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Physics 2112 Unit 20

Physics 2112 Unit 20. Outline:. Driven AC Circuits Phase of V and I Conceputally Mathematically With p hasors. AC Generator . e = V max sin ( w d t ). Driving frequency = natural frequency ( w o ). “Phase” between I and V. Amplitude = V max / R. Simple Case - Resistors.

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Physics 2112 Unit 20

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

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

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

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

  5. Inductors L 90o Amplitude=Vmax/XL where XL=wL is like the “resistance”of the inductor XLdepends on w

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

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

  8. What does this look like together? CapacitorandInductoralways 180o out of phase Capacitor/Inductorand 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 acrossResistor Current and voltage out of phase by unknown phase angle across signal generator (We’ll find this “phase angle” later.)

  10. Reactance Summary R Doesn’t depend on w w goes up, Cc goes down C L w goes up, CL goes up

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

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

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

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

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

  16. CheckPoint1(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. CheckPoint1(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. CheckPoint1(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. 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? What is the maximum current and phase angle if w = 206 rad/sec?

  20. What does this look like graphically?

  21. 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. CheckPoint2(A) A driven RLC circuit is represented by the phasordiagram 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. CheckPoint2(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. CheckPoint2(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. Example 20.3 C V ~ L R 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? • 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 • 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

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