Thevenin theorem with solved problems

Thevenin theorem

What is Thevenin theorem


Many electronic circuits contain a combination of batteries, resistors and make it very complicated. So simplifying these complex circuits we need Thevenin's Theorem. This theorem states that it is possible to simplify any linear circuits, to an equivalent circuit with just a single voltage source and impedance in series with the load, no matter how complex they are.


Thevenin theorem statements


According to this theorem, any two-terminal linear network containing energy sources and impedances can be replaced by an equivalent circuit consisting of a voltage source (VTH) in series with an impedance (RTH), where (VTH) is the open-circuit voltage between the terminals of the network and (RTHis the impedance measured between the terminals with all the energy sources replaced by their internal impedances.

Thevenin's equivalent circuit


To show the Thevenin's equivalent circuit we consider a circuit with a complicated passive network driven by an energy source (Vs). The network contains three resistors (R1, R2, and R3) and they are connected with a load (RL). This circuit will be replaced by an equivalent circuit with a voltage source (VTH) called Thevenin's voltage and impedance (RTH) called Thevenin's impedance.

Thevenin theorem

To calculate the Thevenin's voltage at first remove the load. When the load has removed the voltage across AB is equal to the voltage across the resistor (R2). So the Thevenin's voltage is


Where I = The flow of current through the circuit when the load is removed.

Now to calculate the Thevenin's impedance at first replace the energy sources by their internal impedance and the load (RL) also disconnected.
Note: If the internal impedance of the energy sources is given then it will be added to the resistor network.

Here the internal impedance is zero so the  Thevenin's impedance is



Therefore the Thevenin's equivalent circuit for the above circuit is

Thevenin theorem

Here the load current for this equivalent circuit is




Steps to follow for solving Thevenin's Theorem


Step 1 :
Identify the load (RL).
Step 2 :
Remove the load and calculate the open-circuit voltage (VTH).
Step 3 :
To calculate Thevenin's impedance (RTH), replace the sources by their internal impedance.
Step 4 :
Construct the Thevenin's equivalent circuit by connecting (VTH) in series with (RTH).

Thevenin's Theorem solved problems


Example 1: Calculate the current through the resistor of resistance 6 Ω.

Thevenin theorem

Solution :

To identify the load :
Here the load (RL) = 6 Ω

To calculate (VTH) :
Thevenin theorem

Now remove the load. When the load is removed the open-circuit voltage is the same as that of the voltage across the resistor of resistance 4 Ω.

∴ The current in the circuit is 


∴ The Thevenin's voltage is


To calculate (RTH) :
Thevenin theorem

After replacing the source by their internal impedance the Thevenin's impedance is



Thevenin's equivalent circuit :
Thevenin theorem

∴ The current through the load,



Example 2: Calculate the Thevenin's voltage and Thevenin's resistance.

Thevenin theorem

Solution :

To calculate (VTH) :
Here the open-circuit voltage is the same as that of the voltage across the resistor of resistance 7 Ω.

∴ The current in the circuit is 



∴ The Thevenin's voltage is




To calculate (RTH) :
Thevenin theorem

After replacing the source by their internal impedance the Thevenin's impedance is


Example 3: Calculate the current through the load resistance (RL) = 5 Ω.

Thevenin theorem

Solution :

To identify the load :
Here the load (RL) = 5 Ω

To calculate (VTH) :

Thevenin theorem

Now remove the load. When the load is removed the open-circuit voltage is the same as that of the voltage across the resistor of resistance 10 Ω.

Here the current through the first loop is



Where




And the current through the second loop is




Where




∴ The Thevenin's voltage is




To calculate (RTH) :


Thevenin theorem

After replacing the source by their internal impedance the Thevenin's impedance is



Thevenin's equivalent circuit :
Thevenin theorem

∴ The current through the load,





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