Equivalent Resistance

Use Series, Parallel, and Ohm's law to find Equivalent Resistance

Learn It!

Pre-Requisite Knowledge

To understand this section, you need to know the following concepts:

Voltage and current concepts

Goal

Find the equivalent resistance.

This group of resistors will act like one equivalent resistor. What is its resistance?

Part 1

Redraw The Circuit

Circuit transformation

Circuit diagrams are an abstraction of a real circuit, there are many ways to draw and redraw them. This example shows you how to redraw circuit diagrams to suit out purposes

We want stuff that looks like this. Just trust me. All the resistors are oriented vertically. Terminals exiting from the middle out of the top and bottom.

Ultimately we want only vertical resistors. Our first task is try to resolve this diagonal part on the left side. We don't like diagonals.

Stretch and bend

As our first move, we'll stretch some wire to make that T junction a little nicer to look at.

Rotate

Let's rotate them to get them vertical. We'll make them immediately recognizable later, after we work on other parts of the circuit.

Slide

Now let's clean up the right side a bit. we can slide the

$4 \Omega$ resistor around the corner, YAY!

We can also T-slide points A and B to coincide with the junction of the

$10 \Omega$ resistor

T-Slide

Now we want to work on getting that

$8 \Omega$ resistor in line. First, let's do a T-slide to move that corner to the center of the wire. This gives us terminals poking out the top and bottom.

Now we slide resistor 8 along the wire around the corner onto the vertical segment.

★

Now our circuit is nicely transformed. After doing this a few times, you will be able to do all this in your head. It just takes a little practice.

Are the legal moves 'real'

Part 2

Finding Equivalent Resistance

Use parallel and series formulas to decompose circuits one piece at a time.

Reduce two series resistors on the right

This circuit is nice. The circled segments look like those special patterns mentioned earlier in the example.

These configurations are common and important enough to have names.

The one on the right is called Series

Two resistors are in series when they share a single node that no other element is connected to.

We can replace a series pattern of resistors with a single resistor with the Equivalent resistance denoted with the symbol

$R_{eq}$

$R_{eq} =R_1+R_2$

How to understand and remember this formula?

Proof

$R_{eq1} = R_1 + R_2 = 4+2 = 6$

Now our circuit is nicely transformed. After doing this a few times, you will be able to do all this in your head. It just takes a little practice.

Reduce two parallel resistors on the left

Redraw the diagram with the equivalent resistor.

The circled configuration is also a special configuration. it is called Parallel

Resistors are in parallel if they share the same nodes but go along different wires

It has an equivalent resistance of

$R_{eq} = \dfrac{1} {\dfrac{1}{R_1} + \dfrac{1}{R_2}}$

Another way of representing this is the Product over sum rule

$R_{eq} = \dfrac {R_1 * R_2} {R_1 + R_2}$

How to understand and remember this formula?

Proof

$R_{eq2} = \dfrac{1}{\dfrac{1}{R_1} + {\dfrac{1}{R_2}}} = \dfrac {1} {\dfrac{1}{20} + \dfrac{1}{30}} = 12$

Apply the parallel resistance formula! Note that our answer is reasonable: a parallel configuration reduces resistance.

Reduce the series resistors on the left

$R_{eq3} = 8 +12 = 20$

Apply the series equivalent resistance formula.

Combine three parallel resistors

$R_{eq} = \dfrac{1}{\dfrac{1}{20} + \dfrac{1}{10} + \dfrac{1}{6}} = 3.157 \Omega$

★

$R_{eq} = \dfrac{1}{\dfrac{1}{20} + \dfrac{1}{10} + \dfrac{1}{6}} = 3.157 \Omega$

Apply the parallel resistance formula to three resistors. We have succeeded! The whole mess we started with behaves as though it were a single resistor with a resistance of about

$3.1 \Omega$ .

Can all circuits be broken down with series and parallel?