ECE 252 Introduction to Electrical Engineering

Lesson 2. Resistors in Series and Parallel

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Note that current is defined as the flow of positive charges.

Powers of Ten

Resistance, frequency, power, etc. values are frequently expressed using prefixes which stand for powers of ten. For example, a 12 MΩ resistor has a resistance of 12106 Ω. You should know the designations of powers of ten in the table below.

Prefix Name Value
f femto 10-15
p pico 10-12
n nano 10-9
micro 10-6
m milli 10-3
k kilo 103
M mega 106
G giga 109
T tera 1012
P peta 1015
E exa 1018

Commit them to memory.

Resistor Color Code

Many resistors have colored stripes to indicate the resistance value and the tolerance. Here you will find out how to use the resistor color code: http://micro.magnet.fsu.edu/electromag/java/resistor/. When you need to, you can use the color code to determine the nominal value of the resistor in ohms. Resistor values are not exact; the tolerance of the resistor indicates the accuracy of the resistor value. For example a 100 Ω resistor with a tolerance of 20% could have any value from 80 Ω to 120 Ω. Low tolerance resistors are available for applications requiring high precision.

Resistors in Series

Two elements are in series if they are connected together at one end with no other connection at that end. Use this definition, rather than your intuition, to determine if elements are in series. The following elements are in series:

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These elements are not in series:

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For resistors in series, the net resistance is just the sum of the individual resistances.

REQ = R1 + R2 + ..... Rn

Resistors in Parallel

Two elements are in parallel if both ends of each element are connected together. These elements are in parallel:

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These elements are not in parallel:

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A parallel combination of resistors is found by the equation

1/REQ = 1/R1 + 1/R2 + 1/R3 + ..... 1/Rn

A useful special case is for exactly two resistors in parallel:

REQ = R1R2/(R1 + R2)

The equations for elements in series and parallel are easily derivable from Kirchhoff's Laws.


Before going on, you should complete Tutorial 2 on resistors in series and parallel.


Resistors not in Series or Parallel

Some circuits, such as the one shown below, cannot be simplified by combining elements in series and parallel. When this happens, you just have to grit your teeth and apply Kirchhoff's Laws, or use the delta-wye transformation (discussed in a later section).

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Short Circuits and Open Circuits

Another useful concept in analyzing circuits is that when no current flows through a resistor, capacitor, or inductor, that element can be ignored (removed) when making calculations. A resistor that is connected to the same point on each end is shorted. A resistor with a floating end (not connected to anything) is open circuited.

Voltage Divider

The voltage divider equation will be very useful to you. Consider the figure below.

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It is easily derivable from Kirchhoff's Laws that

V2 = VSR2/(R1 + R2)

Knowing when (and when not) to use the voltage divider equation can save you time and protect you from error. This equation cannot be used if something is connected to V2 that draws current.

Current Divider

The current divider equation may be occasionally useful. Consider the figure below.

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Similar to the voltage divider equation, Kirchhoff's Laws can be used to find that

I2 = ISR1/(R1 + R2)

Notice that the numerator term uses the resistor that the current doesn't go through.


At this time you should complete Tutorial 2A on voltage and current dividers.