CHE3162 Formula Sheet 2017 PDF

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CHE3162 FORMULA SHEET

FO:

Y K  s  1 X

SO:

Y K  2 2 X  s  2s  1

     ; decay ratio overshoot2 overshoot exp  1  2    2 T 1  2 ; tr=T/4 tp=T/2

FOPDT:

  1.5(t63%  t28% );   t63%   ; K= Time domain: y(t) = 0 for t ≤ 0    ( t  )  y (t )  K 1 exp     for t > 0 

AR 

FR: AR 

K (1   2 2 ) K1

  tan 1   K3

K2

1  1 2 2 1  2 2 2 1  3 2 2

  tan 1   1   tan 1   2   tan 1   3   2    tan1  2 2  1   

FVT:

lim f (t )  lim[ sf (s)]

RGA:

11 

t 

AR 

K (1    )  ( 2)2

s 0

1 , 12  1  11  21 K12 K 21 1 K11K 22

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2

2 2

Laplace Transform Functions f(t) δ, unit impulse

f(s) = L [f(t)] 1

U(t), unit step or constant

1/s

t

1/s2

tn

n!/sn+1

e at

1 sa 1 (s  a ) 2 1  s 1 as  1 s ( s  1) 1 ( s  1) n

te at

1



e t /

1

(a   )



e

 t /



1 t n1e  t/  n ( n 1)!  a (  a )   t /  2  3 t e   1 a  2 a t / 1  e e t /2  1( 1  2)  2( 1  2) 1  a  t /  1  2  a  t /  2 1  e e  2  1  2  1

as 1 ( s  1) 2 as 1 ( 1 s 1)( 2 s 1) as  1 s ( 1s 1)( 2 s 1)

 2 s 

sin( t )

2

cos(t )

dy/dt

s s2  2 sy

d2y/dt2

s2y

 1  2  e  t/  sin  t        

 2 s 2  2  s  1

as  1

C

a2

Where C  2



2 a

 1  2

1

 a  2   1  1 tan    a  1   

; and

    



 et / sin(t  )  1   2 2 1   2 2

where   tan

1

(  s 1)( s 2  2 )

  

 1  2  t / e   sin  t   2    1    2 where   tan 1  1        

1

1 s( s  2 s 1)

1

f (t ) 

 0f ( t a)

2 2

e as f ( s)

t a ta

a, ω and τi are real and distinct, 0≤ ξ...