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

134 Chapter 3 The trigonometric functions
Engineeringapplication3.2
Alternatingcurrentwaveformsandtheelectricitysupply
Alternatingcurrentwaveformsareoftenfoundinengineering.Theelectricitysupply
to homes and businesses most often takes the form of an alternating current. This is
becauseitisfareasiertosupplyalternatingcurrentelectricitythandirectcurrentelectricitywhendistributingacrosslongdistances.Energylossesalongthesupplycables
can be reduced by transforming the electricity to high voltages prior to distribution
butelectricity transformersonly work with alternating currents.
Sineandcosinefunctionsareoftenusedtomodelalternatingcurrent(a.c.)waveforms.The
equations foran a.c. waveform are
I =I m
sin(ωt +φ) or I =I m
cos(ωt +φ)
whereI m
=maximumcurrent, ω =angularfrequencyand φ =phaseangle.Inpracticethefunctionscanbeshiftedalongthetimeaxisbygiving
φ anon-zerovalueand
so both the sine and the cosine function can be used to model any a.c. waveform;
which one isused isusually a matter of convenience.
The angular frequency, ω, can be written as ω = 2πf, where f is the frequency
of the waveform in Hertz (Hz). The frequency of the electricity supply in Europe
and across large parts of the world is 50 Hz, while in the Americas and in areas of
Asiaitis60Hz.Alternatingcurrentsuppliesarealsofoundonships,submarinesand
aircraftbut these often use 400 Hz as the operating frequency.
3.7.1 Combiningwaves
There are many situations in which engineers need to combine two or more waves
together to form a single wave. It is possible to make use of trigonometric identities
to calculate the resulting waveform when several waves are combined. Consider the
following example.
Engineeringapplication3.3
Combiningtwosinusoidalvoltagesignals
Two voltage signals, v 1
(t) and v 2
(t), have the following mathematical expressions:
v 1
(t) =3sint
v 2
(t) =2cost
(a) State the amplitude and angular frequency ofthe two signals.
(b) Obtain an expression forthe signal, v 3
(t), given by
v 3
(t) = v 1
(t)+2v 2
(t)
(c) Reduce the expression obtained in part (b) to a single sinusoid and hence state
its amplitude and phase.