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.