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

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

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

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

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

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

Circuit Values Expressed in Rectangular Form 6W 6+j6 W 0+j6 W XL 3-j3 W 0-j6 W XC 6+j0 W 3W

Phasors Expressed in Polar Form 6Ð90 6Ð0 6Ð-90 8.49Ð45 6 4.24Ð-45 Magnitude is followed by the angle. Ð0 means no rotation. Positive angles provide CCW rotation. Negative angles provide CW rotation.

Circuit Values Expressed in Polar Form 6W 8.49Ð45 W 6Ð90 W XL 4.24Ð-45 W XC 6Ð-90 W 6Ð0 W 3W

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±jX = ZÐq Z R X = + 2 First Step: q = æ è ç ö ø ÷ arctan gent X R Second Step:

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

Operations with Complex Expressions Addition (rectangular form) R1+jX1 + R2+jX2 = (R1+R2)+j(X1+X2) Subtraction (rectangular form) R1+jX1 - R2+jX2 = (R1-R2)+j(X1-X2) Multiplication (polar form) Z1Ðq1 ´ Z2Ðq2 = Z1Z2Ð(q1 + q2) Division (polar form)

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

Complex Numbers Applied to a Series-Parallel Circuit VS 6 W 4 W 8 W 2 1 Z x Z EQ + = Z1 = 6+j0 + 0+j8 = 6+j8 W = 10Ð53.1° W Z2 = 4+j0 + 0-j4 = 4-j4 W = 5.66Ð-45° W Z1 + Z2 = 6+j8 + 4-j4 = 10+j4 = 10.8Ð21.8 W Z1 x Z2 = 10Ð53.1° x 5.66Ð-45° = 56.6Ð8.1 W 56.6Ð8.1 W 10.8Ð21.8 W ZEQ = = 5.24Ð-13.7 W

in the Series-Parallel Circuit The Total Current Flow in the Series-Parallel Circuit 24 V 6 W 4 W 8 W 2 1 Z x Z EQ + = 4.58Ð13.7 A 56.6Ð8.1 W 10.8Ð21.8 W ZEQ = = 5.24Ð-13.7 W 24 5.24Ð-13.7 IT = = 4.58Ð13.7 A 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 4 W 8 W 2 1 Z x Z EQ + = 4.58Ð13.7 A W x Vx I x Cos P T 107 972 . 58 4 24 = q

The Branch Dissipations in the Series-Parallel Circuit 6 W 4 W 8 W 24 V 4.58Ð13.7 A 10Ð53.1° I1 = 24 = 2.4Ð-53.1° A 5.66Ð-45° I2 = 24 = 4.24Ð45° A P1 = I2R1 = 2.42 x 6 = 34.6 W P2 = I2R2 = 4.242 x 4 = 71.9 W W x V x I x Cos P T 107 972 . 58 4 24 = q Power check: PT = P1 + P2 = 34.6 + 71.9 = 107 W

Combining the Branch Currents 6 W 4 W 8 W 24 V 4.58Ð13.7 A 10Ð53.1° I1 = 24 = 2.4Ð-53.1° A 5.66Ð-45° I2 = 24 = 4.24Ð45° A Convert branch currents to rectangular form for addition: 2.4Ð-53.1° A = 1.44-j1.92 A 4.24Ð45° A = 3+j3 A IT = 1.44-j1.92 + 3+j3 = 4.44+j1.08 A KCL check: 4.44+j1.08 A = 4.58Ð13.7 A

Branch 1 Voltages KVL check: 8.65-j11.5 + 15.4+j11.5 = 24+j0 V 1 I1 = 6 W 4 W 5.66Ð-45° I2 = 24 = 4.24Ð45° A 8 W 10Ð53.1° I1 = = 2.4Ð-53.1° A 24 V VR1 = 2.4Ð-53.1° x 6Ð0° = 14.4Ð-53.1° V = 8.65-j11.5 V VL1 = 2.4Ð-53.1° x 8Ð90° = 19.2Ð36.9° V = 15.4+j11.5 V KVL check: 8.65-j11.5 + 15.4+j11.5 = 24+j0 V

Branch 2 Voltages KVL check: 12+j12 + 12-j12 = 24+j0 V 2 24 I1 = 10Ð53.1° I1 = 24 = 2.4Ð-53.1° A 6 W 4 W 5.66Ð-45° I2 = = 4.24Ð45° A 8 W 24 V VR2 = 4.24Ð45° x 4Ð0° = 17Ð45° V = 12+j12 V VC1 = 4.24Ð45° x 4Ð-90° = 17Ð-45° V = 12-j12 V KVL check: 12+j12 + 12-j12 = 24+j0 V