1. Chapter 14 Series and Parallel AC Circuits

2. Objectives Become familiar with the characteristics of a series and parallel ac circuit Find the total impedance of a series and parallel ac circuit and sketch the impedance and admittance diagrams Find all the currents and voltages of a series and parallel ac circuit Sketch a phasor diagram for all the voltages and currents of a series or parallel network

3. Objectives Apply Kirchhoff’s current and voltage laws to an ac network Apply the voltage divider rule and current divider rule to ac networks Find the power delivered to any series or parallel ac network and become familiar with the power factor of a network Determine the frequency response of a series or parallel ac network

4. Series Configuration The total impedance of series ac elements is the sum of the individual impedances The current is the same at every point in a series ac circuit Kirchhoff’s voltage law can be applied in the same manner as it is employed for dc circuits

5. Series Configuration If we plot the impedance of each quantity on the same set of axes, we obtain an impedance diagram We can find the total impedance using vector algebra

6. Series Configuration The current is determined by: I S = E/Z T And the voltage across each element by: V R = I S Z R, V L = I S Z L, V C = I S Z C Since power is only dissipated by the resistive element, the total power delivered by the source or absorbed by the circuit is:

7. FIGURE 14-1 Series impedances. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

8. FIGURE 14-2 Series R-L-C ac circuit. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

9. FIGURE 14-3 Impedance diagram for a series R-L-C circuit. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

10. FIGURE 14-4 Finding the total impedance of a series R-L-C circuit in which X L > X C. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

11. FIGURE 14-5 Example 14.1. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

12. FIGURE 14-6 Impedance diagram for Example 14.1. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

13. Average Power and the Power Factor The average or real power to a network composed solely of resistive elements can be found by finding the product of the applied voltage and the resulting current

14. FIGURE 14-9 Series R-L circuit for Example 14.3. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

15. FIGURE 14-10 Applying phasor notation to the network of Fig. 14.9. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

16. FIGURE 14-11 Impedance diagram for the series R-L circuit of Fig. 14.9. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

17. FIGURE 14-13 Plot of e, v R, v L, and i for the circuit of Fig. 14.9. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

18. FIGURE 14-14 Series R-L-C ac circuit for Example 14.4. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

19. FIGURE 14-15 Applying phasor notation to the circuit of Fig. 14.14 Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

20. FIGURE 14-16 Impedance diagram for the series R-L-C circuit of Fig. 14.14. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

21. FIGURE 14-17 Phasor diagram for the series R-L-C circuit of Fig. 14.14. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

22. FIGURE 14-18 Waveforms for the series R-L-C circuit of Fig. 14.14. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

23. Average Power and the Power Factor The introduction of phase shift between the applied voltage and the resulting current will cause a decrease in the power delivered The relationship between the magnitude of the phase angle (between V and I) and the power is: The power factor of the network is:

24. FIGURE 14-19 The voltage and current for a purely resistive load. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

25. FIGURE 14-20 Power curve for the resistive load of Fig. 14.19. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

26. FIGURE 14-21 A circuit with the same magnitude for the total impedance as in Fig. 14.19 but with a phase shift of 30°. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

27. FIGURE 14-22 Power curve for an R-L load that causes a phase shift of 30° between v and i. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

28. Average Power and the Power Factor Power factor gives an indication of whether the network is primarily resistive or reactive in nature Inductive networks have lagging power factors and capacitive networks have leading power factors The angle  is the angle associated with the total impedance of a network 수식 14.8

29. FIGURE 14-26 Example 14.5. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

30. FIGURE 14-27 Example 14.6. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

31. Voltage Divider Rule The basic format for the voltage divider rule in ac circuits is exactly the same as that for dc circuits: where: V X is the voltage across one or more elements in series that have total impedance Z X E is the total voltage appearing across the series circuit and Z T is the total impedance of the series circuit

32. FIGURE 14-28 Circuit for Example 14.8. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

33. FIGURE 14-29 Circuit for Example 14.9. Robert L. Boylestad Essentials of Circuit Analysis Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

34. Frequency Response for Series AC Circuits The total impedance will be frequency dependent The impedance of any one element can be greater than the total impedance of the network The inductive and capacitive reactances are always in direct opposition on an impedance diagram Depending on the frequency applied, the same circuit can be either predominantly inductive or predominantly capacitive

35. Frequency Response for Series AC Circuits At lower frequencies the capacitive elements will usually have the most impact on the total impedance, while at high frequencies the inductive elements will usually have the most impact The magnitude of the voltage across any one element can be greater than the applied voltage

36. Frequency Response for Series AC Circuits The magnitude of the voltage across an element compared to the other elements of the circuit is directly related to the magnitude of its impedance; that is, the larger the impedance of an element, the larger the magnitude of the voltage across the element The voltages across a coil or capacitor are always in direct opposition on a phasor diagram

37. Frequency Response for Series AC Circuits The current is always in phase with the voltage across the resistive elements; lags the voltage across all the inductive elements by 90  ; and leads the voltage across all the capacitive elements by 90  The larger the resistive element of a circuit compared to the net reactive impedance, the closer the power factor is to unity

38. Parallel Configuration For the parallel combination of elements, the total impedance is: For ac networks, the smallest parallel resistor will have the most impact on the resistive component of the total impedance while the smallest reactive element will have the most impact on determining the reactive component of the total impedance

39. Parallel Configuration The applied frequency will affect the terminal characteristics of an ac network with reactive elements –a change in frequency may change the terminal characteristics from inductive to capacitive or vice- versa The total impedance of a parallel ac network will define an equivalent series combination of elements with the same total impedance

40. Parallel Configuration If the parallel network is resistive and inductive, the resulting impedance and series combination of elements will also be resistive and inductive; similarly for parallel resistive and capacitive networks The voltage is the same across the parallel elements and the current through each branch is: Power delivered to the network can be determined by summing the powers delivered to all resistive elements:

41. Current Divider Rule The current divider rule in ac circuits is exactly the same as that for dc circuits –for two parallel branches with impedances Z 1 and Z 2

42. Frequency Response of Parallel Elements The total impedance is frequency dependent The impedance of any one element can be less than the total impedance The smallest parallel resistor or smallest parallel reactance will have the most impact on the real or imaginary component of the total impedance Depending on the frequency applied, the same network can be either predominantly inductive or predominantly capacitive

43. Frequency Response of Parallel Elements At lower frequencies the inductive elements will usually have the most impact on the total impedance, while at higher frequencies the capacitive elements will usually have the most impact The magnitude of the current through any one branch can be greater than the source current

44. Frequency Response of Parallel Elements The magnitude of the current through an element, compared to the other elements of the network, is directly related to the magnitude of its impedance –the smaller the impedance of an element, the larger the magnitude of the current through the element The current through a coil is always in direct opposition with the current through a capacitor on a phasor diagram

45. Frequency Response of Parallel Elements The applied voltage is always in phase with the current through the resistive elements, leads the voltage across all the inductive elements by 90 , and lags the current through all capacitive elements by 90  The smaller the resistive element of a network compared to the net reactive component, the closer the power factor is to unity

46. Phase Measurements For ac parallel networks restricted to resistive loads, the total impedance can be found as with dc circuits, using an ohmmeter For parallel ac networks with reactive elements, the total impedance cannot be measured with an ohmmeter; a dual trace scope can be used –both channels of a dual trace oscilloscope must be connected to the same ground

47. Phase Measurements The oscilloscope will only display voltages versus time, so the peak value of current must be found using Ohm’s law The voltage across and the current through the (small) sensing resistor are in phase

48. Phase Measurements A resistor has been added between the source and the network to permit measuring the current and finding the phase angle between the applied voltage and the source current

49. Admittance and Susceptance Susceptance of an inductor (B L ) is measured in Siemens –the higher the susceptance the greater the susceptability to the flow of current through the element Susceptance of a capacitor (B C ) is measured in Siemens

50. Admittance and Susceptance For a parallel network, the ratio 1/Z is called admittance because it provides an indication of how easily a parallel branch will admit the flow of current The symbol for admittance is Y, and it is also measured in Siemens The larger the admittance the greater the resulting source current for the same applied voltage

51. Admittance and Susceptance The admittance parameter for each element can be defined as:

52. Summary The total impedance of series ac elements is the sum of the individual impedances The current is the same at every point in a series ac circuit The applied voltage equals the sum of the voltage drops around an ac closed path The angle associated with the total impedance is the same as that between the applied voltage and the resulting source current

53. Summary The smaller the angle associated with the total impedance the more resistive the circuit In an impedance diagram the inductive and capacitive reactances are always opposing elements The phasor diagram shows at a glance the relative magnitude of each quantity and the phase relationship between the voltages and current

54. Summary The power to a series circuit can be found by summing the power to all the resistive elements or using the general equation for power that includes the phase angle between the applied voltage and the resulting current For any series ac circuit, the largest resistor or largest reactance will have the most impact on the total resistance or reactance respectively

55. Summary When examining the frequency response, resistance does not change with frequency, while the reactance of an inductor increases linearly with frequency and the reactance of a capacitor decreases exponentially with frequency The total impedance of a parallel ac network is found in exactly the same way as the total resistance for a dc network except for the fact that one is now working with magnitudes and angles and not just magnitudes

56. Summary The total impedance of two parallel impedances is the product divided by their sum The angle associated with the total impedance is the same as that between the applied voltage and the resulting current The smaller the angle associated with the total impedance, the more resistive the network Voltage is the same across parallel ac elements Ohm’s law can be used to find all the currents of a parallel ac network

57. Summary Kirchhoff’s current law can be applied in the same manner as applied to dc circuits The total power to a parallel ac network can be found in the same manner as applied to series ac circuits. The power factor angle for parallel networks will tell at a glance whether the network is resistive or reactive. The larger the power factor the more resistive the total impedance

58. Summary For parallel ac networks the smallest parallel resistive or reactive elements will have the most impact on the total resistance or reactance respectively For the frequency response of parallel elements the response of each element is the same as that for series ac elements. The difference being that the element with the smallest impedance will be the predominant factor

59. Summary When using a dual trace scope always be sure there is a common ground for the two channels A sensing resistor can be placed in series with any inductor or capacitor to determine the phase relationship associated with the current through each, and a voltage or current at some other point in the network - due to the in-phase relationship between the voltage across a resistor and the current through the resistor Always choose a sensing resistor that will have an impedance significantly less than any element in series with it