Physics 212 Lecture 20 AC Circuits Maximum currents

Physics 212 Lecture 20 AC Circuits Maximum currents & voltages Phasors: A Simple Tool Physics 212 Lecture 20, Slide 1

Music Who is the Artist? A) B) C) D) E) Derek Trucks Band Robert Plant & Alison Kraus CCR Emmylou Harris & Gregg Allman Led Zeppelin Theme of the week Bluegrass fiddlers (Mark O’Connor, Alison Kraus) doing something different !! Physics 212 Lecture 20, Slide 2

Exam Tonight • Regular exam 7: 00 • Conflict exam 5: 15 in 100 MSEB – Covers material in Lectures 9 – 18 – Bring your ID: Rooms determined by discussion section (see link) • Final Exam – “Regular”: Fri. May 4, 1: 30; “Conflict”: Fri. May 11, 8: 00 – Sign up for the conflict in the gradebook if you wish Physics 212 Lecture 18, Slide 3

Your Comments “For the first time, I can actually say that I don’t hate circuits!!!! What an accomplishment : )" Change today – driven circuit: Ea. c. “I thought the checkpoints were difficult! I understood the prelecture concept. Everything is pretty much tied to KVR. However I don’t completely get the correlation between maximum voltage or currents across inductors, resistors and capacitors. ” “This stuff is very confusing. I like the phasor diagrams though. ” THE KEY TECHNIQUE: Draw the phasor diagram (and impedance triangle) for each problem! “I don’t get how to read the phasor diagram” “You putting a lecture on the same day as an exam just killed a puppy. ” “I could really use some AC to cool off my brain before this exam. . . ” “Set phasers to maximum stun. ” 05 Physics 212 Lecture 20, Slide 4

Resistors e = Vmaxsin(wt) R I = VR/R = Vmax/R sin(wt) Amplitude = Vmax/R Physics 212 Lecture 20, Slide 5

Capacitors Q = CVmaxsin(wt) I = d. Q/dt e = Vmaxsin(wt) C I = Vmaxw. C cos(wt) Amplitude = Vmax/XC 90 o where XC = 1/w. C: “Reactance” is like the “resistance” of the capacitor XC depends on w Physics 212 Lecture 20, Slide 6

Inductors L d. I/dt = VL = Vmaxsin(wt) e = Vmaxsin(wt) L I = - Vmax/w. L cos(wt) Amplitude = Vmax/XL 90 o where XL = w. L: “Reactance” is like the “resistance” of the inductor XL depends on w Physics 212 Lecture 20, Slide 7

Check. Point 1 a An RL circuit is driven by an AC generator as shown in the figure The A voltages across the resistor and generator are B Always out of phase A) B) Always in phase C Sometimes in and sometimes out of phase C) “Resistor and inductor are out of phase, and generator is determined by vector sum, thus out of phase” “The phasors for resistor and generator are in phase. ” “it depends on the voltage and resistance” Physics 212 Lecture 20, Slide 9

Check. Point 1 a An RL circuit is driven by an AC generator as shown in the figure Draw Voltage Phasors Imax XL emax Imax R The A voltages across the resistor and generator are B Always out of phase A) B) Always in phase C Sometimes in and sometimes out of phase C) Physics 212 Lecture 20, Slide 10

Check. Point 1 b An RL circuit is driven by an AC generator as shown in the figure The voltages across the resistor and inductor are A) Always out of phase B) Always in phase C) Sometimes in and sometimes out of phase “Voltage across resistor always lags the voltage across the inductor by 90 degrees. ” “Their voltage is scaled by the current produced by the generator” “Dependent on time” Physics 212 Lecture 20, Slide 11

Check. Point 1 b An RL circuit is driven by an AC generator as shown in the figure Draw Voltage Phasors Imax XL emax Imax R The voltages across the resistor and inductor are A) Always out of phase B) Always in phase C) Sometimes in and sometimes out of phase Physics 212 Lecture 20, Slide 12

Check. Point 1 c An RL circuit is driven by an AC generator as shown in the figure The phase difference between the CURRENT through the resistor and inductor A) Is always zero B) Is always 90 o C) Depends on the value of L and R D) Depends on L, R and the generator voltage “Current = current, it’s the same current because it’s the same circuit. ” “The phase difference will remain constant at 90 degrees. ” “the phase is dependent on L and R” “current = v/R” Physics 212 Lecture 20, Slide 13

Check. Point 1 c An RL circuit is driven by an AC generator as shown in the figure The CURRENT is THE CURRENT Imax XL f emax Imax R The phase difference between the CURRENT through the f is the phase between resistor and inductor generator and current A) Is always zero B) Is always 90 o C) Depends on the value of L and R D) Depends on L, R and the generator voltage Physics 212 Lecture 20, Slide 14

Review R Imax = Vmax/R VR in phase with I Because resistors are simple C Imax = Vmax/XC XC = 1/w. C “Reactance” L Imax = Vmax/XL X L = w. L “Reactance” VC 90 o behind I Current comes first since it charges capacitor Like a wire at high w VL 90 o ahead of I Opposite of capacitor Like a wire at low w Physics 212 Lecture 20, Slide 15

The Driven RLC Circuit Makes sense to write everything in terms of I since this is the same everywhere in a one-loop circuit: Phasors make this simple to see Imax XL Vmax = Imax XC V 90 o behind I C emax L R Imax R Vmax = Imax XL V 90 o ahead of I Vmax = Imax R V in phase with I Imax XC Always looks the same. Only the lengths will change Physics 212 Lecture 20, Slide 16

Imax XC The voltages still add up C emax Now we are adding three vectors: Imax XL L R Imax XL Imax R emax Imax R Imax XC Imax XL emax Physics 212 Lecture 20, Slide 17

Imax XC Making this simpler… C emax Imax XL L Imax XL R Imax XL emax Imax R Imax XC Physics 212 Lecture 20, Slide 18

Imax XC Making this simpler… C emax L Imax XL R Imax XL emax = Imax Z Imax R Imax(XL-XC) Imax R Imax XC Physics 212 Lecture 20, Slide 19

Imax XC Making this simpler… C emax L Imax XL R Imax R emax = Imax Z Imax(XL-XC) Imax R Physics 212 Lecture 20, Slide 20

Imax XC Making this simpler… C emax Imax XL R emax = Imax Z Imax R Imax(XL-XC) f L Imax R (XL-XC) f Impedance Triangle R Physics 212 Lecture 20, Slide 21

Imax XC Summary: VCmax= Imax XC VLmax= Imax XL C emax L Imax XL R VRmax= Imax R emax = Imax Z Imax = emax / Z (XL-XC) f R Physics 212 Lecture 20, Slide 22

Example: RL Circuit Xc=0 emax L Imax XL R Imax XL emax Imax R Physics 212 Lecture 20, Slide 23

Check. Point 2 a A driven RLC circuit is represented by the phasor diagram below. 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? A) VL = 0 B) VL = VL, max/2 C) VL = VL, max “when current is max the Xl vector is zero” “There is a voltage drop across the resistor too. ” “The current should be max when the voltage is also at max. ” Physics 212 Lecture 20, Slide 24

Check. Point 2 a A driven RLC circuit is represented by the phasor diagram below. 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? A) VL = 0 B) VL = VL, max/2 C) VL = VL, max What does the voltage phasor diagram look like when the current is IXL a maximum? IXe. L IR e IR IXc Physics 212 Lecture 20, Slide 25

Check. Point 2 b A driven RLC circuit is represented by the phasor diagram below. A When the capacitor is fully charged, what is the magnitude of the voltage across the B inductor? C VL = 0 A) B) VL = VL, max/2 C) VL = VL, max “The voltage across the capacitor would be at a maximum, so the voltage across the inductor would be 0” “half voltage because there is a resistor and capacitor” “its negative but the magnitude is the same as its max” Physics 212 Lecture 20, Slide 26

Check. Point 2 b A driven RLC circuit is represented by the phasor diagram below. IXc IXL e What does the voltage phasor diagram look like when the capacitor is fully charged? IR IXc When the capacitor is fully charged, what is the magnitude of the voltage across the e IXL inductor? A) VL = 0 B) VL = VL, max/2 C) VL = VL, max IR Physics 212 Lecture 20, Slide 27

Check. Point 2 c A driven RLC circuit is represented by the phasor diagram below. When the voltage across the capacitor is at its positive maximum, VC = +VC, max, what is the voltage across the inductor ? A) VL = 0 B) VL = VL, max C) VL = -VL, max “all energy in capacitor” “The voltages have to be equal. ” “The capacitor and inductor voltage phasors point in opposite directions by definition. ” Physics 212 Lecture 20, Slide 28

Check. Point 2 c A driven RLC circuit is represented by the phasor diagram below. IXc IXL e What does the voltage phasor diagram look like when the voltage across capacitor is at its positive maximum? IR IXc When the voltage across the capacitor is at its positive maximum, VC = +VC, max, what e is the voltage across the inductor. IX? L A) VL = 0 B) VL = VL, max C) VL = -VL, max IR Physics 212 Lecture 20, Slide 29

Calculation Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 m. A VCmax = 113 V The current leads generator voltage by 45 o L and R are unknown. C V ~ L R 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 Physics 212 Lecture 20, Slide 30

Calculation Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 m. A VCmax = 113 V The current leads generator voltage by 45 o L and R are unknown. C V ~ L R What is XL, the reactance of the inductor, at this frequency? Compare XL and XC at this frequency: (A) XL < XC (B) XL = XC (C) XL > XC (D) Not enough information • This information is determined from the phase – Current leads voltage 45 o VL VL = Imax. XL VC = Imax. XC VR (phase of current) V VC IR V leads Physics 212 Lecture 20, Slide 31

Calculation Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 m. A VCmax = 113 V The current leads generator voltage by 45 o L and R are unknown. C V ~ L R What is XL, the reactance of the inductor, at this frequency? What is Z, the total impedance of the circuit? (A) 70. 7 k. W (B) 50 k. W (C) 35. 4 k. W (D) 21. 1 k. W Physics 212 Lecture 20, Slide 32

Calculation Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 m. A VCmax = 113 V The current leads generator voltage by 45 o L and R are unknown. C V ~ L R Z = 50 k. W What is XL, the reactance of the inductor, at this frequency? sin(45)=. 707 What is R? (A) 70. 7 k. W cos(45)=. 707 (B) 50 k. W (C) 35. 4 k. W (D) 21. 1 k. W • Determined from impedance triangle R 45 o Z=50 k. W (XC-XL) R = Z cos(45 o) = 50 k. W x 0. 707 = 35. 4 k. W Physics 212 Lecture 20, Slide 33

Calculation C Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 m. A VCmax = 113 V The current leads generator voltage by 45 o L and R are unknown. V ~ R Z = 50 k. W What is XL, the reactance of the inductor, at this frequency? (A) 70. 7 k. W (B) 50 k. W (C) 35. 4 k. W We start with the impedance triangle: Z R = 35. 4 k. W (D) 21. 1 k. W XL = X C - R R 45 o L What is XC ? (XC-XL) VCmax = Imax. XC XL = 56. 5 k. W – 35. 4 k. W Physics 212 Lecture 20, Slide 34