Download AC Circuit Phasors

Physics 102: Lecture 13
Exam
III
AC Circuit Phasors
• I = Imaxsin(2pft)
• VR = ImaxR sin(2pft)
• VR in phase with I
• VC = ImaxXC sin(2pft-p/2)
•VC lags I
• VL = ImaxXL sin(2pft+p/2)
•VL leads I
I
L
VR
R
C
t
VC
VL
Peak + RMS values
in AC Circuits (REVIEW)
L
When asking about RMS or Maximum values
relatively simple expresions
C
R
VR ,max  I max R
VC ,max  I max X C
1
XC 
C
VL ,max  I max X L
X L  L
Vgen,max  I max Z
Z  R2  ( X L  X C )2
5
Time Dependence
in AC Circuits
L
R
C
Write down Kirchoff’s Loop Equation:
VG + VL + VR + VC = 0 at every instant of time
However …
VG,max  VL,max+VR,max+VC,max
Maximum reached at different times
for R,L,C
5
A reminder about sines and
y
cosines
Recall: y coordinates
of endpoints are
• asin(q + p/2)
• asin(q)
• asin(q - p/2)
a
qp/2
a
q
x
qp/2
a
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
Phasor Diagrams
• I = Imaxsin(p/6)
• VR = VR,maxsin(p/6)
t = 1 f=1/12
2pft = p/6
p/6
VR,maxsin(p/6)
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
10
Phasor Diagrams
• I = Imaxsin(p/3)
• VR = VR,maxsin(p/3)
t=2
2pft = p/3
VR,maxsin(p/3)
p/3
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Phasor Diagrams
• I = Imaxsin(p/2)
• VR = VR,maxsin(p/2)
VR,max
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
Phasor Diagrams
• I = Imaxsin(4p/6)
• VR = VR,maxsin(4p/6)
t=4
2pft = 4p/6
VR,maxsin(4p/6)
4p/6
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Phasor Diagrams
• I = Imaxsin(p)
• VR = VR,maxsin(p)
VR,maxsin(p)=0
t=6
2pft = p
VR,max
p
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Phasor Diagrams
• I = Imaxsin(8p/6)
• VR = VR,maxsin(8p/6)
t=8
2pft = 8p/6
8p/6
VR,maxsin(8p/6)
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Phasor Diagrams
• I = Imaxsin(10p/6)
• VR = VR,maxsin(10p/6)
t = 10
2pft = 10p/6
10p/6
VR,maxsin(10p/6)
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Drawing Phasor Diagrams
VL
(1) Resistor vector: to the right
•
Length given by VR (or R)
VR
(2) Inductor vector: upwards
•
Length given by VL (or XL)
(3) Capacitor vector: downwards
•
(4)
VC
Length given by VC (or XC)
(coming soon)
VL
VR
(5) Rotate entire thing counter-clockwise
•
Vertical components give instantaneous
voltage across R, C, L
VC
15
Phasor Diagrams
Instantaneous Values:
• I = Imaxsin(2pft)
I X
• VR = ImaxR sin(2pft)
• VC = ImaxXC sin(2pft-p/2)
= -ImaxXC cos(2pft)
max
L
cos(2pft)
ImaxR sin(2pft)
-ImaxXC cos(2pft)
• VL = ImaxXL sin(2pft+ p/2)
= ImaxXL cos(2pft)
Voltage across resistor is always in phase with current!
Voltage across capacitor always lags current!
Voltage across inductor always leads current!
17
Phasor Diagram Practice
Label the vectors that corresponds to
the resistor, inductor and capacitor.
Inductor Leads Capacitor Lags
Which element has the largest voltage
VL
across it at the instant shown?
1) R
2) C
3) L
VR
R: It has largest vertical component
Is the voltage across the inductor
1)increasing or 2) decreasing?
Decreasing, spins counter clockwise
VC
Which element has the largest
maximum voltage across it?
1) R
2) C
3) L
Inductor, it has longest line.
21
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
ImaxR=VR,max
2
ImaxXC=VC,max
(XL  XC)
tan(f ) 
R
“phase angle”
25
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 (or R)
VR
(2) Capacitor vector: Downwards
•
Vgen
Length given by VC (or XC)
(3) Inductor vector: Upwards
•
Length given by VL (or XL)
(4) Generator vector: add first 3 vectors
•
VC
Length given by Vgen (or Z)
VL
VR
Vgen
(5) Rotate entire thing counter-clockwise
•
Vertical components give instantaneous
voltage across R, C, L
VC
27
ACTS 13.1, 13.2, 13.3
VR
VR
VR
time 1
VR
Vgen
time 3
Vgen
Vgen
Vgen
time 4
time 2
When does Vgen = 0 ?
time 2
When does Vgen = VR ?
time 3
30
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
The phase angle is: (1) positive (2) negative (3) zero?
Look at time 1: Vgen is below VR
negative 31
Power P=IV
• The voltage generator supplies power.
– Resistor dissipates power.
– Capacitor and Inductor store and release energy.
• P = IV so sometimes power loss is large, sometimes
small.
• Average power dissipated by resistor:
P = ½ Imax VR,max
= ½ Imax Vgen,max cos(f)
= Irms Vrms cos(f)
34
AC Summary
Resistors:
VRmax=I R
In phase with I
Capacitors:
VCmax =I XC
Lags I
Inductors:
VLmax=I XL
Leads I
Generator:
Vgen,max=I Z
Can lead or lag I
Xc = 1/(2pf C)
XL = 2pf L
Z= sqrt(R2 +(XL-XC)2)
tan(f) = (XL-XC)/R
Power is only dissipated in resistor:
P = ½ImaxVgen,max cos(f)
37
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
R
C
41
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
2
41
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:
33% (1) instantaneous voltages only
ImaxXL=VL,ma
x
32% (2) rms voltages only
35%
f
Imax(XL-XC)
(3) both rms and instantaneous
Rotates Counter Clockwise
ImaxR
ImaxXC = VC,max
Vgen=VL+VR+VC at all times.
Vrms does not!
43
ACT: Voltage Phasor Diagram
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)
46
See You Monday!