PHYSICS 51 CH 31.1

1. PHYSICS 51 CH 31.1

2. HW #12 • Read Chapter 31 • Sections 1-6 • Problems • Ch 31: #4, #14, #20, #35, #51 • Due: Tuesday May 6

3. Goals for Today • Translate Ohms Law to AC circuits • Be able to calculate the impedance of a simple AC series circuit of resistors, capacitors and inductors • Calculate the phase angle between current and voltage in an AC circuit and know what it means • Digression into phasor diagrams

4. Why AC Power What is this

5. US Power Grid High voltage transmission over long distances to minimize loss

6. Ch 31 AC Circuits • AC circuits have the advantage of easy voltage transformation (Transformers) • From Faraday’s law transformers are just coupled coils • Can be designed for very low loss • Allows transmission at very high voltage to allow minimal losses (loss = I2R) • Solving for current, power etc. in AC circuits • The AC equivalent of ohms law • V = IZ • Z = impedance

7. The difficulty with AC circuits • Resistors act similar in DC and AC circuits • Ohm law v(t) = i(t) R • For inductors and capacitors the reactance (like resistance) is a function of the frequency of the drive voltage • Inductors and capacitors have a phase angle between current and voltage • To find the total impedance (and therefore current and power in the circuit) of a simple circuit we need to keep track of the phase • This is why we introduce phasors

8. Graphs (and phasors) of instantaneous voltage and current for a resistor. Resistors in AC circuits Notation:-lower case letters are time dependent and -upper case letters are constant. For example, i(t) is the time dependent current andI is current amplitude; VR is the voltage amplitude = IR

9. Example of a resistive circuit

10. Phasor diagram -- projection of rotating vector (phasor) onto the horizontal axis represents the instantaneous current. Phasor To add two phasors they must have same frequency Pasors add head to tail (like vectors)

11. Graphs (and phasors) of instantaneous voltage and current for an inductor. Inductors in AC circuits i(t) = I cos wt (source) vL(t) = L di / dt vL(t) = L d(I cos wt )/dt vL(t) = -IwL sin wtvL(t) = +IwL cos (wt + 900)where VL = IwL (= IXL)is the voltage amplitude and f = +900 is the PHASE ANGLE(angle between voltage across and current through the inductor).XL =wL ELIVLLI

12. Graphs (and phasors) of instantaneous voltage and current showing phase relation between current (red) and voltage (blue).Remember: “ELI the ICE man”

13. Circuit elements in AC circuits

14. Example of an RL circuit

15. Example of an RLC circuit

16. Instantaneous current and voltage: The average power is half the product of I and the component of V in phase with it. Average power depends on current and voltage amplitudes AND the phase angle f:

17. Resistance and Reactance

18. Graph of current amplitude vs source frequency  for a series RLC circuit with various values of circuit resistance. Resonance

19. Crossover network in a speaker system. Capacitive reactance: XC =1/wCInductive reactance: XL = wL Example: Speaker Circuit

20. Goals for Today AC Circuits • Move from impedance and phase angle to voltage, current, power, and phasor diagrams • Long example • Talk about resonance, cross-over circuits, and frequency filters • Short examples • AC transformers • Quiz (last one ! ? !)

21. The difficulty with AC circuits • Resistors act similar in DC and AC circuits • Ohm law v(t) = i(t) R • For inductors and capacitors the reactance (like resistance) is a function of the frequency of the drive voltage • Inductors and capacitors have a phase angle between current and voltage • To find the total impedance (and therefore current and power in the circuit) of a simple circuit we need to keep track of the phase • This is why we introduce phasors

22. Circuit elements in AC circuits Phase angle j = tan-1(XL-XC/R)

23. More details • Voltage and Current can be written 3 ways • V the amplitude of the oscillating voltage • v(t) = Vcos(wt + j) • VRMS = root mean square of above = V/(2)1/2 Same for I (except no j ) • Always use w not f in impedance calculation • 2pf = w

24. Example R = 2.0 W For the above circuit calculate Total impedance and phase angle Current through the circuit as a function of time Voltage across each element Draw the phasor diagram for the voltages The average power dissipated in the circuit L = 0.20 H V=10V w=10 s-1 C = 0.20 F

25. Phasor diagrams for series RLC circuit (b) XL > XC and (c) XL < XC. Class example like this

26. Graphs of instantaneous voltages for an RLC series circuit: This is one reason we use phasors

27. Resistance and Reactance

28. Graph of current amplitude vs source frequency  for a series RLC circuit with various values of circuit resistance. Resonance

29. Crossover network in a speaker system. Capacitive reactance: XC =1/wCInductive reactance: XL = wL Example: Speaker Circuit

30. Low pass filter R L C Vout Vin Calculate the ratio of Vout/Vin Show that low frequency signals pass through while high frequency signals are blocked

31. Graphs of instantaneous voltage, current, and power for an R, L, C, and an RLC circuit. Average power for an arbitrary AC circuit is 0.5 VI cosf = V rms I rms cos f.

32. TRANSFORMERS can step-up AC voltages or step-down AC voltages. Transformer: AC source is V1 :secondary provides a voltage V2 to a device with resistance R. Key Equations e2 /e1 = N2/N1 V1I1 = V2I1 FB=FB

33. Transformer Example A transformer near your house converts the line voltage from 20,000V to 120V. What is the ratio of wire turns of the primary to secondary coils? If your house draws a maximum of 100A at 120V, how much current does that draw at the line voltage? What is the load (resistance) of your house at maximum current draw (assume resistive load)?