Parallel Resistors

Equivalent Resistance

Circuits Resistance Parallel

Parallel Resistors:

Equivalent Resistance

Introduction

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:

\[{i_1} = \frac{v}{{{R_1}}}\]
(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)
\[v = \frac{1}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}}} \cdot i\]
If we set: \({R_{eq}} = \frac{1}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}}}\) , then we can draw Fig. 1(b) as being equivalent to Fig. 1(a).

Figure 1. Parallel Resistances and equivalent circuit.

Important Generalization

This result can be generalized to a parallel combination of N resistances as follows:

A parallel combination of N resistors R₁, R₂,..., 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

Conductance is an important quantity in circuit design and analysis.

Conductance is simply the reciprocal of resistance, defined as:

(6)
\[G = \frac{1}{R}\]
The unit for conductance is siemens, abbreviated S¹ . Ohm's law, written in terms of conductance, is:

(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.

Conductance of Parallel Resistors

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!

If we write equations (1) in terms of conductance, we get:

(8)
\[\begin{array}{l}{i_1} = {G_1}v\\{i_2} = {G_2}v\end{array}\]
Then equation (3) becomes:

(9)
\[i = \left[ {{G_1} + {G_2}} \right]v\]
(10)
\[i = {G_{eq}}v\]

Where:

(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.

Important Generalization:

This result can be generalized to a parallel combination of N resistances as follows:

A parallel combination of N resistors with conductances G₁, G₂,..., G_N can be replaced with a single equivalent resistance.

\[{G_{eq}} = {G_1} + {G_2} + \cdots + {G_N}\]

Checking Results

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:

The equivalent resistance for a parallel combination of N resistors will always be less than the smallest resistance in the combination. In fact, the equivalent resistance will always obey the following inequalities:

\[\frac{{{R_{\min }}}}{N} \le {R_{eq}} \le {R_{\min }}\]

Where R_min is the smallest resistance value in the parallel combination.

In a parallel combination of resistances, the resistor with the smallest resistance will have the largest current and the resistor with the largest resistance will have the smallest current.

Example 1:

Determine the equivalent resistance seen by the source in the circuit below.

Solution:

All three resistors in the circuit are in parallel, so the equivalent resistance is:

\[{R_{eq}} = \frac{1}{{\frac{1}{{4k\Omega }} + \frac{1}{{4k\Omega }} + \frac{1}{{2k\Omega}}}} = 1k\Omega\]

Alternately, equation (5) can be used to combine the resistors two at a time. The combination of the two 4k resistors is:

\[{R_{eq1}} = \frac{{(4k\Omega )(4k\Omega )}}{{4k\Omega + 4k\Omega }} = 2k\Omega\]

This resistance is in parallel with the 2k resistor, so the overall resistance is:

\[{R_{eq}} = \frac{{(2k\Omega )(2k\Omega )}}{{2k\Omega + 2k\Omega }} = 1k\Omega\]

Which is the same result as before.

Example 2:

Replace any parallel resistors in the circuit below with their equivalent resistance.

Solution:

The 5k and 20k resistors are in parallel, and have an equivalent resistance of:

\[{R_{eq1}} = \frac{{(5k\Omega )(20k\Omega )}}{{5k\Omega + 20k\Omega }} = 4k\Omega\]

The 3k and 6k resistors are also in parallel, and have an equivalent resistance of:

\[{R_{eq1}} = \frac{{(3k\Omega )(6k\Omega )}}{{3k\Omega + 6k\Omega }} = 2k\Omega\]

So the circuit can be converted to:

Test Your Knowledge!

Where it is possible in the circuit below, replace parallel resistances with their equivalent resistance.

Where possible in the circuit below, replace parallel resistances with their equivalent resistance.

Where possible in the circuit below, replace parallel resistances with their equivalent resistance.

Using only the fixed resistors from the Digilent analog parts kit, create resistors with resistances within 5% of the following values. (You may assume that the resistors in the analog parts kit have resistances which are exactly their nominal values.)
- 5kΩ
- 5Ω
- 8kΩ

Selected Answers:

- Two, 10 kΩ resistors in parallel will give 5 kΩ.
- Two, 10 Ω resistors in parallel will give 5 Ω.
- A 10kΩ resistor and a 47 kΩ resistor in parallel are equivalent to about 8.2 kΩ. This is within 5% of the desired 8 kΩ.

¹ Prior to 1971, the siemens was called the mho (and had the symbol ℧). This term may still be found in older reference books.