Driven AC Circuits Phase of V and I

Conceputally Mathematically With phasors

Physics 2112Unit 20

Outline:

Electricity & Magnetism Lecture 20, Slide 1

AC Generator

e = Vmaxsin(wdt)

Electricity & Magnetism Lecture 20, Slide 2

Driving frequency = natural frequency (wo)

“Phase” between I and V

RIR = VR/R

I= Vmax/R sin(wdt)

Amplitude = Vmax/R

Electricity & Magnetism Lecture 20, Slide 3

Voltage goes up current goes up

Simple Case - Resistors

“In phase” Phase angle = 0o

Capacitors

C

I = VmaxwC cos(wt)

where XC = 1/wCis like the “resistance”of the capacitor XC depends on w

Amplitude = Vmax/XC

Q = CV = CVmaxsin(wt)

90o

Unit 20, Slide 4

Inductors

L

where XL = wLis like the “resistance”of the inductor XL depends on w

90o

Electricity & Magnetism Lecture 20, Slide 5

)sin(max tVVdtdIL L w==

)cos(max tL

VI ww

=

Amplitude = Vmax/XL

Phase Summary

L

Electricity & Magnetism Lecture 20, Slide 6

)sin(max tR

VI dw=R

)cos(max tCVI dww=

)90sin(max od

C

tV

= wC

)cos(max tL

VI ww

=

)90sin(max o

L

tV

= w

I “leads” V

V and I “in phase”

I “lags” V

“ELI the ICE man”

What does this look like together?

Electricity & Magnetism Lecture 20, Slide 7

Notice phase relationships

What does this look like together?

Electricity & Magnetism Lecture 20, Slide 8

Capacitor and Inductor always 180o out of phase

Capacitor/Inductor and Resistor always 90o out of phase

Resistor is some unknown phase angle out of phase is signal generator

What about current?

Electricity & Magnetism Lecture 20, Slide 9

Current is always the same through all elements (in series)

Current and Voltage in phase across Resistor

Current and voltage out of phase by unknown phase angle across signal generator

(We’ll find this “phase angle” later.)

Reactance Summary

L

Electricity & Magnetism Lecture 20, Slide 10

RR

CX C w

1=C

wLX L =

w goes up, c goes down

Doesn’t depend on w

w goes up, L goes up

Example 20.1 (Inductor Reactance)

L

A 60Hz signal with a Vmax = 5V is sent through a 50mH inductor. What is the maximum current, Imax, through the inductor?

Electricity & Magnetism Lecture 20, Slide 11

Think of same material graphically using “phasors”

Electricity & Magnetism Lecture 20, Slide 12

Phasors

Phasor just thinks of sine wave as rotating vector

Imax XL

Imax XC

Imax R

VL and VC 180o out of phase

Electricity & Magnetism Lecture 20, Slide 13

Circuit using Phasors

Represent voltage drops across elements as rotating vectors (phasors)

VL and VR 90o out of phase

Remember VR and I in phase

Imax R

emax = Imax Z

Imax(XL XC)

R(X

L XC )

f

f

Impedance Triangle

Make this Simpler

22 )( CL XXRZ =

Electricity & Magnetism Lecture 20, Slide 14

RXX CL =)(tan f

Imax XL

Imax XC

Imax R

L

R

C

emax

fR (X

L XC )

22 )( CL XXRZ =

VCmax = Imax XC

VLmax = Imax XL

VRmax = Imax R

Summary

Imax = emax / Z

emax = Imax Z

Electricity & Magnetism Lecture 20, Slide 15

RXX CL =)(tan f

22CL XXRZ =

Imax XL

Imax XC

Imax R

L

R

C

emax

CheckPoint 1(A)

Electricity & Magnetism Lecture 20, Slide 16

A RL circuit is driven by an AC generator as shown in the figure.

The voltages across the resistor and generator are.

A. always out of phase B. always in phase C. sometimes in phase and sometimes

out of phase

CheckPoint 1(B)

Electricity & Magnetism Lecture 20, Slide 17

A RL circuit is driven by an AC generator as shown in the figure.

The voltages across the resistor and inductor are.

A. always out of phase B. always in phase C. sometimes in phase and sometimes

out of phase

CheckPoint 1(C)

Electricity & Magnetism Lecture 20, Slide 18

A RL circuit is driven by an AC generator as shown in the figure.

The phase difference between the CURRENT through the resistor and inductor

A. is always zero B. is always 90o C. depends on the value of L and R D. depends on L, R and the

generator voltage

Example 20.2 (LCR)

Electricity & Magnetism Lecture 20, Slide 19

C

R

V

In the circuit to the right• L=500mH• Vmax = 6V• C=47uF• R=100W

L

What is the maximum current and phase angle if w = 60rad/sec?

What is the maximum current and phase angle if w = 206 rad/sec?

What is the maximum current and phase angle if w = 400 rad/sec?

What does this look like graphically?

Electricity & Magnetism Lecture 20, Slide 20

Point of confusion??

Electricity & Magnetism Lecture 20, Slide 21

VL + VC + VR + e = 0

VL-max + VC-max + VR-max + e = 0

ZVI max

max = ZV

I

(Add like vectors)

(Imax and Vmax happen at different times.)

CheckPoint 2(A)

A driven RLC circuit is represented by the phasor diagram to the right.

The vertical axis of the phasor diagram represents voltage. When the current through the circuit is maximum, what is the potential difference across the inductor?

A. VL = 0 B. VL = VL-max/2 C. VL = VL=max

CheckPoint 2(B)

Electricity & Magnetism Lecture 20, Slide 23

A driven RLC circuit is represented by the above phasor diagram.When the capacitor is fully charged, what is the magnitude of the voltage across the inductor?

A. VL = 0 B. VL = VL-max/2 C. VL = VL=max

CheckPoint 2(C)

Electricity & Magnetism Lecture 20, Slide 24

A driven RLC circuit is represented by the above phasor diagram.

When the voltage across the capacitor is at its positive maximum, VC = +VC-max, what is the magnitude of the voltage across the inductor?

A. VL = 0 B. VL = VL-max/2 C. VL = VL=max

Conceptual AnalysisThe maximum voltage for each component is related to its reactance and to the

maximum current.The impedance triangle determines the relationship between the maximum

voltages for the components

Strategic AnalysisUse Vmax and Imax to determine ZUse impedance triangle to determine R Use VCmax and impedance triangle to determine XL

~

C

RLV

Example 20.3Consider the harmonically driven series LCR circuit shown. Vmax = 100 VImax = 2 mAVCmax = 113 VThe current leads generator voltage by 45o

L and R are unknown.

What is XL, the reactance of the inductor, at this frequency?

Electricity & Magnetism Lecture 20, Slide 25

Get your

calculators out