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 Z_{L}, can be replaced by a simple network, that has a single current source of I_{N} amperes and a single impedance Z_{eq} in parallel with it, across the two terminals of the load Z_{L}. It is called Norton's current I_{N} and Z_{eq} is the equivalent impedance of given network as viewed through the load terminals, with Z_{L} 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 Z_{eq}".The steps used to convert the simple circuit into Norton's circuit using theorem are:
- Short the branch, through which the current is to be calculated
- Obtain the current through this short circuited branch, using any of the network simplification techniques. This current is Norton's current.
- Draw the Norton's equivalent across the terminals, with current source I_{N}, with impedance Z_{eq} parallel with it. The current through the branch
I = I_{N }$\times$ $\frac{Z_{eq}}{Z_{eq} + Z_L}$
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 G_{n}, where i_{sc} 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.
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:
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 I_{1 }+ 2 - 0.1 I_{1} = 0
$\therefore$ 10.1 I_{1} = 2
$\therefore$ I_{1} = 0.198 A
-20 (I_{1} - I_{N}) - 0.2 (I_{1} - I_{N}) - 4 = 0
$\therefore$ - 20.2 I_{1 }+ 20.2 I_{N} = 4
$\therefore$ I_{N} = 0.396 A
Step 3: Calculate R_{eq} 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 I_{L} is in downward direction while the flow of Norton current I_{N }is in upward direction
Step 5 : I_{L} = I_{N }$\times$ $\frac{R_{eq}}{R_{eq} + R_L}$
= 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 I_{N }= 2 A, equivalent resistance R_{eq} = 100 $\Omega$, Load resistance R_{L}= 10 $\Omega$
Hence the load current is
I_{L} = I_{N} $\times$ $\frac{R_{eq}}{R_{eq} + R_L}$
= 2 A $\times$ $\frac{100 \Omega}{100 \Omega + 10 \Omega}$
= 1.818 A .