1. RLC Circuits Physics 102 Professor Lee Carkner Lecture 25

2. Three AC Circuits   V max = 10 V, f = 1Hz, R = 10   V rms = 0.707  V max = (0.707)(10) = 7.07 V  R = 10   I rms =  V rms /R = 0.707 A  I max = I rms /0.707 =  Phase Shift =  When V = 0, I =   V max = 10 V, f = 1Hz, C = 10 F   V rms = 0.707  V max = (0.707)(10) = 7.07 V  X C = 1/(2  fC) = 1/[(2)(  )(1)(10)] =  I rms =  V rms /X C =  I max = I rms /0.707 =  Phase Shift = ¼ cycle (-  /2)  When V = 0, I = I max = 625 A

3. Three AC Circuits   V max = 10 V, f = 1Hz, L = 10 H   V rms = 0.707  V max = (0.707)(10) = 7.07 V  X L = 2  fL = (2)(  )(1)(10) =  I rms =  V rms /X L =  I max = I rms /0.707 =  Phase Shift = ¼ cycle (+  /2)  When V = 0, I = I max = 0.16 A

4. For capacitor, V lags IFor inductor, V leads I

5. Solving RLC Circuits   = 2  f  The frequency determines the degree to which capacitors and inductors affect the flow of current  X C = 1/(  C) X L =  L

6. Current and Power  We use the reactances to find the impedance, which can be used in the modified version of Ohm’s law to find the current from the voltage Z = (R 2 + (X L - X C ) 2 ) ½  V = IZ   We then can find the degree to which the total voltage is out of phase with the current by finding the phase angle  The phase angle is also related to the power P av = I rms V rms cos 

7. RLC Circuit

8. Frequency Dependence   X L depends directly on  and X C depends inversely on    High f means rapid current change, means strong magnetic inductance and large back emf  High f means capacitors never build up much charge and so have little effect

9. High and Low f  For “normal” 60 Hz household current both X L and X C can be significant   For high f the inductor acts like a very large resistor and the capacitor acts like a resistance-less wire   At low f, the inductor acts like a resistance- less wire and the capacitor acts like a very large resistor 

10. High and Low Frequency

11. LC Circuit  Suppose we connect a charged capacitor to an inductor with no battery or resistor   The inductor keeps the current flowing until the other plate of the capacitor becomes charged   This process will cycle over and over 

12. LC Resonance

13. Oscillation Frequency  The rate at which the charge moves back and forth depends on the values of L and C  Since they are connected in parallel they must each have the same voltage IX C = IX L  = 1/(LC) ½  This is the natural frequency of the LC circuit

14. Natural Frequency   Example: a swing  If you push with the same frequency as the swing (e.g., every time it reaches the end) it will go higher  If you push the swing at all different random times it won’t   If you connect it to an AC generator with the same frequency it will have a large current

15. Resonance  This condition is known as resonance  Will happen when Z is a minimum  Z = (R 2 + (X L - X C ) 2 ) ½  To minimize Z want X L = X C   Frequencies near the natural one will produce large current

16. Impedance and Resonance

17. Resonance Frequency

18. Resistance and Resonance   The smallest you can make Z is Z = R   If we change R we do not change the natural frequency, but we do change the magnitude of the maximum current  Peak becomes shorter and also broader 

19. Next Time  Read 22.1-22.4, 22.7  Homework, Ch 21, P 71, Ch 22, P 3, 7, 8