Thevenin & Norton Theorem

 Thevenin Theorem

 Suppose we have to calculate the electric current through any particular branch in a circuit. This branch is connected with rest of the circuit at its two terminals. Due to active sources in the circuit there is one potential difference between the points where the said branch is connected. The current through the said branch is caused by this potential difference appears across the terminals. So rest of the circuit can be considered as a single voltage source, whose voltage is nothing but the open circuit voltage between the terminals where the said branch is connected and the internal resistance of the source is nothing but the equivalent electrical resistance of the circuit looking back into the terminals where the branch is connected. So the Thevenin theorem can be stated as follows,

1) When a particular branch is remove from a circuit, the open circuit voltage appears across the terminals of the circuit, is Thevenin equivalent voltage and


2) the equivalent resistance of the circuit network looking back into the terminals, is Thevenin equivalent resistance.


3) If we replace the rest of the circuit network by a single voltage source, then the voltage of the source would be Thevenin equivalent voltage and internal resistance of the voltage source would be Thevenin equivalent resistance which would be connected in series with the source as shown in the figure below.

For better understanding Thevenin theorem, we have shown the circuit below,

Here two resistors R₁ and R₂ are connected in series and this series combination is connected across one voltage source of emf E with internal resistance R_i as shown. One resistive branch of R_L is connected across the resistance R₂ as shown. Now we have to calculate the current through R_L.

First we have to remove the resistor R_L from the terminals A and B.

Second we have to calculate the open circuit voltage or Thevenin equivalent voltage V_T across the terminals A and B.

The current through resistance R₂,

Hence voltage appears across the terminals A and B i.e.

 

 

 Third, for applying Thevenin theorem, we have to determine the Thevenin equivalent electrical resistance of the circuit and for that first we have to replace the voltage source from the circuit leaving behind only its internal resistance R_i. Now view the circuit inwards from the open terminals A and B. It is found the circuits now consists of two parallel paths – one consisting of resistance R₂ only and the other consisting of resistance R₁ and R_i in series.

Thus the Thevenin equivalent resistance R_T as viewed from the open terminals A and B is given as. As per Thevenin theorem when resistance R_L is connected across terminals A and B the network behaves as a source of voltage V_T and internal resistance R_T and this is called Thevenin equivalent circuit. The current through R_L is given as

Thevenin Equivalent Circuit

 Norton theorem
 In this theorem the circuit network is reduced into a single constant current source in which the equivalent internal resistance is connected in parallel with it. Every voltage source can be converted to equivalent current source.
Suppose in complex network we have to find out the electrical current through a particular branch. If the network has one of more active sources then it will supply current through the said branch. As the said branch current comes from the network, it can be considered the network itself is a current source. So in Norton theorem the network with different active sources is reduced to single current source whose internal resistance is nothing but the looking back resistance connected in parallel to the derived source. The looking back resistance of a network is the equivalent electrical resistance of the network when someone looks back into the network from the terminals where said branch is connected. During calculating this equivalent resistance, all sources are removed leaving their internal resistances in the network. Actually in Norton Theorem, the branch of the network through which we have to find out the current, is removed from the network. After removing the branch we short circuit the terminals where the said branch was connected. Then we calculate the short circuit current flows between the terminals. This current is nothing but Norton equivalent current I_N of the source. The equivalent resistance between the said terminals with all sources removed leaving their internal resistances in the circuit is calculated and say it is RN. Now we will form a current source whose current is I_N A and internal shunt resistance is R_N Ω.
For getting more clear concept of this theorem, we have explained it by the following example,
In the example two resistors R₁ and R₂ are connected in series and this series combination is connected across one voltage source of emf E with internal resistance R_i as shown. Series combination of one resistive branch of R_L and another resistance R₃ is connected across the resistance R₂ as shown. Now we have to find out the current through R_L by applying Norton theorem,
First we have to remove the resistor R_L from terminals A and B and make the terminals A and B short circuited by zero resistance.

Second we have to calculate the short circuit current or Norton equivalent current I_N through the points A and B.


 For determining of internal resistance or Norton equivalent resistance R_N of the network under consideration. Remove the the branch between A and B and also replace the voltage source by its internal resistance. Now the equivalent resistance as viewed from open terminals A and B is R_N,


As per Norton theorem when electrical resistance R_L is reconnected across terminals A and B, the network behaves as a source of constant current I_N with shunt connected internal resistance R_N and This is Norton Equivalent Circuit.

 

Norton Equivalent Circuit