5.1 Nodal analysis

Node Analysis Fundamentals

Nodal analysis is a systematic technique for solving circuits by finding the voltage at every node. Instead of chasing individual branch currents, you write equations based on Kirchhoff's Current Law (KCL) at each node, then solve for the unknown voltages. Once you have the node voltages, finding any current or power in the circuit becomes straightforward with Ohm's Law.

This method scales well to complex circuits with many components, which is why it's one of the most widely used analysis techniques in electrical engineering.

Nodes and the Reference Node

A node is any point in a circuit where two or more elements connect. In a circuit diagram, nodes are often drawn as dots or junctions where wires meet.

Before you start writing equations, you need to pick one node as the reference node (also called the ground node). This node is assigned a voltage of 0 V, and every other node voltage is measured relative to it.

• You can choose any node as the reference, but a good rule of thumb is to pick the node with the most connections. This tends to simplify your equations.
• The reference node is marked with the standard ground symbol in circuit diagrams.

Node Voltage and Branch Current

Node voltage is the electric potential at a given node, measured with respect to the reference node. If you know the node voltages at two nodes, the voltage across any element connecting them is simply the difference between those two node voltages.

For example, if $V_1 = 8 \text{ V}$ and $V_2 = 3 \text{ V}$, the voltage across a resistor connected between nodes 1 and 2 is $V_1 - V_2 = 5 \text{ V}$.

Branch current is the current flowing through a specific element. You find it by applying Ohm's Law once you know the voltage across that element:

$I = \frac{V_1 - V_2}{R}$

The direction of the current depends on which reference direction you choose. Just stay consistent throughout your analysis.

Kirchhoff's Current Law

KCL Statement

Kirchhoff's Current Law (KCL) states that the algebraic sum of all currents entering and leaving a node equals zero:

$\sum_{k=1}^{n} I_k = 0$

This comes directly from conservation of charge: charge can't pile up at a node, so everything flowing in must flow out.

To apply KCL at a node:

1. Pick a sign convention. A common choice is to treat currents leaving the node as positive and currents entering as negative (or vice versa). Either works as long as you're consistent.
2. Write an expression for each branch current using Ohm's Law and the node voltages.
3. Sum all the currents at that node and set the sum equal to zero.

Conductance

Conductance ($G$) is the reciprocal of resistance. It measures how easily current flows through an element:

$G = \frac{1}{R}$

Conductance is measured in siemens (S). Using conductance, Ohm's Law becomes:

$I = GV$

Why bother with conductance? In nodal analysis, you're constantly dividing by resistance. Writing equations in terms of $G$ avoids fractions and makes the math cleaner, especially when you're setting up matrix equations.

Solving Nodal Equations

Setting Up the System of Equations

Each non-reference node in the circuit gets its own KCL equation. So if your circuit has $n + 1$ nodes total (including the reference), you'll write $n$ equations with $n$ unknown node voltages.

Here's the step-by-step process:

1. Identify all nodes in the circuit and choose one as the reference node (ground, 0 V).
2. Label the unknown node voltages as $V_1$, $V_2$, etc.
3. Apply KCL at each non-reference node. For every branch connected to that node, express the branch current in terms of node voltages and conductances (or resistances).
4. Solve the resulting system of equations. For two or three unknowns, substitution or elimination works fine. For larger circuits, use matrix methods like Gaussian elimination or matrix inversion.

The General Nodal Equation

The KCL equation at node $i$ takes this general form:

$\sum_{j=1}^{n} G_{ij}(V_i - V_j) + I_i = 0$

where:

• $G_{ij}$ is the conductance of the element connecting nodes $i$ and $j$
• $V_i$ and $V_j$ are the voltages at nodes $i$ and $j$
• $I_i$ is the net current injected into node $i$ by any current sources (positive if the source pushes current into the node)

Each term $G_{ij}(V_i - V_j)$ represents the current flowing from node $i$ toward node $j$ through the connecting element. The equation simply says: all the currents leaving node $i$ through resistors, plus any current source contributions, must sum to zero.

Once you solve for all the node voltages, you can go back and find any branch current using $I = G(V_i - V_j)$ or equivalently $I = \frac{V_i - V_j}{R}$.