Question 2.34: Using the superposition theorem, find the power delivered by......

Let,I_S= Current in the series branch with the 20 V source and 33 W in series.

I_S^{\prime}= Current in the series branch when the 20 V source alone is acting.

I_S^{\prime\prime}= Current in the series branch when the 10 V source alone is acting.
Now, by the superposition theorem,

I_S=I_S^{\prime}+I_S^{\prime \prime}

Note : Since power is proportional to the square of voltage, it is not a linear quantity. So, power cannot be determined directly by the superposition theorem.

To find the response I_S^{\prime} when the 20 V source is acting alone

The 10 V source is replaced with a short circuit as shown in Fig. 2.
The delta-connected resistances 22 W, 47 W and 68 W in Fig. 2 are converted into a star-connected resistance as shown in Fig. 3.

\begin{aligned}& R _1=\frac{22 \times 68}{22+47+68}=10.9197 \Omega \\ & R _2=\frac{22 \times 47}{22+47+68}=7.5474 \Omega \\& R _3=\frac{47 \times 68}{22+47+68}=23.3285 \Omega \end{aligned}

With reference to Fig. 5, by Ohm’s law,

I_S^{\prime}=\frac{20}{7.5474+12.8524+33}=0.3745 A

To find the response I_S^{\prime\prime} when the 10 V source is acting alone

The 20 V source is replaced with a short circuit as shown in Fig. 6. Now the current I_S^{\prime\prime} is solved by mesh analysis.

With reference to Fig. 6, the mesh basis matrix equation is,

Let us define two determinants \Delta \text { and } \Delta_1 as shown below and mesh current I_1 is solved by Cramerr’s rule.

To find the total response I_S and power

By the superposition theorem,

=0.3745+(-0.0411)=0.3334 A

Power delivered by 20 V source =20 \times I_1=20 \times 0.3334 = 6.668 \ W