AC Circuit Phasors

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AC Circuit Phasors
Physics 102 Lecture 13

• I Imaxsin(2pft)
• VR ImaxR sin(2pft)
• VR in phase with I

I
VR

• VC ImaxXC sin(2pft-p/2)
• VC lags I

t
VL

• VL ImaxXL sin(2pftp/2)
• VL leads I

VC
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Peak RMS values in AC Circuits (REVIEW)
When asking about RMS or Maximum values
relatively simple expresions
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Time Dependence in AC Circuits

• Write down Kirchoffs Loop Equation
• VG VL VR VC 0 at every instant of time

I
VR
However VG,max ? VL,maxVR,maxVC,max Maximum
reached at different times for R,L,C
t
VL
VC
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Here is a problem that we will now learn how to
solve

• 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.

Example
41
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A reminder about sines and cosines
y

• Recall y coordinates of endpoints are
• asin(q p/2)
• asin(q)
• asin(q - p/2)

x
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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)
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Phasor Diagrams A Detailed Example

• I Imaxsin(p/6)
• VR VR,maxsin(p/6)

t 1 f1/12 2pft p/6
VR,max
VR,maxsin(p/6)
p/6
Length of vector Vmax across that
component Vertical component instantaneous
value of V
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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
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Phasor Diagrams

• I Imaxsin(p/2)
• VR VR,maxsin(p/2)

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
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Phasor Diagrams

• I Imaxsin(4p/6)
• VR VR,maxsin(4p/6)

t 4 2pft 4p/6
VR,max
VR,maxsin(4p/6)
4p/6
Length of vector Vmax across that
component Vertical component instantaneous
value of V
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Phasor Diagrams

• I Imaxsin(p)
• VR VR,maxsin(p)

t 6 2pft p
p
VR,maxsin(p)0
Length of vector Vmax across that
component Vertical component instantaneous
value of V
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Phasor Diagrams

• I Imaxsin(8p/6)
• VR VR,maxsin(8p/6)

t 8 2pft 8p/6
8p/6
VR,max
VR,maxsin(8p/6)
Length of vector Vmax across that
component Vertical component instantaneous
value of V
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Phasor Diagrams

• I Imaxsin(10p/6)
• VR VR,maxsin(10p/6)

t 10 2pft 10p/6
10p/6
VR,maxsin(10p/6)
VR,max
Length of vector Vmax across that
component Vertical component instantaneous
value of V
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Drawing Phasor Diagrams
(4) (coming soon)
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Phasor Diagrams
Instantaneous Values

• I Imaxsin(2pft)
• VR ImaxR sin(2pft)

• VC ImaxXC sin(2pft-p/2)
• -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!
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Phasor Diagram Practice
Example

• 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 1)increasing
  or 2) decreasing?
• Which element has the largest maximum voltage
  across it?
• 1) R 2) C 3) L

Inductor Leads Capacitor Lags
VR
VL
R It has largest vertical component
VC
Decreasing, spins counter clockwise
Inductor, it has longest line.
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KVL Impedance Triangle

• Instantaneous voltage across generator (Vgen)
  must equal sum of voltage across all of the
  elements at all times

ImaxXLVL,max
Vmax,genImaxZ
Vgen (t) VR (t) VC (t) VL (t)
Imax(XL-XC)
f
Vgen,max Imax Z
ImaxRVR,max
ImaxXCVC,max
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Phase angle f
I Imaxsin(2pft) Vgen ImaxZ sin(2pft f)
f is positive in this particular case.
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Drawing Phasor Diagrams
VL
VR

• (5) Rotate entire thing counter-clockwise
• Vertical components give instantaneous voltage
  across R, C, L

VC
Rotates Counter Clockwise
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ACTS 13.1, 13.2, 13.3
When does Vgen 0 ?
time 2
When does Vgen VR ?
time 3
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ACTS 13.1, 13.2, 13.3
When does Vgen 0 ?
time 2
When does Vgen VR ?
time 3
The phase angle is (1) positive (2) negative
(3) zero?
negative
Look at time 1 Vgen is below VR
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Power PIV

• 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)

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AC Summary

• Resistors VRmaxI R
• In phase with I
• Capacitors VCmax I XC Xc 1/(2pf C)
• Lags I
• Inductors VLmaxI XL XL 2pf L
• Leads I
• Generator Vgen,maxI Z Z sqrt(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)

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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.

Example
41
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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.

Example
Imax Vgen,max /Z
Imax 2.5/2.76 .91 Amps
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ACT/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 (1) instantaneous voltages
only (2) rms voltages only (3) both rms
and instantaneous
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ImaxXLVL,max
Vgen,max
Imax(XL-XC)
Rotates Counter Clockwise
f
ImaxR
VgenVLVRVC at all times. Vrms does not!
ImaxXC VC,max
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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)
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See You Monday!