1. Physics 2112 Unit 20 Outline: Driven AC Circuits Phase of V and I
Conceputally
Mathematically
With phasors

2. e = Vmaxsin(wdt) Driving frequency = natural frequency (wo)
AC Generator
e = Vmaxsin(wdt)
Driving frequency
= natural frequency (wo)

3. IR = VR/R “Phase” between I and V Simple Case - Resistors
Voltage goes up  current goes up
“In phase”  Phase angle = 0o
I= Vmax/R sin(wdt)
Amplitude = Vmax/R

4. Q = CV = CVmaxsin(wt)  I = VmaxwC cos(wt) Capacitors C
Amplitude = Vmax/XC
90o
where XC = 1/wC
is like the “resistance” of the capacitor XC depends on w

5. Inductors L Amplitude = Vmax/XL where XL = wL
is like the “resistance” of the inductor XL depends on w

6. V and I “in phase” I “leads” V I “lags” V Phase Summary R C L
“ELI the ICE man”

7. What does this look like together?
Notice phase relationships

8. 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

9. 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.)

10. Reactance Summary Doesn’t depend on w w goes up, Cc goes downL
w goes up, CL goes up

11. Example 20.1 (Inductor Reactance)
A 60Hz signal with a Vmax = 5V is sent through a 50mH inductor. What is the maximum current, Imax, through the inductor?

12. Think of same material graphically using “phasors”
Phasor just thinks of sine wave as rotating vector

13. Remember VR and I in phase
Circuit using Phasors
Represent voltage drops across elements as rotating vectors (phasors)
Imax XL
Imax XC
Imax R
VL and VC 180o out of phase
VL and VR 90o out of phase
Remember VR and I in phase

14. emax = Imax Z emax Make this Simpler f C L R Imax XC Imax XL
Imax R L R C
emax
Imax R
emax = Imax Z
Imax(XL - XC) f R
(XL - XC)
Impedance Triangle

15. emax emax = Imax Z Summary f VCmax = Imax XC C VLmax = Imax XL L
Imax R L R C
emax
VCmax = Imax XC
VLmax = Imax XL
VRmax = Imax R
emax = Imax Z
Imax = emax / Z f R
(XL - XC)

16. 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.
always out of phase
always in phase
sometimes in phase and sometimes out of phase

17. CheckPoint 1(B)
A RL circuit is driven by an AC generator as shown in the figure.
The voltages across the resistor and inductor are.
always out of phase
always in phase
sometimes in phase and sometimes out of phase

18. CheckPoint 1(C)
A RL circuit is driven by an AC generator as shown in the figure.
The phase difference between the CURRENT through the resistor and inductor
is always zero
is always 90o
depends on the value of L and R
depends on L, R and the generator voltage

19. In the circuit to the right
Example 20.2 (LCR)
In the circuit to the right
L=500mH
Vmax = 6V
C=47uF
R=100W V C L R
What is the maximum current and phase angle if w = 60rad/sec?
What is the maximum current and phase angle if w = 400 rad/sec?
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.
What is the maximum current and phase angle if w = 206 rad/sec?

20. What does this look like graphically?
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.

21. VL-max + VC-max + VR-max + e = 0
Point of confusion??
VL + VC + VR + e = 0
VL-max + VC-max + VR-max + e = 0
(Add like vectors)
(Imax and Vmax happen at different times.)

22. 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?
VL = 0
VL = VL-max/2
VL = VL=max

23. 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?
VL = 0
VL = VL-max/2
VL = VL=max

24. 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?
VL = 0
VL = VL-max/2
VL = VL=max

25. Get your calculators out
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? ~ C R L V
Conceptual Analysis
The 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
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.
Strategic Analysis
Use Vmax and Imax to determine Z
Use impedance triangle to determine R
Use VCmax and impedance triangle to determine XL
Get your calculators out