8. AC POWER

Electronics - Devices

Nov 24, 2013 (4 years and 5 months ago)

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8. AC POWER

CIRCUITS by Ulaby &
Maharbiz

Overview

Linear Circuits at ac

Instantaneous power

Average power

)
(
)
(
)
(
t
i
t
t
p

Power
at any instant of time

Average
of instantaneous power
over one period

T
dt
t
p
T
P
0
)
(
1

Power delivery (utilities)

Electronics (laptops, mobile phones, etc.)

Logic circuits

Power is critical for many

reasons:

Note:

Power
is not a linear function, cannot
apply superposition

Instantaneous Power for Sinusoids

Power depends on
phases
of voltage and current

i
i
t
t
I
V
t
p
t
i
t
t
p
t
I
t
i
t
V
t

cos
cos
)
(
)
(
)
(
)
(
cos
)
(
cos
)
(
m
m
m
m

B
A
B
A
B
A

cos
cos
2
1
cos
cos

i
i
t
I
V
t
p

2
cos
cos
2
1
)
(
m
m
Trig
.
Identity:

Constant in
time
(dc term)

ac
at 2

Average Value

Sine wave

Truncated
sawtooth

Average Value for

These properties hold true
for any values of
φ
1

and
φ
2

Effective or RMS Value

Equivalent Value That Delivers Same
Average Power
to Resistor as in dc case

For current given by

Effective value is the (square)
R
oot

of the
M
ean

of the
Square

of the
periodic signal, or
RMS

value

T
T
dt
i
T
R
dt
R
i
T
P
0
2
0
2
1
R
I
P
2
eff

T
dt
i
T
I
0
2
eff
1

T
dt
T
V
0
2
eff
1

t
I
t
i

cos
m

2
cos
1
m
0
2
2
m
rms
I
dt
t
I
T
I
T

Hence:

Similarly,

Average Power

i
i
t
t
I
V
t
p
t
i
t
t
p
t
I
t
i
t
V
t

cos
cos
)
(
)
(
)
(
)
(
cos
)
(
cos
)
(
m
m
m
m

i
i
t
I
V
t
p

2
cos
cos
2
1
)
(
m
m

B
A
B
A
B
A

cos
cos
2
1
cos
cos
Note dependence on
phase difference

Average Power

Since

and a similar relationship applies to
I,

Power factor angle:

0 for a resistor

= 90 degrees for inductor

90 degrees for capacitor

ac Power
Capacitors

2

i
dt
d
C
i
C
C

C
C
CV
j
I

2
C
C

CV
I
Capacitors (ideal) dissipate
zero average power

2
2
2
cos
2
1
2
cos
cos
2
1
m
m
m
m

t
I
V
t
p
t
I
V
t
p
i
i
= 0

ac Power
Inductors

2

i
dt
di
L
L
L

L
L
LI
j
V

2
L
L

i
LI
V
Inductors (ideal) dissipate
zero average power

2
2
2
cos
2
1
2
cos
cos
2
1
m
m
m
m

i
i
i
t
I
V
t
p
t
I
V
t
p
= 0

Complex Power

Phasor

form defining “real” and “reactive” power

Inductive/capacitive loads will require more from the
power supply than the average power being consumed

Power supply needs to supply
S

in order to deliver
P
av

Power factor relates
S

to
P
av

Power Factor

Power Factor Compensation

Introduces reactive elements to increase Power Factor

Example 8
-
6:

pf

Compensation

Maximum Power Transfer

Max
power

is delivered
to

is equal
to
Thévenin

equivalent

*
s
s
s
L
L
L
Z
jX
R
jX
R
Z

Max power
transfer when

Set derivatives equal to zero

0
L

X
P
0
L

R
P
s
2
Th
max
8
R
V
P

Example 8
-
7:
Maximum Power

Cont.

Example 8
-
7:
Maximum Power

Three Phase

Y & Delta

Y
-
Source Connected to a Y
-