Node-voltage analysis: foundations

A branch-current approach could assign a separate unknown current to every element and then add enough Kirchhoff equations to relate them. Nodal analysis instead asks whether those currents can be written directly from a smaller set of node voltages.

For the circuit in Figure 1.1, the source fixes the input node at 12 V and the chosen reference node is 0 V. Only the output-node voltage remains unknown. Once that one number is known, Ohm's law determines all three resistor currents.

This is the central economy of the method: do not make a branch current an independent unknown if an element law already expresses it from node voltages.

Electric potential is meaningful through differences. We therefore choose one node and define its potential as zero. This is the reference node. Circuit diagrams often mark it with a ground symbol, but that symbol does not automatically mean the node is physically connected to Earth.

If V denotes the output-node potential relative to the reference node, then

Choosing a different reference changes the numerical labels assigned to individual node potentials, but it does not change any physical voltage difference, branch current or power. The reference is a coordinate choice.

Consider a resistor R connecting node a at voltage Va to node b at voltage Vb. If current is defined positive from a to b, the branch voltage in that direction is Va−Vb. Ohm's law gives

This single expression is the workhorse of resistive nodal analysis. It contains both magnitude and sign. If Va<Vb, the calculated current is negative, correctly indicating that the physical current flows from b to a.

For a branch from the unknown node V to ground through R2,

For a branch from the unknown node V toward a known 12 V node through R1, when we deliberately define current as leaving V,

Nothing special happens because the known node is “a source node.” It is simply a node whose voltage is already known.

Use one sign rule and keep it. In Figure 1.3 all three branch currents are defined as leaving the unknown node. Kirchhoff's current law states that the algebraic sum of signed currents at the node is zero:

Substitute the resistor relations directly:

This is the nodal equation. There is no separate current-division formula and no need to guess which current is “incoming.” The chosen references and algebra handle the signs.

Use the values in the figures: R1=3 kΩ, R2=6 kΩ, RL=6 kΩ and the known input node at 12 V. The nodal equation is

Multiply the entire equation by 6 kΩ:

Expand and collect the terms containing V:

Now recover the branch currents from the definitions used in Figure 1.3:

The negative sign on I1 means the real current flows opposite its chosen reference: 2 mA actually enters the unknown node from the 12 V side. KCL is then physically transparent: 2 mA enters, while 1 mA leaves through each lower branch.

DC-03 gives an independent route. The two 6 kΩ lower resistors are in parallel:

The circuit therefore becomes a 3 kΩ upper resistor in series with a 3 kΩ lower equivalent across 12 V. A symmetric voltage divider gives

The nodal result and the reduction result agree exactly. This cross-check matters: systematic analysis should reproduce the simpler shortcut whenever the shortcut is actually valid.

The real current through R1 is 2 mA from the 12 V node toward the 6 V node. The resistor drops 6 V, so

Each 6 kΩ lower resistor has 6 V across it and carries 1 mA:

Total resistor absorption is therefore 24 mW. The ideal 12 V source supplies 2 mA:

The total signed power is zero. A nodal voltage that satisfies KCL but produces an inconsistent power balance would be a strong signal to recheck signs or arithmetic.

Suppose one unknown node V is connected only through positive resistors to several neighboring nodes whose voltages V1,…,VN are already known. Writing all resistor currents as leaving V gives

Define conductance Gk=1/Rk. Rearranging gives

So this purely resistive node voltage is a conductance-weighted average of the known neighboring voltages. With all conductances positive, V must lie between the smallest and largest neighboring voltages.

For the worked circuit, the neighboring voltages are 12 V, 0 V and 0 V, with conductances 1/3 kΩ, 1/6 kΩ and 1/6 kΩ. The weighted average is 6 V.

Any node can be the mathematical reference, but some choices make the equations simpler. A useful reference often touches many branches, turns several branch voltages directly into node voltages, or coincides with the return node already emphasized by the circuit.

For an ordinary resistive network, after one reference is chosen, each remaining unknown node voltage usually contributes one KCL equation. A node whose voltage is fixed directly by an ideal voltage source to the reference does not need its own unknown.

A special case appears when an ideal voltage source lies directly between two unknown nonreference nodes. The source fixes their voltage difference but does not provide a resistor-like current expression. We do not invent one. Chapter 03 will introduce the supernode, which preserves KCL while adding the voltage-source constraint correctly.

1. A node V connects to 10 V through 2 kΩ and to ground through 3 kΩ. Write the nodal equation and find V.
2. A node V connects to 8 V through 4 kΩ, to 2 V through 2 kΩ and to ground through 4 kΩ. Find V and verify that it lies between the neighboring voltages.
3. Repeat the worked 12 V example but replace RL by 3 kΩ. Find V and all branch currents.
4. A student writes (12−V)/3k + V/6k + V/6k = 0 while claiming that all currents are defined as leaving V. Diagnose the sign inconsistency.
5. Why is choosing a different reference node allowed even though all numerical node-voltage labels change?
6. An ideal 5 V source is connected directly between two unknown nonreference nodes. Why can you not write its branch current as a voltage difference divided by resistance?

Chapter 02 will keep exactly the same logic but scale it to several unknown nodes, where the equations naturally organize themselves into a linear system and eventually a conductance matrix.