Physics 2112Unit 21

Resonance Power/Power Factor Q factor

What is means A useful approximation

Transformers

Outline:

From last time…..

XL

XC

R

“Ohms Law” for each element

ZIVgen max

RIVresistor max

Linductor XIV max

Ccapacitor XIV max Z

VI genmax

22CL XXRZ

LX L

CXC

1

where:(impedance)

(inductive reactance)

(capacitive reactance)

w of generator

L

R

C

This means…..

L

R

C22max)( CL

gen

XXR

VI

LC

Current is largest when

CL XX LC

d

1or

Where have we seen this before?

LCo

1 (Natural Frequency)

When wd = wo resonance

Resonance

R is independent of w

Frequency at which voltage across inductor and capacitor cancel

Resonance in AC Circuits

frequency

Impe

danc

e

RXC

Z

XL

w0

Z = R at resonance

XL increases with wLLX

XC increases with 1/w

CXC

1

is minimum at resonance

22 )( CL XXRZ

Resonance: XL = XC LC

10

Example 21.1 (Tuning Radio)

L

R

CA radio antenna is hooked to signal filter consisting of a resistor, a variable capacitor and a 50uH inductor.

If we would like to get maximum current for the signal from our favorite country music station, US 99 (99.5MHz), what should we set the capacitor to?

amplifier

Example 21.2 (Peak Current)

A generator with peak voltage 15 volts and angular frequency 25 rad/sec is connected in series with an 8 Henry inductor, a 0.4 mF capacitor and a 50 ohm resistor. What is the peak current through the circuit?

L

R

C

Imax XL

Imax XC

Imax R

Case 1

Imax XL

Imax XC

Imax R

Case 2

Resonance: XL = XC

Z = R

Same since R doesn't change

CheckPoint 1(A)

Consider two RLC circuits with identical generators and resistors. Both circuits are driven at the resonant frequency. Circuit II has twice the inductance and 1/2 the capacitance of circuit I as shown above.

Compare the peak voltage across the resistor in the two circuits.

A. VI > VII B. VI = VII C. VI < VII

Imax XL

Imax XC

Imax R

Case 1

Imax XL

Imax XC

Imax R

Case 2

CheckPoint 1(B)

Consider two RLC circuits with identical generators and resistors. Both circuits are driven at the resonant frequency. Circuit II has twice the inductance and 1/2 the capacitance of circuit I as shown above.

Compare the peak voltage across the inductor in the two circuits

A. VI > VII B. VI = VII C. VI < VII

Voltage in second circuit will be twice that of the first because of the 2L compared to L.

Imax XL

Imax XC

Imax R

Case 1

Imax XL

Imax XC

Imax R

Case 2

CheckPoint 1(C)

Consider two RLC circuits with identical generators and resistors. Both circuits are driven at the resonant frequency. Circuit II has twice the inductance and 1/2 the capacitance of circuit I as shown above.

Compare the peak voltage across the capacitor in the two circuits

A. VI > VII B. VI = VII C. VI < VII

The peak voltage will be greater in circuit 2 because the value of XC doubles.

Imax XL

Imax XC

Imax R

Case 1

Imax XL

Imax XC

Imax R

Case 2

CheckPoint 1(D)

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

“

Resonance….what it means…..

frequency of the signal generator is the same as the natural frequency of the circuit.

Imax = e/RVoltage drop across the resistor is maximumXL = XC

Z has minimum valuephase angle, f, is zerovoltage and current are in sync at the signal

generator.

A circuit is in resonance when….

Power

What is the power put into circuit by signal?

L

R

C

At any instant P=VI varies with time.

But V and I not in sync at genarator

Power

PIN = POUT

L

R

C

Changes electrical energy into thermal

energy

RMS = Root Mean Square2/peakrms II

RIRIP peakrmsAVG )2/( 22

cosrmsrmsAVG VIP

cosrmsrms

rms

VR

Z

VRI

Example 21.3 (Power lost)

How much electrical energy was turned into thermal energy every minute in the three situations we had in example 20.2?

Recall w=60rad/sec f =-72o

w=400rad/sec f=55.7o

w =206rad/sec f = 0o

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

Example 21.4

A bright electric light bulb and a small window air conditioner both are plugged into a wall socket with puts out Vrms=120V. Both draw Irms= 1amp of current. The light bulb has a power factor of 1 and the AC unit has a power factor of 0.85.

How much electrical energy do they consume in 1 hour?

Power Line Calculation

If you want to deliver 1,500 Watts at 100 Volts over transmission lines w/ resistance of 5 Ohms. How much power is lost in the lines?

Current Delivered: I = P/V = 15 Amps Loss = IV (on line) = I2 R = 15*15 * 5 = 1,125 Watts!

If you deliver 1,500 Watts at 10,000 Volts over the same transmission lines. How much power is lost?

Current Delivered: I = P/V = .15 Amps Loss = IV (on line) = I

2R = 0.125 Watts

Lower, but not zero!!!

Cost to you

Utility companies charge a surcharge for industrial customers with large power factors (e.g. large AC loads).

Customers can correct power factor with capacitors.

Assumed power factor for residential > 0.95.

Warning!!!!

Electricity & Magnetism Lecture 21, Slide 18

About to talk about “Quality Factor”, Q

Just to confuse you….we’re also going to be talking about the charge on a capacitor, Q

We’ll go through the math quickly….

Hang in there. We’ll show what it means conceptually and do an example problem in the end.

Recall Damped Harmonic Motion

Unit 19, Slide 19

)'cos( teQQ tMax

L

R

2

222' o

Damping factor

Damped oscillation frequency

LC

1

Natural oscillation frequency

What does Q mean algebraically….

Unit 19, Slide 20

R

L

LCo

1

Energy Charge2

)//(2)( RLtt ee Energy Lost

orms

rms

RI

LIQ

2

2

Define Quality Factor oQ

RI

fLI

rms

orms2

2 )2(

Q = 2p

(Maximum energy Stored)

(Energy Lost per Cycle)

Second way to look at Q

Unit 19, Slide 21

How far you are from resonance

o

x

22max)( CL

gen

XXR

VI

2

2

22 )1(

1

1

xx

QR

Vgen

How fast you fall away from current at resonance

Bigger Q, fall off faster

What does Q mean graphically…..

Q=0.5

Q=5.0 Q=20.0

x=w/wo

x=w/wo

Q=1.0

x=w/wo

x=w/wo

I ma

x/(e

/R)

I ma

x/(e

/R)

I ma

x/(e

/R)

I ma

x/(e

/R)

Useful approximation from Q

Often used approximation for Q>2

FWHMQ o

Q=5.0

“Full Width at Half Maximum”

Example 21.5 (FWHM)

Electricity & Magnetism Lecture 21, Slide 24

An electronic filter is made up of an inductor, L=0.5H, a capacitor, C=2.0uF and resistor, R=20W. It has an input signal with Vmax = 0.1V.

1) What is the resonant frequency for this filter?

2) What is the average output power at this frequency?

3) What is the quality factor Q for this filter?

4) Use this Q to estimate FWHM.

5) What is the average power output at a frequency of wo + FWHM/2?

Transformers

DVP DIP

DBP

DFS

DVS

DIS

DBS

DFP

Primary (P) Secondary (S)

Difficult to calculate step by

step

Transformers

looploopSP

S

S

P

P

NN

PP

SS N

N

PS

PSPS I

N

NIPP

Note by energy

conservation

Example 21.6 (Coil in your car)

In the old days, the high voltage needed to fire the spark plugs in your car was created by “the coil”. On a typical coil, there were 200 turns the primary side and 400,000 turns on the secondary side.

If the car’s alternator provided 12 volts to the coil, what was the voltage provided to the spark plugs?

Uniy 21, Slide 27