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# Norton's Theorem

We could see the Thevenin theorem that has two-terminal active network is converted into a voltage source and an equivalent series resistance across the load where current would be calculated. Here there is another method of analyzing a network called Norton's theorem. Here the two terminal network with current and voltage source is converted into a constant current source and a parallel resistance and is connected across the load through which the current is to be calculated.

Norton circuit

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## What is Norton's Theorem?

Norton's theorem, as Thevenin's theorem is the way that is used to solve the complex circuits to represent control devices. It was developed by American scientist E.L. Norton, is generally used to reduce the complicated circuit network. It is the alternate to thevenin's theorem to analyze the network that has a simple current source and single parallel resistor. It states that
"Any combination of linear bilateral circuit containing network elements and active sources, regardless of the connection to a given load ZL, can be replaced by a simple network, that has a single current source of IN amperes and a single impedance Zeq in parallel with it, across the two terminals of the load ZL. It is called Norton's current IN and Zeq is the equivalent impedance of given network as viewed through the load terminals, with ZL removed and all the active sources are replaced by their internal impedances. If the internal impedances are unknown then the independent voltage sources must be replaced by short circuit while the independent current sources must be replaced by open circuit, while calculating Zeq".
The steps used to convert the simple circuit into Norton's circuit using theorem are:
1. Short the branch, through which the current is to be calculated
2. Obtain the current through this short circuited branch, using any of the network simplification techniques. This current is Norton's current.
3. Draw the Norton's equivalent across the terminals, with current source IN, with impedance Zeq parallel with it. The current through the branch
I = IN $\times$ $\frac{Z_{eq}}{Z_{eq} + Z_L}$

## Norton Equivalent Circuit

An American engineer, E.L. Norton at Bell telephone laboratories, proposed an equivalent circuit the current source and a equivalent resistance. This circuit is related to the Thevenin equivalent circuit by a source transformation. Hence a source transformation converts a Thevenin equivalent circuit into a Norton's equivalent circuit or vice versa. Norton published this method in 1926, 43 years after Thevenin.

The Norton's equivalent circuit replaces the simple circuit by a parallel combination of an ideal current source isc and a conductance Gn, where isc is the short circuit at the two terminals and Gn is the ratio of the short circuit current to the open-circuit voltage at the terminal pair.

## Difference between Thevenin Theorem and Norton's Theorem

Even though Thevenin's and Norton's theorem can be derived from each other and their resistance are equal in magnitude. There are some differences that rule them out:

 Thevenin's theorem Norton's theorem 1 The Thevenin's theorem is derived without referring any theorem It is the converse of Thevenin's theorem derived by referring it 2 It is the theorem where we get the simple circuit from the  complicated circuit that has voltage source VTH, resistance RTH   and load RL. It is the theorem where we just follow the similar steps like Thevenin's but  some parameters to determine are different (resistance RN, current IN) 3 Here Voltage source VTH is used in the circuit Here current source IN is used in the circuit instead of voltage source 4 The equivalent resistance RTH is in series with the source The equivalent resistance RN is in parallel with the source

Thevenin and norton equivalent circuit

## Norton's Theorem Examples

Lets see some examples on Norton's theorem:
Example 1:

Calculate the current through the branch be using Norton's theorem

Step 1 : Short the branch be

Step 2: Calculate the short circuit current using Kirrchhoff's laws

Apply KVL to two loops,

- 10 I1 + 2 - 0.1 I1 = 0

$\therefore$ 10.1 I1 = 2

$\therefore$ I1 = 0.198 A

-20 (I1 - IN) - 0.2 (I1 - IN) - 4 = 0

$\therefore$ - 20.2 I1 + 20.2 IN = 4

$\therefore$ IN = 0.396 A

Step 3: Calculate Req shorting voltage sources by simplifying the circuit, we get

Req = 20.2 || 10.1 = 6.733 $\Omega$

Step 4: Norton equivalent is shown in the figure gives the idea that flow of load current IL is in downward direction while the flow of Norton current IN is in upward direction

Step 5 : IL = IN $\times$ $\frac{R_{eq}}{R_{eq} + R_L}$

=  0.396 $\times$ $\frac{6.733}{6.733 + 5}$

= 0.2272 A
Example 2:

In a Norton's circuit there would be a flow of the Norton's current of 2 A having a equivalent resistance of 100 $\Omega$ and carries load resistance of 10 $\Omega$. What would be the load current in the circuit?

Given:

Norton current IN = 2 A, equivalent resistance Req = 100 $\Omega$, Load resistance RL= 10 $\Omega$

IL =  IN $\times$ $\frac{R_{eq}}{R_{eq} + R_L}$
=  2 A $\times$ $\frac{100 \Omega}{100 \Omega + 10 \Omega}$