Combining Resistors

Project Title

Combining Resistors

Author

Images & Exploration Peter Rapp '99
Idea by Mr. Charles Coe
Updated 8/2/99

Issue:

In simple circuit problems you will often be asked to find currents and voltages with the fundamental "V=IR" relationship. Unless you can effectively combine resistors in series and parallel, you will have a hard time finding the "R" (Equivalent resistance) for the whole circuit. Given the schematic below, what is the equivalent resistance?

Discussion:

Equivalent resistance can be thought of as the question "How can I combine these two resistors into one resistor and not effect the behavior of the circuit?" There are two basic configurations for resistors: series and parallel.
Simple Series Circuit

Resistors in series can simply be "added up." For example, the diagram to the left shows an original circuit with three resistors: A, B, and C. If you wanted to combine these, the equivalent resistor would simply be A+B+C ohms.
The colors are drawn in as part of "Mr. Bourne's Color Coding System™" For those curious, take a marker, start at the battery, and trace any wire you come across until you hit another component (like a resistor). If you have to cross a resistor, change colors. Series resistors suddenly become more obvious. A (Red-Green) is in series with B (Green-Blue) which is in series with C (Blue-Yellow). This system works well with simple circuits and helps you get the hang of it.

Simple Parallel Circuit Resistors in parallel, such as those in the diagram to the right, can be combined with Mr. duBourg's "Flip-Add-Flip" trick. But before we get to that, notice Mr. Bourne's color coding here. Again, we started from the battery with a red marker, traced all of the wires, and changed color when we "crossed" something. Resistors A (Red-Blue) and B (Red-Blue) will be combined in parallel because they have the same color relations. Now onto combining in parallel:

Again, let's take the parallel circuit in the diagram. Step 1: "Flip" the values. For example: "A" becomes "1/A" and "B" becomes "1/B." (Reciprocals) Step 2: "Add" the fractions. Step 3: "Flip" again! Here, let's try this numerically, combining two resistors, 3-ohm and 6-ohm:

Flip
Add
Flip

There! A 3-ohm and 6-ohm in parallel can be combined to form a 2-ohm. Now, let's go back to the original question. How can be combine this circuit. Hint: start at the right side, furthest away from the battery. Mentally color code if that helps too:

Original Circuit

You should have seen the 6-ohm, 1-ohm, and 3-ohm on the right, and identified that they are in series. They can therefore be combined like so:

Step 1

Now look at the circuit. The 10-ohm that you just "made" is in parallel with the other 10-ohm. Flip-Add-Flip!

Step 2

Already, the circuit is looking simpler. Keep simplifying in this manner until you have only one resistor and the battery. You should have calculated an equivalent resistance of 15-ohms.

Resources Window

Circuit pictures were made in Adobe Photoshop, and converted to gif files.

Text versions of the equation work are made with Microsoft Equation Editor and are embedded in Microsoft Word documents.