# Nodal Analysis

Nodal Analysis (aka Node Voltage Analysis) is a systematic method for determining all of the node voltages in a circuit where KCL is applied at each of the principle nodes and each current is expressed in terms of the voltages and branch resistances using Ohm's Law.

The steps are as follows:

1. Indicate all of the nodes in the circuit, draw current directions with all voltage polarities shown and select a reference node.

2. Write nodal equations (apply KCL) at each of the nodes (except the reference node) and express each of the currents in terms of voltage and resistance using Ohm's Law.

3. Solve the resulting equations to determine the node voltages.

Note that if there are n nodes in a circuit then writing n - 1 nodal equations is enough to determine all of the voltages and currents in the circuit. For example, if a circuit contains 2 nodes, one of which is the reference node (GND), then we only need to write 1 nodal equation.

Nodal analysis is a very powerful and efficient technique commonly used by circuit simulators to solve circuits and is used extensively in circuit analysis so learn it well!

### Examples:

1. a) Calculate all of the node voltages in the following circuit using nodal analysis:

The first step is to highlight all of the nodes in the circuit (indicating the reference node) and assigning currents and voltages to the circuit:We can now write a nodal equation at node Va via KCL: Notice that all the terms were brought to the left hand side in the above, this simplifies the mathematics and is a useful quick optimization as a first step. We can then express the currents in terms of voltage and resistance by applying Ohm's Law to each current as below: Which we can then rearrange to solve for Va:

b) Use the node voltage to determine all of the currents.

As we now know the voltage at node Va, we can simply apply Ohm's law to determine each branch current:

c) What is the power provided by the voltage sources V1 and V2?

As we know the currents flowing through V1 and V2 we can calculate the power provided by each source:

2. a) Calculate all of the node voltages in the following circuit using nodal analysis:

Highlighting all of the nodes in the circuit (indicating the reference node) and assigning currents and voltages to the circuit: For this circuit we have two principle nodes: Va and Vb and one reference node. This means we have two equations to solve to determine all voltages and currents in the circuit. Writing a nodal equation at Va: Writing a nodal equation at Vb: From here we have two equations with two unknowns which we can solve for Va by adding  and : Substituting this back into  to determine Vb:

b) Determine all of the currents.

c) Determine the power dissipated by R4.

### Questions:

1. a) Calculate all of the node voltages in the following circuit using nodal analysis:

Va=2.5V

b) Calculate all of the currents:

I1=750uA, I2=1.750mA, I3=2.499mA

c) What is the power delivered by voltage sources V1 and V2?

P1=3mW, P2=10.5mW

d) Calculate the power dissipated through R3

PR3=6.25mW

2. a) Calculate all of the node voltages in the following circuit using nodal analysis:

Va=489.691mV, Vb=463.918mV

b) Calculate all of the currents:

I1=5mA, I2=4.536mA, I3=51.546uA, I4=51.546uA, I5=4.897mA, I6=4.639mA

c) What is the power delivered by current source I1 and voltage source V2?

PI1=27.448mW, PV2=22.68mW

d) Calculate the power dissipated through R4