Theory

To Verify Millman's Theorem.  

This theorem is a combination of Thevenin's and Norton's theorems. When a number of voltage sources (`V<sub>1</sub>`,`V<sub>2</sub>`,...,`V<sub>n</sub>`) are in parallel having internal resistances (`R<sub>1</sub>`,`R<sub>2</sub>`,...,`R<sub>n</sub>`) respectively, the arrangement can be replaced by a single equivalent voltage source `V<sub>m</sub>` in series with an equivalent series resistance `R<sub>m</sub>` as given below in Fig.1 and Fig.2.

     

As per Millman's Theorem ,

$$V_m = \frac{(+- V_1G_1 +- V_2G_2 +-.........+- V_nG_n)}{(G_1+G_2+.........+G_n)}$$

where $$G_i = \frac{1}{R_i}$$,    $$i = 1,2,....,n$$

$$R_m = \frac{1}{G} = \frac{1}{(G_1+G_2+.........+G_n)}$$

This voltage represents the Millman's equivalent voltage `V<sub>m</sub>`. The resistance `R<sub>m</sub>` can be found , as usual , by replacing each voltage source by a short circuit. If there is a load resistance `R<sub>L</sub>` across the terminals A and B , then the load current `I<sub>L</sub>` is given by

$$I_L = \frac{V_m}{(R_m + R_L)}$$