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
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.