AC Circuits

1
AC Circuits

• Physics 102
• Professor Lee Carkner
• Lecture 23

2
PAL 23 Alternating Current

• 240 W lightbulb, Vrms 120 V, 60 Hz
• the rms current
• Vrms IrmsR, Irms Vrms/R 120/240 0.5A
• the maximum current
• Imax (2)½Irms (2)½(0.5) 0.707 A
• the maximum power
• Pmax I2maxR (0.707)2(240) 120 W
• the average power
• Pav I2rmsR (0.5)2(240) 60 W
• the power at time equals 1/120 second
• I Imax sinwt Imax sin(2pft) Imax sin
  (2)(p)(60)(120)-1 Imax sin (p) 0
• P 0
• Completed 1/2 cycle, I back to zero

3
AC Circuit Elements

• In an AC circuit we get resistance-like effects
  from three different elements
• Capacitors (Reactance, XC)
• We can combine them together to get the impedance
  (Z)
• We can then use Ohms Law to find the current
• For AC circuits we also define 3 different values
  of V and I
• The instantaneous (I Imax sinwt)
• The rms (Irms 0.707 Imax)

4
AC Circuit with Resistor
5
AC and Capacitors

• The resistance of a capacitor is the reactance,
  XC
• XC 1/(wC)
• High frequency and large capacitance means less
  reactance
• The voltage and the current across the capacitor
  are not in phase
• Shift the current sine wave ¼ cycle backwards
  from the in-phase situation

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AC Capacitor Phase Lag
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Inductive Reactance

• We can define the way in which an inductor
  impedes the current with the inductive reactance
• XL wL
• Creating a rapidly changing magnetic field and
  thus a strong back emf
• DVL IXL

8
Inductors and Phase

• What is the phase shift between V and I?
• look at the slope of the current sine wave
• The induced voltage is zero when the current is a
  maximum (since that is where the current is not
  changing)
• The voltage leads the current by 90 degrees (V is
  max 1/4 cycle before I)

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AC Circuit With Inductor
10
Reactance and Frequency

• Resistor
• Capacitor
• Inductor
• Low current at high frequency

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RCL and AC

• Lets combine all three elements together
• If you combine a resistor, capacitor and an
  inductor into one series circuit, they all will
  have the same current but all will have
  difference voltages at any one time
• Voltages are all out of phase with each other

12
RLC Circuit
13
RLC Impedance

• Called the impedance (Z)
• Z (R2 (XL - XC)2)½
• The voltages for the inductor and capacitor are
  180 degrees opposed and so subtract
• The total voltage is
• Can think of Z as a generalized resistance for
  any AC circuit

14
Phase Angle and Power Factor

• They are separated by a phase angle f, defined
  as
• cos f IR/IZ R/Z
• We know that power can be written P IV
• Can write power as
• Pav IrmsVrms cos f
• Note that only the resistor dissipates power

15
Next Time

• Read 22.1-22.4, 22.7
• Homework Ch 21, P 64, 65, Ch 22, P 3, 7

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• Consider a sinusoidally varying current with a
  maximum value of 1 A. What is the value of the
  current at ¼, ½ and ¾ of the cycle?
• ¼, ½, ¾
• 0, -1, 1
• 1, 0, -1
• 0, 1, 0
• 1, 1, 1

17

• Consider a sinusoidally varying current with a
  maximum value of 1 A and an angular frequency of
  p. What is the value of the current at time
  equals ½ second and one second?
• ½, 1
• 1, 2
• 0, 1
• 1, 0
• 0, 0

18

• Consider two sine waves with a phase shift of p
  radians. When one wave is at its maximum value,
  the other is at,
• its minimum value
• 0
• its maximum value
• v2 times its maximum value
• p times its maximum value