Parallel Combinations of Resistors

Introduction

As the name suggests, a resistor is an electrical device used to resist current flow to the desired extent. This resistance helps control the electrical circuit and offers better operation with increased protection. However, when a single resistor alone does not prove to be efficient in resisting the flow of current, two or more combinations of resistance are used to control the current flow.

Accordingly, there are two primary ways in which resistors are connected. The first is a series combination of resistors, and the second is a parallel combination of resistors. Both these combinations have several differences and are used according to the objective of resistance. For example, the total net resistance offered in the case of a series combination is generally more than that offered by a parallel combination. On the other hand, the parallel combination offers better electrical control when compared with the series combination.

So, let us look at the complete theory behind parallel combinations of resistors.

Parallel Combinations of Resistors

When two or more resistors are connected to have the same starting and ending point, the connection is known as a parallel combination of resistors. Also, each resistor connected in parallel directly connects to the battery, which is not the case with a series combination. Every resistor has a start, and endpoint in a series combination, and not every resistor is directly connected to the battery.

In addition to the standard starting and ending point, several other properties of the parallel connection make it different from the series combination. All the essential properties of a parallel combination with the derivation on determining its total net resistance are discussed in detail in the following sections.

Properties of a Parallel Combination

• Multiple paths for current to flow

A parallel combination of resistors offers multiple paths for electrical current to flow. Accordingly, each resistor has a flow path with a different current flow rate. Although each connection has a different flow of current, the total net current flow through the resistance is the same as the overall flow of current coming from the source.

Also, the total equivalent current flow is equal to the sum of all individual flows through each path and is denoted by Ieq. 

For example, if there are three resistors connected in parallel, there are a total of three flow paths with the current flow as I1, I2 , and I3 . 

Then, the total current passed through this circuit will be given as follows:

Total current, Ieq = I1 + I2  + I3 

• Constant potential differences throughout the circuit

As explained earlier, each resistor in parallel is directly connected to the battery. And for the same reason, the potential difference is constant throughout the electrical circuit. This means that each resistor has the same potential difference as its batteries. The potential difference is the battery’s voltage used in the electrical circuit, denoted by V.

• More effortless operation of the electrical circuit

As there are multiple paths available in the parallel combination of resistance, the overall operation of the circuit becomes easier. For example, even if one of the paths for current flow is disturbed, for some reason, it will not affect the operation of the entire circuit, making it more manageable. On the other hand, a series connection does not offer this feature as there is a single path for the current to flow.

• Total or equivalent combinations of resistance are always less than all individual resistances

In parallel combinations, the total resistance offered is always less than that of the individual resistors. A complete derivation of the formula to determine equivalent resistance for parallel combinations of resistors is explained further in the article. Also, the reciprocal of net resistance is the sum of the reciprocals of all resistances offered by each resistor.

Derivation of the Formula determining Total Resistance Offered in Parallel Combination

According to Ohm’s law, 

V = I x R

Therefore,

I = V/R 

If three resistors R1, R2, and R3 are connected in a parallel circuit.

Then the total current passed through the circuit (I) will be equal to

I = I1 + I2 + I3 … (1)

Where, 

I1 = V/R1

I2 = V/R2

I3 = V/R3

Substituting all values in equation (1),

V/R = V/R1 + V/R2 + V/R3

Taking out V common,

V(1/R) = V(1/R1) + V(1/R2) + V(1/R3)

V(1/R) = V (1/R1 + 1/R2 + 1/R3) 

Cancelling out V from both sides

1/R = 1/R1 + 1/R2 + 1/R3

Thus, the reciprocal of the equivalent resistance offered equals the sum of the reciprocals of the resistance offered by each resistor connected in a parallel combination.