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Unformatted text preview: Physics 212
Lecture 20
AC Circuits
Maximum currents & voltages
Phasors: A Simple Tool
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Avg = 2.6 Confident Physics 212 Lecture 20, Slide 1 Music
Who is the Artist?
A)
B)
C)
D)
E) Professor Longhair
Johnny Adams
David Egan
Dr. John
Allen Toussaint classics
classics The Theme of the week:
New Orleans piano players
New A great piano CD Physics 212 Lecture 20, Slide 2 Physics 212
Lecture 20
AC Circuits
Maximum currents & voltages
Phasors: A Simple Tool
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Avg = 2.6 Confident Physics 212 Lecture 20, Slide 3 Resistors ε = εmaxsin(ωt) R I = VR/R = Vmax/R sin(ωt) Amplitude = Vmax/R
Amplitude Physics 212 Lecture 20, Slide 4 Capacitors
Q = VC = Cεmaxsin(ωt)
I = dQ/dt
dQ/dt ε = εmaxsin(ωt) C I = VmaxωC cos(ωt) Amplitude = Vmax/XC
Amplitude
where XC = 1/ωC
is like the “resistance”
is
of the capacitor
of
XC depends on ω Physics 212 Lecture 20, Slide 5 Inductors
dI/dt = VL = εmaxsin(ωt) ε = εmaxsin(ωt) L I =  Vmax/ωL cos(ωt) Amplitude = Vmax/XL
Amplitude
where XL = ωL
is like the “resistance”
is
of the inductor
of
XL depends on ω Physics 212 Lecture 20, Slide 6 RL ACT
An RL circuit is driven by an AC generator as shown in the
An
figure.
figure. L BB Imax = Vmax/XL
XL = ωL R
For what driving frequency ω of the generator will the
For
of
current through the resistor be largest
A) ω large
A)
B) Current through R doesn’t depend on ω
C) ω small
small
Physics 212 Lecture 20, Slide 7 Summary
R Imax = Vmax/R V in phase with I
in
Because resistors are simple C Imax = Vmax/XC
XC = 1/ωC L Imax = Vmax/XL
XL = ωL V 90o behind I
behind
Current comes first since it
charges capacitor
Like a wire at high ω V 90o ahead of I
ahead Opposite of capacitor
Like a wire at low ω
Like
Physics 212 Lecture 20, Slide 8 Makes sense to write everything in
Makes
terms of I since this is the same
since
everywhere in a oneloop circuit:
everywhere Phasors make this
make
simple to see
simple
Imax XL Vmax = Imax XC
V 90o behind I
behind Imax R C εmax L
R Vmax = Imax XL
V 90o ahead of I
ahead
Imax XC Vmax = Imax R
V in phase with I
in Prelecture animation Always looks the same.
Only the lengths will
Only
change
change
Physics 212 Lecture 20, Slide 9 Imax XC The voltages still add up
The
C εmax But now we are adding
But
vectors: Imax XL L
R Imax R Imax XL εmax
Imax R Imax R Imax XC Imax XC Imax XL Imax R
Imax XC Imax XL εmax
Physics 212 Lecture 20, Slide 10
10 Imax XC Making this simpler…
C εmax
Imax XL L Imax XL R
Imax R Imax XL εmax
Imax R Imax XC Imax R Imax XC
Physics 212 Lecture 20, Slide 11
11 Imax XC Making this simpler…
C εmax L Imax XL R
Imax R Imax XL εmax = Imax Z
max
Imax R Imax(XLXC)
Imax R Imax XC
Physics 212 Lecture 20, Slide 12
12 Imax XC Making this simpler…
C εmax L Imax XL R
Imax R εmax = Imax Z
max
Imax(XLXC)
Imax R Physics 212 Lecture 20, Slide 13
13 Imax XC Making this simpler…
C εmax Imax XL R εmax = Imax Z
max Imax R Imax(XLXC) φ L Imax R (XLXC) φ
R
Impedance Triangle X L − XC
tan (φ ) =
R
Physics 212 Lecture 20, Slide 14
14 Imax XC Summary:
C VCmax= Imax XC
max εmax VLmax= Imax XL
max L Imax XL R VRmax= Imax R
max εmax = Imax Z
max Imax R Imax = εmax / Z
max
Z = R + ( X L − XC )
2 X L − XC
tan (φ ) =
R 2 (XLXC) φ
R Physics 212 Lecture 20, Slide 15
15 Example: RL Circuit Xc=0
Example: εmax L Imax XL R
Imax R Imax XL εmax
Imax R Physics 212 Lecture 20, Slide 16
16 Preflight 2
BB Draw Voltage Phasors
Imax XL εmax
Imax R A
B
C 60
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17 Preflight 4 BB Draw Voltage Phasors
Imax XL εmax
Imax R A
B
C 60
50
40
30
20
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18 Preflight 6 BB The CURRENT is THE CURRENT
Imax XL φ
A
B
C
D εmax
Imax R φ is the phase between
generator and current
50
40
30
20
10
0 Physics 212 Lecture 20, Slide 19
19 Preflight 8
BB A
B
C What does the voltage
phasor diagram look
like when the current
IXL
is a maximum? IXL
ε IR ε 50
40 IR 30
20 IXc 10
0 IXc Physics 212 Lecture 20, Slide 20
20 Preflight 10 IXL ε
IR IR A
B
C IXc BB 40 IXc ε 30
20
10 IXL What does the voltage
phasor diagram look
like when the capacitor
is fully charged? 50 0 Physics 212 Lecture 20, Slide 21
21 Preflight 12 IXL ε
IR IR A
B
C IXc BB 40 IXc ε 30
20
10 IXL What does the voltage
phasor diagram look
like when the voltage
across capacitor is at
its positive maximum? 50 0 Physics 212 Lecture 20, Slide 22
22 Calculation C Consider the harmonically driven series LCR circuit shown.
Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown. V~ L
R What is XL, the reactance of the inductor, at this frequency? • Conceptual Analysis
–
– The maximum voltage for each component is related to its
reactance and to the maximum current.
The impedance triangle determines the relationship between the
maximum voltages for the components • Strategic Analysis
–
–
– Use Vmax and Imax to determine Z
Use impedance triangle to determine R
Use VCmax and impedance triangle to
determine XL
Physics 212 Lecture 20, Slide 23
23 Calculation C Consider the harmonically driven series LCR circuit shown.
Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown. V~ L
R What is XL, the reactance of the inductor, at this frequency? Compare XL and XC at this frequency:
(A) XL < XC
(B) XL = XC
(C) XL > XC BB (D) Not enough information • This information is determined from the phase
– Current leads voltage 45ο VL VL = ImaxXL
VC = ImaxXC VR (phase of current)
V
VC IR
V leads
Physics 212 Lecture 20, Slide 24
24 Calculation C Consider the harmonically driven series LCR circuit shown.
Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown. V~ L
R What is XL, the reactance of the inductor, at this frequency? What is Z, the total impedance of the circuit?
(A) 70.7 kΩ
(B) 50 kΩ
(C) 35.4 kΩ BB (D) 21.1 kΩ Vmax 100V
Z=
=
= 50k Ω
I max 2mA Physics 212 Lecture 20, Slide 25
25 Calculation C Consider the harmonically driven series LCR circuit shown.
Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown. V~ What is XL, the reactance of the inductor, at this frequency? What is R?
(A) 70.7 kΩ L
R
Z = 50kΩ sin(45)=.707
cos(45)=.707 (B) 50 kΩ (C) 35.4 kΩ (D) 21.1 kΩ • Determined from impedance triangle
R
45ο Z= 50k Ω (XCXL) R
cos(45) =
Z R = Z cos(45) BB = 50k Ω (.707 )
= 35.4 kΩ
Physics 212 Lecture 20, Slide 26
26 Calculation C Consider the harmonically driven series LCR circuit shown.
Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown. V~
R Z = 50kΩ What is XL, the reactance of the inductor, at this frequency? (A) 70.7 kΩ (B) 50 kΩ We start with the
impedance triangle: R
45ο Z (C) 35.4 kΩ XC − X L
= tan 45° = 1
R L R = 35.4kΩ (D) 21.1 kΩ X L = XC − R BB What is XC ? (XCXL) VC max = I max X C
X L = 56.5kΩ − 35.4kΩ 113
XC =
= 56.5kΩ
2 Physics 212 Lecture 20, Slide 27
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This note was uploaded on 09/22/2011 for the course PHYSICS 212 taught by Professor Selig during the Fall '10 term at University of Illinois, Urbana Champaign.
 Fall '10
 Selig
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