Electrical circuits lecture notes b. Tech

 
Compensation Theorem Statement: 
In a linear, bilateral, time invariant network when the resistance (R) of an 
uncoupled branch, carrying a current (I), is changed by (ΔR). The currents in all the branches 
would change and can be obtained by assuming that an ideal voltage source of (VC) has been 
connected such that VC = I (ΔR) in series with (R + ΔR) when all other sources in the network 
are replaced by their internal resistances. 
In Compensation Theorem, the source voltage (V
C
) opposes the original current. 
In simple words compensation theorem can be stated as – the resistance of any network can be 
replaced by a voltage source, having the same voltage as the voltage drop across the resistance 
which is replaced. 

---

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.