Electrical Network Analysis

(EE-151)

Lecture-07
Network Theorems
(Millman’s and Tellegen’s Theorems)

Dr. M. N. Alam, EED, NITW

 Outlines
©2023 Dr. M. N. Alam

This Lecture covers:

• Millman’s theorem
• Tellegen’s theorem

10-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 2

 Millman’s Theorem
©2023 Dr. M. N. Alam
• It states that “if there are 𝑛 voltage sources 𝑉1 , 𝑉2 , … , 𝑉𝑛 with internal
resistances 𝑅1 , 𝑅2 , … , 𝑅𝑛 respectively connected in parallel then these
voltage sources can be replaced by a single voltage source 𝑉𝑚 and a single
series resistance 𝑅𝑚 .”

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 Millman’s Theorem
©2023 Dr. M. N. Alam

• Where

𝑉1 𝐺1 + 𝑉2 𝐺2 + ⋯ + 𝑉𝑛 𝐺𝑛
𝑉𝑚 =
𝐺1 + 𝐺2 + ⋯ + 𝐺𝑛

𝑉1 /𝑅1 + 𝑉2 /𝑅2 + ⋯ + 𝑉𝑛 /𝑅𝑛

=
1/𝑅1 + 1/𝑅2 + ⋯ + 1/𝑅𝑛

1 1
𝑅𝑚 = =
𝐺𝑚 𝐺1 + 𝐺2 + ⋯ + 𝐺𝑛

1
=
1/𝑅1 + 1/𝑅2 + ⋯ + 1/𝑅𝑛

10-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 4

 Millman’s Theorem
©2023 Dr. M. N. Alam
Explanation of Millman’s theorem:
• By source transformation, each voltage source in series with a resistance can be
converted to a current source in parallel with a resistance as shown below

• Let 𝐼𝑚 be the resultant current of the parallel current sources and 𝑅𝑚 be the
equivalent resistance
𝑉1 𝑉2 𝑉𝑛
𝐼𝑚 = 𝐼1 + 𝐼2 + ⋯ + 𝐼𝑛 = + + ⋯+ = 𝑉1 𝐺1 + 𝑉2 𝐺2 + ⋯ + 𝑉𝑛 𝐺𝑛
𝑅1 𝑅2 𝑅𝑛
1 1 1 1
= + +⋯+ = 𝐺1 + 𝐺2 + ⋯ + 𝐺𝑛 = 𝐺𝑚
𝑅𝑚 𝑅1 𝑅2 𝑅𝑛

10-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 5

 Millman’s Theorem
©2023 Dr. M. N. Alam
• So, we get an equivalent current source as an equivalent parallel resistance

𝐼𝑚 = 𝑉1 𝐺1 + 𝑉2 𝐺2 + ⋯ + 𝑉𝑛 𝐺𝑛

1
𝑅𝑚 =
𝐺1 + 𝐺2 + ⋯ + 𝐺𝑛

• By source transformation, Millman’s equivalent network is obtained as shown below

𝑉1 𝐺1 +𝑉2 𝐺2 +⋯+𝑉𝑛 𝐺𝑛
𝑉𝑚 = 𝐼𝑚 𝑅𝑚 =
𝐺1 +𝐺2 +⋯+𝐺𝑛

10-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 6

 Millman’s Theorem
©2023 Dr. M. N. Alam
Dual of Millman’s Theorem:
• It states that “if there are 𝑛 current sources 𝐼1 , 𝐼2 , … , 𝐼𝑛 with internal
resistances 𝑅1 , 𝑅2 , … , 𝑅𝑛 respectively connected in series then these current
sources can be replaced by a single current source 𝐼𝑚 and a single parallel
resistance 𝑅𝑚 .”

• Where

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 Millman’s Theorem
©2023 Dr. M. N. Alam
Steps to be Followed in Millman’s Theorem
1. Remove the load resistance 𝑅𝐿
2. Find Millman’s voltage across points A and B
𝑉1 𝐺1 + 𝑉2 𝐺2 + ⋯ + 𝑉𝑛 𝐺𝑛
𝑉𝑚 =
𝐺1 + 𝐺2 + ⋯ + 𝐺𝑛
3. Find Millman’s resistance 𝑅𝑚 between points A and B
1
𝑅𝑚 =
𝐺1 + 𝐺2 + ⋯ + 𝐺𝑛
4. Replace the network voltage by a voltage source 𝑉𝑚 in series with
resistance 𝑅𝑚
5. Find the current through 𝑅𝐿 using ohm’s law
𝑉𝑚
𝐼𝐿 =
𝑅𝑚 + 𝑅𝐿

10-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 8

 Millman’s Theorem
©2023 Dr. M. N. Alam

Numerical Example-17:
• Find current through 10 Ω resistor using Millman’s theorem.

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 Millman’s Theorem
©2023 Dr. M. N. Alam

Solution:
Step I: Calculation of 𝑉𝑚

1 1 1
𝑉1 𝐺1 + 𝑉2 𝐺2 + 𝑉3 𝐺3 5 − 10 + 15
2 4 6
𝑉𝑚 = = = 2.73 𝑉
𝐺1 + 𝐺2 + 𝐺3 1 1 1
2+4+6

Step II: Calculation of 𝑅𝑚

1 1 1
𝑅𝑚 = = = = 1.09 Ω
𝐺𝑚 𝐺1 + 𝐺2 + 𝐺3 1 + 1 + 1
2 4 6

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 Millman’s Theorem
©2023 Dr. M. N. Alam
Step III: Millman’s Equivalent

• Step IV: Calculation of 𝐼𝐿

𝑉𝑚 2.73
𝐼𝐿 = = = 0.25 𝐴
𝑅𝑚 + 𝑅𝐿 1.09 + 10

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 Millman’s Theorem
©2023 Dr. M. N. Alam

Numerical Example-18:
• Find Millman’s equivalent for the left of the terminals A-B

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 Millman’s Theorem
©2023 Dr. M. N. Alam

Solution:
• By source transformation,

Step I: Calculation of 𝑉𝑚

1 1
𝑉1 𝐺1 + 𝑉2 𝐺2 6 + 2
4 2
𝑉𝑚 = = = 3.33 𝑉
𝐺1 + 𝐺2 1 1
+
4 2

10-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 13

 Millman’s Theorem
©2023 Dr. M. N. Alam
Step II: Calculation of 𝑅𝑚

1 1 1
𝑅𝑚 = = = = 1.33 Ω
𝐺𝑚 𝐺1 + 𝐺2 1 + 1
4 2

Step III: Millman’s Equivalent

10-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 14

 Tellegen’s Theorem
©2023 Dr. M. N. Alam
• Tellegen’s theorem can be stated as
• In any linear, non-linear, passive, active, time-variant or time-invariant
network, the summation of power (instantaneous or complex power for ac
sources) is zero.
• In other words, if a network is having n branches
𝑛

෍ 𝑉𝑘 𝐼𝑘 = 0
𝑘=1
• In order to justify this theorem in electrc network, the first step is to find the
branch voltage drops and the corresponding branch currents using
conventional analysis.
• Telegen’s theorem can then be justified by summing all products of branch
voltages and currents.

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 Tellegen’s Theorem
©2023 Dr. M. N. Alam

HW Problem:
• Justify Tellegen’s theorem in the network shown below.

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