Review

1. Review Part 3 of Course

2. Passive Circuit Elements i i i + -

3. Energy stored in the capacitor The instantaneous power delivered to the capacitor is The energy stored in the capacitor is thus

4. Energy stored in the capacitor Assuming the capacitor was uncharged at t = -, and knowing that represents the energy stored in the electric field established between the two plates of the capacitor. This energy can be retrieved. And, in fact, the word capacitor is derived from this element’s ability (or capacity) to store energy.

5. Parallel Capacitors Thus, the equivalent capacitance of N capacitors in parallel is the sum of the individual capacitances. Capacitors in parallel act like resistors in series.

6. Series Capacitors The equivalent capacitance of N series connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitors.Capacitors in series act like resistors in parallel.

7. Energy stored in an inductor The instantaneous power delivered to an inductor is The energy stored in the magnetic field is thus

8. Series Inductors The equivalent inductance of series connected inductors is the sum of the individual inductances. Thus, inductances in series combine in the same way as resistors in series.

9. Parallel Inductors The equivalent inductance of parallel connected inductors is the reciprocal of the sum of the reciprocals of the individual inductances. Thus inductances in parallel combine like resistors in parallel.

10. Complex Numbers Complex Plane imag A A real j Euler's equation: j measured positive counter-clockwise Note:

11. Comparing Sinusoids Note: positive angles are counter-clockwise

12. A phasor is a complex number that represents the amplitude and phase of a sinusoid. AC Circuits and Phasors XM j Recall that when the assumed form of the current was substituted into the differential equations, the cancelled out. We were then left with just the phasors

13. - VR + + VL - Impedance Impedance Units = ohms Note that impedance is a complex number containing a real, or resistive component, and an imaginary, or reactive, component.

14. - VR + + VL - Admittance Admittance Units = siemens susceptance conductance

15. Im V V I I Re I in phase with V Im V I V Re I lags V I Im V I V I Re I leads V

16. We see that if we replace Z by R the impedances add like resistances. Impedances in series add like resistors in series Impedances in parallel add like resistors in parallel

17. Voltage Division But Therefore

18. Instantaneous Power Note twice the frequency

19. Average Power Purely reactive circuit Purely resistive circuit

20. Effective or RMS Values We define the effective or rms value of a periodic current (voltage) source to be the dc current (voltage) that delivers the same average power to a resistor. root-mean-square

21. Effective or RMS Values and Using

22. Ideal Transformer - Voltage The input AC voltage, v1, produces a flux f This changing flux through coil 2 induces a voltage, v2 across coil 2

23. Ideal Transformer - Current Magnetomotive force, mmf f The total mmf applied to core is For ideal transformer, the reluctance R is zero.

24. Ideal Transformer - Impedance Load impedance Input impedance Turns ratio

25. Ideal Transformer - Power Power delivered to primary Power delivered to load Power delivered to an ideal transformer by the source is transferred to the load.

26. Force on current in a magnetic field Force on moving charge q -- Lorentz force Current density, j, is the amount of charge passing per unit area per unit time. N = number of charges, q, per unit volume moving with mean velocity, v. j Force per unit length on a wire is

27. B Force out commutator i + - i brushes Force in X Rotating Machine

28. Back emf X X B B Force out r i a q l + - b i Force in

29. B Force out commutator i + - i brushes Force in X Back emf

30. X X X X Armature with four coil loops S N

31. Motor Circuit Power and Torque

32. AC Nodal Analysis

33. For steady-state AC circuits we can use same the method of writing nodal equations by inspection that we learned for resistive circuits. To do so, we must replace resistances with impedances.

34. We solved Problem 4.31 in Class: Change impedances to admittances

35. Nodal Analysis for Circuits Containing Voltage Sources That Can’t be Transformed to Current Sources • Assume temporarily that the current across each voltage source is known and write the nodal equations in the same way we did for circuits with only independent voltage sources. • Express the voltage across each independent voltage source in terms of the node voltages and replace known node voltages in the equations. • Rewrite the equations with all unknown node voltages and source currents on the l.h.s. of the equality and all known currents and voltages on the r.h.s of the equality.

36. We solved #4.33 in text in Class: Note: V2 = 10 assume I2

37. AC Mesh Analysis

38. For steady-state AC circuits we can use same the method of writing mesh equations by inspection that we learned for resistive circuits. To do so, we must replace conductances with admittances.

39. We solved Problem 4.38 in text in Class: Find I1 and I2:

40. What happens if we have independent current sources that can’t be transformed in the circuit? • Assume temporarily that the voltage across each current source is known and write the mesh equations in the same way we did for circuits with only independent voltage sources. • Express the current of each independent current source in terms of the mesh currents and replace known mesh currents in the equations. • Rewrite the equations with all unknown mesh currents and voltages on the left hand side of the equality and all known voltages and currents on the r.h.s of the equality.

41. Note I2 = -2 We solved Problem 4.40 in text in Class: Find I0: Assume you know V2

42. Matlab Solution: Know how to program

43. AC Thevenin's Theorem

44. AC Thevenin's Theorem Thevenin’s theorem states that the two circuits given below are equivalent as seen from the load ZL that is the same in both cases. VTh = Thevenin’s voltage = Vab with ZL disconnected (= ) = the open-circuit voltage = VOC

45. AC Thevenin's Theorem ZTh = Thevenin’s impedance = the input impedance with all independent sources turned off (voltage sources replaced by short circuits and current sources replaced by open circuits). This is the impedance seen at the terminals ab when all independent sources are turned off.

46. We solvedProblem 4.40 using Thevenin's Thm. in Class:

47. AC Superposition

48. Superposition Principle The superposition principle states that the voltage across (or the current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.

49. Steps in Applying the Superposition Principle • Turn off all independent sources except one. Find the output (voltage or current) due to the active source. • Repeat step 1 for each of the other independent sources. • Find the total output by adding algebraically all of the results found in steps 1 & 2 above.

50. Example Done in Class: Note that the voltage source and the current source have two different frequencies. Thus, if we want to use phasors, the only way we've solved sinusoidal steady-state problems, we MUST use superposition to solve this problem. We considered each source acting alone, and then found v0(t) by superposition. Remember that