why study AC circuits? You probably live in a house or
appartment with sockets that deliver AC. Your radio, television and portable
phone receive it, using (among others) circuits like those below. As for the
computer you're using to read this, its signals are not ordinary sinusoidal AC,
but, thanks to Fourier's theorem, any varying signal may be analysed in
terms of its sinusoidal components. So AC signals are almost everywhere. And you
can't escape them, because even the electrical circuits in your brain use
capacitors and resistors.
Some terminology
For brevity, we shall refer to electrical potential difference as voltage.
Throughout this page, we shall consider voltages and currents that vary
sinusoidally with time. We shall use lower case v and i for the voltage and
current when we are considering their variation with time explicitly. The
amplitude or peak value of the sinusoidal variation we shall
represent by Vm and Im, and we shall use V = Vm/21/2
and I = Im/21/2 without subscripts to refer to the RMS
values. For an explanation of RMS values, see
Power and RMS values.
For the origin of the sinusoidally varying voltage in the mains supply, see
Motors and generators.
So for instance, we shall write:
v = v(t) = Vm sin (ωt + φ)
i = i(t) = Im sin (ωt).
where ω is the angular frequency. ω = 2πf, where f is the ordinary or
cyclic frequency. f is the number of complete oscillations per second. φ is the
phase difference between the voltage and current. We shall meet this and
the geometrical significance of ω later.
Resistors and Ohm's law in AC circuits
The voltage v across a resistor is proportional to the current i travelling
through it. (See the page on
drift velocity and Ohm's
law.) Further, this is true at all times: v = Ri. So, if the current in a
resistor is
i = Im . sin (ωt) , we write:
v = R.i = R.Im sin (ωt)
v = Vm. sin (ωt) where
Vm = R.Im
So for a resistor, the peak value of voltage is R times the peak value of
current. Further, they are in phase: when the current is a maximum, the voltage
is also a maximum. (Mathematically, φ = 0.) The first animation shows the
voltage and current in a resistor as a function of time.
The rotating lines in the right hand part of the animation are a very simple
case of a phasor diagram (named, I suppose, because it is a vector
representation of phase). With respect to the x and y axes, radial vectors or
phasors representing the current and the voltage across the resistance rotate
with angular velocity ω. The lengths of these phasors represent the peak current
Im and voltage Vm. The y components are Im sin (ωt)
= i(t) and voltage Vm sin (ωt)= v(t). You can compare i(t) and v(t)
in the animation with the vertical components of the phasors. The animation and
phasor diagram here are simple, but they will become more useful when we
consider components with different phases and with frequency dependent
behaviour.
(For a comparison of simple harmonic motion and circular motion, see
Physclips.)
What are impedance and reactance?
Circuits in which current is proportional to voltage are called linear
circuits. (As soon as one inserts diodes and transistors, circuits cease to
be linear, but that's another story.) The ratio of voltage to current in a
resistor is its resistance. Resistance does not depend on frequency, and
in resistors the two are in phase, as we have seen in the animation. However,
circuits with only resistors are not very interesting.
In general, the ratio of voltage to current does depend on frequency
and in general there is a phase difference. So impedance is
the general name we give to the ratio of voltage to current. It has the
symbol Z. Resistance is a special case of impedance. Another special case is
that in which the voltage and current are out of phase by 90�: this is an
important case because when this happens, no power is lost in the circuit.
In this case where the voltage and current are out of phase by 90�, the
ratio of voltage to current is called the reactance, and it has the
symbol X. We return to summarise these terms and give expressions for them
below in the section
Impedance of
components, but first let us see why there are frequency dependence and
phase shifts for capacitors and for inductors.
Capacitors and charging
The voltage on a capacitor depends on the amount of charge you store on its
plates. The current flowing onto the positive capacitor plate (equal to that
flowing off the negative plate) is by definition the rate at which charge is
being stored. So the charge Q on the capacitor equals the integral of the
current with respect to time. From the definition of the capacitance,
vC = q/C, so
Now remembering that the integral is the area under the curve (shaded blue), we
can see in the next animation why the current and voltage are out of phase.
Once again we have a sinusoidal current i = Im . sin (ωt), so
integration gives
(The constant of integration has been set to zero so that the average charge
on the capacitor is 0).
Now we define the capacitive reactance XC as the ratio of
the magnitude of the voltage to magnitude of the current in a capacitor. From
the equation above, we see that XC = 1/ωC. Now we can rewrite the
equation above to make it look like Ohm's law. The voltage is proportional to
the current, and the peak voltage and current are related by
Vm = XC.Im.
Note the two important differences. First, there is a difference in phase:
the integral of the sinusoidal current is a negative cos function: it reaches
its maximum (the capacitor has maximum charge) when the current has just
finished flowing forwards and is about to start flowing backwards. Run the
animation again to make this clear. Looking at the relative phase, the voltage
across the capacitor is 90�, or one quarter cycle, behind the current. We can
see also see how the φ = 90� phase difference affects the phasor diagrams at
right. Again, the vertical component of a phasor arrow represents the
instantaneous value of its quanitity. The phasors are rotating counter clockwise
(the positive direction) so the phasor representing VC is 90�
behind the current (90� clockwise from it).
Recall that reactance is the name for the ratio of voltage to current
when they differ in phase by 90�. (If they are in phase, the ratio is called
resistance.) Another difference between reactance and resistance is that the
reactance is frequency dependent. From the algebra above, we see that the
capacitive reactance XC decreases with frequency . This is shown in
the next animation: when the frequency is halved but the current amplitude kept
constant, the capacitor has twice as long to charge up, so it generates twice
the potential difference. The blue shading shows q, the integral under the
current curve (light for positive, dark for negative). The second and fourth
curves show VC = q/C . See how the lower frequency leads to a larger
charge (bigger shaded area before changing sign) and therefore a larger VC.
Thus for a capacitor, the ratio of voltage to current decreases with
frequency. We shall see later how this can be used for filtering different
frequencies.
Inductors and the Farady emf
An inductor is usually a coil of wire. In an ideal inductor, the resistance of
this wire is negligibile, as is its capacitance. The voltage that appears across
an inductor is due to its own magnetic field and Faraday's law of
electromagnetic induction. The current i(t) in the coil sets up a magnetic
field, whose magnetic flux φB is proportional to the field strength,
which is proportional to the current flowing. (Do not confuse the phase φ with
the flux φB.) So we define the (self) inductance of the coil thus:
φB(t) = L.i(t)
Faraday's law gives the emf EL = - dφB/dt. Now this emf is
a voltage rise, so for the voltage drop vL across the inductor, we
have:
Again we define the inductive reactance XL as the ratio of
the magnitudes of the voltage and current, and from the equation above we see
that XL = ωL. Again we note the analogy to Ohm's law: the voltage is
proportional to the current, and the peak voltage and currents are related by
Vm = XL.Im.
Remembering that the derivative is the local slope of the curve (the purple
line), we can see in the next animation why voltage and current are out of phase
in an inductor.
Again, there is a difference in phase: the derivative of the sinusoidal
current is a cos function: it has its maximum (largest voltage across the
inductor) when the current is changing most rapidly, which is when the current
is intantaneously zero. The animation should make this clear. The voltage across
the ideal inductor is 90� ahead of the current, (ie it reaches its peak one
quarter cycle before the current does). Note how this is represented on the
phasor diagram.
Again we note that the reactance is frequency dependent XL
= ωL. This is shown in the next animation: when the frequency is halved but the
current amplitude kept constant, the current is varying only half as quickly, so
its derivative is half as great, as is the Faraday emf. For an inductor, the
ratio of voltage to current increases with frequency, as the next animation
shows.
Impedance of components
Let's recap what we now know about voltage and curent in linear components. The
impedance is the general term for the ratio of voltage to current.
Resistance is the special case of impedance when φ = 0, reactance the special
case when φ = � 90�. The table below summarises the
impedance of the different components. It is easy to remember that the voltage
on the capacitor is behind the current, because the charge doesn't build
up until after the current has been flowing for a while.
The same information is given graphically below. It is easy to remember the
frequency dependence by thinking of the DC (zero frequency) behaviour: at DC, an
inductance is a short circuit (a piece of wire) so its impedance is zero. At DC,
a capacitor is an open circuit, as its circuit diagram shows, so its impedance
goes to infinity.
RC Series combinations
When we connect components together, Kirchoff's laws apply at any instant. So
the voltage v(t) across a resistor and capacitor in series is just
vseries(t) = vR(t) + vC(t)
however the addition is complicated because the two are not in phase. The next
animation makes this clear: they add to give a new sinusoidal voltage, but the
amplitude is less than VmR(t) + VmC(t). Similarly,
the AC voltages (amplitude times 21/2) do not add up. This may seem
confusing, so it's worth repeating:
vseries = vR + vC but
Vseries > VR + VC.
This should be clear on the animation and the still graphic below: check that
the voltages v(t) do add up, and then look at the magnitudes. The amplitudes and
the RMS voltages V do not add up in a simple arithmetical way.
Here's where phasor diagrams are going to save us a lot of work. Play
the animation again (click play), and look at the projections on the vertical
axis. Because we have sinusoidal variation in time, the vertical component
(magnitude times the sine of the angle it makes with the x axis) gives us v(t).
But the y components of different vectors, and therefore phasors, add up simply:
if
rtotal = r1 + r2,
then
ry total = ry1 + ry2.
So v(t), the sum of the y projections of the component phasors, is just the y
projection of the sum of the component phasors. So we can represent the three
sinusoidal voltages by their phasors. (While you're looking at it, check the
phases. You'll see that the series voltage is behind the current in phase, but
the relative phase is somewhere between 0 and 90�, the exact value depending on
the size of VR and VC. We'll discuss phase below.)
Now let's stop that animation and label the values, which we do in the
still figure below. All of the variables (i, vR, vC, vseries)
have the same frequency f and the same angular frequency ω, so their phasors
rotate together, with the same relative phases. So we can 'freeze' it in
time at any instant to do the analysis. The convention I use is that the x
axis is the reference direction, and the reference is whatever is common in
the circuit. In this series circuit, the current is common. (In a parallel
circuit, the voltage is common, so I would make the voltage the horizontal
axis.) Be careful to distinguish v and V in this figure!
(Careful readers will note that I'm taking a shortcut in these
diagrams: the size of the arrows on the phasor diagrams are drawn the same as
the amplitudes on the v(t) graphs. However I am just calling them VR,
VC etc, rather than VmR, VmR etc. The reason is
that the peak values (VmR etc) are rarely used in talking about AC:
we use the RMS values,
which are peak values times 0.71. Phasor diagrams in RMS have the same shape as
those drawn using amplitudes, but everything is scaled by a factor of 0.71 = 1/21/2.)
The phasor diagram at right shows us a simple way to calculate the series
voltage. The components are in series, so the current is the same in both.
The voltage phasors (brown for resistor, blue for capacitor in the
convention we've been using) add according to vector or phasor addition, to
give the series voltage (the red arrow). By now you don't need to look at
v(t), you can go straight from the circuit diagram to the phasor diagram,
like this:
From Pythagoras' theorem:
V2mRC = V2mR + V2mC
If we divide this equation by two, and remembering that the
RMS value V = Vm/21/2,
we also get:
Now this looks like Ohm's law again: V is proportional to I. Their ratio is the
series impedance, Zseries and so for this series circuit,
Note the frequency dependence of the series impedance ZRC: at low
frequencies, the impedance is very large, because the capacitive reactance 1/ωC
is large (the capacitor is open circuit for DC). At high frequencies, the
capacitive reactance goes to zero (the capacitor doesn't have time to charge up)
so the series impedance goes to R. At the angular frequency ω = ωo =
1/RC, the capacitive reactance 1/ωC equals the resistance R. We shall show this
characteristic frequency on all graphs on this page.
Remember how, for two resistors in series, you could just add the
resistances: Rseries = R1 + R2 to get the
resistance of the series combination. That simple result comes about because
the two voltages are both in phase with the current, so their phasors are
parallel. Because the phasors for reactances are 90� out of phase with the
current, the series impedance of a resistor R and a reactance X are given by
Pythagoras' law:
Zseries2 = R2 + X2 .
Ohm's law in AC. We can rearrange the equations above to obtain the
current flowing in this circuit. Alternatively we can simply use the Ohm's Law
analogy and say that I = Vsource/ZRC. Either way we get
where the current goes to zero at DC (capacitor is open circuit) and to V/R at
high frequencies (no time to charge the capacitor).
So far we have concentrated on the magnitude of the voltage and current. We
now derive expressions for their relative phase, so let's look at the phasor
diagram again.
From simple trigonometry, the angle by which the current leads the voltage is
tan-1 (VC/VR) = tan-1 (IXC/IR)
= tan-1 (1/ωRC) = tan-1 (1/2πfRC).
However, we shall refer to the angle φ by which the voltage leads the current.
The voltage is behind the current because the capacitor takes time to
charge up, so φ is negative, ie
φ = -tan-1 (1/ωRC) = tan-1
(1/2πfRC).
(You may want to go back to the
RC animation
to check out the phases in time.)
At low frequencies, the impedance of the series RC circuit is dominated by the
capacitor, so the voltage is 90� behind the current. At high frequencies, the
impedance approaches R and the phase difference approaches zero. The frequency
dependence of Z and φ are important in the applications of RC circuits. The
voltage is mainly across the capacitor at low frequencies, and mainly across the
resistor at high frequencies. Of course the two voltages must add up to give the
voltage of the source, but they add up as vectors.
V2RC = V2R + V2C.
At the frequency ω = ωo = 1/RC, the phase φ = 45� and the voltage
fractions are VR/VRC = VC/VRC = 1/2V1/2
= 0.71.
So, by chosing to look at the voltage across the resistor, you select mainly the
high frequencies, across the capacitor, you select low frequencies. This brings
us to one of the very important applications of RC circuits, and one which
merits its own page:
filters, integrators and differentiators where we use sound files as
examples of RC filtering.
RL Series combinations
In an RL series circuit, the voltage across the inductor is aheadof the
current by 90�, and the inductive reactance, as we saw before, is XL
= ωL. The resulting v(t) plots and phasor diagram look like this.
It is straightforward to use Pythagoras' law to obtain the series impedance and
trigonometry to obtain the phase. We shall not, however, spend much time on RL
circuits, for three reasons. First, it makes a good exercise for you to do it
yourself. Second, RL circuits are used much less than RC circuits. This is
because inductors are always* too big, too expensive and the wrong value, a
proposition you can check by looking at an electronics catalogue. If you can use
a circuit involving any number of Rs, Cs, transistors, integrated circuits etc
to replace an inductor, one usually does. The third reason why we don't look
closely at RL circuits on this site is that you can simply look at RLC circuits
(below) and omit the phasors and terms for the capacitance.
* Exceptions occur at high frequencies (~GHz) where only small value Ls are
required to get substantial ωL. In such circuits, one makes an inductor by
twisting copper wire around a pencil and adjusts its value by squeezing it
with the fingers.
RLC Series combinations
Now let's put a resistor, capacitor and inductor in series. At any given
time, the voltage across the three components in series, vseries(t),
is the sum of these:
vseries(t) = vR(t) + vL(t) + vC(t),
The current i(t) we shall keep sinusoidal, as before. The voltage across the
resistor, vR(t), is in phase with the current. That across the
inductor, vL(t), is 90� ahead and that across the capacitor, vC(t),
is 90� behind.
Once again, the time-dependent voltages v(t) add up at any time, but the RMS
voltages V do not simply add up. Once again they can be added by phasors
representing the three sinusoidal voltages. Again, let's 'freeze' it in time for
the purposes of the addition, which we do in the graphic below. Once more, be
careful to distinguish v and V.
Look at the phasor diagram: The voltage across the ideal inductor is
antiparallel to that of the capacitor, so the total reactive voltage (the
voltage which is 90� ahead of the current) is VL - VC, so
Pythagoras now gives us:
V2series = V2R + (VL
- VC)2
Now VR = IR, VL = IXL = ωL and VC =
IXC= 1/ωC. Substituting and taking the common factor I gives:
where Zseries is the series impedance: the ratio of the voltage to
current in an RLC series ciruit. Note that, once again, reactances and
resistances add according to Pythagoras' law:
Zseries2 = R2 + Xtotal2
= R2 + (XL - XC)2.
Remember that the inductive and capacitive phasors are 180� out of phase, so
their reactances tend to cancel.
Now let's look at the relative phase. The angle by which the voltage
leads the current is
φ = tan-1 ((VL - VC)/VR).
Substiting VR = IR, VL = IXL = ωL and VC
= IXC= 1/ωC gives:
The dependence of Zseries and φ on the angular frequency ω is
shown in the next figure. The angular frequency ω is given in terms of a
particular value ωo, the resonant frequency (ωo2 = 1/LC),
which we meet below.
(Setting the inductance term to zero gives back the equations we had above for
RC circuits, though note that phase is negative, meaning (as we saw above) that
voltage lags the current. Similarly, removing the capacitance terms gives the
expressions that apply to RL circuits.)
The next graph shows us the special case where the frequency is such that VL
= VC.
Because vL(t) and vC are 180� out of phase, this means
that vL(t) = - vC(t), so the
two reactive voltages cancel out, and the series voltage is just equal to that
across the resistor. This case is called series resonance, which is our next
topic.
Resonance
Note that the expression for the series impedance goes to infinity at high
frequency because of the presence of the inductor, which produces a large emf if
the current varies rapidly. Similarly it is large at very low frequencies
because of the capacitor, which has a long time in each half cycle in which to
charge up. As we saw in the plot of Zseriesω above, there is a
minimum value of the series impedance, when the voltages across capacitor and
inductor are equal and opposite, ie vL(t) = - vC(t)
so VL(t) = VC, so
ωL = 1/ωC so the frequency at which this occurs is
where ωo and fo are the angular and cyclic frequencies of
resonance, respectively. At resonance, series impedance is a minimum, so the
voltage for a given current is a minimum (or the current for a given voltage is
a maximum).
This phenomenon gives the answer to our teaser question
at the beginning. In an RLC series circuit in which the inductor has
relatively low internal resistance r, it is possible to have a large
voltage across the the inductor, an almost equally large voltage across
capacitor but, as the two are nearly 180� degrees out of phase, their
voltages almost cancel, giving a total series voltage that is quite
small. This is one way to produce a large voltage oscillation with only
a small voltage source. In the circuit diagram at right, the coil
corresponds to both the inducance L and the resistance r, which is why
they are drawn inside a box representing the physical component, the
coil. Why are they in series? Because the current flows through the coil
and thus passes through both the inductance of the coil and its
resistance.
You get a big voltage in the circuit for only a small voltage
input from the power source. You are not, of course, getting
something for nothing. The energy stored in the large
oscillations is gradually supplied by the AC source when you
turn on, and it is then exchanged between capacitor and inductor
in each cycle. For more details about this phenomenon, and a
discussion of the energies involved, go to
LC
oscillations.
You have perhaps been looking at these phasor diagrams, noticing that they are
all two-dimensional, and thinking that we could simply use the complex plane.
Good idea! But not original: indeed, that is the most common way to analyse such
circuits.
The only difference from the presentation here is to consider cosusoids,
rather than sinusoids. In the animations above, we used sin waves so that
the vertical projection of the phasors would correspond to the height on the
v(t) graphs. In complex algebra, we use cos waves and take their projections
on the (horizontal) real axis. The phasor diagrams have now become diagrams
of complex numbers, but otherwise look exactly the same. They still rotate
at ωt, but in the complex plane. The resistor has a real impedance R, the
inductor's reactance is a positive imaginary impedance
XL = jωL
and the capacitor has a negative imaginary impedance
XC = -j.1/ωC = 1/jωC.
Consequently, using bold face for complex quantities, we may write:
Zseries = (R2 + (jωL + 1/jωC)2)1/2
and so on. The algebra is relatively simple. The magnitude of any complex
quantity gives the magnitude of the quantity it represents, the phase angle its
phase angle. Its real component is the component in phase with the reference
phase, and the imaginary component is the component that is 90� ahead.
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So, by chosing to look at the voltage across the resistor, you select mainly the
high frequencies, across the capacitor, you select low frequencies. This brings
us to one of the very important applications of RC circuits, and one which
merits its own page:
filters, integrators and differentiators where we use sound files as
examples of RC filtering.
.gif)
Look at the phasor diagram: The voltage across the ideal inductor is
antiparallel to that of the capacitor, so the total reactive voltage (the
voltage which is 90� ahead of the current) is VL - VC, so
Pythagoras now gives us:
2.gif)
Because vL(t) and vC are 180� out of phase, this means
that vL(t) = - vC(t), so the
two reactive voltages cancel out, and the series voltage is just equal to that
across the resistor. This case is called series resonance, which is our next
topic.
