Superposition theorem

             Superposition Theorem

 According to this theorem,"In any linear network containing linear impedances and two or more than two potential sources, the current flowing in any element is the algebraic sum of current that would flow in that element by each potential source, all other sources being replaced at that time by their internal impedances".

Let the current due to E1 and E2 acting together are I1 and I2 (Fig. a) and let the current due to e.m.f. E1 acting alone is I1’ and I2’ (Fig. b) and due to E2 acting alone is I1” and I2”.

Applying Kirchoff’s second law to the mesh of fig. (a) we have

E1 = I1 (Z1 + Z3) + I2Z3    ...............(1)

  And              E2 = I2 (Z2 + Z3) + I1Z3    ..............(2)  

When E1 is considered to act alone, mesh of fig. (b) gives

                        E1 = I1’ (Z1 + Z3) + I2’Z3    .............(3)

                        0 = I2’ (Z2 + Z3) + I1’ Z3      ..............(4)

And, when E2  considered alone mesh if fig. (c ) gives

                        0 = I1” (Z1 + Z3) + I2” Z3     ...............(5)

                         E2 = I2” (Z2 + Z3) + I1” Z3   ...............(6)

 adding equation (3) and (5), we get—

          E1 + 0 =  I1’ (Z1 + Z3) + I2’Z3  +  I1” (Z1 + Z3) + I2” Z3

          E1 = (I1’ + I1”) (Z1 + Z3) + (I2’ + I2”)Z3     .......…..…(7)

Adding equation (4) and (6), we get ------

          0 + E2 = I2’ (Z2 + Z3) + I1’ Z3  + I2” (Z2 + Z3) + I1” Z3

          E2 = (I2’ + I2”) (Z2 + Z3) + ( I1’ + I1”)Z3   ..….....…    (8)

equation (7) and (8) are identical with equation (1) and (2), respectively

  If                          

                     I1 = (I1’ + I1”)          and I2 = (I2’ + I2”)

 This proves the superposition theorem.

                                                                          http://youtu.be/O3D2belxKBs