Lecture 5 - Download as PowerPoint

							   Physics 212
     Lecture 20

      AC Circuits
Maximum currents & voltages
  Phasors: A Simple Tool




                        Physics 212 Lecture 20, Slide 1
                              Main Point 1




First, we introduced the driven LCR circuit by adding a generator which serves to
eliminate the resistive damping by sustaining voltage and current oscillations at the
driving frequency of the generator.
                                                           Physics 212 Lecture 20, Slide 2
                             Main Point 2




Second, we introduced phasors, a graphical representation of the voltages and
phases in the circuit that allows us to efficiently solve for the maximum current
and the phase between the driving voltage and the current in the one loop circuit.
In particular, we represent the maximum voltage across each element as a vector
that rotates counterclockwise at the driving frequency with fixed relative phases
as shown. Both the phase between the driving voltage and the current, f, and the
maximum value of the current, Im, can then be obtained from the impedance
triangle as shown.

                                                          Physics 212 Lecture 20, Slide 3
                  Resistors



e = emaxsin(wt)       R       I = VR/R= Vmax/R sin(wt)




                                   Physics 212 Lecture 20, Slide 4
                  Capacitors
                               Q = VC = Cemaxsin(wt)
                               I = dQ/dt


e = emaxsin(wt)       C        I = VmaxwC cos(wt)




                                   Physics 212 Lecture 20, Slide 5
Physics 212 Lecture 20, Slide 6
                  Inductors
                              LdI/dt = VL =   emaxsin(wt)

e = emaxsin(wt)           L    I = - Vmax/wL cos(wt)




                                  Physics 212 Lecture 20, Slide 7
Physics 212 Lecture 20, Slide 8
                               RL ACT

 An RL circuit is driven by an AC generator as shown in the
figure.



                                             L         Imax = Vmax/XL
                                                          XL = wL



                                   R

 For what driving frequency w of the generator will the
 current through the resistor be largest
 A) w large
 B) Current through R doesn’t depend on w
 C) w small


                                                    Physics 212 Lecture 20, Slide 9
Physics 212 Lecture 20, Slide 10
    Summary:
                       How do I remember all this ?
R    Imax = Vmax/R          V in phase with I
                          Because resistors are simple




C    Imax = Vmax/XC        V 90o behind I
      XC = 1/wC          Current comes first since it
                         charges capacitor
                         Like a wire at high w




    L Imax = Vmax/XL        V 90o ahead of I
                            Opposite of capacitor
        XL = wL             Like a wire at low w
                                      Physics 212 Lecture 20, Slide 11
Makes sense to write everything in          Phasors make this
terms of I since this is the same           simple to see
everywhere in a one-loop circuit:               Imax XL
       Vmax = Imax XC
       V 90o behind I



            C                                                  Imax R
                    L    Vmax = Imax XL
emax                     V 90o ahead of I
                R
                                                Imax XC
         Vmax = Imax R
         V in phase with I                  Always looks the same.
                                            Only the lengths will
                                            change

                                                Physics 212 Lecture 20, Slide 12
Physics 212 Lecture 20, Slide 13
 Summary:                                   Imax XC


 VCmax= Imax XC                       C
                                                L      Imax XL
 VLmax= Imax XL           emax
                                            R
 VRmax= Imax R
 emax = Imax Z                        Imax R



 Imax = emax / Z

                                                       (XL-
Z  R   X L  XC 
       2              2
                                    f                  X C)
           X L  XC                     R
tan f  
               R

                                 Physics 212 Lecture 20, Slide 14
    Checkpoint 1a




A
B
C




                    Physics 212 Lecture 20, Slide 15
Physics 212 Lecture 20, Slide 16
    Checkpoint 1b




A
B
C




                    Physics 212 Lecture 20, Slide 17
    Checkpoint 1c




A
B
C
D




                    Physics 212 Lecture 20, Slide 18
Physics 212 Lecture 20, Slide 19
    Checkpoint 2a




A
B
C




       Physics 212 Lecture 20, Slide 20
Physics 212 Lecture 20, Slide 21
    Checkpoint 2b




A
B
C




       Physics 212 Lecture 20, Slide 22
    Checkpoint 2c




A
B
C




       Physics 212 Lecture 20, Slide 23
Physics 212 Lecture 20, Slide 24
                              Calculation                                  C
Consider the harmonically driven series LCR circuit shown.
Vmax = 100 V
Imax = 2 mA                                                     V ~            L
VCmax = 113 V
The current leads generator voltage by 45o                                 R
L and R are unknown.

What is XL, the reactance of the inductor, at this frequency?

Conceptual Analysis




Strategic Analysis




                                                       Physics 212 Lecture 20, Slide 25
Physics 212 Lecture 20, Slide 26
Physics 212 Lecture 20, Slide 27