1. Physics 212
Lecture 21
Resonance and power in AC circuits

2. emax emax = Imax Z f C L R Imax XL Imax XC Imax R Imax XL Imax XCw
XL = wL
XC = 1/wC
Imax R
emax = Imax Z
Imax(XL-XC) R f
XL-XC

3. Peak AC Problems C L Problem R “Ohms” Law for each element07 L R C
“Ohms” Law for each element
PEAK values
Vgen = Imax Z
VResistor = Imax R
Vinductor = Imax XL
VCapacitor = Imax XC
Problem
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? XL XC R

4. Peak AC Problems Typical Problem “Ohms” Law for each element12 L R C
“Ohms” Law for each element
Vgen = I Z
VResistor = I R
Vinductor = I XL
VCapacitor = I XC
Typical Problem
A generator with peak voltage 15 volts and angular frequency 25 rad/sec is connected in series with an 8 Henry inductor, a Farad capacitor and a 50 ohm resistor. What is the peak current through the circuit? XL XC R
Which element has the largest peak voltage across it?
A) Generator E) All the same.
B) Inductor
C) Resistor
D) Capacitor

5. Peak AC Problems Typical Problem “Ohms” Law for each element12 L R C
“Ohms” Law for each element
Vgen = I Z
VResistor = I R
Vinductor = I XL
VCapacitor = I XC
Typical Problem
A generator with peak voltage 15 volts and angular frequency 25 rad/sec is connected in series with an 8 Henry inductor, a Farad capacitor and a 50 ohm resistor. What is the peak current through the circuit? XL XC R
Which element has the largest peak voltage across it?
A) Generator E) All the same.
B) Inductor
C) Resistor
D) Capacitor

6. Peak AC Problems Typical Problem “Ohms” Law for each element14 L R C
“Ohms” Law for each element
Vgen = I Z
VResistor = I R
Vinductor = I XL
VCapacitor = I XC
Typical Problem
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? XL XC R XL XC R
What happens to the impedance if we decrease the angular frequency to 20 rad/sec?
A) Z increases
B) Z remains the same
C) Z decreases
Z25
Z20
(XL-XC): ( )  ( )

7. Resonance
Light-bulb Demo

8. Resonance Resonance: XL = XC
Frequency at which voltage across inductor and capacitor cancel R XC Z XL w0
Z = R at resonance
R is independent of w
XL increases with w
XC increases with 1/w
is minimum at resonance
To my surprise many people got this preflight question wrong (61% correct)
Resonance: XL = XC 10

9. Same since R doesn't change
Checkpoint 1a
Imax XL
Imax XC
Imax R
Case 2
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
Resonance: XL = XC
Z = R
Same since R doesn't change

10. Checkpoint 1b Imax XL Imax XL Imax R Imax R Imax XC Case 1 Case 2
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
Imax XL
Imax XC
Imax R
Case 1
Light Demo here
Voltage in second circuit will be twice that of the first because of the 2L compared to L

11. Checkpoint 1c Imax XL Imax XL Imax R Imax R Imax XC Case 1 Case 2
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
Imax XL
Imax XC
Imax R
Case 1
Light Demo here
The peak voltage will be greater in circuit 2 because the value of XC doubles.

12. Checkpoint 1d Imax XL Imax XC Imax R Case 2 Imax XL Imax XC Imax R
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.
At the resonant frequency, which of the following is true?
A. Current leads voltage across the generator
B. Current lags voltage across the generator C. Current is in phase with voltage across the generator
Imax XL
Imax XC
Imax R
Case 1
Light Demo here
The voltage across the inductor and the capacitor are equal when at resonant frequency, so there is no lag or lead.

13. Umax = max energy stored in one cycle at resonance
Quality factor Q Z
In general
Umax = max energy stored
DU = energy dissipated
in one cycle at resonance

14. Larger Q, sharper resonance

15. X X Series LCR circuit ~ (A) (B) (C) (D)V ~
1. The current leads generator voltage by 45o.
2. Assume V = Vmaxsinwt
What does the phasor diagram look like at t = 0?
(A) (B) (C) (D) VL VR VC V X X
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.
V = Vmax sinwt  V is horizontal at t = 0 (V = 0)
VL < VC if current leads generator voltage

16. How should we change w to bring circuit to resonance?
Series LCR circuit C R L V ~
The current leads generator voltage by 45o
How should we change w to bring circuit to resonance?
(A) decrease w (B) increase w (C) Not enough info VL VR VC V
At resonance (w0)
Original w VL VR VC V f
At resonance XL = XC
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.
XL increases
XC decreases
Increase w

17. By what factor should we increase w to bring circuit to resonance w0 ?
Series LCR circuit C R L V ~
1. The current leads generator voltage by 45o.
2. Suppose XC = 2XL at frequency .
By what factor should we increase w
to bring circuit to resonance w0 ?
(A)
(B)
(C)
(D)
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. f

18. If we change , what is the maximum current that can flow?
Series LCR circuit C R L V ~
Consider the harmonically driven series LCR circuit shown.
At some frequency , XC = 2XL = R
Vmax = 100 V
R = 50 k
If we change , what is the maximum current that can flow?
(A)
(B)
(C)
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.
At resonance XL = XC

19. Instantaneous power P(t) = I(t)V(t) C L R
P(t) varies in time. We want the average power.
Average, over one cycle, of any periodic function :
Circuit element:
For harmonically-varying functions like sin t or cos t , T = 2π/ω .

20. Define root mean square (RMS)
For L or C, voltage and current are /2 out of phase:
I  sin t, V  cost t
Average power for R is not zero since V and I are in phase. R f
XL-XC Z
Define root mean square (RMS)

21. f Average power dissipated in R for LCR circuit Z XL-XC R
Maximum power delivered when  = 0 … at resonance !
Imax 0L
Imax / 0C
Imax R = max
At resonance,
0L = 1/0C
 = 0

22. Power Transmission
If you want to deliver 1500 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 from power line = I2 R = 15*15 * 5 = 1,125 Watts!
If you deliver 1500 Watts at 10,000 Volts over the same transmission lines. How much power is lost?
Current Delivered: I = P/V = 0.15 Amps
Loss from power line = I2R = 0.15*0.15*5 = Watts
Volts = 1,125 lost in lines
K Volts = .125 Lost in lines
DEMO

23. Flux per turn = B (Area) = B
How do we transform 10,000 V into 120 V for your toaster ?
Transformer
Time-varying voltage in primary causes
time-varying B in iron-core.
Flux per turn = B (Area) = B
Iron core maintains same B throughout.
7200 V / 240 V
Primary coil captures flux P = NP B
Secondary coil captures flux S = NS B
Faraday’s Law:
VS = d S /dt = NS B
VP = d P /dt = NP B
Physics 212 Lecture 21, Slide 23 23