Electrical Engineering ⇒ Topic : Millmans Theorem

MILLMAN'S THEOREM

The chief utility of this theorem is that any number of parallel voltage sources can be replaced by an equivalent one.

This theorem states that

Any number of parallel voltage sources V₁, V₂, ...... V_N," having internal resistances R₁, R₂, ..... .R_rv,, respectively can be replaced by a single equivalent voltage source V in series with an equivalent series resistance R. G stands for conductance.

Figure 1 (b) represents the equivalent voltage V and resistance R of the original circuit shown in Figure 1 (a).

                                                                                   figure (1)

                                              ..........  (1)

                                                                                                                 ............ (2)

Millman's theorem is used to replace a number of parallel-connected a.c. voltage/current sources by a single equivalent a.c. voltage/current source

(1) For parallel-connected a.c. current sources, Millman's theorem may be stated as under :

 Any number of parallel-connected a.c. current sources can be replaced by a single equivalent a.c. current source . This single equivalent a.c. current source consists of an ideal a.c. current source and a parallel equivalent source impedance

The current of the equivalent a.c. current source is equal to the phasor sum of individual source currents. The parallel equivalent source impedance is the parallel combination of individual source impedances

(2) For parallel-connected a.c. voltage sources, Millman's theorem may be stated as under :

  Any number of parallel-connected a.c. voltage sources can be replaced by a single equivalent a.c. voltage source. This single equivalent a.c. voltage source consists of voltage V_m in series with equivalent source impedance Z_m whose values are given by

                                      ........

Note: If the circuit has a combination of parallel a.c. voltage and current sources, each parallel-connected a.c. voltage source is converted into equivalent a.c. current source. The result is a set of parallel-connected a.c. current sources and we can replace them by a single equivalent a.c. current source. Alternatively, each parallel connected a.c. current source can be converted into an equivalent a.c. voltage source and the set of parallel connected a.c. voltage sources can be replaced by a single equivalent a.c. voltage source.

Millman's Theorem

This theorem enables a number of voltage (or current) source to be combined into a single voltage (or current) source.

Suppose there are 3 voltage sources E₁, E₂ and E₃ of internal impedances Z₁, Z₂ and Z₃, respectively, connected between a and b as shown in Fig. (1).

                                          FIGURE (1) Circuit to illustrate Millman's theorem

Then, according to this theorem, these voltage source between a and b can be replaced by a single voltage source E' in series with an impedance Z' where

                                       ........... (1)

or, in general terms

                                        ..........  (2)

The proof of this theorem is given as follows.

Using Norton's theorem, the constant voltage source E and the series impedance Z (Fig. 1 (a)) can be converted into an equivalent current source I, where I= E/Z = EY in parallel with an admittance Y = 1/Z. All the voltage sources (see Fig. 1 (a)) can be connected into an equivalent current sources as shown in Fig. 2 (a) and connected across ab. The equivalent current source is shown in Fig. 2 (b), in which I = (I₁ + I₂ + I₃) with an admittance Y = Y₁ + Y₂ + Y₃ connected across it. The current source in Fig. 2 (b) can be converted into an equivalent voltage source as shown in Fig. 2 (c), in which

                             ............... (3)

                              ................ (4)

Millman's theorem is useful in calculating the voltage of the neutral point in 3 phase ac systems when the load is unbalanced as discussed in Example 3.20 below.

                              FIGURE (2) Circuit 2 to illustrate Millman's theorem