1. Complex Numbers for AC Circuits Topics Covered in Chapter 24 24-1: Positive and Negative Numbers 24-2: The j Operator 24-3: Definition of a Complex Number 24-4: How Complex Numbers Are Applied to AC Circuits 24-5: Impedance in Complex Form Chapter 24 © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

2. Topics Covered in Chapter 24  24-6: Operations with Complex Numbers  24-7: Magnitude and Angle of a Complex Number  24-8: Polar Form for Complex Numbers  24-9: Converting Polar to Rectangular Form  24-10: Complex Numbers in Series AC Circuits  24-11: Complex Numbers in Parallel AC Circuits  24-12: Combining Two Complex Branch Impedances  24-13: Combining Complex Branch Currents  24-14: Parallel Circuit with Three Complex Branches McGraw-Hill© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

3. 24-1: Positive and Negative Numbers  Our common use of numbers as either positive or negative represents only two special cases.  In their more general form, numbers have both quantity and phase angle.

4. 24-1: Positive and Negative Numbers Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-1: In Fig. 24-1, positive and negative numbers are shown corresponding to the phase angles of 0° and 180°, respectively.

5. 24-2: The j Operator  The operator of a number can be any angle between 0° and 360°.  Since the angle of 90° is important in ac circuits, the factor j is used to indicate 90°.  In Fig. 24-2, the number 5 means 5 units at 0°, the number −5 is at 180°, and j5 indicates the number 5 at the 90° angle. Fig. 24-2

6. 24-2: The j Operator Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-3:  The angle of 180° corresponds to the j operation of 90° repeated twice.  This angular rotation is indicated by the factor j 2.  Note that the j operation multiplies itself, instead of adding.

7. 24-3: Definition of a Complex Number  The combination of a real and an imaginary term is called a complex number.  Usually, the real number is written first.  As an example, 3 + j4 is a complex number including 3 units on the real axis added to 4 units 90° out of phase on the j axis.  Complex numbers must be added as phasors.

8. 24-3: Definition of a Complex Number Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-4  Phasors for complex numbers are shown in Fig. 24-4  The phasors are shown with the end of one joined to the start of the next, to indicate addition.  Graphically, the sum is the hypotenuse of the right triangle formed by the two phasors.

9. 24-4: How Complex Numbers Are Applied to AC Circuits  Applications of complex numbers are a question of using a real term for 0°, +j for 90°, and −j for −90°, to denote phase angles.  An angle of 0° or a real number without any j operator is used for resistance R.  An angle of 90° or +j is used for inductive reactance X L.  An angle of −90° or −j is used for X C.

10. 24-4: How Complex Numbers Are Applied to AC Circuits Circuit Values Expressed in Rectangular Form 6+j0  6+j6  3−j3  0+j6  XLXL 0−j6  XCXC 66 66 33 33 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

11. 24-5: Impedance in Complex Form  The rectangular form of complex numbers is a convenient way to state the impedance of series resistance and reactance.  The general form of stating impedance is Z = R ± jX.  If one term is zero, substitute 0 for this term to keep Z in its general form.  This procedure is not required, but there is usually less confusion when the same form is used for all types of Z.

12. 24-6: Operations with Complex Numbers  Real numbers and j terms cannot be combined directly because they are 90° out of phase.  For addition or subtraction, add or subtract the real and j terms separately.  To multiply or divide a j term by a real number, multiply or divide the numbers. The answer is still a j term.  To multiply or divide a real number by a real number, just multiply or divide the real numbers, as in arithmetic.

13. 24-6: Operations with Complex Numbers  To divide a j term by a j term, divide the j coefficients to produce a real number; the j factors cancel.  To multiply complex numbers, follow the rules of algebra for multiplying two factors, each having two terms.  To divide complex numbers, the denominator must first be converted to a real number without any j term.  Converting the denominator to a real number without any j term is called rationalization.

14. 24-7: Magnitude and Angle of a Complex Number  In electrical terms, the complex impedance (4 + j3) means 4 Ω of resistance and 3 Ω of inductive reactance with a leading phase angle of 90°.  The magnitude of Z is the resultant of 5 Ω.  Finding the square root of the sum of the squares is vector or phasor addition of two terms in quadrature, 90° out of phase.  The phase angle of the resultant is the angle whose tangent is 0.75. This angle equals 37°

15. 24-7: Magnitude and Angle of a Complex Number Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-8:

16. 24-8: Polar Form for Complex Numbers  Calculating the magnitude and phase angle of a complex number is actually converting to an angular form in polar coordinates.  The rectangular form 4 + j3 is equal to 5 in polar form.  In polar coordinates, the distance from the center is the magnitude of the phasor Z.  Its phase angle Θ is counterclockwise from the 0° axis.  To convert any complex number to polar form,  Find the magnitude by phasor addition of the j term and real term.  Find the angle whose tangent is the j term divided by the real term.

17. 24-8: Polar Form for Complex Numbers 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 

18. 24-9: Converting Polar to Rectangular Form  Complex numbers in polar form are convenient for multiplication and division, but cannot be added or subtracted if their angles are different because the real and imaginary parts that make up the magnitude are different.  When complex numbers in polar form are to be added or subtracted, they must be converted into rectangular form.

19. 24-9: Converting Polar to Rectangular Form Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-9:  Consider the impedance Z in polar form.  Its value is the hypotenuse of a right triangle with sides formed by the real and j terms.  In Fig. 24-9, note the polar form can be converted to rectangular form by finding the horizontal and vertical sides of the right triangle. Real term for R = Z cos Θ j term for X = Z sin Θ

20. 24-10: Complex Numbers in Series AC Circuits  Refer to Fig. 24-10 (next slide).  Although a circuit like this with only series resistances and reactances can be solved graphically with phasor arrows, the complex numbers show more details of the phase angles.  The total Z T in Fig. 24-10 (a) is the sum of the impedances: Z T = 2 + j4 − j12 = 6 − j8  Convert Z T to polar and divide into V T to determine I.

21. 24-10: Complex Numbers in Series AC Circuits Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-10:

22. 24-11: Complex Numbers in Parallel AC Circuits  A useful application is converting a parallel circuit to an equivalent series circuit.  See Fig. 24-11 (next slide), with a 10-Ω X L in parallel with a 10-Ω R.  In complex notation, R is 10 + j0 and X l is 0 + j10.  Their combined parallel impedance Z T equals the product divided by the sum.  Z T in polar form is

23. 24-11: Complex Numbers in Parallel AC Circuits Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-11: The rectangular form of Z T means that a 5-Ω R in series with a 5-Ω X L is the equivalent of a 10-Ω R in parallel with a 10-Ω X L, as shown in Fig. 24-11.

24.  Recall the product over sum method of combining parallel resistors:  The product over sum approach can be used to combine branch impedances: 24-12: Combining Two Complex Branch Impedances Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-12: R EQ = R 1 × R 2 R 1 + R 2 Z EQ = Z 1 × Z 2 Z 1 + Z 2

25. 24-12: Combining Two Complex Branch Impedances Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Z T = Z 1 × Z 2 Z 1 + Z 2 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 × Z 2 = 10  53.1° x 5.66  45° = 56.6  56.6  10.8  21.8  ZT =ZT = = 5.24  Fig. 24-12:

26. 24-12: Combining Two Complex Branch Impedances Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-12: 4.58  13.7  A 56.6  8.1  10.8  21.8  Z T = = 5.24  13.7  24 5.24  13.7  I T = = 4.58  13.7  A Z T = Z 1 × Z 2 Z 1 + Z 2 Note: The circuit is capacitive since the current is leading by 13.7°. V A = 24 V

27. 24-13: Combining Complex Branch Currents Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-13: 4.58  13.7  A Adding the branch currents, I T = I 1 + I 2 = (6 + j6) + (3 − j4) = 9 + j2 A In polar form, the I T of 9 + j2 is calculated as the phasor sum of the branch currents.

28. 24-14: Parallel Circuit with Three Complex Branches  Because the circuit in Fig. 24-14 (next slide) has more than two complex impedances in parallel, use the method of branch currents.  Convert each branch impedance to polar form.  Convert the individual branch currents from polar to rectangular form so they can be added for I T.  Convert I T from rectangular to polar form.  Z T can remain in polar form with its magnitude and phase angle or can be converted to rectangular form for its resistive and reactive components.

29. 24-14: Parallel Circuit with Three Complex Branches Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 24-14: