Download AC Circuits - Cedarville University

AC Circuits
See online explanation at
http://www.physclips.unsw.edu.au/jw/AC.html
Total Energy
• Total energy of the system
is the sum of the electric
and magnetic field energy.
++++
E
L
----
2
m
Q
2
UC 
cos t 
2C
LI m2
UL 
sin 2 t 
2
Qm2 LI m2
U  UC  U L 

2C
2
C
I
L
B
C
RLC Circuit
R
• After the switch is closed
Kirchoff’s rules gives
2
d Q
dQ Q
L 2 R
 0
dt
dt C
• Solution:
Q  Qm e - Rt / 2 L cos d t 
• Critical Damping:
4L
Rc 
C
+
I
L
++++C
----
-
 1  R
d  
- 
 LC  2 L 
2 1/ 2



AC Source
• The voltage of the power Vm
v
supply follows that of a
cosine function.
• Represent the source as a -V
m
vector rotating in the
complex plane.
• Real component of vector
is the instantaneous
voltage.
t
Im
t
v
Vm
Re
v(t )  Vm cost 
AC & Resistors
v
i
From
Kirchoff v
R
v -i  R  0
v(t )  Vm cost 
Vm
i (t ) 
cost 
R
• Voltage and current
are in phase, f = 0.
Im
t
iv
Re
Vm
Power
• Power used by a resistor is
2
m
V
2
P(t )  v  i 
cos t 
R
• Average power is
2
Vm Vm  I m
P

2R
2
• Effective voltage and current are
Vrms
Vm

2
I rms
Im

2
 P  Vrms  I rms
AC & Inductors
f
From
Kirchoff v
di
v-L 0
dt
v(t )  Vm cost 
v
L
i
Vm
Im
i (t ) 
cost - p2 
L
t
• Voltage leads current Reactance
by 90°, f = p/2.
X L  L
i
v Re
Vm
AC & Capacitors
f
From
Kirchoff v
v
C
i
1
v -  i  dt  0
C
v(t )  Vm cost 
i(t )  CVm cost  p2 
Im
t
• Voltage lags current Reactance
1
XC 
by 90°, f = -p/2.
C
i
Vm
v
Re
LRC Circuits & AC
• In a series circuit current is
the same everywhere, so
add voltages as phasors.
• Phasors are rotating vectors
in the complex plane.
R
v
L
Im
C
vL
• Assume we are at t = 0s.
v  vR  jvL - jvC
vC
i
vR
Re
Impedance
• Express voltage as current
times impedance.
v  iR   j iX L - j iX C 
v  iR  j  X L - X C   iZ
Z  R  X L - X C 
2
 X L - XC 
f  tan 

R


-1
2
f
vC
Im
vS
f vL
vR
Re
ELI the ICE man
• At resonance, 0, XL=XC and f = 0.
0 
1
LC
 X L - XC 
f  tan 

R


-1
• For <0, the circuit is capacitive and
current leads voltage. Also f < 0.
• For >0, the circuit is inductive and
voltage leads current. Also f > 0.
Quality Factor
• As the AC frequency approaches resonance,
0, the current increases.
• Quality factor indicates how sharp the
current peak is around resonance.
Q0 
0L
High Q
Im
R
Low Q
0
Transformers
• A transformer consists of two
sets of coils which share the
same magnetic flux.
• AC current through the primary
generates a changing magnetic
flux, which generates a changing
voltage on the secondary.
NS
VS 
VP
NP
VS I S  VP I P
Primary
v
Secondary
RL
Power Lines
• Ex. How much power is lost if 120V is
delivered through a 10W transmission line
to a 10W load?
Compare that to transmission at 12,000 V
and then stepped down to 120V at the load.
RT
RT
120V RL
12,000V
120V
RL
Power
Transmission
Less resistive loss
Through Town:
at low current.
20 kV
To Home:
240 V
Long Distance:
230 kV