Dorf Appendix C

8/9/2019 Dorf Appendix C

1/22

Symbols, Units,

and Conversion Factors

Table C.1 Symbols and Units

Parameter or Variable Name Symbol SI Eng

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

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-l

rad>Nm

rad>s2

lbft>sNm>s

ft-lb

rad>Nm

rad>s

A P P E N D I X

C


2/22

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 3.281 102

cm in. 0.3937cm3 ft3 3.531 105

deg (angle) rad 0.01745deg/s rpm 0.1667dynes g 1.020 103

dynes lb 2.248 106

dynes N 105

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 1.286 103

ft-lb/s hp 1.818 103

ft-lb/s kW 1.356 103

20.11

g dynes 980.7

g lb 2.205 103

g-cm2 oz-in2 5.468 103

g-cm oz-in. 1.389 102

g-cm ft-lb 1.235 105

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

in. meters 2.540 102

in. cm 2.540

J Btu 9.480 10

4

J ergs 107

J ft-lb 0.7376J W-hr 2.778 104

kg lb 2.205kg slugs 6.852 10

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

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

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

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 57.30

rad 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

s (angle) rad 4.848 10

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

W Nm/s 1.0Wh Btu 3.413

oz-in.

rpm

ft-lb

rad>s

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


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Laplace Transform Pairs

A P P E N D I X

DTable D.1F(s) f(t), t 0

1. 1 d(t0), unit impulse at t t02. 1/s 1, unit step

3. tn

4. e

at

5. tn 1eat

6. 1 eat

7. (eat ebt)

8. [(a a)eat (a b)ebt]

9. 1 eat ebt

10.

11.

12. a eat ebt

13. sin vt

14. cos vt0s

s2 + v2

v

s2 + v2

a1a - b21b - a2

b1a - a21b - a2

ab1s + a2s1s + a2 1s + b2

1a - c2e-ct1a - c2 1b - c2

1a - b2e-bt1c - b2 1a - b2

1a - a2e-at1b - a2 1c - a2

s + a1s + 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

a

1b - a

2

b

1b - a

2

ab

s

1s + a

2 1s + b

2

1

1b - a2s + a

1s + a2 1s + b2

1

1b - a21

1s + a2 1s + b2

a

s1s + a2

1

1n - 12!1

1s + a2n

1

1s + a2

n!

sn+1

Table D.1 contin


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Table D.1 Continued

F(s) f(t), t 0

15. sin (vt f), f tan1 v/a

16. eatsin vt

17. eatcos vt

18. [(a a)2 v2]1/2 eatsin (vt f),

f tan1

19. ezvntsin vn t, z 1

20. eatsin (vt f),

f tan1

21. 1 ezvntsin ,

f cos1 z, z 1

22. 1/2

eatsin (vt f),

f tan1 tan1

23. ,f tan1v

c - a

e-atsin1vt+ f2v1c - a2

2

+ v2

1

>2

e-ct

1c - a22

+ v2

1

1s + c21s + a22

+ v2

v

-av

a - a

1

vc 1a - a2

2 + v2

a2 + v2da

a2 + v21s + a2

s1s + a22 + v2

1vn21 - z2t+ f2121 - z2

v2n

s1s2 + 2Zvns + v2n2

v-a

1

v2a2 + v21

a2 + v21

s1s + a22 + v2

21 - z2vn

21 - z2v2n

s2 + 2Zvns + v2n

v

a - a

1

v

s + a1s + a22 + v2

1s + a21s + a22 + v2

v

1s + a22 + v2

2a2 + v2v

s + as2 + v2

4 Appendix D Laplace Transform Pairs


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An Introduction

to Matrix Algebra

E.1 DEFINITIONS

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

(E

is known as a matrix. The numbers aijare called elements of the matrix, with the sscript i denoting the row and the subscriptjdenoting the column.

A matrix with m rows and n columns is said to be a matrix oforder (m, n) orternatively called an m n (m-by-n) matrix.When the number of the columns equthe number of rows (m n), the matrix is called a square matrix of order n. It is cmon 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 column matrix or,more commonly, a column vector. We will represent a column vtor with boldfaced lowercase letters as

(E

Analogously, a row vector is an ordered collection of numbers written in a rowthat is, a 1 n matrix. We will use boldfaced lowercase letters to represent vectTherefore a row vector will be written as

(E

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

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

z = 3z1 z2 p zn 4 ,

y = Dy1y2oym

T

A =

Da11

a21

o

am1

a12

a22

o

am2

pp

p

a1n

a2n

o

amnT

A P P E N D I X

E


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(E.4

If all the elements of a diagonal matrix have the value 1, then the matrix is known athe 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, onull 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 elementofA are aijand the elements ofB are bij, and if

C A B, (E.7

then the elements ofC 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 thprocess 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.Therefore we can write

(A B) C A (B C).

A B B A.

C = C210

1

-16

0

3

2

S + C814

2

3

2

1

0

1

S = C1024

3

2

8

1

3

3

S .

H =

C3

-2

1

-26

4

1

4

8S

I = D10o0

01

o0

pppp

00

o1T .

B = Cb1100

0

b22

0

0

0

b33

S .6 Appendix E An Introduction to Matrix Algebra


7/22

(E

Then to carry out a subtraction operation, we use a 1,and A is obtained by mtiplying each element ofA by 1. For example,

(E

E.3 MULTIPLICATION OF MATRICES

The multiplication of two matrices AB requires that the number of columns obe equal to the number of rows ofB. Thus ifA is of order m n and B is of ord q, then the product is of order m q. The elements of a product

C AB (E

are found by multiplying the ith row ofA and thejth column ofB and summing thproducts to give the element cij. That is,

(E

Thus we obtain c11, the first element ofC, by multiplying the first row ofA by the fcolumn ofB and summing the products of the elements.We should note that, in geral, matrix multiplication is not commutative; that is

(E

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

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

(E

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

x1 (a11y1 a12y2 a13y3) (Eis the first element obtained by multiplying the first row ofA by the first (and oncolumn ofy.

Another example, which the reader should verify, is

(EC = AB = B 2-1

-12RB 3

-12

-2R = B 7

-56

-6R .

x = Ay = Ba11a21

a12

a22

a13

a23RCy1y2

y3

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

AB BA.

cij = ai1b1j + ai2b2j + p + aiqbqj = aq

k=1aikbkj.

C = B - A = B24 12R - B63 11R = B-41 01R .

aA = D aa11aa12o

aam1

aa12

aa22

oaam2

pp

p

aa1n

aa2n

oaamn

T .Section E.3 Multiplication of Matrices


8/22

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 o

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

(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 osimultaneous 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 bthe 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 inter

changing the rows and columns ofA. For example, if

then

A = C 61-2

0

4

3

2

1

-1S ,

xy = 3x1 x2 p xn 4

D

y1

y2

o

ynT= x1y1 + x2y2 + p + xnyn.

A = C324

2

1

-1

1

6

2S .

x = Cx1x2x3

S and u = Cu1u2u3

S .

8 Appendix E An Introduction to Matrix Algebra


9/22

(E

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

(EBecause xT is a row vector, we obtain a matrix multiplication ofxTby x as follow

(E

Thus the multiplication xTx results in the sum of the squares of each element ofThe transpose of the product of two matrices is the product in reverse orde

their transposes, so that

(E

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

(E

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

(E

If the determinant ofA is equal to zero, then the determinant is said to be singuThe value of a determinant is determined by obtaining the minors and cofactorthe determinants. The minor of an element aijof a determinant of order n is a deminant of order (n 1) obtained by removing the row i and the columnjof the oinal determinant.The cofactor of a given element of a determinant is the minor of element with either a plus or minus sign attached; hence

cofactor ofaij aij (1)ijMij,

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

(E

is

(E

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

a23 = 1-125M23 = - 2a11a31

a12

a322 .

detA =3a11

a21

a31

a12

a22

a32

a13

a23

a33 3

detA = 2a11a21

a12 2a21

= a11a22 - a12a21.

tr A a11 a22 ann.

(AB)T BTAT.

xTx = 3x1 x2 p xn 4 Dx1x2oxn

T = x21 + x22 + p + x2n.x

T

= 3x1 x2 p xn 4 .

AT = C602

1

4

1

-23

-1S .

Section E.4 Other Useful Matrix Operations and Definitions


10/22

(E.32

The general nth-order determinant has a value given by

with i chosen for one row, (E.33

or

withjchosen for one column. (E.33

That is, the elements aijare 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 specifi3 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 a

by the cofactor aijand transposing. Therefore

(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

The inverse of a matrix A is

(E.37A-1 =adjointofA

detA

adjointA = Da11

a21o

an1

a12

a22o

an2

p

p

p

a1n

a2no

annTT

= Da11

a12o

a1n

a21

a22o

a2n

p

p

p

an1

an2o

annT .

= 21-12- 1-52+ 2132= 9,= 2 20

1

1

02 - 1 23

1

5

02 + 2 23

0

5

12

det

A = detC

2

12

3

01

5

10S

detA = ani=1

aijaij

detA = anj=1

aijaij

2a11a21

a12

a222 = 1a11a22 - a21a122.



11/22

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

(E

and the det A a11a22 a12a21. Consider the matrix

(E

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

(E

In a similar manner we obtain

(E

E.6 MATRICES AND CHARACTERISTIC ROOTS

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

y Ax, (E

where the y vector can be considered as a transformation of the vector x. We mask whether it may happen that a vector y may be a scalar multiple ofx. Try

y lx, where l is a scalar, we havelx Ax. (E

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

lx Ax (lI A)x 0, (E

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

(E

This determinant is called the characteristic determinant ofA. Expansion of the

terminant of Eq. (E.45) results in the characteristic equation. The characteristic eqtion is an nth-order polynomial in l. The n roots of this characteristic equation called the characteristic roots. For every possible value li (i 1 , 2 , . . . , n) of the norder characteristic equation, we can write

(liI A)xi 0. (E

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

det (lI A) 0.

A-1 =adjointA

detA = a- 17b

C3

-2

-2

-51

1

11

2

-5S.

a11 = 1-122 2 -1-1 41 2 = 3.A = C

1

2

0

2

-1-1

3

4

1S .adjointA = B a22

-a21

-a12a11

R,Section E.6 Matrices and Characteristic Roots


12/22

(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 chosen

equal 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 thmatrix.

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 bshown to be convergent for all square matrices.Also a matrix exponential that is function of time is defined as

(E.53

If we differentiate with respect to time, then we have

(E.54d

dt1eAt2= AeAt.

eAt = a

k=0A

k

tk

k!.

expA = eA = I + A1!

+A2

2!+ p +

Ak

k!+ p = a

k=0Ak

k!,

d

dtA1t2 = Cda111t2>dto

dan11t2>dtda121t2>dt

odan21t2>dt

p

p

da1n1t2>dto

dann1t2>dtS .

xT3 = 32 3 -1 4 .

xT2 30 1 -1 4 ,

31 -1 0 4T1

detC1l - 22

-21

-1

1l - 321-1

-41l + 22S = 1-l3

+ 3l2

+ l - 32= 0.

A = C 22-1

1

3

-1

1

4

-2S .



13/22

Therefore for a differential equation

(E

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

(E

or

AeAt AeAt, (E

and we have in fact satisfied the relationship, Eq. (E.55). Then the value ofc is sply x(0), the initial value ofx, because when t 0, we have x(0) c. Thereforesolution to Eq. (E.55) is

(Ex(t) eAtx(0).

dxdt

= Ax,

dx

dt= Ax,

Section E.7 The Calculus of Matrices


14/22


15/22

Decibel Conversion

Table F.1

M 0 1 2 3 4 5 6 7 8

0.0 m 40.00 33.98 30.46 27.96 26.02 24.44 23.10 21.94 20

0.1 20.00 19.17 18.42 17.72 17.08 16.48 15.92 15.39 14.89 14

0.2 13.98 13.56 13.15 12.77 12.40 12.04 11.70 11.37 11.06 10

0.3 10.46 10.17 9.90 9.63 9.37 9.12 8.87 8.64 8.40 8

0.4 7.96 7.74 7.54 7.33 7.13 6.94 6.74 6.56 6.38 6

0.5 6.02 5.85 5.68 5.51 5.35 5.19 5.04 4.88 4.73 40.6 4.44 4.29 4.15 4.01 3.88 3.74 3.61 3.48 3.35 3

0.7 3.10 2.97 2.85 2.73 2.62 2.50 2.38 2.27 2.16 2

0.8 1.94 1.83 1.72 1.62 1.51 1.41 1.31 1.21 1.11 1

0.9 0.92 0.82 0.72 0.63 0.54 0.45 0.35 0.26 0.18 0

1.0 0.00 0.09 0.17 0.26 0.34 0.42 0.51 0.59 0.67 0

1.1 0.83 0.91 0.98 1.06 1.14 1.21 1.29 1.36 1.44 1

1.2 1.58 1.66 1.73 1.80 1.87 1.94 2.01 2.08 2.14 2

1.3 2.28 2.35 2.41 2.48 2.54 2.61 2.67 2.73 2.80 2

1.4 2.92 2.98 3.05 3.11 3.17 3.23 3.29 3.35 3.41 3

1.5 3.52 3.58 3.64 3.69 3.75 3.81 3.86 3.92 3.97 4

1.6 4.08 4.14 4.19 4.24 4.30 4.35 4.40 4.45 4.51 41.7 4.61 4.66 4.71 4.76 4.81 4.86 4.91 4.96 5.01 5

1.8 5.11 5.15 5.20 5.25 5.30 5.34 5.39 5.44 5.48 5

1.9 5.58 5.62 5.67 5.71 5.76 5.80 5.85 5.89 5.93 5

2. 6.02 6.44 6.85 7.23 7.60 7.96 8.30 8.63 8.94 9

3. 9.54 9.83 10.10 10.37 10.63 10.88 11.13 11.36 11.60 11

4. 12.04 12.26 12.46 12.67 12.87 13.06 13.26 13.44 13.62 13

5. 13.98 14.15 14.32 14.49 14.65 14.81 14.96 15.12 15.27 15

6. 15.56 15.71 15.85 15.99 16.12 16.26 16.39 16.52 16.65 16

7. 16.90 17.03 17.15 17.27 17.38 17.50 17.62 17.73 17.84 17

8. 18.06 18.17 18.28 18.38 18.49 18.59 18.69 18.79 18.89 18

9. 19.08 19.18 19.28 19.37 19.46 19.55 19.65 19.74 19.82 190. 1. 2. 3. 4. 5. 6. 7. 8.

Decibels 20 log10 M.

A P P E N D I X

F


16/22


17/22

Complex Numbers

G.1 A COMPLEX NUMBER

We all are familiar with the solution of the algebraic equation

x2 1 0, (G

which isx 1. However, we often encounter the equation

x2 1 0. (G

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

x2 1, (G

and we denote the solution of Eq. (G.3) by the use of an imaginary numberj1that

j2 1, (G

and

(G

An imaginary number is defined as the product of the imaginary unit jwith a number.Thus we may, for example,write an imaginary number asjb.A complex nuber is the sum of a real number and an imaginary number, so that

(G

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

Re{c} a, (Gand

Im{c} b. (G

c = a + jb

j=4

-1.

A P P E N D I X

G


18/22

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, ashown 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

(G.9

where

r (a2 b2)1/2, (G.10

and

u tan1(b/a). (G.11

Note that a rcos u and b rsin u.The number ris also called the magnitude ofc, denoted as c.The angle u can also

be denoted by the form . Thus we may represent the complex number in polaform 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 fin

ras

r (42 32)1/2 5,

and u as

u tan1(3/4) 36.9.

c = c2lu = rlu.lu

c reju,

18 Appendix G Complex Numbers

Imaginary axis

b

0 aReal axis

cajb

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

crej

Im

Re

3

r

0 4

Im

Re


19/22

The exponential form is then

c 5ej36.9.

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

In polar form we have

(G

To add or subtract two complex numbers,we add (or subtract) their real parts their imaginary parts. Therefore ifc a jb and d fjg, then

c d (a jb) (fjg) (a f) j(b g). (G

The multiplication of two complex numbers is obtained as follows (notej2 1

(G

Alternatively we use the polar form to obtain

(G

where

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

(G

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

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

c

d=

r1lu1r2lu2

=r1

r2lu1 - 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


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EXAMPLE G.2 Complex number operations

Find c d, c d, cd,and c/d when c 4 j3 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

c

d =5l36.9

22l-45 = 522l81.9.

cd = 15l36.92 122l-452= 522l-8.1.

c = 5l36.9, and d = 22l-45.

20 Appendix G COMPLEX NUMBERS

Table G.1 Useful

Relationships for

Complex Numbers

(1) j

(2) (j)( j) 1

(3) j2 1(4) 1 j

(5) ck rklk

l>2

1

j


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z-Transform Pairs

Table H.1

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

1. d(t) 1 1

2. d(t kT) ekTs zk

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

4. t 1/s2

5. t2 2/s3

6. eat

7. 1 eat

8. teat

9. t2eat

10. bebt aeat

11. sin vt

12. cos vt

13. eat

sin vt 1ze-aTsinvT

2z2 - 2ze-aTcosvT+ e-2aTv

1s + a22 + v2

z1z - cosvT2z2 - 2zcosvT+ 1ss2 + v2

zsinvTz2 - 2zcosvT+ 1vs2 + v2

zz1b - a2- 1be-aT - ae-bT21z - e-aT2 1z - e-bT2

1b - a2s1s + a2 1s + b2

T2e-aTz1z + e-aT21z - e-aT23

2

1s + a23

Tze-aT

1z - e-aT

22

1

1s + a22

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

a

s1s + a2

z

z - e-aT1

s + a

T2z1z + 121z - 123

Tz

1z - 122

z

z

-1

e 10

t= kT,t kT

e10

t= 0,t= kT,k 0

A P P E N D I X

H


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Table H.1 (continued)

14. eatcos vt

15. 1 eat

A 1 eaTcos bT eaTsin bT

B e2aT eaTsin bT eaTcos bTa

b

a

b

z1Az + B21z - 12z2 - 2e-aT1cosbT2z + e-2aT

a2 + b2

s1s + a22 + b2acosbt+a

bsinbtb

z2 - ze-aTcosvTz2 - 2ze-aTcosvT+ e-2aTs + a1s + a22 + v2

22 Appendix H Z-TRANSFORM PAIRS

Documents

Dorf Appendix C