First Order Functions
• Time Domain
dy
+ y = Kx
dt
• Laplace Domain
Response of First Order
Transfer Functions
K
G( s) =
τs + 1
X (s )
τ
K
Y ( s)
= G( s) =
τs + 1
X ( s)
Y (s )
Input Functions (i.e., X(s))
Step
u (s ) =
Ramp
Rectangular pulse
Triangular pulse
Sine wave
Impulse
M
s
a
u (s ) = 2
s
h
u (s ) = 1 − e −tw s
s
2 ⎛ 1 − 2e − t w s / 2 + e − t w s ⎞
⎟⎟
u (s ) = ⎜⎜
tw ⎝
s2
⎠
(
Aω
s2 + ω2
u( s ) = a
u( s) =
)
(5-6)
(5-8)
(5-11)
(5-13)
(slope=tw/2)
(5-15)
Y ( s ) = X ( s )G ( s )
M
X ( s) =
s
K
G( s) =
τs + 1
KM
Y ( s) =
s (τs + 1)
y (t ) = KM (1 − e − t /τ )
(5-18)
p. 76
Response to Ramp
Y ( s ) = X ( s )G ( s )
a
X ( s) = 2
s
K
G( s) =
τs + 1
Ka
Y ( s) = 2
s (τs + 1)
Response to Step
Response to Sine Wave
Y ( s ) = X ( s )G ( s )
Aω
s + ω2
K
G( s) =
τs + 1
bs + c ⎤
KAω
⎡ a
= KAω ⎢
+ 2
Y ( s) = 2
2
2
(s + ω )(τs + 1)
⎣τs + 1 s + ω ⎥⎦
X ( s) =
y (t ) = Kaτ (e − t /τ − 1) + Kat
(5-22)
2
y (t ) =
Ka
ω 2τ 2 + 1
(ωτe
−t / τ
− ωτ cos ωt + sin ωt )
(5-25)
1
Example: FOPDT
Time Delays (θ)
In time domain:
• Replace t with (t-θ) and multiply by S(t-θ)
• This is a response of a first order model to a step
function M
• First order with a step function is:
KM
y (t ) = KM 1 − e − t /τ
Y ( s) =
f (t − θ ) ⋅ S (t − θ )
(
In Laplace domain
• Multiply by e-θs
)
s (τs + 1)
• Now add time delay
y (t ) = KM (1 − e − ( t −θ ) /τ )⋅ S (t − θ )
e −θs F (s )
Y ( s) =
Integrating Process
KM ⋅ e −θs
s (τs + 1)
Pumped Tank Example
• Pumped Tank
dh
= qi − q
dt
sAH ′( s ) = Qi′( s ) − Q ′( s )
H ′( s ) 1
=
Qi′( s ) sA
A
H ′( s )
1
1
=−
[Qi′( s ) − Q′( s )]
Q′ ( s)
sA
sA
• This is not a first order model
• Called an integrating process (no steady-state gain)
• Step function in q or qi results in ramp in h!!
H ′( s ) =
Qi′( s ) =
M
s
H ′( s ) =
M
s2 A
Problem 4.7
Assumptions
H, L, and V are molar flow rates
y’s and x’s are mole fractions in vapor and liquid
• Wanted:
X 1′(s )
X 0′ (s )
X 1′(s ) Y1′(s )
Y2′(s ) X 0′ (s )
Y1′(s )
Y2′(s )
• Molar holdup H is constant
dH
=0
dt
• Given:
Stirred tank blending system
(or stage on distillation column)
dH
= L0 + V2 − (L1 + V1 )
dt
dx1H
= x0 L0 + y2V2 − ( x1L1 + y1V1 )
dt
y1 = a0 + a1 x1 + a2 x12 + a3 x13
Vapor pressure correlation
• Constant molal overflow
L0 = L1
V1 = V2
• Simplification: only use L and V
(no subscripts)
2