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APPENDIX

C Table C.1

Symbols, Units, and Conversion Factors

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

Nm rad>s

ft-lb rad>s

Friction, translational

b

N m>s

lb ft>s

Inertia, rotational

J

Nm rad>s2

rad>s2

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

1

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2

Appendix C

Symbols, Units, and Conversion Factors

Table C.2

Conversion Factors

To Convert

Into

Multiply by

Btu Btu Btu/hr Btu/hr Btu/min Btu/min Btu/min

ft-lb J ft-lb/s W hp kW W

778.3 1054.8 0.2162 0.2931 0.02356 0.01757 17.57

cal cm cm cm3

J ft in. ft3

4.182 3.281 102 0.3937 3.531 105

deg (angle) deg/s dynes dynes dynes

rad rpm g lb N

0.01745 0.1667 1.020 103 2.248 106 105

ft/s ft/s ft-lb ft-lb ft-lb/min ft-lb/s ft-lb/s ft-lb rad>s

miles/hr miles/min g-cm oz-in. Btu/min hp kW oz-in. rpm

0.6818 0.01136 1.383 104 192 1.286 103 1.818 103 1.356 103

g g g-cm2 g-cm g-cm

dynes lb oz-in2 oz-in. ft-lb

980.7 2.205 103 5.468 103 1.389 102 1.235 105

hp hp hp hp

Btu/min ft-lb/min ft-lb/s W

42.44 33,000 550.0 745.7

in. in.

meters cm

2.540 102 2.540

J J J J

Btu ergs ft-lb W-hr

9.480 104 107 0.7376 2.778 104

20.11

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

kg kg To Convert

lb slugs Into

2.205 6.852 102 Multiply by

kW kW kW

Btu/min ft-lb/min hp

56.92 4.462 104 1.341

miles (statute) mph mph mph mils mils min (angles) min (angles)

ft ft/min ft/s m/s cm in. deg rad

5280 88 1.467 0.44704 2.540 103 0.001 0.01667 2.909 104

Nm Nm Nms

ft-lb dyne-cm W

0.73756 107 1.0

oz oz-in. oz-in2 oz-in. oz-in.

g dyne-cm g-cm2 ft-lb g-cm

28.349527 70,615.7 1.829 102 5.208 103 72.01

lb(force) lb/ft3 lb-ft-s2

N g/cm3 oz-in2

4.4482 0.01602 7.419 104

rad rad rad rad/s rad/s rad/s rpm rpm

deg min s deg/s rpm rps deg/s rad/s

57.30 3438 2.063 105 57.30 9.549 0.1592 6.0 0.1047

s (angle) s (angle) slugs (mass) slug-ft2

deg rad kg km2

2.778 104 4.848 106 14.594 1.3558

W W W W W Wh

Btu/hr Btu/min ft-lb/min hp Nm/s Btu

3.413 0.05688 44.27 1.341 103 1.0 3.413

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APPENDIX

D

Laplace Transform Pairs

Table D.1 F(s)

f(t), t 0

1. 1 2. 1/s n! 3. n+1 s 1 4. 1s + a2 1 5. 1s + a2 n a 6. s1s + a2 1 7. 1s + a2 1s + b2 s+a 8. 1s + a2 1s + b2 ab 9. s1s + a21s + b2

d(t0), unit impulse at t t0 1, unit step

10. 11. 12.

1 1s + a2 1s + b21s + c2 s+a 1s + a2 1s + b21s + c2 ab1s + a2

s1s + a21s + b2 v 13. 2 s + v2 s 14. 2 s + v2

tn eat 1 tn 1eat 1n - 12! 1 eat 1 (eat ebt) 1b - a2 1 [(a a)eat (a b)ebt] 1b - a2 a b 1 eat ebt 1b - a2 1b - a2

e-at e-bt e-ct 1b - a21c - a2 1c - a21a - b2 1a - c21b - c2 1a - a2e-at

1a - b2e-bt

1a - c2e-ct

1b - a21c - a2 1c - b21a - b2 1a - c21b - c2 b1a - a2 at a1a - b2 bt a e e 1b - a2 1b - a2 sin vt cos vt0

Table D.1 continued

3

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4

Appendix D

Table D.1

Laplace Transform Pairs

Continued

F(s)

f(t), t 0

s+a s2 + v2 v 16. 1s + a2 2 + v2

2a2 + v2 sin (vt f), f tan1 v/a v

15.

17. 18.

19. 20.

21.

22.

23.

1s + a2

1s + a2 2 + v2

s+a 1s + a2 2 + v2 v2n 2

s + 2Zvns + v2n 1 s1s + a2 2 + v2

v2n

s1s + 2Zvns + v2n 2 2

1s + a2

s1s + a2 2 + v2

1

1s + c21s + a2 2 + v2

eat sin vt eat cos vt 1 [(a a)2 v2]1/2 eat sin (vt f), v v f tan1 a-a vn n ezv t sin vn 21 - z2 t, z 1 21 - z2 1 1 eat sin (vt f), a2 + v2 v2a2 + v2 v f tan1 -a 1 n 1 ezv t sin 1vn 21 - z2 t + f2 , 21 - z2 f cos1 z, z 1 a 1 1a - a2 + v 1/2 at c d e sin (vt f), a2 + v2 v a2 + v2 v v f tan1 tan1 a-a -a e-at sin 1vt + f2 e-ct v , f tan1 2 2 2 2 1>2 c a 1c - a2 + v v1c - a2 + v 2

2

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APPENDIX

An Introduction to Matrix Algebra

E E.1 DEFINITIONS

In many situations, we must deal with rectangular arrays of numbers or functions. The rectangular array of numbers (or functions) a11 a12 a a22 A = D 21 o o am1 am2

p p p

a1n a2n T o amn

(E.1)

is known as a matrix. The numbers aij are called elements of the matrix, with the subscript 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 alternatively called an m n (m-by-n) matrix.When the number of the columns equals the number of rows (m n), the matrix is called a square matrix of order n. It is common 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 a column matrix or, more commonly, a column vector. We will represent a column vector with boldfaced lowercase letters as y1 y y = D 2T o ym

(E.2)

Analogously, a row vector is an ordered collection of numbers written in a rowâ€” that is, a 1 n matrix. We will use boldfaced lowercase letters to represent vectors. Therefore a row vector will be written as z = 3z1

z2

p

zn 4,

(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 matrix would be

5

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6

Appendix E

An Introduction to Matrix Algebra

b11 B= C 0 0

0 b22 0

0 0 S.

(E.4)

b33

If all the elements of a diagonal matrix have the value 1, then the matrix is known as the identity matrix I, which is written as 1 0 I=D o 0

0 1 o 0

p p p p

0 0 T. o 1

(E.5)

When all the elements of a matrix are equal to zero, the matrix is called the zero, or null 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 3 H = C -2 1

1 4S 8

-2 6 4

(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 sum of two matrices is obtained by adding the corresponding elements.Thus if the elements of 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: 2 C = C1 0

1 -1 6

0 8 3S + C1 2 4

2 3 2

1 10 3 0S = C 2 2 1 4 8

1 3S. 3

(E.9)

From the operation used for performing the operation of addition, we note that the process is commutative; that is, A B B A.

(E.10)

Also we note that the addition operation is associative, so that (A B) C A (B C).

(E.11)

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

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Section E.3

7

Multiplication of Matrices

aa11 aa12 aa aa22 aA = D 12 o o aam1 aam2

p p p

aa1n aa2n T. o aamn

(E.12)

Then to carry out a subtraction operation, we use a 1, and A is obtained by multiplying each element of A by 1. For example, 2 4

C=B-A=B

1 6 R-B 2 3

1 -4 R=B 1 1

0 R. 1

(E.13)

E.3 MULTIPLICATION OF MATRICES The multiplication of two matrices AB requires that the number of columns of A be 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 these products to give the element cij. That is, q

cij = ai1b1j + ai2b2j + p + aiqbqj = a aikbkj.

(E.15)

k=1

Thus we obtain c11, the first element of C, by multiplying the first row of A by the first column of B and summing the products of the elements. We should note that, in general, matrix multiplication is not commutative; that is AB BA.

(E.16)

Also we note that the multiplication of a matrix of m n by a column vector (order n 1) results in a column vector of order m 1. A specific example of multiplication of a column vector by a matrix is a x = Ay = B 11 a21

a12 a22

y1 a13 1a y + a12y2 + a13y3 2 R C y2 S = B 11 1 R. a23 1a21y1 + a22y2 + a23y3 2 y3

(E.17)

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

C = AB = B

3 -1 RB 2 -1

2 7 R=B -2 -5

6 R. -6

(E.19)

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8

Appendix E

An Introduction to Matrix Algebra

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 following 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 x1 x = C x2 S x3

u1 u = C u2 S. u3

and

(E.21)

Then we can write the matrix equation Ax u,

(E.22)

where 3 2 A = C2 1 4 -1

1 6 S. 2

We immediately note the utility of the matrix equation as a compact form of a set of simultaneous equations. The multiplication of a row vector and a column vector can be written as xy = 3x1 x2

p

y1 y2 xn 4 D T = x1 y1 + x2 y2 + p + xn yn. o yn

(E.23)

Thus we note that the multiplication of a row vector and a column vector results in a 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 by the 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 notation A' for AT in the literature. The transpose of a matrix A is obtained by interchanging the rows and columns of A. For example, if 6 A= C 1 -2 then

0 4 3

2 1 S, -1

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Section E.4

9

Other Useful Matrix Operations and Definitions

6 1 AT = C 0 4 2 1

-2 3 S. -1

(E.24)

Therefore we are able to denote a row vector as the transpose of a column vector and write xT = 3x1

x2

p

xn 4.

(E.25)

Because xT is a row vector, we obtain a matrix multiplication of xT by x as follows: xTx = 3x1

x2

p

x1 x2 xn 4 D T = x21 + x22 + p + x2n. o xn

(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 (AB)T BTAT.

(E.27)

The sum of the main diagonal elements of a square matrix A is called the trace of A, written as tr A a11 a22 â€Ś ann.

(E.28)

The determinant of a square matrix is obtained by enclosing the elements of the matrix A within vertical bars; for example, det A = 2

a11 a12 2 = a11a22 - a12a21. a21 a21

(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 of the determinants. The minor of an element aij of a determinant of order n is a determinant of order (n 1) obtained by removing the row i and the column j of the original determinant.The cofactor of a given element of a determinant is the minor of the element with either a plus or minus sign attached; hence cofactor of aij aij (1)ijMij, where Mij is the minor of aij. For example, the cofactor of the element a23 of a11 det A = 3 a21 a31

a12 a22 a32

a13 a23 3 a33

(E.30)

is a23 = 1-12 5M23 = - 2

a11 a31

a12 2. a32

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

(E.31)

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10

Appendix E

An Introduction to Matrix Algebra

a11 a21

2

a12 2 = 1a11a22 - a21a12 2. a22

(E.32)

The general nth-order determinant has a value given by n

det A = a aijaij

with i chosen for one row,

(E.33)

with j chosen for one column.

(E.33)

j=1

or n

det A = a aijaij i=1

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 specific 3 3 determinant is 2 det A = det C 1 2 =2 2

0 1

3 0 1

5 1S 0

1 3 2 -1 2 0 1

5 3 2 +2 2 0 0

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

5 2 1 (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 a11 a adjoint A = D 21 o an1

a12 a22 o an2

p p p

a1n T a2n T = o ann

a11 a21 a a22 D 12 o o a1n a2n

p p p

an1 an2 T. o ann

(E.35)

E.5 MATRIX INVERSION The inverse of a square matrix A is written as A1 and is defined as satisfying the relationship A1A AA1 I.

(E.36)

adjoint of A det A

(E.37)

The inverse of a matrix A is A-1 =

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Section E.6

11

Matrices and Characteristic Roots

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

a22 -a21

-a12 R, a11

(E.38)

and the det A a11a22 a12a21. Consider the matrix 1 A = C2 0

2 -1 -1

3 4 S. 1

(E.39)

The determinant has a value det A 7. The cofactor a11 is a11 = 1-12 2 2

-1 -1

4 2 = 3. 1

(E.40)

In a similar manner we obtain -1

A

3 -5 adjoint A 1 = = a- b C -2 1 det A 7 -2 1

11 2 S. -5

(E.41)

E.6 MATRICES AND CHARACTERISTIC ROOTS A set of simultaneous linear algebraic equations can be represented by the matrix equation y Ax,

(E.42)

where the y vector can be considered as a transformation of the vector x. We might ask 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 det (lI A) 0.

(E.45)

This determinant is called the characteristic determinant of A. Expansion of the determinant of Eq. (E.45) results in the characteristic equation. The characteristic equation is an nth-order polynomial in l. The n roots of this characteristic equation are called the characteristic roots. For every possible value li (i 1, 2, . . . , n) of the nthorder 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

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12

Appendix E

An Introduction to Matrix Algebra

2 A= C 2 -1

1 3 -1

1 4 S. -2

(E.47)

The characteristic equation is found as follows: 1l - 22 det C -2 1

-1 1l - 32 1

-1 -4 S = 1-l3 + 3l2 + l - 32 = 0. 1l + 22

(E.48)

Ax1 l1x1,

(E.49)

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 and we have xT1 k 31 -1 equal to 1. Similarly, we find

04 , where k is an arbitrary constant usually chosen xT2 30

1

-14,

and xT3 = 32

3

(E.50)

-14.

E.7 THE CALCULUS OF MATRICES The derivative of a matrix A A(t) is defined as d A1t2 = C dt

da11 1t2>dt da12 1t2>dt o o dan1 1t2>dt dan2 1t2>dt

p p

da1n 1t2>dt o S. dann 1t2>dt

(E.51)

That is, the derivative of a matrix is simply the derivative of each element aij(t) of the matrix. The matrix exponential function is defined as the power series exp A = eA = I +

A A2 p Ak p Ak + + + + = a , 1! 2! k! k=0 k!

(E.52)

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

eAt = a k=0

Aktk . k!

(E.53)

If we differentiate with respect to time, then we have d At 1e 2 = AeAt. dt

(E.54)

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Section E.7

13

The Calculus of Matrices

Therefore for a differential equation dx = Ax, dt

(E.55)

we might postulate a solution x eAtc fc, where the matrix f is f eAt, and c is an unknown column vector. Then we have dx = Ax, dt

(E.56)

AeAt AeAt,

(E.57)

or

and we have in fact satisfied the relationship, Eq. (E.55). Then the value of c is simply x(0), the initial value of x, because when t 0, we have x(0) c. Therefore the solution to Eq. (E.55) is x(t) eAtx(0).

(E.58)

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APPENDIX

F

Decibel Conversion

Table F.1 M 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2. 3. 4. 5. 6. 7. 8. 9.

0 m 20.00 13.98 10.46 7.96 6.02 4.44 3.10 1.94 0.92 0.00 0.83 1.58 2.28 2.92 3.52 4.08 4.61 5.11 5.58 6.02 9.54 12.04 13.98 15.56 16.90 18.06 19.08 0.

1 40.00 19.17 13.56 10.17 7.74 5.85 4.29 2.97 1.83 0.82 0.09 0.91 1.66 2.35 2.98 3.58 4.14 4.66 5.15 5.62 6.44 9.83 12.26 14.15 15.71 17.03 18.17 19.18 1.

2 33.98 18.42 13.15 9.90 7.54 5.68 4.15 2.85 1.72 0.72 0.17 0.98 1.73 2.41 3.05 3.64 4.19 4.71 5.20 5.67 6.85 10.10 12.46 14.32 15.85 17.15 18.28 19.28 2.

3 30.46 17.72 12.77 9.63 7.33 5.51 4.01 2.73 1.62 0.63 0.26 1.06 1.80 2.48 3.11 3.69 4.24 4.76 5.25 5.71 7.23 10.37 12.67 14.49 15.99 17.27 18.38 19.37 3.

4 27.96 17.08 12.40 9.37 7.13 5.35 3.88 2.62 1.51 0.54 0.34 1.14 1.87 2.54 3.17 3.75 4.30 4.81 5.30 5.76 7.60 10.63 12.87 14.65 16.12 17.38 18.49 19.46 4.

5 26.02 16.48 12.04 9.12 6.94 5.19 3.74 2.50 1.41 0.45 0.42 1.21 1.94 2.61 3.23 3.81 4.35 4.86 5.34 5.80 7.96 10.88 13.06 14.81 16.26 17.50 18.59 19.55 5.

6 24.44 15.92 11.70 8.87 6.74 5.04 3.61 2.38 1.31 0.35 0.51 1.29 2.01 2.67 3.29 3.86 4.40 4.91 5.39 5.85 8.30 11.13 13.26 14.96 16.39 17.62 18.69 19.65 6.

7 23.10 15.39 11.37 8.64 6.56 4.88 3.48 2.27 1.21 0.26 0.59 1.36 2.08 2.73 3.35 3.92 4.45 4.96 5.44 5.89 8.63 11.36 13.44 15.12 16.52 17.73 18.79 19.74 7.

8 21.94 14.89 11.06 8.40 6.38 4.73 3.35 2.16 1.11 0.18 0.67 1.44 2.14 2.80 3.41 3.97 4.51 5.01 5.48 5.93 8.94 11.60 13.62 15.27 16.65 17.84 18.89 19.82 8.

9 20.92 14.42 10.75 8.18 6.20 4.58 3.22 2.05 1.01 0.09 0.75 1.51 2.21 2.86 3.46 4.03 4.56 5.06 5.53 5.98 9.25 11.82 13.80 15.42 16.78 17.95 18.99 19.91 9.

Decibels 20 log10 M.

15

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APPENDIX

G

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) may be written as x2 1,

(G.3)

and we denote the solution of Eq. (G.3) by the use of an imaginary number j1, so that j 2 1,

(G.4)

j = 4-1.

(G.5)

and

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

(G.6)

where a and b are real numbers. We designate a as the real part of the complex number and b as the imaginary part and use the notation Re{c} a,

(G.7)

Im{c} b.

(G.8)

and

17

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18

Appendix G

Complex Numbers

G.2 RECTANGULAR, EXPONENTIAL, AND POLAR FORMS The complex number a jb may be represented on a rectangular coordinate place called a complex plane. The complex plane has a real axis and an imaginary axis, as shown in Fig. G.1. The complex number c is the directed line identified as c with coordinates 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 from the origin and the angle u, as shown in Fig. G.2. The exponential form is written as c re ju,

(G.9)

r (a2 b2)1/2,

(G.10)

u tan1(b/a).

(G.11)

where

and 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 lu. Thus we may represent the complex number in polar form as c = c2 lu = rlu.

(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 tan1(3/4) 36.9°.

Imaginary axis

Im

Im c a jb

b

c re j

b

j3 r

0

a

Real axis

FIGURE G.1 Rectangular form of a complex number.

0

a

Re

FIGURE G.2 Exponential form of a complex number.

0

4

Re

FIGURE G.3 Complex plane for Example G.1.

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Section G.3

19

Mathematical Operations

The exponential form is then c 5e j36.9째. The polar form is c = 5l36.9째.

G.3 MATHEMATICAL OPERATIONS The conjugate of the complex number c a jb is called c* and is defined as c* a jb.

(G.13)

c* = rl-u.

(G.14)

In polar form we have

To add or subtract two complex numbers, we add (or subtract) their real parts and their 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 j 1): 2

cd = 1a + jb21f + jg2 = af + jag + jbf + j2bg = 1af - bg2 + j1ag + bf2.

(G.16)

Alternatively we use the polar form to obtain cd = 1r1lu1 21r2lu2 2 = r1r2lu1 + u2,

(G.17)

where c = r1lu1,

and

d = r2lu2.

Division of one complex number by another complex number is easily obtained using the polar form as follows: c r1lu1 r1 = = lu1 - u2. d r2lu2 r2

(G.18)

It is easiest to add and subtract complex numbers in rectangular form and to multiply and divide them in polar form. A few useful relations for complex numbers are summarized in Table G.1.

ph_dorf_app 4/24/01 4:54 AM Page 20

20

Appendix G

COMPLEX NUMBERS

Table G.1

Useful Relationships for Complex Numbers (1)

1 j j

(2) (j)( j) 1 (3) j2 1 (4) 1l >2 j (5) ck rklk

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 c = 5l36.9°,

and

d = 22l-45°.

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

cd = 15l36.9°21 22l-45°2 = 522l-8.1°.

For division we have 5l36.9° c 5 l81.9°. = = d 22l-45° 22

ph_dorf_app 4/24/01 4:54 AM Page 21

APPENDIX

H

z-Transform Pairs

Table H.1 x(t) 1 t = 0, 1. d(t) e 0 t = kT, k 0 1 t = kT, 2. d(t kT) e 0 t kT

X(s)

X(z)

1

1

ekTs

zk

3. u(t), unit step

1/s

4. t

1/s2

5. t2

2/s3

6. eat

1 s+a

7. 1 eat

a s1s + a2

8. teat

1 1s + a2 2

9. t2eat

2 1s + a2 3

10. bebt aeat 11. sin vt 12. cos vt 13. eat sin vt

1b - a2s

1s + a21s + b2 v s + v2 s s2 + v2 2

v 1s + a2 2 + v2

z z-1 Tz 1z - 12 2

T2z1z + 12 1z - 12 3 z z - e-aT

11 - e-aT 2z

1z - 121z - e-aT 2 Tze-aT 1z - e-aT 2 2

T2e-aTz1z + e-aT 2 1z - e-aT 2 3

zz1b - a2 - 1be-aT - ae-bT 2 1z - e-aT 21z - e-bT 2

z sin vT z - 2z cos vT + 1 z1z - cos vT2 2

z2 - 2z cos vT + 1 1ze-aT sin vT2

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

21

ph_dorf_app 4/24/01 4:54 AM Page 22

22

Appendix H

Z-TRANSFORM

PAIRS

Table H.1 (continued) 14. eat cos vt 15. 1 eat acos bt +

a sin btb b

s+a 1s + a2 2 + v2 a2 + b2 s1s + a2 2 + b2

z2 - ze-aT cos vT z - 2ze-aT cos vT + e-2aT z1Az + B2 2

1z - 12z2 - 2e-aT 1cos bT2z + e-2aT a A 1 eaT cos bT eaT sin bT b a B e2aT eaT sin bT eaT cos bT b

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