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Init ial ndFinalValueTheoremsFinalValueTheorem determineshesteady-statealueof thesystem esponse ithout indinghe nversetransform.

Procedure: 1.) find he transferunction (s)2.) mul tiply (s)bys3.) take he imitof sX(s)as Sgoes o zero4.) result s valueofx(t)when =infinityl im x( t) = l im s.X(s)

{ - - @ S-> 0

Initial alue heoremdetermineshevalue f he imefunction hen =0withoutindinghe nverseransformProcedure: 1.) find he ransferunction (s)2.) multiply (s)bys3.) take he imit f sX(s) sSgoes o infinity4.)results value fx(t)when =0

l im x( t ) = l im s.X(s)t - .0 s-> oo

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Example: Find he nitial alue f he ransferunctionx(s)= z ; tot i

s.(s ' * 2.s r 101tx(oi=l , ,* r tXCt)= l ,nt t sx(s) : ) ,mi KBoV* T-70 S +oc S-+oo V(S.+ZSttot)l ,^ ,1 _ BoE Lar-c1r a-| ,ue=,ds n,okc-S--+o0'J;Zg+E de^o^rruo$uc Ter- fc,trq > sx(o).J:,1;r#, . S gExample: Find he inal alue f the ransferunctionX(s)above.

F^*- V*t*-

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Using aplaceorsolvingmechanicalystems

Write hedifferentialquationsor heabove ystemwith espectopositionnd solvehemusing aplacetransform ethods.Assume(t;= Fand hat hemassslides n a frictionlessurface. (0)=gf ( t ) =M. x(t)+Bhx(t)*K'x(t )

r{x(t)+8.:-x( t )*K.x( t )dtF=M'TakeLaplaceransformfboth idesl- 6: -=M.sz.X(s) B.s.X(s) . K.X(s)S Solveor heposition (s)

=X(s)

d2dt2

d2dt2

t_2s\M's +Et438a-6.mcd 7

\B.s , K)

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LetM= 1,F= 5,B = 4 andK= 5. SolvethissingLaplacendpartialraction xpansion.= X(s) Use uadraticformulato actor enominator

a b:= 4 c: :5

/. ts( 1.s ' + 4.s + 5)

' 2.aS 1 = -2 +iFactoredormof unction

5

s2s2

2-a=-2- i

=X(s)s.(s + (2 r j ) ) ' (s r (2

Usepartialraction xpansionr ) - s

))

5 .B, , (stx)\, -c . s+ ,_{ )Fre+tr)(so(b6rr

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I

Mechancicalystem olution: ontinued

S 5 r - Rdi ' g**pleX--; : : -' L = (-z+)(.i) - -1i -L Conr,\,.q,_1es\ - ' ( ' t ( '^ ( - _t_ -+: 'L--\ \ o r -r [ i l iS \ i ' t I

r'fffil=JLtj-@J/FTieL u ' ' t (- l L 1 ' ----Et438a-6.mcd9

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Mechanicalofution- (at j ) f s I - (z-*) t\+ry1 vzFTt.f, s :,i :z+^+(te*+l_z-?l*e\

8.-1,o- .)tzeJJr ' n^N Jocs

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Plotof themechanicalystemesponsex(t) : : 1- e-2 ' t . (cos(t ) 2.s in( t ) )-2 relateso damping f system ecrease ndseeeffectsxt( t ) : 1- e '2 ' t ' (cos(t ) 2 's in( t ) )

t ,=0 ,0.25.10

x( )x1(t) 0

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Transfer unctionsInpuUoutputelationshipsora mathematicalodel sually ivenby he atio f wopolynominalsf hevariableDefinitionsPoles roots f hedenominatorolynomial.alueshatcausetransferunction agnitudeogo o nfinity.Zeros roots f henumerator olynomial.alueshatcause hetransferunctionogo o 0.eigenvaluesCharacteristicesponsesf a system.Roots f the

denominatorolynomial.lleigenvaluesustbenegativeor a systemransientnaturalesponse)to decay ut.

r ight holfplone

X's ndicateocationfpole.0 is ocationf zeroCloser oles o maginaryxis toweresponse.omptexootsappearnconjugateairset438a-7.MCD

lef t hol fprqne

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Examples

TransferFunctionisa gain sa functionfS

R(') f; x( ')--f G(s)|->'r linput - outputX(s)=G(s).R(s)

ffi=G(s)Passive owpassifter intergrator

ui t)

fv;(r)=R. i ( t )I I i ( t ) tt / l JTakeLapfaceVi(s)=R. (s) *

1Vo(s)=C.s ' l (s)*r(s)

Voltagedivider'V;(s) formuta

V;(s)1 = (s)Rn C.s

Remember1

So n.vo(s)==;.V;(s)=f tn- . -C.sR'C's 1

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1RCs+

Transferunctionsf OPAMPcircuitsPractical ifferentiator-ctive ighpass ilterwithdifiniteowfrequencyutoff.

Draw sa block iagramRC s imeconstantfsystem.System as1poleat -1lRC ndnozerosLargerRCsloweresponse

TakeLaplace fcomponentsnd reat ikeimpedances

vo General ain ormulaV,CI ^\

Simplify uR ;'C's n 1

-Rf

Transferunctionhas1 zeroat s=0and1 poleats = -1lRiC

-z g(s) V o(s)Au(t) =,( r ) v i l tz;(s)=Ri.a: z1(s)=Rf Av(s)=

R, * 1' C.sA u(s)=

- R1C's v o(s)V;(s)

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Transferunctionor Practical ifferentiator%(')

BlockAlgebraor ransferunctions-ascadedlocksSeries onnectedmultiplyransferunctionsNote: onotcancel ommonermsromnumeratornd

-RtCsR'Cs+

X, ' (s)l\ tc1(s) tr2 (si

X r (s)=R(s) 'G1(s) (1)X(s)=G1 s) 'c2(s) 'R(s)X(s)=G2(s).X1(s) (2)5+=G1(s) 2(s)R(s) ' \

Substitute1) nto 2)andsimplifyogetoverall ain

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Example ithOPAMPs0.1 uF\J 1

c 1'sAv1 s)=-,

v,,( t )vo(t)

First tage-ntegatorV 1(s)ct(s)=v.r( i lForstage_z1(s)A v1 s)=; iu z i (s)=Ri z1(s)

Second tage-practical ifferentiatorV o(s) TakeLaplace fGz(s)=V 1G) ::Tff.entsand sev' general ain ormula

-1

c 1's-1Simplify u1ogetG1(s) G 1(s)=Ri 'c1Gz(s)romPreviousxamPleA y(s)=

- RgC 2's =G2(s)R1'C2'st 1v o(s) _1 -R1C2's Negativeigns ancelVi(s) R; 'C1's R1'C2's+ 1

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v o(s) R1C2'sV;(s) - (RSimplifiedorm

1.c2.s + 1)(Ri .C 1s)Plug ngiven aluesor hecomponentymbols ndcomputeparametersFi ,= 10000 C1

R1:=25000 RfRi 'C1=0.001

Above reall imeconstantsor hesystemFinalransferunction

0.1.10-6 C2,=0.05.10-6100000 R 1'C2= 0.001

Rt 'C2 =0.005

Function as1 zeros = 0 and wopoleso(s)V;(s) 0.005.sat

(0.001.s 1 .(0.001.s) = -110.0011000ands=0Parallel locksadd ransferunctions

et438a-7.MCD