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5.9 
mILLman’S theorem
Consider a set of n independent voltage sources, each 
in series with a resistance representing its internal 
resistance. Assume that these n sources are connected 
in parallel and are supplying a common load. We wish 
to replace these n sources by a single independent 
voltage source in series with a resistance. The relevant 
circuit is shown in Fig. 5.9-1.
We can achieve our objective by applying Source 
Transformation Theorem repeatedly. Each voltage 
source is replaced by a current source in parallel with 
a resistor across the terminal pair a
-
b. The current 
source will have a value equal to the voltage of the voltage source multiplied by the conductance of 
resistor in series with it. The circuit after this replacement is shown in Fig. 5.9-2.
a
b
G
n
G
2
G
1
G
1
V
1
G
2
V
2
G
n
V
n
Fig. 5.9-2 
Circuit in Fig. 5.9-1 after source transformation
G
1
, G
2
,
…,
G
n
are the conductances of the resistors. Current sources in parallel can be replaced by 
a single current source with a source function equal to the sum of source functions of each current 
source. Conductances in parallel can be replaced by a single conductance of value equal to the sum of 
conductances. This circuit reduction process results in a single current source in parallel with a single 
resistance. See Fig. 5.9-3 (a). This can be replaced by a single voltage source in series with a single 
resistance as shown in Fig. 5.9-3 (b).
a
b
(a)
I
eq
R
eq
= 1
G
eq
V
eq
= 
R
eq
I
eq
a
b
(b)
–
+
R
eq
Fig. 5.9-3 
(a) Current source equivalent (Norton’s equivalent) (b) Voltage source 
equivalent (Thevenin’s equivalent) for the circuit in Fig. 5.9-1
a
b
V
1
V
2
–
+
–
+
–
+
R
2
R
n
R
1
V
n
Fig. 5.9-1 
Practical voltage sources 
connected in parallel

---

Summary 
5.37
Thus, n ideal independent voltage sources of voltage values V
1
, V
2
,…, V
n
each in series with a 
resistance, delivering power to a common load in parallel, can be replaced by a single ideal 
independent voltage source in series with a resistance. The value of voltage source
V
G V
G
R
G
G
R
i
eq
i i
i
n
i
i
n
i
i
n
i
i
=
=
=
=
=
=
=
∑
∑
∑
1
1
1
1
1
;
eq
where
for
to n.
This is known as Millman’s Theorem. This theorem is only a restatement of Source Transformation 
Theorem that is valid under a special context.