Physics 2112 Unit 21  Resonance  Power/Power Factor  Q factor  What is means  A useful approximation  Transformers Outline:

From last time….. XLXL XCXC R “Ohms Law” for each element where: (impedance) (inductive reactance) (capacitive reactance)  of generator L R C

This means….. L R C L C Current is largest when or Where have we seen this before? (Natural Frequency) When  d =  o  resonance

Resonance R is independent of  Frequency at which voltage across inductor and capacitor cancel R XCXC Z XLXL 00 Z = R at resonance X L increases with  X C increases with 1  is minimum at resonance Resonance: X L  X C

Example 21.1 (Tuning Radio) L R C A 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

I max X L I max X C I max R Case 1 I max X L I max X C I max R Case 2 Resonance: X L  X C 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.V I > V II B.V I = V II C.V I < V II

I max X L I max X C I max R Case 1 I max X L I max X C I max 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.V I > V II B.V I = V II C.V I < V II Voltage in second circuit will be twice that of the first because of the 2L compared to L.

I max X L I max X C I max R Case 1 I max X L I max X C I max 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.V I > V II B.V I = V II C.V I < V II The peak voltage will be greater in circuit 2 because the value of X C doubles.

I max X L I max X C I max R Case 1 I max X L I max X C I max 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.  I max =  /R  Voltage drop across the resistor is maximum  X L = X C  Z has minimum value  phase angle, , is zero  voltage 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 P IN = P OUT L R C Changes electrical energy into thermal energy RMS  Root Mean Square

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  =60rad/sec  =-72 o  =400rad/sec  =55.7 o  =206rad/sec  = 0 o In the circuit to the right L=500mH V max = 6V C=47uF R=100 

Example 21.4 A bright electric light bulb and a small window air conditioner both are plugged into a wall socket with puts out V rms =120V. Both draw I rms = 1amp of current. The light bulb has a power factor of 1 and the AC unit has a power factor of 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)  I 2 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 2 R  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 Damping factor Damped oscillation frequency Natural oscillation frequency

What does Q mean algebraically…. Unit 19, Slide 20 Energy Charge 2 Energy Lost Define Quality Factor Q = 2  (Maximum energy Stored) (Energy Lost per Cycle)

Second way to look at Q Unit 19, Slide 21 How far you are from resonance 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=  /  o Q=1.0 x=  /  o I max /(  /R)

Useful approximation from Q Often used approximation for Q>2 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=20 . It has an input signal with V max = 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  o + FWHM/2?

Transformers VP  IPVP  IP  BP BP   S  VS VS  IS IS  BS BS   P Primary (P) Secondary (S) Difficult to calculate step by step

Transformers 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