Millman’s Theorem

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Millman’s Theorem statement

The millman’s theorem states that, when a circuit has n number of voltage sources such as V₁, V₂, V₃⋅⋅⋅⋅⋅⋅⋅V_n along with their internal resistances R₁, R₂, R₃, ⋅⋅⋅⋅⋅⋅⋅R_n respectively are connected in parallel as shown in the below figure, then the circuit can be replaced by a single
equivalent voltage source V in series with the equivalent resistance R.

Where,
Equivalent voltage source V is given by,
V=(V₁+V₂+V₃+…..V_n)/(G₁+G₂+G₃+…...G_n) (V)

Equivalent Resistance R is given by,
R=1/(G₁+G₂+G₃+…...G_n) (Ω)

G₁, G₂, G₃,…..G_n are conductance which are equal to 1/R₁, 1/R₂, 1/R₃ …. 1/R_n respectively.

Circuit diagrams for Millman's Theorem

The below two figures shows the actual circuit and the equivalent circuit obtained after applying the Millman's theorem,

Actual circuit for applying Millman's Theorem

Equivalent circuit after applying Millman's theorem

The theorem can also applied to the parallel connected current sources by transforming them to voltage sources using source transformation.

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Read Also
Norton's Theorem
Superposition Theorem
Reciprocity Theorem
Thevenins Theorem