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).

v(t )  Vmax sin(t )
1 2
T 
f

Consider a sinusoidal voltage source:
v(t )  Vmax sin(t )
This source can be represented graphically as a vector called a phasor:
v(t)
Vmax
b
b
Vmax
v(t)
t
T
a
c
-Vmax
e
d
t
c
a e
d
v - vR  0
Vmax sin t  vR
iR 
I max
vR Vmax

sin t  I max sin t
R
R
Vmax

vR  I max R sin t
R
Since both have the
same arguments in
the sine term, iR and
vR are in line, they
are in phase.
iR  I max sin t
P (t )  RiR  RI max sin 2 t
2
2
P (t )  RiR  RI max sin 2 t
2
Pav  RI max
I rms
2
2
1
2
   I rms R
2
I max

 0.707 I max
2
Vrms
Vmax

 0.707Vmax
2
v - vL  0
Vmax sin t  L
di 
iL 
di
dt
Vmax
sin tdt
L
iL   di  -
Vmax
cos t
L
For a sinusoidal voltage,
the current in the inductor
always lags the voltage
across it by 90°.
Vmax


sin t - 
L
2

I max 
Vmax Vmax

L
XL
X L  L
Inductive
Reactance
vL  - I max X L sin t
v  vC  Vmax sin t
I max  CVmax 
q  CVmax sin t
dq
 CVmax cos t
dt


iC  CVmax sin  t  
2

iC 
For a sinusoidal voltage,
the current in the
capacitor always leads
the voltage across it by
90°.
XC 
1
C
Vmax
XC
Capacitive
Reactance
vC  I max X C sin t
v  Vmax sin t
i  I max sint -  
vR  I max R sin t  VR sin t


vL  I max X L sin t    VL cos t
2



vC  I max X C sin t -   -VC cos t
2

v  vR  vL  vC
Vmax  VR  VL - VC 
2
2
I max R 2  I max X L - I max X C 2
Vmax 
Vmax  I max R 2   X L - X C 
2
I max 
Vmax
R2  X L - X C 
2
Z  R2  X L - X C 
2
Vmax  I max Z
 X L - XC 

R


  tan -1 
Impedance
R = 425 W
L = 1.25 H
C = 3.5 mF
 = 377 s-1
Vmax = 150 V

vR  124V sin 377t
1
1

 758W
C 377 s -1 3.5mF 


Z  R2  X L - X C  
2
I max 
425W2  471W - 758W2
Vmax 150V

 0.292 A
Z
513W
 X L - XC
R

  tan -1 

-1  471W - 758W 
  tan 
  -34
425
W



VL  I max X L  0.292 A471W  138V
VC  I max X C  0.292 A758W  221V

X L  L  377s -1 1.25H   471W
XC 
VR  I max R  0.292 A425W  124V
 513W
vL  138V cos 377t
vC  -221V cos 377t
P  iv  I max sint -  Vmax sin t
P  I max Vmax sin t cos  - cos t sin  sin t
P  I max Vmax sin 2 t cos  - I max Vmax sin t cos t sin 
1
Pav  I max Vmax cos 
2
Pav  I rms Vrms cos 
vR  Vmax cos   I max R
 Vmax
Pav  I rms 
 2
I max R
 I max R
 I rms

2
 Vmax
Pav  I rms R
2
No power loss occurs in
an ideal capacitor or
inductor.
For a purely resistive load, =0
Pav  I rms Vrms
A circuit is in resonance when its current is maximum.
I rms
Vrms

Z
I rms 
Vrms
R2  X L - X C 
2
Irms=max when XL-XC=0
X L  XC
1
0 L 
0 C
0 
1
LC
Resonance Frequency
Pav  I rms
2



V
2
rms
R
R
Z2
X L - X C 
2
Pav 
Pav,max
2

1 
L2 2

2
  L   2  - 0
C  

Vrms 2 R 2

R   L  - 0
2
Vrms 2 R
2
R2  X L - X C 
2
2
2

Vrms 

R
0 0 L
Q


R
2

2 2
=0
Quality Factor

2
V1  - N1
d B
dt
d B
V2  - N 2
dt
N2
V2 
V1
N1
I1V1  I 2 V2
V1
I1 
Req
V2
I2 
RL
2
 N1 
 RL
Req  
 N2 
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
20 106W
I

 87 A
3
V 230 10 V
For 22kV transmission:
P  I 2 R  87 A 2W  15kW
Total Cost = $4100
Pg
P g= 20 MW
Cost = 10¢/kWh
2
TotalCost  15kW 24h$0.1  $36



Midterm 2 Review on Tuesday
Midterm 2 on Wednesday
Reading Assignment for Thursday
 Chapter 34 – Electromagnetic Waves

WebAssign: Assignment 11 (due Tuesday, 5
PM)