Norton’s Theorem Formula and Equivalent Circuit

Before learning about Norton’s theorem formula, let us know the uses of Norton Theorem first.

The uses of Norton Theorem are to:

• Simplify a complex circuit into a simple equivalent circuit.
• The equivalent circuit can be used repeatedly even though the load is changed without doing calculation from the beginning.

What is Norton’s Theorem

Norton’s theorem statement is

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.

This concludes that any linear electrical circuit can be simplified into an equivalent circuit with an ideal current source in parallel with an equivalent resistor and observed terminal or element.

Norton’s theorem affirms that any linear electrical circuit is equivalent to an ideal current source in parallel with an equivalent resistor.

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 as we do in Thevenin’s theorem. Since Thevenin and Norton are equivalent, hence

Norton’s Theorem Formula

This is basically the source transformation theorem. Because of this, source transformation is also known as Thevenin-Norton transformation. Since V_Th, I_N, 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:

The Norton’s theorem formula is

Norton’s Current Formula

Below is the step of Norton’s current formula.

1. Find and determine terminal a-b where a parameter is observed.

2. 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.

3. 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.

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

5. 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).

6. Redraw the Norton equivalent circuit consisting of the Norton equivalent current source, Norton equivalent resistance, and the component we remove in Step.(2).

How to Find Norton Equivalent Circuit

Assume that we have an electrical circuit and we have to find the value of a variable in the circuit. For a start, we are asked to find the Norton equivalent circuit at terminals a-b before everything.

Keep in mind that like Thevenin’s theorem, everything in the circuit will be simplified into an equivalent circuit except the element in question. Let us use a resistor at terminals a-b to make things simpler.

The procedure to find the Norton’s equivalent circuit at terminals a-b is:

1. Remove the observed resistor.
2. Make the terminals a-b as a short circuit.
3. Calculate the short circuit current or Norton current (I_sc = I_N).
4. Replace the voltage source with a short circuit.
5. Replace the current source with an open circuit.
6. Make the terminals a-b as an open circuit.
7. Calculate the equivalent resistance in the circuit (R_N).
8. Draw the Norton equivalent circuit where I_N current source, R_N, and the observed resistor all in parallel.

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 I₁:

From loop I₂:

From loop I₃:

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

How to find i norton?

The norton current or I_N is the current flowing through a short circuit terminal where we remove the resistor. 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

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. Calculate the norton current for the circuit below!

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 Problems with Solutions

We can observe the Norton theorem examples with solutions below:

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 I_N 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,