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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

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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

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Circuit Values Expressed in Polar Form 6 XLXL XCXC

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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 ) Division (polar form)

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

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Complex Numbers Applied to a Series-Parallel Circuit VSVS ZZ x Zx ZZ Z EQ Z 1 = 6+j0 + 0+j8 = 6+j8 = ° Z 2 = 4+j0 + 0-j4 = 4-j4 = ° Z 1 + Z 2 = 6+j8 + 4-j4 = 10+j4 = Z 1 x Z 2 = ° x ° = Z EQ = = 5.24

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The Total Current Flow in the Series-Parallel Circuit Z EQ = = I T = = A Note: The circuit is capacitive since the current is leading by 13.7° A 24 V ZZ x Zx ZZ Z EQ

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The Total Power Dissipation in the Series-Parallel Circuit WxxVx I x CosP T V ZZ x Zx ZZ Z EQ A

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The Branch Dissipations in the Series-Parallel Circuit WxxV x I x CosP T ° I 1 = 24 = ° A ° I 2 = 24 = 4.24 ° A P 1 = I 2 R 1 = x 6 = 34.6 W P 2 = I 2 R 2 = x 4 = 71.9 W Power check: P T = P 1 + P 2 = = 107 W V A

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Combining the Branch Currents ° I 1 = 24 = ° A ° I 2 = 24 = 4.24 ° A Convert branch currents to rectangular form for addition: ° A = 1.44-j1.92 A 4.24 ° A = 3+j3 A I T = 1.44-j j3 = 4.44+j1.08 A V A KCL check: 4.44+j1.08 A = A

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Branch 1 Voltages ° I 2 = 24 = 4.24 ° A ° I 1 = 24 = ° A 24 V V R 1 = ° x 6 ° = ° V = 8.65-j11.5 V V L 1 = ° x 8 ° = 19.2 ° V = 15.4+j11.5 V KVL check: 8.65-j j11.5 = 24+j0 V 1

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Branch 2 Voltages ° I 1 = 24 = ° A ° I 2 = 24 = 4.24 ° A 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+j j12 = 24+j0 V 2

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