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

82 Chapter 2 Engineering functions
Table2.4
The values ofthe exponential
function f (x) = e −x for
various values ofx.
e –x 1
x
e −x
−3 20.086
−2 7.389
−1 2.718
0 1
1 0.368
2 0.135
3 0.050
Figure2.28
Graph ofy = e −x showing exponential decay.
x
Engineeringapplication2.8
Dischargeofacapacitor
The capacitor is another of the three fundamental electronic components. It is a
device that is used to store electrical charge. It consists of two parallel conducting
platesseparatedbyaninsulatingmaterial,knownasadielectric.Abuildupofanet
positive charge on one plate and a net negative charge on the other plate creates an
electricfieldacrossthedielectric,allowingelectricalenergytobestoredthathasthe
potential to do useful work. The symbol for a capacitor is two parallel lines that are
perpendicular tothe conductors inthe circuit.
Consider thecircuitofFigure2.29.Beforetheswitchisclosed,thecapacitorhas
avoltageV acrossit.Supposetheswitchisclosedattimet = 0.Acurrentthenflows
in the circuit and the voltage, v, across the capacitor decays with time. The voltage
across the capacitor isgiven by
{
V t < 0
v =
Ve −t/(RC) t 0
ThequantityRC isknown asthetimeconstantofthecircuitand isusuallydenoted
byτ.So
{
V t < 0
v =
Ve −t/τ t 0
v
V
t larger
C
Figure2.29
Circuit to discharge acapacitor.
R
Figure2.30
Thecapacitor takes longerto discharge fora
largercircuit time constant, τ.
If τ is small, then the capacitor voltage decays more quickly than if τ is large.
This isillustratedinFigure 2.30.
t