A Faster Way to Form Linear Circuit Equations — Extended Millman’s Theorem

Kryvochyi

July 09, 2025

Linear circuits are typically analyzed using nodal or loop analysis. However, forming these equations can be tedious and error-prone.
This post shows a quicker, more intuitive method based on an extended form of Millman’s Theorem.

Standard Nodal Analysis (Quick Recap)

Consider a node connected to others by resistors:

The standard nodal analysis steps are:

Apply Kirchhoff’s Current Law (KCL): sum currents entering and leaving the node:

$I_{1} - I_{2} \dots+ I_{N} = 0$
Substitute each current using Ohm’s Law:

$\frac{V_{R1}}{R_{1}} - \frac{V_{R2}}{R_{2}} \dots+ \frac{V_{RN}}{R_{N}} = 0$
Express currents in terms of node voltage differences:

$\frac{(V_{1} - V_{o})}{R_{1}} - \frac{(V_{o} - V_{2})}{R_{2}} \dots+ \frac{(V_{N} - V_{o})}{R_{N}} = 0$

Every node (except the ground reference) needs such an equation. Solving them involves basic linear algebra, but setting them up is repetitive.

The Voltage Averaging Insight

When you look at the nodal equations, the signs of the KCL current terms and Ohm’s Law potential differences BOTH depend on denoted current direction. Therefore, if you consistently express all currents with the same sign convention, you will find that the effects of denoted current directions nullify themselves.
If you rearrange the equations to isolate for the potentials of the respective KCL nodes, you find that each node’s voltage is actually a conductance-weighted average of its connected nodes’ voltages — this is Millman’s Theorem in disguise.
Rearranged:

${V_{o}}^{= \frac{V_{1} G_{1} + V_{2} G_{2} \dots+ V_{N} G_{N}}{G_{1} + G_{2} \dots+ G_{N}}}$

Extending Millman’s Theorem

Current Sources:

If a branch has a current source pointing into the node:

Its value simply adds to the numerator. The denominator stays the same:

${V_{o}}^{= \frac{V_{1} G_{1} + V_{2} G_{2} \dots+ V_{N} G_{N} + I_{x}}{G_{1} + G_{2} \dots+ G_{N}}}$

If the current source is facing away from the node, subtract it.
Also, it is interesting to note that this equation does not depend on the node at the other side of the current source.

Voltage Sources:

For a voltage source in series with a resistor:

If its positive terminal faces the node, add its value to the potential of the connected node at the source’s other side in the average. If it faces away, subtract it.

This shifts the connected node’s voltage accordingly:

${V_{o}}^{= \frac{V_{1} G_{1} + V_{2} G_{2} \dots+ V_{N} G_{N} + (V_{x} + V_{offset}) G_{x}}{G_{1} + G_{2} \dots+ G_{N} + G_{x}}}$

Capacitors and Inductors:

Use Laplace-domain admittances:

Capacitors: $C s$
Inductors: $\frac{1}{L s}$

This works for general transient analysis; in steady-state AC, simply set s = jω.

Forming the System Quickly

Instead of doing full KCL for each node, just write the extended Millman form for each.
Multiply both sides by the denominator (total conductance) to get the standard matrix form. No more tedious sign-checking or expansions.

Why Use This?

This extended form:

Speeds up forming nodal equations
Reduces algebraic mistakes
Works for resistive, reactive, and source elements

It’s a smart shortcut that frees you to focus on design, not tedious algebra.

That’s it! A faster way to form your circuit equations — ready for your next analysis.