Laplace Transform.pdf

634
S
ECTION
4
PLC Process
Applications
Industrial Text & Video Company 1-800-752-8398
www.industrialtext.com
C
HAPTER
14
Process Responses
and Transfer Functions
First-Order Equations
DESCRIPTION
Any Function
f
Unit Step Input
Step Input
Delay (Dead Time) to
Step Input
First-Order Response
First-Order Response with
Lag
First-Order Response with
Lag plus Dead Time
LAPLACE
TRANSFORM
F
(
s

)
1
s
A
s
Ae
−
t
d
s
A
s
+
a
A
τ
s
+ 1
A
1
A
2
s
+
a
e
−
t
d
s

A
1
A
2
τ
s
+ 1
e
−
t
d
s
TIME
FUNCTION
f
(
t
)
u
(
t
)
A
(
t
)
A
(
t
−

t
d
)
Ae
−
at
A
τ
e
−
t
τ
AAe t t
at
d
12
−
≥for
AAe t t
t
d
12
−
≥
τ
for
A
01 2
t
a

=1
a
=2
..
..
..
ττ
==
=
=
=
12
0 1 000 0 500
1 0 368 0 303
2 0 135 0 184
t
t
t
aa
tt
tt
tt
d
d
d
==
=
=+
=+
12

1 000 1 000
1 0 368 0 135
2 0 135 0 018
..
..
..
A
(
t
) = 0
t
<
t
0
A
(
t
) =
At
≥
t
0
u
(
t
) = 0
t
<
t
0

u
(
t
) = 1
t
≥
t
0
A
(
t
−
t
d
) = 0
t
<
t
d
A
(
t
−
t
d
) =
At
≥
t
d

t
0
1
t
0
A
t
0
t
d
A
Table 14-2. Laplace transforms.
PROCESS
RESPONSE
A
A
2
01 2
t
τ =1
τ
=2
A
1
A
2
0
t
d
t

d
+1
t
A
1
A
2
0
t
t
d
t
d
+1
A
1
A
2
2
..
..
..
ττ
==
=
=
=
12
0 1 000 1 000
1 0 368 0 135

2 0 135 0 018
t
t
t
ττ
==
=
=+
=+
12
1 000 0 500
1 0 368 0 135
2 0 135 0 018
tt
tt
tt
d
d
d
..
..
..
For
A
=1
For
A
=1
For
A

1
A
2
=1
For
A
1
A
2
=1
First-Order Response plus
Dead Time
635
C
HAPTER
14
Process Responses
and Transfer Functions
Industrial Text & Video Company 1-800-752-8398
www.industrialtext.com
S
ECTION
4
PLC Process
Applications
DESCRIPTION
LAPLACE
TRANSFORM
TIME
FUNCTION

First-Order Response to Step
Input (
A
1
/
s
) with Lag
AA e
t
12
1()−
−
τ
A
1
A
2
s
(
τ
s
+ 1)
τ
= 1
τ
= 2
t
= 0 0.000 0.000
t
= 1 0.632 0.393

t
= 2 0.865 0.632
First-Order Response to Step Input
(
A
1
/
s
) with Lag plus Dead Time
A
1
A
2
A
3
s
(
τ
s
+ 1)
e
−
t
d
s
AA A e t t
t
d
12 3
1()−≥

−
τ
for
Second-Order Transfer Function
(
Hp
(
s
)
) for ζ < 1 (Underdamped)
A
ω
n
2
s
2
+ 2
ζω
n
s
+
ω
n
2
Second-Order Transfer Function
(
Hp
(
s
)

) for ζ = 1 (Critically Damped)
A
(
τ
s
+ 1)
2
Second-Order Step Response
(
A
1
/
s
) for ζ = 1 (Critically Damped)
A
1
A
2
1−
τ
−
t
τ
e
−
t
τ
⎛
⎝
⎜

⎞
⎠
⎟
A
1
A
2
s
(
τ
s
+ 1)
2
Second-Order Step Response
(
A
1
/
s
) for ζ > 1 (Overdamped)
AA
ee
tt
12
12
21
1
22
+
−

−
−−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
ττ
ττ
τ
τ
A
1
A
2
s
(
τ
1
s
+ 1) (
τ
2
s
+ 1)
Second-Order Step Response
(

A
1
/
s
) for ζ < 1 (Underdamped)
A
1
A
2
ω
n
2
s
(
s
2
+ 2
ζω
n
s
+
ω
n
2
)
A
1
A
2
1+

e
−
ζω
n
t
1−
ζ
2
sin
ω
n
1−
ζ
2
t
−
ψ
⎛
⎝
⎞
⎠
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
where

ψ
= tan
−1
1−
ζ
2
−
ζ
(0 <
ψ
<
π
)
Second-Order Equations
At
τ
2
e
−
t
τ
Second-Order Transfer Function
(
Hp
(
s
)
) for ζ > 1 (Overdamped)
A
τ

1
−
τ
2
(
e
−
t
τ
1
−
e
−
t
τ
2
)
A
(
τ
1
s
+ 1) (
τ
2
s
+ 1)
A
ω
n

e
−
ζω
n
t
1−
ζ
2
sin
ω
n
1−
ζ
2
t
⎛
⎝
⎞
⎠
Table 14-2 continued.
PROCESS
RESPONSE
A
1
A
2
A
3
0
t

t
d
t
d
+1
A
1
A
2
01 2
t
In Out
Out
In
Hp
(s)
Hp
(s)
=
2π
ω
n
1
f
n
or
A
1
A
2

Underdamped ζ <1
Overdamped ζ >1
Critically damped ζ =1
t
ττ
==
=
=+
=+
12
1 000 1 000
1 0 632 0 393
2 0 865 0 632
tt
tt
tt
d
d
d
..
..
..
For
A
1
A
2
=1
For
A

1
A
2
A
3
=1