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Parallel Circuit

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.

 figure776
Figure 1.10: Parallel connection 

 figure842
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 tex2html_wrap_inline5513. Since all voltages across the resistors are equal to V, we have tex2html_wrap_inline5517, and the equivalent conductance as seen from the left end is tex2html_wrap_inline5519, as shown in figure 1.11. Note that the expression tex2html_wrap_inline5521 can be written as
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For example, if there are only two resistors, the equivalent resistance of the parallel circuit is
displaymath5466

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 tex2html_wrap_inline5523 to replace the circuit such that the input current and voltage are unaffected.

 figure919
Figure 1.12: Series/parallel reduction process 

First of all, we observe that tex2html_wrap_inline5541 and tex2html_wrap_inline5543 are in parallel and can be replaced by an equivalent resistance R' which is given by
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This R' is connected in series with tex2html_wrap_inline5549, and the resulting sub-circuit is connected in parallel with tex2html_wrap_inline5551. Thus, using the series formula, followed by the parallel formula, we get an equivalent resistance R'' which represents the part of the circuit covering tex2html_wrap_inline5549, tex2html_wrap_inline5551, tex2html_wrap_inline5541 and tex2html_wrap_inline5543, i.e.,
displaymath5468
Finally, adding tex2html_wrap_inline5563 to R'' yields the equivalent resistance tex2html_wrap_inline5523 as required.
displaymath5892


next up previous
Next: Ladder Circuit Up: Series and Parallel Circuits Previous: Series Circuit

Michael Tse
Tue Mar 10 13:15:28 HKT 1998