complex numbers for ac circuits topics covered in chapter 24 24-1: positive and negative numbers...

Complex Numbers Complex Numbers for AC Circuitsfor 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

ChapterChapter2424

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Topics Covered in Chapter 24Topics 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.

24-1: Positive and Negative 24-1: Positive and Negative NumbersNumbers

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.

24-1: Positive and Negative 24-1: Positive and Negative NumbersNumbers

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.

24-2: The24-2: The j j OperatorOperator

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

24-2: The24-2: The j j OperatorOperator


Fig. 24-3:

The angle of 180° corresponds to the j operation of 90° repeated twice. This angular rotation is indicated by the factor j2. Note that the j operation multiplies itself, instead of adding.

24-3: Definition of a 24-3: Definition of a Complex NumberComplex 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.

24-3: Definition of a 24-3: Definition of a Complex NumberComplex Number


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.

24-4: How Complex Numbers Are 24-4: How Complex Numbers Are Applied to AC CircuitsApplied 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 XL.

An angle of −90° or −j is used for XC.

24-4: How Complex Numbers Are 24-4: How Complex Numbers Are Applied to AC CircuitsApplied to AC Circuits

Circuit Values Expressed in Rectangular Form

6+j0

6+j6

3−j3

0+j6 XL

0−j6 XC

6 6

3 3


24-5: Impedance in Complex 24-5: Impedance in Complex FormForm

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.

24-6: Operations with Complex 24-6: Operations with Complex NumbersNumbers

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.

24-6: Operations with Complex 24-6: Operations with Complex NumbersNumbers

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.

24-7: Magnitude and Angle 24-7: Magnitude and Angle of a Complex Numberof 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°

24-7: Magnitude and Angle 24-7: Magnitude and Angle of a Complex Numberof a Complex Number


Fig. 24-8:

24-8: Polar Form for 24-8: Polar Form for Complex NumbersComplex 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.

37

24-8: Polar Form for 24-8: Polar Form for Complex NumbersComplex 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

8.496

64.24

24-9: Converting Polar to 24-9: Converting Polar to Rectangular FormRectangular 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.

24-9: Converting Polar to 24-9: Converting Polar to Rectangular FormRectangular Form


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 Θj term for X = Z sin Θ

24-10: Complex Numbers in 24-10: Complex Numbers in Series AC CircuitsSeries 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 ZT in Fig. 24-10 (a) is the sum of the impedances:

ZT = 2 + j4 − j12

= 6 − j8

Convert ZT to polar and divide into VT to determine I.

24-10: Complex Numbers in 24-10: Complex Numbers in Series AC CircuitsSeries AC Circuits


Fig. 24-10:

24-11: Complex Numbers in 24-11: Complex Numbers in Parallel AC CircuitsParallel 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 XL in parallel with a 10-Ω R.

In complex notation, R is 10 + j0 and Xl is 0 + j10.

Their combined parallel impedance ZT equals the product divided by the sum.

ZT in polar form is 7.04 45

24-11: Complex Numbers in 24-11: Complex Numbers in Parallel AC CircuitsParallel AC Circuits


Fig. 24-11:

The rectangular form of ZT means that a 5-Ω R in series with a 5-Ω XL is the equivalent of a 10-Ω R in parallel with a 10-Ω XL, as shown in Fig. 24-11.

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 24-12: Combining Two Complex Branch ImpedancesBranch Impedances


Fig. 24-12:

REQ = R1 × R2

R1 + R2

ZEQ = Z1 × Z2

Z1 + Z2



ZT = Z1 × Z2

Z1 + Z2

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 × Z2 = 1053.1° x 5.6645° = 56.6

56.610.821.8ZT = = 5.24

Fig. 24-12:



Fig. 24-12:

4.5813.7A

56.68.110.821.8ZT = = 5.2413.7

24

5.2413.7IT = = 4.5813.7A

ZT = Z1 × Z2

Z1 + Z2

Note: The circuit is capacitive since the current is leading by 13.7°.

VA = 24 V

24-13: Combining Complex 24-13: Combining Complex Branch CurrentsBranch Currents


Fig. 24-13:

4.5813.7A

Adding the branch currents,

IT = I1 + I2

= (6 + j6) + (3 − j4) = 9 + j2 A

In polar form, the IT of 9 + j2 is calculated as the phasor sum of the branch currents.

2 2T

I

T

I = 9 2 = 85

= 9.22 A

2tan = = 0.222

9 = arctan (0.22)

= 12.53

Therefore, I is 9 j2 A in rectangular form

or 9.22 12.53 in polar form.

24-14: Parallel Circuit with Three 24-14: Parallel Circuit with Three Complex BranchesComplex 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 IT.

Convert IT from rectangular to polar form.

ZT can remain in polar form with its magnitude and phase angle or can be converted to rectangular form for its resistive and reactive components.

24-14: Parallel Circuit with Three 24-14: Parallel Circuit with Three Complex BranchesComplex Branches


Fig. 24-14:

complex numbers for ac circuits topics covered in chapter 24 24-1: positive and negative numbers...

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