Free-Ed: Basic Electronics Fundamentals

Basic Circuit Analysis
Parallel Circuits

Section 2-2 Voltage in Parallel Circuits

Recall a Definition

A parallel circuit has more than one path for current flow.

Equation

Total Resistance of a Parallel Circuit

R_T =	1
	1	+	1	+	1	+ ... +	1
	R₁		R₂		R₃		R_n

Where

R_T = Total resistance of the parallel circuit

R₁, R₂, R₃, R_n = Resistance of the individual resistors

This formal equation for the total resistance of a parallel circuit looks horribly complicated ... at least at first glance. In the days of pencil-and-paper calculation, this equation was generally considered impossible to use. With today's calculator technology, the 1/x function key renders the process of solving the parallel resistance equation fairly straightforward.

Example

Suppose there is a parallel resistor circuit with values of 10W, 15W, and 25W.

1. Set up the equation:

R_T =

1

1
+
1
+
1

R₁ R₂ R₃

2.Subsititue the given values

R_T =

1

1
+
1
+
1

10 15 25

3. Solve with keystrokes:

10 1/x + 15 1/x + 25 1/x = 1/x

4.838709677

4. Present the solution:

R_T= 4.8W

More Examples

Endless Examples & Exercises

Given the value of all the resistors in a parallel circuit, calculate the total resistance. Do these exercises until you are confident you can handle them.

In practical electronic design and troubleshooting, it is sometimes necessary to adjust the total resistance of a parallel circuit. Suppose, for instance, a circuit is a simple 180W resistor. For some good reason, you fine it necessary to lower the resistance to 120W by adding another resistor in parallel. The problem, of course, is to determine the value of that resistor.

Using the product-over-sum equation, you are given the desired total resistance (R_T) and the value of one of the two resistors (R₁). Solve the equation for R₂.

Procedure

1. Begin with the product-over-sum equation:

R_T =

R₁R₂

R₁ + R₂

2. Solve it for R₂:

R₂ =

R_TR₁

R₁ � R_T

3. Substitute the known values and do the math:

R₂ =

(120)(180)
=360

180 � 120

4. Present the solution:

R₂ = 360W

Some Notes About Parallel Resistance

The total resistance of a parallel circuit is always less than the value of the smallest resistor.

When all resistors in a parallel circuit have the same value, the total resistance is equal to the common value divided by the number of resistors. Example: Three 330W resistors in parallel provide a total resistance of 110W.

When a resistor is removed from a parallel circuit, (1) current continues to flow through the remaining resistors, and (2) the total resistance increases

When a resistor is shorted in a parallel circuit, the total resistance falls to zero.