1. AC Circuits Physics 102 Professor Lee Carkner Lecture 23

2. PAL #23 Alternating Current  240  lightbulb, V rms = 120 V, 60 Hz  the rms current  V rms = I rms R, I rms = V rms /R = 120/240 = 0.5A  the maximum current  I max = (2) ½ I rms = (2) ½ (0.5) = 0.707 A  the maximum power  P max = I 2 max R = (0.707) 2 (240) = 120 W  the average power  P av = I 2 rms R =(0.5) 2 (240) = 60 W  the power at time equals 1/120 second  I = I max sin  t = I max sin(2  ft) = I max sin [(2)(  )(60)(120) -1 ] = I max sin (  ) = 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, X C )   We can combine them together to get the impedance (Z)  We can then use Ohm’s Law to find the current   For AC circuits we also define 3 different values of V and I   The instantaneous (I = I max sin  t)  The rms (I rms = 0.707 I max )

4. AC Circuit with Resistor

5. AC and Capacitors  The ”resistance” of a capacitor is the reactance, X C X C = 1/(  C)  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

6. AC Capacitor Phase Lag

7. Inductive Reactance  We can define the way in which an inductor impedes the current with the inductive reactance: X L =  L   Creating a rapidly changing magnetic field and thus a strong back emf   V L = IX L

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)

9. AC Circuit With Inductor

10. Reactance and Frequency  Resistor   Capacitor   Inductor  Low current at high frequency

11. RCL and AC  Let’s 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 = (R 2 + (X L - X C ) 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  defined as: cos  = IR/IZ = R/Z  We know that power can be written P = IV   Can write power as: P av = I rms V rms cos   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

16. 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? A)¼, ½, ¾ B)0, -1, 1 C)1, 0, -1 D)0, 1, 0 E)1, 1, 1

17. Consider a sinusoidally varying current with a maximum value of 1 A and an angular frequency of . What is the value of the current at time equals ½ second and one second? A)½, 1 B)1, 2 C)0, 1 D)1, 0 E)0, 0

18. Consider two sine waves with a phase shift of  radians. When one wave is at its maximum value, the other is at, A)its minimum value B)0 C)its maximum value D)√2 times its maximum value  times its maximum value