The load current
I
in the circuit shown above is given by,
𝐼 =
𝑉
𝑇𝐻
𝑅
𝑇𝐻
+𝑅
𝐿
The power delivered by the circuit to the load:
𝑃 = 𝐼
2
𝑅 =
𝑉
𝑇𝐻
2
(𝑅
𝑇𝐻
+𝑅
𝐿
)
2
𝑅
𝐿
The condition for maximum power transfer can be obtained by differentiating the above
expression for power delivered with respect to the load resistance (Since we want to find out the
value of
R
L
for maximum power transfer) and equating it to zero as :
𝜕𝑃
𝜕𝑅
𝐿
= 0 =
𝑉
𝑇𝐻
2
(𝑅
𝑇𝐻
+𝑅
𝐿
)
2
−
2𝑉
𝑇𝐻
2
(𝑅
𝑇𝐻
+𝑅
𝐿
)
3
𝑅
𝐿
= 0
Simplifying the above equation, we get:
(𝑅
𝑇𝐻
+ 𝑅
𝐿
) − 2𝑅
𝐿
= 0 ⟹
𝑅
𝐿
= 𝑅
𝑇𝐻
Under the condition of maximum power transfer, the power delivered to the load is given by :
𝑃
𝑀𝐴𝑋
=
𝑉
𝑇𝐻
2
(𝑅
𝐿
+𝑅
𝐿
)
2
× 𝑅
𝐿
=
𝑉
𝑇𝐻
2
4𝑅
𝐿
Under the condition
of maximum power transfer, the efficiency
𝜼
of
the network is then given
by:
𝑃
𝐿𝑂𝑆𝑆
=
𝑉
𝑇𝐻
2
(𝑅
𝐿
+𝑅
𝐿
)
2
× 𝑅
𝑇𝐻
=
𝑉
𝑇𝐻
2
4𝑅
𝐿
𝜼 =
output
input
=
𝑉
𝑇𝐻
2
4𝑅
𝐿
(
𝑉
𝑇𝐻
2
4𝑅
𝐿
+ 𝑉
𝑇𝐻
2
4𝑅
𝐿
)
= 0.50
For maximum power transfer the load resistance should be equal to the Thevenin equivalent
resistance ( or Norton equivalent resistance) of the network to which it is connected . Under the
condition of maximum power transfer the efficiency of the system is 50 %.
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