Basic ElectronicsNinth Edition

Basic ElectronicsNinth Edition

©2002 The McGraw-Hill Companies

GrobSchultz

GrobSchultz

Basic ElectronicsNinth Edition

Basic ElectronicsNinth Edition

©2003 The McGraw-Hill Companies

25CHAPTER

Complex Numbers for AC Circuits

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+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.

Circuit Values Expressedin Rectangular Form

6+j0

6+j6

3-j3

0+j6 XL

0-j6 XC

6 6

3 3

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

8.496

6

4.24

Circuit Values Expressedin Polar Form

6XL

6 6

3 3

6

XC 6

8.49

4.24

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 = Z

arctangent

X

R• Second Step:

Z R X 2 2• First Step:

Polar-to-Rectangular Conversion

• General expression for the conversion:

ZR±jX

X Z sin• Second Step:

R Z cos• First Step:

Operations with Complex Expressions

• Addition (rectangular form) R1+jX1 + R2+jX2 = (R1+R2)+j(X1+X2)

• Subtraction (rectangular form) R1+jX1 R2+jX2 = (R1R2)+j(X1X2)

• Multiplication (polar form) Z11Z22 = Z1Z21 + 2)

• Division (polar form)

VS 6 4

8 4

Complex Numbers Appliedto a Series-Parallel Circuit

Recall the product over sum methodof combining parallel resistors:

21

21

RR

x RRR

EQ

The product over sum approach canbe used to combine branch impedances: 21

21

ZZ

x ZZZ

EQ

Complex Numbers Appliedto a Series-Parallel Circuit

VS 6 4

8 4 21

21

ZZ

x ZZZ

EQ

Z1 = 6+j0 + 0+j8 = 6+j8 = 1053.1°

Z2 = 4+j0 + 0-j4 = 4-j4 = 5.6645°

Z1 + Z2 = 6+j8 + 4-j4 = 10+j4 = 10.821.8

Z1 x Z2 = 1053.1° x 5.6645° = 56.6

56.610.821.8ZEQ = = 5.24

The Total Current Flowin the Series-Parallel Circuit

56.68.110.821.8ZEQ = = 5.2413.7

245.2413.7

IT = = 4.5813.7ANote: The circuit is capacitive since the current is leading by 13.7°.

4.5813.7A

24 V 6 4

8 4 21

21

ZZ

x ZZZ

EQ

The Total Power Dissipationin the Series-Parallel Circuit

WxxVx I x CosPT

107972.058.424

24 V 6 4

8 4 21

21

ZZ

x ZZZ

EQ

4.5813.7A

The Branch Dissipationsin the Series-Parallel Circuit

WxxV x I x CosPT

107972.058.424

1053.1°I1 =

24= 2.453.1° A

5.6645°I2 =

24= 4.24° A

P1 = I2R1 = 2.42 x 6 = 34.6 W

P2 = I2R2 = 4.242 x 4 = 71.9 W

Power check: PT = P1 + P2 = 34.6 + 71.9 = 107 W

6 4

8 4

24 V

4.5813.7A

Combining the Branch Currents

1053.1°I1 =

24= 2.453.1° A

5.6645°I2 =

24= 4.24° A

Convert branch currents to rectangular form for addition:

2.453.1° A = 1.44-j1.92 A

4.24° A = 3+j3 A

IT = 1.44-j1.92 + 3+j3 = 4.44+j1.08 A

6 4

8 4

24 V

4.5813.7A

KCL check: 4.44+j1.08 A = 4.5813.7A

Branch 1 Voltages

6 4

5.6645°I2 =

24= 4.24° A8

4

1053.1°I1 =

24= 2.453.1° A

24 V

VR1 = 2.453.1° x 6° = 14.453.1° V = 8.65-j11.5 V

VL1 = 2.453.1° x 8° = 19.2° V = 15.4+j11.5 V

KVL check: 8.65-j11.5 + 15.4+j11.5 = 24+j0 V

1

Branch 2 Voltages

1053.1°I1 =

24= 2.453.1° A6 4

5.6645°I2 =

24= 4.24° A8

4 24 V

VR2 = 4.24° x 4° = 17° V = 12+j12 V

VC1 = 4.24° x 4° = 17° V = 12-j12 V

KVL check: 12+j12 + 12-j12 = 24+j0 V

2