In this unit you’ll begin learning how to analyze circuits. Circuit
analysis means looking at a schematic diagram for a circuit
and computing the voltage, current, or power for any component in that
circuit. Closely related to the task of circuit analysis is the task
of troubleshooting, which means figuring out what
is wrong in a circuit that is not working correctly. Analyzing and
troubleshooting go hand in hand; when a circuit is not working correctly,
the easiest way to figure out what’s wrong is usually to measure voltages
in the circuit and compare those measured values to the values that
the voltages should have (which you compute by analyzing the circuit).
The rest of this course will concentrate on analyzing and troubleshooting resistive
circuits (circuits that contain only resistors in addition
to power supplies). In later courses you’ll learn how to analyze and
troubleshoot circuits that contain other components, such as capacitors,
inductors, diodes, and transistors.
This unit covers just about everything you could ever want to know about
the simplest type of circuit, which is called a series resistive circuit. Some
of the things you’ll learn here apply only to series resistive circuits,
but other things apply to any kind of circuit. For instance, the rule
called Kirchhoff’s Voltage Law applies to all circuits,
and is therefore a very important general rule of circuit analysis.
Unit 6 Review
This unit will build on material that you studied in Unit
6. So let’s begin by taking this self-test to review what you
learned in that unit.
First, we’ll define some terms: series connection, series
path, and series circuit. And even before
we define those terms, we’ll define what we mean by saying that two
components are connected to
Defining these terms carefully will be a big help later when we get
to more complicated series-parallel circuits.
Two components are said to be connected to each other
when there is a path of zero resistance joining a terminal of one component
to a terminal of the other component.
Basically, this means that either the components are connected directly
to each other, or there’s a conductor (such as a wire, or a trace on
a circuit board) that connects them together.
Examples: In the circuit shown below,
VS is connected to R1 at one point;
R2 and R3 are connected to each other at two points;
VS and R2 are not connected to each other.
Now that we’ve defined what we mean by saying that two components
are connected to each other, we can define what we mean by a series
Two components are connected in series if they are connected
to each other at exactly one point and no other component is connected
to that point.
Notice that there are two halves to this definition, both of which
must be met in order to have a series connection.
First, the components must be directly or indirectly connected at
exactly one point, no more and no less.
Second, no other component can be connected to that point where
the other two components meet.
The most common mistake that
students make here is to remember the first half of the definition
but forget about the second half. For example, in the circuit shown
below, R1 and R2 are connected to each other at exactly one point,
but the voltage source is also connected to that same point. Therefore,
R1 and R2 are not connected in series.
Series-Connected Components Have the Same Current
The most important property of series connections is that the
current is the same in every series-connected component.
Example: In the circuit shown below, VS and R1
are connected in series, so we know that the current through VS must
be the same as the current through R1. But R1 and R3 are not connected
in series, so we cannot assume that the current through R1 is equal
to the current through R3.
Now that we’ve defined what we mean by a series connection, we can
define what we mean by a series path.
A series path is a group of connected components in which
each connection is a series connection.
As noted above, components connected in series have the same current.
Therefore, all of the components in a series path must have the
same current as each other
Now that we’ve defined what we mean by a series path, we can define
what we mean by a series circuit.
A series circuit is a (complete) circuit that consists exclusively of
one series path. In other words, it’s a complete circuit in which every
connection is a series connection.
As noted above, components connected in series have the same current
as each other. Therefore, all of the components in a series circuit
must have the same current as each other.
In fact, this feature of having the same current is how some textbooks define
series circuits: "A
series circuit provides only one path for current between two points
so that the current is the same through each series component."
Current in a Series Circuit
We’ve already said this once or twice, but it’s so important that
we’ll repeat it: the current is the same everywhere in a series
So, for example, in the series circuit shown below, if you measure
or compute the current through any one of the components, then you
immediately know the current through each of the other components.
All of the components have the same current.
By the way, this is true for all series circuits,
not just series resistive circuits. In the circuit shown below, if
we replaced R2 with a diode, inductor, or other component, we would
still be able to say that that component has the same current as R1,
R3, and the voltage source.
Total Series Resistance
The total resistance of resistors connected in series is simply
equal to the sum of the individual resistance values.
Example: if a 1.2 kΩ resistor is connected in series with
a 4.7 kΩ resistor, then the total resistance is 5.9 kΩ.
We’ll use the symbol RT to stand
for total resistance. So for the case above, we could write
RT = 5.9 kΩ.
The general formula for finding the total resistance of resistors
connected in series is
RT = R1 + R2 + R3 +
… + Rn
An Animated Lesson from our Friends in Wisconsin
Instructors in the Wisconsin Technical College System have created
a library of short online animations and quizzes to help students learn
electronics. I’ll include links to some of these "learning objects." Whenever
you see the icon below, click it to see a learning object on the material
you’re studying. The Wisconin learning object will open in a new window;
close the winow when you’re finished and want to return to this lesson.
This first one will give you more practice calculating total series
Analyzing Series Resistive Circuits
We now know enough to be able to find currents and voltage drops
in a series resistive circuit. There are four basic steps.
Find the total resistance by adding all of the individual resistances:
RT = R1 + R2 + R3 +
… + Rn
Apply Ohm’s law in the form I = V ÷ R to
the entire circuit. In words, the total current produced by the
voltage source is equal to the source voltage divided by the total
resistance. In symbols,
IT = VS ÷ RT
Recall that in a series circuit, every component has the same
current. Therefore, each resistor’s current is equal to the total
current. In symbols,
IT = I1 = I2 = I3 =
… = In
Use Ohm’s law in the form V = I × R to
find the voltage drop across each resistance. In symbols,
V1 = I1 × R1 V2 = I2× R2 V3 = I3 × R3
and so on for each of the resistors.
These four steps do not apply to all circuits. In particular,
Steps 1 and 3 do not work for circuits that aren’t series circuits.
(But Steps 2 and 4 do work for all resistive circuits.)
Consider the series circuit shown below:
We want to determine the current through each resistor and the voltage
across each resistor.
Our analysis of this circuit has four steps:
Add the resistor values together to find the total resistance RT.
= R1 + R2
= 100 Ω + 200 Ω
= 300 Ω
Use Ohm’s law on the entire circuit to find the circuit’s total
= VS ÷ RT
= 12 V ÷ 300 Ω
= 40 mA
Recall that since this is a series circuit, current is the same
everywhere. Therefore, each resistor receives the total current:
= 40 mA
= 40 mA
Use Ohm’s law on each resistor to find the voltage drops V1 and V2 across
= I1 × R1
= 40 mA × 100 Ω
= 4 V
= I2 × R2
= 40 mA × 200 Ω
= 8 V
Current Direction and Voltage Polarity
When we analyze a circuit to figure out the current, we’re interested
not only in the current’s magnitude (or size), but also in its direction.
For example, in the series circuit shown below, the current has a magnitude of
1.93 mA. What is the current’s direction? The current’s direction
is clockwise around the circuit, because current comes out of the voltage
source’s positive terminal (represented by the longer line in the symbol
for a voltage source) and goes into the voltage source’s negative terminal.
Also, when we analyze a circuit to figure out a particular voltage,
we’re interested not only in the voltage’s magnitude (or size),
but also in its polarity. Polarity means which end of the voltage
is positive and which end is negative. Here’s how to figure out the
polarity of the voltage across a resistor:
The end of a resistor into which current enters is the positive
The end of a resistor from which current leaves is the negative
For example, in the circuit shown above, we’ve determined that current
flows clockwise around the circuit, which means that it flows into
the left-hand end of R1 and out of the right-hand end of R1. Therefore,
the polarity of R1’s voltage is: positive on the left-hand end, and
negative on the right-hand end.
The diagram below uses + and – signs to show the polarity of each
voltage in the circuit we’ve been discussing:
Notice an important difference between voltage sources and resistors:
Current flows out of a voltage source’s positive end and
into its negative end.
Current flows into a resistor’s positive end and out of
its negative end.
Voltage Drops and Voltage Rises
Some circuit-analysis techniques (including one called Kirchhoff’s
Voltage Law that we’ll introduce soon) require you to take an imaginary
journey around a circuit, keeping track of the voltage changes as you
As you mentally move through the circuit, if you pass through a component
from its − end to its + end, we’ll call that a voltage rise.
On the other hand, if you mentally pass through a component from
its + end to its − end, we’ll call that a voltage drop.
Example: In the circuit shown below, suppose you decide to "travel" around
the circuit in a clockwise direction. Then you’ll encounter voltage drops as
you pass through R1, R2, and R3, and you’ll encounter a voltage rise as
you pass through the voltage source.
But, continuing the same example, suppose you now decide to "travel" around
the circuit in a counter-clockwise direction. Then you’ll encounter
voltage rises as you pass through R1, R2, and R3, and you’ll
encounter a voltage drop as you pass through the voltage source.
So a particular voltage can be considered as either a voltage drop
or a voltage rise, depending on the direction of your imaginary trip
around the circuit.
Voltage Sources in Series
Sometimes you’ll encounter a circuit with two or more voltage sources
connected in series.
A common example is a flashlight. As you probably know, most flashlights
require two batteries. Each battery is a voltage source, and when you
load the batteries into the flashlight, they go in end-to-end, so they’re
connected in series.
It’s pretty easy to analyze voltage sources connected in series.
You just have to be careful to notice whether the voltage sources are
trying to drive current in the same direction or in opposite directions.
We call these two cases series-aiding voltage sources
and series-opposing voltage sources.
Series-Aiding Voltage Sources
If two series-connected voltage sources are connected so as to produce
current in the same direction, they are said to be series-aiding.
The net effect on the circuit is the same as that of a single source
whose voltage equals the sum of the two voltage sources.
Example: in the circuit shown below, VS1 and VS2 are
both connected so as to push current in a clockwise direction around
the circuit. Therefore, they are series-aiding, and they combine to
have the same effect as a 22-V source.
Series-Opposing Voltage Sources
When two series-connected sources are connected so as to produce
current in opposite directions, they are said to be series-opposing.
The net effect on the circuit is the same as that of a single source
equal in magnitude to the difference between the source voltages
and having the same polarity as the larger of the two.
Example: in the circuit shown below, VS1 tries
to push current in a clockwise direction around the circuit, but VS2 tries
to push current in a counter-clockwise direction. Therefore, they are
series-opposing, and they combine to have the same effect as a 2-V
source with the polarity of VS1 (pushing current
clockwise around the circuit).
Double-Subscript Notation for Voltages
You’re familiar with the concept of the voltage across
a resistor or other component. When we wish to refer to the voltage
across a particular resistor, we usually use a notation such as V1,
which means the voltage across resistor R1. In this notation, we write
a capital V with a single number as a subscript: this number
is the number of the resistor whose voltage we’re talking about.
Sometimes, when we want to talk about a voltage between two points
in a circuit, we’ll use a different notation, which has a capital V with
two letters as a subscript, such as VAB.
In such cases, the points A and B would be labeled
in the circuit ‘s schematic diagram to identify them, as in the diagram
VAB means the voltage between points A and B,
with point A regarded as + and point B regarded as −.
In other words, it’s the voltage that you would measure with a voltmeter
if you placed the meter’s positive (red) lead at point A and
the meter’s negative (black) lead at point B.
On the other hand, VBA means the voltage between
points A and B, but with B regarded as + and A regarded
as −. In other words, it’s the voltage that you would measure
with a voltmeter if you placed the meter’s positive (red) lead at point B and
the meter’s negative (black) lead at point A.
So in any given circuit, VAB and VBA will
have the same magnitude, but one of them will have a negative sign
and the other will not. For example, in the circuit shown above, VAB = −12 V
and VBA = 12 V.
Kirchhoff’s Voltage Law
Kirchhoff’s Voltage Law says that the sum of the voltage drops
around any closed loop in a circuit equals the sum of the voltage
rises around that loop.
We use the abbreviation KVL as a shorthand way of
referring to Kirchhoff’s Voltage Law.
KVL in Series Resistive Circuits
When applied to a complete series resistive circuit with a single
voltage source, KVL says that if you add the voltages across all of
the resistors, the sum must be equal to the value of the source voltage.
For example, consider the circuit shown below, which shows the polarities
of the voltages across the source and across the resistors.
If we "travel" clockwise around the circuit, then voltages V1, V2,
and V3 are voltage drops, and voltage VS is
a voltage rise. Since KVL says that the sum of the voltage drops
must equal the sum of the voltage rises, we know that
V1 + V2 + V3 = VS
On the other hand, if we "travel" counter-clockwise
around the circuit, then voltages V1, V2,
and V3 are voltage rises, and voltage VS is
a voltage drop. Since KVL says that the sum of the voltage drops
must equal the sum of the voltage rises, we know that
VS= V1 + V2 + V3
Either way, we reach the same conclusion: the sum of the resistor
voltages is equal to the source voltage. So in the circuit shown
above, we know that if we add together the voltages across the
three resistors, we’ll get a sum of 10 V.
KVL in Other Circuits
KVL is a general rule that applies in all circuits,
not just series circuits and not just circuits containing resistors.
In more complicated circuits, it can get tricky to apply KVL correctly,
but when applied correctly it is a powerful tool. We’ll see this in
A string of series resistors is often called a voltage divider because
the total voltage across the entire string is divided among the various
resistors in direct proportion to the resistance of each one.
For example, if you have two resistors in series and one resistor
is twice as large as the other one (for example,
suppose that one is 20 kΩ and the other is 10 kΩ),
then there will be twice as much voltage across
the larger resistor as there is across the smaller one.
On the other hand, if one of the series resistors is three
times as large as the other one (say, 30 kΩ and
10 kΩ), then there will be three times as
much voltage across the larger resistor as there is across the
The Voltage-Divider Rule
The voltage-divider rule is a shortcut rule that you can use to find
the voltage drop across a resistor in a series circuit.
The rule says that the voltage across any resistance in a series
circuit is equal to the ratio of that resistance to the circuit’s
total resistance, multiplied by the source voltage.
In equation form, this rule is expressed as:
Vx = (Rx ÷ RT) × VS
Here x is a variable representing the number of the resistor
that you’re interested in.
For instance, if you’re trying to find the voltage across resistor
R1, you would replace x with 1 to get:
V1 = (R1 ÷ RT) × VS
On the other hand, applying the rule to resistor R4 in a series
circuit gives us:
V4 = (R4 ÷ RT) × VS
Of course, you can also find these voltage drops using the procedure
we used earlier: first find total resistance, then use Ohm’s law to
find the current, and then use Ohm’s law to find the voltage drop that
you’re interested in. Doing it this way will give you the same answer
that you get by using the voltage-divider rule.
Potentiometer as a Voltage Divider
3 you learned that a potentiometer is a type
of variable resistor with three terminals, represented by the following
Recall that when you adjust a potentiometer you are moving the middle
terminal (called the wiper terminal) toward one end or the other
of the resistor. The resistance between the two end terminals stays
constant, but the resistance between the wiper and either end terminal
will change as you adjust the potentiometer.
In effect, what you have here is an adjustable voltage divider. In
other words, it’s like having two resistors in series whose total resistance
is constant, but you can adjust the relative sizes of the individual
resistors by moving the wiper in one direction or the other.
Power in a Series Circuit
In Unit 1 of this course you learned three formulas for computing
the power dissipated in a resistor:
P = I2 × R
P = V × I
P = V2 ÷ R
Recall that in each of these equations, R is the resistor’s
resistance, V is the voltage across the resistor, and I is
the current through the resistor.
These formulas let you find a resistor’s power in any kind of circuit,
including a series resistive circuit. But you need to be a little careful. The
most common mistake that students make when using these formulas
is to use the wrong value for V. In particular, students often
mistakenly use the value of the source voltage when they should use
the value of a single resistor’s voltage.
In the circuit shown below, for example, the source voltage (VS)
equals 12 V, and R1’s voltage (V1) equals
4 V. Here’s one correct way to find the power dissipated in R1:
We’ve been talking about the power dissipated in a single resistor.
Not surprisingly, we use the symbol P1 for the
power dissipated in resistor R1,and the symbol P2 for
the power dissipated in resistor R2, and so on. We can also talk about
the total power dissipated in an entire circuit, for
which we use the symbol PT.
Here are two ways to compute total power in a resistive circuit.
You’ll get the same answer either way:
You can find the power for each resistor, and then add these
PT = P1 + P2 + P3 +
… + Pn
Or you can apply any one of the power formulas
to the entire circuit:
PT = IT2 × RT
PT = VS × IT
PT = VS2 ÷ RT
These are the same power formulas from above, except that now we’re
applying them to the entire circuit, instead of to a single resistor.
Voltage Relative to Ground
Earlier in this unit you learned about the double-subscript notation
that we use to talk about a voltage between two points in a circuit.
For instance, if we have points labeled A and B in
a circuit, we use the symbol VAB to refer to the
voltage between those two points. As mentioned earlier, this is the
voltage that you would measure with a voltmeter by placing the meter’s
positive (red) lead at point A and the meter’s negative (black)
lead at point B.
In many cases, we’re interested in knowing the voltage at a point
in a circuit relative to the circuit’s ground. In such cases we use
a notation that has a capital V with one letter as a subscript,
such as VA. This is the voltage at point Arelative
to ground, which simply means the voltage that you would measure
with a voltmeter by placing the meter’s positive (red) lead at point A and
the meter’s negative (black) lead at the circuit’s ground point.
Troubleshooting a non-working circuit means finding
the problem that is preventing the circuit from working correctly.
The two most common types of problems are open circuits and short
An open circuit, or "open," is a break in a circuit
For example, when a light bulb burns out, it causes an open. A resistor
or other component can also fail by becoming open. This can happen,for
instance, if too much current passes through a resistor, causing it
to "burn out."
A circuit containing an open is said to be an open circuit,
or to be open-circuited.
Current Through an Open
The most important thing to remember about opens is that no current
can flow through an open.
Therefore, no current can flow anywhere in a series circuit containing
Since no current flows through an open, you can think of the open
as having infinite resistance (R = ∞).
Voltage Across an Open
A common mistake is to believe
that since the current through an open is zero, the voltage across
the open must also be zero.
Usually, an open will not have a voltage drop of 0 V.
In fact, in a series circuit that contains an open, the entire
source voltage will appear across the open, and no voltage will appear
across any of the other resistors.
So if you measure the voltage between any two points in a series
circuit containing an open, you’ll measure 0 V if the two points
are on the same side of the open, but you’ll measure the entire source
voltage if the points are on opposite sides of the open.
For example, suppose R3 is open in the circuit shown below. Then
there will be 0 V across R1, across R2, and across R4. Also, Vab = 0 V.
But there will be 9 V across R3. Also, Vac = 9 V,
and Vbc = 9 V.
A short circuit, or "short," is a path of zero resistance
connecting two points in a circuit that are not supposed to be connected.
For example, a wire clipping or a loose lump of solder can accidentally
touch the leads of two resistors, thereby connecting those resistors
to each other.
Voltage Across a Short
Since a short has zero resistance, the voltage across it must be
zero. This follows from Ohm’s law, V = I × R.
Current Through a Short
A component is said to be short-circuited, or "shorted out," when
there is a short circuit connected across it. No current flows
through a short-circuited component. Instead, current is diverted
through the short itself.
For example, suppose that in the circuit shown below there is a
short between points a and b, perhaps caused by
a loose wire clipping that connects
these two points. Then R2 is short-circuited. No current will flow
through R2; instead, current will follow the path of zero resistance
through the short itself (the wire clipping).
A short in a series circuit reduces the circuit’s total resistance,
causing more current to flow out of the voltage source.
For example, in the circuit shown above, if R2 is short-circuited
by a wire clipping that connects points a and b,
then R2’s resistance disappears from the circuit, and the circuit’s
total resistance is equal to R1 + R3 + R4.
Typical Causes of Opens and Shorts
The following animated lesson shows some of the real-world conditions
that typically cause shorts or opens in circuits. It’s got some good
practical examples, so be sure no to skip it.
Unit 7 Review
This e-Lesson has covered several important topics, including:
series connections and series paths
voltage drops and voltage rises
voltage sources in series
Kirchhoff’s Voltage Law (KVL)
voltage dividers and the voltage-divider rule
power in series circuits
open circuits and short circuits.
To finish the e-Lesson, take this self-test to check your understanding
of these topics.
Congratulations! You’ve completed the e-Lesson for this unit.