Circuits which consist of resistors connected in parallel can be simplified. Consider the resistive circuit shown in Fig. 1(a). The resistors are connected in parallel, so both resistors have a voltage difference of V.
Ohm's law, applied to each resistor results in:
(1)
\[{i_2} = \frac{v}{{{R_2}}}\]Applying KCL at node “a”:
(2)
\[i = {i_1} + {i_2}\]Substituting equations (1) into equation (2):
(3)
\[i = \left[ {\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}} \right]v\](4)
\[v = \frac{1}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}}} \cdot i\]This result can be generalized to a parallel combination of N resistances as follows:
A parallel combination of N resistors R_{1}, R_{2},..., R_{N} can be replaced with a single equivalent resistance:
(5)
\[{R_{eq}} = \frac{1}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \cdots\frac{1}{{{R_N}}}}}\]The equivalent circuit can be analyzed to determine the voltage across the parallel combination of resistors.
One final comment about notation: two parallel bars are commonly used as shorthand notation to indicate that two circuit elements are in parallel. For example, the notation R1||R2 indicates that the resistors R1 and R2 are in parallel. The notation R1||R2 is often used as shorthand notation for the equivalent resistance of the parallel resistance combination, in lieu of equation (5).
Conductance is an important quantity in circuit design and analysis.
(6)
\[G = \frac{1}{R}\](7)
\[i(t) = Gv(t)\]Some circuit analyses can be performed more easily and interpreted more readily if the elements' resistance is characterized in terms of conductance.
Determining the conductance of a set of parallel resistors can be easier than determining equivalent resistance of the combination. In fact, determining the c onductance of parallel resistors is analogous to determining the resistance of a set of series resistors!
(8)
\[\begin{array}{l}{i_1} = {G_1}v\\{i_2} = {G_2}v\end{array}\](9)
\[i = \left[ {{G_1} + {G_2}} \right]v\](10)
\[i = {G_{eq}}v\](11)
\({G_{eq}} = {G_1} + {G_2}\)So, the equivalent conductance of a pair of parallel resistors is the sum of the conductances of the individual resistors.
This result can be generalized to a parallel combination of N resistances as follows:
A parallel combination of N resistors with conductances G_{1}, G_{2},..., G_{N} can be replaced with a single equivalent resistance.
Calculating the equivalent resistance of a set of parallel resistors is typically more prone to errors than determining the equivalent resistance of a set of series resistors. There are a couple of ways to double-check your results when calculating the equivalent resistance of parallel resistors:
Determine the equivalent resistance seen by the source in the circuit below.
Replace any parallel resistors in the circuit below with their equivalent resistance.