Basic Electronics Ninth Edition Grob Schultz 2002 The

Basic Electronics Ninth Edition CHAPTER 25 Complex Numbers for AC Circuits © 2003 The

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

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

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

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

Circuit Values Expressed in Rectangular Form 6 W XL 6 W 0+j 6 W 6+j 0 W XC 0 -j 6 W 3 W 3 W 6+j 6 W 3 -j 3 W

Phasors Expressed in Polar Form 6Ð 90 8. 49Ð 45 6 6 6Ð 0 6Ð-90 4. 24Ð-45 • Magnitude is followed by the angle. l Ð 0 means no rotation. • Positive angles provide CCW rotation. • Negative angles provide CW rotation.

Circuit Values Expressed in Polar Form 6 W XL 6 W 6Ð 90 W 8. 49Ð 45 W 6Ð 0 W XC 6Ð-90 W 3 W 3 W 4. 24Ð-45 W

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).

Rectangular-to-Polar Conversion • General expression for the conversion: R±j. X = ZÐq • First Step: 2 2 = + Z R X æ Xö • Second Step: q = arctangent çè ÷ø R

Polar-to-Rectangular Conversion • General expression for the conversion: ZÐq = R±j. X • First Step: R = Z cos q • Second Step: X = Z sin q

Operations with Complex Expressions • Addition (rectangular form) R 1+j. X 1 + R 2+j. X 2 = (R 1+R 2)+j(X 1+X 2) • Subtraction (rectangular form) R 1+j. X 1 - R 2+j. X 2 = (R 1 -R 2)+j(X 1 -X 2) • Multiplication (polar form) Z 1Ðq 1 ´ Z 2Ðq 2 = Z 1 Z 2Ð(q 1 + q 2) • Division (polar form)

Complex Numbers Applied to a Series-Parallel Circuit VS 6 W 8 W 4 W 4 W R 1 x R 2 Recall the product over sum method REQ = + of combining parallel resistors: R 1 R 2 Z 1 x Z 2 The product over sum approach can ZEQ = + Z 1 Z 2 be used to combine branch impedances:

Complex Numbers Applied to a Series-Parallel Circuit VS 6 W 8 W 4 W 4 W ZEQ = Z 1 x Z 2 Z 1 + Z 2 Z 1 = 6+j 0 + 0+j 8 = 6+j 8 W = 10Ð 53. 1° W Z 2 = 4+j 0 + 0 -j 4 = 4 -j 4 W = 5. 66Ð-45° W Z 1 + Z 2 = 6+j 8 + 4 -j 4 = 10+j 4 = 10. 8Ð 21. 8 W Z 1 x Z 2 = 10Ð 53. 1° x 5. 66Ð-45° = 56. 6Ð 8. 1 W ZEQ = 10. 8Ð 21. 8 W = 5. 24Ð-13. 7 W

The Total Current Flow in the Series-Parallel Circuit 6 W 24 V 8 W 4 W 4 W ZEQ = Z 1 x Z 2 Z 1 + Z 2 4. 58Ð 13. 7 A 56. 6Ð 8. 1 W ZEQ = 10. 8Ð 21. 8 W = 5. 24Ð-13. 7 W IT = 24 = 4. 58Ð 13. 7 A 5. 24Ð-13. 7 Note: The circuit is capacitive since the current is leading by 13. 7°.

The Total Power Dissipation in the Series-Parallel Circuit 24 V 6 W 8 W 4 W 4 W ZEQ = Z 1 x Z 2 Z 1 + Z 2 4. 58Ð 13. 7 A PT = Vx I x Cosq = 24 x 4. 58 x 0. 972 = 107 W

The Branch Dissipations in the Series-Parallel Circuit 6 W 24 V 8 W 4. 58Ð 13. 7 A 4 W 4 W I 1 = I 2 = 24 10Ð 53. 1° = 2. 4Ð-53. 1° A 24 5. 66Ð-45° = 4. 24Ð 45° A P 1 = I 2 R 1 = 2. 42 x 6 = 34. 6 W P 2 = I 2 R 2 = 4. 242 x 4 = 71. 9 W PT = V x I x Cosq = 24 x 4. 58 x 0. 972 = 107 W Power check: PT = P 1 + P 2 = 34. 6 + 71. 9 = 107 W

Combining the Branch Currents 4. 58Ð 13. 7 A 6 W 24 V 8 W 4 W 4 W I 1 = 24 10Ð 53. 1° = 2. 4Ð-53. 1° A 24 = 4. 24Ð 45° A I 2 = 5. 66Ð-45° Convert branch currents to rectangular form for addition: 2. 4Ð-53. 1° A = 1. 44 -j 1. 92 A 4. 24Ð 45° A = 3+j 3 A IT = 1. 44 -j 1. 92 + 3+j 3 = 4. 44+j 1. 08 A KCL check: 4. 44+j 1. 08 A = 4. 58Ð 13. 7 A

Branch 1 Voltages 1 6 W 24 V 8 W 4 W 4 W I 1 = 24 10Ð 53. 1° = 2. 4Ð-53. 1° A 24 = 4. 24Ð 45° A I 2 = 5. 66Ð-45° VR = 2. 4Ð-53. 1° x 6Ð 0° = 14. 4Ð-53. 1° V = 8. 65 -j 11. 5 V 1 VL = 2. 4Ð-53. 1° x 8Ð 90° = 19. 2Ð 36. 9° V = 15. 4+j 11. 5 V 1 KVL check: 8. 65 -j 11. 5 + 15. 4+j 11. 5 = 24+j 0 V

Branch 2 Voltages 2 6 W 24 V 8 W 4 W 4 W 24 = 2. 4Ð-53. 1° A I 1 = 10Ð 53. 1° 24 = 4. 24Ð 45° A I 2 = 5. 66Ð-45° VR = 4. 24Ð 45° x 4Ð 0° = 17Ð 45° V = 12+j 12 V 2 VC = 4. 24Ð 45° x 4Ð-90° = 17Ð-45° V = 12 -j 12 V 1 KVL check: 12+j 12 + 12 -j 12 = 24+j 0 V