Lect13 handout

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()