lect12 handout
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Physics 102: Lecture 12, Slide 1
Physics 102: Lecture 12
L
R
C
AC Circuits
Physics 102: Lecture 12, Slide 2
Review: Generators and EMF
vv
•
x
r
= A B sin() = A B sin(t) = Vmax sin(t)
t
1
2
Frequency = How fast its spinning
Amplitude = Maximum voltage
Voltage across generator:
Vmax
-Vmax
20
Physics 102: Lecture 12, Slide 3
AC Source
0.25
0.52f t = 8tf = 4 HzT=(1/4)seconds/cycle
+24
-24
V(t) = Vmax sin(t)=Vmax sin(2f t)
Vmax = maximum voltage
f = frequency (cycles/second)
V(t) = 24 sin(8 t)
RMS: Root Mean Square Vrms=Vmax/√2
Physics 102: Lecture 12, Slide 4
RMS?
RMS: Root Mean Square Vrms=Vmax/√2
+Vma
x
-Vmax
V(t) = Vmax sin(2f t)
Vmax2
Square: Vmax2 / 2
Mean:
square Root: Vmax /
√2
Physics 102: Lecture 12, Slide 5
Preflight 12.1, 12.2
I(t) = 10 sin(377 t)
Find Imax
Find Irms
LR
C
Well… We know that the maximum value sine is 1. So the maximum current is 10!
Imax = 10 A
Just like Vrms=Vmax/√2 …
Irms=Imax/√2 =10/√2 A = 7.07 A
Physics 102: Lecture 12, Slide 6
Resistors in AC circuit
VR = I R always true – Ohm’s Law
• VR,max = ImaxR
R
• Voltage across resistor is “IN PHASE” with current.
– VR goes up and down at the same times as I does.
I
t
t
VR
FrequencyResis
tan
ce
(R)
Frequency does not affect Resistance!
Physics 102: Lecture 12, Slide 7
C
Capacitors in AC circuit
I
t
• Voltage across capacitor “LAGS” current.
– VC goes up and down just after I does.
t
VC
FrequencyReacta
nce
(XC)
Frequency does affect Reactance!
VC = Q/C always true
• VC,max = ImaxXC
• Capacitive Reactance: XC = 1/(2fC)
Physics 102: Lecture 12, Slide 8
Inductors in AC circuit
I
t
• Voltage across inductor “LEADS” current.
– VL goes up and down just before I does.
t
VL
L
FrequencyReacta
nce
(XL) Frequency does
affect Reactance!
VL = -L(I)/(t) always true
• VL,max = ImaxXL
• Inductive Reactance: XL = 2fL
Physics 102: Lecture 12, Slide 9
ACT/Preflight 12.4, 12.5
The capacitor can be ignored when…(a) frequency is very large(b) frequency is very small
LR
C
The inductor can be ignored when…(a) frequency is very large(b) frequency is very small
Physics 102: Lecture 12, Slide 10
AC Circuit VoltagesAn AC circuit with R= 2 , C = 15 mF, and L = 30 mH
has a current I(t) = 0.5 sin(8t) amps. Calculate the maximum voltage across R, C, and L.
VR,max = Imax R = 0.5 2 = 1 Volt
= 0.5 1/(80.015) = 1.33 Volts
= 0.5 80.03 = 0.38 Volts
VC,max = Imax XC
VL,max = Imax XL
LR
C
Physics 102: Lecture 12, Slide 11
ACT: AC Circuit VoltagesAn AC circuit with R= 2 , C = 15 mF, and L = 30 mH
has a current I(t) = 0.5 sin(8t) amps. Calculate the maximum voltage across R, C, and L.
Now the frequency is increased so I(t) = 0.5 sin(16t). Which element’s maximum voltage decreases?
1) VR,max
2) VC,max
3) VL,max
LR
C
Physics 102: Lecture 12, Slide 12
Summary so far…L
R
C
• I = Imaxsin(2ft)
• VR = ImaxR sin(2ft)
• VR in phase with I
• VC = ImaxXC sin(2ft–)
• VC lags I
• VL = ImaxXL sin(2ft+)
• VL leads I
I
t
VL
VC
VR
𝑋𝐶 = 12𝜋𝑓𝐶= 1𝜔𝐶
𝑋𝐿 = 2𝜋𝑓𝐿= 𝜔𝐿
Physics 102: Lecture 12, Slide 13
Kirchhoff: generator voltage
Write down Kirchhoff’s Loop Equation:
Vgen(t) = VL(t) + VR(t) + VC(t) at every instant of time
LR
C
However …Vgen,max VL,max+VR,max+VC,max
Maximum reached at different times for R, L, C
I
t
VL
VC
VR
We solve this using phasors
Vgen
Physics 102: Lecture 12, Slide 14
A reminder about sines and cosines
Recall: y coordinates of endpoints are
• asin( + /2)
• asin()
• asin( – /2)
a a
a
x
y
Physics 102: Lecture 12, Slide 15
L
R
C
I = Imaxsin(2ft) ( = 2ft)
VL = ImaxXL sin(2ft + )
VR = ImaxR sin(2ft)
VC = ImaxXC sin(2ft – )
Graphical representation of voltages
ImaxXL
ImaxR
ImaxXC
Physics 102: Lecture 12, Slide 16
Phasor Diagrams: A Detailed Example
• I = Imaxsin(2ft)
• VR = VR,maxsin(2ft)
VR,maxsin()
t = 1 f=1/122ft = /6
Length of vector = Vmax across that componentVertical component = instantaneous value of V
V R,max
Physics 102: Lecture 12, Slide 17
Phasor Diagrams
VR,maxsin()
t = 22ft = /3
V R,m
ax
• I = Imaxsin(/3)
• VR = VR,maxsin(/3)
Length of vector = Vmax across that componentVertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 18
Phasor Diagrams
VR,maxsin()=V0
t = 32ft = /2
VR
,max
• I = Imaxsin(/2)
• VR = VR,maxsin(/2)
Length of vector = Vmax across that componentVertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 19
Phasor Diagrams
VR,maxsin(4)
t = 42ft = 4/6
VR
,max
• I = Imaxsin(4/6)
• VR = VR,maxsin(4/6)
Length of vector = Vmax across that componentVertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 20
Phasor Diagrams
t = 62ft =
VR,max
• I = Imaxsin()
• VR = VR,maxsin()
VR,maxsin()=0
Length of vector = Vmax across that componentVertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 21
Phasor Diagrams
VR,maxsin(8)
t = 82ft = 8
VR
,max
• I = Imaxsin(8/6)
• VR = VR,maxsin(8/6)
Length of vector = Vmax across that componentVertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 22
Phasor Diagrams
VR,maxsin(10)
t = 102ft = 10
VR
,max
• I = Imaxsin(10/6)
• VR = VR,maxsin(10/6)
Length of vector = Vmax across that componentVertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 23
AC circuit summary
LR
C
VR in phase with I
VC lags I
VL leads I
I
t
VL
VC
VR
Kirchoff’s Loop Equation always holds true:
Vgen = VL + VR + VC
However, Vgen,max VL,max+VR,max+VC,max
Maximum reached at different times for R, L, C
Phasors represent instantaneous voltages