Physics 102: Lecture 12, Slide 1

Physics 102: Lecture 12

L

R

C

AC Circuits

Physics 102: Lecture 12, Slide 2

Review: Generators and EMF

vv

•

x

r

= A B sin() = A B sin(t) = Vmax sin(t)

t

1

2

Frequency = How fast its spinning

Amplitude = Maximum voltage

Voltage across generator:

Vmax

-Vmax

20

Physics 102: Lecture 12, Slide 3

AC Source

0.25

0.52f t = 8tf = 4 HzT=(1/4)seconds/cycle

+24

-24

V(t) = Vmax sin(t)=Vmax sin(2f t)

Vmax = maximum voltage

f = frequency (cycles/second)

V(t) = 24 sin(8 t)

RMS: Root Mean Square Vrms=Vmax/√2

Physics 102: Lecture 12, Slide 4

RMS?

RMS: Root Mean Square Vrms=Vmax/√2

+Vma

x

-Vmax

V(t) = Vmax sin(2f t)

Vmax2

Square: Vmax2 / 2

Mean:

square Root: Vmax /

√2

Physics 102: Lecture 12, Slide 5

Preflight 12.1, 12.2

I(t) = 10 sin(377 t)

Find Imax

Find Irms

LR

C

Well… We know that the maximum value sine is 1. So the maximum current is 10!

Imax = 10 A

Just like Vrms=Vmax/√2 …

Irms=Imax/√2 =10/√2 A = 7.07 A

Physics 102: Lecture 12, Slide 6

Resistors in AC circuit

VR = I R always true – Ohm’s Law

• VR,max = ImaxR

R

• Voltage across resistor is “IN PHASE” with current.

– VR goes up and down at the same times as I does.

I

t

t

VR

FrequencyResis

tan

ce

(R)

Frequency does not affect Resistance!

Physics 102: Lecture 12, Slide 7

C

Capacitors in AC circuit

I

t

• Voltage across capacitor “LAGS” current.

– VC goes up and down just after I does.

t

VC

FrequencyReacta

nce

(XC)

Frequency does affect Reactance!

VC = Q/C always true

• VC,max = ImaxXC

• Capacitive Reactance: XC = 1/(2fC)

Physics 102: Lecture 12, Slide 8

Inductors in AC circuit

I

t

• Voltage across inductor “LEADS” current.

– VL goes up and down just before I does.

t

VL

L

FrequencyReacta

nce

(XL) Frequency does

affect Reactance!

VL = -L(I)/(t) always true

• VL,max = ImaxXL

• Inductive Reactance: XL = 2fL

Physics 102: Lecture 12, Slide 9

ACT/Preflight 12.4, 12.5

The capacitor can be ignored when…(a) frequency is very large(b) frequency is very small

LR

C

The inductor can be ignored when…(a) frequency is very large(b) frequency is very small

Physics 102: Lecture 12, Slide 10

AC Circuit VoltagesAn AC circuit with R= 2 , C = 15 mF, and L = 30 mH

has a current I(t) = 0.5 sin(8t) amps. Calculate the maximum voltage across R, C, and L.

VR,max = Imax R = 0.5 2 = 1 Volt

= 0.5 1/(80.015) = 1.33 Volts

= 0.5 80.03 = 0.38 Volts

VC,max = Imax XC

VL,max = Imax XL

LR

C

Physics 102: Lecture 12, Slide 11

ACT: AC Circuit VoltagesAn AC circuit with R= 2 , C = 15 mF, and L = 30 mH

has a current I(t) = 0.5 sin(8t) amps. Calculate the maximum voltage across R, C, and L.

Now the frequency is increased so I(t) = 0.5 sin(16t). Which element’s maximum voltage decreases?

1) VR,max

2) VC,max

3) VL,max

LR

C

Physics 102: Lecture 12, Slide 12

Summary so far…L

R

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

VL

VC

VR

𝑋𝐶 = 12𝜋𝑓𝐶= 1𝜔𝐶

𝑋𝐿 = 2𝜋𝑓𝐿= 𝜔𝐿

Physics 102: Lecture 12, Slide 13

Kirchhoff: generator voltage

Write down Kirchhoff’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 12, Slide 14

A reminder about sines and cosines

Recall: y coordinates of endpoints are

• asin( + /2)

• asin()

• asin( – /2)

a a

a

x

y

Physics 102: Lecture 12, Slide 15

L

R

C

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

VL = ImaxXL sin(2ft + )

VR = ImaxR sin(2ft)

VC = ImaxXC sin(2ft – )

Graphical representation of voltages

ImaxXL

ImaxR

ImaxXC

Physics 102: Lecture 12, Slide 16

Phasor Diagrams: A Detailed Example

• I = Imaxsin(2ft)

• VR = VR,maxsin(2ft)

VR,maxsin()

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

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

V R,max

Physics 102: Lecture 12, Slide 17

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

Physics 102: Lecture 12, Slide 18

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

Physics 102: Lecture 12, Slide 19

Phasor Diagrams

VR,maxsin(4)

t = 42ft = 4/6

VR

,max

• I = Imaxsin(4/6)

• VR = VR,maxsin(4/6)

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

Physics 102: Lecture 12, Slide 20

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

Physics 102: Lecture 12, Slide 21

Phasor Diagrams

VR,maxsin(8)

t = 82ft = 8

VR

,max

• I = Imaxsin(8/6)

• VR = VR,maxsin(8/6)

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

Physics 102: Lecture 12, Slide 22

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

Physics 102: Lecture 12, Slide 23

AC circuit summary

LR

C

VR in phase with I

VC lags I

VL leads I

I

t

VL

VC

VR

Kirchoff’s Loop Equation always holds true:

Vgen = VL + VR + VC

However, Vgen,max VL,max+VR,max+VC,max

Maximum reached at different times for R, L, C

Phasors represent instantaneous voltages