THEVENIN'S AND NORTON'S THOEREMS

SUPERPOSITION THEOREM

MAXIMUM POWER TRANSFER THEOREM

This page describes application of network theorems to ac circuits. There is no significant difference between the application of network theorems to ac and dc circuits, except in the case of the maximum power transfer theorem. Hence the application of maximum power transfer is explained in detail, whereas the other theorems have either a brief or no coverage.

These two theorems can be applied to ac networks also and the technique to be used is the same as that outlined for resistive circuits. The only difference is that the quantities being computed are phasors and complex numbers. The application of these theorems is illustrated through worked examples presented later.

A circuit is linear when superposition theorem can be used to obtain its currents and voltages. When this theorem is applied to an ac circuit, it has to be remembered that the voltage and current sources are in the phasor form and the passive elements are impedances.

**WORKED EXAMPLE 1: **

A circuit is presented in Fig. 30. The task is to find the current through the load current, using superposition theorem.

The values of components are specified by equation (99). We can find the contribution due to one voltage source, by replacing the other voltage by a short circuit. Only the ideal part of the source is to be replaced by a short-circuit, and its internal impedance should be left in the circuit. For example, we can view **Z**_{1} and **Z**_{2} as the internal impedance of sources, **V**_{A} and **V**_{B} respectively. The application of superposition theorem is illustrated by the sketch in Fig. 31.

The equations obtained are shown below. The current supplied source can be obtained after determining the impedance seen by the source. The impedance seen by source **V**_{A} is the sum of impedance **Z**_{1} and the parallel value of **Z**_{2} and **Z**_{L}. Once the source current is determined, current **I**_{L1} through impedance **Z**_{L} can be found out using the current division rule. In the same way, we can find current **I**_{L2} through impedance **Z**_{L} due to source **V**_{B} acting alone. Using superposition theorem, we get the load current as the phasor sum of **I**_{L1} and **I**_{L2}.

The answer can be obtained much more easily by using nodal analysis. The voltage across the load **Z**_{L} can be found out by solving a single KCL equation. The purpose here is to illustrate the application of superposition theorem. Its direct application is not always handy, bit its importance lies in that the entire analysis of linear circuits rests on it.

In some low power circuits, the need for maximum power transfer tends to be more important than the need for efficiency. For example, extraction of maximum signal from a weak source such as a transducer may be necessary. Another example is the case of impedance matching, where the output impedance of an amplifier is to be matched with that of the load such as a loudspeaker. On the other hand, efficiency is the important criterion for a voltage source with low output impedance. For example, efficiency is of utmost importance in electrical power generation and transmission, and it is over 80% for electrical power generation units. Under the maximum power transfer condition, efficiency is no more than 50 %.

When maximum power transfer is to be brought about in ac circuits, the result depends upon type of load. At first, a circuit with resistive load is taken up for study. In the circuit in Fig 32a, the load is purely resistive, designated as * R* . The source has source impedance containing source resistance,

When the load is purely resistive, maximum power transfer to the load resistance occurs when the load resistance equals the magnitude of the source impedance. This point is proved as follows.

We consider the case when the load is purely resistive. From Fig. 32a, current supplied by the source can be obtained as shown by equation (105). Then power delivered to the load is obtained as shown by equation (106). In equation (106), the current is expressed as the ratio of source voltage over impedance. A factor of half is present, because the source voltage is represented by its peak value. To find the value of load resistance at which maximum power transfer occurs, equation (106) is differentiated with respect to load resistance *R*_{L}, and the derivative is equated to zero. Then the value of load resistance *R*_{L} at maximum power transfer can be obtained.

From equation (106), the derivative obtained is shown by equation (107). The formula used for obtaining the derivative is shown below equation (108). On setting the derivative to zero, we obtain the value of load resistance *R*_{L} as shown by equation (108). When the load resistance equals the magnitude of source impedance, the power transferred to the load is maximum. That the power transferred is the maximum and not the minimum can be verified by evaluating the second derivative with respect to load resistance *R*_{L} and finding it to be negative. Equation (108) states the condition for maximum power transfer when the load is purely resistive.

When the load impedance is complex, the condition for maximum power transfer is obtained as follows.

From Fig. 32b, the load current can be obtained as shown by equation (109). Then power delivered to the load is obtained as shown by equation (1110). First let us consider the case when the load resistance *R*_{L} is fixed and the load reactance *X*_{L} is variable.

When the load resistance *R*_{L} is fixed and the load reactance *X*_{L} is variable, maximum power transfer occurs when equation (111) is satisfied. In this case, the maximum power transferred is obtained from equation (112). If the source impedance has inductive reactance, then the load impedance should contain capacitive reactance. In this case, suffix L is a pointer to load reactance. It is not a pointer to inductive reactance.

If both the load resistance *R*_{L} and the load reactance *X*_{L} are variable, one of the conditions for maximum power transfer is specified by equation (113). Then the power delivered to the load is obtained as shown by equation (114). When load resistance *R*_{L} is variable, equation (114) is differentiated with respect to load resistance *R*_{L}. The derivative obtained is shown next.

The derivative obtained is expressed by equation (115). As shown by equation (116), the derivative is zero, when the load resistance equals the source resistance. The maximum power transfer to load occurs when equation (116) is satisfied. In this case, the maximum power transferred to the load is obtained from equation (117).

The condition for maximum power transfer is specified by equation (118), when both the load resistance *R*_{L} and the load reactance *X*_{L} are variable . That is, the load impedance is the conjugate of the source impedance and the power factor of the circuit is unity. Since the source impedance normally has inductive reactance, the load impedance should contain capacitive reactance. In this case, suffix L is a pointer to load reactance. It is not a pointer to inductive reactance.

**WORKED EXAMPLE 2:**

An example is presented now to illustrate the application of maximum power transfer theorem.

For the circuit in Fig. 33, find the value of load resistance at which maximum power transfer occurs and find the maximum power transferred to the load. The values of components used are specified below.

The solution is as follows:

At first, the circuit containing the source, the resistors, *R*_{1} and *R*_{2} and the capacitance should be replaced by its Thevenin's equivalent circuit. Then the results of maximum power transfer theorem can be applied. In short, the procedure is as follows. Get Thevenin's Equivalent Circuit. Then load resistance *R*_{L} equals the magnitude of Thevenin's impedance.

From the circuit in Fig. 33, the open-circuit voltage is the voltage across the capacitor, as shown by equation (120). Next we find the Thevenin's impedance by replacing source voltage *V*_{S} by a short-circuit. Then the Thevenin's impedance is obtained as shown by equation (121). The load resistance equals the magnitude of Thevenin's impedance and its value is shown by equation (122).

The Thevenin's equivalent circuit is shown by the circuit in Fig. 34. Note that Thevenin's equivalent circuit does not contain the load resistance. Some worked examples are presented next, to illustrate the use of network theorems and the use of mesh and nodal analysis.