How to Perform Nodal Analysis on an Electrical Circuit

Things you need to perform nodal analysis:

Pencil or pen
Paper
Calculator
Basic understanding of algebra
Method for solving a system of equations

Terms you need to know:

element: an object within a circuit
matrix: a series of entries that represent a system of equations
current (i): the flow of charge in a circuit, measured in Amps (A)
voltage (V): the measure of energy transfer through an element, measured in volts (V)
resistance (R): the opposition of an element to allow current to flow through it, measured in Ohms (Ω)

Circuit element symbols

voltage source: circle with a plus (+) and minus (-) sign contained within the circle
resistor: zig-zag line
wire: straight lines that connect the circuit elements

What is nodal analysis?

Nodal analysis is a circuit analysis technique that can be applied to any circuit. The nature of nodal analysis allows you to perform circuit analysis on a circuit that has been generalized (no values, just variables).

The benefit of nodal analysis allows you to solve a circuit once and place any values for the variables that you want within the equations. This method is very efficient when trying to find optimal values for circuits because you can adjust values of variables without having to completely resolve the circuit.

In this Instructable, you will learn how to perform nodal analysis on the simple circuit (resistors only and a single voltage source) shown above in figure 1. The circuit can not be completely solved without values, so this Instructable is more about the process rather than application.

Step 1: Identify All of the Nodes in the Circuit and Select a Reference Node

1. Circle and label all of the nodes that you do not know the nodal voltage for.

A node is a conductor that connects multiple circuit elements together in a circuit. In the circuit diagrams, nodes take the form of wires between the circuit elements.

Figure 2 shows the sample circuit with the nodes circled and labeled.

2. Select a reference node and place a ground symbol at the node.

A reference node is the node on the circuit where the nodal voltage is 0 V.
It will be helpful if you pick a node connected to a voltage source in a circuit, so that you can eliminate an unknown nodal voltage.

Figure 3 shows the selections of node D as the reference node. The ground symbol, denoted by GND, has been placed at node D as well. Since node D was chosen as the reference node, the nodal voltage of node A is equal to V1. This only leaves node B and C with unknown nodal voltages.

Step 2: Write a Kirchhoff's Current Law Equation for Any Unknown Nodal Voltages

For nodal analysis, Kirchhoff's current law (KCL) states that the currents entering or leaving a node must sum to zero. In terms of a current leaving a node, that means the current flows from the node you are analyzing to another node across a resistor.

1. Assign a current through each element in whatever direction that you want.

When you are writing a KCL equation, the direction that you choose for the current does not matter, but it needs to be consistent in each node.

Figure 4 shows nodes B and C with the currents leaving and entering the nodes. Node B currents are shown in black, and node C currents are shown in red.

Figure 5 shows the general KCL equation and the KCL equations for the two nodes. The currents in the equations are written so that they describe the current's direction. For example, i_BA is the current that flows from node B to node A.

Step 3: Use Ohm's Law to Rewrite the Currents in Terms of Nodal Voltages

Ohm's law shows the relationship that exists between voltage, current, and resistance. Ohm's law is useful for solving for an unknown voltage, current, or resistance. Additionally, Ohm's law allows you to solve KCL equations for unknown nodal voltages.

Figure 6 shows two forms of Ohm's law. The first form, V=iR, allows you to solve for voltage if you know the current and resistance. The second form, i=V/R, allows you to solve for current if you know the voltage and resistance.

1. Use your KCL equations to determine the currents that you need to write as nodal voltages.

Figure 7 shows the currents that need to be rewritten as nodal voltages circled in the color associated with the respective node.

2. Use the second form of Ohm's law to rewrite the currents as nodal voltages.

Figure 8 shows the current from node B to node A rewritten in terms of Ohm's law. The notation of the current tells you the order for the nodal voltages in the numerator, and the denominator is dependent on the resistor between the two nodes.

Step 4: Substitute the Currents From Step 3 Into the KCL Equations From Step 2

1. Use simple substitutions to replace the currents in your KCL equations with the unknown nodal voltages that you determined.

By doing this, you create a system of equations that you will be able to use to solve for the unknown nodal voltages.

Figure 9 shows the original KCL equation for node B and then the KCL equation rewritten using the unknown nodal voltages.

Remember that the nodal voltage of the reference node is 0 V, so you can leave that voltage out of your final KCL equations. Also, in this example, the voltage of node A is equal to the voltage of the voltage source, so it becomes a constant.

Step 5: Solve for the Unknown Voltages

1. Create a matrix using your substituted KCL equations from step 4.

Figure 10 shows the matrix that you get using the KCL equations. On the left hand side, you place the coefficients of the unknown nodal voltages, and on the right hand side, constants of the equations. In this example, the nodal voltage of node A is on the right hand side because the nodal voltage of node A is equal to the voltage of the voltage source.

The first column should contain the coefficients of the nodal voltage for node B, and the second column should contain the coefficients of the nodal voltage for node C from both equations.

2. Solve the matrix using a method of your choice.

The simplest way to solve the matrix is to have your calculator solve it.
Here are some links to solve a matrix with a graphing calculator:
1. TI-89: http://bit.ly/2dosQaf
2. TI-84: http://bit.ly/2dorD2F
3. TI-83: http://bit.ly/2djLrIs
Here is a link on how to solve a matrix by hand: http://bit.ly/2dwFyaW
The solution to your matrix contains the values of the unknown voltages. In this example:
1. The first row will contain the value for the nodal voltage of node B.
2. The second row will contain the value of the nodal voltage of node C.