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Unformatted text preview: Physics 212
Lecture 20
AC Circuits
Maximum currents & voltages Phasors: A Simple Tool
40 30 20 10 0 Confused Avg = 2.6 Confident Physics 212 Lecture 20, Slide 1 Physics Resistors ε = εmaxsin(ωt) R I = VR/R = Vmax/R sin(ωt) Amplitude = Vmax/R Physics 212 Lecture 20, Slide 2 Physics Capacitors
Q = VC = C I = dQ/dt εmaxsin(ωt) ε = εmaxsin(ωt) C I = VmaxωC cos(ωt) Amplitude = Vmax/XC where XC = 1/ωC is like the “resistance” of the capacitor XC depends on ω Physics 212 Lecture 20, Slide 3 Physics Inductors
dI/dt = VL = εmaxsin(ωt) ε = εmaxsin(ωt) L I =  Vmax/ωL cos(ωt) Amplitude = Vmax/XL where XL = ωL is like the “resistance” of the inductor XL depends on ω Physics 212 Lecture 20, Slide 4 Physics RL ACT
An RL circuit is driven by an AC generator as shown in the figure.
BB L Imax = Vmax/XL XL = ωL R
For what driving frequency ω of the generator will the current through the resistor be largest A) ω large B) Current through R doesn’t depend on ω C) ω small
Physics 212 Lecture 20, Slide 5 Physics Summary
R
Imax = Vmax/R V in phase with I
Because resistors are simple C Imax = Vmax/XC XC = 1/ωC V 90o behind I
Current comes first since it charges capacitor Like a wire at high ω L Imax = Vmax/XL XL = ωL Opposite of capacitor Like a wire at low ω
Physics 212 Lecture 20, Slide 6 Physics V 90o ahead of I Makes sense to write everything in terms of I since this is the same everywhere in a oneloop circuit: Vmax = Imax XC V 90o behind I Phasors make this simple to see Imax XL C εmax L R Vmax = Imax XL V 90o ahead of I Imax XC Imax R Vmax = Imax R V in phase with I Always looks the same. Only the lengths will change
Physics 212 Lecture 20, Slide 7 Physics Prelecture animation The voltages still add up
C
But now we are adding vectors: Imax XL Imax XC εmax
Imax XL L R Imax XL Imax R εmax
Imax R Imax R Imax XL Imax R Imax XC Imax XC Imax XC εmax
Physics 212 Lecture 20, Slide 8 Physics Making this simpler…
C Imax XC εmax
Imax XL Imax XL L R Imax XL Imax R εmax
Imax R Imax R Imax XC Imax XC
Physics 212 Lecture 20, Slide 9 Physics Making this simpler…
C Imax XC εmax
Imax XL L R Imax XL Imax R εmax = Imax Z
Imax R Imax(XLXC) Imax R Imax XC
Physics 212 Lecture 20, Slide 10 Physics Making this simpler…
C Imax XC εmax L R Imax XL Imax R εmax = Imax Z
Imax(XLXC) Imax R Physics 212 Lecture 20, Slide 11 Physics Making this simpler…
C Imax XC εmax εmax = Imax Z
φ L R Imax XL Imax(XLXC) Imax R Imax R φ
Impedance Triangle R (XLXC) X L − XC tan (φ ) = R
Physics 212 Lecture 20, Slide 12 Physics Summary:
VCmax= Imax XC VLmax= Imax XL VRmax= Imax R C Imax XC εmax L R Imax XL εmax = Imax Z
Imax = εmax / Z
Z = R + ( X L − XC )
2 2 Imax R φ
R (XLXC) X L − XC tan (φ ) = R Physics 212 Lecture 20, Slide 13 Physics Example: RL Circuit Xc=0 L R
Imax R Imax XL εmax
Imax XL εmax
Imax R Physics 212 Lecture 20, Slide 14 Physics Preflight 2
BB Draw Voltage Phasors Imax XL εmax
Imax R A B C “Since VR and VL are out of phase from each other and the voltage of the generator is the vector sum of the voltages, VR and Vgenerator are out of phase. “ “The presence of the inductor has nothing to do with the fact that Vg and Vr will ALWAYS be in phase. “ 60 50 40 30 20 10 0 Physics 212 Lecture 20, Slide 15 Physics Preflight 4
Imax XL BB Draw Voltage Phasors εmax
Imax R A B C 70 60 50 40 30 20 10 0 Physics 212 Lecture 20, Slide 16 Physics Preflight 6
Imax XL BB The CURRENT is THE CURRENT εmax
Imax R φ
A B C D “current is current NUFF SAID” “The resistor will be in phase with the current, whereas the inductor will lead the current by 90 degrees.” φ is the phase between generator and current
50 40 30 20 10 Physics 212 Lecture 20, Slide 17 Physics 0 Preflight 8
BB A B C What does the voltage phasor diagram look like when the current IXL is a maximum? IXL ε IR ε
IR IXc IXc 50 40 30 20 10 0 Physics 212 Lecture 20, Slide 18 Physics Preflight 10
BB A B C IXL IXc ε
IR
50 40 What does the voltage phasor diagram look like when the capacitor is fully charged? IR ε IXc 30 20 10 IXL Physics 212 Lecture 20, Slide 19 Physics 0 Preflight 12
BB A B C IXL IXc ε
IR
60 50 IXL What does the voltage phasor diagram look like when the voltage across capacitor is at its positive maximum? IR ε IXc 40 30 20 10 Physics 212 Lecture 20, Slide 20 Physics 0 Calculation
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? V~ C L R • 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
Physics 212 Lecture 20, Slide 21 Physics Calculation
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? V~ C L R Compare XL and XC at this frequency: (A) XL < XC (B) XL = XC (C) XL > XC
– (D) Not enough information BB • This information is determined from the phase
Current leads voltage IR
45ο VL VL = ImaxXL VC = ImaxXC
VC VR (phase of current) V leads V Physics 212 Lecture 20, Slide 22 Physics Calculation
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? V~ C L R What is Z, the total impedance of the circuit? (A) 70.7 kΩ (B) 50 kΩ (C) 35.4 kΩ BB (D) 21.1 kΩ Vmax 100V = = 50k Ω Z= I max 2mA Physics 212 Lecture 20, Slide 23 Physics Calculation
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? V~ C L R Z = 50kΩ sin(45)=.707 cos(45)=.707 What is R? (A) 70.7 kΩ (B) 50 kΩ (C) 35.4 kΩ (D) 21.1 kΩ • Determined from impedance triangle R
45ο 50k Ω Z= (XCXL) R cos(45) = Z R = Z cos(45)
= 50k Ω (.707 )
= 35.4 kΩ BB Physics 212 Lecture 20, Slide 24 Physics Calculation
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? V~ C L R Z = 50kΩ R = 35.4kΩ (A) 70.7 kΩ (B) 50 kΩ (C) 35.4 kΩ (D) 21.1 kΩ We start with the impedance triangle: R XC − X L = tan 45° = 1 R X L = XC − R
What is XC ? BB 45ο Z (XCXL) VC max = I max X C X L = 56.5kΩ − 35.4kΩ
113 XC = = 56.5kΩ 2 Physics 212 Lecture 20, Slide 25 Physics ...
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