Question

Find the equivalent capacitances of the combinations shown in figure (31-E15) between the indicated points.

Answer 1

(a)




Applying star-delta conversion in the part indicated in the diagram.

The capacitance of the C₁ is given by
$$\begin{gathered} C_1\text{'}=\frac{C_2{C}_3}{C_1+C_2+C_3} \\ {C}_2\text{'}=\frac{C_1{C}_3}{C_1+C_2+C_3} \\ {C}_3\text{'}=\frac{C_1{C}_2}{C_1+C_2+C_3} \\ \Rightarrow C_1\text{'}=\frac{3\times 4}{1+3+4}=\frac{12}{8}\mathrm{\mu F} \\ \Rightarrow C_2\text{'}=\frac{1\times 3}{1+3+4}=\frac{3}{8}\mathrm{\mu F} \\ \Rightarrow C_3\text{'}=\frac{1\times 4}{1+3+4}=\frac{4}{8}\mathrm{\mu F} \end{gathered}$$
Thus, the equivalent circuit can be drawn as:


Therefore, the equivalent capacitance is given by
C_eq = $\frac{3}{8}+\left[\frac{\left(3+{\displaystyle \frac{1}{2}}\right)\times \left({\displaystyle \frac{3}{2}}+1\right)}{\left(3+{\displaystyle \frac{1}{2}}\right)+\left({\displaystyle \frac{3}{2}}+1\right)}\right]=\frac{3}{8}+\frac{35}{24}=\frac{9+35}{24}=\frac{11}{6}\mathrm{\mu F}$


(b) By star-delta conversion,






$1+1=2\mu \mathrm{F}$
(c)
It is a balanced bridge.
Therefore, the capacitor of capacitance 5 μF can be removed.

The capacitance of the two branches are:
$$\begin{gathered} C_1=\frac{2\times 4}{2+4}=\frac{4}{3}\mathrm{\mu F} \\ {C}_2=\frac{4\times 8}{4+8}=\frac{8}{3}\mathrm{\mu F} \end{gathered}$$

∴ Equivalent capacitance $=\frac{4}{3}+\frac{8}{3}+4$$=8\mu \mathrm{F}$

(d)

It is also a balanced diagram.



It can be observed that the bridges are balanced.
Therefore, the capacitors of capacitance 6 μF between the branches can be removed

The capacitances of the four branches are:
$$\begin{gathered} C_1=\frac{2\times 4}{2+4}=\frac{4}{3}\mathrm{\mu F} \\ {C}_2=\frac{4\times 8}{4+8}=\frac{8}{3}\mathrm{\mu F} \\ {\mathrm{C}}_3=\frac{4\times 8}{4+8}=\frac{8}{3}\mathrm{\mu F} \\ {\mathrm{C}}_4=\frac{2\times 4}{2+4}=\frac{4}{3}\mathrm{\mu F} \end{gathered}$$
∴ Equivalent capacitance

$$\begin{gathered} =\frac{4}{3}+\frac{8}{3}+\frac{8}{3}+\frac{4}{3} \\ =8\mathrm{\mu F} \end{gathered}$$