Video Transcript
In this lesson, we’re going to
learn about impedance and reactance, which generalize the concept of resistance for
resistors to capacitors and inductors in alternating current circuits. Since our goal is to generalize
resistance, let’s review some of the properties of resistors.
In a resistor, the resistance 𝑅
provides the constant of proportionality between the current 𝐼 and the voltage drop
across the resistor 𝑉. We can state this relationship as
Ohm’s law, 𝑉 equals 𝐼𝑅, which is true whether the current is alternating or
direct. What’s special about resistors is
that we don’t need to know anything else about the current or the voltage in the
circuit to understand how the resistor will affect the relationship. It is always true that 𝑉 equals
𝐼𝑅.
Specifically, a resistor’s effect
in a circuit doesn’t depend on the frequency of the alternating current. On the other hand, the effect of
capacitors and inductors on a circuit does depend on the frequency of the
alternating current. Furthermore, the current through a
resistor is in phase with the voltage across it, but the current through capacitors
and inductors is out of phase with the voltage across them.
We therefore need to define a new
kind of quantity called the reactance, which is similar to resistance but takes into
account phase and frequency. We’ll use the Greek letter 𝜒 to
denote reactance and the subscript 𝐶 to denote the reactance of the capacitor and
the subscript 𝐿 to denote the reactance of an inductor. Remember that we don’t need to
consider phase or frequency when dealing with only resistors, because the effect of
resistors doesn’t depend on either phase or frequency. Before actually working out these
reactances, it’s worth noting how capacitors and inductors behave in direct current
circuits.
We’ve drawn two similar circuits,
one with a capacitor and one with an inductor. Each circuit also has a cell with
voltage 𝑉 and a switch to turn it on. As soon as we close the switch in
the circuit with the capacitor, the electromotive force of the cell starts driving
current through the circuit to deposit charge on the two plates of the
capacitor. However, as the capacitor continues
to charge, it develops its own growing voltage that opposes the emf of the cell. This results in a diminishing
current that eventually disappears entirely when the voltage across the capacitor is
equal to the voltage of the cell. In other words, shortly after the
switch is closed, the capacitor acts like a conducting wire. But a long time after the switch is
closed, the capacitor acts like an open circuit.
The opposite effect occurs in a
direct current circuit with an inductor. When the switch is closed, the emf
from the cell initially acts to create a magnetic field inside of the inductor. As the strength of this magnetic
field grows, more and more of the emf drives current through the inductor instead of
being devoted to establishing the magnetic field. So when the switch is first closed,
the inductor acts like an open circuit. But after a long time when the
magnetic field has reached full strength, the inductor acts like a conducting
wire. As these examples illustrate, the
ability of an electromotive force to drive current through a capacitor or an
inductor can change.
On the other hand, as we saw
through Ohm’s law, an electromotive force always has the same ability to drive
current through a resistor. The changing effects of inductors
and capacitors in a circuit will be the driving principle behind defining inductive
and capacitive reactances. Let’s start by working out the
frequency dependence of inductive and capacitive reactance. Recall that as we just saw for
capacitors experiencing a constant electromotive force, the more time the capacitor
spends charging, the more the capacitor will oppose and reduce the current in the
circuit. In other words, the more the
capacitor charges, the larger its reactance, since larger reactances oppose current
more.
However, if we have attached our
capacitor to an alternating voltage source, after the maximum voltage has been
reached, the electromotive force will start to decrease. This will result in the capacitor
beginning to discharge, and so there will again be a flow of charge and thus a
current in the circuit. Eventually, when the electromotive
force reverses direction, the capacitor will again start to charge, but this time
with opposite charges on each plate. Once the maximum emf in the other
direction has been reached, the capacitor will again begin to discharge. The faster the electromotive force
reaches its maximum value, the less time the capacitor will have to charge before
the emf changes direction and the capacitor starts to discharge. This means that the capacitor will
provide less opposition to current the faster the electromotive force changes
direction.
In technical terms, we see that the
capacitive reactance gets larger at smaller frequencies and gets smaller at larger
frequencies. In this case, we’ve used 𝜔, the
angular frequency of the electromotive force, which is defined as two 𝜋 radians
times the regular frequency or the number of cycles per second. We use angular frequency instead of
regular frequency simply because there’re certain calculations which are beyond the
scope of this lesson that are much neater when expressed in terms of 𝜔 instead of
𝐹. The other factor that we expect to
affect the capacitive reactance is the actual capacitance of the capacitor. Since more capacitance means that a
capacitor will fully charge more slowly, we expect that a capacitor will have a
larger reactance with a smaller capacitance and a smaller reactance with a larger
capacitance.
These two results are
characteristic of an inverse proportionality relationship, where one quantity
getting larger or smaller corresponds to the other quantity getting smaller or
larger. As a formula, the size of the
capacitive reactance is equal to one divided by the frequency of the current or the
emf times the capacitance. It’s worth mentioning that in an AC
circuit, both the current and the emf have the same frequency. In an inductive circuit, as we saw
before, as time goes on, more magnetic field builds up in the inductor. However, unlike a capacitor which
opposes current more the longer it’s been charging, an inductor opposes current less
the longer the magnetic field has been building up. So an inductor becomes less
reactive the longer an emf maintains similar values.
As the emf changes direction, the
direction of the magnetic field will change as well. So the inductor will oppose current
less, the slower the emf varies. In other words, the inductive
reactance will decrease as angular frequency decreases. Furthermore, the smaller the
inductance of the inductor, the smaller the maximum magnetic field. The smaller the maximum magnetic
field, the faster this field can be built up, and thus the smaller the opposition to
the current. So the inductive reactance will be
smaller for smaller inductances. For both of these relationships, a
smaller value of one quantity corresponds to a smaller value of another quantity and
vice versa. This is characteristic of a
directly proportional relationship, and so we have the size of the inductive
reactance is equal to the angular frequency present times the inductance.
Now that we’ve incorporated
frequency into the size of the reactance, let’s see how to incorporate phase. We’ve drawn two graphs showing
electromotive force and current with respect to time, one for a capacitor and one
for an inductor. In the capacitor graph, we can see
that the solid green line representing the current reaches a maximum one-quarter of
a period before the dashed blue line representing the electromotive force. So we see that the current leads
the emf by one-quarter of a period or 90 degrees. We could also say that the current
has been phase shifted back relative to the emf. For the inductor, it is the emf
that leads the current by one-quarter of a period. And in this case, the current has
been phase shifted forward. Note that these phase shifts don’t
affect the magnitude of the current just its relationship in time to the emf.
So we can write full expressions
for the capacitive and inductive reactances as the size of the reactance and the
phase shift. When we say that the phase of the
capacitive reactance is negative 90 degrees, what we really mean is that the phase
shift is backward relative to the emf as opposed to the inductive phase shift, which
is positive or forward relative to the emf. Let’s take a moment to interpret
what these reactances mean. Like resistance, reactance is meant
to provide a connection between the emf and the associated current. However, unlike resistance,
reactance also introduces a phase shift. The size of the reactance behaves
just like the resistance in terms of relating the maximum current to the maximum
emf.
So, for example, the maximum emf
through the inductor is 𝜔𝐿 times the maximum current. However, these two maxima don’t
occur at the same time. This is precisely the information
that is carried in the phase portion of the reactance. In an inductor, the current peaks
one-quarter of a period after the emf. And in a capacitor, the current
peaks one-quarter of a period before the emf. If a circuit had both inductors and
capacitors, the total reactance would be a combination of the inductive and
capacitive reactances.
If we look at our two graphs, we
see that the current through the capacitor and the current through the inductor
always have opposite signs. Where the capacitor current is
positive, the inductor current is negative. And where the inductor current is
positive, the capacitor current is negative. This is because the inductor
introduces a 90-degree phase shift forward and the capacitor introduces a 90-degree
phase shift backward. But the net effect of this is a
180-degree phase shift or half a period between the current through the inductor and
the current through the capacitor. And for a sinusoid, like the
currents and emfs that we’re dealing with, shifting by half a period is equivalent
to changing the sign.
To account for this phase shift
between the inductor and capacitor driven by the same electromotive force, to
combine the reactances, we subtract the capacitive reactance from the inductive
reactance. This total reactance will still
carry with it a 90-degree phase shift of the current relative to the emf. However, which direction this phase
shift occurs will be determined by the sign of the difference between the inductive
and capacitive reactances. If the inductive reactance is
larger, the difference will be positive and the phase shift will be forward. If the capacitive reactance is
larger, the difference will be negative and the phase shift will be backward. So the larger of the two reactances
determines the phase shift for the current.
Okay, let’s now see how to combine
reactance and resistance into impedance. To deal with a circuit that also
has a resistor in addition to a capacitor and an inductor, we can’t simply add the
resistance to the reactance because the reactance involves a phase shift and the
resistance doesn’t. The correct combination of
resistance and reactance is known as impedance. And we’re going to be interested in
the size of the impedance. We can perform this combination
geometrically using the Pythagorean theorem for right triangles. If one leg of the triangle has a
length equal to the resistance in the circuit and the other leg has a length equal
to the combination of inductive and capacitive reactances, then the length of the
hypotenuse will be the size of the impedance usually given the symbol 𝑍.
The reason we’re using a right
triangle to do this calculation is precisely because the resistance introduces no
phase shift to the current, but the reactance introduces a 90-degree phase shift to
the current in either direction. In fact, although we’re only going
to calculate the size of the impedance, the new phase shift associated with the
impedance is given by this angle of the triangle here. Anyway, the Pythagorean theorem
tells us that the sum of the squares of the two legs of a right triangle is always
equal to the square of the hypotenuse. We therefore have that resistance
squared plus inductive reactance minus capacitive reactance as a quantity squared is
equal to impedance squared.
We can now solve for impedance by
taking the square root of both sides. And thus, we get an expression for
the size of the impedance in terms of the resistances and reactances in a
circuit. It’s worth mentioning that if we
work out the units of 𝜔𝐿 and 𝜔𝐶, the inductive and capacitive reactances, we
find that those units are equivalent to ohms. This is important because
resistance also has units of ohms. So if we look at our expression, we
see that resistance and reactance both have units of ohms and therefore the
impedance must also have units of ohms.
We can therefore write at long last
as an analog to Ohm’s law for alternating current circuits with inductors and
capacitors, voltage is equal to current times impedance. If we use only the size of the
impedance, this equation tells us the relationship between the maximum voltage and
maximum current in the circuit. If we include the new angle for
impedance, this equation also tells us about the phase difference between the
current and the voltage.
Okay, now that we’ve seen how
reactance and impedance generalize the concept of resistance, let’s work through an
example.
Which of the following graphs
correctly shows how the reactance of a capacitor varies with the frequency of the
alternating voltage source that the capacitor is connected to?
For our answer choices, we have
four different graphs. Each graph has frequency in hertz
on the horizontal axis and reactance in ohms on the vertical axis. The blue curve on each graph shows
a possible way that reactance might change with frequency. Our task for this question is to
choose the graph that correctly shows the relationship between reactance and
frequency for a capacitor hooked up to an alternating voltage source.
Recall that for a capacitor
connected to an alternating voltage source, the capacitor first charges with, say,
positive charge on the top plate and negative charge on the bottom plate. Then, after the emf reaches a
maximum value, the capacitor begins to discharge and then charge again, but this
time with negative charges on the top plate and positive charges on the bottom
plate. After the electromotive force
reaches its maximum value in the other direction, the capacitor again discharges and
the cycle repeats itself. The more charged the capacitor is,
the more it opposes current, so the larger its reactance. However, the faster the
electromotive force changes directions, the less time the capacitor has to charge
before it begins to discharge. This means that the capacitive
reactance gets smaller as 𝜔, the angular frequency of the voltage source, gets
larger.
The converse is also true. The capacitive reactance is larger
for smaller angular frequencies. If we write this as a formula, we
can write that the capacitive reactance is equal to one divided by the angular
frequency of the voltage source times the capacitance of the capacitor. To use this formula, we’ll need to
convert between 𝜔, the angular frequency, and 𝐹, the regular frequency. This is easily accomplished with
the simple relationship 𝜔 equals two 𝜋𝐹. So our relationship between
reactance and frequency is that capacitive reactance is equal to one divided by two
times 𝜋 times the frequency of the alternating voltage source times the capacitance
of the capacitor.
Since our formula with frequency
and our formula with angular frequency both have the same basic form with frequency
and angular frequency both appearing in the denominator, we can clearly see that the
capacitive reactance will get smaller as the regular frequency gets larger and vice
versa, just like with angular frequency. All of the graphs pictured show a
larger reactance at lower frequencies and a smaller reactance at higher
frequencies. So we need some other way to
distinguish between these graphs. If we look back at our formula, we
can see that the reactance will always be greater than zero as long as the frequency
is finite. The reactance may get very, very,
very small, but the right-hand side of this formula is never zero.
Looking back at our graphs, we can
eliminate choices (b) and (c) because both of these show a reactance that reaches
zero at some frequency. Both graphs (a) and (d) show a
reactance that gets smaller with increasing frequency but never actually reaches
zero. Since these two graphs have the
same behavior at high frequency, let’s see what happens at low frequency. As the frequency gets lower and
lower, the denominator of this fraction gets smaller and smaller, so the overall
fraction gets larger and larger. And since we can make the frequency
as close as we want to zero, the capacitive reactance should increase without limit
as the frequency gets closer and closer to zero.
Of choices (a) and (d), only choice
(a) shows a reactance that appears to be increasing without limit for very low
frequencies. On the other hand, the graph in
choice (d) appears to be tapering off and will not increase without limit. So the correct answer is the graph
shown in choice (a). It turns out we actually didn’t
need to analyze all these graphs in detail to get this answer. Both our qualitative and
quantitative relationships between reactance and frequency tell us that reactance
and frequency are inversely proportional. This means that as one of them gets
larger, the other gets smaller and as one gets smaller, the other gets larger. A graph showing the relationship
between two inversely proportional quantities always has the same basic shape, and
that’s the shape of the graph in (a).
All right, now that we’ve seen an
example, let’s review the key points we learned in this lesson. In this video, we learned about the
reactance of capacitors and inductors. For a capacitor, the reactance is
given by one divided by the angular frequency of the voltage and current in the
circuit times the capacitance. For an inductor, the reactance is
given by the angular frequency of the voltage and current times the inductance. The reactance is meant to be an
analog of resistance for inductors and capacitors, so logically, the units for both
of these quantities work out to ohms. An important difference between the
resistance and the reactance, though, is that resistance doesn’t depend on the
frequency of the current and voltage in the circuit.
On the other hand, capacitors and
inductors do oppose the current differently depending on the frequency of the
current and voltage. We can see this in the formulas
where the capacitive reactance decreases with increasing frequency and the inductive
reactance increases with increasing frequency. This is because the faster the
voltage switches, the less time a capacitor has to charge, and the less charge the
capacitor is, the less it opposes current. On the other hand, the less time an
inductor has to build up a magnetic field, the more it opposes current. So higher frequencies lead to
larger reactances.
When both inductors and capacitors
are present in a circuit, the total reactance is the difference between the
inductive and capacitive reactances. The reason we subtract instead of
add the reactances is because inductors and capacitors shift the phase of the
current in different directions. If a circuit includes resistors and
capacitors or inductors, the analog of resistance is called the impedance. Due to the 90-degree phase shift
introduced by reactance, the size of the impedance is the length of the hypotenuse
of a right triangle whose legs are the total reactance and the resistance. Using the Pythagorean theorem, we
then find an expression for the size of the impedance in terms of the resistance and
the total reactance. Finally, impedance replaces
resistance in Ohm’s law for alternating current circuits that include capacitors and
inductors.
