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

12.2 Maximum points and minimum points 411
is a minimum. At x = −2, y ′′ = −3 which is negative and so the point is a
maximum. ⎧
⎪⎨ −1 t<0
(d) y ′ = 1 t>0
⎪⎩
undefined att = 0
Since y ′ (0) is undefined, we use the first-derivative test. This was employed in
Example 12.1.
Engineeringapplication12.1
Risetimeforasecond-orderelectricalsystem
Consider the electrical system illustrated in Figure 12.7. The input voltage, v i
, is
appliedtoterminalsa--b.Theoutputfromthesystemisavoltage, v o
,measuredacross
the terminals c--d. The easiest way to determine the time response of this system to
a particular input istouse the technique of Laplace transforms (seeChapter 21).
When a step input is applied to the system, the general form of the response dependsonwhetheraquantitycalledthedampingratio,
ζ,issuchthat ζ > 1,ζ = 1or
ζ < 1.Thequantity ζ itselfdependsuponthevaluesofL,CandR.Thisisillustrated
inFigure12.8.Ifthedampingratio, ζ < 1,then v o
overshootsitsfinalvalueandthe
systemissaidtobe underdamped. For thiscase itcan beshown that
(
v o
=U −Ue −αt cos(βt)+ αsin(βt) )
for t > 0 (12.1)
β
whereU isthe height ofastepinputapplied att = 0,and
α = R 2L
(12.2)
ω r
= 1 √
LC
resonant frequency (12.3)
β =
√
ωr 2 − α2 naturalfrequency (12.4)
Engineers are often interested in knowing how quickly a system will respond to a
particular input. For many systems this is an important design criterion. One way of
characterizingthespeedofresponseofthesystemisthetimetakenfortheoutputto
reach a certain level in response to a step input. This is known as the rise time and
is often defined as the time taken for the output to rise from 10% to 90% of its final
value.However,bylookingattheunderdampedresponseillustratedinFigure12.8it
is clear that the time,t m
, required for the output to reach its maximum value would
also provide an indicator of system response time. As the derivative of a function is
zero atamaximum point itispossible tocalculate thistime.
Differentiating Equation (12.1) and usingthe product rule,
dv o
dt
= d dt
( (U −Ue −αt cos(βt)+ αsin(βt)
β
))
( ) e −αt αsin(βt)
t > 0
=0−U d dt (e−αt cos(βt)) −U d dt
β
➔