Proof of Thevenin's Theorem

Proof of Thevenin’s Theorem|Definition|Circuit diagram for Thevenin’s Theorem|Application

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Thevenin’s theorem is an important tool to avoid lengthy and difficult calculation in network analysis.

Thevenin’s Theorem Definition

According to Thevenin’s theorem,” any two-terminal linear network can be replaced by an equivalent circuit consisting of a voltage source (V) in series with the impedance (Z).“

The value of V is one which is presented to two terminals, when the external load is removed i.e. when the output current is zero. This voltage is known as open circuit voltage.

The value of Z is one which is presented to the output terminals, when all the sources of voltage are short circuited.

Circuit diagram for Thevenin’s Theorem

Network shown in figure (a) can be represented by a single voltage source (V) in series with the impedance Z (equal to the impedance of the two terminals network) as shown in figure (b).

The combination of voltage source (V) in series with impedance (Z) makes an equivalent voltage source known as Thevenin’s equivalent.

Proof of Thevenin's Theorem

Proof of Thevenin’s Theorem

Consider a two terminal network as shown in figure (a) below. ZL is the load impedance. Thevenin’s equivalent network of the network shown in figure (a) is represented as shown in figure (b) below.

Proof of Thevenin's Theorem

Current (I2) through the load impedance ZL in the network can be calculated by using Kirchoff’s voltage law.

Applying Kirchoff’s voltage law to closed loop I, we get

Applying Kirchoff’s voltage law to closed loop II, we get

Using eqn. (ii) in eqn. (i), we get

However, I2 (i.e current through ZL) can be easily calculated using Thevenin’s equivalent circuit.

Determination of open circuit voltage V

Open circuit voltage V can be calculated by finding voltage across the terminals 1 and 2, when ZL is disconnected as shown in the fig…

Proof of Thevenin's Theorem

In this case, there is no voltage drop across Z3 as Z3 is in the open circuit. Therefore, current in the given circuit is given by

Therefore, open circuit voltage is

By using eqn. (iv) we get eqn. (v)

Determination of equivalent impedance (Z)

Impedance (Z) can be calculated by locking back from the terminals 1 and 2 with all voltage sources short circuited as shown in the fig…

Proof of Thevenin's Theorem

Since Z1 and Z2 are in parallel, therefore, equivalent impedance of Z1 and Z2 is given by

Z’ and Z3 are in series , therefore, the equivalent impedance (Z) of the circuit is given by

Now from Thevenin’s equivalent of the network

Put values of eqns. (v) and (vi) in eqn. (iii), we get

Proof of Thevenin's Theorem

Since eqns. (vii) and (viii) are identical, so current through the load resistance can be calculated easily by Thevenin’s equivalent circuit.

Thevenin’s Theorem Applications

  • Use to find current through the load impedance.
  • For the determination of open circuit voltage.
  • For the determination of equivalent impedance.

Why thevenin theorem is used?

Thevenin’s theorem is an important tool use to avoid lengthy and difficult calculation in network analysis.

What is thevenin’s theorem statement?

Thevenin’s theorem state that,” any two-terminal linear network can be replaced by an equivalent circuit consisting of a voltage source (V) in series with the impedance (Z).“