Dorf

1

Symbols, Units,and Conversion Factors

Table C.1 Symbols and Units

Parameter or Variable Name Symbol SI English

Acceleration, angular a(t) rad/s2 rad/s2

Acceleration, translational a(t) m/s2 ft/s2

Friction, rotational b

Friction, translational b

Inertia, rotational J

Mass M kg slugs

Position, rotational u(t) rad rad

Position, translational x(t) m ft

Speed, rotational v(t) rad/s rad/s

Speed, translational v(t) m/s ft/s

Torque T(t) Nm ft-lb

ft-lb

rad>s2

Nm

rad>s2

lb

ft>sN

m>s

ft-lb

rad>sNm

rad>s

A P P E N D I X

C

ph_dorf_app 4/24/01 4:54 AM Page 1

2 Appendix C Symbols, Units, and Conversion Factors

Table C.2 Conversion Factors

To Convert Into Multiply by

Btu ft-lb 778.3Btu J 1054.8Btu/hr ft-lb/s 0.2162Btu/hr W 0.2931Btu/min hp 0.02356Btu/min kW 0.01757Btu/min W 17.57

cal J 4.182cm ft 3.281 � 10�2

cm in. 0.3937cm3 ft3 3.531 � 10�5

deg (angle) rad 0.01745deg/s rpm 0.1667dynes g 1.020 � 10�3

dynes lb 2.248 � 10�6

dynes N 10�5

ft/s miles/hr 0.6818ft/s miles/min 0.01136ft-lb g-cm 1.383 � 104

ft-lb oz-in. 192ft-lb/min Btu/min 1.286 � 10�3

ft-lb/s hp 1.818 � 10�3

ft-lb/s kW 1.356 � 10�3

20.11

g dynes 980.7g lb 2.205 � 10�3

g-cm2 oz-in2 5.468 � 10�3

g-cm oz-in. 1.389 � 10�2

g-cm ft-lb 1.235 � 10�5

hp Btu/min 42.44hp ft-lb/min 33,000hp ft-lb/s 550.0hp W 745.7

in. meters 2.540 � 10�2

in. cm 2.540

J Btu 9.480 � 10�4

J ergs 107

J ft-lb 0.7376J W-hr 2.778 � 10�4

kg lb 2.205kg slugs 6.852 � 10�2

To Convert Into Multiply by

kW Btu/min 56.92kW ft-lb/min 4.462 � 104

kW hp 1.341

miles (statute) ft 5280mph ft/min 88mph ft/s 1.467mph m/s 0.44704mils cm 2.540 � 10�3

mils in. 0.001min (angles) deg 0.01667min (angles) rad 2.909 � 10�4

Nm ft-lb 0.73756Nm dyne-cm 107

Nms W 1.0

oz g 28.349527oz-in. dyne-cm 70,615.7oz-in2 g-cm2 1.829 � 102

oz-in. ft-lb 5.208 � 10�3

oz-in. g-cm 72.01

lb(force) N 4.4482lb/ft3 g/cm3 0.01602lb-ft-s2 oz-in2 7.419 � 104

rad deg 57.30rad min 3438rad s 2.063 � 105

rad/s deg/s 57.30rad/s rpm 9.549rad/s rps 0.1592rpm deg/s 6.0rpm rad/s 0.1047

s (angle) deg 2.778 � 10�4

s (angle) rad 4.848 � 10�6

slugs (mass) kg 14.594slug-ft2 km2 1.3558

W Btu/hr 3.413W Btu/min 0.05688W ft-lb/min 44.27W hp 1.341 � 10�3

W Nm/s 1.0Wh Btu 3.413

oz-in.rpm

ft-lbrad>s

gram (g), joule (J), watt (W), newton (N), watt-hour (Wh)


3

Laplace Transform Pairs

A P P E N D I X

DTable D.1

F(s) f(t), t � 0

1. 1 d(t0), unit impulse at t � t0

2. 1/s 1, unit step

3. tn

4. e�at

5. tn � 1e�at

6. 1 � e�at

7. (e�at � e�bt)

8. [(a � a)e�at � (a � b)e�bt]

9. 1 � e�at � e�bt

10. � �

11. � �

12. a � e�at � e�bt

13. sin vt

14. cos vt0s

s2 + v2

v

s2 + v2

a1a - b21b - a2

b1a - a21b - a2

ab1s + a2s1s + a2 1s + b2

1a - c2e-ct

1a - c2 1b - c21a - b2e-bt

1c - b2 1a - b21a - a2e-at

1b - a2 1c - a2s + a

1s + a2 1s + b2 1s + c2

e-ct

1a - c2 1b - c2e-bt

1c - a2 1a - b2e-at

1b - a2 1c - a21

1s + a2 1s + b2 1s + c2

a1b - a2

b1b - a2

abs1s + a2 1s + b2

11b - a2

s + a1s + a2 1s + b2

11b - a2

11s + a2 1s + b2

as1s + a2

11n - 12!

11s + a2n

11s + a2

n!

sn+1

Table D.1 continued


Table D.1 Continued

F(s) f(t), t � 0

15. sin (vt � f),f � tan�1 v/a

16. e�at sin vt

17. e�at cos vt

18. [(a � a)2 � v2]1/2 e�at sin (vt � f),

f � tan�1

19. e�zvnt sin vn t, z� 1

20. � e�at sin (vt � f),

f � tan�1

21. 1 � e�zvnt sin ,

f � cos�1 z, z� 1

22. �1/2

e�at sin (vt � f),

f � tan�1 � tan�1

23. � ,f � tan�1 v

c - a

e-at sin 1vt + f2v�1c - a22 + v2�1>2

e-ct

1c - a22 + v2

1

1s + c2�1s + a22 + v2�

v

-av

a - a

1vc 1a - a22 + v2

a2 + v2 da

a2 + v2

1s + a2s�1s + a22 + v2�

1vn21 - z2 t + f2121 - z2

v2n

s1s2 + 2Zvns + v2n2

v

-a

1

v2a2 + v2

1a2 + v2

1

s�1s + a22 + v2�

21 - z2vn21 - z2

v2n

s2 + 2Zvns + v2n

v

a - a

1v

s + a1s + a22 + v2

1s + a21s + a22 + v2

v

1s + a22 + v2

2a2 + v2

v

s + as2 + v2

4 Appendix D Laplace Transform Pairs


5

An Introduction to Matrix Algebra

E.1 DEFINITIONS

In many situations, we must deal with rectangular arrays of numbers or functions.The rectangular array of numbers (or functions)

(E.1)

is known as a matrix. The numbers aij are called elements of the matrix, with the sub-script i denoting the row and the subscript j denoting the column.

A matrix with m rows and n columns is said to be a matrix of order (m, n) or al-ternatively called an m � n (m-by-n) matrix.When the number of the columns equalsthe number of rows (m � n), the matrix is called a square matrix of order n. It is com-mon to use boldfaced capital letters to denote an m � n matrix.

A matrix comprising only one column, that is, an m � 1 matrix, is known as acolumn matrix or, more commonly, a column vector. We will represent a column vec-tor with boldfaced lowercase letters as

(E.2)

Analogously, a row vector is an ordered collection of numbers written in a row—that is, a 1 � n matrix. We will use boldfaced lowercase letters to represent vectors.Therefore a row vector will be written as

(E.3)

with n elements.A few matrices with distinctive characteristics are given special names.A square

matrix in which all the elements are zero except those on the principal diagonal, a11,a22, . . . , ann, is called a diagonal matrix. Then, for example, a 3 � 3 diagonal matrixwould be

z = 3z1 z2p zn 4 ,

y = D y1

y2

oym

T

A = D a11

a21

oam1

a12

a22

oam2

pp

p

a1n

a2n

oamn

T

A P P E N D I X

E


(E.4)

If all the elements of a diagonal matrix have the value 1, then the matrix is known asthe identity matrix I, which is written as

(E.5)

When all the elements of a matrix are equal to zero, the matrix is called the zero, ornull matrix. When the elements of a matrix have a special relationship so that aij �aji, it is called a symmetrical matrix. Thus, for example, the matrix

(E.6)

is a symmetrical matrix of order (3, 3).

E.2 ADDITION AND SUBTRACTION OF MATRICES

The addition of two matrices is possible only for matrices of the same order.The sumof two matrices is obtained by adding the corresponding elements.Thus if the elementsof A are aij and the elements of B are bij, and if

C � A � B, (E.7)

then the elements of C that are cij are obtained as

cij � aij � bij. (E.8)

For example, the matrix addition for two 3 � 3 matrices is as follows:

(E.9)

From the operation used for performing the operation of addition, we note that theprocess is commutative; that is,

(E.10)

Also we note that the addition operation is associative, so that

(E.11)

To perform the operation of subtraction, we note that if a matrix A is multipliedby a constant a, then every element of the matrix is multiplied by this constant.There-fore we can write

(A � B) � C � A � (B � C).

A � B � B � A.

C = C210

1-1

6

032S + C8

14

232

101S = C10

24

328

133S .

H = C 3-2

1

-264

148S

I = D10o0

01o0

pppp

00o1

T .

B = Cb11

00

0b22

0

00

b33

S .

6 Appendix E An Introduction to Matrix Algebra


(E.12)

Then to carry out a subtraction operation, we use a� �1, and �A is obtained by mul-tiplying each element of A by �1. For example,

(E.13)

E.3 MULTIPLICATION OF MATRICES

The multiplication of two matrices AB requires that the number of columns of Abe equal to the number of rows of B. Thus if A is of order m � n and B is of order n� q, then the product is of order m � q. The elements of a product

C � AB (E.14)

are found by multiplying the ith row of A and the jth column of B and summing theseproducts to give the element cij. That is,

(E.15)

Thus we obtain c11, the first element of C, by multiplying the first row of A by the firstcolumn of B and summing the products of the elements.We should note that, in gen-eral, matrix multiplication is not commutative; that is

(E.16)

Also we note that the multiplication of a matrix of m � n by a column vector (ordern � 1) results in a column vector of order m � 1.

A specific example of multiplication of a column vector by a matrix is

(E.17)

Note that A is of order 2 � 3, and y is of order 3 � 1. Therefore the resulting matrixx is of order 2 � 1, which is a column vector with two rows. There are two elementsof x, and

x1 � (a11y1 � a12y2 � a13y3) (E.18)

is the first element obtained by multiplying the first row of A by the first (and only)column of y.

Another example, which the reader should verify, is

(E.19)C = AB = B 2-1

-12R B 3

-12

-2R = B 7

-56

-6R .

x = Ay = Ba11

a21

a12

a22

a13

a23R Cy1

y2

y3

S = B1a11y1 + a12y2 + a13y321a21y1 + a22y2 + a23y32R .

AB � BA.

cij = ai1b1j + ai2b2j + p + aiqbqj = aq

k=1 aikbkj.

C = B - A = B24

12R - B6

311R = B-4

101R .

aA = D aa11

aa12

oaam1

aa12

aa22

oaam2

pp

p

aa1n

aa2n

oaamn

T .

Section E.3 Multiplication of Matrices 7


For example, the element c22 is obtained as c22 � �1(2) � 2(�2) � �6.Now we are able to use this definition of multiplication in representing a set of

simultaneous linear algebraic equations by a matrix equation. Consider the follow-ing set of algebraic equations:

3x1 � 2x2 � x3 � u1,2x1 � x2 � 6x3 � u2,4x1 � x2 � 2x3 � u3. (E.20)

We can identify two column vectors as

(E.21)

Then we can write the matrix equation

Ax � u, (E.22)

where

We immediately note the utility of the matrix equation as a compact form of a set ofsimultaneous equations.

The multiplication of a row vector and a column vector can be written as

(E.23)

Thus we note that the multiplication of a row vector and a column vector results ina number that is a sum of a product of specific elements of each vector.

As a final item in this section, we note that the multiplication of any matrix bythe identity matrix results in the original matrix, that is, AI � A.

E.4 OTHER USEFUL MATRIX OPERATIONS AND DEFINITIONS

The transpose of a matrix A is denoted in this text as AT. One will often find the no-tation A' for AT in the literature. The transpose of a matrix A is obtained by inter-changing the rows and columns of A. For example, if

then

A = C 61

-2

043

21

-1S ,

xy = 3x1 x2p xn 4 Dy1

y2

oyn

T = x1 y1 + x2 y2 + p + xn yn.

A = C324

21

-1

162S .

x = Cx1

x2

x3

S and u = Cu1

u2

u3

S .



(E.24)

Therefore we are able to denote a row vector as the transpose of a column vector andwrite

(E.25)

Because xT is a row vector, we obtain a matrix multiplication of xT by x as follows:

(E.26)

Thus the multiplication xTx results in the sum of the squares of each element of x.The transpose of the product of two matrices is the product in reverse order of

their transposes, so that

(E.27)

The sum of the main diagonal elements of a square matrix A is called the traceof A, written as

(E.28)

The determinant of a square matrix is obtained by enclosing the elements of thematrix A within vertical bars; for example,

(E.29)

If the determinant of A is equal to zero, then the determinant is said to be singular.The value of a determinant is determined by obtaining the minors and cofactors ofthe determinants. The minor of an element aij of a determinant of order n is a deter-minant of order (n � 1) obtained by removing the row i and the column j of the orig-inal determinant.The cofactor of a given element of a determinant is the minor of theelement with either a plus or minus sign attached; hence

cofactor of aij � aij � (�1)i�jMij,

where Mij is the minor of aij. For example, the cofactor of the element a23 of

(E.30)

is

(E.31)

The value of a determinant of second order (2 � 2) is

a23 = 1-125M23 = - 2a11

a31

a12

a322 .

det A = 3a11

a21

a31

a12

a22

a32

a13

a23

a33

3

det A = 2a11

a21

a12 2a21

= a11a22 - a12a21.

tr A � a11 � a22 � … � ann.

(AB)T � BTAT.

xTx = 3x1 x2p xn 4 Dx1

x2

oxn

T = x21 + x2

2 + p + x2n.

xT = 3x1 x2p xn 4 .

AT = C602

141

-23

-1S .

Section E.4 Other Useful Matrix Operations and Definitions 9


(E.32)

The general nth-order determinant has a value given by

with i chosen for one row, (E.33)

or

with j chosen for one column. (E.33)

That is, the elements aij are chosen for a specific row (or column), and that entire row(or column) is expanded according to Eq. (E.33). For example, the value of a specific3 � 3 determinant is

(E.34)

where we have expanded in the first column.The adjoint matrix of a square matrix A is formed by replacing each element aij

by the cofactor aij and transposing. Therefore

(E.35)

E.5 MATRIX INVERSION

The inverse of a square matrix A is written as A�1 and is defined as satisfying the re-lationship

A�1A � AA�1 � I. (E.36)

The inverse of a matrix A is

(E.37)A-1 =adjoint of A

det A

adjoint A = Da11

a21

oan1

a12

a22

oan2

pp

p

a1n

a2n

oann

TT

= Da11

a12

oa1n

a21

a22

oa2n

pp

p

an1

an2

oann

T .

= 21-12 - 1-52 + 2132 = 9,

= 2 201

102 - 1 23

1502 + 2 23

0512

det A = det C212

301

510S

det A = an

i=1 aijaij

det A = an

j=1 aijaij

2a11

a21

a12

a222 = 1a11a22 - a21a122.



when the det A is not equal to zero. For a 2 � 2 matrix we have the adjoint matrix

(E.38)

and the det A � a11a22 � a12a21. Consider the matrix

(E.39)

The determinant has a value det A � �7. The cofactor a11 is

(E.40)

In a similar manner we obtain

(E.41)

E.6 MATRICES AND CHARACTERISTIC ROOTS

A set of simultaneous linear algebraic equations can be represented by the matrixequation

y � Ax, (E.42)

where the y vector can be considered as a transformation of the vector x. We mightask whether it may happen that a vector y may be a scalar multiple of x. Trying y � lx, where l is a scalar, we have

lx � Ax. (E.43)

Alternatively Eq. (E.43) can be written as

lx � Ax � (lI � A)x � 0, (E.44)

where I � identity matrix. Thus the solution for x exists if and only if

(E.45)

This determinant is called the characteristic determinant of A. Expansion of the de-terminant of Eq. (E.45) results in the characteristic equation. The characteristic equa-tion is an nth-order polynomial in l. The n roots of this characteristic equation arecalled the characteristic roots. For every possible value li (i � 1, 2, . . . , n) of the nth-order characteristic equation, we can write

(liI � A)xi � 0. (E.46)

The vector xi is the characteristic vector for the ith root. Let us consider the matrix

det (lI � A) � 0.

A-1 =adjoint A

det A= a- 1

7b C 3

-2-2

-511

112

-5S .

a11 = 1-122 2-1-1

412 = 3.

A = C120

2-1-1

341S .

adjoint A = B a22

-a21

-a12

a11R,

Section E.6 Matrices and Characteristic Roots 11


(E.47)

The characteristic equation is found as follows:

(E.48)

The roots of the characteristic equation are l1 � 1, l2 � �1, l3 � 3. When l �l1 � 1, we find the first characteristic vector from the equation

Ax1 � l1x1, (E.49)

and we have x � k , where k is an arbitrary constant usually chosenequal to 1. Similarly, we find

and

(E.50)

E.7 THE CALCULUS OF MATRICES

The derivative of a matrix A � A(t) is defined as

(E.51)

That is, the derivative of a matrix is simply the derivative of each element aij(t) of thematrix.

The matrix exponential function is defined as the power series

(E.52)

where A2 � AA, and, similarly, Ak implies A multiplied k times. This series can beshown to be convergent for all square matrices. Also a matrix exponential that is afunction of time is defined as

(E.53)

If we differentiate with respect to time, then we have

(E.54)ddt1eAt2 = AeAt.

eAt = a�

k=0 Aktk

k!.

exp �A� = eA = I +A1!

+A2

2!+ p +

Ak

k!+ p = a

�

k=0 Ak

k!,

ddt

�A1t2� = Cda111t2>dto

dan11t2>dt

da121t2>dto

dan21t2>dt

p

p

da1n1t2>dto

dann1t2>dtS .

xT3 = 32 3 -1 4 .

xT2 30 1 -1 4 ,

31 -1 0 4T1

det C 1l - 22-2

1

-11l - 32

1

-1-4

1l + 22S = 1-l3 + 3l2 + l - 32 = 0.

A = C 22

-1

13

-1

14

-2S .



Therefore for a differential equation

(E.55)

we might postulate a solution x � eAtc � fc, where the matrix f is f � eAt, and c isan unknown column vector. Then we have

(E.56)

or

AeAt � AeAt, (E.57)

and we have in fact satisfied the relationship, Eq. (E.55). Then the value of c is sim-ply x(0), the initial value of x, because when t � 0, we have x(0) � c. Therefore thesolution to Eq. (E.55) is

(E.58)x(t) � eAtx(0).

dxdt

= Ax,

dxdt

= Ax,

Section E.7 The Calculus of Matrices 13



15

Decibel Conversion

Table F.1

M 0 1 2 3 4 5 6 7 8 9

0.0 m � �40.00 �33.98 �30.46 �27.96 �26.02 �24.44 �23.10 �21.94 �20.920.1 �20.00 �19.17 �18.42 �17.72 �17.08 �16.48 �15.92 �15.39 �14.89 �14.420.2 �13.98 �13.56 �13.15 �12.77 �12.40 �12.04 �11.70 �11.37 �11.06 �10.750.3 �10.46 �10.17 �9.90 �9.63 �9.37 �9.12 �8.87 �8.64 �8.40 �8.180.4 �7.96 �7.74 �7.54 �7.33 �7.13 �6.94 �6.74 �6.56 �6.38 �6.200.5 �6.02 �5.85 �5.68 �5.51 �5.35 �5.19 �5.04 �4.88 �4.73 �4.580.6 �4.44 �4.29 �4.15 �4.01 �3.88 �3.74 �3.61 �3.48 �3.35 �3.220.7 �3.10 �2.97 �2.85 �2.73 �2.62 �2.50 �2.38 �2.27 �2.16 �2.050.8 �1.94 �1.83 �1.72 �1.62 �1.51 �1.41 �1.31 �1.21 �1.11 �1.010.9 �0.92 �0.82 �0.72 �0.63 �0.54 �0.45 �0.35 �0.26 �0.18 �0.091.0 0.00 0.09 0.17 0.26 0.34 0.42 0.51 0.59 0.67 0.751.1 0.83 0.91 0.98 1.06 1.14 1.21 1.29 1.36 1.44 1.511.2 1.58 1.66 1.73 1.80 1.87 1.94 2.01 2.08 2.14 2.211.3 2.28 2.35 2.41 2.48 2.54 2.61 2.67 2.73 2.80 2.861.4 2.92 2.98 3.05 3.11 3.17 3.23 3.29 3.35 3.41 3.461.5 3.52 3.58 3.64 3.69 3.75 3.81 3.86 3.92 3.97 4.031.6 4.08 4.14 4.19 4.24 4.30 4.35 4.40 4.45 4.51 4.561.7 4.61 4.66 4.71 4.76 4.81 4.86 4.91 4.96 5.01 5.061.8 5.11 5.15 5.20 5.25 5.30 5.34 5.39 5.44 5.48 5.531.9 5.58 5.62 5.67 5.71 5.76 5.80 5.85 5.89 5.93 5.982. 6.02 6.44 6.85 7.23 7.60 7.96 8.30 8.63 8.94 9.253. 9.54 9.83 10.10 10.37 10.63 10.88 11.13 11.36 11.60 11.824. 12.04 12.26 12.46 12.67 12.87 13.06 13.26 13.44 13.62 13.805. 13.98 14.15 14.32 14.49 14.65 14.81 14.96 15.12 15.27 15.426. 15.56 15.71 15.85 15.99 16.12 16.26 16.39 16.52 16.65 16.787. 16.90 17.03 17.15 17.27 17.38 17.50 17.62 17.73 17.84 17.958. 18.06 18.17 18.28 18.38 18.49 18.59 18.69 18.79 18.89 18.999. 19.08 19.18 19.28 19.37 19.46 19.55 19.65 19.74 19.82 19.91

0. 1. 2. 3. 4. 5. 6. 7. 8. 9.

Decibels � 20 log10 M.

A P P E N D I X

F



17

Complex Numbers

G.1 A COMPLEX NUMBER

We all are familiar with the solution of the algebraic equation

x2 � 1 � 0, (G.1)

which is x � 1. However, we often encounter the equation

x2 � 1 � 0. (G.2)

A number that satisfies Eq. (G.2) is not a real number. We note that Eq. (G.2) maybe written as

x2 � �1, (G.3)

and we denote the solution of Eq. (G.3) by the use of an imaginary number j1, sothat

j 2 � �1, (G.4)

and

(G.5)

An imaginary number is defined as the product of the imaginary unit j with a realnumber.Thus we may, for example, write an imaginary number as jb.A complex num-ber is the sum of a real number and an imaginary number, so that

(G.6)

where a and b are real numbers.We designate a as the real part of the complex num-ber and b as the imaginary part and use the notation

Re{c} � a, (G.7)

and

Im{c} � b. (G.8)

c = a + jb

j = 4-1.

A P P E N D I X

G


G.2 RECTANGULAR, EXPONENTIAL, AND POLAR FORMS

The complex number a � jb may be represented on a rectangular coordinate placecalled a complex plane. The complex plane has a real axis and an imaginary axis, asshown in Fig. G.1. The complex number c is the directed line identified as c with co-ordinates a, b.The rectangular form is expressed in Eq. (G.6) and pictured in Fig. G.1.

An alternative way to express the complex number c is to use the distance fromthe origin and the angle u, as shown in Fig. G.2. The exponential form is written as

(G.9)

where

r � (a2 � b2)1/2, (G.10)

and

u � tan�1(b/a). (G.11)

Note that a � r cos u and b � r sin u.The number r is also called the magnitude of c, denoted as �c�.The angle u can also

be denoted by the form . Thus we may represent the complex number in polarform as

(G.12)

EXAMPLE G.1 Exponential and polar forms

Express c � 4 � j3 in exponential and polar form.Solution First sketch the complex plane diagram as shown in Fig. G.3.Then find

r as

r � (42 � 32)1/2 � 5,

and u as

u � tan�1(3/4) � 36.9°.

c = �c2lu = rlu.

lu

c � re ju,

18 Appendix G Complex Numbers

Imaginary axis

b

0 aReal axis

c � a � jb

FIGURE G.1 Rectangular form ofa complex number.

FIGURE G.2 Exponential form ofa complex number.

FIGURE G.3 Complex plane forExample G.1.

b

0 a

c � re j�

Im

Re�

j3

r

0 4

Im

Re�


The exponential form is then

c � 5e j36.9°.

The polar form is

G.3 MATHEMATICAL OPERATIONS

The conjugate of the complex number c � a � jb is called c* and is defined as

(G.13)

In polar form we have

(G.14)

To add or subtract two complex numbers, we add (or subtract) their real parts andtheir imaginary parts. Therefore if c � a � jb and d � f � jg, then

c � d � (a � jb) � (f � jg) � (a � f) � j(b � g). (G.15)

The multiplication of two complex numbers is obtained as follows (note j2 � �1):

(G.16)

Alternatively we use the polar form to obtain

(G.17)

where

Division of one complex number by another complex number is easily obtained usingthe polar form as follows:

(G.18)

It is easiest to add and subtract complex numbers in rectangular form and tomultiply and divide them in polar form.

A few useful relations for complex numbers are summarized in Table G.1.

cd

=r1lu1

r2lu2=

r1

r2 lu1 - u2.

c = r1lu1, and d = r2lu2.

cd = 1r1lu12 1r2lu22 = r1r2lu1 + u2,

= 1af - bg2 + j1ag + bf2. = af + jag + jbf + j2bg

cd = 1a + jb2 1f + jg2

c* = rl-u.

c* � a � jb.

c = 5l36.9°. �

Section G.3 Mathematical Operations 19


EXAMPLE G.2 Complex number operations

Find c � d, c � d, cd, and c/d when c � 4 � j 3 and d � 1 � j.Solution First we will express c and d in polar form as

Then, for addition, we have

c � d � (4 � j3) � (1 � j) � 5 � j2.

For subtraction we have

c � d � (4 � j3) � (1 � j) � 3 � j4.

For multiplication we use the polar form to obtain

For division we have

cd

=5l36.9°22l-45°

=522

l81.9°. �

cd = 15l36.9°2 122l-45°2 = 522l-8.1°.

c = 5l36.9°, and d = 22l-45°.

20 Appendix G COMPLEX NUMBERS

Table G.1 Useful Relationships forComplex Numbers

(1) � �j

(2) (�j)( j) � 1

(3) j2 � �1

(4) 1 � j

(5) ck � rklk

l>2

1j


21

z-Transform Pairs

Table H.1

x(t) X(s) X(z)

1. d(t) � 1 1

2. d(t � kT) � e�kTs z�k

3. u(t), unit step 1/s

4. t 1/s2

5. t2 2/s3

6. e�at

7. 1 � e�at

8. te�at

9. t2e�at

10. be�bt � ae�at

11. sin vt

12. cos vt

13. e�at sin vt1ze-aT sin vT2

z2 - 2ze-aT cos vT + e-2aT

v

1s + a22 + v2

z1z - cos vT2z2 - 2z cos vT + 1

s

s2 + v2

z sin vT

z2 - 2z cos vT + 1v

s2 + v2

z�z1b - a2 - 1be-aT - ae-bT2�1z - e-aT2 1z - e-bT2

1b - a2s1s + a2 1s + b2

T2e-aTz1z + e-aT21z - e-aT23

21s + a23

Tze-aT

1z - e-aT221

1s + a22

11 - e-aT2z1z - 12 1z - e-aT2

as1s + a2

z

z - e-aT

1s + a

T2z1z + 121z - 123

Tz

1z - 122

zz - 1

e10

t = kT,t � kT

e10

t = 0,t = kT, k � 0

A P P E N D I X

H


Table H.1 (continued)

14. e�at cos vt

15. 1 � e�at

A � 1 � e�aT cos bT � e�aT sin bT

B � e�2aT � e�aT sin bT � e�aT cos bTab

ab

z1Az + B21z - 12�z2 - 2e-aT1cos bT2z + e-2aT�

a2 + b2

s�1s + a22 + b2�acos bt +

ab sin btb

z2 - ze-aT cos vT

z2 - 2ze-aT cos vT + e-2aT

s + a

1s + a22 + v2

22 Appendix H Z-TRANSFORM PAIRS


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