Sometimes it is desirable to determine a particular branch current in a circuit as the resistance of that branch is changing while all other resistances and voltages remain constant. For instance, in the circuit shown in Fig. 1.23, it may be required to determine the current through RL for five values of RL, let that R1, R2, R3, and E remain constant. In such condition, the *solution can be obtained readily by using Thevenin’s theorem stated below :

**Thevenin’s theorem state that** **Any two-terminal network having a number of e.m.f. sources and resistances can be changed by an equivalent series circuit using a voltage source E0 in series with a resistance R0 where,**

E0= open-circuited voltage between the two terminals.

R0= the resistance between two terminals of the circuit find by looking “in” at

the terminals with removed load and voltage sources changed by their internal resistances if any.

To understand and learn the usage of this theorem, consider the two-terminal circuit shown in Fig. 1.23. The circuit enclosed in the box can be replaced or changed by one voltage E0 in series with resistance R0 as shown in Fig. 1.24. The response at the terminals AB and A′B′ are the same for the two circuits, independent of the RL values connected across the terminals.

(i) Finding E0. Eo is the voltage which is between terminals A and B of the circuit when load RL is remove. Fig. 1.25 shows the circuit with no load. The voltage drop across R2 is the required voltage E0

.

(ii) Finding R0. R0 is the resistance between terminals A and B with No load and e.m.f. decreased to zero (see Fig. 1.26).

Thus, the value of R0 is found. Once the values of E0 and R0 are found, then the current passes through the resistance of load RL can be determined easily (Refer to Fig. 1.24).

**Procedure for Finding Thevenin Equivalent Circuit**

- (i) Open the two terminals (i.e. remove any load) between which you need to determine the Thevenin equivalent circuit.
- (ii) determine the open-circuit voltage between the two open terminals. It is known as Thevenin voltage E0.
- (iii) Find the resistance between the two open terminals and must all ideal voltage sources shorted and all ideal current sources must be opened (a non-ideal source is replaced by its internal resistance). It is known as Thevenin resistance R0.
- (iv) Connect E0 and R0 in series to develop the Thevenin equivalent circuit between the two terminals under consideration.
- (v) Put the same load resistor removed in step (i) across the terminals of the Thevenin equivalent circuit. The load current can now be determined by using only Ohm’s law and it has the exact same value as the load current in the original circuit.

Example 1.8. Using Thevenin’s theorem, determine the current passes through 100 Ω resistance connected across terminals A and B in the circuit of Fig. 1.27.

Solution.

(i) Finding E0. Eo is the voltage across terminals A and B and removes the 100 Ω resistance as shown in Fig. 1.28.

(ii) Finding R0. R0 is the resistance between terminals A and B with 100 Ω resistance removed and all voltage source are short-circuited as shown in Fig. 1.29.

R0= Resistance at terminals A and B in Fig. 1.29

Therefore, Thevenin’s equivalent circuit will be as shown in Fig. 1.30. Now, the current passes through 100 Ω resistance connected across terminals A and B can be determined by applying Ohm’s law.

Example 1.9. Find the Thevenin’s equivalent circuit for Fig. 1.31.

Solution. The Thevenin’s voltage E0 is the voltage between the terminals of A and B. This voltage is equal to the voltage across R3 because of parallel. It is due to terminals A and B are open-circuited and there is zero current flowing through R2 and hence zero voltage drop across it.

The Thevenin’s resistance R0 is the resistance measured between terminals A and B with load removed (i.e. open at terminals A and B) and voltage source changed by a short circuit.

Therefore, Thevenin’s equivalent circuit will be as shown in Fig. 1.32.

Example 1.10. Calculate the value of load resistance RL to which maximum power may be delivered from the circuit shown in Fig. 1.33 (i). Also determine the maximum power.

Solution. First we determine Thevenin’s equivalent circuit to the left of terminals AB in Fig. 1.33 (i)

The Thevenin’s equivalent circuit to the left of AB terminals in Fig. 1.33 (i) is E0 (= 40 V) in series with R0 (= 73.33 Ω). When RL is connected between A and B terminals, the circuit will as shown in Fig. 1.33 (ii). It is clear that maximum power will be transferred or delivered when

Example 1.11. Calculate the current in the resistor 50 Ω in the network shown in Fig. 1.34.

Solution. We will simplify the given circuit shown in Fig. 1.34 by the multi-use of Thevenin’s theorem. First, we determine Thevenin’s equivalent circuit to the left of *XX.

We can again replace or change the circuit to the left of YY in Fig. 1.35 by its Thevenin’s equivalent circuit. Therefore, the original circuit decreases to that shown in Fig. 1.36.

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