Physics 102: Lecture 13, Slide 1
AC Circuit Phasors
Physics 102: Lecture 13
• 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
LR
C
Physics 102: Lecture 13, Slide 2
Peak & RMS values in AC Circuits (REVIEW)
LR
CWhen asking about RMS or Maximum values relatively simple expressions
𝑋𝐶 = 12𝜋𝑓𝐶= 1𝜔𝐶
𝑋𝐿 = 2𝜋𝑓𝐿= 𝜔𝐿
VR,max = ImaxR
VC,max = ImaxXC
VL,max = ImaxXL
Physics 102: Lecture 13, Slide 3
Time Dependence in AC Circuits
Write down Kirchoff’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 13, Slide 4
I = Imaxsin(2ft) ( = 2ft)
VL = ImaxXL sin(2ft + )
VR = ImaxR sin(2ft)
VC = ImaxXC sin(2ft – )
Graphical representation of voltages
ImaxXL
ImaxR
ImaxXC
L
R
C
Physics 102: Lecture 13, Slide 5
Drawing Phasor Diagrams
VL,max
(2) Inductor vector: upwards• Length given by VL,max (or XL)
VC,max
(3) Capacitor vector: downwards• Length given by VC,max (or XC)
VR,max
(1) Resistor vector: to the right• Length given by VR,max (or R)
VC(t)
VR(t)VL(t)
(5) Rotate entire thing counter-clockwise• Vertical components give instantaneous
voltage across R, C, L
(4) Generator vector (coming soon)
Physics 102: Lecture 13, Slide 6
Phasor Diagrams
• I = Imaxsin(2ft)• VR = ImaxR sin(2ft) I max
RImaxR sin(2ft)
• VC = ImaxXC sin(2ft–)= –ImaxXC cos(2ft)
• VL = ImaxXL sin(2ft + )= ImaxXL cos(2ft)
ImaxXL cos(2ft)
-ImaxXC cos(2ft)
Im
ax XL
Im
ax XC
Voltage across resistor is always in phase with current! Voltage across capacitor always lags current! Voltage across inductor always leads current!
Instantaneous Values:
Physics 102: Lecture 13, Slide 7
Phasor Diagram PracticeLabel 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 1) increasing or 2) decreasing?
Which element has the largest maximum voltage across it?
1) R 2) C 3) L
VL
VC
VR
Inductor Leads Capacitor Lags
R: It has largest vertical component
Decreasing, spins counter clockwise
Inductor, it has longest line.
Physics 102: Lecture 13, Slide 8
VL,max-VC,max
Kirchhoff: generator voltage• Instantaneous voltage across generator (Vgen) must
equal sum of voltage across all of the elements at all times:
VL,max=ImaxXL
VC,max=ImaxXC
VR,max=ImaxR
V gen,max=I max
Z
Vgen (t) = VR (t) +VC (t) +VL (t)
“phase angle”
Define impedance Z: Vgen,max ≡ Imax Z
2 2L C( )Z R X X L C( )
tan( )X X
R
“Impedance Triangle”
𝑉gen,max =ට𝑉R,max2 +(𝑉L,max −𝑉C,max )2
tan𝜙 = 𝑉L,max −𝑉C,max𝑉R,max
Physics 102: Lecture 13, Slide 9
Phase angle
2ftImax
I = Imaxsin(2ft)
Vgen = ImaxZ sin(2ft + )ImaxZ
2ft +
is positive in this particular case.
Physics 102: Lecture 13, Slide 10
Drawing Phasor Diagrams
VL,max
(2) Capacitor vector: Downwards• Length given by VC,max (or XC)
VC,max(3) Inductor vector: Upwards• Length given by VL,max (or XL)
VR,max
(1) Resistor vector: to the right• Length given by VR,max (or R)
(4) Generator vector: add first 3 vectors• Length given by Vgen,max (or Z)
Vgen,max
VC
VR
VL
(5) Rotate entire thing counter-clockwise• Vertical components give instantaneous
voltage across R, C, L
Vgen
Physics 102: Lecture 13, Slide 11
time 1 time 2
time 3 time 4
ACTS 13.1, 13.2, 13.3
When does Vgen = VR ?
When does Vgen = 0 ?
The phase angle is: (1) positive (2) negative (3) zero?
Physics 102: Lecture 13, Slide 12
Problem Time!
An AC circuit with R= 2 , C = 15 mF, and L = 30 mH is driven by a generator with voltage V(t)=2.5 sin(8t) Volts. Calculate the maximum current in the circuit, and the phase angle.
2 2( )L CZ R X X
2 212 (8 .030 ) 2.76
8 .015Z
Imax = 2.5/2.76 = .91 Amps
tan( ) L CX X
R
1(8 .030 )
8 .015 43.52
Imax = Vgen,max /ZL
R
C
Physics 102: Lecture 13, Slide 13
ACT: Voltage Phasor DiagramI m
ax X
L
I max
XC
I max
R
Vge
n,m
ax
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() 3) VR = ImaxR cos()
Physics 102: Lecture 13, Slide 14
Resonance and the Impedance Triangle
R
(XL-XC) Z
XL and XC point opposite. When adding, they tend to cancel!
When XL = XC they completely cancel and Z = R. This is resonance!
Vmax,gen = Imax Z
Imax(XL-XC)
ImaxXL
ImaxXC
ImaxR
V gen,max
LR
C
Physics 102: Lecture 13, Slide 15
Resonance
Resonance in AC Circuits
frequency
Impe
danc
e
R is independent of f Resonance in AC Circuits
frequency
Impe
danc
e
Resonance in AC Circuits
frequency
Imp
edan
ce
Resonance in AC Circuits
frequency
Imp
edan
ceR
XL increases with f
XL
XC decreases with f
XC
Z: XL and XC subtract
ZXC = 1/(2fC)
XL = 2fL
2 2( )L CZ R X X
Resonance: XL = XC
f0
Z is minimum at resonance frequency!
𝑓0 = 12𝜋ξ𝐿𝐶
Physics 102: Lecture 13, Slide 16
Resonance in AC Circuits
frequency
ResonanceR is independent of fXL increases with f
XC decreases with f
Z: XL and XC subtract
ZXC = 1/(2fC)
XL = 2fL
2 2( )L CZ R X X
Resonance: XL = XC
f0
𝑓0 = 12𝜋ξ𝐿𝐶
Current
Imax = Vgen,max/Z
Current is maximum at resonance frequency!
Physics 102: Lecture 13, Slide 17
ACT: Resonance
The AC circuit to the right is being driven at its resonance frequency. Compare the maximum voltage across the capacitor with the maximum voltage across the inductor.
1) VC,max > VL,max
2) VC,max = VL,max
3) VC,max < VL,max
4) Depends on R
LR
C
Physics 102: Lecture 13, Slide 18
Summary of Resonance
• At resonance– Z is minimum (=R)– Imax is maximum (=Vgen,max/R)– Vgen is in phase with I– XL = XC VL(t) = -VC(t)
• At lower frequencies– XC > XL Vgen lags I
• At higher frequencies– XC < XL Vgen lead I
Imax(XL-XC)
ImaxXL
ImaxXC
ImaxR
V gen,max
Physics 102: Lecture 13, Slide 19
Power in AC circuits
• The voltage generator supplies power. – Only resistor dissipates power.
– Capacitor and Inductor store and release energy.
• P(t) = I(t)VR(t) oscillates so sometimes power loss is large, sometimes small.
• Average power dissipated by resistor:
P = ½ Imax VR,max
= ½ Imax Vgen,max cos()
= Irms Vgen,rms cos()
Physics 102: Lecture 13, Slide 20
AC Summary
Resistors: VR,max=Imax R In phase with I
Capacitors: VC,max =Imax XC Xc = 1/(2f C)Lags I
Inductors: VL,max=Imax XL XL = 2f LLeads I
Generator: Vgen,max=Imax Z Z = √R2 +(XL -XC)2
Can lead or lag I tan() = (XL-XC)/R
Power is only dissipated in resistor:
P = ½ImaxVgen,max cos()