Download I max R

Physics 102: Lecture 13
AC Circuit Phasors
L
R
C
Physics 102: Lecture 13, Slide 1
Review: AC Circuit
L
• I = Imaxsin(2pft)
• VR = ImaxR sin(2pft)
• VR in phase with I
R
C
I
• VC = ImaxXC sin(2pft–p/2)
•VC lags I
• VL = ImaxXL sin(2pft+p/2)
•VL leads I
Physics 102: Lecture 13, Slide 2
VR
t
VC
VL
Peak & RMS values
in AC Circuits (REVIEW)
L
When asking about RMS or Maximum values
relatively simple expressions
C
VR,max = ImaxR
VC,max = ImaxXC
1
1
𝑋𝐶 =
=
2𝜋𝑓𝐶 𝜔𝐶
VL,max = ImaxXL
𝑋𝐿 = 2𝜋𝑓𝐿 = 𝜔𝐿
Physics 102: Lecture 13, Slide 3
R
Time Dependence
in AC Circuits
L
Vgen
R
Write down Kirchoff’s Loop Equation:
Vgen(t) = VL(t) + VR(t) + VC(t) at every instant of time
However …
Vgen,max  VL,max+VR,max+VC,max
I
VC
Physics 102: Lecture 13, Slide 4
VR
t
Maximum reached at different
times for R, L, C
We solve this using phasors
C
VL
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
q+p/2
L
R
ImaxR
q
q-p/2
C
Physics 102: Lecture 13, Slide 5
ImaxXC
Drawing Phasor Diagrams
(1) Resistor vector: to the right
•
VL,max
Length given by VR,max (or R)
VR,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)
(4) Generator vector (coming soon)
VL(t)
(5) Rotate entire thing counter-clockwise
•
Vertical components give instantaneous
voltage across R, C, L
Physics 102: Lecture 13, Slide 6
VR(t)
VC(t)
Phasor Diagrams
Instantaneous Values:
• I = Imaxsin(2pft)
• VR = ImaxR sin(2pft)
ImaxXL cos(2pft)
• VC = ImaxXC sin(2pft–p/2)
= –ImaxXC cos(2pft)
• VL = ImaxXL sin(2pft + p/2)
= ImaxXL cos(2pft)
ImaxR sin(2pft)
-ImaxXC cos(2pft)
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 7
Phasor Diagram Practice
Label the vectors that corresponds to
the resistor, inductor and capacitor.
Inductor Leads & Capacitor Lags
Which element has the largest voltage
across it at the instant shown?
VL
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
Which element has the largest
maximum voltage across it?
1) R
2) C
3) L
Inductor, it has longest line.
Physics 102: Lecture 13, Slide 8
VC
Kirchhoff: generator voltage
• Instantaneous voltage across generator (Vgen) must
equal sum of voltage across all of the elements at
all times:
Vgen (t) = VR (t) +VC (t) +VL (t)
𝑉gen ,max =
VL,max=ImaxXL
2
𝑉R,max
+ (𝑉L,max − 𝑉C,max )2
f
𝑉L,max − 𝑉C,max
tan 𝜙 =
𝑉R,max
VR,max=ImaxR
“phase angle”
VC,max=ImaxXC
Define impedance Z: Vgen,max ≡ Imax Z
Z  R2 + ( X L - X C )2 tan(f ) 
Physics 102: Lecture 13, Slide 9
VL,max-VC,max
(XL - XC)
R
“Impedance Triangle”
Phase angle f
I = Imaxsin(2pft)
Vgen = ImaxZ sin(2pft + f)
ImaxZ
Imax
2pft
f is positive in this particular case.
Physics 102: Lecture 13, Slide 10
2pft + f
Drawing Phasor Diagrams
(1) Resistor vector: to the right
•
VL,max
VR,max
Length given by VR,max (or R)
(2) Capacitor vector: Downwards
•
Vgen,max
Length given by VC,max (or XC)
VC,max
(3) Inductor vector: Upwards
•
Length given by VL,max (or XL)
(4) Generator vector: add first 3 vectors
•
Length given by Vgen,max (or Z)
VL
VR
Vgen
(5) Rotate entire thing counter-clockwise
•
Vertical components give instantaneous
voltage across R, C, L
Physics 102: Lecture 13, Slide 11
VC
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
The phase angle is: (1) positive (2) negative (3) zero?
Physics 102: Lecture 13, Slide 12
Vgen is clockwise from VR
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
Physics 102: Lecture 13, Slide 13
R
2
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
Physics 102: Lecture 13, Slide 14
2) VR = ImaxR sin(f)
3) VR = ImaxR cos(f)
L
Resonance and the
Impedance Triangle
R
C
Vgen,max = Imax Z
ImaxXL
(XL-XC)
f
f
Imax(XL-XC)
ImaxR
ImaxXC
Physics 102: Lecture 13, Slide 15
R
XL and XC point opposite. When adding,
they tend to cancel!
When XL = XC they completely cancel
and Z = R. This is resonance!
Resonance
R is independent of f
XC decreases with f
XC = 1/(2pfC)
Z: XL and XC subtract
Z  R + ( X L - XC )
2
2
Resonance: XL = XC
Physics 102: Lecture 13, Slide 16
Z is minimum at
resonance frequency!
Impedance
Impedance
Impedance
XL increases with f
XL = 2pfL
Resonance
Circuits
Resonance
ACCircuits
Circuits
Resonance in AC
Resonance
AC
Circuits
Z
XC
XL
f0
𝑓0 =
R
frequency
frequency
frequency
frequency
1
2𝜋 𝐿𝐶
Resonance
R is independent of f
Resonance in AC Circuits
XL increases with f
XL = 2pfL
Current is maximum at
resonance frequency!
XC decreases with f
XC = 1/(2pfC)
Z
Z: XL and XC subtract
Current
Z  R2 + ( X L - X C )2
Resonance: XL = XC
Physics 102: Lecture 13, Slide 17
Imax = Vgen,max/Z
f0
𝑓0 =
frequency
1
2𝜋 𝐿𝐶
L
R
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)
2)
3)
4)
VC,max > VL,max
VC,max = VL,max
VC,max < VL,max
Depends on R
At resonance XL = XC. Since
everything has the same
current we can write
XL = XC
XLImax = XCImax
Physics 102: Lecture 13, Slide 18
C
VL,max = VC,max
Also VGen is in
phase with
current!
Summary of Resonance
• At resonance
–
–
–
–
ImaxXL
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
f
ImaxR
– XC > XL Vgen lags I
• At higher frequencies
– XC < XL Vgen lead I
Physics 102: Lecture 13, Slide 19
ImaxXC
Imax(XL-XC)
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(f)
= Irms Vgen,rms cos(f)
Physics 102: Lecture 13, Slide 20
AC Summary
Resistors:
VR,max=Imax R
In phase with I
Capacitors:
VC,max =Imax XC
Xc = 1/(2pf C)
Lags I
Inductors:
VL,max=Imax XL
XL = 2pf L
Leads I
Generator:
Vgen,max=Imax Z
Z = √R2 +(XL -XC)2
Can lead or lag I
tan(f) = (XL-XC)/R
Power is only dissipated in resistor:
P = ½ImaxVgen,max cos(f)
Physics 102: Lecture 13, Slide 21
See You Monday!
Physics 102: Lecture 13, Slide 22