AC Circuit Phasors

Physics 102: Lecture 13Exam III

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C

• 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

VLVC

VR

Peak + RMS values in AC Circuits (REVIEW)

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5

,max maxC CV I X

,max maxRV I R

,max maxL LV I X

,max maxgenV I Z

When asking about RMS or Maximum values relatively simple expresions

1CX

C

LX L

2 2( )L CZ R X X

Time Dependence in AC Circuits

Write down Kirchoff’s Loop Equation:

VG + VL + VR + VC = 0 at every instant of time

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C

5

However …VG,max VL,max+VR,max+VC,max

Maximum reached at different times for R,L,C

A reminder about sines and cosines

Recall: y coordinates of endpoints are

• asin( + /2)• asin()• asin( - /2)

a a

a

x

y

L

RC

I = Imaxsin(2ft) ( = 2ft)

VL = ImaxXL sin(2ft + )

VR = ImaxR sin(2ft)

VC = ImaxXC sin(2ft - )

Graphical representation of voltages

ImaxXL

ImaxR

ImaxXC

Phasor Diagrams

• I = Imaxsin(/6)

• VR = VR,maxsin(/6)

VR,maxsin()

t = 1 f=1/122ft = /6

Length of vector = Vmax across that componentVertical component = instantaneous value of V

10

V R,max

Phasor Diagrams

VR,maxsin()

t = 22ft = /3

V R,m

ax

• I = Imaxsin(/3)

• VR = VR,maxsin(/3)

Length of vector = Vmax across that componentVertical component = instantaneous value of V

Phasor Diagrams

VR,maxsin()=V0

t = 32ft = /2

VR

,max

• I = Imaxsin(/2)

• VR = VR,maxsin(/2)

Length of vector = Vmax across that componentVertical component = instantaneous value of V

Phasor Diagrams

VR,maxsin(4)

t = 42ft = 4/6

VR,m

ax

• I = Imaxsin(4/6)

• VR = VR,maxsin(4/6)

Length of vector = Vmax across that componentVertical component = instantaneous value of V

Phasor Diagrams

t = 62ft =

VR,max

• I = Imaxsin()

• VR = VR,maxsin()

VR,maxsin()=0

Length of vector = Vmax across that componentVertical component = instantaneous value of V

Phasor Diagrams

VR,maxsin(8)

t = 82ft = 8

V R,m

ax

• I = Imaxsin(8/6)

• VR = VR,maxsin(8/6)

Length of vector = Vmax across that componentVertical component = instantaneous value of V

Phasor Diagrams

VR,maxsin(10)

t = 102ft = 10

VR,max

• I = Imaxsin(10/6)

• VR = VR,maxsin(10/6)

Length of vector = Vmax across that componentVertical component = instantaneous value of V

Drawing Phasor DiagramsVL

(2) Inductor vector: upwards• Length given by VL (or XL)

VC(3) Capacitor vector: downwards• Length given by VC (or XC)

VR

(1) Resistor vector: to the right• Length given by VR (or R)

VC

VRVL

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

R, C, L

(4) (coming soon)

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Phasor Diagrams

• I = Imaxsin(2ft)

• VR = ImaxR sin(2ft) I maxR ImaxR 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:

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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

VRInductor Leads Capacitor Lags

R: It has largest vertical component

Decreasing, spins counter clockwise

Inductor, it has longest line. 21

Imax(XL-XC)

KVL: Impedance Triangle• Instantaneous voltage across generator

(Vgen) must equal sum of voltage across all of the elements at all times: ImaxXL=VL,max

ImaxXC=VC,max

ImaxR=VR,max

V max,gen=I maxZ

Vgen (t) = VR (t) +VC (t) +VL (t)

“phase angle”

Vgen,max = Imax Z

2 2L C( )Z R X X

L C( )tan( )

X XR

25

Phase angle

2ftImax

I = Imaxsin(2ft)

Vgen = ImaxR sin(2ft + )ImaxR

2ft +

is positive in this particular case.

Drawing Phasor DiagramsVL

(2) Capacitor vector: Downwards• Length given by VC (or XC)

VC(3) Inductor vector: Upwards• Length given by VL (or XL)

VR

(1) Resistor vector: to the right• Length given by VR (or R)

(4) Generator vector: add first 3 vectors• Length given by Vgen (or Z)

Vgen

VC

VRVL

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

R, C, L

Vgen

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time 1 time 2time 3 time 4

ACTS 13.1, 13.2, 13.3

When does Vgen = VR ?

When does Vgen = 0 ?

VgenVgen

VgenVgen

VR

VRVR

VR

time 2

time 3

30

time 1 time 2time 3 time 4

ACTS 13.1, 13.2, 13.3

The phase angle is: (1) positive (2) negative (3) zero?

When does Vgen = VR ?

When does Vgen = 0 ?

Look at time 1: Vgen is below VR

negative

time 3

time 2

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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()

= Irms Vrms cos()

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AC SummaryResistors: VRmax=I R

In phase with ICapacitors: VCmax =I XC Xc = 1/(2f C)

Lags IInductors: VLmax=I XL XL = 2f L

Leads IGenerator: Vgen,max=I Z Z= sqrt(R2 +(XL-XC)2)

Can lead or lag I tan() = (XL-XC)/R

Power is only dissipated in resistor: P = ½ImaxVgen,max cos()

37

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.

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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 ) 2.768 .015

Z

Imax = 2.5/2.76 = .91 Amps

tan( ) L CX XR

1(8 .030 )8 .015 43.5

2

Imax = Vgen,max /ZL

R

C

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Imax(XL-XC)

Preflight 13.1

ImaxXL=VL,max

ImaxXC = VC,max

ImaxR

V gen,max

Rotates Counter Clockwise

Vgen=VL+VR+VC at all times. Vrms does not!

33%32%35%

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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ACT: Voltage Phasor DiagramI m

ax X

L

I max

XC

I max

R

Vge

n ,m

a x

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() 46

See You Monday!