) is the case v(0⁺) = 0: v(t) = v(∞)·(1 − e^(−t/τ)). Both are special cases of the same unified formula, which shows that the apparent distinction between them is superficial.

Question 4

To find the initial condition v(0⁺) for a first-order transient circuit, you should analyze the circuit in its new configuration — after the switch has changed — and compute the initial value from that.

• T. True
• F. False

Answer: False

Initial conditions must be taken from the circuit state just before switching (t = 0⁻), not after. Capacitor voltage and inductor current cannot change instantaneously (continuity conditions), so v_C(0⁺) = v_C(0⁻) and i_L(0⁺) = i_L(0⁻). The pre-switching circuit (which may be a completely different topology) gives you the value of the stored energy. Using the post-switching circuit to find initial conditions is the common error — it would give you the final (t → ∞) value, not the initial one, completely inverting the transient response.

Question 5

What three values do you need to apply the shortcut formula v(t) = v(∞) + [v(0⁺) − v(∞)]·e^(−t/τ) for a DC-forced first-order circuit? Explain how to find each one.

Model answer: The three values are: (1) v(0⁺): the initial condition, found by analyzing the circuit just before switching (t = 0⁻) and using continuity — capacitor voltage cannot jump, so v(0⁺) = v(0⁻). (2) v(∞): the final DC steady-state value, found by treating the capacitor as an open circuit (or inductor as a short circuit) at t → ∞ and solving the resulting resistive circuit. (3) τ: the time constant, found by computing the Thevenin resistance seen by the storage element with all independent sources zeroed — τ = R_th × C for RC, τ = L / R_th for RL.

The power of this shortcut formula is that it converts a differential equation problem into three straightforward circuit analysis steps. Once you have the three numbers, you substitute and the exponential trajectory is fully determined. The formula unifies all DC-forced first-order responses — natural, step, and mixed — into a single expression. The key errors are: (1) reading the initial condition from the wrong circuit instant, (2) computing τ from nominal component values rather than the Thevenin equivalent, and (3) finding the final value incorrectly by forgetting to treat C as open or L as short at DC steady state.