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AC Voltages and Phasors Resistors, Inductors and Capacitors in
AC Circuits RLC Circuits Power and Resonance Transformers
AC sources have voltages and currents that vary sinusoidally.
The current in a circuit driven by an AC source also oscillates at the same frequency.
The standard AC frequency in the US is 60 Hz (337 rad/s).
)sin()( max tVtv
21
fT
t
Vmax
-Vmax
v(t)
Consider a sinusoidal voltage source: )sin()( max tVtv
This source can be represented graphically as a vector called a phasor:
t T
Vmax v(t)
b
c
b
c
d
e
d
a a e
R
R
vtV
vv
-
sin
0
max
tItR
V
R
vi RR sinsin max
max
R
VI max
max
Since both have the
same arguments in
the sine term, iR and
vR are in line, they
are in phase.
tRIvR sinmax
tIiR sinmax
tRIRit R 22
max
2sin)( P
tRIRit R 22
max
2sin)( P
maxmax
22
max
707.02
2
1
II
I
RIRI
rms
rmsav
P
maxmax 707.02
VV
Vrms
dt
diLtV
vv L
-
sin
0
max
tdtL
Vdi sinmax
tL
VdiiL
cosmax
-
-
2sinmax
tL
ViL
LX
V
L
VI maxmax
max
LX L
tXIv LL sinmax-
Inductive
Reactance
For a sinusoidal voltage,
the current in the inductor
always lags the voltage
across it by 90°.
tVvv C sinmax
tVCq sinmax
2sin
cos
max
max
tVCi
tVCdt
dqi
C
C
CX
VVCI max
maxmax
CXC
1
Capacitive
Reactance
tXIv CC sinmax
For a sinusoidal voltage,
the current in the
capacitor always leads
the voltage across it by
90°.
tVv sinmax
- tIi sinmax
tVtRIv RR sinsinmax
tVtXIv LLL
cos2
sinmax
tVtXIv CCC
cos2
sinmax -
-
CLR vvvv
22
maxmax
2
maxmax
2
maxmax
22
max
CL
CL
CLR
XXRIV
XIXIRIV
VVVV
-
-
-
22
maxmax
CL XXR
VI
-
22
CL XXRZ - Impedance
ZIV maxmax
- -
R
XX CL1tan
R = 425 W
L = 1.25 H
C = 3.5 mF
= 377 s-1
Vmax = 150 V
W
-758
5.3377
111 FsC
XCm
W - 47125.1377 1 HsLX L
WW-WW- 5137584714252222
CL XXRZ
AV
Z
VI 292.0
513
150maxmax
W
-
W
W-W
- -- 34
425
758471tantan 11
R
XX CL
VARIVR 124425292.0max W
VAXIV LL 138471292.0max W
VAXIV CC 221758292.0max W
tVvC 377cos221-
tVvL 377cos138
tVvR 377sin124
tVtIvi sinsin maxmax -P
sincossincossin
sinsincoscossin
maxmax
2
maxmax
maxmax
ttVItVI
tttVI
-
-
P
P
cos2
1maxmax VIav P
cosrmsrmsav VI P
RIVvR maxmax cos
22
max
max
maxmax RII
V
RIVI rmsrmsav
P
RIrmsav
2P
For a purely resistive load, =0
rmsrmsav VI P
No power loss occurs in
an ideal capacitor or
inductor.
Z
VI rms
rms
22
CL
rmsrms
XXR
VI
-
A circuit is in resonance when its current is maximum.
Irms=max when XL-XC=0
CL
XX CL
0
0
1
LC
10 Resonance Frequency
22
2
2
22
CL
rmsrmsrmsav
XXR
RVR
Z
VRI
-
P
22
0
2
2
22
2 1
-
--
L
CLXX CL
22
0
2222
22
-
LR
RVrmsavP
R
Vrmsav
2
max,
P =0
R
L00
Q Quality Factor
Power is transmitted in high voltage over long distances and then gradually
stepped down to 240V by the time it gets to a house.
22 kV 230 kV 240 V
P g= 20 MW
Cost = 10¢/kWh
AV
W
VI
g87
10230
10203
6
P
kWARI 1528722 WP
36$1.0$2415 hkWTotalCost
For 22kV transmission:
Total Cost = $4100