Download Chapter 31: AC Circuits

Chapter 31
1
An alternator
V (t )  Vmax sin d t 
2
The Great Divide: 60 Hz vs 50 Hz

 is an angular frequency.


In the US and the rest of the Western
Hemisphere, the frequency is 60 Hz


=2pf where f is the frequency in Hertz (Hz)
=2p(60)=377 s-1 or rad/s
In Europe and Asia, 50 Hz or 314 s-1
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3 Simple Circuits-- Resistor
This is
sometimes
called a
“resistive load”
on the circuit
VR
V  VR  0 and V  Vmax sin d t 
because
VR  iR , Vmax sin d t   iR 
Vmax
sin d t   i
R
Vmax
R
i  I R sin d t 
Let
IR 
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Phasor Diagrams



Phasor Diagrams
 A method to analyze the
relative phase between
voltage and current
 Of great concern since
power maximizes when
voltage and phase are in
phase
A convention is that the
driving frequency, d , is
measured from the +x axis
and rotates counterclockwise
around the origin
In the case to the right, the
phase angle between VR and
IR is zero.
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3 Simple Circuits– Capacitive Load
V  VC  0 and V  Vmax sin d t 
q
q
, Vmax sin d t    CVmax sin d t   q
C
C
because
VC 
Let ic 
dq
 d CVmax cosd t 
dt
For symmetry reasons, we introduce
“capacitive reactance”, XC
VC
XC 
ic 
if
1
d C

Vmax
V
cosd t   max sin d t  900
XC
XC
ic  I C sin d t   
  900

Vmax=ICXC
Capacitive reactance has
units of ohms
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ICE—For capacitors, current leads
EMF
If you monitor iC, the current will reach its maximum before the
voltage across the capacitor, VC
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3 Simple Circuits– Inductive Load
For symmetry reasons, we introduce “inductive
reactance”, XL
V  VL  0 and V  Vmax sin d t 
because
VL  L
Let iL  
VL
di
di V
di
, Vmax sin d t   L  max sin d t  
dt
dt
L
dt
Vmax
cosd t 
d L

Let
X L  d L  cosd t   sin d t  900
iL  
Vmax
sin d t  900
XL


and I L 

(  900 )
Vmax
XL
Vmax=ILXL
Inductive reactance has
units of ohms
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ELI—For inductors, the EMF leads
current
If you monitor iL, the voltage reaches its maximum value before
the current in the circuit.
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ELI the ICEman
For resistors, current and EMF are in
phase
 For inductors, the EMF leads the current
(ELI)
 For capacitors, the current leads the
EMF (ICE)
 ELI the ICEman will always help you
from getting killed…

10
RLC Circuit
VL
VR
VR
dt
VL-VC
VC
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RLC Phasor
From Pythagorean theorem,
V
VL-VC
VR
V  VL  VC   VR2
2
2
V I
2
I
2
 X
2


X

R
L
C
2

V
X L  X C 
2
 R2
2
2


Z  X L  XC  R
VL  VC  X L  X C 
tan  

VR
R
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Impedance, Z

The impedance of the circuit is the effective
resistance


Has units of ohms
Consists of





Total resistance, R
Total capacitance, C
Total inductance, L
Frequency of the electric field, d
V=IZ, amplitude of voltage across AC circuit
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Something missing?
If there is no resistor, set R=0 and solve
for Z and 
 If there is no inductance, set L=0 and
solve for Z and 
 If there is no capacitor, set C=∞, and
solve for Z and 
 If DC then Z=R

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Resonance
Driving
Frequency, d
Natural frequency of swing is
=1/(LC)
When the driving frequency is equal to the natural frequency, then
energy transfer is maximized.
d= or XC= XL
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Power in AC circuits

P=i2R where i=I sin(dt-)


P=I2R* sin2(dt-)
Since P=i2R, then average power is




P=i2R/2 and if R is constant,
Irms=I/sqrt(2)
Or Vrms= V/sqrt(2)
RMS means root mean square value


Typical AC voltage values measured by voltmeter
Pav=IrmsVrms cos




For pure resistive load, 0
For pure inductive load, 900
For pure capacitive load, 900
For RLC load, cosR/Z
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Transformers



A method of stepping up
or stepping down the
voltage or current in an
AC circuit
A wire is wrapped
around an iron core, the
current in the wire
induces a magnetic flux
in the iron core
A second wire is
wrapped around the
core. The flux in the core
is then induces EM in
the second wire.
Symbol
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Transformers cont’d
V2 N 2

V1 N1
V2  V1
N2
N1
P1  P2
V1 I1  V2 I 2
V2 I1 N 2
 
V1 I 2 N1
I 2  I1
N1
N2
V2
N
V N
 I1 1  1 2
R
N 2 R N1
V1 N 22
I1 
R N12
1 is called “Primary” and 2 is called “Secondary”
The current, I1 is the current drawn by a load, R,
placed on transformer which has a turns ratio of
N2/N1 with primary voltage V1
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