Network theory - Norton's theorem
Norton's theorem is similar to Thevenin's theorem. It indicates that any two-terminal linear network or circuit can be represented with a network or equivalent circuit, which consists of a current source in parallel with a resistor. It is called Norton equivalent circuit . A linear circuit can contain independent sources, dependent sources and resistors.
If a circuit has multiple independent sources, dependent sources, and resistors, then the answer in one element can be easily found by replacing the entire network to the left of that element with a circuit equivalent of Norton .
The response in an element can be the voltage across that element, the current flowing through that element, or power dissipated through that element.
This concept is illustrated in the following figures.
Norton's equivalent circuit looks like a convenient current source, therefore, it has a current source in parallel with a resistor.
- The current source present in the Norton equivalent circuit is called Norton equivalent current or simply Norton current I _{ N }.
- The resistance present in the Norton equivalent circuit is called Norton equivalent resistance or simply Norton resistance R _{ N }.
Methods of finding the Norton equivalent circuit
There are three methods to find a Norton equivalent circuit. Depending on the type of sources present in the network, we can choose any of these three methods. Now let's discuss these three methods one by one.
Method 1
Follow these steps to find Norton's equivalent circuit, when only type independent sources are present.
- Step 1 - Consider the circuit diagram by opening the terminals relative to which the equivalent Norton circuit is located.
- Step 2 - Find the current I_{N} of the Norton by shorting them two open terminals of the above circuit.
- Step 3 - Find the resistance of the Norton R_{N} at the open terminals of the circuit considered in Step 1 by eliminating independent sources present there. Norton R_{N} resistance will be the same as Thevenin R _{ Th } resistance.
- Step 4 - Draw the Norton equivalent circuit by connecting the current IN of a Norton in parallel with the resistance of Norton R _{ N }.
Now we can find the answer in an element which is on the right side of the Norton equivalent circuit.
Method 2
Follow these steps in order to find the equivalent Norton circuit, when sources of both independent type and dependent type are present.
- Step 1 - Consider the electrical diagram by opening the terminals with respect to which the equivalent circuit of Norton is located.
- Step 2 - Find the open circuit voltage V _{ OC } on the open terminals of the circuit above.
- Step 3 - Find current I N by shorting the two open terminals of the above circuit.
- Step 4 - Find the resistance of Norton R _{ N } using the following formula .
$$ R_N = frac {V_ {OC}} {I_N} $$
- Step 5 - Draw the Norton equivalent circuit by connecting the current of a Norton I _{ N } in parallel with the resistor by Norton R _{ N }.
Now we can find the answer in an item which is on the right side of Norton Equivalent Circuit.
Method 3
This is another method to find an equivalent Norton circuit.
- Step 1 - Find an equivalent Thevenin circuit between the two desired terminals. We know that it consists of a Thevenin voltage source, V _{ Th }, and Thevenin resistance, R _{ Th }.
- Step 2 - Apply the source transformation technique to the equivalent circuit of Thevenin above. We will get the equivalent circuit from Norton. Here,
Current Norton,
$$ I_N = frac {V_ {Th}} {R_ {Th}} $$
La Norton resistance,
$$ R_N = R_ {Th} $$
This concept is illustrated in the following figure.
Now we can find the answer in an element by placing the Norton equivalent circuit to the left of that element.
Note - Similarly, we can find Thevenin's equivalent circuit by finding the equivalent circuit of a Norton sapinst, and then apply the source transform technique to it. This concept is shown in the figure next.
Here is method 3 to find an equivalent circuit of Thevenin.
Example
Find the current through the 20 Ω resistor by first finding a Norton equivalent circuit to the left of terminals A and B.
Let's solve this problem using the Method 3 .
Step 1 - In the previous chapter, we calculated Thevenin's equivalent circuit on the left side of terminals A and B. We can use this circuit now. It is shown in the following figure.
Here, Thevenin voltage, $ V_ {Th} = frac {200} {3} V $ and Thevenin resistance, $ R_ {Th} = frac {40} {3} Omega $
Step 2 - Apply source transformation technique to the equivalent Thevenin circuit above. Substitute the values of V _{ Th } and R_{Th} in the following current Norton formula.
$$ I_N = frac {V_ {Th}} {R_ {Th}} $$
$$ I_N = frac {frac {200} {3}} {frac {40} {3}}= 5A $$
Therefore, Norton's current I_{N} is 5 A .
We know that Norton's resistance, R_{N} is the same as that of Thevenin's resistance R _{ Th } .
$$ mathbf {R_N = frac {40} {3} Omega} $$
The Norton equivalent circuit corresponding to Thevenin equivalent circuit above is shown in the following figure .
Now place the Norton equivalent circuit to the left of terminals A and B of the given circuit.
Using the principle of , the current through the 20 Ω resistor will be
$$ I_ {20 Omega} = 5 lgroup frac {frac {40} {3}} {frac {40} { 3} + 20} rgroup $$
$$ I_ {20 Omega} = 5lgroup frac {40} {100} rgroup = 2A $$
Therefore, the current through the 20 Ω resistor is 2 A .