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Contents

PREFACE ix 

Chapter  TRIGONOMETRY AND CYCLIC FUNCTIONS 1 Uni t s o f Measurement ,  1  

Funct ions of Angles , 2

Definit ions,   3  

Q u a d r a n t s ,   3

Frequency of Cycl ing,  5Sine of the Sum of Tw o Ang les ,  7

Cosine of a Sum ,  8

Sine of a Difference Between Two Angles,  9

Cosine of a Difference,  10

Potent ia l ly Useful Rela t ionships ,  10

Chapter 2 DIFFERENTIAL CALC ULUS 15 Conce p t of A ppro ach ing a L imit ,  16

Proced ure for De te rm in ing a Der iva t ive ,  19

Chapter 3 INTEGRAL CALC ULUS 29 Prob lem Areas , 30

Practical Uses of Integrat ion,  31

Integrat ion Over a Speci f ied Range, 39

Table of Basic Integrals,  48

Chapter 4 INFINITE SER IES 49 Power Ser ies , 50

The n th Term, 50Test for Convergence,  51

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vi Table of ontents

M aclaur in s Seri es, 54

Taylo r s Ser ies, 58

Cha pters COMPLEX QUANTITIES 61  Ba c k g r o u n d ,  61

Graph ica l Represen ta t ion ,  63  

The Comp lex Var iab le , 64

Tr igonomet r i c and Exponen t i a l Func t ions ,  65

Separa t ing the Real and Imag inary Par t s ,  67

Chapter 6 DIFFERENTIAL EQUATIONS 71 In t roduc t ion ,  71

Phi losophy ,   72

Definit ions,   72Appl i ca t ion ,  74

Differential Equations of the First Order and First Degree,   74

Linear Dif ferentia l E qua t ions w i th C on stan t Coeffic ients,  80  

Second O rde r Linear Dif ferentia l E qua t ion w i th Co nsta nt Coeffic ients,  80

The Osci l la tory Case,  83

The Constant of In tegrat ion,  84

Uni t s , 89

Part ial Differential Equations,  91

Chapter 7 LAPLACE TRANS FORM S 93  

His to ry ,  93

Transforms of Der ivat ives ,  95

The Osci l la tory Case,  103

Chapter 8 FREQUENCY RESPONSE ANALY SIS 107 Background , 107

The Bode Diagram , 108

Frequency R esponse o f a T ime Cons tan t E lement ,  109

Frequency Response of a Dead Time Element ,   109Co m b i n a t io n s of Co m p o n e n t s ,  111

Period of Oscil lat ion,  112

S u m m a r y ,  114  

Chapter 9 TRANSFE R FUNCTIONS AND BLOCK DIAGRAMS 117 Ba c k g r o u n d ,  117

Transfer Funct ions,  118

The Step Input Funct ion,  118  

T ime Cons tan t s ,  119

Dead T ime ,  120The Value of the Transfer Fu nct ion,  120

Block D iagram s ,  123

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Tab le of onten ts vii

Con di t ions fo r Con t in uou s Osc i l la t ion ,  125

The Transfer Funct ion of a Closed Lo op,   127

Evaluat ing the Closed Loop Transfer Funct ion,  127

Chapter 10 THE Z -N APPROXIMATION 131 Histor ical ,  131

T h e Z - N A p p r o x i m a t i o n ,  136  

Est imat ing the Frequency of Osci l la t ion,  141  

Values for the Dead Time and Time Constant ,   141

Jus t How Good I s the Approx imat ion? ,   142

M aking a P rocess Reac t ion Cu rve ,  143

Chapter  UNITS BEST VALU ES FORM ULAS AND OTHER GOOD

STUFF 151 True Value ,  151

Errors ,  152

Er ro r s in Com bina t ions of Quan t i t ie s ,  152

Correct ion Factor ,  153

Signif icant Figures,  153

Conversion of Uni ts ,  155

Conver t ing Formulas to New Uni t s ,  157

The M ost Represen ta t ive Value (MRV), 162

Pred ic t ing Fu tu re Va lues ,  165

How Much Conf idence in the Most Represen ta t ive Value? ,  167

The S tandard Dev ia t ion , 168

Curve Fi t t ing ,  169

INDEX 173