Chapter 31
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Transcript of Chapter 31
1
Chapter 31
2
An alternator
tVtV dsin)( max
3
The Great Divide: 60 Hz vs 50 Hz
is an angular frequency. =2f where f is the frequency in Hertz (Hz)
In the US and the rest of the Western Hemisphere, the frequency is 60 Hz=2(60)=377 s-1 or rad/s
In Europe and Asia, 50 Hz or 314 s-1
4
3 Simple Circuits-- Resistor
VR
tIiR
VILet
itR
ViRtViRVbecause
tVVandVV
dR
R
ddR
dR
sin
sinsin,
sin0
max
maxmax
max
This is sometimes called a “resistive load” on the circuit
5
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.
6
3 Simple Circuits– Capacitive Load
0
0maxmax
90
sin
90sincos
1
tIiif
tX
Vt
X
Vi
CX
dCc
dC
dC
c
dC
VC
For symmetry reasons, we introduce“capacitive reactance”, XC
tCVdt
dqiLet
qtCVC
qtV
C
qVbecause
tVVandVV
ddc
ddC
dC
cos
sinsin,
sin0
max
maxmax
max
Vmax=ICXC
Capacitive reactance has units of ohms
7
ICE—For capacitors, current leads EMF
If you monitor iC, the current will reach its maximum before the voltage across the capacitor, VC
8
3 Simple Circuits– Inductive Load
VL
For symmetry reasons, we introduce “inductive reactance”, XL
LLd
LL
dddL
dd
L
ddL
dL
X
VIandt
X
Vi
ttLXLet
tL
ViLet
dt
dit
L
V
dt
diLtV
dt
diLVbecause
tVVandVV
max0max
00
max
maxmax
max
90sin
)90(90sincos
cos
sinsin,
sin0
Vmax=ILXL
Inductive reactance has units of ohms
9
ELI—For inductors, the EMF leads current
If you monitor iL, the voltage reaches its maximum value before the current in the circuit.
10
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…
11
RLC Circuit
VR
VC
VLVR
VL-VCdt
12
RLC Phasor
VR
VL-VC
From Pythagorean theorem,
R
XX
V
VV
RXXZ
RXX
VI
RXXIV
VVVV
CL
R
CL
CL
CL
CL
RCL
tan
22
22
2222
222V
13
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
14
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
15
Resonance
Natural frequency of swing is=1/(LC)
Driving Frequency, d
When the driving frequency is equal to the natural frequency, then energy transfer is maximized. d= or XC= XL
16
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, For pure inductive load,
For pure capacitive load,
For RLC load, cosR/Z
17
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
21
221
1
1
21
2
11
2
2
112
1
2
2
1
1
2
2211
21
1
212
1
2
1
2
N
N
R
VI
N
N
R
V
N
NI
R
V
N
NII
N
N
I
I
V
V
IVIV
PP
N
NVV
N
N
V
V
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