Ch. 31 Alternating Current Circuits

piquantdistractedElectronics - Devices

Oct 5, 2013 (3 years and 8 months ago)

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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



牭r


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.