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

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

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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

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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

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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

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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.

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

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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

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

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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).

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

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

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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)

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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

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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

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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

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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

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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

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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

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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

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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

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