element
CIRCUIT ELEMENTS IN AC CIRCUITS
Resistor
At this point, you’ve learned how resistors, capacitors and
inductors act when in a DC circuit. It’s time to take a look
at how they might act in an AC circuit. Specifically, we
want to know what the net resistive nature (i.e., it’s
tendency to resist current flow) is for each element.
symbol
R
units
ohms
C farads
Capacitor
resistive nature
“resistor-like” resistance R (ohms)
frequency-dependent capacitive
reactance in an RC circuit:
XC =
To make this quick and dirty, I’m going to put all the cogent
information into a table. During class, I’ll zero in on the
parts that really matter.
Filter?
phase relationship
no
current in phase
with voltage
across element
high
pass
1
ohms
2!"C
L
henrys
low
pass
X L = 2!"L ohms
!
radians
2
“resistor-like” resistance rL
impedance
RLC ckt.
Z
1.
Summary of the Table
ohms
!
radians
2
with minimal
resistance-like
resistance in
circuit, current
LEADS voltage
by
frequency-dependent inductive
reactance in an RL circuit:
Inductor
with minimal
resistance-like
resistance in
circuit, current
LAGS voltage by
resonant frequency
1/2
2
*
1 ' 2
$
Z = ,( R + rL ) + & 2!"L #
)( /
%
2!"C
,+
/.
!=
1
1
2" LC
2.
Capacitor:
Resistors:
1.) Capacitors do not have a “resistor-like” resistive nature to them.
They do have a resistive nature, though, that is depends upon the
frequency being impressed upon them by the AC source.
1.) Resistors have a resistive nature that is characterized as
“resistance” and that has the units of ohms.
2.) This resistive nature, called the capacitive reactance, has the
units of ohms (it DOES identify resistance to current flow) and, as
has been said above, is frequency dependent.
2.) The resistive nature (i.e., the resistance) of a resistor is not
frequency dependent. That is, it doesn’t matter at what frequency
the AC power supply is acting, the resistive nature of the resistor
will always be the same.
C
3.) To see why a capacitor
has this a frequencydependent resistive nature,
consider the RC circuit to the
right.
3.) Because the resistive nature of the resistor is not frequency
dependent, it doesn’t act as a frequency filter in any frequency
range.
3.
R
V0 sin ( 2!"t )
4.
3.) Why frequency-dependent resistive nature?
3.) Why frequency-dependent resistive nature?
a.) At low frequency, the voltage
across the AC source will vary
slowly and will look like the
waveform shown to the right.
d.) At any given instant, the sum
of the voltage across the
capacitor and voltage across the
load resistor must equal the net
voltage across the AC source.
b.) The voltage across a
capacitor is a function of the
charge on its plates.
e.) That means that at low frequency,
the voltage across the capacitor is
predominately high and the voltage
across the resistor is low.
C
c.) As the AC voltage is high
most of the time in the circuit,
the charge on the plates will
be high most of the time in
the circuit and the voltage
across the capacitor must
also be high most of the time.
R
V0 sin ( 2!"t )
R
V0 sin ( 2!"t )
6.
3.) Why frequency-dependent resistive nature?
g.) The waveform for a high
frequency signal coming out of
an AC source is shown to the
right.
h.) At high frequency, the AC
source’s voltage AVERAGES TO
ZERO (take a very short time interval
and you find as much up as down).
f.) As the voltage across the resistor is
proportional to the current through the
resistor, a near zero resistor voltage
means there is essentially no current
in the circuit.
5.
3.) Why frequency-dependent resistive nature?
C
j.) With essentially no net voltage
drop across the capacitor, all the
voltage drop must be across the load
resistor.
k.) As always, the voltage drop across
a resistor is proportional to the current
through the resistor.
C
i.) That means the alternating charge
polarity across the capacitor’s plates
changes back and forth so fast over
time that it’s AVERAGE VOLTAGE
over time turns out to be ZERO.
R
C
l.) So in this case, the high voltage
across the load resistor signifies the
fact that at high frequencies, there is
current in the circuit.
R
m.) This is why capacitors are
sometimes called high pass filters.
V0 sin ( 2!"t )
7.
V0 sin ( 2!"t )
8.
4.) How big, quantitatively, is the capacitor’s resistive nature?
4.) How big, quantitatively, is the capacitor’s resistive nature (it’s
capacitive reactance?
C
C
a.) If we write out Kirchoff’s loop equation
for this circuit, we get:
!
q
! iR + Vo sin ( 2"#t ) = 0
C
b.) The relationship between the amount
of charge on the capacitor plates q and
the current in the circuit i is i = dq/dt.
R
1
ohms
2!"C
V0 sin ( 2!"t )
R
V0 sin ( 2!"t )
e.) This is the “frequency dependent” resistive nature capacitors
exhibit. It is called capacitive reactance and it’s symbol is XC.
c.) Substitute this into the Kirchoff
relationship, solve for q, then taking q’s
derivative to get i (none of this is
particular easy--you won’t be asked to
do any of it) and you end up with an
interesting relationship.
9.
f.) Note that this relationship suggests that at high frequency with
big !, the cap’s resistive nature is small. Small resistive nature
means big current . . . as predicted earlier. Capacitors allow high
frequency signals to pass while resisting (i.e., not allowing) the
passage of low frequency signals.
10.
3.) Why frequency-dependent resistive nature?
Inductors:
1.) Inductors do have a “resistor-like” resistance to them that is due
to the resistance inherent within the wire making up the inductor’s
coil. That resistance is characterized as rL . In addition, they also
have a frequency-dependent resistive nature.
2.) The inductor’s frequencydependent resistive nature is called
the inductive reactance. As do all
resistive natures, it has the units of
ohms.
d.) From Ohm’s Law, we know that in
general, i=V/R. It isn’t surprising, then, to
find that the resulting expression for i will
have a voltage term in its numerator and
a resistance term in its denominator.
What’s important is that that denominator
term will be:
L, rL
a.) As you know, coils don’t like
a changing magnetic flux
through their cross section.
b.) As an AC source drives an
alternating current, a coil
(inductor) will constantly be
producing an induced EMF that
opposes the changes the AC
source is attempting to impress
on the circuit.
L, rL
R
3.) To see why an inductor has this
frequency-dependent resistive
nature, consider the RL circuit to the
right.
c.) At low frequency, the change of
current is slow and the induced EMF
is small (remember, the EMF is
related the how fast the B-field down
the axis of the coil is changing).
V0 sin ( 2!"t )
11.
R
V0 sin ( 2!"t )
12.
3.) Why frequency-dependent resistive nature?
3.) Why frequency-dependent resistive nature?
d.) As a consequence, at low
frequency there is very little
frequency-dependent voltage
across the inductor.
e.) But if the voltage across the
inductor is small at low frequencies,
then the voltage across the load
resistor must be high at low frequency.
g.) This is why inductors are sometimes
called low pass filters.
h.) Continuing, the waveform for a high
frequency signal out of the AC source is
shown to the right.
i.) At high source frequency, the coil is
being forced to deal with a fast changing
magnetic field down its axis. That, in turn,
creates a huge back-EMF across the coil.
L, rL
f.) And as the voltage across a resistor
is proportional to the current through
the resistor, high voltage across the
resistor at low frequency means there
is current in the circuit.
R
V0 sin ( 2!"t )
13.
a.) Again, if we write out Kirchoff’s loop
equation for this circuit, we get:
L, rL
di
!i ( R + rL ) ! L + Vo sin ( 2"#t ) = 0
dt
b.) Solving this equation for i (again,
this requires solving a first-order
differential equation--this is not
something I’d expect you to do)
produces an interesting relationship.
R
j.) With this big voltage drop across the
inductor at high frequency, there will be
essentially no voltage drop across the
load resistor. That, in turn, means no
current in the circuit.
4.) How big, quantitatively, is the
capacitor’s resistive nature?
4.) How big, quantitatively, is the inductor’s resistive nature?
R
V0 sin ( 2!"t )
14.
L, rL
d.) As was the case with the capacitor,
the resulting expression for i will have a
voltage term in its numerator and, in this
case, two resistance terms in its
denominator. One of those resistance
terms will be:
2!"L ohms
V0 sin ( 2!"t )
L, rL
R
V0 sin ( 2!"t )
e.) This is the “frequency dependent” resistive nature that inductors
exhibit. It is called inductive reactance and it’s symbol is X L.
f.) Note that this relationship suggests that at low frequency with
low !, the inductor’s resistive nature is small. Small resistive
nature means big current . . . as predicted earlier. Inductors allow
low frequency signals to pass while resisting (i.e., not allowing) the
passage of high frequency signals.
15.
16.
RLC Circuits:
3.) You might expect that there would be NO frequency that might
produce current in the circuit, but that’s not the case. In fact, there will
be one frequency whereupon the inductor and capacitor will exactly
cancel one another out leaving only the resistor-like resistance in the
circuit to limit current.
1.) There is only one more bit of amusement to deal with. What
happens when we have an inductor, capacitor and resistor all in the
same AC circuit?
2.) All hell breaks loose! The inductor
tries to make the current lead the
voltage and fights to suppress the AC
source signal if it happens to be high
frequency (remember, inductors don’t
pass high frequency, only low
frequency), and the capacitor tries to
make the current lag the voltage and
fights to suppress the AC source
signal if it happens to be low
frequency (remember, capacitors
don’t pass low frequency, only high
frequency).
C
C
4.) To see how this works, we have to
go back to Kirchoff’s loop equation for
this circuit. It is:
R
L, rL
!i ( R + rL ) !
q
di
! L + Vo sin ( 2"#t ) = 0
C
dt
R
L, rL
5.) If you thought the previous two
differential equations were a mess,
you’ll agree that this one is a beauty (in
fact, it’s actually a second order
differential equation in q).
V0 sin ( 2!"t )
V0 sin ( 2!"t )
17.
6.) As before, though, solving for i leads
to a voltage term in the numerator and
several resistance terms in the
denominator. What’s more, the
frequency-dependent part of that
resistance term will be found to be:
L, rL
1/2
2
*
1 ' 2
$
,( R + rL ) + & 2!"L #
/
)
%
2!"C ( /.
,+
C
b.) Things to notice:
C
R
18.
V0 sin ( 2!"t )
ii.) If there is no capacitor or inductor
in the circuit, the impedance simply
becomes the resistance R in the
circuit. In other words, the version of
Ohm’s Law you have come to know
and love is a special case of this
more expanded version.
R
1/2
2
*
1 ' 2
$
,( R + rL ) + & 2!"L #
) /
%
2!"C ( /.
,+
L, rL
V0 sin ( 2!"t )
1 '
$
iii.) There is a frequency at which the & 2!"L #
)
%
2!"C (
a.) This overall “frequency dependent” resistive nature for the RLC
circuit is called impedance and it is given the symbol Z.
part of the equation goes to zero. That is, when 2!"L =
b.) There are several things to notice about this relationship.
or when:
i.) Ohm’s Law still works, it is now simply written as V = i Z,
where the impedance Z is the net resistive nature of the circuit.
!=
1
,
2!"C
1
1
2" LC
This is called the resonance frequency of the circuit, and it is at this
frequency that the AC source signal DOES NOT DIE in the circuit.
19.
20.
C
c.) When the AC source is set to the
circuit’s resonance frequency, the only
resistance in the circuit is resistor-like
resistance and the signal proliferates.
C
R
i.) What this means is that the
frequency response curve for a typical
RLC circuit looks like:
L, rL
c.) This frequency-response characteristic
is going to be very important in the radio
circuit.
R
L, rL
d.) And lastly, just as a point of order,
you now know what the impedance
label on the back of your stereo
V0 sin ( 2!"t )
speakers means. It is telling the you
speaker’s net, overall resistance to
charge flow due to all of the resistors, capacitors or inductors that exist
in the speaker circuit.
V0 sin ( 2!"t )
amplitude
In other words, when a speaker says, Impedance: 8 ohms, it means
that for the circuit at a standard frequency (I don’t know that the
industry standard is, but they have one):
1/2
!=
1
1
2" LC
2
*
1 ' 2
$
Z = ,( R + rL ) + & 2!"L #
/
)
%
2!"C ( /.
,+
frequency
21.
= 8 0.
22.