PlanetPhysics/Capacitor Networks

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Capacitors in networks cannot always be grouped into simple series or parallel combinations. As an example, the figure shows three capacitors , , and in a delta network , so called because of its triangular shape. This network has three terminals , , and and hence cannot be transformed into a sinle equivalent capacitor. \begin{figure}

\includegraphics{circuit1.eps} \caption{The delta network}

\end{figure} It can be shown that as far as any effect on the external circuit is concerned, a delta network is equivalent to what is called a Y network . The name "Y network" also refers to the shape of the network. \begin{figure}

\includegraphics{circuit2.eps} \caption{The Y network}

\end{figure} I am going to show that the transformation equations that give , , and in terms of , , and are

The potential difference must be the same in both circuits, as must be. Also, the charge that flows from point along the wire as indicated must be the same in both circuits, as must . Now, let us first work with the delta circuit. Suppose the charge flowing through is and to the right. According to Kirchoff's first rule: Lets play with the equation a little bit.. From Kirchoff's second law: Therefore we get the equation:

Similarly, we apply the rule to the right part of the circuit: We then get the second equation

Solving (1) and (2) simultaneously for and , we get: Keeping these in mind, we proceed to the Y network. Let us apply Kirchoff's second law to the left part: From conservation of charge, Simplifying the above equation yields: Similarly for the right part: The coefficients of corresponding charges in corresponding equations must be the same for both networks. i.e. we compare the equations for and for both networks. Immediately by comparing the coefficient of in we get: Now compare the coefficient of : Substitute the expression we got for , and solve for to get: Now we look at the coeffcient of in the equation for : Again substituting the expression for and solving for we get: We have derived the required transformation equations mentioned at the top.