CIRCUIT ELEMENTS IN AC CIRCUITS

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.

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.

1.

element symbol units resistive nature Filter? phase relationship

R

C

L

ohms

farads

henrys

Resistor

Capacitor

Inductor

“resistor-like” resistance R (ohms)

frequency-dependent capacitive

reactance in an RC circuit:

XC=

1

2!"C ohms

frequency-dependent inductive

reactance in an RL circuit:

XL= 2!"L ohms

“resistor-like” resistance rL

no

high

pass

low

pass

current in phase

with voltage

across element

with minimal

resistance-like

resistance in

circuit, current

LAGS voltage by!

2 radians

with minimal

resistance-like

resistance in

circuit, current

LEADS voltage

by!

2 radians

2.

RLC ckt.

impedance

Z ohmsZ = R + r

L( )2

+ 2!"L #1

2!"C$%&

'()2*

+,,

-

.//

1/2

resonant frequency

! =1

2"

1

LC

Summary of the Table

Resistors:

1.) Resistors have a resistive nature that is characterized as

“resistance” and that has the units of ohms.

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.

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.

Capacitor:

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.

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.

3.) To see why a capacitor

has this a frequency-

dependent resistive nature,

consider the RC circuit to the

right.

V0sin 2!"t( )

R

C

4.

a.) At low frequency, the voltage

across the AC source will vary

slowly and will look like the

waveform shown to the right.

3.) Why frequency-dependent resistive nature?

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.

5.

b.) The voltage across a

capacitor is a function of the

charge on its plates.

V0sin 2!"t( )

R

C

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.

3.) Why frequency-dependent resistive nature?

6.

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.

e.) That means that at low frequency,

the voltage across the capacitor is

predominately high and the voltage

across the resistor is low.

V0sin 2!"t( )

R

C

g.) The waveform for a high

frequency signal coming out of

an AC source is shown to the

right.

3.) Why frequency-dependent resistive nature?

7.

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).

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.

V0sin 2!"t( )

R

C

3.) Why frequency-dependent resistive nature?

8.

k.) As always, the voltage drop across

a resistor is proportional to the current

through the resistor.

j.) With essentially no net voltage

drop across the capacitor, all the

voltage drop must be across the load

resistor.

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.

m.) This is why capacitors are

sometimes called high pass filters.

V0sin 2!"t( )

R

C

4.) How big, quantitatively, is the capacitor’s resistive nature (it’s

capacitive reactance?

9.

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.

a.) If we write out Kirchoff’s loop equation

for this circuit, we get:

!q

C! iR +Vo sin 2"#t( ) = 0

b.) The relationship between the amount

of charge on the capacitor plates q and

the current in the circuit i is i = dq/dt.V0sin 2!"t( )

R

C

4.) How big, quantitatively, is the capacitor’s resistive nature?

10.

e.) This is the “frequency dependent” resistive nature capacitors

exhibit. It is called capacitive reactance and it’s symbol is .

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:1

2!"C ohms

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.

!

XC

V0sin 2!"t( )

R

C

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 . In addition, they also

have a frequency-dependent resistive nature.

2.) The inductor’s frequency-

dependent resistive nature is called

the inductive reactance. As do all

resistive natures, it has the units of

ohms.

3.) To see why an inductor has this

frequency-dependent resistive

nature, consider the RL circuit to the

right.

11.

L, rL

V0sin 2!"t( )

R

rL

a.) As you know, coils don’t like

a changing magnetic flux

through their cross section.

3.) Why frequency-dependent resistive nature?

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).

12.

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

V0sin 2!"t( )

R

d.) As a consequence, at low

frequency there is very little

frequency-dependent voltage

across the inductor.

3.) Why frequency-dependent resistive nature?

13.

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.

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.L, r

L

V0sin 2!"t( )

R

h.) Continuing, the waveform for a high

frequency signal out of the AC source is

shown to the right.

3.) Why frequency-dependent resistive nature?

14.

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.

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.

g.) This is why inductors are sometimes

called low pass filters.

L, rL

V0sin 2!"t( )

R

4.) How big, quantitatively, is the inductor’s resistive nature?

15.

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.

a.) Again, if we write out Kirchoff’s loop

equation for this circuit, we get:

!i R + rL( ) ! L

di

dt+V

osin 2"#t( ) = 0

L, rL

V0sin 2!"t( )

R

4.) How big, quantitatively, is the

capacitor’s resistive nature?

16.

e.) This is the “frequency dependent” resistive nature that inductors

exhibit. It is called inductive reactance and it’s symbol is .

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

!

XL

L, rL

V0sin 2!"t( )

R

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.

RLC Circuits:

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).

17.

V0sin 2!"t( )

L, rL

C

R

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.

4.) To see how this works, we have to

go back to Kirchoff’s loop equation for

this circuit. It is:

18.

V0sin 2!"t( )

L, rL

C

R

!i R + rL( ) !q

C! L

di

dt+Vo sin 2"#t( ) = 0

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).

19.

a.) This overall “frequency dependent” resistive nature for the RLC

circuit is called impedance and it is given the symbol Z.

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:

R + rL( )

2

+ 2!"L #1

2!"C$%&

'()2*

+,,

-

.//

1/2

V0sin 2!"t( )

L, rL

C

R

b.) There are several things to notice about this relationship.

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.

20.

R + rL( )

2

+ 2!"L #1

2!"C$%&

'()2*

+,,

-

.//

1/2

V0sin 2!"t( )

L, rL

C

R

b.) Things to notice:

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.

iii.) There is a frequency at which the 2!"L #1

2!"C$%&

'()

part of the equation goes to zero. That is, when ,2!"L =1

2!"C

or when:! =

1

2"

1

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.

21.

V0sin 2!"t( )

L, rL

C

R

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.

i.) What this means is that the

frequency response curve for a typical

RLC circuit looks like:

frequency

amplitude

! =1

2"

1

LC

22.

V0sin 2!"t( )

L, rL

C

R

c.) This frequency-response characteristic

is going to be very important in the radio

circuit.

d.) And lastly, just as a point of order,

you now know what the impedance

label on the back of your stereo

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.

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):

Z = R + rL( )

2

+ 2!"L #1

2!"C

$%&

'()

2*

+,,

-

.//

1/2

= 8 0.