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AC Circuits
PH 203
Professor Lee Carkner
Lecture 23
Alternating Current Issues
Voltage and current vary sinusoidally with
time

Voltage and current will have a frequency and
angular frequency

w in radians per second

Need to create a more general form of resistance
Circuits have natural oscillation frequencies
May get resonance
Generators
We can use induction to produce a
continuous source of current

Turn a wire loop in a magnetic field with an
external source of work to make a generator
Produces alternating current

Thus the current flows in one direction and then
the other
Characterized with angular frequency wd
Alternating Current

e = emax sin wdt
As the loop makes one
complete rotation (wt goes
from 0 to 2p radians) the
emf goes from 0, to
maximum +, to maximum -,
and back to zero again

The current through the loop
goes one way and then the
other, sometimes is weak
and sometimes is strong
Frequency

The number of these cycles made per second is
the frequency f = wd/2p

We can also refer to the period, the number of
seconds per cycle
T = 1/f = 2p/wd

Note that if wd is similar to the w of the circuit we
have resonance and large current amplitudes are
possible
AC Circuit Elements

Resistors (Resistance, R)
Capacitors (Reactance, XC)
Inductors (Reactance XL)

Current and voltage may not be in phase
Resistors and AC
Resistor connected to an AC generator


Voltage through the circuit varies as the emf of
the generator does
vR = VR sin wdt

We can then find the maximum current with
Ohm’s Law
VR = IRR
Resistors and Phase

When v is max, i is
max

iR = IR sin wdt
Capacitors and AC

As charge builds up on the capacitor, the potential
difference across it increases, decreasing the
current

The capacitor is constantly being charged
and discharged

In a AC circuit the current will vary with some
average rms value that depends on the voltage
and the capacitance
Capacitive Reactance

We define the capacitive reactance to symbolize
this “resistance”:
XC = 1/(wdC)

VC = ICXC
Note that the current depends on the frequency
At high frequency the capacitor never gets much charge
on it
Capacitors and Phase

 When the voltage is zero, the
current is a maximum

 As voltage increases current
decreases

 We say the voltage lags the
current by 90 degrees or ¼ cycle
 The equation for current is then
iC = IC sin (wdt + p/2)
Capacitor Power

Since P = vi, we can see if the power is positive
or negative based on the sign of i and v
P is positive half the time and negative half the
time

The capacitor draws energy from and returns
energy to the generator in equal measure
Inductive Reactance

XL = wdL
The inductor is most “effective” at high frequency
due to the rapid current changes

The maximum voltage is again the generator
emf and is related to the maximum current by:
VL = ILXL
Inductors and Phase

 look at the slope of the current sine
wave

 The induced voltage is zero when
the current is a maximum (since
that is where the current is not
changing)

 The current at any time is:
iL = IL sin (wdt – p/2)
Reactance and Frequency
Resistor

Capacitor
Low current at
low frequency
Inductor

Next Time
Read 31.9-31.11
Problems: Ch 31, P: 32, 33, 34, 41, 53
For an LC oscillator when the current
through the inductor is zero, the
charge on the capacitor is,
A)
B)
C)
D)
E)
Maximum
Zero
½ maximum
p maximum
2p maximum
For an LC oscillator the direction of
current in the circuit is,
A) Always in one direction
B) In one direction for one complete cycle
and then the other direction for the
next complete cycle
C) In one direction for ½ of the cycle
and then the other direction for the
other ½ of the cycle
D) Changing direction 2p times per cycle
E) Fluctuating unpredictably
For an LC oscillator when the energy in
the capacitor is ½ maximum the
energy in the inductor is,
A)
B)
C)
D)
E)
Maximum
Negative maximum
Zero
½ maximum
2p maximum