It then
follows that the entire network of parallel branches may be simplified
into a circuit consisting of just a voltage source Vm in series with a
single resistor Rm, where Rm = 1/(1/R1 + 1/R2 + 1/R3 + ... + 1/Rn).
The current Im through this simple circuit is given by: Im = Vm / Rm, or
Im = V1/R1 + V2/R2 + V3/R3 + ... + Vn/Rn.
Millman's
Theorem, therefore, simplifies the computation of the voltage across the
parallel branches of a circuit. Note that Millman's Theorem is
only applicable to circuits that can be redrawn as a network of parallel
branches,
with each branch consisting of a resistor or a resistor in series
with a voltage source or current source.
Millman's
Theorem may also be written in terms of the impedance Z or admittance Y
(Z = 1/Y) through each branch. Thus,
Vm = (V1Y1
+ V2Y2 + V3Y3 + ... + VnYn) / (Y1 + Y2 + Y3 + ... + Yn);
Zm = 1 /
(Y1 + Y2 + Y3 + ... + Yn); and
Im = V1Y1 +
V2Y2 + V3Y3 + ... + VnYn.
Figure 1.
Illustration of Millman's Theorem
In summary,
Millman's Theorem is just a tool used for the quick computation of the
voltage across and the current through a circuit that consists of
nothing but parallel branches of a resistor in series with a voltage
source.
See Also:
Kirchhoff's Voltage Law;
Kirchhoff's Current Law;
Thevenin's Theorem;
Norton's Theorem
