Download AC Circuit Phasors

Physics 102: Lecture 13
AC Circuit Phasors
• Today’s lecture will cover Textbook
Section 21.5
• I = Imaxsin(2pft)
• VR = ImaxR sin(2pft)
• VC = ImaxXC sin(2pft-p/2)
• VL = ImaxXL sin(2pft+ p/2)
Physics 102: Lecture 13, Slide 1
L
R
C
A reminder about sines and
cosines
Recall: y coordinates
of endpoints are
• asin(q + p/2)
• asin(q)
• asin(q - p/2)
a
q+p/2
a
q
q-p/2
a
Physics 102: Lecture 13, Slide 2
Graphical representation of
voltages
I = Imaxsin(2pft) (q = 2pft)
VL = ImaxXL sin(2pft + p/2)
VR = ImaxR sin(2pft)
VC = ImaxXC sin(2pft - p/2)
ImaxXL
L
ImaxR
q
R
C
q+p/2
q-p/2
ImaxXC
Peak values
in AC Circuits
voltage
VR ,max  I max R
reactance or
impedance
L
R
C
VC ,max  I max X C
1
XC 
C
(capacitive reactance)
VL ,max  I max X L
X L  L
(inductive reactance)
Vgen,max  I max Z
Z  R2 + ( X L - X C )2 (impedance)
Physics 102: Lecture 13, Slide 4
Phasor Diagrams
• I = Imaxsin(p/6)
• VR = VR,maxsin(p/6)
f =1/12
t=1
2pft = p/6
L
R
p/6
VR,maxsin(p/6)
C
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 13, Slide 5
Phasor Diagrams
• I = Imaxsin(p/3)
• VR = VR,maxsin(p/3)
f =1/12
t=2
2pft = p/3
VR,maxsin(p/3)
p/3
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 13, Slide 6
Phasor Diagrams
• I = Imaxsin(p/2)
• VR = VR,maxsin(p/2)
Vmax
t=3
2pft = p/2
VR,maxsin(p/2)=V0
p/2
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 13, Slide 7
Drawing Phasor Diagrams
VL
(1) Resistor vector: to the right
•
Length given by VR
VR
(2) Inductor vector: upwards
•
Length given by VL
(3) Capacitor vector: downwards
•
Length given by VC
(4) Rotate entire thing counter-clockwise
•
VC
VL
VR
Vertical components give instantaneous
voltage across R, C, L
VC
Physics 102: Lecture 13, Slide 8
Phasor Diagrams
Instantaneous Values:
I X
• I = Imaxsin(2pft)
sin(2pft+p/2)
• VR = ImaxR sin(2pft)
• VC = ImaxXC sin(2pft-p/2)
max
ImaxR sin(2pft)
L
• VL = ImaxXL sin(2pft+p/2)
ImaxXC
sin(2pft-p/2)
Voltage across resistor is always _______ with current!
Voltage across capacitor always _______ current!
Voltage across inductor always ________ current!
Physics 102: Lecture 13, Slide 9
Phasor Diagrams
Instantaneous Values:
I X
• I = Imaxsin(2pft)
sin(2pft+p/2)
• VR = ImaxR sin(2pft)
• VC = ImaxXC sin(2pft-p/2)
max
L
• VL = ImaxXL sin(2pft+p/2)
ImaxR sin(2pft)
ImaxXC
sin(2pft-p/2)
Voltage across resistor is always in phase with current!
Voltage across capacitor always lags current!
Voltage across inductor always leads current!
Physics 102: Lecture 13, Slide 10
Phasor Diagram Practice
Label the vectors that corresponds to
the resistor, inductor and capacitor.
Which element has the largest voltage
across it at the instant shown?
1) R
2) C
3) L
Is the voltage across the inductor
increasing or decreasing?
Which element has the largest
maximum voltage across it?
Physics 102: Lecture 13, Slide 11
Phasor Diagram Practice
Label the vectors that corresponds to
the resistor, inductor and capacitor.
Inductor Leads Capacitor Lags
VR
Which element has the largest voltage VL
across it at the instant shown?
1) R
2) C
3) L
R: It has largest vertical component
Is the voltage across the inductor
increasing or decreasing?
Decreasing, spins counter clockwise
Which element has the largest
maximum voltage across it?
Inductor, it has longest line.
VC
KVL: Impedance Triangle
• Instantaneous voltage across generator
(Vgen) must equal sum of voltage across all
of the elements at all times:
I X =V
max
L
L,max
f
Imax(XL-XC)
Vgen (t) = VR (t) +VC (t) +VL (t)
Vgen,max = Imax Z
Z=
Physics 102: Lecture 13, Slide 13
ImaxR=VR,max
ImaxXC=VC,max
KVL: Impedance Triangle
• Instantaneous voltage across generator
(Vgen) must equal sum of voltage across all
of the elements at all times:
I X =V
max
L
L,max
f
Imax(XL-XC)
Vgen (t) = VR (t) +VC (t) +VL (t)
Vgen,max = Imax Z
Z  R + (XL - XC )
2
(XL - XC)
tan(f ) 
R
“phase angle”
ImaxR=VR,max
2
ImaxXC=VC,max
Phase angle f
I = Imaxsin(2pft)
Vgen = ImaxR sin(2pft + f)
ImaxR
Imax
2pft
f is positive in this particular case.
2pft + f
Drawing Phasor Diagrams
VL
(1) Resistor vector: to the right
•
Length given by VR
VR
(2) Capacitor vector: downwards
•
Vgen
Length given by VC
(3) Inductor vector: upwards
•
Length given by VL
(4) Generator vector: add first 3 vectors
•
VC
Length given by Vgen
VL
VR
Vgen
(5) Rotate entire thing counter-clockwise
•
Vertical components give instantaneous
voltage across R, C, L, gen
Physics 102: Lecture 13, Slide 16
VC
ACTS 13.1, 13.2, 13.3
f
time 1
time 3
time 2
When does Vgen = 0 ?
When does Vgen = VR ?
Is the phase angle positive or negative?
Physics 102: Lecture 13, Slide 17
time 4
ACTS 13.1, 13.2, 13.3
f
time 1
time 3
time 4
time 2
When does Vgen = 0 ?
time 2
When does Vgen = VR ?
time 3
Is the phase angle positive or negative?
Look at time 1: Vgen is below VR
Physics 102: Lecture 13, Slide 18
negative
Power P=IV
• The voltage generator supplies power.
– Resistor dissipates power.
– Capacitor and Inductor store and release energy.
• P = IV so power loss is sometimes large, sometimes
small.
• Average power dissipated by resistor:
P = ½ Imax VR,max
= ½ Imax Vgen,max cos(f)
= Irms Vgen,rms cos(f)
Physics 102: Lecture 13, Slide 19
Power P=IV
• The voltage generator supplies power.
– Resistor dissipates power.
– Capacitor and Inductor store and release energy.
• P = IV so power loss is sometimes large, sometimes
small.
• Average power dissipated by resistor:
P = ½ Imax VR,max
= ½ Imax Vgen,max cos(f)
= Irms Vgen,rms cos(f)
Physics 102: Lecture 13, Slide 20
AC Summary
Resistors:
VR=I R
In phase with I
Capacitors:
VCmax =Imax XC
Xc = 1/(2pf C)
Lags I
Inductors:
VLmax=Imax XL
XL = 2pf L
Leads I
Generator:
Vgenmax=Imax Z
Z = R2 +(XL-XC)2
Can lead or lag I
tan(f) = (XL-XC)/R
Power is only dissipated in resistor:
P = IrmsVrms cos(f)
Physics 102: Lecture 13, Slide 21
Problem Time!
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH is
driven by a generator with voltage V(t)=2.5 sin(8pt) Volts.
Calculate the maximum current in the circuit, and the
phase angle.
Imax = Vgen,max /Z
L
R
Z=
C
Physics 102: Lecture 13, Slide 22
Problem Time!
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH is
driven by a generator with voltage V(t)=2.5 sin(8pt) Volts.
Calculate the maximum current in the circuit, and the
phase angle.
L
Imax = Vgen,max /Z
R
Z  R2 + ( X L - X C )2
1
C
Z  2 + (8p  .030 )2  2.76W
8p  .015
Imax = 2.5/2.76 = .91 Amps
1
(8p  .030 )
X L - XC
8p  .015    -43.5
tan(f ) 

2
R
Physics 102: Lecture 13, Slide 23
2
Preflight 13.1
The statement that the voltage across the generator equals the sum
of the voltages across the resistor, capacitor and inductor is
true for:
ImaxXL=VL,ma
x
f
(1)
instantaneous voltages only
(2)
rms voltages only
(3)
both rms and instantaneous voltages
Imax(XL-XC)
ImaxR
ImaxXC = VC,max
Preflight 13.1
The statement that the voltage across the generator equals the sum
of the voltages across the resistor, capacitor and inductor is
true for:
ImaxXL=VL,ma
x
f
(1)
instantaneous voltages only
(2)
rms voltages only
(3)
both rms and instantaneous voltages
Imax(XL-XC)
Rotates Counter Clockwise
ImaxR
ImaxXC = VC,max
Vgen=VL+VR+VC at all times.
Vrms does not!
Voltage Phasor Diagram
Rotates Counter Clockwise
Imax XL=VL,max
f
ImaxR=VR,max
Imax XC=VC,max
Voltage Phasor Diagram
Rotates Counter Clockwise
ACT: Voltage Phasor Diagram
Rotates Counter Clockwise
At this instant, the voltage
across the generator is
maximum.
What is the voltage across the resistor at this instant?
1) VR = ImaxR
2) VR = ImaxR sin(f)
3) VR = ImaxR cos(f)
ACT: Voltage Phasor Diagram
Rotates Counter Clockwise
At this instant, the voltage
across the generator is
maximum.
What is the voltage across the resistor at this instant?
1) VR = ImaxR
2) VR = ImaxR sin(f)
3) VR = ImaxR cos(f)
See you next time!
Read Sections 21.6, 22.1, 4-5, 9
Physics 102: Lecture 13, Slide 30