When devices are connected such that one terminal of each device is connected to a node of the circuit while the other terminals of the elements are connected to another node of the circuit, the resulting circuit is said to be a parallel circuit.

**Figure 1.10:** Parallel connection

**Figure 1.11:** Equivalent conductance for *n* conductors in parallel

A parallel circuit consisting of *n* resistors is shown in
figure 1.10. Note that we use conductance instead of
resistance in this case. Clearly, KCL gives .
Since all voltages across the resistors are equal to *V*, we have
, and the equivalent conductance as seen
from the left end is , as shown in
figure 1.11. Note that the expression
can be written as

For example, if there are only two resistors, the equivalent
resistance of the parallel circuit is

*Remarks* -- Although the choice between using resistance and
conductance in analysis is arbitrary, it is preferable to perform
calculations in terms of resistance for the case of series connection,
but in terms of conductance in the case of parallel connection.
As seen from the above derivation, algebraic brevity is an obvious
advantage of making this choice.

**Example 1.1: Illustration of series/parallel reduction** -- It
is possible to reduce an assembly of resistors whose configuration is
based on series and parallel connections. Referring to the circuit of
figure 1.12, we can use an equivalent resistance
to replace the circuit such that the input current and voltage
are unaffected.

**Figure 1.12:** Series/parallel reduction process

First of all, we observe that and are in parallel
and can be replaced by an equivalent resistance *R*' which is
given by

This *R*' is connected in series with , and the resulting
sub-circuit is connected in parallel with . Thus, using
the series formula, followed by the parallel formula, we get
an equivalent resistance *R*'' which represents the part of the
circuit covering , , and , i.e.,

Finally, adding to *R*'' yields the equivalent resistance
as required.

Tue Mar 10 13:15:28 HKT 1998