Circuit Analysis: AC Circuits

Alternating-current (AC) circuit analysis in steady state is built on a simple observation: many electrical systems operate with voltages and currents that are sinusoidal at a fixed frequency. When the waveform is a sine (or cosine), the circuit’s response is also sinusoidal at the same frequency, differing only in amplitude and phase. This allows a powerful shortcut: represent sinusoids as complex numbers (phasors) and replace capacitors and inductors with frequency-dependent impedances. The result is algebra instead of differential equations, plus a clear framework for real and reactive power using the power triangle.

Steady-State Sinusoidal Signals and Phasors

A typical sinusoidal voltage can be written as:

$v(t)=V_m\cos (\omega t+\theta )$

where $V_m$ is peak amplitude, $\omega =2\pi f$ is angular frequency, and $\theta$ is phase angle. In AC power and circuit calculations, root-mean-square (RMS) values are commonly used because they relate directly to heating and average power in resistive loads:

$V_{\text{rms}}=\frac{V_m}{\sqrt{2}}$ and $I_{\text{rms}}=\frac{I_m}{\sqrt{2}}$

A phasor represents the magnitude and phase of a sinusoid while omitting the time-dependent part. Using RMS magnitude, the phasor form is:

$\underline{V}=V_{\text{rms}}\angle \theta$

Similarly, $\underline{I}=I_{\text{rms}}\angle \varphi$. Once signals are in phasor form, circuit relationships use complex arithmetic at a single frequency.

Impedance: The AC Version of Resistance

In AC analysis, the ratio of phasor voltage to phasor current defines impedance:

$\underline{Z}=\frac{\underline{V}}{\underline{I}}$

Impedance generalizes resistance to include phase shift. For the three basic elements:

Resistor

A resistor has no phase shift between voltage and current:

${\underline{Z}}_R=R$

Voltage and current are in phase, so the power factor is 1.

Inductor

An inductor opposes changes in current. In sinusoidal steady state:

${\underline{Z}}_L=j\omega L$

Current lags voltage by $90^{\circ }$ in a purely inductive element.

Capacitor

A capacitor opposes changes in voltage. In sinusoidal steady state:

${\underline{Z}}_C=\frac{1}{j\omega C}=-\frac{j}{\omega C}$

Current leads voltage by $90^{\circ }$ in a purely capacitive element.

These impedances depend on frequency, which is why AC circuits can behave very differently at 50/60 Hz compared with audio or radio frequencies.

Applying Ohm’s Law, KCL, and KVL with Phasors

Once every element is expressed as an impedance, familiar DC methods carry over:

• Ohm’s law becomes $\underline{V}=\underline{I}\underline{Z}$
• Kirchhoff’s Current Law (KCL) applies to phasor currents
• Kirchhoff’s Voltage Law (KVL) applies to phasor voltages

Series and parallel combinations work the same way, using complex values:

• Series: ${\underline{Z}}_{eq}={\underline{Z}}_1+{\underline{Z}}_2+\cdots$
• Parallel: $\frac{1}{{\underline{Z}}_{eq}}=\frac{1}{{\underline{Z}}_1}+\frac{1}{{\underline{Z}}_2}+\cdots$

A practical interpretation is that the real part of impedance dissipates power (resistive behavior), while the imaginary part stores and releases energy each cycle (reactive behavior).

Power in AC Circuits: Real, Reactive, and Apparent

Power analysis is where AC circuit concepts become especially useful. In DC, power is simply $P=VI$. In AC, if voltage and current are out of phase, not all the apparent power becomes useful work.

Complex Power

Using RMS phasors, complex power is defined as:

$$\underline{S}=P+jQ=\underline{V}{\underline{I}}^{\ast }$$

where ${}^{\ast }$ denotes complex conjugate, $P$ is real (average) power in watts, and $Q$ is reactive power in vars. Apparent power magnitude is:

$|\underline{S}|=S=V_{\text{rms}}{I}_{\text{rms}}$ in volt-amperes (VA)

Power Factor

Power factor (PF) quantifies how effectively current is converted into real power:

$\text{PF}=\cos \phi =\frac{P}{S}$

where $\phi$ is the phase angle between voltage and current. A lagging power factor indicates inductive behavior (current lags). A leading power factor indicates capacitive behavior (current leads).

Low power factor matters in real systems because it increases current for a given real power, raising conductor losses ($I^{2}R$), voltage drop, and equipment loading.

The Power Triangle

The relationship among $P$, $Q$, and $S$ is often visualized with the power triangle, which follows:

$$S^2=P^2+Q^2$$

This geometry provides quick insight: reducing $|Q|$ improves power factor and lowers required apparent power for the same real power.

Resonance in RLC Circuits

Resonance occurs when inductive and capacitive reactances cancel. In many AC circuits, this creates a strong frequency-selective behavior.

Series Resonance

For a series RLC circuit, the impedance is:

$\underline{Z}=R+j\left(\omega L-\frac{1}{\omega C}\right)$

Resonance happens when the imaginary part is zero:

${\omega }_{0}L=\frac{1}{{\omega }_{0}C}$

so:

$${\omega }_0=\frac{1}{\sqrt{LC}},f_0=\frac{1}{2\pi \sqrt{LC}}$$

At $f_0$, the impedance is minimum (approximately $R$), and the current can become large compared with off-resonant conditions. Although the source voltage may be moderate, the inductor and capacitor voltages can be much larger in magnitude due to energy exchange between $L$ and $C$. This is a real design concern in filters and power electronics.

Parallel Resonance

In a parallel RLC configuration, resonance corresponds to a condition where the net susceptance is zero, often yielding a maximum impedance at resonance. Practically, a parallel resonant circuit can “reject” current at a specific frequency, making it useful in tuning and frequency selection.

Practical Workflow for AC Circuit Analysis

A reliable approach for steady-state AC problems looks like this:

1. Convert sources to phasor form using RMS values and reference angles.
2. Replace $R$, $L$, and $C$ with ${\underline{Z}}_R$, ${\underline{Z}}_L$, and ${\underline{Z}}_C$ at the given frequency.
3. Solve the circuit using KCL/KVL, nodal analysis, mesh analysis, or Thevenin/Norton equivalents, all with complex numbers.
4. Convert phasor results back to time domain if needed:

$i(t)=I_m\cos (\omega t+\varphi )$, with $I_m=\sqrt{2}{I}_{\text{rms}}$

1. Compute power using complex power:

$\underline{S}=\underline{V}{\underline{I}}^{\ast }$ and interpret $P$, $Q$, $S$, and PF.

Why These Concepts Matter

Phasors and impedance are not just mathematical convenience. They explain everyday engineering tradeoffs. Inductive motors draw lagging reactive power and often require power factor correction. Capacitive elements can compensate, improving system efficiency. Resonance can be harnessed for filters and tuning, or it can unintentionally amplify currents and voltages. And the power triangle provides a clean language for sizing equipment and diagnosing why a system draws more current than expected.

Steady-state sinusoidal analysis is the backbone of much of power engineering and analog circuit design. Once you are fluent with phasors, impedance, power factor, and resonance, AC circuits become predictable, calculable, and far less mysterious.