**What is Norton’s Theorem**

Norton’s theorem states that:

A linear two-terminal electrical circuit can be simplified into a circuit consisting of a current source I

_{N}connected parallel with an equivalent resistance, R_{N}with the observed terminal.

The purpose of this Norton analysis is to make an equivalent circuit with a current source connected parallel with its equivalent resistance.

Equation below can help us on how to find Norton current:

**How to Find Norton Equivalent Circuit**

Assume that we have a linear as shown below:

We can redraw the circuit above into Norton equivalent circuit example:

Our next main focus are finding the value of:

- Norton equivalent resistance (R
_{N}) - Norton equivalent current (I
_{N})

If you have learned about the source transformation theorem, we will know that Thevenin equivalent resistance, R_{Th} and Norton equivalent resistance, R_{N} are equivalent because they don’t affect the voltage-current relationship since they are used on linear circuits.

In order to calculate the Norton equivalent current I_{N}, we calculate the current flowing through short-circuited terminal *a-b*.

The short-circuit current flowing from terminal *a* to *b* is short circuit current (i_{sc}) is the same with Norton equivalent current (I_{N}). Hence,

Keep in mind that we treat the independent and dependent source the same with we do in Thevenin’s theorem. Since Thevenin and Norton are equivalent, hence

This is basically the source transformation theorem. Because of this, source transformation is also known as Thevenin-Norton transformation. Since V_{Th}, IN, R_{Th} are related each other, we conclude that we need:

- Open circuit voltage across terminal
*a-b*, v_{oc} - Short circuit current at terminal
*a-b*, i_{sc} - Equivalent resistance at terminal
*a-b*when all independent sources are turned off, R_{N}

Using the basic Ohm’s Law we can use the equations below:

**Procedure of Norton’s Theorem**

Below is the step of Norton circuit analysis.

- Find and determine terminal
*a-b*where a parameter is observed. - Remove the component on that terminal, make it short circuit to the terminal
*a-b*, and calculate the current at that point*a-b*(I_{ab}=I_{sc}=I_{N}). This is known as*I*Norton or Norton equivalent current. - If all the sources are independent sources, then find the equivalent resistance when all the sources are turned off and replaced by their inner resistances (R
_{ab}=R_{N}=R_{Th}):- Independent voltage source is replaced by a short circuit.
- Independent current source is replaced by an open circuit.

- If there is a dependent source, to find the Norton equivalent resistance we can use:

- In order to find the V
_{oc}at terminal*a-b*, make that terminal open circuit and find the voltage across that terminal (V_{ab}=V_{oc}). - Redraw the Norton equivalent circuit consisting of the Norton equivalent current source, Norton equivalent resistance, and the component we remove in Step.(2).

**Norton’s Theorem with Independent Sources**

1. Find the value of *i* with Norton’s theorem!

Solution:

Determine the point *a-b* on *R* where *i* is observed. Calculate the i_{sc}=I_{N} when R=4Ω is removed:

With mesh analysis:

From loop I_{1}:

(1)

From loop I_{2}:

(2)

From loop I_{3}:

(3)

Substitute Equation.(2) into Equation.(3):

Thus,

Find R_{N} when all of the independent sources are turned off (replaced with their inner resistances), from the perspective of point *a-b*:

The Norton equivalent resistance is

Norton equivalent circuit:

The value of *i* is

2. Find the value of *V* with Norton’s theorem!

Solution:

We remove the 40Ω resistor and make it short-circuit

Find the equivalent resistance of the parallel resistors:

The voltage across terminal *a-b* is equal to the voltage across the R_{p}:

Find i_{sc}:

Find R_{N} at point *a-b*

The Norton equivalent resistance is

The Norton equivalent circuit is

The value of *V* is

Thus

3. Find the value of *i* with Norton’s theorem!

Solution:

We remove the voltage source:

Find i_{sc}

Thus

Turn off all the independent sources to find the R_{N}:

The Norton equivalent resistance R_{N}

The Norton equivalent circuit:

Thus

**Norton’s Theorem with Dependent Sources**

1. Find the value of *i* with Norton’s theorem!

Solution:

We remove the observed component

The value of i_{sc}:

In order to find R_{N}, we need to find V_{oc} first

The value of V_{oc} is

Hence the Norton equivalent resistance is

The Norton equivalent circuit is

Thus

2. Find the value of *i* with Norton’s theorem!

Solution:

Remove the observed component

Find the i_{sc}

Find the R_{N} from V_{ab} when point *a-b* is open circuit:

The V_{ab} is

The Norton equivalent resistance, R_{N}

The Norton equivalent circuit:

Thus,

3. Find the voltage value *V* with Norton’s theorem!

Solution:

Remove the observe component

Find i_{sc}

Find the V_{ab}

Thus

Hence the Norton equivalent resistance

The Norton equivalent circuit

Thus

**Norton’s Theorem Examples**

1. Redraw the circuit below into its Norton equivalent circuit at terminals *a-b*.

Solution:

Replace all the independent sources with their inner resistances.

From this circuit we will get Norton resistance

In order to find the Norton current I_{N}, we make terminal *a-b* short circuit as shown in circuit below,

We ignore the 5Ω resistor because it is parallel with a short circuit. Using mesh analysis, we get

From equations we obtained above, we get

Because Norton is equivalent to Thevenin, we can get Norton current I_{N} from V_{Th}/R_{Th}. We can get V_{Th} if we open circuit the terminal *a-b* as shown below.

Using mesh analysis, we get

And

Thus

The Norton equivalent circuit is

2. Find Norton resistance R_{N} and Norton current IN using Norton’s theorem from the circuit below at terminal *a-b*.

Solution:

We replace the independent voltage source with short circuit and connect terminal *a-b* with voltage source v_{o} = 1 V as shown below,

Ignore the 4Ω resistor because it is in parallel with a short circuit. Hence voltage source v_{o}, dependent current source, and 5Ω resistor are in parallel. Thus, i_{x} = 0.

From node *a*,

And

In order to get Norton equivalent current I_{N}, we make terminal *a-b* short circuit to find the current i_{sc} as shown below

From the circuit above all the components are in parallel. Thus,

At node *a*, use KCL

Hence,

## Frequently Asked Questions

### What does Norton theorem states?

Norton’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor

### What is Norton’s theorem formula?

Norton’s theorem formula consists of a equivalent current source connected parallel with the equivalent resistance and desired component. We can use the Ohm’s Law.

### What is Norton theorem with example?

A linear two-terminal electrical circuit can be simplified into a circuit consisting of a current source I_{N} connected parallel with an equivalent resistance, R_{N} with the observed terminal.

### Is Norton theorem same as Thevenin?

If you have learned about the source transformation theorem, we will know that Thevenin equivalent resistance, R_{Th} and Norton equivalent resistance, R_{N} are equivalent because they don’t affect the voltage-current relationship since they are used on linear circuits.

### How do I get Norton equivalent?

Remove the component on that terminal, make it short circuit to the terminal *a-b*, and calculate the current at that point *a-b* (I_{ab}=I_{sc}=I_{N}). This is known as *I* Norton or Norton equivalent current.

How to find V thevenin in the last excerise?