Electrical Network Analysis

(EE-151)

Lecture-05
Network Theorems
(Maximum Power Transfer Theorem)

Dr. M. N. Alam, EED, NITW

 Outlines
©2023 Dr. M. N. Alam

This Lecture covers:

• Maximum power transfer theorem
• Numerical examples

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

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam
• It states that “the maximum power is delivered from a source to a load when
the load resistance is equal to the source resistance.”
Proof:
• Current flowing in the circuit
𝑉
𝐼=
𝑅𝑆 + 𝑅𝐿
• Power delivered to the load 𝑅𝐿
𝑉2
𝑃= 𝐼 2 𝑅𝐿 = 2 𝑅𝐿
𝑅𝑆 + 𝑅𝐿
• To determine the value of 𝑅𝐿 for maximum power to be transferred to the load,
𝑑𝑃
=0
𝑑𝑅𝐿

𝑑𝑃 𝑑 𝑉2 𝑉 2 𝑅𝑆 + 𝑅𝐿 2
− 2𝑅𝐿 𝑅𝑆 + 𝑅𝐿
= 2 𝑅𝐿 =
=0
𝑑𝑅𝐿 𝑑𝑅𝐿 𝑅𝑆 + 𝑅𝐿 𝑅𝑆 + 𝑅𝐿 4

04-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 3

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam
𝑅𝑆 + 𝑅𝐿 2 − 2𝑅𝐿 𝑅𝑆 + 𝑅𝐿 = 0

𝑅𝑆2 + 𝑅𝐿2 + 2𝑅𝑆 𝑅𝐿 − 2𝑅𝐿 𝑅𝑆 − 2𝑅𝐿2 = 0

𝑅𝑆2 − 𝑅𝐿2 = 0

𝑅𝑆 = 𝑅𝐿
• Hence, the maximum power will be transferred to the load when load resistance is
equal to the source resistance.

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

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam

During maximum power transfer:

• Current flowing in the circuit
𝑉 𝑉
𝐼= =
𝑅𝑆 + 𝑅𝐿 2𝑅𝑆

• Power delivered to the load 𝑅𝐿

2
𝑉 𝑉2
𝑃𝐿,𝑚𝑎𝑥 = 𝐼2 𝑅𝐿 = 𝑅 =
2 𝑆
2𝑅𝑆 4𝑅𝑆
• Power supplied by the source
𝑉2 𝑉2
𝑃𝑇𝑜𝑡𝑎𝑙 = 𝐼2 𝑅𝐿 + 𝐼2 𝑅𝑆 = 2
(2𝑅𝑆 ) =
2𝑅𝑆 2𝑅𝑆
• Efficiency during maximum power transfer,
𝑃𝑜𝑤𝑒𝑟 𝑑𝑒𝑙𝑖𝑣𝑒𝑟𝑒𝑑 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑜𝑎𝑑 𝑉 2 /4𝑅𝑆 1
𝜂= = = = 0.5 (𝑜𝑟 50 %)
𝑃𝑜𝑤𝑒𝑟 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑠𝑜𝑢𝑟𝑐𝑒 𝑉 2 /2𝑅𝑆 2
04-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 5
 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam
Steps to be Followed in Maximum Power Transfer Theorem
1. Remove the variable load resistance 𝑅𝐿
2. Find open circuit voltage 𝑉𝑇ℎ across A and B
3. Find the resistance 𝑅𝑇ℎ as seen from points A and B
4. Find resistance 𝑅𝐿 for maximum power transfer
𝑅𝐿 = 𝑅𝑇ℎ
1. Find the maximum power,
𝑉𝑇ℎ 𝑉𝑇ℎ
𝐼𝐿 = =
𝑅𝑇ℎ + 𝑅𝐿 2𝑅𝑇ℎ

2 2
𝑉𝑇ℎ 𝑉𝑇ℎ
𝑃𝑚𝑎𝑥 = 𝐼𝐿2 𝑅𝐿 = 2 𝑅𝑇ℎ =
4𝑅𝑇ℎ 4𝑅𝑇ℎ

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

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam

Numerical Example-13:
• For the value of resistance 𝑅𝐿 for maximum power transfer and calculate the
maximum power.

04-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 7

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam

Solution:
Step I: Calculation of 𝑉𝑇ℎ ,
• By star-delta transformation;
100 100
𝐼= = = 2.08 𝐴
5 + 5 + 20 + 9 + 9 48

• Writing the 𝑉𝑇ℎ equation,

100 − 5𝐼 − 𝑉𝑇ℎ − 9𝐼 = 0
𝑉𝑇ℎ = 100 − 14𝐼
= 100 − 14 2.08
= 70.88 𝑉
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 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam
Step II: Calculation of 𝑅𝑇ℎ ,

𝑅𝑇ℎ = 23.92 Ω

04-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 9

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam
Step III: Calculation of 𝑅𝐿 ,
• For maximum power transfer,
𝑅𝐿 = 𝑅𝑇ℎ = 23.92 Ω

Step IV: Calculation of 𝑃𝑚𝑎𝑥 ,

• The maximum power transfer,
2
𝑉𝑇ℎ
𝑃𝑚𝑎𝑥 =
4𝑅𝑇ℎ
70.882
=
4 × 23.92
= 52.51 𝑊

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

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam

Numerical Example-14:
• For the value of resistance 𝑅𝐿 for maximum power transfer and calculate the
maximum power.

04-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 11

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam

Solution:
Step I: Calculation of 𝑉𝑇ℎ ,

• From the figure, we can write that;

𝐼𝑥 = 𝐼2 … . (𝑖)
𝐼1 = 10 𝐴 … . (𝑖𝑖)
• Applying KVL to Mesh 2,
2𝐼𝑥 − 6𝐼2 − 4 𝐼2 − 𝐼1 = 0
2𝐼2 − 6𝐼2 − 4 𝐼2 − 𝐼1 = 0
4𝐼1 − 8𝐼2 = 0 … . (𝑖𝑖𝑖)

04-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 12

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam
• Solving Eqs (ii) and (iii),
𝐼1 = 10 𝐴
𝐼2 = 5 𝐴
• Writing the 𝑉𝑇ℎ equation,
6𝐼2 − 0 − 𝑉𝑇ℎ = 0
𝑉𝑇ℎ = 6𝐼2
= 30 𝑉
Step II: Calculation of 𝐼𝑁 ,

• From the figure, we can write that;

𝐼𝑥 = 𝐼2 − 𝐼3 … . (𝑖𝑣)
𝐼1 = 10 𝐴 … . (𝑣)
04-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 13
 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam
• Applying KVL to Mesh 2,
2𝐼𝑥 − 6 𝐼2 − 𝐼3 − 4 𝐼2 − 𝐼1 = 0
2 𝐼2 − 𝐼3 − 6 𝐼2 − 𝐼3 − 4 𝐼2 − 𝐼1 = 0
4𝐼1 − 8𝐼2 + 4𝐼3 = 0 … . (𝑣𝑖)
• Applying KVL to Mesh 3,
−3𝐼3 − 6 𝐼3 − 𝐼2 = 0
6𝐼2 − 9𝐼3 = 0 … . (𝑣𝑖𝑖)
• Solving Eqs (v), (vi) and (vii),
𝐼1 = 10 𝐴
𝐼2 = 7.5 𝐴
𝐼3 = 5 𝐴
𝐼𝑁 = 5 𝐴
Step III: Calculation of 𝑅𝑇ℎ ,
𝑉𝑇ℎ 30
𝑅𝑇ℎ = = =6Ω
𝐼𝑁 5

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

 Maximum Power Transfer Theorem
©2023 Dr. M. N. Alam
Step IV: Calculation of 𝑅𝐿 ,
• For maximum power transfer,
𝑅𝐿 = 𝑅𝑇ℎ = 6 Ω

Step V: Calculation of 𝑃𝑚𝑎𝑥 ,

• The maximum power transfer,
2
𝑉𝑇ℎ
𝑃𝑚𝑎𝑥 =
4𝑅𝑇ℎ
302
=
4×6
= 37.5 𝑊

04-04-2023 EE151 - Dr. M. N. Alam, EED, NITW 15