Download Inductors and AC Circuits

Physics 102: Lecture 12
Inductors and AC
Circuits
• Today’s lecture will cover Textbook
Sections 20.9, 21.1, 3-4
L
R
Physics 102: Lecture 12, Slide 1
C
Mutual Inductance
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•
•
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AC Generator
Changing current in P
Changing B-field thru P
Changing B-field thru S
Changing  thru S
– S proportional to IP:
N s  s  MI p
• Induced EMF (voltage) in S
– Recall Faraday’s law:

f  i
 

t
t f  ti
Physics 102: Lecture 12, Slide 2
Primary Secondary
Coil
Coil
“Mutual
Inductance”
 s  M
I p
t
Self Inductance – Single Coil
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•
•
•
AC Generator
Changing current
Changing B-field
Changing 
–  proportional to I:
N  LI
• Induced EMF (voltage)
– Recall Faraday’s law:
• Direction

f  i
 

t
t f  ti
– Opposes change in current
Physics 102: Lecture 12, Slide 3
“Inductance”
I
  L
t
Physical Inductor
Inductor resists current change!
N
L
I
Recall: =BA
l
A
L
Recall: B=m
nI
L o
L
Energy stored:
U=
Physics 102: Lecture 12, Slide 4
(# turns)
= (# turns/meter) x (# meters)
L
L
N = nl
Physical Inductor
Inductor resists current change!
N
L
I
Recall: =BA
NBA
L
I
l
A
Recall: B=monI
Nm 0 nIA
L
I
Energy stored:
U = ½ LI2
Physics 102: Lecture 12, Slide 5
L  m o nNA
(# turns) = (# turns/meter) x (# meters)
L  m o n 2lA
N=nl
ACT
Compare the inductance of two solenoids, which are identical
except solenoid 2 has twice as many turns/meter as solenoid
1.
N  LI
L  N / I
1) L2=L1
Physics 102: Lecture 12, Slide 6
2) L2 = 2 L1
3) L2 = 4 L1
ACT
Compare the inductance of two solenoids, which are identical
except solenoid 2 has twice as many turns/meter as solenoid
2
1.
L  m o n lA
N  LI
L  N / I
1) L2=L1
2) L2 = 2 L1
3) L2 = 4 L1
One factor of 2 for N in equation, another factor of 2 for
doubling B field inside solenoid. ( = B A)
Physics 102: Lecture 12, Slide 7
Review: Generators and EMF
Voltage across generator:
1
•
w
v
 = w A B sin(wt)
 = Vmax sin(wt)
2
x
q
v
r
Vmax
Frequency = How fast its spinning
Amplitude = Maximum voltage
Physics 102: Lecture 12, Slide 8
-Vmax
t
AC Source
V(t) = Vmax sin(2pf t)
Vmax = maximum voltage
f = frequency (cycles/second)
V(t) = 24 sin(8p t)
+24
Find ƒ…
-24
0.25
0.5
RMS: Root Mean Square Vrms=Vmax/sqrt(2)
Physics 102: Lecture 12, Slide 9
AC Source
V(t) = Vmax sin(2pf t)
Vmax = maximum voltage
f = frequency (cycles/second)
V(t) = 24 sin(8p t)
2pf t = 8pt
f = 4 Hz
T= (1/4) seconds
+24
-24
0.25
0.5
RMS: Root Mean Square Vrms=Vmax/sqrt(2)
Physics 102: Lecture 12, Slide 10
Preflight 12.1, 12.2
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I(t) = 10 sin(377 t)
C
Find Imax
Find Irms
Physics 102: Lecture 12, Slide 11
Preflight 12.1, 12.2
L
R
I(t) = 10 sin(377 t)
C
Find Imax
Well… We know that the maximum value
of sine is 1. So the maximum current is 10!
Imax = 10 A
Find Irms
Just like Vrms=Vmax/sqrt(2)…
Irms=Imax/sqrt(2) =10/sqrt(2) A = 7.07 A
Physics 102: Lecture 12, Slide 12
Resistors in AC circuit
VR = I R
always true - Ohm’s
Law
R
• VR,max = ImaxR
• Voltage across resistor is “__________” with current.
Resistance (R)
– VR goes up and down at the
same times as I does.
I
t
VR
t
Frequency
Physics 102: Lecture 12, Slide 13
Resistors in AC circuit
VR = I R
always true - Ohm’s
Law
R
• VR,max = ImaxR
• Voltage across resistor is “IN PHASE” with current.
Resistance (R)
– VR goes up and down at the
same times as I does.
I
t
VR
Frequency does not
affect Resistance!
Frequency
Physics 102: Lecture 12, Slide 14
t
Capacitors in AC circuit
VC = Q/C
always true
• VC,max = ImaxXC
• Capacitive Reactance:
C
XC =
• Voltage across capacitor “_______” current.
Reactance (XC)
– VR goes up and down
after I does.
I
t
VC
t
Frequency
Physics 102: Lecture 12, Slide 15
Capacitors in AC circuit
VC = Q/C
always true
• VC,max = ImaxXC
• Capacitive Reactance:
C
XC = 1/(2pfC)
• Voltage across capacitor “LAGS” current.
Reactance (XC)
– VR goes up and down
after I does.
Frequency does
affect Reactance!
Frequency
Physics 102: Lecture 12, Slide 16
I
t
VC
t
Inductors in AC circuit
VL = -L(I)/(t) always true
• VL,max = ImaxXL
• Inductive Reactance:
L
XL =
• Voltage across inductor “_______” current.
Reactance (XL)
– VR goes up and down
before I does.
I
t
VL
t
Frequency
Physics 102: Lecture 12, Slide 17
Inductors in AC circuit
VL = -L(I)/(t) always true
• VL,max = ImaxXL
• Inductive Reactance:
L
XL = 2pfL
• Voltage across inductor “LEADS” current.
Reactance (XL)
– VR goes up and down
before I does.
I
t
VL
Frequency does
affect Reactance!
Frequency
Physics 102: Lecture 12, Slide 18
t
Generators in AC circuit
L
VG+VL+VR+VC = 0 always true.
Reactance (Z)
• Vmax = Imax Z
• Total Reactance:
Z=
• Voltage across generators sometimes leads and
sometimes lags current. (It depends on the
frequency.)
Frequency
Physics 102: Lecture 12, Slide 19
R
C
Generators in AC circuit
L
VG+VL+VR+VC = 0 always true.
R
• Vmax = Imax Z
Reactance (Z)
• Total Reactance:
Z = R2+ (XL- XC)2
• Voltage across generators sometimes leads and
sometimes lags current. (It depends on the
frequency.)
Frequency does
affect Impedance!
Frequency
C
Preflight 12.4, 12.5
The capacitor can be ignored when…
frequency is very large
frequency is very small
The inductor can be ignored when…
frequency is very large
frequency is very small
Physics 102: Lecture 12, Slide 21
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C
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Preflight 12.4, 12.5
R
C
The capacitor can be ignored when…
frequency is very large
XC
frequency is very small
w
very large w gives very small XC
The inductor can be ignored when…
frequency is very large
frequency is very small
XL
w
very small w gives very small XL
Physics 102: Lecture 12, Slide 22
ACT: AC Circuit Voltages
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH has
a current I(t) = 0.5 sin(8p t) amps. Calculate the
maximum voltage across R, C, and L.
L
R
VR,max = Imax R =
VC,max = Imax XC =
VL,max = Imax XL =
Now the frequency is increased so I(t) = 0.5 sin(16p t).
Which element’s maximum voltage decreases?
1) VR,max
2) VC,max
3) VL,max
Physics 102: Lecture 12, Slide 23
C
ACT: AC Circuit Voltages
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH has
a current I(t) = 0.5 sin(8p t) amps. Calculate the
maximum voltage across R, C, and L.
L
VR,max = Imax R = 0.5  2 = 1 Volt
VC,max = Imax XC = 0.5  1/(8p0.015) = 1.33 Volts
VL,max = Imax XL = 0.5  8p0.03 = 0.38 Volts
Now the frequency is increased so I(t) = 0.5 sin(16p t).
Which element’s maximum voltage decreases?
1) VR,max Stays same: R doesn’t depend on f
2) VC,max Decreases: XC = 1/(2pfC)
3) VL,max Increases: XL = 2pf L
Physics 102: Lecture 12, Slide 24
R
C
See you later!
• Read Section 21.5
Physics 102: Lecture 12, Slide 25