Phasors and AC
(sec. 31.1)
Resistance and reactance
(sec. 31.2)
RLC series circuit
(sec. 31.3)
Power in AC circuits
(sec. 31.4)
Resonance in AC circuits
(sec. 31.5)
Transformers
(sec. 31.6)
Alternating Current Ch. 31
C 2012 J. F. Becker
Learning Goals

we will learn: ch 31
• How
phasors
make it easy to describe
sinusoidally varying quantities.
• How to
analyze RLC series circuits
driven
by a sinusoidal
emf
.
• What determines the amount of
power
flowing into or out of an AC circuit.
• How an RLC circuit responds to
emfs
of
different frequencies
.
Phasor diagram

projection of rotating vector
(phasor) onto the
horizontal
axis represents the
instantaneous
current.
Graphs (and
phasors
) of instantaneous
voltage
and
current
for a
resistor
.
i(t) = I cos
w
琠⡳潵牣攩
v
R
(t) = i(t) R
v
R
(t) = IR cos
w
t
where V
R
= IR is
the voltage amplitude.
V
R
= IR
Notation:

lower case letters
are time dependent
and

upper case letters
are constant.
For example,
i(t) is the time
dependent current
and
I is current
amplitude;
V
R
is the voltage
amplitude (= IR ).
Graphs of
instantaneous voltages
for RLC series circuit.
(The phasor diagram is much simpler.)
Graphs (and phasors) of instantaneous voltage and
current for an
inductor
.
i(t) = I cos
w
琠⡳潵牣攩
v
L
(t) = L di / dt
v
L
(t) = L d(I cos
w
琠⤯摴t
v
L
(t) =

I
w
䰠獩渠
w
t
v
L
(t) = +I
w
䰠捯猠L
w
琠⬠㤰
0
)
where V
L
= I
w
䰠⠽⁉X
L
)
is the voltage amplitude
and
f
㴠⬹=
0
is the
PHASE ANGLE
(angle between voltage
across and current
through the inductor).
X
L
=
w
L
E L I
V
L
L I
Graphs (and phasors) of instantaneous voltage and
current showing phase relation between
current
(
red
)
and
voltage
(
blue
).
Remember: “ELI the ICE man”
Crossover network in a speaker system.
Capacitive reactance: X
C
=1/
w
C
Inductive reactance: X
L
=
w
L
Phasor diagrams for series RLC circuit
(b) X
L
> X
C
and (c) X
L
< X
C
.
Graphs of
instantaneous voltages
for RLC series circuit.
(The phasor diagram is much simpler.)
Graphs of
instantaneous
voltage, current, and power for an R,
L, C, and an RLC circuit.
Average power for an arbitrary
AC circuit is 0.5
VI
cos
f
㴠
嘠
牭r
䤠
牭猠捯猠
f
.
The
average power
is half the product of
I
and the
component of
V
in phase with it.
Instantaneous
current and voltage:
Average power
depends on current
and voltage
amplitudes AND
the phase angle
f
:
Graph of current amplitude I
vs
source frequency
w
for a series RLC circuit
with various values of circuit resistance.
The resonance
frequency is at
w
= 1000 rad / sec
(
where the current
is at its maximum
)
AMPLITUDE MODULATION (AM) of CARRIER WAVE
resonance frequency (fo)
Electric field
amplitude
AM modulated
Electric field
amplitude
FREQUENCY MODULATION (AM) of CARRIER WAVE
resonance frequency (fo)
Electric field
amplitude
FM modulated
Electric field
amplitude
A radio tuning circuit at resonance. The circles denote
rms current and voltages.
Transformer: AC
source is V
1
and secondary provides a
voltage V
2
to a device with resistance R.
TRANSFORMERS
can step

up AC
voltages or step

down AC voltages.
e
2
/
e
1
= N
2
/N
1
V
1
I
1
= V
2
I
1
F
B
=
F
B
e
=

d
F
B
/ dt
(a) Primary P and secondary S windings in a transformer.
(b) Eddy currents in the iron core shown in the cross

section AA. (c) Using a laminated core reduces the
eddy currents.
Figure 32.2b
Large step

down
transformers
at power stations are
immersed in tanks of oil for insulation and cooling.
Figure 31.22
Figure 31.23
A full

wave diode rectifier circuit. (LAB)
A mathematical model of Earth's
magnetic field near the core.
(Courtesy: Gary Glatzmaier)
See
www.physics.sjsu.edu/becker/physics51
Review
C 2012 J. F. Becker
PREPARATION FOR FINAL EXAM
At a minimum the following should be reviewed:
Gauss's Law

calculation of the magnitude of the electric field caused by
continuous distributions of charge starting with Gauss's Law and completing all the
steps including evaluation of the integrals.
Ampere's Law

calculation of the magnitude of the magnetic field caused by
electric currents using Ampere's Law (all steps including evaluation of the integrals).
Faraday's Law and Lenz's Law

calculation of induced voltage and current,
including the direction of the induced current.
Calculation of integrals
to obtain values of electric field, electric potential, and
magnetic field caused by continuous distributions of electric charge and current
configurations (includes the Law of Biot and Savart for magnetic fields).
Maxwell's equations

Maxwell's contribution and significance.
DC circuits

Ohm's Law, Kirchhoff's Rules, series

parallel combinations, power.
Series RLC circuits

phasors, phase angle, current, power factor, average power.
Vectors

as used throughout the entire course.
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