Let
us assume a load R
L
be connected to a DC source network whose Thevenin’s equivalent
gives V
0
as the Thevenin’s voltage and R
TH
as the Thevenin’s resistance as shown in the figure
below.
Here,
I =
V
0
R
TH
+R
L
… … … … … . . (1)
Let the load resistance RL be changed to (RL + ΔRL). Since the
rest of the circuit remains
unchanged, the Thevenin’s equivalent network remains the same as shown in the circuit diagram
below
Here,
I
′
=
V
0
R
TH
+ (R
L
+
Δ
R
L
)
… … … … … … … . . (2)
The change of current being termed as ΔI Therefore,
Δ
I = I
′
− I … … … … … … … … … . . (3)
Putting the value of I’ and I from the equation (1) and (2) in the equation (3) we will get the
following equation.
Δ
I =
V
0
R
TH
+ (R
L
+
Δ
R
L
)
−
V
0
R
TH
+ R
L
ΔI
=
V
0
{(R
TH
+R
L
)−(R
TH
+(R
L
+ΔR
L
)}
(R
TH
+(R
L
+ΔR
L
))×(R
TH
+R
L
)
ΔI = − [
V
0
R
TH
+R
L
]
R
TH
R
TH
+(R
L
+
Δ
R
L
)
……..(4)
Now, putting the value of I from the equation (1) in the equation (4), we will get the following
equation.
I = −
IR
TH
R
TH
+(R
L
+
Δ
R
L
)
… … … … . (5)
As we know, V
C
= I Δ RL and is known as compensating voltage. Therefore, the equation (5)
becomes.
ΔI =
−V
C
R
TH
+ (R
L
+
Δ
R
L
)
Hence, Compensation Theorem tells that with the
change of branch resistance, branch currents
changes and the change is equivalent to an ideal compensating voltage source in series with the
branch
opposing the original current, all other sources in the network being replaced by their
internal resistances.
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