Download pptx - Driven AC Circuits Phase of V and I Conceputally Mathematically With phasors Physics 2112 Unit 20 Outline: Electricity & Magnetism Lecture 20,

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)


Driving frequency = natural frequency (wo)

“Phase” between I and V

R

IR = VR/R

I= Vmax/R sin(wdt)

Amplitude = Vmax/R


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


)sin(max tVVdt

dIL L

)cos(max tL

VI

Amplitude = Vmax/XL

Phase Summary

L


)sin(max tR

VI dR

)cos(max tCVI d

)90sin(max od

C

tV

C

)cos(max tL

VI

)90sin(max o

L

tV

I “leads” V

V and I “in phase”

I “lags” V

“ELI the ICE man”

What does this look like together?


Notice phase relationships

http://courses.physics.illinois.edu/phys212/sp2013/simulations/Circuit/Circuit.html

What does this look like together?


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?


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


RR

CX C

1C

LX L

w goes up, Cc goes down

Doesn’t depend on w

w goes up, CL 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?


Think of same material graphically using “phasors”


Phasors

Phasor just thinks of sine wave as rotating vector

http://vnatsci.ltu.edu/s_schneider/physlets/main/phasor1.shtml

Imax XL

Imax XC

Imax R

VL and VC 180o out of phase


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


R

XX CL )(tan

Imax XL

Imax XC

Imax R

L

R

C

emax

f

R (XL - X

C )

22 )( CL XXRZ

VCmax = Imax XC

VLmax = Imax XL

VRmax = Imax R

Summary

Imax = emax / Z

emax = Imax Z


R

XX CL )(tan

22CL XXRZ

Imax XL

Imax XC

Imax R

L

R

C

emax

CheckPoint 1(A)


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)



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)



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)


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?


http://vnatsci.ltu.edu/s_schneider/physlets/main/rlc.shtml

Point of confusion??


VL + VC + VR + e = 0

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

Z

VI max

max Z

VI

(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)


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?


CheckPoint 2(C)


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?


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

R

LV

Example 20.3

Consider the harmonically driven series LCR circuit shown. Vmax = 100 V

Imax = 2 mA

VCmax = 113 V

The current leads generator voltage by 45o

L and R are unknown.

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


Get your

calcu

lators out