1. Basic Electronics Ninth Edition Basic Electronics Ninth Edition ©2002 The McGraw-Hill Companies Grob Schultz

2. Basic Electronics Ninth Edition Basic Electronics Ninth Edition ©2003 The McGraw-Hill Companies 25 CHAPTER Complex Numbers for AC Circuits

3. Topics Covered in Chapter 25 Positive and Negative Numbers The j Operator Definition of a Complex Number Complex Numbers and AC Circuits Impedance in Complex Form

4. Topics Covered in Chapter 25 (continued) Operations with Complex Numbers Magnitude and Angle of a Complex Number Polar Form Converting Polar to Rectangular Form Complex Numbers in Series AC Circuits

5. Topics Covered in Chapter 25 (continued) Complex Numbers in Parallel AC Circuits Combining Two Complex Branch Impedances Combining Complex Branch Currents Parallel Circuit with Three Complex Branches

6. Phasors Expressed in Rectangular Form 6+j0 0+j6 0-j6 6+j6 3-j3 The j-operator rotates a phasor by 90°. j0 means no rotation. +j means CCW rotation. -j means CW rotation.

7. Circuit Values Expressed in Rectangular Form 6+j0 6+j6 3-j3 0+j6 XLXL 0-j6 XCXC 6 6 3 3

8. Phasors Expressed in Polar Form Magnitude is followed by the angle. 0 means no rotation. Positive angles provide CCW rotation. Negative angles provide CW rotation. 6 6 6 8.49 6 6 4.24

9. Circuit Values Expressed in Polar Form 6 XLXL 6 6 3 3 6 XCXC 6 8.49 4.24

10. Why Different Forms? Addition and subtraction are easier in rectangular form. Multiplication and division are easier in polar form. AC circuit analysis requires all four (addition, subtraction, multiplication, and division).

11. Rectangular-to-Polar Conversion General expression for the conversion: R±jX = Z arctangent X R Second Step: ZRX 22 First Step:

12. Polar-to-Rectangular Conversion General expression for the conversion: Z R±jX XZ sin Second Step: RZ cos First Step:

13. Operations with Complex Expressions Addition (rectangular form) R 1 +jX 1 + R 2 +jX 2 = (R 1 +R 2 )+j(X 1 +X 2 ) Subtraction (rectangular form) R 1 +jX 1 R 2 +jX 2 = (R 1 R 2 )+j(X 1 X 2 ) Multiplication (polar form) Z 1 1 Z 2 2 = Z 1 Z 2 1 + 2 ) Division (polar form)

14. VSVS 6 4 8 4 Complex Numbers Applied to a Series-Parallel Circuit Recall the product over sum method of combining parallel resistors: 21 2 1 RR x RR R EQ The product over sum approach can be used to combine branch impedances: 21 21 ZZ x Zx ZZ Z EQ

15. Complex Numbers Applied to a Series-Parallel Circuit VSVS 6 4 8 4 21 21 ZZ x Zx ZZ Z EQ Z 1 = 6+j0 + 0+j8 = 6+j8 = 10 53.1° Z 2 = 4+j0 + 0-j4 = 4-j4 = 5.66 45° Z 1 + Z 2 = 6+j8 + 4-j4 = 10+j4 = 10.8 21.8 Z 1 x Z 2 = 10 53.1° x 5.66 45° = 56.6 56.6 10.8 21.8 Z EQ = = 5.24

16. The Total Current Flow in the Series-Parallel Circuit 56.6 8.1 10.8 21.8 Z EQ = = 5.24 13.7 24 5.24 13.7 I T = = 4.58 13.7 A Note: The circuit is capacitive since the current is leading by 13.7°. 4.58 13.7 A 24 V 6 4 8 4 21 21 ZZ x Zx ZZ Z EQ

17. The Total Power Dissipation in the Series-Parallel Circuit WxxVx I x CosP T 107972.058.424 24 V 6 4 8 4 21 21 ZZ x Zx ZZ Z EQ 4.58 13.7 A

18. The Branch Dissipations in the Series-Parallel Circuit WxxV x I x CosP T 107972.058.424 10 53.1° I 1 = 24 = 2.4 53.1° A 5.66 45° I 2 = 24 = 4.24 ° A P 1 = I 2 R 1 = 2.4 2 x 6 = 34.6 W P 2 = I 2 R 2 = 4.24 2 x 4 = 71.9 W Power check: P T = P 1 + P 2 = 34.6 + 71.9 = 107 W 6 4 8 4 24 V 4.58 13.7 A

19. Combining the Branch Currents 10 53.1° I 1 = 24 = 2.4 53.1° A 5.66 45° I 2 = 24 = 4.24 ° A Convert branch currents to rectangular form for addition: 2.4 53.1° A = 1.44-j1.92 A 4.24 ° A = 3+j3 A I T = 1.44-j1.92 + 3+j3 = 4.44+j1.08 A 6 4 8 4 24 V 4.58 13.7 A KCL check: 4.44+j1.08 A = 4.58 13.7 A

20. Branch 1 Voltages 6 4 5.66 45° I 2 = 24 = 4.24 ° A 8 4 10 53.1° I 1 = 24 = 2.4 53.1° A 24 V V R 1 = 2.4 53.1° x 6 ° = 14.4 53.1° V = 8.65-j11.5 V V L 1 = 2.4 53.1° x 8 ° = 19.2 ° V = 15.4+j11.5 V KVL check: 8.65-j11.5 + 15.4+j11.5 = 24+j0 V 1

21. Branch 2 Voltages 10 53.1° I 1 = 24 = 2.4 53.1° A 6 4 5.66 45° I 2 = 24 = 4.24 ° A 8 4 24 V V R 2 = 4.24 ° x 4 ° = 17 ° V = 12+j12 V V C 1 = 4.24 ° x 4 ° = 17 ° V = 12-j12 V KVL check: 12+j12 + 12-j12 = 24+j0 V 2