After studying this unit, you should be able to:

■ Discuss the characteristics of a parallel circuit.

■ State three rules for parallel circuits.

■ Solve the missing values in a parallel circuit

using the three rules and Ohm’s Law.

**Parallel circuits** are probably the type of circuit with which most people are familiar. Most devices such as lights and receptacles in homes and office buildings are connected in parallel. Imagine if the lights in your home were wired in series. All the lights in the home would have to be turned on in order for any light to operate, and if one were to burn out, all the lights would go out. The same is true for receptacles. If receptacles were connected in series, some device would have to be connected into each receptacle before power could be supplied to any other device.

*Note: Parallel Circuit are circuits that have more than one path for current flow.*

## Parallel Circuit Values

**Total Current**

Parallel circuits are circuits that have more than one path for

current flow, Figure 6-1. If it is assumed that current leaves terminal A

and returns to terminal B, it can be seen that there are three separate paths that can be taken by the electrons. In Figure 6-1, 3 A of current leaves terminal A. One amp flows through resistor R1 and 2 A flows to resistors R2 and R 3. At the junction of resistors R2 and R3, 1 A flows through resistor R2 and 1 A flows to resistor R 3. Notice that the power supply, terminals A and B, must furnish all the current that flows through each individual resistor or circuit branch. One of the rules for parallel circuits states that the total current flow in the circuit is equal to the sum of the currents through all of the branches. This is known as current adds.

**Circuit Branch:** *The total current flow in the circuit is equal to the sum of the currents through all of the branches; this is known as current adds. *

*The voltage drop across any branch of a parallel circuit is the same as the applied voltage.*

## Voltage Drop

Figure 6-2 shows another parallel circuit and gives the values of voltage and current for each individual resistor or branch. Notice that the voltage drop across each resistor is the same. If the circuit is traced, it can be seen that each resistor is connected directly to the power source. The second rule for parallel circuits states that the voltage drop across any branch of a parallel circuit is the same as the applied voltage. For this reason, most electrical circuits in homes are connected in parallel. Each lamp and receptacle is supplied with 120 V, Figure 6-3.

## Total Resistance

In the circuit shown in Figure 6-4, three separate resistors have values of 15, 10 , and 30 . The total resistance of the circuit, however, is 5. The total resistance of a parallel circuit is always less than the resistance of the lowest-value resistor, or branch, in the circuit. This is because there is more than one path for current flow. Each time

Another element is connected in parallel, there is less opposition to the flow of current through the entire circuit. Imagine a water system consisting of a holding tank, a pump, and return lines to the tank, Figure 6-5. Although large return pipes have less resistance to the flow of water than small pipes, the small pipes do provide a return path to the holding tank. Each time another return path is added, regardless of size, there is less resistance to flow and the rate of flow increases.

*Note: The total resistance of a parallel circuit is always less than the resistance of the lowest-value resistor, or branch, in the circuit.*

This concept often causes confusion concerning the definition of load among students of electricity. Students often think that an increase of resistance constitutes an increase of load. The reason for this is that in a laboratory exercise, students often see the circuit current increase each time a resistive element is connected to the circuit. They conclude that an increase of resistance must, therefore, cause an increase of current. This conclusion is, of course, completely contrary to Ohm’s Law, which states that an increase of resistance must cause a proportional decrease of current. The false concept that an increase of resistance causes an increase of current can be overcome once the student understands that if the resistive elements are being connected in parallel, the circuit resistance is actually being decreased and not increased.

*load: When all resistors are of equal value, the total resistance is equal to the value of one individual resistor, or branch, divided by the number (N) of resistors, or branches*

## Parallel Resistance Formulas

## Resistors of Equal Value

Three formulas can be used to determine the total resistance of a parallel circuit. The first formula shown can be used only when all the resistors in the circuit are of equal value. This formula states that when all resistors are of equal value, the total resistance is equal to the

value of one individual resistor, or branch, divided by the number (N) of resistors, or branches.

Assume that three resistors, each having a value of 24 are connected in parallel, Figure 6-6. The total resistance of this circuit can be found by dividing the resistance of one single resistor by the total number of resistors.

## Product over Sum

The second formula used to determine the total resistance in a parallel circuit divides the product of two resistors by their sum. This is commonly referred to like the product over the sum method.

In the circuit shown in Figure 6-7, three branches having single resistors with values of 20, 30 , and 60 are connected in parallel. To find the total resistance of the circuit using the previous formula, find the total resistance of the first two branches in the circuit, Figure 6-8

The total resistance of the last two resistors in the circuit is 20 This 20 however, is connected in parallel with a 20 resistor. The total resistance of the last two resistors is now used to substitute for the value of R2 in the formula, and the value of the next resistor is used to substitute for the value of R1, Figure 6-9.

## Reciprocal Formula

The third formula used to find the total resistance of a parallel circuit is often referred to as the reciprocal formula.

Notice that this formula actually finds the reciprocal of the total resistance, instead of the total resistance. To make the formula equal to the total resistance it can be rewritten as follows

The value RN stands for R number and means the number of resistors in the circuit. If the circuit has twenty-five resistors connected in parallel,

for example, the last resistor in the formula would be R25. This formula is referred to as the reciprocal formula because the reciprocal of any number is that number divided into 1. The reciprocal of 4, for example, is 0.25 because of 1/4= 0.25. A third rule of parallel circuits is **the total resistance of a parallel circuit is the reciprocal of the sum of the reciprocals of the individual branches.**

Before the invention of handheld calculators, the slide rule was often employed to help with the mathematical calculations in electrical work.

At that time, the product over the sum method of finding total resistance was the most popular. Since the invention of calculators, however, the reciprocal formula has become the most popular because scientific calculators have a reciprocal key (1/X), Figure 6-10, which makes computing total resistance using the reciprocal method very easily. In Figure 6-11, three resistors having values of 150 , 300 , and 100 are connected in parallel. The total resistance will be found using the reciprocal formula.

To find the total resistance of the circuit in Figure 6-11 using a scientific calculator press the keys in the sequence shown in Figure 6-12. Notice in Figure 6-13 that the sequence of key punches exactly follows the formula.