Electrical Engineering ⇒ Topic : Millmans Theorem

Samual
 
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 V_{1}, V_{2}, ...... V_{N}," having internal resistances R_{1}, R_{2}, ..... .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)  
 
Mason
 
Millman's theorem is used to replace a number of parallelconnected a.c. voltage/current sources by a single equivalent a.c. voltage/current source (1) For parallelconnected a.c. current sources, Millman's theorem may be stated as under : Any number of parallelconnected 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 parallelconnected a.c. voltage sources, Millman's theorem may be stated as under : Any number of parallelconnected 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 parallelconnected a.c. voltage source is converted into equivalent a.c. current source. The result is a set of parallelconnected 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.  
 
William
 
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 E_{1}, E_{2} and E_{3 }of internal impedances Z_{1}, Z_{2 }and Z_{3}, 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 = (I_{1 }+ I_{2} + I_{3}) with an admittance Y = Y_{1} + Y_{2} + Y_{3 }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  
 
!! OOPS Login [Click here] is required for more results / answer