Course Updateshttp://www.phys.hawaii.edu/~varner/PHYS272-Spr10/physics272.html

Reminders:

1) Assignment #11 posted, due next Wednesday

2)Quiz # 5 today (Chap 29, 30)

3)Complete AC Circuits

Last time: LRC Circuits with phasors…

⇒

The phasor diagram gives us graphical solutions for φ and I:

LX L ω≡

where . . .

CX C ω

1≡

RXX CL −=φtan

( )( )2222CLm XXRI −+=ε

( ) IZXXRI CLm =−+= 22ε

⇓

( )22CL XXRZ −+≡

IR

I XL

I XC

εm

φεm

IR

I XL I XC

LC

∼

R

ε

φ

( ) ( )( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )φωεφωε

ωωω

ωω

ω

ωω

+=+==+=+=

−=−=

==

ttIZtvttLItIXtv

tC

ItIXtv

tIRtvtIti

mad

LL

CC

R

coscos)(90cos90cos

90cos190cos

coscos

LRC series circuit; Summary of instantaneousCurrent and voltages

( ) ( )22 - /1tan CLRR

CL XXXZR

CLV

VV+=

−=

−=

ωωφ

CC

LL

R

IXVIXVIRV

===

Lagging & LeadingThe phase φ between the current and the driving emf depends on the relative magnitudes of the inductive and capacitive reactances.

RXX CL −=φtan

ZI mε

=LX L ω≡

CX C ω

1≡

φR

XL

XC

Z

XL > XCφ > 0

current LAGS

applied voltage

φ R

XL

XC

Z

XL < XCφ < 0

current LEADS

applied voltage

XL = XCφ = 0

current IN PHASE WITH applied voltage

R

XL

XC

Z

Impedance, Z

“ Impedance Triangle” CL XXI −IZ

RIφ| |

( )22mL C

m

Z R X XIε

≡ = + −

• From the phasor diagram we found that the current amplitude I was related to the drive voltage amplitude εm by

m mI Zε =

• Z is known as the “impedance”, and is basically the frequency dependent equivalent resistance of the series LRC circuit, given by:

orcos( )

RZφ

=

• Note that Z achieves its minimum value (R) when φ = 0. Under this condition the maximum current flows in the circuit.

m

CL XX −Z

Rφ| |

Resonance• For fixed R, C, L the current I will be a maximum at the

resonant frequency ω which makes the impedance Zpurely resistive (Z = R). i.e.,

This condition is obtained when:

⇒C

Lω

ω 1=

LC1

=ω

• Note that this resonant frequency is identical to the natural frequency of the LC circuit by itself!

• At this frequency, the current and the driving voltage are in phase:

0tan =−

=R

XX CLφ

( )22CL

mmm

XXRZI

−+==

εε

reaches a maximum when:L CX X=

ResonancePlot the current versus ω, the frequency of the voltage source:

• For ω very large, XL >> XC, φ 90°, Im 0

• For ω very small, XC >> XL, φ -90°, Im 0

00 2ωω

Rmε

Im

Example: vary RV=100 vω=1000 rad/sR=200, 500, 2000 ohmL=2 HC=0.5 μC

Consider a general AC circuit containing a resistor, capacitor, and inductor, driven by an AC generator.

As the frequency of the circuit is either raised above or lowered below the resonant frequency, the impedance of the circuit ________________.a) always increases b) only increases for lowering the frequency below resonance c) only increases for raising the frequency above resonance

Question 1

Consider a general AC circuit containing a resistor, capacitor, and inductor, driven by an AC generator.

As the frequency of the circuit is either raised above or lowered below the resonant frequency, the impedance of the circuit ________________.a) always increases b) only increases for lowering the frequency below resonance c) only increases for raising the frequency above resonance

Question 1

Consider a general AC circuit containing a resistor, capacitor, and inductor, driven by an AC generator.

At the resonant frequency, which of the following is true?

a) The current leads the voltage across the generator. b) The current lags the voltage across the generator. c) The current is in phase with the voltage across the generator.

Question 2

Consider a general AC circuit containing a resistor, capacitor, and inductor, driven by an AC generator.

At the resonant frequency, which of the following is true?

a) The current leads the voltage across the generator. b) The current lags the voltage across the generator. c) The current is in phase with the voltage across the generator.

Question 2

Changing the frequency away from the resonant frequency will change both the reductive and capacitive reactance such that XL - XC is no longer 0. This, when squared, gives a positive term to the impedance, increasing its value. By definition, at the resonance frequency, Imax is at its greatest and the phase angle is 0, so the current is in phase with the voltage across the generator.

• Impedance = Z = sqrt( R2 + (XL-XC)2)• At resonance, (XL-XC) = 0, and the impedance has its minimum value: Z = R

• As frequency is changed from resonance, either up or down, (XL-XC) no longer is zero and Z must therefore increase.

Fill in the blank. This circuit is being driven __________ its resonance frequency.

a) above b) below c) exactly at

Question 3

Fill in the blank. This circuit is being driven __________ its resonance frequency.

a) above b) below c) exactly at

Question 3

The generator voltage ________________ the current.

a) leadsb) lagsc) is in phase with

Question 4

The generator voltage ________________ the current.

a) leadsb) lagsc) is in phase with

Question 4

• At resonance, XL = XC. • Here, XL > XC• Therefore, need to reduce XL=ωL and increase XC = 1/(ωC)• Therefore, lower ω !!

• From diagram, ε leads IR (rotation = ccw)

Power in LRC circuit

The instantaneous power delivered to L-R-C is:

We can use trig identities to expand the above to,

( ) ( ) ( ) ( ) ( )tItVtvtitP ad ωφω coscos +==

( ) ( ) ( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )φφω

φωωφω

φωωφω

ωφωφω

cos21coscos

sincossincoscos

sincossincoscoscossinsincoscos

2

2

2

⎟⎠⎞

⎜⎝⎛==

−==

−=

−=

VItVI

ttVItVItPP

ttVItVItIttVtP

ave

( ) ( ) ( ) ( )φωω +== tVtvttIti ad cos)(;cos

( ) ( ) ( )

( )φ

φφ

cos

cos22

cos21

RMSRMS

ave

IV

IVVItPP

=

===

Power in LRC circuit, continued

( ) ( ) ( )φφ coscos21

RMSRMSave IVVItPP ===

Example; 100Watt light bulb plugged into 120V house outlet,Pure resistive load (no L and no C), ϕ = 0.

AVP

I

PV

R

RV

VIP

rms

averms

ave

rms

rmsrmsrms

83.0120100

144100

12022

2

===

Ω===

==

General result. VRMS is voltage across element, I RMS is currentthrough element, and ϕ is phase angle between them.

Question: What is PAVE for an inductor or capacitor?

Note; 120V house voltage is rms and has peak voltage of 120 √2 = 170V

If you wanted to increase the power delivered to this RLC circuit, which modification(s) would work?

a) increase R

b) increase C

c) increase C,L

d) decrease R

e) decrease C,L

Question 5

If you wanted to increase the power delivered to this RLC circuit, which modification(s) would work?

a) increase R

b) increase C or L

c) increase C,L

d) decrease R

e) decrease C,L

Question 5

Would using a larger resistor increase the current?

a) yes b) no

Question 6

Would using a larger resistor increase the current?

a) yes b) no

Question 6

Since power peaks at the resonant frequency, try to get XL and XC to be equal. Power also depends inversely on R, so decrease R to increase Power.

• Power ~ Icosφ ~ (1/Z)(R/Z) = (R/Z2)

• To increase power, want Z to decrease:• L: decrease XL ⇒ decrease L• C: increase XC ⇒ decrease C• R: decrease Z ⇒ decrease R

Summary• Power

• Driven Series LRC Circuit:• Resonance condition

• Resonant frequency

LC1

=ω

( )22CL XXRZ −+≡

mrms εε2

1≡

mrms II2

1≡

( )2

( ) cosrms rms

rms

P t I

I R

ε φ⟨ ⟩ =

=

“power factor”{

RXX CL −=φtan

For next time

• Homework #11 posted, due next Wed.

• Quiz now: Faraday’s Law, Inductance and Inductors, RLC circuits