“In a network of linear resistances containing more than
one generator,the current which flows at any point is the sum of all the
currents which would flow at that point if each generator where considered separately
and all the other generators replaced for the time being by resistances equal
to their internal resistances ”
Explanation
This theorem is only applicable to the linear networks where
current is linearly related to voltage as per ohm’s law. This theorem states
that if some emf’s act simultaneously in any linear bilateral network, than
each emf will act freely. The amount of current in any conductor is the
algebraic sum of the voltages that each emf would have produced while
circulating freely.
In other words, any conductor of the network is obtained by
superimposing the currents and a voltage due to each emf is present in the
network. Figure (a) shows the conditions obtained when left hand side battery
acts alone whereas the figure (b) shows that the right hand side battery acts
alone in the circuit.
By combining the current values of Figure a and Figure b, we
get
I1=I1’-I1’’;
I2=I2’’-I2; I=I’+I’’
Example
By using Superposition Theorem, find the current in resistance R shown
in Figure.
R1 = 0.005 Ω, R2 = 0.004 Ω, R = 1 Ω, E1 = 2.05 V, E2 = 2.15 V
Internal resistances of cells are
negligible.
Solution.
In Fig. (b) E2 has been removed.
Resistances of 1 Ω and 0.04 Ω are in parallel
across poins A and C. RAC = 1 || 0.04 = 1 × 0.04/1.04 = 0.038 Ω. This resistance is in
series with = I1 + I2 = 0.896 + 1.16 = 2.056 A.
0.05 Ω. Hence, total resistance offered to battery E1 = 0.05 + 0.038 = 0.088 Ω. I
= 2.05/0.088 = 23.3 A.
Current through 1-Ω resistance, I1 = 23.3 × 0.04/1.04 = 0.896 A from C to A.
When E1 is removed, circuit becomes as shown in Fig.(c).
Combined resistance of paths CBA
and CDA is = 1 || 0.05 = 1 × 0.05/1.05 = 0.048 Ω.
Total resistance offered to E2 is = 0.04 + 0.048 = 0.088 Ω.
Current I
= 2.15/0.088 = 24.4 A.
Again, I2 = 24.4 × 0.05/1.05 = 1.16 A.
To current through 1-Ω resistance when both batteries are present