Questions: Phasors and Sinusoidal Steady-State Analysis

Question 1

A sinusoidal source has been driving an RC circuit for a very long time. A student wants to find the current amplitude and phase. Which approach is appropriate?

• A. Solve the full differential equation including both transient and steady-state terms
• B. Phasor analysis — the transient has decayed and only the sinusoidal steady-state response remains
• C. Phasor analysis combined with Laplace transforms to capture the complete response
• D. DC analysis using the source's peak voltage as a constant

Answer: (B)

After a circuit has run for a long time, transients (the natural response driven by initial energy storage) have decayed to zero. What remains is exactly the sinusoidal steady-state response — which phasors are designed to compute. There is no need for a full differential equation or Laplace analysis. Option A is correct in general but unnecessary here. Options C and D are incorrect: Laplace is needed for the complete response including transients; DC analysis ignores reactance entirely.

Question 2

Why does phasor analysis convert differential equations into algebraic equations?

• A. Phasors average over time, so the time derivative vanishes
• B. In steady state, voltages and currents are constant, so their derivatives are zero
• C. Differentiation in the time domain corresponds to multiplication by jω in the phasor domain
• D. Complex numbers encode phase information, eliminating the need to solve for initial conditions

Answer: (C)

The algebraic key is that d/dt[Re{V·e^(jωt)}] = Re{jω·V·e^(jωt)}. Differentiation in time maps to multiplication by jω on the phasor. This means the voltage-current relationship for a capacitor (i = C·dv/dt) becomes I = jωC·V in the phasor domain — an algebraic equation. Similarly, inductors become V = jωL·I. Every reactive element gets an impedance Z = V/I, and the entire DC analysis toolkit (Kirchhoff's laws, superposition, Thevenin) applies directly. Options A and B are wrong: phasors don't average over time, and in sinusoidal steady state, voltages are not constant — they oscillate.

Question 3

Phasor analysis gives the complete response of a circuit, capturing both the transient behavior immediately after switching and the long-term steady-state behavior.

• T. True
• F. False

Answer: False

Phasors yield only the sinusoidal steady-state (particular) solution — the response after all transients have decayed. The complete response is the sum of the particular solution (phasor) and the homogeneous solution (natural response, which decays over time). Immediately after a source is switched on, energy stored in capacitors and inductors drives transient currents that are not captured by phasors. Using a phasor solution to describe circuit behavior right after switching is a consequential error.

Question 4

A capacitor with impedance 1/(jωC) presents lower opposition to current at higher frequencies, behaving more like a short circuit as frequency increases.

• T. True
• F. False

Answer: True

The magnitude of the capacitor's impedance is |Z_C| = 1/(ωC). As frequency ω increases, this magnitude decreases toward zero — a short circuit. Intuitively, a capacitor blocks DC (ω = 0, infinite impedance) but passes high-frequency signals easily. This frequency-dependent behavior is why capacitors are used in filters: they block low frequencies and pass high ones. The dual behavior holds for inductors: Z_L = jωL increases with frequency, so inductors short low frequencies and block high ones.

Question 5

A student analyzes an RC circuit with a phasor method immediately after a switch is closed at t = 0. What is wrong with this approach, and under what conditions would phasor analysis give correct results?

Model answer: Phasors describe only the sinusoidal steady-state response — the behavior after all transients have decayed. Immediately after the switch closes, the capacitor has an initial voltage (or zero charge) that drives a transient current governed by the circuit's time constant τ = RC. This transient is the homogeneous solution to the circuit's differential equation and is not captured by phasors. Phasor analysis gives correct results only after t >> τ, when the transient has decayed to negligible amplitude and the circuit's response is dominated by the forced sinusoidal response.

The complete response is: v(t) = v_transient(t) + v_steady-state(t). Phasors compute only the second term. For many engineering applications — power systems at 60 Hz operating in steady state, audio circuits processing continuous signals — the transient is brief and phasors are sufficient. But for circuits that switch on and off repeatedly, or for precise timing applications, the transient response must be computed separately.