AC Circuit Analysis: Sinusoidal Steady State

The vast majority of the electrical world runs on alternating current (AC), from the power grid that charges your devices to the radio signals that connect them. To analyze these systems, we cannot use simple DC laws because capacitors and inductors behave fundamentally differently when voltages and currents constantly change. Sinusoidal steady-state analysis provides the powerful mathematical framework to solve these AC circuits by transforming complex differential equations into manageable algebra, unlocking the ability to design and troubleshoot everything from audio amplifiers to power distribution networks.

The Foundation: Sinusoids and the Steady State

A sinusoidal function, like a sine or cosine wave, is defined by its amplitude, frequency, and phase. In AC circuit analysis, we assume our independent sources (voltage or current) are perfect sinusoids at a single frequency, often denoted as $f$ (Hz) or its angular equivalent $\omega =2\pi f$ (radians/second). When such a source is applied to a linear circuit—one composed of resistors, capacitors, and inductors—every voltage and current in the circuit will eventually become sinusoidal at that same frequency after any initial transient behavior decays.

This final condition is the sinusoidal steady state. It is the circuit's "particular solution," the sustained response to the driving source. In this state, the only differences between signals at various points in the circuit are their amplitudes and their phases (i.e., their timing shift relative to the source). The entire challenge of AC analysis reduces to finding these amplitudes and phases. Solving the circuit's differential equations directly for the steady state is laborious. This is precisely the problem the phasor transform solves.

The Transform: Phasor Representation

A phasor is a complex number that represents the amplitude and phase of a sinusoid. It is not the signal itself, but a simplified, frequency-domain representation of it. The transformation from a time-domain sinusoid to a phasor is defined as follows:

Given a sinusoidal voltage: $v(t)=V_m\cos (\omega t+\varphi )$

Its phasor representation is: $\mathbf{V}=V_m\angle \varphi$ or, in rectangular form, $\mathbf{V}=V_m(\cos \varphi +j\sin \varphi )=V_m{e}^{j\varphi }$.

The critical insight is that the frequency $\omega$ is implicit and the same for all phasors in a given circuit. This transform converts the calculus operations required for inductors and capacitors into simple algebra with the imaginary unit $j$ (where $j^2=-1$). The derivative $d/dt$ in the time domain becomes multiplication by $j\omega$ in the phasor domain. Integration becomes division by $j\omega$. This is the engine that simplifies analysis.

The Tool: Complex Impedance and Admittance

Impedance, denoted by $\mathbf{Z}$, is the phasor-domain extension of resistance. It is a complex number that generalizes the opposition a circuit element presents to sinusoidal current. Its real part is resistance, and its imaginary part is reactance.

The impedance of the three passive elements is derived from their voltage-current relationships in the phasor domain:

• Resistor: ${\mathbf{Z}}_R=R$ (purely real)
• Inductor: ${\mathbf{Z}}_L=j\omega L=j{X}_L$ (purely imaginary, positive)
• Capacitor: ${\mathbf{Z}}_C=\frac{1}{j\omega C}=-j\frac{1}{\omega C}=-j{X}_C$ (purely imaginary, negative)

Here, $X_L$ and $X_C$ are inductive reactance and capacitive reactance, respectively. The reciprocal of impedance is admittance, $\mathbf{Y}=1/\mathbf{Z}$, measured in siemens (S).

With impedance defined, all the techniques you learned for DC circuit analysis become applicable in the phasor domain. You can use series/parallel combinations, voltage and current division, nodal analysis, mesh analysis, Thévenin's theorem, and Norton's theorem—with the crucial difference that you are now performing arithmetic with complex numbers.

Applying the Method: A Worked Example

Let's analyze a simple series RLC circuit to see the complete process. A voltage source $v_s(t)=10\cos (1000t)$ V is connected in series with a $5\Omega$ resistor, a $20mH$ inductor, and a $100\mu F$ capacitor.

Step 1: Convert to Phasor Domain.

• Source: $v_s(t)=10\cos (1000t+0^{\circ })$ V $\to$ ${\mathbf{V}}_s=10\angle 0^{\circ }$ V.
• Angular frequency: $\omega =1000$ rad/s.
• Impedances:
• ${\mathbf{Z}}_R=5\Omega$
• ${\mathbf{Z}}_L=j\omega L=j(1000)(0.020)=j20\Omega$
• ${\mathbf{Z}}_C=\frac{1}{j\omega C}=\frac{1}{j(1000)(100\times 10^{-6})}=\frac{1}{j0.1}=-j10\Omega$

Step 2: Solve the Phasor Circuit. The total series impedance is: ${\mathbf{Z}}_{total}={\mathbf{Z}}_R+{\mathbf{Z}}_L+{\mathbf{Z}}_C=5+j20-j10=5+j10\Omega$. In polar form: ${\mathbf{Z}}_{total}=\sqrt{5^2+10^2}\angle \arctan (10/5)\approx 11.18\angle 63.43^{\circ }\Omega$.

Using Ohm's Law in phasor form: $\mathbf{I}=\frac{{\mathbf{V}}_s}{{\mathbf{Z}}_{total}}=\frac{10\angle 0^{\circ }}{11.18\angle 63.43^{\circ }}\approx 0.894\angle -63.43^{\circ }$ A.

Step 3: Convert Back to Time Domain. The current as a function of time is: $i(t)=0.894\cos (1000t-63.43^{\circ })$ A.

We could now find the voltage phasor across any element (e.g., ${\mathbf{V}}_C=\mathbf{I}\cdot {\mathbf{Z}}_C$) and convert it back to $v_C(t)$.

Common Pitfalls

1. Mixing Time-Domain and Phasor-Domain Quantities: You cannot add a time-domain voltage $v(t)$ to a phasor voltage $\mathbf{V}$. You must perform all calculations entirely within one domain before converting the final answer back. Always explicitly draw the phasor-domain equivalent circuit with impedances and phasor sources before beginning analysis.

1. Inconsistent Phase Reference: The phase angle in a phasor is relative. It is critical to define a reference sinusoid (typically a source with $\varphi =0$). All other calculated phases are relative to this reference. Forgetting this can lead to incorrect interpretations of whether a current leads or lags a voltage.

1. Misinterpreting Impedance Magnitude: The magnitude of an impedance $|\mathbf{Z}|$ is not simply the sum of $R$ and $X$. It is the Euclidean distance in the complex plane: $|\mathbf{Z}|=\sqrt{R^2+X^2}$. Adding a $5\Omega$ resistor to a $j5\Omega$ inductor gives an impedance magnitude of $5\sqrt{2}\approx 7.07\Omega$, not $10\Omega$.

1. Handling the $j$ Operator Incorrectly in Algebra: Remember that $1/j=-j$. A common error is to leave $j$ in the denominator of a complex number. Always express impedance and other complex quantities in standard rectangular ($a+jb$) or polar ($A\angle \theta$) form for clarity and to avoid mistakes in addition/multiplication.

Summary

• Sinusoidal steady-state analysis finds the sustained response of a linear circuit to a sinusoidal source, where all voltages and currents are sinusoids at the source frequency, differing only in amplitude and phase.
• The phasor transform converts a sinusoid from the time domain to a complex-number representation ($\mathbf{V}=V_m\angle \varphi$), implicitly containing the frequency. This transforms circuit differential equations into algebraic equations.
• Impedance $\mathbf{Z}$ generalizes resistance to the AC domain, allowing the use of all DC circuit analysis techniques (Ohm's Law, series/parallel, network theorems) with complex numbers.
• The impedance of an inductor is $j\omega L$ and of a capacitor is $1/(j\omega C)$. The imaginary component represents energy storage and causes phase shifts between voltage and current.
• The complete analysis workflow is: 1) Convert sources and elements to phasors/impedances, 2) Solve the algebraic phasor circuit, 3) Convert the resulting phasors back to time-domain sinusoidal functions.