# 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:

**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.

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

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

**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 [1] and [2]: Substituting this back into [2] 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:

#### Answer

**b)** Calculate all of the currents:

#### Answer

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

#### Answer

**d)** Calculate the power dissipated through R3

#### Answer

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

#### Answer

**b)** Calculate all of the currents:

#### Answer

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

#### Answer

**d)** Calculate the power dissipated through R4

#### Answer

## Lab:

Verify the answers for each of the above questions by drawing the circuits in LTSpice :)