AC Circuit Analysis — Impedance and Phasors

AC circuit analysis uses impedance and phasors to solve sinusoidal circuits at a fixed frequency. If the circuit is in sinusoidal steady state, you can replace resistors, inductors, and capacitors with complex impedances and solve for magnitude and phase with algebra instead of differential equations.

The key idea is simple: Ohm's law and Kirchhoff's laws still work, but now the quantities are complex so they can track phase shift as well as size.

What Impedance Means In AC Circuits

In DC, a resistor is often enough to describe opposition to current. In AC, capacitors and inductors also matter because they store and release energy during each cycle. That introduces phase shift, so a single real number is no longer enough.

Impedance handles both effects. For a sinusoidal signal at angular frequency $\\omega$,

$$Z_R = R$$ $$Z_L = j\\omega L$$ $$Z_C = \\frac{1}{j\\omega C} = -\\frac{j}{\\omega C}$$

Here $j^2 = -1$. The resistor has no phase shift in this ideal model, the inductor gives positive imaginary impedance, and the capacitor gives negative imaginary impedance.

If the frequency changes, the inductive and capacitive impedances change too. That is why AC circuit analysis always has to name the frequency.

What A Phasor Represents

A phasor is a compact way to represent a sinusoid with its magnitude and phase. Instead of carrying around the full time function at every step, you work with its complex amplitude.

For example, a source written as

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

can be represented by a phasor with magnitude and angle. The exact numerical magnitude depends on whether you choose peak or RMS form, but the phase relationships stay the same as long as you stay consistent.

That consistency rule matters. Do not mix peak values and RMS values in one calculation.

Why Impedance And Phasors Simplify The Math

Once everything is in phasor form, Ohm's law keeps the same structure:

$$\\tilde{V} = \\tilde{I} Z$$

Kirchhoff's voltage and current laws also keep the same logic. The difference is that voltages, currents, and impedances are now complex quantities, so you must add them with both real and imaginary parts.

This is why phasor analysis is useful in filters, power circuits, and basic RLC problems. It turns time-shifted sinusoids into quantities you can combine directly.

Worked Example: Series RC Circuit

Suppose a series RC circuit has

$$R = 100\\ \\Omega$$ $$C = 100\\ \\mu\\mathrm{F}$$ $$f = 50\\ \\mathrm{Hz}$$

and the source phasor is

\\tilde{V} = 10\\angle 0^\\circ\\ \\mathrm{V}

using RMS values. Find the current phasor.

Start with the angular frequency:

$$\\omega = 2\\pi f = 2\\pi(50) \\approx 314\\ \\mathrm{rad/s}$$

Then find the capacitor impedance:

$$Z_C = -\\frac{j}{\\omega C} = -\\frac{j}{314(100 \\times 10^{-6})} \\approx -j31.8\\ \\Omega$$

Now add the series impedances:

$$Z = R + Z_C = 100 - j31.8\\ \\Omega$$

Convert that result into magnitude and angle:

$$|Z| = \\sqrt\{100^2 + 31.8^2\} \\approx 104.9\\ \\Omega$$

and its phase is

\\angle Z = \\tan^{-1}\\left(\\frac{-31.8}{100}\\right) \\approx -17.7^\\circ

Now use phasor Ohm's law:

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

The current magnitude is

$$|\\tilde\{I\}| = \\frac\{10\}\{104.9\} \\approx 0.095\\ \\mathrm\{A\}$$

and the phase is

\\angle \\tilde{I} = 0^\\circ - (-17.7^\\circ) = 17.7^\\circ

Therefore,

\\tilde{I} \\approx 0.095\\angle 17.7^\\circ\\ \\mathrm{A}

The current leads the source voltage by about 17.7^\\circ. That matches the physical picture: in a circuit with a capacitive effect, current leads voltage.

Common Mistakes In AC Circuit Analysis

Using the method outside its condition

Phasor analysis is for sinusoidal steady state at a fixed frequency. If the circuit is switching, starting up, or driven by a non-sinusoidal waveform, this is not the whole story.

Adding reactances like ordinary positive resistances

Inductive and capacitive parts carry signs in the imaginary direction. If you drop the sign, you can predict the wrong phase and the wrong current.

Mixing peak and RMS values

Either convention can work, but one calculation must use one convention consistently.

Forgetting that impedance is complex

In AC analysis, you usually cannot add element magnitudes alone. You must add the complex impedances before taking a magnitude.

Where Impedance And Phasors Are Used

This approach shows up in power systems, filter design, resonance problems, signal processing, and electronics where sinusoidal response matters. Even when the final system is more complicated, impedance and phasors are often the first clean model that makes the behavior understandable.

Try A Similar Problem

Try your own version of the example by keeping $R$ fixed and doubling the frequency. Since $Z_C = -j/(\omega C)$, the capacitor's impedance magnitude becomes smaller, so the current magnitude increases and the phase angle moves closer to $0^\circ$.

If you want to explore another case with different values, solve a similar phasor problem with GPAI Solver.