082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

9 Complexnumbers
Contents 9.1 Introduction 324
9.2 Complexnumbers 325
9.3 Operationswithcomplexnumbers 328
9.4 Graphicalrepresentationofcomplexnumbers 332
9.5 Polarformofacomplexnumber 333
9.6 Vectorsandcomplexnumbers 336
9.7 Theexponentialformofacomplexnumber 337
9.8 Phasors 340
9.9 DeMoivre’stheorem 344
9.10 Lociandregionsofthecomplexplane 351
Reviewexercises9 354
9.1 INTRODUCTION
Complex numbers often seem strange when first encountered but it is worth persevering
with them because they provide a powerful mathematical tool for solving several
engineeringproblems.Oneofthemainapplicationsistotheanalysisofalternatingcurrent
(a.c.) circuits. Engineers are very interested in these because the mains supply is
itself a.c., and electricity generation and transportation are dominated by a.c. voltages
and currents.
A great deal of signal analysis and processing uses mathematical models based on
complex numbers because they allow the manipulation of sinusoidal quantities to be
undertakenmoreeasily.Furthermore,thedesignoffilterstobeusedincommunications
equipment relies heavily on their use.
One area of particular relevance is control engineering -- so much so that control
engineers often prefer to think of a control system in terms of a ‘complex plane’ representation
rather than a ‘time domain’ representation. We will develop these concepts in
this and subsequent chapters.