Week 5 — Sinusoidal Steady State: Phasors, Impedance, and AC Power

Overview

Once the transient of Week 4 dies away, a circuit driven by a sinusoid settles into a sinusoidal steady state: every voltage and current is a sinusoid at the same frequency, differing only in amplitude and phase. The phasor method is the great simplification that exploits this — it replaces calculus (the differential \(i\)–\(v\) relations) with algebra (complex impedance), turning a differential equation into Ohm’s law for complex numbers. This is arguably the single most important computational tool in analog electronics, and it works because of linearity (Week 2’s superposition) and the special property that the derivative of a complex exponential is just a scaled complex exponential.

On the bench, this is the week you start thinking in amplitude and phase. You will drive RC and RLC circuits with a sine source and use the scope’s two channels to measure both the amplitude ratio and the phase shift between input and output, confirming the complex impedances \(Z_C=1/j\omega C\) and \(Z_L=j\omega L\) are real, measurable things. The AC power half — real, reactive, and apparent power, power factor, and RMS — is where you learn why your true-RMS Fluke matters and why reactive elements store-and-return energy without dissipating it.

This week rests on Week 3’s element relations and Week 4’s notion of steady state, and it is the immediate foundation for Week 6 (frequency response is impedance as a function of \(\omega\)) and Week 8 (phasors are the single-frequency case of the Fourier transform).

Readings

CAD Ch. 7: Sinusoidal signals, the phasor transform, impedance and admittance of R/L/C, and AC nodal/mesh analysis. Extract: the phasor procedure (time → phasor → solve algebraically → phasor → time) and the impedance of each element.
CAD Ch. 8: Instantaneous and average power, complex power \(S=P+jQ\), power factor, and RMS values. Extract: why reactive power does no net work, and the power triangle.
O-S&S Ch. 3 (intro / sinusoids & complex exponentials): The signals-and-systems framing of \(e^{j\omega t}\) as an eigenfunction of LTI systems. Extract: the deep reason phasors work — complex exponentials are eigenfunctions, and impedance is the eigenvalue.

Key Concepts

The phasor transform

A sinusoid \(v(t)=V_m\cos(\omega t+\phi)\) is represented by the complex phasor \(\mathbf{V}=V_m e^{j\phi}=V_m\angle\phi\), with the convention \(v(t)=\mathrm{Re}\{\mathbf{V}e^{j\omega t}\}\). The magic: differentiation in time becomes multiplication by \(j\omega\) in phasor land,

\[ \frac{d}{dt}\ \longleftrightarrow\ j\omega. \]

So the capacitor relation \(i=C\,dv/dt\) becomes \(\mathbf{I}=j\omega C\,\mathbf{V}\), and the inductor relation \(v=L\,di/dt\) becomes \(\mathbf{V}=j\omega L\,\mathbf{I}\). Differential equations collapse into algebraic ones. All this is valid only in steady state at a single frequency \(\omega\).

Why it works: eigenfunctions

The complex exponential \(e^{j\omega t}\) is an eigenfunction of any LTI system: feed it in, and the output is the same exponential scaled by a (complex) constant — the system’s frequency response at \(\omega\). Impedance is exactly that eigenvalue for a single element. This is the conceptual bridge to Week 8: the Fourier transform decomposes any signal into these eigenfunctions, and the system acts on each independently. Phasor analysis is the \(n=1\) case.

Impedance and admittance

Impedance \(Z=\mathbf{V}/\mathbf{I}\) (units Ω) generalizes resistance to complex, frequency-dependent form:

\[ Z_R=R,\qquad Z_L=j\omega L,\qquad Z_C=\frac{1}{j\omega C}=\frac{-j}{\omega C}. \]

Write \(Z=R+jX\): \(R\) is resistance (dissipative), \(X\) is reactance (energy-storing). Inductors have positive reactance (\(X_L=\omega L\), voltage leads current by 90°); capacitors have negative reactance (\(X_C=-1/\omega C\), current leads voltage by 90°). The mnemonic ELI the ICE man: in an inductor (L) voltage E leads current I; in a capacitor (C) current I leads voltage E. Impedances combine with the same series/parallel rules as resistors — that is the entire payoff, because all of Week 2’s machinery (dividers, nodal, Thévenin) now works unchanged with complex numbers.

AC power: real, reactive, apparent

For \(v(t)=V_m\cos(\omega t)\) and \(i(t)=I_m\cos(\omega t-\theta)\), the average (real) power is

\[ P=\tfrac12 V_m I_m\cos\theta = V_\text{rms}I_\text{rms}\cos\theta, \]

where \(\cos\theta\) is the power factor and \(\theta\) is the impedance angle. Complex power:

\[ S=\mathbf{V}_\text{rms}\mathbf{I}_\text{rms}^{*}=P+jQ,\quad P=|S|\cos\theta\ \text{(W, real)},\quad Q=|S|\sin\theta\ \text{(VAR, reactive)}. \]

Reactive power \(Q\) flows back and forth into reactive elements each cycle but does no net work — a purely reactive element (ideal L or C) dissipates zero average power. Apparent power \(|S|=V_\text{rms}I_\text{rms}\) (VA) is what the wiring must carry regardless. This is why power utilities care about power factor: low PF means large current for little real power.

RMS and why true-RMS matters

The RMS value \(V_\text{rms}=\sqrt{\frac1T\int_0^T v^2\,dt}\) is the DC-equivalent heating value; for a sinusoid \(V_\text{rms}=V_m/\sqrt2\). The Fluke 117 is true-RMS, meaning it computes this integral and reads correctly for non-sinusoidal waveforms; cheaper “average-responding” meters assume a sine and misread distorted signals. You’ll exploit this when measuring non-sinusoidal sources later.

Theory Exercises

Show that \(\frac{d}{dt}\{V_m\cos(\omega t+\phi)\}\) corresponds to multiplying the phasor by \(j\omega\), and derive \(Z_C=1/j\omega C\) and \(Z_L=j\omega L\) from the element relations.
Prove \(e^{j\omega t}\) is an eigenfunction of an LTI system and identify impedance as the eigenvalue for a single element.
For a series RC driven at frequency \(\omega\), derive the magnitude and phase of \(Z\), and the phase of the output across \(C\) relative to the input. Evaluate at \(\omega=1/RC\).
Derive \(P=\tfrac12 V_m I_m\cos\theta\) from the time average of \(v(t)i(t)\), and show an ideal capacitor dissipates zero average power.
Define complex power \(S=P+jQ\) and draw the power triangle for an RL load. Compute the power factor for a load with \(Z=R+j\omega L\).
Show \(V_\text{rms}=V_m/\sqrt2\) for a sinusoid, and explain why a true-RMS meter is required for a square or triangle wave.

Lab / Bench Work

Impedance magnitude & phase: Drive a series RC (and separately RL) with a sine at several frequencies spanning \(\omega=1/RC\). On the Siglent, use CH1 for the input and CH2 for the output (across one element). Measure the amplitude ratio and the time delay between zero-crossings; convert delay to phase \(\phi=2\pi f\,\Delta t\). Confirm magnitude and phase match \(|Z|\) and \(\angle Z\).

Verify 90° relations: Show experimentally that across a capacitor the current (sensed via a small series resistor) leads the voltage by ~90°, and across an inductor it lags — confirming ELI/ICE.

AC power & power factor: For an RL or RC load, measure \(V_\text{rms}\) and \(I_\text{rms}\) (true-RMS Fluke + sense resistor) and the phase angle on the scope; compute \(P\), \(Q\), \(|S|\), and power factor. Confirm the reactive element’s average power is ~0.

Qucs-S (ngspice): Build the circuit and run an .ac analysis, confirming the simulated magnitude/phase at your test frequencies match the scope measurements.

Measurement Methodology

Phase from the scope: measure \(\Delta t\) between the same feature (e.g. rising zero-crossing) on the two channels; \(\phi^\circ = 360\,f\,\Delta t\). Ensure both probes share a ground reference and are compensated. The scope’s built-in phase measurement is convenient but verify it against the cursor method once.
Common-ground caution: both scope channel grounds are tied together and to mains earth. Choose the measurement node so you are not accidentally shorting a node to ground through the probe; rearrange the circuit if needed (put the grounded element at the bottom).
Frequency accuracy: confirm the actual source frequency (scope frequency measurement) rather than trusting the dial.
Reconcile: measured \(|Z|\), \(\angle Z\), \(P\), PF vs hand-calculated phasor values vs Qucs-S .ac. Residuals come from component tolerance, ESR (Week 3), and source impedance.

Expected baselines: At \(\omega=1/RC\) the RC output magnitude is \(1/\sqrt2\) (−3 dB) of input with 45° phase — a result you’ll formalize next week. Phase measurements within a few degrees. Reactive-element average power within measurement noise of zero.

Connections

Impedance as a function of \(\omega\) is precisely the subject of Week 6 (frequency response and filters) — next week we sweep \(\omega\) and plot \(|H(j\omega)|\) and \(\angle H\). The eigenfunction view of \(e^{j\omega t}\) is the seed of Week 8’s Fourier analysis: a general signal is a superposition of these, and the system’s response is found frequency-by-frequency. The −3 dB / 45° result at \(\omega=1/RC\) links straight back to Week 4’s time constant (\(\omega_c=1/\tau\)). RMS and power-factor reasoning carry into any real power-delivery or signal-amplitude budgeting.