For most of this course you’ve studied circuits with resistors and
DC voltage sources. In Unit 13 we added a new
component, the capacitor.
This unit will add another new component, the inductor. As you’ll
see, the techniques and equations for analyzing circuits with inductors
are very similar to the techniques and equations that you’ve learned
Recall that resistors oppose the flow of current. And capacitors store
charge. What about inductors? They oppose changes in
current. What does that mean? Suppose that a circuit with an inductor
has a certain current flowing through it. If you try to increase or
decrease that current, then the inductor will fight against you, and
will try to keep the current at its initial value. Eventually the inductor
will lose this fight, and the current will change, but this will take
some time to happen. This is different from what happens in a circuit
with no inductor: if there’s no inductor, then the amount of current
can increase or decrease immediately. If there is an inductor,
changes in current take a while to happen.
Unit 13 Review
This unit will build on material that you studied in Unit
13. So let’s begin by taking this self-test to review what you
learned in that unit.
Preview of Electromagnetism
In a future course you’ll study two related phenomena called electromagnetism
and electromagnetic induction. These two principles are key to understanding
how an inductor operates, so let’s take a quick look at them.
In 1820, Hans Oersted discovered that electrical current creates
a magnetic field. This phenomenon is called electromagnetism.
Oersted also realized that the you can increase the strength of
the magnetic field surrounding a current-carrying wire by winding
the wire into a series of closely spaced loops. A wire that is looped
in this way is called a coil.
A few years later, Michael Faraday discovered that a voltage
is induced in a wire whenever there’s a change in the size of the
magnetic field surrounding the wire. This phenomenon is
induction. Also, the induced voltage will be greater if you
use a coil of wire rather than a straight piece of wire.
Now here comes the part that we’re really interested in. Suppose
you’ve got some current running through a coil of wire. According
to the principle of electromagnetism, this current creates a magnetic
field around the coil. What will happen if you change the size of
the current? Well, that will change the strength of the magnetic field.
But according to the principle of electromagnetic induction, when
the magnetic field surrounding a wire changes, a voltage will be induced
across that wire.
And it turns out that this voltage will always oppose the change
you’re making to the current. In other words, if you increase the
current, then a voltage will be induced that will try to decrease the
current. On the other hand, if you decrease the current, then
a voltage will be induced that will try to increase the current.
The bottom line is: Whenever the current in a coil increases or
decreases, a voltage is induced in the coil, and this induced voltage
opposes the change in current.
This is called self-inductance; as we’ve seen, it’s the result
of electromagnetism and electromagnetic induction working at the same
Inductance of a Coil
The size of the voltage induced in a coil depends on a property
of the coil called its self-inductance (or simply inductance).
The symbol for a coil’s inductance is L.
The unit of inductance is the henry, abbreviated H.
An inductor is a device designed to have a certain amount
Here’s the schematic symbol for an inductor:
Most of the inductors in our labs look similar to this:
Typical inductors found in electronic equipment are in the microhenry
(μH) or millihenry (mH) range. Recall that micro- means
10-6 and milli- means 10-3.
Usually we treat wire as having zero resistance, but in reality
wire does have some resistance. And the longer and thinner a piece
of wire is, the greater its resistance.
An inductor is simply a coiled piece of very long, very thin wire.
Therefore an inductor will have some resistance, which we call the
inductor’s winding resistance.
The symbol for winding resistance is RW, and
it is measured in ohms.
You can measure an inductor’s winding resistance simply by connecting
an ohmmeter to its two leads, just as you would measure a resistor’s
This winding resistance can be fairly large–for example, it’s not
unusual to have an inductor whose resistance is 50 Ω or
more. But 50 Ω is still not huge, and we might be able
to ignore it if the inductor is in a circuit whose resistors are much
For example, suppose you’ve got an inductor whose winding
resistance is 50 Ω, and suppose this inductor is
in a series circuit whose total resistance is 2 kΩ.
Then the inductor’s winding resistance is only 2.5% of the
circuit’s total resistance, small enough that you can probably
ignore it when you’re calculating the circuit’s current and
As a general rule of thumb, if an inductor’s winding resistance
is less than about 5% of the resistance that it’s in series
with, then you can ignore it.
An ideal inductor has no winding resistance. In other words, RW = 0 Ω for
an ideal inductor. In most of the following discussion we’ll assume
that inductors are ideal, but in a few places we’ll mention the effect
of a real inductor’s winding resistance.
Inductors are classified by the materials used for their cores.
Common core materials are air, iron, and ferrites.
Variable inductors are also available. The schematic symbol
has an arrow to show that the component’s value can be adjusted:
Chokes and Coils
Inductors used in high-frequency AC circuits are often called chokes,
or simply coils.
Energy Stored in an Inductor
Recall that resistors dissipate energy as heat,
but that capacitors store energy.
Like a capacitor, an inductor stores energy, which can later be
returned to the circuit.
An ideal inductor (with zero winding resistance)
doesn’t dissipate any energy as heat.
Since RW ≠ 0 Ω for
a real inductor, a real inductor does dissipate
some energy as heat, but generally it’s small enough to ignore.
We’ll return to this point later when we discuss power in an
A capacitor stores energy in the electric field that
exists between the positive and negative charges stored on its opposite
plates. But an inductor stores energy in the magnetic field that
is created by the current flowing through the inductor.
The energy W stored by an inductance L is given
W = ½ LI 2
where I is the current through the inductor.
Inductors in Series
Suppose you have two or more inductors connected in series, as
in the picture above. The total inductance is equal to the sum of
the individual inductances:
LT = L1 + L2 + …
So inductors in series combine like resistors in series.
Want more pratice finding total inductance of inductors in series?
Here are some more practice problems:
Inductors in Parallel
Suppose you have two or more inductors connected in parallel, as
in the picture above. To find the total inductance, use the reciprocal
LT = 1 ÷ (1÷L1 +
1÷L2 + … + 1÷Ln)
So inductors in parallel combine like resistors in parallel.
more pratice finding total inductance of inductors in parallel? Here
Shortcut Rules for Inductors in Parallel
In Unit 8 you learned two shortcut rules that you can use
for special cases of parallel resistors. The same basic rules apply
to inductors in parallel.
The first shortcut rule said that if you have just two resistors
in parallel, you can find their total resistance using the product-over-sum
rule: RT = (R1 × R2) ÷ (R1 +
R2). Similarly, if you have just two inductors
in parallel, you can find their total inductance using the use the
LT = (L1 × L2) ÷ (L1 +
The second shortcut rule applied to several parallel resistors that
all have the resistance. This rule said that if you have n parallel
resistors, each with resistance R, the total resistance is
given by RT = R ÷ n. Similarly, if
you have n parallel inductors, each with inductance L,
the total inductance is given by:
LT = L÷ n
Of course, the reciprocal formula given earlier applies to all cases
of inductors in parallel, so it will give the same answer that the
shortcut rules give for these special cases.
As you can probably guess, when you have series-parallel combinations of
inductors, you find the total equivalent inductance by combining the
rule for inductors in series with the rule for inductors in parallel.
How about some more pratice problems? (Some of these are a little tricky, so be sure to try them all.) Remember, practice makes perfect!
Series RL Network
A resistor and inductor connected in series are called a series
DC RL Circuit
An RL circuit is any circuit containing, in addition to a
power supply, just resistors and inductors.
In this course we’ll restrict our attention to RL circuits
containing DC voltage sources. We’ll refer to such circuits as DC RL circuits.
Examples: A very simple DC RL circuit just has a resistor,
an inductor, and a voltage source in series:
Here’s a more complicated DC RL circuit:
Behavior of Inductors in DC Circuits
Just as we have rules of thumb that let us analyze the behavior
of capacitors when they’re fully charged or fully discharged, we have
similar rules for inductors.
Here’s an important
rule of thumb that you must memorize:
When an inductor with no current flowing through it is first
switched into a circuit, it behaves like an open circuit.
So to find currents and voltages in a DC RL circuit whose inductors
have just been switched into the circuit, replace all inductors with
open circuits. Then you’ll be left with a circuit containing just
a power supply and resistors, which you can analyze using the skills
you learned earlier in this course.
Here’s another important
rule of thumb:
When a constant, unchanging current is flowing through an
ideal inductor, the inductor behaves like a short circuit.
So to find currents and voltages in a DC RL circuit whose inductors
are carrying a constant, unchanging current, replace all inductors
with short circuits (in other words, with wires). Then you’ll be left
with a circuit containing just a power supply and resistors, which
you can analyze using the skills you learned earlier in this course.
Notice that this second rule of thumb applies to ideal inductors
(with zero winding resistance). On the other hand, when a constant,
unchanging current is flowing through a real inductor
(with RW ≠ 0 Ω), the inductor
behaves like a resistor whose resistance is equal to RW.
But as we’ve seen, this RW is often small enough
that we can ignore it and treat the inductor as a short.
Initial, Transient, Steady-State
In most practical DC RL circuits, the values of current and voltage
change with time as the current through each inductor changes. Typically
such circuits contain a switch that is initially open, and you’re
interested in finding the values of voltage and current after the
switch has been closed.
To remind ourselves of this fact, we often include an open switch
in schematic drawings of DC RL circuits, as in the following picture:
Just as with DC RC circuits, we distinguish three time periods in
the behavior of any DC RL circuit:
the initial period
the transient period
the steady-state period
During the transient period, the circuit’s currents and voltages
are changing from their initial values to their final (steady-state)
Initial Currents and Voltages
The currents and voltages in a circuit at the instant when a switch
is first closed are called the initial currents and initial
In most cases, at this initial instant the circuit’s inductors have
no current flowing through them. Therefore, using the first rule of
thumb you learned above, you’ll find
the circuit’s initial values of voltage and current by replacing the
inductors with opens.
This is the opposite of capacitors, which initially
behave like short circuits (assuming that they start out being fully
discharged, which is normally the case).
Steady-State Currents & Voltages
When the switch in a DC RL circuit has been closed for a long time,
currents and voltages have reached their steady-state values.
According to the second rule of thumb you learned above,
an ideal inductor (with zero winding resistance) behaves like a short
circuit in the steady state. So you find steady-state currents and
voltages in an RL circuit by replacing all ideal inductors with short
Usually we treat inductors as being ideal. But if you want to
take a real inductor’s winding resistance into account, then to
find steady-state values you should replace the inductor with a
resistor whose resistance is equal to RW, instead
of replacing the inductor with a short.
Again, this is the opposite of capacitors, which
behave like open circuits in the steady state.
Transient Currents & Voltages
When a switch is first closed (or opened) in a DC RL circuit, currents
and voltages change for a short time from their initial values to
their steady-state values. This is very similar to what happens
in a DC RC circuit, and the equations are also similar.
Transients: Closing Switch
In the circuit shown below, if we close the switch at time t = 0,
the current will gradually increase from its initial value (zero)
to its steady-state value (which is equal to VS÷R).
We wish to be able to calculate i, vR,
and vL. In other words, we want to be able to
calculate the current, the resistor’s voltage drop, and the inductor’s
voltage drop at any particular time after the switch has been closed.
In the series RL circuit shown above, after the switch is closed
at time t = 0, the current is given by the equation:
i = (VS÷R) (1 − e−t÷(L÷R))
The quantity L÷R in the equation above is called
the time constant of the series RL network. It is represented
by the Greek letter τ, and
it is measured in seconds (s):
τ = L ÷ R
So we can rewrite our equation for i in a slightly simpler
i = (VS÷R) (1 − e−t÷τ)
Be careful not to confuse t with τ.
Remember, for any particular circuit, τis
a constant that depends on the size of the inductors and resistors,
but t is a variable that represents time.
How Long to Reach Steady State?
The time constant τ is
an indicator of how long i takes to increase from zero to
its steady-state value.
Here is a useful rule of thumb:
For most practical purposes, we may assume that all quantities
in a DC RL circuit have reached their steady-state values after
five time constants.
So if a circuit has a time constant of 1 millisecond, then it will
take about 5 milliseconds for the circuit’s currents and voltages
to reach their steady-state values.
Since one time constant is equal to L÷R, we can
write this rule of thumb as an equation:
Time to reach steady state ≈ 5×L÷R
Notice that in this equation I used a "squiggly equals sign" ≈ to
indicate that this is an approximation. Actually, after five time
constants the current will have risen to about 99.3% of its steady-state
value. For most practical purposes, that’s close enough.
Calculating vR and vL
Let’s continue our analysis of a simple series DC RL circuit.
We’ve seen that the current in this circuit, after the switch is
closed at time t = 0, is given by the equation:
i = (VS÷R) (1 − e−t÷τ)
In the same circuit, the voltage drop across the resistor is given
by the equation:
vR = VS (1 − e−t ÷τ)
and the voltage drop across the inductor is given by:
vL = VS e−t ÷ τ
Don’t think of these as three separate equations that you
have to remember. Once you’ve got the equation for i,
you can easily use Ohm’s Law to derive the expression for vR.
Remember, Ohm’s Law says that a resistor’s voltage is
equal to its current times its resistance. Do you see that
if you take the expression above for i and multiply
it by R, you’ll get the expression above for vR?
And once you’ve got the equation for vR, you
can easily use Kirchhoff’s Voltage Law (KVL) to derive the expression
Remember, KVL says that the sum of the voltage drops around
a loop must equal the sum of the voltage rises around that
loop. Applying KVL to our simple series DC RL circuit gives
us VS = vR + vL.
Do you see how this lets you derive the expression above for vL from
the expression above for vR?
More Exponential Curves
If we plot these equations for i, vR,
and vL, we get exponential curves similar to the
curves we saw in the previous unit for DC RC circuits.
In the plots below, the values on the horizontal and vertical
axes will change depending on the values of resistance, inductance,
and source voltage in a particular circuit, but the shape of
the curves will be the same for all series DC RL circuits.
For example, the current in a DC RL circuit starts
at 0 and rises to its final value:
Ohm’s law tells us that a resistor’s voltage is directly proportional
at all times to its current. So we know that a graph of the resistor’s
voltage has the same shape as the graph of the current:
On the other hand, the inductor’s voltage starts at its maximum
value and then decreases to 0:
Notice again that, in each of these graphs, the values change very
quickly at first, and then gradually approach a final value.
De-Energizing an Inductor
Up to now we’ve been talking about energizing an inductor. Similar
comments, but in reverse, apply to the case of discharging a
capacitor. In this case we get what’s called an inductive kick, which
has some interesting practical applications , as you’ll read about
in this animation:
When an inductor is connected to a series-parallel resistor network,
the equations given above for transient current and voltages still
work, as long as you first replace the series-parallel network with
its Thevenin equivalent.
For example, in the series-parallel circuit shown below, which you
will analyze in the Self-Test questions, we have an inductor connected
to a network of three resistors and a voltage source. Thevenin’s theorem
lets you "collapse" those three resistors and the voltage
source down to a single resistor and voltage source connected in series
with the inductor. You can then use the equations you’ve learned above
to analyze that "collapsed" circuit, and the results you
get from this analysis will be correct for the original circuit as
Unit 14 Review
This e-Lesson has covered several important topics, including:
types of inductors
energy stored in an inductor
inductors in series, in parallel, and in series-parallel
calculating initial values, steady-state values, and transient
values in a DC RL circuit
time constant of a DC RL circuit.
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. This concludes
your study of DC circuits. Congratulations on making it through
to the end! For a good review, I suggest that you go back and
re-take each of the Unit Review self-tests (located at the end of the
Of course, there’s plenty more to learn. In this course we’ve concentrated
on analyzing circuits that contain DC voltage sources and
DC current sources. In later courses, you’ll learn about circuits with
AC sources instead of (or in addition to) DC sources. To get a head
start on these topics, take a look at the material that you’ll study
in EET 1155.