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

674 Chapter 21 The Laplace transform
The pole atthe origincontributes a gain of
−20log 10
|jω|
This is plotted as a straight line with no start or end points, with a slope of −20dB
per decade passing through ω = 1.
The two other poles contribute a gain of
∣ ∣ ∣∣∣ jω
−20log 10
+1
−p ∣ = −20log jω ∣∣∣
10 ∣
1
100 +1
and
∣ ∣ ∣∣∣ jω
−20log 10
+1
−p ∣ = −20log jω ∣∣∣
10 ∣
2
500 +1
respectively.Thefirstoftheseisapproximatedbyalinewithslope−20dBperdecade
startingat ω = 100, the second has the same gradient and startsat ω = 500.
TheindividualcontributionstotheoverallplotareshowninFigure21.22.Notice
thestartingpointsoftheasymptotes--thebreakpoints--andtheircorrespondenceto
the positions of the poles and zeros.
60
40
20 log 10 jv
3
dB
20
0
–20
–40
–20 log 10 |jv|
20 log 10 |0.3|
+ 1 –20 log 10 jv
500 + 1
–20 log 10 jv
100 + 1
–60
–80
0.1 1 10 100 1000 10 4
Figure21.22
Contributionsofthe individual terms to the Bode plot.Dottedlinesare the asymptotic
approximations.
v
The asymptotes can now be added together to form a combined graph which is
shown in Figure 21.23. Also plotted on the same graph is an exact Bode plot of the
transferfunction obtained usingacomputer.
Notice that the largest error in the approximation occurs at the breakpoints,
whereas the two graphs agree well at other frequencies. The reason for the error
is most apparent when we consider the case where ω = −z 1
, for example. The gain
atthisbreak frequency according tothe lineplottedshould be
20log 10
∣ ∣∣∣ −jz 1
−z 1
∣ ∣∣∣
= 20log 10
|j| = 0