1.4 Steady-state AC circuit analysis techniques

AC circuit analysis techniques let you break down complex networks into manageable pieces using systematic methods. Nodal analysis, mesh analysis, superposition, and equivalent circuits all extend directly from the DC methods you already know, but now every quantity is a complex phasor.

Circuit Analysis Techniques

Fundamental Laws and Node-Based Analysis

Kirchhoff's laws work the same way in AC steady-state analysis as in DC, except every voltage and current is now a phasor (complex number with magnitude and phase).

• Kirchhoff's Current Law (KCL): The sum of phasor currents entering any node equals the sum leaving it. In phasor form: $\sum \mathbf{I}_{in} = \sum \mathbf{I}_{out}$
• Kirchhoff's Voltage Law (KVL): The sum of phasor voltages around any closed loop equals zero: $\sum \mathbf{V} = 0$

Nodal analysis uses KCL to solve for unknown node voltages. Here's the process:

1. Choose a reference node (ground). Pick the node with the most connections.

2. Label the voltage at every other node as $\mathbf{V}_1, \mathbf{V}_2, \ldots$

3. Write a KCL equation at each non-reference node. Express branch currents in terms of node voltages and impedances: $\mathbf{I} = \frac{\mathbf{V}_a - \mathbf{V}_b}{\mathbf{Z}}$

4. If a voltage source connects two non-reference nodes, use a supernode: enclose both nodes and write KCL for the supernode boundary, plus the constraint equation from the source voltage.

5. Solve the resulting system of linear (complex) equations for the node voltages.

Nodal analysis is especially efficient when the circuit has many elements in parallel or when voltage sources are present (handled via supernodes).

Mesh and Superposition Techniques

Mesh analysis uses KVL and is the dual of nodal analysis. It works on planar circuits (circuits you can draw flat without crossing wires).

1. Identify each independent mesh (a loop that doesn't contain another loop inside it).

2. Assign a mesh current $\mathbf{I}_1, \mathbf{I}_2, \ldots$ to each mesh, all flowing in the same direction (typically clockwise).

3. Apply KVL around each mesh. For a shared impedance between meshes $m$ and $n$, the voltage drop is $\mathbf{Z}(\mathbf{I}_m - \mathbf{I}_n)$.

4. If a current source sits in only one mesh, that mesh current is known directly. If a current source is shared between two meshes, form a supermesh: write KVL around the combined loop and add the constraint from the current source.

5. Solve the system of complex equations for the mesh currents.

Mesh analysis tends to be more convenient when the circuit is series-heavy or has current sources.

Superposition applies to any linear circuit with multiple independent sources:

1. Turn on one source at a time. Deactivate the others: replace voltage sources with short circuits ($\mathbf{Z} = 0$) and current sources with open circuits ($\mathbf{Z} = \infty$).
2. Solve for the desired phasor voltage or current due to that single source.
3. Repeat for every independent source.
4. Add all individual phasor contributions to get the total response.

A key detail: you must add the phasor quantities (magnitude and phase), not just magnitudes. If two sources operate at different frequencies, you cannot add their phasors directly. Instead, solve each frequency separately and combine the results in the time domain.

Equivalent Circuits

Thévenin and Norton Equivalents

These theorems let you replace everything except your load with a simple two-terminal equivalent. In AC circuits, resistances become impedances.

Thévenin equivalent: a phasor voltage source $\mathbf{V}_{th}$ in series with an impedance $\mathbf{Z}_{th}$.

1. Remove the load from terminals A-B.
2. Find the open-circuit voltage $\mathbf{V}_{th} = \mathbf{V}_{AB,\text{open}}$ using any analysis method.
3. Find $\mathbf{Z}_{th}$: deactivate all independent sources and calculate the impedance looking into terminals A-B. If dependent sources exist, apply a test source instead.

Norton equivalent: a phasor current source $\mathbf{I}_N$ in parallel with $\mathbf{Z}_N$.

1. $\mathbf{I}_N$ is the short-circuit current through terminals A-B.
2. $\mathbf{Z}_N = \mathbf{Z}_{th}$ (they're always equal).
3. The two equivalents are related by source transformation: $\mathbf{V}_{th} = \mathbf{I}_N \cdot \mathbf{Z}_{th}$

You can freely convert between Thévenin and Norton forms. Use whichever makes the next step of your analysis easier.

Power Transfer and Circuit Optimization

The maximum power transfer theorem for AC circuits states that maximum average power is delivered to the load when:

$\mathbf{Z}_{load} = \mathbf{Z}_{th}^*$

where $\mathbf{Z}_{th}^*$ is the complex conjugate of the Thévenin impedance. If $\mathbf{Z}_{th} = R_{th} + jX_{th}$, then the optimal load is $\mathbf{Z}_{load} = R_{th} - jX_{th}$.

This is called conjugate matching. The reactive parts cancel, and the resistive parts are equal, so maximum power flows to the load.

• Maximum power transfer does not mean maximum efficiency. At the conjugate match point, efficiency is only 50% because the source impedance dissipates the same power as the load.
• Conjugate matching matters most in RF systems, antenna design, and audio amplifiers where extracting maximum power from a source is the priority.
• In power distribution (utility grids), efficiency matters more, so you do not design for maximum power transfer.

AC Circuit Parameters

Impedance and Admittance

Impedance $\mathbf{Z}$ is the AC generalization of resistance. It captures both energy dissipation (resistance) and energy storage (reactance) in a single complex number:

$\mathbf{Z} = R + jX$

• $R$ is the real part (resistance, in ohms)
• $X$ is the imaginary part (reactance, in ohms)
• The magnitude $|\mathbf{Z}| = \sqrt{R^2 + X^2}$ tells you the overall opposition to current
• The phase angle $\theta = \arctan(X/R)$ tells you how much the current leads or lags the voltage

Admittance $\mathbf{Y}$ is the reciprocal of impedance:

$\mathbf{Y} = \frac{1}{\mathbf{Z}} = G + jB$

• $G$ is conductance (real part, in siemens)
• $B$ is susceptance (imaginary part, in siemens)

Admittance is particularly useful for parallel circuits. When elements are in parallel, their admittances simply add: $\mathbf{Y}_{total} = \mathbf{Y}_1 + \mathbf{Y}_2 + \cdots$, just like conductances add in DC parallel circuits.

Reactance and Susceptance

Reactance $X$ quantifies opposition to current from energy-storing elements:

• Inductive reactance: $X_L = \omega L$, where $\omega = 2\pi f$. It increases with frequency because an inductor resists rapid changes in current. Voltage leads current by 90°.
• Capacitive reactance: $X_C = \frac{1}{\omega C}$. It decreases with frequency because a capacitor passes high-frequency signals more easily. Voltage lags current by 90°.

The net reactance in a series RLC circuit is $X = X_L - X_C = \omega L - \frac{1}{\omega C}$. When $X_L = X_C$, the circuit is at resonance and behaves as a pure resistance.

Susceptance $B$ is the imaginary part of admittance:

• Capacitive susceptance: $B_C = \omega C$ (positive)
• Inductive susceptance: $B_L = -\frac{1}{\omega L}$ (negative)

Note the sign convention: in the $\mathbf{Z} = R + jX$ framework, inductive reactance is positive and capacitive reactance is negative. In the $\mathbf{Y} = G + jB$ framework, the signs flip: capacitive susceptance is positive and inductive susceptance is negative.