Cs Unit III Full

8/4/2019 Cs Unit III Full

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, C '" H ' A 'P 'T E R ' 3 '_.: I -. "-, . I, I e: I " . ' _ _ ' . - : l . FRE ,QUEN . 'CY

ANALYSISRES~PONSE

.1 1 SINUSO.I DAL TRANSFER FUNCTION AN'D FREQUENCY RESPONSE '. . ""-"':. . _. .. . . . ..The response. of' .a system fo:r the sinusoidel input iscalled sinusoidal response., The ratio of

inusoidal response and sinusoidal input is called sinusoiilal transfer /unctin'n of the system and in .n eral, it is d en ote d by Tr(jm).The sinusoidal t ransf er func tion is th e frequency domain representation of ,

he system , and" so it is also caBedfrequency d oma in tr an sfe r functiO _n. ' 'The sinusoidal transfer, function TOtp) can b,Jobtained as shown' below.. ', " " .~,,' , 'L Contruct a physicalmodel of a system using bas ic elements/parameters.2. Determine the differential equations governing th~ syst,emfrorn' the physical model of the"system, ,

3.' Take Laplace transformofdifferential equations in o rder toconven them to s-domain equation. .. ." : . .. ". .-4. Determine s-dornain transfer function, 'I'(s), which is ratio of s-domain output and input,5. Determine the frequency domaintransferfunctlon, T O m ) by replacing s by jo inthe s-doU;a intransfer function, T(s).

Note : If the s-domain transfer function, 1 '(8) is known, then frequency .domain, transfer Ifunction. T aco) can be obtained directlyfr-om. Tts) by replacing s by [co. '

i.e., 'T(s) s-jcu 0' T U r o ) i.' I~----~---.~--------~.-----,----~~~----~--------"--~----~~Consider. a linear time' invariant system with frequency domaintransfer function , TGm} shown ing 3.i-,et th e system be excited by a sinusoidal signalfrequency oJ"ampiitudeA, and phase e. N ow the '.esponse or output w i n also. be a sinusoidal signalof same frequency ffi,but the amplitude arid phase of ,sponse win be modified by amplitude and phase , of the transferfunction respectively. '

. . . .Now, 'the amplitude of th e response is , given b y th e product. of the 'amplitude of th e input and .function ..The phase of the response is given by the sum of the phase of the input and transferuoc~a ' . , ."

'Let" ' TOO) =IT(jro)i LTGro)'wher~" iT(j.ro)1~ Magnitude of TO(O),.and~" LTum) = Phase of TO co)..

, ,Let" Input; r{ t) - A sin (c ot+ ,H)=A ,L H, ., where, A=Amplitude of input , o= Frequency of input, and ' e ~ Phase of input.

Now, Response.en) =r(t)x,TOruJ =ALH x'lT(jro)ILTGm)'=Ax~T(joiji L(O+LTGro)) ~BL~' '.where" B =AxiT(jro ) 1 . = Magnitude of response, and, ~ = =9+ LTU ro) =Phase o f resp on se. "r{t) 1 . . . . . , I ' c(t) ctt) = BL$ret) = A sine rot + 8) = A Lo S TUm) = 1 TOm ) 1 LT(jro): ' . :~where~ B ,;:=A x jTO ' ,? )\ . . .. . , . ; h si .,]' . .." .. . . . = e + LT(Jro) ,Fig ~.1:System wU sinusoidal transferfunction TOm), ' "

, , . . ,


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3.2F R E Q U E N C Y R E S P O N S E

. . . The fr equency- domain 'tra nsfe rfu nctio n TU ! ;o )is a complex function of e i .Hence it ca n b e se pe ra te dinto magnitude function and pha se function.Now, the magnitude and phase functionswilfbe real functions .'of 0), and they are caUedfl'equencyresp onse. . .'

.' .' .The frequencyresponse can be .evaluated -for open loop system and dosed loop system ..Thefrequency domain. transfer function of open' loop andelosed loop systems can-be obtainedfrom th es-domain transfer function by replacing s by jm shownbelow, . .'. Ope. loep- transfer. function ::G(s) '. s"'"jm:. G(j(l)) "'"1 0 0 ( 0 ) 1 LG(jO)) ' (3.]).Loop transfer f-unction .' _:G(s)H(s). s=j@ . G(j@Jl -fGw)""' IG( jm}HGm)I!LGGro)HGro) . .. .. (3.2)C losed loop transfer function: M(s) . s""jro ) MOm) = I M , ( jr o . ) j LMO ro) ,. .~.(33)

where, IG(jro}~,IM(jfD)l,G(joo.)H(joo)l are Magnitude functionsLG(jro), LM(jm), LO(jro} HOoo) are Phase. functions,

~N~te z F~r unity feedback system~ H(s) =1and open loop_.and loop' tr~nsferfunctions are same. I T he advantages of frequency resp onse analysis are the following.- 1 . Th~absolu te and relat ive s tabil ity-of the c lo sed loop system can b e estim ated from the know ledge .:.:of~beir open l_oopfrequency response, . _


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Re sonan t F requen cy ( < D , )The' frequency at which the resonant peak occurs is called reSQri~n.1frequency; (Or" This is relatedto the frequency of osciltation in the step response 'and .thus it is. indicative of the speed of transient

response.Bandwidth ' ( '1 .) . . ' . ' , . ,. . . : b .

. . ' : T h e Bandwidth is ' t h e' range: o f frequencie s ' f o r which normalized gadnofthe system is more tharl '-Jdb.The frequency atwhich the gain is~J'db'iscaUe9cut-off.frequencY, Bandwidth i s uSuallY'defined:forclosed loop system and. it transmits the signals whosefrequenciesare less than the cut-off'frequeney, '.he Bandwidth isa measure of the ability of a feedback system to reproduce the input signal," noiserejection characteri st ics a r l d - rise time, A la rge band'\vidth corresponds to a sm:aH :r i s e time-or fast response.(ut-Qff Rate.

The slope of the log-magnitude curve near the cut off frequency is called cut-off r a t e . The cur-off ..ate indicates the ability of the. SY$~m to distinguish t h e signal from noise, "~ait11:Margin ,I(g " ' . ..

,The gain margin, Kg is defined as the value of gain, to. be added t o system, in.order to b r in g th ,e 'system to the verge of Instability, '.. ' "'. "... . ' The gain margin, K!!:s giv~n by the reciprocal oftbe ma~ituqe ofopen loop. transfer funetienm .

phase cross over frequency The frequency at which th e phase of open loop _ransfer function is ] 8 .00-is . .the phase crnss-overfreqaency, rope.; .

.. , . '. .. .', t. . .(Gam Margin, Kg: _,.'.'.,. . ..,.... 3 .4). ' ' . ' . jG ( J < O pe ) 1 , .The gain . margin i n db .ean be expressed as.. .'

.Kg 'in.db =,20 'log Kg =20 log 1..' IG(je) ) 1, . pc.':Note. :~IG(j(t)rJl is th e magn~Iu d'e a / G a m ) ' a t 0)= r o p e . i

. .. .. . : (1 .5). .

The Gain margin 'in db is .given by' the negative ,of the 'db magnitude of G G r o ) ' at phase cto~s-Q ver "requency, The gain margin indicates the additional gain that can be provid~d'to .system .wi~hout affecting ..the stability of the system, .Pha~e,~a,'!Ii.n.(1 '), .

, The p h a s e , margin r, is defined a s the additional phase lag: to' he added ?t the gain CfOSSOYler'frequency in order' to bring the system to the verge of instability. The gain cross. o v e r 'frequency r o ~ c istherequency at whichthe magnitude of the open loop transfer function Is unity (or it is. jhe frequency athich the, db magnitude is zero). .' . .

The phase margin y, is obtained by adding 180('0 the p h a s e a n g l e '< p of t h e open loop trarisfeiuCtloil at th~' gain cross over frequency . " . ,

. ."'~

.... Phase margin,'. Y= 1800+$g ; . ~where, 4 l : = L G ( jO ) I, E .C . . , IP

' . .:. .,. .. (3 _ 6 ) 1 .

",- - - - - --Note : : ,LGOU)gc) is the phase angle ?!GOm) at ro =0)g'1 .


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3.4. . .The phase m argin _indicates the' additional phase lag that can be provided tothe system w ith ou tstability . . . .~.-- .-

.FREQUENCVDOMAlN.S; ,P;EC1IF j ICATIONS OF SeC-oNDORDJiEF tSYSTEMONANT wPEA~ .(M )" ... : .C o usidcr th e closed'. loop transfer function of second order system"

'. .' C(s) .,'. .' 002. _ . _ _ . = = ' M { : = : :) =, ?,' .,.Il , , '. R.(s) . s- + 2~(l)n S+00; .. .The s inusoidal tr ansfer function Mfjw).is obtainedby l " eU ipg s =jm,

. ,2.' 00,.: . M C J m ) . ; ; e o .( " . ')i , 2 'r' (")" 2. .: Jro ' +~ron J"ffi+(I)n

.....(1.7)

., ~ (3 .8.)

2 . 2 1(0 . { I J : ) I "2 ~ ..,~ n ,+ ' 2 = .(. 2 n ) = (' J 2 . .(i)I T~mnW fin 200.. ill ,I. . 'cJ . . '0)(!)n, ~ _2," "+ 12'~".-,'+ t, 1- I - ' - . ' +j21;-' -',.. . ' , 0 0 , (i)~. , 'i';' - r . ". . n.: .... .'-'-'n' . \)J.n

.Let, Normalized frequency, U~(:J::. M Ow) =;: . 2i~... '. :(1 -U ")+JZ t:;;!U '

-. - "_a "= .P hase of closed 1 , 0 0 1 ' trans fe r functi on .. [ \ . . . ' y l . I\ 12 " '.'M=IM(jrol= ' ,.' "',2 : : : : : [ o ' - u2 i + 4 S ? u :; 1 " '2 . '. " O -M 2i+ (2,S lJ .I). _ . ' ". . .... . . .~! 2~u C " 10)a"'" L.M(Jf:t}) ;;;:,-.tan ~,,; ... .J . . .l~u. ':

The resonant peak is the maximum v~lue ofM. The co nd itio n for maximum value of Mean beby differentiating the equation o f M w ith respect to u and letting dM ldu= 0 . w h e n u =u,

. . ,~,. . (3 .9)

,I. {[ll) .. ' .', ,.,' . ",...,' " '_'. '_, _',: .where, u, = ._ =No rm alized reso naa t frequency'0)' ,On . d if fe rentia ting' equati on (3.9) with respect to u we get,.. .. , -..' .'.' ..' 3 . .

dM . d [ .'2. 2 : . . 2,2]--2' [[. (1' 2)2 Arl 2 ] - - - 2 1 [ ' 2 " C 1' ,2. (. 2") +"Qr2 ' J '- ='-,' Cl-'u) .+~" ill = ~-, -'l!.. + ~ lIl, '. ~j ._U J -u ~.u .du du ., . 2 .. . ' I=-[~4u._(L~ u~)+ : S Z ; 2 U ) = .'4u(1~. U ) , 2 ~ ~zu .

] . J 2 1 [ ( 1- u2i+ _ 4 S 2 U :l ] '2 . 2 [ O ~ U2)2 -+ 4 S 2u2 p -

, R ep lac-e u by Ur in . equation (3:1 1 ) a n d equate to zero, .4u~ .(1- U; }_&;2U, .='0. .' "1' . .. 2 [ ( 1 - u;i~+4S1u; ] i .


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.5. The equation (3:.12)will be zero if numerator is zero ..Hence, on equating numerator to zero we get,

4 uj[( 1 -:- u;) - S S2U r_= 0 .. => 4~ _4U;_&;2U T = 0 . -"u3 - 4 ' U, _llr2u l. "It 'r ~ r. ~ "r =r=. (3.13)

Ther~fore, the resonant peak occurs when u, =:J " l - 2 ( /Put this condition in the equation for M and solve for MI,

M 1:.r =---~--:--c:-Ih. ('(1- u2')?~'4;2U2F. ji u =Ur. f

1=----~-~_ ", 1[ (1- u~/+4;2u ; . y .I .

1- .,',:Resonant peak, Mr =-~=! 2 s j l - ~ z ....~.t3..]4)

F R E O U E N C Y (m l ,,\'. \\'.

Normali~dres~n.antfr,eque;~qr, ur= ( ) l ~ j " _ = = ~i':_2~2' .. . .". . . \ (j:)j'] .\.

The resonant frequency, "', ='"~~1-2(,'

.....(3.15J.....(3.16)

\' ..'Let, Normalized bandwidth, Ub = rol),.. r o DWhe~u =Ub; th~ magnitude M, of the c l o s ed loopsyst~m i~ IIJ2 (or ,-3db). .Hen~ein' the equation for .M(equation 3.9), put u =~ and equate to IfJi..

1 .]~'.M ;;;;.. .' '. . ".!: -J2 ' .[ ( 1 - U~)l +~2U~]2 . -.On' squaring andcross multiplying we get,. . .

...:.(3.17)

(1- u2)2 +~2U~ =2. . b ... ! P. .u ! ~ 2 u ; (1-~2) -1 .~ ()

Let, x "'"u~; :. xl - 2(1- 2 i; ;? )x -: ] "': '0:..X= 2b-2,~2) ~4(1~'2r/i+4.'= 2(1~2(/)2J(1+4l;4 _4~2) + ~

2 2Let us take.only the positive sign,

.. ' ~ 2 r2 J . Ar2 ' Ar4:.x=.I!--:- '-.; +2-~ + " " " " ' ; . , . . . .'1 ' 0)bA so U=-b r o n


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3-.6 .....(3.18)

ASE . MARGI~ (y )The open loop transfer function of second .order system,

. - 2;- wG(15) =, II'. .' .s(s+2l;cofjJ ....(3.19)

- .The sinusoidal transfer function G(jro) is obtained by letting s =jro._ 2 : - _ . 2

O(jW ) "", . O J [I . _.. _"" _. . c o It. = _ . _ 1jrn O w + 2C ;wn ) '. m i r ( J . i W . : J

1O O : n ( - 2 L ; +jC O J . '.j"W _ ( 2 ~ +j_. m ~ ' ) . 1 .ill ,'" ro , Will., . n,. .n. "..: ]I. n

. - Let [I ,annal ize? frequency, 1 1 = w/ill n .. On substituting u= w/(j)g inequation ( 3 , 20Ywe get-

G(jm) = . (2~ . . ) -Jll .' +JU'MagnitudteofGU (i)I)=~GG(())i'~ ,1 _= .. 1:.'. _ . - . .UJ.1,t;,2+U2 ~U4 +~2U2_

Phase of GUm) = -90~~ tan-i2~Atthe gain cross-over. frequ~ncy co) ie" the magnitude of O(](I)) is unity.Let normalized sain cross over f re quency ; U ==(0 -1 m. . ' , ' . i l l : C . 1 1 ' ; 1 1 .- .On substituting u bYl i l~c infue'equation(i2~) and equating to unity, we get,

:. A tu= u,o> lG (jro~ =~~4+~i u' =.1."," u;, +4~2,,:, = 1 ~ u:, + 4l;2u~ ~ 1= 0. gc .. gc

.....(3.20)

. .. . ; (3 . 2 2}

..... (3.23)

Let, x ""- U ! c ; : . XL +4 {;?x-1 ~ ()'., . ~~2 fl6S4-+4 . . :r-,--.. x = 1 J ' . ' . = ~2r/ ' J 4 S 4 1 +1, " 2 '. _. .:

Let ustake only th e positive sign,. :. x ;= :~2S2 +~ 4S 4 .+1

. : . . . ( 3 ' .24)

The phasemargin, . y ~ 180+LOOm) i O J = ~gc, U"'Ug~. Substituting for LGOro) from equation (3.23) in equation (3.25) we. get, .

. . - ~,

. . .... 1]' ... " . i [ _ 2 1 ; 2 + ~4S4 1 + ] J -"t =' 180+(~90~_tan-1t \ c ) =90-tan-1 '. . ... - .. ' .. . . 2~ .1 . 2 s

...,,(J.25)

....:...(3.26)

N ote: T he,gain m arg in of second order system is in ji:nite.


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. The correlation betw een tim e andfrequency response h a S anexplicit-formonly fo r first andsecondd er sy stems. The .correlatlon for second-order' system is discU5!s~d b~re... . .Cans id er the magnitude and phase Qfoados .ed Loop'secondordersystem as a function of'no rm alized .

given by equations (3.9) and (:1 .1 0)_ .Magnitude of closed loopsystem, M = I M O r o ) l : ; : " - . , ! '. 2':':" . ' . . ' . . J { l - u") + (2t;u). . j~': .. .. ' ' ]2< :uP hase of dosed loop system, a = LM(jm) ~ -tan- .'-._.-2- .1-u .The magnitude and.ph~e.~gle ,charn~teristics for normalized frequency' u, for certainvalues of~ ..

re shown ill fig 3~2 and 3.3, The-frequency at which M has a peakvalue is known as the _resonantrequency. The peak value of the magnnudeis the resonantpeak M. At this f re quency :th e. si9pe. of theagnitude curve is zero, Tile 'frequency corresponding to Mr isur~ which is the normalized resonant :.tequency, ... . . . .

F r om equatio ns ( 3 . . . 1 4 ) -and (3 , . .1 5 ) w e get,. . - 1Resonan t-peak ; M. = = . ,r:::i.. '' . . 2 s , , ' 1 ~~2 .Re sonan t fr equency, 0); =(on~r-l-- '2i;-.: '-2

. W hen t:=.o.';; ~1 . .] \ . 1 = = = 00'. r 2 r ; 4 J _ . t : ; , 2 . ~... ~

. M~,1 :.4 . '1.21.0-----0..80.6 .. .~.

0:4 I . - : ' 1 : :l0;2 ~. I. _.,.0' u, u.

Flg 3.2 : l'\4 agn it it de ,.M as a jimct.iorJoi U~

-._.(327). (3 ..28)

0: ,

o . t .Fig 3.j : Phase, o:as function of u.

2

From equations (1.27) 'and. (3.28), it 'is clear .that as ~. tendsto zero, r o r approaGhe~. mri , : ,and 'Mr .pproaches infinity.' ... ', . . .Whery'I-2~ = = 1 0 , (Or ~ 0, ~hich means there is no resonant peak. at this condition.

1~=-' .. -/2F~r .10< ~ :S 1I.n the resonant freq uency alw ays h a s a value iess. than I D Il~ and the resonant peak:

as a value greater than one .:


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3..8Fo~ S> 1I fi, , the ~cbi1di tion(dlv l ldu) . =0, willnot be ' S t i t i s f i e d fo f a n y r e a r ~ a l lJ e G fm ~.' .~ : . ~: ' .

". ' . 1 }:nc'~ w ben '>1~.J2 tbemagnitwi M de.cr~M~slRo~~Ic;anY -h ~Q 1 M '-:-:I. ,at u ~ . 0 ~i~h_ -' -. - . - .. - - . . . . . . . , . - - . . . 'in crea sin g u, It fo llow s th at'fo t- "~;llJitbere:is0 ' resonant p~ak;m:dhegP eaitestV'alue ..o-f.M equa l s oae,. The frequency at which M h~~a~lue'of ~lfJi"i-s efspecia1. ,signincance and iscaI1ed)he cut-offfrequenq .coc : ' The signal frequ~ncies ' a ; o o l i e ' cllt~ff are ~ly : a tt e tW . m e : d 'OIR,paSsmg:thro;ngh a system~. For feedback ~ontrol & Y i ~ , t h e ' m n g e offrequeneiesover which M';:::l/Ji is defm~d'as"baoowidthI D : o : ' 'C~ntroi ,$ .Y,stembcing1ow~paS$:mtefSJ~~terij,freqUency M=' f l " lhe: !b~dtb J ' b : i~equal '1o'cUt~off

~Al,t1e1i~~,mc' " '.' .,," , > : ' ~ . ' , ' . ' , . " . . , . ' " , " , . . . . . c .. ': ". :: - ". - In : general thebandwidthof'ncourrol system 'indicates the ' i lo is ( } ; . ft fte r l r ig :chara c te , r is t i rcs ' o fthe ;~ystem,.Also, bandwidthgi'l/S a t measure 'df the ttansi!entrespensre. ": , '. : :. ' . ' - . .-,:~~ vJ i ,.. P. '"M r does not :ex iSf1md' t l I e -correlation ' . 1 . ' 0down. This is:not a ser i l )m ~m a s fOr th ise:oft;,."thestePresponseoscmatioos.are~weU damped. 0.5:a a d 'M .is negligible. ' '. ' . . . .

. , p .' . . .' .. The comparison of the equatien ofcoralld (Odrev eals th at there exists a defini~e.orrelation betweenhem ., The sketch of ( i ) r /m" with respect t o , t; 'is'shownm" F ig '3 .4 : NofRwlis .ed bandw,id th as afonctioo ,of~'~g1.6. .' '. '.

. .'." .

r. . . , ..5 O~707 ".1

W r~.

1J) ,

ro d

, 2~O

_ ., .'. '

o " _ (is 0 707 '1 ~ . .. F ,ig .3.5 : M ; . a i J - P M p as ajunclitm ofs. ~


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.5 .FREQU.ENCY,RESP:QNSEPl,01S, .' -~':- :: ," '~. . " . ~. . .Frequencyresponse analYsts cfocntrolsystems- can b carried either analytically or' graphically>he.various graphical techniqaes available for . frequeney respeaseanalysis are, . . .

". :.1. ..>Bodeplot. ; .. ,': ':.... ' - . ,C' .. . 4. Mand'N:cirles:. "",: . , ~ 2. . : p o . n a r : p m , Q t :( o r :N Y q E I li s tp l n t ) 5.. Nicholschart ';... ' .~ .. ,-; ,. ; ' . . 3 .. .N ~ '~ ilJolsplot. . . - '. -v-: .. , _ . ...-, ~. ~t ' .,-- ", ", ;:.,'. ...; . - ' , " T h e ) a ~ . d ~lot~;POllarplQ,t8.1!dNic]jols plot are.usuaijy.d~~ f o i open.loop system s ..From t h t ; : . Q pen .oopresponse plot;' th e performance a n d stabilityofclosed loop .system are estimated. Th~ M. a n d . N .:

and: N icho ls, c fu~ . ~:~,~ed J O 'gntp:h,ically determine . i f l A f ! ' .freqll,el: l}cy response of unity feedback- ' , - - ; " ~< - . ~- ~ ." " '- . , ' _ : : _ loop system' from th e knowtedgeofjrpen loop respoq~e..'. .. .... .'. . .' '.. .' r

- ~. p _ ~ ~ _ _ ~ _ . - . T h e ~ q u e n c y r e s p o n s e p I n t s " ~ a r _ e lUsedtodetenjlinetbe ~equericy-don1a in spec if icat ions:" tos tudyte s tab~1 i tyofthesysteins and to . adjuS:ttil 'egain.oftlie~System~Q s@; tisZY: the.dt? s_ iiedspec ifications. ...6": BODE'PLOT, > .... .-.. _..... .. . . . _ . I.. l .. The Bode plot is a frequencyresponse plot of the sinusoidaltransferfunction of a system.A Bodecoin:sistS o i ' t w o g f a : p b s . On~f 's ~~pldt()f the magnimde of a .8 jb ii$Q; j\da li .t ra ii sfe: rf il lic ti~n verses ' log . 6khe other is a plot ofthe phase angle ofa sinusoidal transfer-function versus logm. ., .. . . . .. ,

The Bode'plotcanbe drawn ferbathopen loop and closed loop system. Usuallythe bode plot israwn for open loop system, T h e : standard representation of the logarithmic magnitude' of open .loopan sfer. fu nctio n o f Gum) is 20. i l o g _ ~ " I G ( j r o ) 1 where thebase of thy Iogari thm is 1 0. The unit' used .in thisp re senta tio rr o f th e magnitude is the decibel.usually tlbb~evja~~A. db, _ '1becurVes am drawn OR ~ e m H l J gper, using the .log.scale (abcissa) for frequency and the:lin~ar~sRal~{Or~ina~t1)oreither ~gfl~tude (in.

Or phase angle (in de~es):" "_- - ~ ' .:.. .' .~ . . ~~ i" ".. The main advantage.ofthe bode plot is that ri1U ltiplicatio,1fll."jO :finagnitU des;.canbeconv.ert~d into,Also a sim ple m ethod for sketching an:approximate log -m egn imde cu rve is available, .. ,... -. -. .~ _::::.~~ -~ ; ~- . :.

.- C onsider the open loop transfer function, G(s) ~ _. ',.'J


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.3."10. . . [ . , . .. '.. K. .2 ... = 20 log r:x ~l+'ro J t x.(0 ..

, . .

.

'.' K 1 2 . . . . 2 . . . . . I' ... : 1"'"20 l?g .:-:+20,:lBg VI+a:c 1 m +20 log ....' . . . . . . ' . _ +20 log . '. :-. '.t . .00 - -; .' '. 1 / 1 + 0),2Ti - ". , J l + r o ZT i'~201og~K+20-I~g Jl+ro1;-20 lo~ 4 1 ~(02Tff ~2~I~g 4 1 +ro21 j: ~ . ...:.(3.29) G . . '. .- . .. ' -F rom the e#lu 'on (3 ~29) it is c le ar th at,w h en th e ~~itude is. e xpre ssed in 4 b , _ f u b multip~icationconve rte dto add ition. H encein ~.~tlide p lo t, th e d b magnitudes 9fmdividtm1 fa cto rs o f G(jrn) can be~de(L . . - .., .. ... ',

f .. There fo re . o , . sketch the m~gnimd~plo~a knowledge oith~' I1l8gnibide variations ofindividua]is essential, -The 'magh iU l de~ -p l e t and p b a s e ;pb)t ofvarioas _ f a c t o r s : o t , ' O O c c i } a r e ~ x p i ; a r n e d ' in.theoU 9W ir.lg section, . .' '. - .~.- ~.: .... . :G:. - ~ -' -'. - .

" . . . . . ,... . ",' -,-

.-

, , The~b a s i c factors that v ~ z y t t e q u e r i t l y occur in a typical t f a o s f e r 'function O O m ) are, -,_o ~.~. C o nstant gain, K

.:.: .. _" . K._ K' _- :2. .]nte~l factor,-..-' or _._...0.'" _ ' j ( 1 ' J , (jroJn .- .".3.- Deriv ativ e facto r" Kx jioo,nrK x ( jj:i) )n.'. . .. - -.. 1 . 14. FIrStorder factor m denominator, . . " . or ..,..... --..m

. . - r + J ,roT ([ + JmT).-5..- First order factor in numerator, (1~+jcoT) r {1+j m . t i ', . .'. -. ..' '-' --.. . ~ - .' '.. .'. -,;--. . [. 1 ] ! I. ', 6 . . Quadraticfactor m denominetor; . '. - . . . . . , . . . - '. . ' -. - :2

. '.J' ..... . .I2t'; (jro J ( O I I J +Ow Imo).. . I . . ,- . ;.',. s, ,~- . .' '. .. . \

.7: Qua d ra tic fa c lrm " ~ , [ . ! 1+2l; ( . j m ' . ) . I : + ( ' j ( O ) ! ' ] i .' . ' . I ro o ro n .' to I . . " . .

'.Lef;-G(s) =K .'

. .~._0(0) = K =K Loai-t .f.d b : -.

K = '. 1 \ . . 0 -A =IG( j - r o )nn db =:.21 0log:K .~: .. "

4 1 ~ LOGm) =.00The ma~~tud~p~o~. foraconstant gain K is ahorizontal straight line at themagnitude o f 20:' lo g K .dlb .he phase p lo t is straight line atO -. '

Ain O'I-l--~----.....;...-db O


11/99

:2 t~'!71". ~ + < '. : a ~ ~ .

' . : A " = l G O r o ) l in:db=2tJlog (KIm). -: . - -.".., ;.:' '. '.: 0 .: ..,'~ . ~. LO(jOO) = --90

I . .

Derivativ.eFitctor... .' , . .. . . '- . . . . . ; . .~ " ;: .. . . . . ..., GGro}=K J O ) =Kro.L90

') A=IG(j;m)! FJ idb ' "" , 20; log (Kro)" ' : : "

Let, G(s) ;;='K$

! " -. - . , . = LG(Jro) = +900 .(WheH,{t) "'"0; 11K, --A=20 leg(Q.;1)=_;20 db

. - 'I ,. ., ....Whefl.ro= ilK, A'=201og:1=Odb .. ::+90~~.-'---:-...;...;,. . . .. .' -__ ; .. . , ;; . ;. ._ "_,, ;; ;, . . .. . . .. . . .W h t m ~ - HlIK,"~ t ~O"~10= +20"". .1 . 0" I-~~ .~....,.:----,.._.::-,.---..;.:...."....._.,..,._, '. ,'_';.:L~From ."the above analysis it Isevident that the .$ , .' I .

m a g n i t u d e plot of the derivativefaetoris a straight linc'L;", ""_'_~~~'_''''''1'-. - ' : ' _ 1 ~O " " " ' _ _ _ ; _ - - ; ' : " " - 'i--. .....,,;.::...with a slQpe of +2n .d b/d ec and p assin g th ro ugh zero d~ ~1. K. K . roqog~ciller:~w h e n 0) = :l!K. Since the LGJjro) is .a constant and . . '. . '.. ' .' ... denen f . " .' .'- .... .' F i g 3..10.:B o d e p lo t o f.der tva ti ve f a c t Q ! ! ; . K x , j ( f J . :independent 0 o, the.phase plet is a straight line at +90P. . .

. " ', ,, -i . '. .. "

" .. - .; ; ,":_


12/99

3.12"'"When.d6ivatlve factor:_h~ muitipOOity.~of nthen, .. t . " ..-,..-- .0; ..

G(s)~K S O " : ' " , , . " A 'I +~ ...",":" "," .::~~. , 'O( jw) '= 'Kf joo)""" K r onL 9 0 J i i o - ~ -x'. ~ n : . ~'4 . !!

'" '' .A .== 'IO( j ( o ) (i it . .db~2Qlog:(~ID". .: . db:... :20rr : '~ \.O '9 .~ ~ , '.'. ~ 2 0 logtKli'li ( 0 ) ' 1 1 ~2O--tt log(KiI~ill) .. .' ,.

".

. - , = = LGGCl ) =90n'? ." ' .::" . : . '""t:'c iw ~e ~itudeplot Of~deri~ '~' ' r ~ " " ' - - , . . . . . . . . .. . . . . .~ '' _ ' _ : : . ~ ~ - '. . .: . . .' . . . .- - . . _ ' _ _a sl_!a.Ig~tllne WIth a slope of + 20n dbfde~;-and,passIEIg .~ . 0 . : ; : .4-when w = lIKUII~The phase plot is a .fine'et +90n. . .. .- -, ." ~.~ ; A . i f e n ( }minator.. . . . . .

0.1 . 1 ' . 10 .-:.(I 1 ). : . m l o g ' : s c a l e . '-1 -11. ;(. - . : >"K"' .;KJi, KfI. : " ; " '~ c " , . ; . 'ilg' J.11}jJodeltiot/oj~rtvdtiv~:a : e t o r ; . K O r o l ' .. /..' . - ':", -~.'


13/99

. '.Thephase angle of the fa cto 'r, ..1/(1tjmT)~- . v a r i e S from. a!?~to--:,9no: ~,.(i). is-varied froM zero:'~ti:iufinity.The phase'p,lot is a curve passing thro-~gh-450_'at Ol~. . .Wh en. th efirsrordee factor-in th e .denom inator hasa multip licity of m , ..then,. _ . . .' .:'. . .."1 ... . ': 1 IO (s )"""-- '-" G O m :} = ". O + sT)w ." .. _ .: . (1+jroT)mc.( '/' . ) m . .,'"!Jl+0l2T2,,>~m _tIm:lmT , . ' .:~.. .

. ,,~

'. -:~~

i - 45" ' . ' _ _ ; _0 1 . } ~ ~ .. . ; . . . : . . . . ~ .: : : : : ~ . =-_ ___::_'f I' III I~ I

,'.'

. , 1 '1'0 O J,m . ;: ~ (log - 1)~..~.:T T. '..... ~ca :C . '

fig-3:.:12':' B ( ) d e p!otoft~/acto,.' J ' . 1 . _ T. . +](J.lN ow th e magnitude plot of th e factor: :l!(l+ ]mT)'" c a n be-approximetedby ' two straightlines, one i s : ~a straight H ue at zero db for the frequ ency rang e; 0


14/99

3.14When the flsst order factor in the numerator hasa multiplicity' , 'of m,'then,

O(s) = ( 1 + s T ) mG U r o ) =(1+ imT)'" ~Ul~ID'TTLm~C1IDT ..~A ~.!?{j~)I'n db= 2Qlog ( J l +OO2T2 J ~= 20milogJ'l+ olT2 .

'. :~."

. ~ - .. , + . . /GG")" ...",..;,-1 Tr = ,.f-': '0) i ;i::m' H l ! J ' . r I (i) ij"

1 , . Appro~dmate plot '-It': .'~.. 1. . +20 ---~------1--,,~~--~60 ..+3 ' -~~~:t-~I~'!~"""""'k1;rt) ~ . .A . a ~._... ....-."-.,, ~ . I' ,,in' .: ..d b + 90 0 r~~"'"---'r---~:::;;;;::;;:i;: = ' " " ' : ' " " " " " " ' " - - : : : ' , ' 7 " . -.' """'1 ' "'," .- r.:

1 .," . I- .:~., '1'.

. .: : : , " : " ' ,

_',' 1. . . . . . : .

Le t, A =I GOm ) [i n d b . . .," '( , . 1 , . ' 1 - ' '00,',2.'. ' J 2 + . ~ .1 f. .; .~:.2= ~201og'. ,..,

, (02. ' . 0 : , 1'. .~n.... , .', .-,It

=-2Q[log . '~4 .; ro2' '0:/ . ,." .1+ - ~2-+lr_2 - . ""'.-2010g',4' 2'~' 2 '.O l n C O n . C O n


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3 . ' ._. -" ~' .. '

'.,_ . ", _' ~ ' : ' - .~ ' ~ := - - : - - - . ' - . : . "

. - . .At very .low freq 't lrencies ' when " ( 0 , ,(j); ,,i the- m'Sgnimde is" .'. " . . . ' _ . . . . : . . . . n . _ .',... " .0)2 ' 2m 4 . . .A - = -:-20log. ; J : - - - - r - z (2~4l;-) +~4 ;::,-20 log 1= 0. . ill 'ro .n n

At very high frequencies, when { G , m~,,tbe: inag i l l tude Is, ... .. :

A = -20 log 1 - : : (2~ ~ z ; : : , . ~ 2 0 l o g . 1 S f = ~ 2 0 I ~ : ~ = -20 l o g ( ' : J - 211 n V ~ . , n .. n. .' m:.A= -40 log-'. . ..~ . IDa. At O)"'".~"'A~ ,. ;"cr 'l :e .gl= 'Odb, . .. .~ .. -. ..'At (9 ~ 10(U~A~'40 lagrO=~ 40dh ., . .. - . '-, . F rom tl1 f~ atr6 ve'analysis: itis ev:ident tlmt 't lle magnitude'plOt ():fthe-:quaaraticfactorinthe: den~Jmiriator ' .can -be approxi~ated by two strai :gbtH:nes , one is a straight fine 'atO .db for th e fre qu en cy ran ge 0 < . ro < ; 0)" ..' , . _ ._. . . . i.. ..... . '. - '. - . . ,.,.....


16/99

3,'l6A . ' 1 jJ - t : . .i.2 S , tan-I.' '90':" s o o =(J) - - L a n _=-' . 1 , 00 = = .

.. . '. [l' _. . .O . ." '.As (0 ~ 0, q , -0'. . A s ro 'rl:~ ~':4.,UK;

,":', . . ',;:'

The phase angle of.&cqUadm& "factol' v:arie$:.from(l:to~]80o~.ro isv aried fr0 iJ 1 .0 ':to 00. The .lhas~ plot is a curve passingthrough ~ 9:O t'It ~.ctA tth e OOT Iierfrequency p hase angle. is:-:-9 00 a I ! ! d i n d : e p .e n d e n t .f'~..but at all other frequenCy ~td~pends: D l l t ; . , . ' . ' . . . . . . . : , _ . . . :_ ., ' - _ . . . . . -,'.

. . . . . . . . ( " ' ) ' ( ) ' 2. '" .' 2 .. S +2~ID.:.S+O) . . ~S [, I ' , ' . S.' : 1 ' .G(s)=. .' It. .n = 1+2


17/99

17I P R O C iE D U R E F O R M A G N I T U D E P L O T O F B O D E P L O T

From the analysis of previous sections the following conclusions e~b~ obtained.I,. The constant gainK, in te gra l a nd d eriv ativ e fa cto rs contclbut~ g a i n (magnitude) a t al l frequen.cies;-2. In .approximete plot the first, quadratic and higher order factors contribute gain (magnitude)

only when the frequency is greater than the comer frequency. ~ _. - . :He~ce the,low f r e q ~ e n c y r e s p o n s e upto the l o w ~ . _ o < ? r r i e r ~ q 4 ~ n ~ Yi s decided by K orKj O i c i , t ~ r term . T hen at-ev ery comer frequency the slope of the m a g n i t U d e plot is alteredby th e first, quadfati .ndhigher order terms. Therefore the magnitudeplot c a n b e s t a r i ~ . ~ t ~ : I < : o~:KI(j(Oro-r )({jrof-tef.rrt~d _en the d b magnitude o f every f i r s , ' a n d higher order terms are added one by one in th e increasingorderf the cornerfrequency, -, - , .

_This is illustrated Inthe follow in g ex am ple .. . .. K (]+ST\2Let, G(s) = 2. .. . .1 ' ; . . - .. s- (1 +ST2) (1 :TsT3)

- . K _ . , ( ~ +J~""T,2,:.00(0) ~ G i(l~T.~(ll" ,. t... ..0) I +J i m , 2; 1,+Jm' 3Let , T 2 : < T . : > , < T1.. . , _ _ . ' . - . ; _ . 1 . 1 1-The comer frequencies are, atl::::::i: T ' ~ : ' ffig:2 = : : ; : . - . " -Wc3 = - T . ,. .. _ - 1 - ~23Let, fl:)cl - c We:! (1+ j ~ T ' + - : ; +j2l; :J

.._Step 2: List thecom~r. Irequeaciea inthe i~easi~i{ota~and prepare a table a s shown below.. . . - - . ' .. . . ~: ..,_ - . . . . - - .. - . . ,-

I .Term ... Corner frequencyrad/~~~ -S~lopt--::.'t:haDge~hisiopedb/il~ . diJ/dec

I .....

, . : r

; .

" ". . .., .....' . . -n .... . -:. ... - . :.:' .. '--In the above table enter K .orK f (jrot or K O r o l . 3 ;S the.firstterm and the o t h e r : tet:tn:s_jp.~eincreasingorder of comer frequencies, ,Then~nty:r.the_,cQmerfrequency, s-lope contributed . .by each term a n d change in s l ope at every c o m e r : - t r e q u e n c y . , - - . . . . ,Step 3.: .Choose an arbitraryfrequency I D i which is lesser thanthe lowest.corner frequency,Calculate

.the dbmagnitude of Kor lK/iGcotor K U r o ) 1 1 : a t ror8!nd~t the lowest corner frequency. .


18/99

" ,3.,18

I ~ .- , : , t < ::Magnnude , plot! o f the t e rm , Owl+db'

.... ~. .-0 ' db L , - - - - - " , , '~ ' - ~ . : ; , . . , ' " " " " ~ - ~ ~--;---:--;;-~ -f-~ -'-~ ~~ _---'---ooo-'" .. .". ., ,--'-':-. '-db

. '.

. ".lj!, .I.';

_,i _;- .

,~ 1M a g n i t u d e , p , " , JotQltheterril (' ", ' , ' , , 'T ' - , : ) ' :. , ' - , 1 " , " Jm ! 3,. - '.. .- . -, ,".

'!' -, , . L~ L - - - - ~ - - ~ , ~ - ' ~ " ~ '~ ~ ~ _ _- -~ - -~ ~ - -~ ~ ~ '~ . '~ ~ ~ ~ ~ - - - - - - - -db . _, '. _ ,

' " J . . ', .-:db / , , ..,: ; '1 ' ,": _Ma9~itudlplotofthe ten~'j " " , : 1 " . ) ,'. ',-, ",- " - (1+ J002+db:

. . . . . , f l . . . . , :- . . . C !lV .. . . . ; -' '0':&0-: ,Odb / 4~~ ". ..~~ .- . . . . -: . - . ., . . . .,"-db- 'II ' ...'

, ," ./ ~. . ""

" ':,1

. . '.. .- - ..~. .a

" -~e3

" ' , ' , :" ,-' , " '-', . " , ,-- K{l+jwT;/' 'Fig s.t 8,:1Wagnit ,Jd~ploi , a/bode plo~0''G O m _ ) - , (j~)4;(1+jw ! 2 ) (1+jr o 73)',


19/99

!. : 1 , - _ - - ~ _ . , _ .Step 4 ~~ ' 'Ihencaleulatetae 'gain (db ' m agnitude) at everycomer frequency one by one by using-..

file formula, ,'G ain at O J : = ch ange. ingain fr-om O l to ill' + Gain. atm : '.. . ' ;I - ' ,. . . ,,t y" . x . "

. . _ - - ; ; ; ; : ' i S lope from "'. to-e ,Kog'" ]:.+G~in._-mili~'1'.,' . (Ox , .

, .'.,' .

xA . : ' . . .... ' :IIJ. ' L ,p_ ,- - 1 ~ ' . , ~ . - .A , J . . ~ " , , , ~ . -- " : ' " ; " y _

, - .~.. . .- .'0

-...: .. .. ~

-,. "" '- ., A r - - " " '. . ; ; - , . - , . , . . .

'.

Sti!P -5:Choose.fl1l a r p c i t r a r Y (re que lllc y. O Jhw b j c I l , is .~:?:tI!(~~~~_.~:~ghest. 9 0 l 1 1 e F f r e q u e n C y ;. Calculate the gain at (Oh , by us ing . the formulain step 4. ',~'.'r .' '~. .step ~ : - In a semjlog, graph sheet - r n ~ kthe required range o{ft~quency onx-axis (log s~ie).~4 ':. the.range of-db-.magUi,mde-p11- -y-axi_s(ordinary scale) aft:er-cilOosing proper unilt$~ ' . _ . .

Siep7,: . Mark a l l t h ~ points obt~~.ed in steps 3,,4~ ,ami 5 on the ~phand join th e p d i r i t s : " b ystra ig h! T in es. Ma rk the slope a t every part of the graph . '. . ' . .Note: The ~~ainit~deplot obtained above i s an approxiwuitep!ot_ lfan' exaczpiotis .'need ed the n" a p p r o p r l a i e . c o r r e c t l o n s " i f i o ; u I J b e m a 1 l e a t e v : e f j ; - : c . o m e r f r e q u - e n c f~ s - : . ' ' . '

F O R - PHASE-I~lnt O :F O O O E P L O T: T he p hase 'p J~ ot:s ' a n eX3dtplui-and no approximations' are made w . h : i l i : e - 'drawing the phase plot.the e xac t p hase -a ng le s o fG fjo o) areeomputed f q J f . :Vjjd~~-Y;~lu~ofui;'and tabu1ated~:-The choice ofpreferably the-frequencies cb~sel'ffor r;nagnf~d~:p_IQL Us.u~l]yhe m agnim .de 'p1ot .and

se plot are drawnina single seg]ilt?g ,_;'Sh~et on acommon frequency scale ..'. .' ..' '. ... '-. . .Tak e' an oth er y -ax is in: diegmpb: where th e magnitude plot i s. d rawn and in th is y-axismark the

esired ran ge o~.phase angles.' after .choosing proper units...From the tabulated valuesof m and. phase.ngles, mark all the points on the graph; Jointhe points by a smooth.curve,J : E T E R :M I N A I IO N ' O F G A I N . MARG IN A :N I ) P H A S E ,M A R G I N m O M B O D E P lJO T ... - The gain margin ill db :is given by the , n eg ativ e _of db rn;agni1;udeof GUO)) at the phase cross-over.

pc 'The ropei s th e fWquen~y ~fwhich phase of G(j0))is-180o If the dbmagnsude of G ( j! ; ! ) _ .(Ope is negative then gain margin is positive and vice versa. . . . ':, ... . -. . Let.~ be thephase angle of G(jro) at gait}!cress over' fteq)uency @~. The _m g < : . i s - t ~ e f r e q u e n c y at .

the db magnitude of.G(jro)_ is zero. N ow the phase margin, ' " ' I - i : s given by, y =1 800 +" 4 l ' g < ; ; . : If t P g c is less' .egative than -[800 then phase margin i s positive and viceversa, - . ' . '. - -.'.


20/99

.. The positive and.negativegaln.marginsandphase margins areillustrated in fig 3J9~

~Oog scale),,':'Fig.3~19:BoJeptQt~howi.~g phase margin (Pl,d),arJ[i~ai~margin (GM).. :~~."~.:~;


21/99

. : : uEXAMPLE '3..1 .

. .:...... "..: : , . ...:.~

. S k ero h B od e p lo t furth ef6 m o win g tla n sfe ltfun ctio l1 a n d determilleth e systBmgain K fo rm e 'g a!inc ro s s o ve r 1 ife qu e~ c y:o b e 5 ra d /s e c . ", . '. .: ......: . . ".'. ." " .. '; ,. 'K ' ~2. .."sG{s) = . . . . . . , " . _ . ' ."(1+ 0.25 ) (1+~102s)'.

SOLUTIONThe s i n uS iJ id a l t ra n s f e r f imot ion Gam) is P!bta inedby r e p l a c i ng s by j ro i n ilIhegiVen s-do~ain1 ran~rfunction.

: . 2. . K { Jm }. . : . G U m ) . " " Sl+0.2iro)!t~t)~02jm)

. ( ' ) 2. , ' . ' . '. , J m .LetK=t,'. :.G(JOO}= ' . ' -. '. . ,:''. '(1+ jO.2w)(1+jO.02ro)-_ '. . '!" .

. .. ,", '. 1 . . . l' ... ., . The ~merfrequencies are, m c1 = = 0', 1 . ,2 ' . ; ;; 5 red looc and ID "22 ~ .~--; - = 50md , sec:0.02.

T h eva r io us te m i ls o f G fj " a re lis te d in T a b le -1 t n t t i le increasing o r d e r ~ ft h e ir c o ~ e r freQU9RCy. A l s o the table s h o w s .th e s i o p e con t r tbu t~ by each te ,nnand : the cbange in s lo p e a t tt1e ,c c in ie r fre que n c ,y~ .~.' ~ ..' .' :" .. :ABLE-:i

. . : 1 ' Te rn l l i'

", .

. : .. ' .. - .. CQ~~~rfieq~ency:T

.' C h.a rngie . in .slope. . . . . : _ . : . " . ,= . . . ". ..db/de'c

Slo'pedbfd"c:

! : . _+40 .. . .~ . . . ; < ~ - ; . , . " ' . . . . . ' . .-;20 ' . : 40 - '20 = 20' .. . . . . . . . . . r f . 7 / f ' . . . . . . .

......,.- ,..~".!~.-..;:... ,..".

. : ' ' Om '1.

1 + jO .2m1

. 1 + jO .Q 2ro ,

_.", .

20 ' ., . . 20 .,; .20 = 0 -' ~ ~ ' : ' ' ' . ' '. . . . . , . .; ~;;~. . . , . , .I.C h o o s e . a , l ow f ; r eguency (OJ 'S lich thatto"


22/99

leti1.hepo in ts e, b , c and d be "e pa,intS oomeS .pon d ing i to f r equendes . ro l, mel ,00 c2and (tIiti respect ive ly on th e magn i tudeo t . In S I s em :i~ o ggr ap h s li1 ee tc h oo s e a s e ale o f' 1 1 1 n i t =10db ony",.axis. The f r equenc ies a re m a rk ed ln de ca de s fro m 0.1 tot 00d /s e c o n , loga~,th l l1. i~sca lesin:xc;a~i ; . ; : "OEi*: the:P-Oi~.~,, ; .D,.andden tlle'.grqph;~ J ,Q i l l . 1 l ; t h e - PQm . j S b y st(a Ig l ; l~: l ine l : i~Od .ma t k t h eo pe o n tile : r~p~cl1v:e r e g . i a n ; - . . - . . - ; - . . - - - " . . . - .HASE. PLOT .

The phase :ang ' le ~o t~G( j r o ) a s e r f ! L 1 n c t ! o n t i f "m is 'g uv en b y, .. -,-" . - . . - ..

. ' ~ LG(1m ) ~ 1 8 1 0 _ -='tafl~ 1 9,2m ., tan-1o . o e r o.. ~. . . .. .. - - .' - . . . .. ",~:.' . : , : : .Th e p ha se a r i g l e ' o f G ( j m ) ' a r e ' c a : J c u l ? t e c t fOI.vadous V a l u e s o f w.and I:Isted in~!e-:2: . .~. . . . . - - .

"

- : ' 0 :: . . ._"~';..".--'~.. I:'~ - :- :. -~ .. : i' " .. -' .~ . - "~.-r' -, . .: -: --'~

__ L':' . . p < ~ ., " ; - ' .

,;,..

-~ ~'-

-[tati~I-:0:Zij)-..-: : _ : : _ : t , a n ~ c . l . 0 , . , 0 2 . ( 1 ) . ' "I - . . -. ..:: :_"'-::d~,g'-.:.':.'_;.-.~- > ~ _ _> - < : . ' ~'~g;_ . . , J.,',~9j.~'.' .:'Poi:ntin .. '.- phase p l d~ : .:..... - '.-:. .--:

'. ,. - -. - .. 1 1 - . 3 -45' 5..7

. ' ' . - 1 1 " 3 " ' ; ":::11. ... .. - r . ..:.- " -.\: I'. --6 ":3 - :.'. ----:....51)". . .U'"t.

1 : 0 0 I' " 1 i - . __.- , :'.}" . ._' .' - " 8 7 ; , 1 : :. . '.-. '6 3 4.. } .. -., .- .. _ ..

. :: ..

l67.6~168t29.3~130-_105;3~'1U6 '; ~,-. - , .

. .. :.-

. ". -.' e, .' -.:-.~ .-. ' .,- f

9 .: h.'

-i _0.7_ ~ '5 0

. . ..:-- ~~'.-' O n 'th e s a m e s e m i r O g : g r a p h ~ e . e t ~ e ~ e - . a : ~ l e Qf- l l .m i t ; 2d" ,~onthe y, ,~~n l h e r l g~ t s !deQ f : s em i i oggm p h s he et.

g raph s~ j o i i r i the. poh l lts b : / a - s i r J o o t h o u F V 1 e . : . . . . .K _ . -.:-- ...

. Given th~~:tOe:9ajil ( ~ ~ t ~ ~ i l ~ 1 _ $ ; 5 ~ 9 t ~ ~ - : ~ t ~ o ; , ~ . 5 ~dJseeail~Q ai!l.1~ ~8,db,I f - l J ~ J n~ffss~verft~I~~ncyjS ..radtsecthe:n~,f~at'fueq4rency,ithe'BltgaittsneE tret~:Zf:jfo~ i ~ n c e ilO,'eveiy:po'iin~);tf~ag~iludep . I Q t a l dp~ga, i r i { jJ- : " ' "28t!p-shouldd e d . .The,t3dditL0:nl;Of.~2~db~_~~t~.~]~~;e.comr~,~-:r]*~f1itad~~~:iSt~bta.iijed1Qyshii~g1;lJe!:p!otWith.='1~by _ ; 2 8 d . b (1OWJ:iw~rds:.T~etij-agfji~.ide-;tarr~ct{Oitiisf , n d ~ p e l i 1 d l i l i l t 9t : f1~; t fuenq< - J H l e r ' c , ~ t l t J ~ J t1ag i1 it ude . ij (- .; 28db i sterril KThevahi'e ofiKisfCai~p~'i~ts;I~byqll.l;:J t ir i ig~2Q :lp -g,K19 ;~28db . .. '.. '- . - .... . ,-. .:'. : . ..- -.~ ' .-:. < . ~ . . . . , . . _ . . . . . . ~ - ,r, 2 0 log K ~ . . . , . 2 8d b ' . . - . - - . . , ~ . - ' . ~ . . ' ,

IO~K=:~~~8: - 'K '~ ' 1 0 ~ ( :U~ - O .O ~ g 8 - , - ' " .. 20 -, ';"- -, '..''. .

.,. _ N o t e ; ' Thebjfq'liencyeiJ =5fJiJI$~ i i l 0 f P f # e - rf re q u e r ; c y i .; i i en l i e i i T _ i ! i e _ ; e X a c tj J l tJ U / 1 e d / J z g a i ' ! ' a~:~ ~51J1d1sec:

will be 3db l~tI1an;tn~~ctUalPk 1~:~:7J;e~~~iiii:6pC lPlqtlhii2tJ loifIfwil!'konhtbllle'8cjj:;in:t)t7~5 d~''- ..: 20lQQK=.-25 ,~.~,..... .. .

lo g' K ;::.:2~.;;~~t~I~(:)""..0562 .. ' 2 . 0 - ' . - ' . . ' ' .


23/99

- .:

._:

- Q. . . . . ON:'1 f II I I II I I I . . .. . .. .I~


24/99

EXAMPLE 3.2: .- ~..' ..' ~ .'.'. S k e tc h the bode- l l lo t fOr the fQl l pw i ' l l gJran5 te r l f l \ JnOt ion anadet i :mTI inep~$e.. . r iJ i : la l i j in a:ndgaj~.margtn.... " ~ $ ;~ , ..1 5 ~ !+ _0.2~ )-- - -. ~ : , ,-. ' < . ; : . ~ . ~ - - - ' . - . . . ; - < - : , : ' - : - . '

S'(S2 + 1 6 5 + t 'O O ) ;. ' , _ -r.. - _ ' . . ; , .

: 'd ''-.,an '(1) .' '', !1. "S2 '1 a s ' l iO "o '52 2r; . '2_--."...++ .Y-. =: . + : ~ r o w > ~ +. '~n

' OnGompa r i n g we ge t, ."," ~-.. -: '-:',,: ... . ,

",

. , , : , _ k e t IJ~H~O'ny.eltt u ; l 1 i Q i v e n ~ ~ o m ~ E ntra~~(.'~Q~~ni:ntP,b:QQe'_;fo~,q;r.tim~,C9ti~tanUQ.lm"..:.., -.. . . '. :-" ' _ -', -'~~ -'.~".'.,.-- .--." '.: .~- .r.--:.~.:: - ..... : _ ._..:..~-_ : ~:.' :.-:~'; ... ,,-~-"':T~:"'-i"~~';-'-:'~ : ' ' . , , ' ' : - . . . ~ 1 5 i 1 . f . ' ( j l 2 1 3 : } - :.c-';; ... / ' 7 S ' ( l 4 ' l pt s : j ' . '. - - - . - : " ' ' _ ' i i k i ' . . -(+"'a:!~'~..., .- '. .-"-'- ' ,,-'-,':-c . '. - ..

":_-'" .: .G ( S ~ ~ : S . f S ~ - . ; : t 6 i . ~ ' t h o r - : - - - - - - : r s ~ < -1&'~~= ~ ~ ( i f . a " i j f s 2 ~ ~ b ~ f 6 s j ' < - < : - : :' - : . . . . .' ; . ( : . : , , : : ' ; ~ ~ - : ~ - ' ~ .,- - ,- - -- - - . - - - -: .. ' - ' : _!:D dOO t 1 - O P +~l'OO -+ 1 ,- . '. _ - .: -' :- '.~ '_' ' : , ~ " , , ~ _ , - ; - > - < - " :

.,-" .Thesinuidaltrart$fe:rfun'tiot.tGijttl}i$,OtJt;iinedj)y:~jpla~ng-s:by.j(,l)in::G(5)_' '. '.:~.~:.:i-~\>, ,,'-:.. '. . . ' .._:,:.. .. :. . . . . . . . . ,

-.. G{j~~)'~' , -';O?Sf~+,,~.~j~!. Q . 1 5 J 1 t ! : :p " ~ ~ L _ > ;' - - . - - f . . / " : " - _ .' , :"',"_.> :.-::-: " ,.;~. . " . : _ j ~ ( 1 + tJ.~01(j(!j~~:RlI6j~) . j r o , (1.~-OA;1ro~t+jO~1&). _.. '_ . ~ '.~"';- - '!""-,-:,~:, . ~_. I -_ "' :

:A G N , '" i j U D E P 'L O '. T '"". . If I: '.' . "., .". ..... .. .- - , 1 i > ' - , . " ..T h e c om e r frequencies.are, ffii:l =~~:= . ,5 , : i ad l sec' 'and '@a:~ Q )~ if~ 1 . ! ~ _ i a iJ I s e c .- : , '- . ' .'. . , - 0.2: .....

The various-te:iTnso f G O r e ; ) afe :fisileciintabl~ it ilnthei~ndr.easing-order(jfilfU~ij(oomerfr-equei1cies_Alsoh~t;~[~~~~contributed b y eaeh t esm and1tu:~, .-ch~n.g8J in~- Iepea tti:ie 'cem erfr :'E ~que tiic i . - ..: .. ' ,-~ ' - -: -.' .. . --_ ,. .

....',;: ' . . - - - :::' . ":.:"

- ~ . ..' - _.

. , ' . : -

$ i : o p e . . . ' : - . ' _ , :Ghar imr .~(I$! I !Q,pe : ! ,_ , : ~ . _ . : < - .-'db/dec': .~.. dl,tiMec - 1 - ' , "0 0 . 1 5 .

j Q )

. , --COm : e r tr e( )u ~J lic y _ '.. ." - '\ -~.' .: _ .:--",ra dlsec.. . ... _-' -~~, ' -

; _ . _ . 1 - , -- i .,j~':I 'II

", - ~':.:1 + jO . 2 ( .1 )0- . ' - 1 ,I," ... _ : r o 'c2 = : ' r o i l " , , 1 0 - -

GhOG.$ea,low tr;~~u~~Cyml.su~,~.tjhal~:r~i:ia~d.cpoo.s_e~l,hig~~_gl[J~nr.-,C)tl~)~tl ,tha~:,~~>l)~~ .: .. '''-:'Let , ( J ) ; = 0 : . 5 r ed / s ec arid f f i l l : ; 2 0 n : t d / s e : 6 . . : . ' - . - : . - . " , ' . _ - : . : , '"1 > ... -> : . ,"~. let,; A ;;;: i G U m ) l . i n db .. .; :-. " .- - : . . .'

,/


25/99

. .' ~ . [ , I ..e . .... roh]; ..... .' . :A= Islope from ' m c 2 to -oJh x log-' .+A(i5i ._.):: .. ' . . . '..~ (j) ...' . . .' . . 1-IJ.Oh2 :".:' AI .,'. .. '. .' .' '.Th e phase a o g h : ; 'o f G um ) a re calculatea fo r v a rlo u s ~ ik le s o f , i N : BtJd l iS te9 .in t ~bl e. .2 . ' ..'.' -" _..' . . . .. . .

.." '_'

_ ," ] . _,',

4 .5'92.46.89 0

. ., 46.8+l80=133,..2

" -;~

. 5;]11..345 . '.

0.5' .151020

. 1

',- ;

I. i :

i .. I i ' . . 75:~, I . . -18.4+180=16t.6 .. - 9 r2 + 18 0= 1 7Q ..8 '.

.. ,"

84.:3'100

' , , . . . _ ' , ; : - : .

. +=LGoD)deg -

Points inphase. plot

e.fg.hi

" j. k

,",-- _;,

- 8g,~9~ ~88 .:- 87 .9'.~.,.88- 9 1.,8 ~ -9 :Z-116.6 ~-116: "'147 ..3 ~~148.,..167.3~ -168-r73.7~':"~174

I ',. ';

".)'. ::-~

,_.~

. .. -~ .' . . . ~ . . . '.' . -: . '. .' - . . . . ..~. " On 1!b esam e s e m U o g gra ph shee t choose a sca le o f 1un~=2 t J90 i l , t h~y -ax i s "O~ 'tfle rightsid O f sem i log ' graph sheet.~Ma r _ J


26/99

3.26

"e\a.+-, .. . . . : ,% . c5.~ . --.",~ +"~,j

" '"" l1 + e~ . . . . .Gl'''1 . c : :r-,Q I .

'-.:..~~. . . , .Ij- - - , _: s :~\j;C_, . , .~~; : : :~~~s : :~-~.'Q .. . . . ."Q .. . . . . .. . . .. . . . . . .~Q~" ' . ,N " " - 1 '~~:.~~.

< : : : > on "" 0 . . . , 0 ." V 0 V> 1,00 VlI - . . . . . \C J N N "M ('


27/99

.Le t l jlQcbe t t te phaseoo fG( j ) at gain cross-.overfreque!i i lOY:.Ol '~,F ro m th e fig ~l2. ,1,w e ge~".~= .8 80

. .. .,.;Pll'a!S'e!~Jlgitl\l,g=t8tt +.p:~:.180.~88= = 9 2 < > .- .. "

Th e p h~ Se p l : 6 t J ? f O S S e s ' - '180:~-o~iy: ta tn f i i 1 J l t y . - Th e IG(jP D~ I.atnf inly ' is~ 'o o db:.. H e ~ l C e , ; g : a i i 1 1marg in : ; s+~oo .

. .' .EXAMPLEJ.3 . - -. .

. _'. -- .. ' -,. .Ke~l_~ . '.~ '.' _.' ' . . ~ ~:GlVen,G{s) =: '. . .... . . . IF i nd Ksolhatthi~system:Is..stabr,e w irth . _.' - '~(sd-:2) (.s+8) . . -'. , .- ~ . . ,- .".... rd :~ c.

. C~oQ'sea lo w frequency roll suc h tlilat (()~ Ci)~.Let; (0~= = 0 . 5 rad l sec a nd (i)'h ~ ' 5 ' b i ra~;I:sse, .. ~'. "'let," f : \ =!G( j~)11n' db.. Let u s calculateAat , m 1 , ro 'C I ; r u e2 and illh ,

~.:

:- "


28/99

. . . - 1 r u J 6 2 5 t . . 0 , 0 6 2 : 5 . :At,~ = @J'. A= 20Jog~ '. ... = '2 0 !o Q . . i = -l8~b-' . . . " '1 L J " " . " 05' ," "ll~ .!"_'. .... _ .. ..' '...~ {j]06iZ51'. .. ' 0 . 0 6 2 5 . , . . . .A t (iJ ::::;:o 1 A ::;:: ZO;_ [o . " ." ; " " . ': . 1 : :: .2(1'tQ g. =-:3.Qdb .' ..' . c '-. ' . ."' .. jm~' ... ' : 2 . .. . - - . ' " , : _

'..28

. ~ . AtW;=OOc2', A = [ . .~ i[ O p e f [ ~ tr H u 1 : ~ f f i , ( Q - x l a g :@ C 2 ] + A / a t . : . ' ) ': ; ' : --40Xlag,8+{-30.}:~._:54db. . . ..', '. . . -. (j)-c1 , \,G.C'"{I) c1 . . . 2 : . ' . . . . . . . ' .At'" = ~ . . A= f ~ e ~ ~ " I o < n . X i Q ~ : : }Ai1lt';"'..J~-6(hto9 5: ~(~4) ~-10~b .

' L euh~ p o in ts a ,b ,c .d betheiPOjn~:oorres~,lifig tofrequendesrnj 'cf' 1 m - a a n d 91'hes pec t i ve l yo l ii lt b ema g f1J lk 1de~p l o t . .s e !T IJ Jo gijf~gfu:shee:tcrboQsea-sqajt:.(I\fWnj;t=':lQ'mb..o1Tl y" , :ax i s .T .be f fe \q tJa iTo ie s ;anHEa~ked indecades . - - f r om Q o : 1 to .to o ~ .o r:il~ og~ lith m ic .S ca Je -f.itK ~ F ~e :p tii l1 ts ,a ;: ~';'C; cm r: Jd ol1lthe-,gt:aJ)h;J~ojr.l' l lepl)ints bystJraigbt l i~ a l\dm 8d c .th e .9pao fltJ 'iEHrespec ti"e :nagiom . .... ,:' .:., '. " . " .'," ., ' ". ' ..HASEPlor'' .:, ..T fue phase .ang le ofG(irola,aafu_'_llciieJ~}ei ( J ) is g 1 v e n b y ;

. ... . ...' :. _ . . .. ' 18()"' ". '. . . " ' . ; ; - Q 2 fi>. x _.,_:_'"_.9 ~ ' ; _ ' ' ' ; . ' ' ' ' . n . . ; l , o : c :- ,~ta,'n""10 1 ' ' ' ' > ; 1 ; : . , , v , ..... , , 1 J I = . . II .fI!.Ct .... ~ ..... ..::;~.'1t , . .... ..

.,.

1 1 1 e p h aSe a n g le : ofG i j( O } are 'aiqJf la le r l : . fo t va ) 10 lJ s 'v a lt le 5 , a fr o a f :d l is t e d . im t ab le - 2~ ,. " . '- ..- - - . . --. . -

"', '. . : .. ; A , .2 .....l1,o.n0 ~_,-'). .... .ta_-,1:.I'\\-I:::' .,,;;w U U \ ~u: r-Jt ":.'. ~ I-I ..,~.y.:


29/99

I IIII

0 ~ 0 ON 000 0-III,('::, .IJ) \Q r-. l'- t'-oOO .0-J I i I ! i " .!I . . . . . .I

o00.0 """".......N! I. I


30/99

,',20109K=30 . K =: 1 o-:mo.Th e mag.fl'iit~depbtwitttK7_:t$.i~::ail]!:l31~ a re s l hGWf r in f ig ; } , .$ ; t, . t ' =. ,- .. . - -. . _ . - - . "- . - . . -'-~ ~~" .-.-~"- .. . .. . - ':: . -. . . '_ '." , _- . _" ~ .. ._ " ~~- ",, _ .. . - .

xAMPLE 3.,4 - :, ._,._~:':..i;-... . ::,- , .. " '.' " . . :". ."~. - "' '.~

.The~~n~soidattranSfe~furittion~OfG~)~Jlbta i ineaby~pladnit~'byjd!tJin the g ive 'n tm~n~ r . 'EU nc t io rn , :; .., ,' : , ' ? < i " > l : j~!1'i~t~1~~~~}.:_ .---- t ~ , - _ ; . : - . . . : ' - . : : -' . - - - - . . ' , . " : . . . . :" . - ~ - _ ' - . . . . ' , : = .. . . ' ' ; , ' : _ " ' , - , ~ . . .

i': ...-:" ." ,

_ . '_ " -.. _ . . - - . : . _ : : : - ::.:'.~. . . , : .. : . : ;: .= :- ',_ - '. _. ' .~.' -: - .. '...",", -. -T h e C om e rfi 'e Q u~ n d ~ s a ~ ,- . - , ".. "1'.' .,,.'. ~ ., ;, .-, l' C '.

(i)c1""~' '~'..::= .2.Sract/sec ' a n d (i)~:=--'_',= 10raft! s e e ' ,'0 .4 , . .: ,., 0_1 .. ~ , . . .: " \,~- The V~1iOUS'~rinsofG(jml' a!ie;iist~intabJ~tirii ite inoreaslrig or~efo~ttelrcOm~rfrequencfes~A.fso:tlie til'bleShovts : .he.SIOiJe~Xintrib~ted.byeach:re~and-the~nge.jfrtsIQpeattheCome:rfireque.nCY,;' , . , .", .' .A s L E - 1 " '. ._ . , . , . " . : . ' . , ~ :'> .:'.,""" , .. ' .:' .. ". ~ :',', ." . " . , ...; .... .r:.

' _. ~ .. . ' . .. ' . . . . . . _ .. __ c . ~ _ .: "., - .- ..::. - . __ ~ " ,- '.: ',--: :.> .. . . .~ ~ { . ' - . ~ : ; . , : r - : ~ ~ ~ < r 1 ' : , ; " ~ t { -: , ' " < _ . " ' , < : . " ; ~ / - : : , - " : , . ~ '. - . ' , .

" . . I.:'.., '."'" 2 0" ~.,j,,' ,,. ..:. ., ..::",: =..:. A =20'109~~ ~~tO ~:4Gdb- .' :.. < - ' . . . . . . . . . .. . : - , > C C . , ' : i '

., . , ' : , ' . ' , . . - , . . , : : . : r - , . , ' . _ : . . . . - . , . . ~ . ~ ~ . : . ; ' , ' , : ' . ' . ' . , ' . , . : : , \ ' " '~ '_ " ' > ' : " " . " , - ' : ' . . - ' , ~ - ' : " '~ . 'A1 00l;ID c1 iA .=?,cJtog+ J:= ,201 0S 25 '",,1 2db - ,~ ::' , , '. , , _ , : ' ~ i : . : : ', , " I I J w , 1 , ' " . . ; . . ' , . : " . ~ ' . : '.At!)). =~c2' A.=:[ . .~ : I~~e r rO~ ~~1~ C ) :~ ' X l ~9f J C 2 : ] ' . +A ("a t ., .. )'~:40X:lil921:05:~~:2.:f..1~db. ' ' ..', '. . ...... ,'.' .... ... " . ro,e1" .' .(O .. G(l~' .. , : . ,.', . ,.':' ......' .A~@ :: W I. . ."A' =:;[$JO pefrO~ '~a'tO"roh x log ( D , ~ j+Ai9t ' . =1," ') '.;~~60xiog5?~,~.{'12J::=:'~54~db.'.. i l . . . . . . . ' ~ ( ; 2 ] . , , , , \\Ic2.' .: ' : 10 . . .' . "

'.


31/99

_ . _~. :

L,I

I""0' 1 '..ro{"),I'c'I',O '"..0~.. .. ..-4 ,~~I !.~


32/99

. '...'. . LeU h e p o in ts a " P. C ~ o o .d b e i hepp j lF l t$ tQaspQf lQ tn .g~ f r~uen .Q ie .$ -q l l ' 9 ? ~ 1 " ( l ) c2and W " b r~ ly o n ltI~1fl9~~ .d e, p l o t t fl .a .sem Iloggraph ~~eet - c b p ( ) $ . e p i$cale ' ;Qf 1lJr l i i t; : 1! 'pdb ony..,axi-s. T he ~ u~ n~ PiremJirted ~n,(ilecapes1mm 0 .1 to100 (ad I.s e~Qo 100~r: i thrni~~~r~j4X""~ f~tbep~$is 'aJ,b; c .an iJd ont tw@ :i.pJi . JOIDJt l$.Ppints QY'~ st r~jgbt ! jn( i ji a;n tJ.m.an,..... , . .... '.' . . . .. '~.'., ". '.' ..' . . ~: 'JP HA S.E IP LP T .....

. T r i~~pha~ j l~ng~ ~GGO}a$ ,a f~ci i .On9 f.i 'PiSQiv~n by.. ~ ; : : : . , . . 9 0 : 0 " ' "tall"';l ~.4o. ; 1 B ~ - 1 .O > ~ ;@ .

U te p h a s e ' anglof"Gi j ( l )aJ~' I~~~elf ( ) ryat iou$ v a t \ : l a i o . f iJ) @ l ld 1Ii.oin1ijl;!if;.2. .' . .

'm . !: :tan-1 ~O~4m.~. . ."::~:..:: . ,'. -;,.Il;"':A ...~p .

. . .~." -- , ~-1 (Lt .',0)" c J ~ t J . . . .__ - ., .- T.

. . . -. _ - - - - _... .. _ . _ . _ .. . _ ._ . . . . . - . . . . . . -... ~ .~~.Qll .'. ". :~l1~l;njj'~j .. . I . ~~~,$e: ~~(j~ ..

0..1'1

. .2 .54''10''20

. ~~~9.2'1.00.45.;0 $ 7 . : 9 9" 7 5 . ; 9 6- a 2 , 8 7 . . . . . - ._ .. - . , . . . - .

e..57. '1I '.' 5 ; . ] ' 1

. 1 4 . ( 1 '~.21.S45 .0 .~A3

,~92.$6 :I: e-'9~.. = 1 1 1 ; $ * , . , . . 1 1 3 "'":"l4~;'.~,;,..50. ~169.7$Ri~17tl., ....Hl.96;:.." i210. . , . .23~t3.~236

e.f .jl.

tr. . . j

.:', Q n$& ;S am esem ilo g g,r;apb~ttBet~ .. a~~ le . o .f1u fi lf to : ;: :O~-On1t t~.~o~; tbef l , !Jh t s i a e - P t ~e:mIlQgg.raph~A ~t. . ;M.a rk1 t te , f iIC IJ la t f*1nase;aIlgle;o~1n~'Ii1!;Pa:~b~.J9in~.p()imS.bYaS f':lo o ~ c~N e ... . ..... ' ." . ,.. '. .. - ' . ' : ", '.' - . -' . " .

. ~ t h e magn i iUde~n9pha~~p t ( ) $a l 1 ( t $hqw l l ' i i hf i g : ;3 . 4 , . 1 . ..1=i-Bmthegraph. the:gaiH af l :d lPm ase , , ' 0 . 3 3 3 ra d I_ se c.

. . . '.: . '. ... .. .i-~ ,. . . .;3 . . . . '.. .T he va no l,J ste i'fT l$ ;o fG (j.m ) .a re lj~e(f:iffta"l~l in 1be ' i l1crea~ngl


33/99

.... ~,::.~:_.;,'=;'i,:~l~~i>.! ~~~O~)I " - .% - , - , . .~ ~ ~ _ . . . . . .~_,:,-lj-.:y

~-. . ' -, ". .. ",.... .... .vr-." - ..-c, . _" -" ,-, -. "---:"

TAHLE~l' ...".'I ., .', ~ :T er i'n , . C o~ li'ie r :tr e qj~ ;e n~ cY '

rad/sec S l : o p e.. db/dec 'Cl idng~' inslope'dbidlec,20 .

j m .1 : . .

,1+j~,1

,1, . 02ro , ",,-=..5,cl '4 ...

" i

.~hoose:a \ f iequeI i lCYO)~6Udh I f at .a , ' I I < { O c : lan~ ~hoosea h i i g h : fr~quenCyIDh ~ ~Ch i I : f i J a t ' c Q h > m~,Let . : C J 1 = .O. 15 r a d /s .e c i ~ n d :(l}h== 1Itadfsec~

, 'Let, A - = I G O m ) : ! i l r i l db .. : .. '_ .' ' "

Atc o ~ m .~ , A'" [S!~P~ i ~ om~ d ta (j){2 ~ i o 9 r u ( Q ] t'A(1lt~ ..C ) el) :.. ..,' Well .'" .

... 0.33,,.. .... ,.= 40* l:Og_.._._. +3a ~',33'db .. 0.2:5At r o = ill,Ii': A= [Slope 'from {nato {Oh x log~I'I.., ]". :- A{at'iil "''''a) .

. {I},rQ ~, . ,. ,.. . ' . 1 . . .= -60 x:log ,-._.- +33 =4 dbO.~ J .. ' , L e t t n ,~:p O i~ ts a ,.b , c a n d ( U ~e tn e points corre~pondi~glo fr e qu e n c ie s o ) ~ , OOt 1 'OO~ an d ro n r especWe: ly~ t i i . h~m~~~ im i ~l :o t .. 1~ : :a s em i log lJm ph s h e e t dho ; q5~a s ~~~1a f 1.uni. t.=10 :d b o i i ly-ax js . I lw~q lJe :n ,c i ' l i iSa re ma~t ; ! t ; l J . t1e c ad e s fro m 0 . 0 1 1 to0 r a d / s e c . o F 1 l l o g a l T i t h m : i c scaies ' o n } i : ., a o c i s , . F f > r t l le l l ( j, in ; 1 s a , b , c and d o n t h e gra~hshe~tjoi'be po in t s b y ' : a s t r a i g h t fine andik th e s lo pe in th e r espec l ive re g io n . . . ' . ... ., .ASE pLOT

J .:

T h e p h a s e a n g le I : > f G U 0 ') ' ,~ :::~~O,( l . . . .an-1 3m - tan"l 4 0 0T h e phase . a , n i g ' l e o f : O O e o } ~ r e ;cah:~JlatedC i r , v a r i o . u s v~lu'es6f r o : : fi ~ d : l is t e d i m : ta b J . e - 2 . .

.,.. . ~..

:" ~ .

e

P oin ts h ill. .p h . fl s e .:p t q t~~--~--+---~----~~'~r-------------~'_~~----~~---+--~'~'--~'~~--~0.15

0.20.250..330.61

. 24k 22

. 3O ~96.36.8644 .760 .1471.56,

30 ,9638.66~ ,O ,. ',52.867.3875.96,

~145.18~-146~t59.61~-t60 .

- :--_171- .85 ; : :717~. .. ' -1 ' 8 7. 5 ;:. : -1 '8 8-218,32~-218-237..56~-238.

. rg

I

j


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3.34.

0\QC}f"'-


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.K _ _.__ ' ." -.;-- ......CA .... ', .( . __ Z : : : : ; ; : O Z w . ; ( : s c i . . . . "'. !_~~:t:'-.:._ . _ c : - . . . . . . , _ . _ . - _ _ . ' .. _ _ . . ~ . . ._ . .. _ . . ~ . . : - ' ; . ~ - . . . 1 !1 !. - . - . , - '1. .O nrth e s am e ~ em ilo g' ,graiph s h e , e t c ho os e ~ s ca le o f 1 u n i t = :20" o l l1~ the :y~ o n th e nght s id e o fs em ilo ggi'a p !h ~ he _ et . .- : '

a rk t h e c a lQ tH~ te d p h a se'e J ]g l.e o n ~hegraph sheet ,Jo, i~, - t i1eo i n t s b y : a :$ !1 1 J ,m t h c U ' l ' v e : : r h ~ m~gnitude qnd : p i h a s e pJot$~ r e ' .i i9-~


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ts

M !t .~.~ ..~.~.~ ... '.

. ~.'


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..". " -.~

ASE PLOT', ,. 'The p h a s e ~ngleowGgro- ) , t = tan:' l2{!)} - : t an-1(4ru) _ $11-i(I~2.5ro). - - ._ " " . - -

Ihe phase angie o f GU ( ! ) ' ) ,8 .l 'e GS ! !ou la t ed 'fo r v a rIOUS 'va iues o f ID anc!"J is" ted :~~t J h e t t t b i E ; l - 2 ' "l'ABLE-2 " . . ; . - .

. (I)

. - .

0.25 i0.5.24i O

' . I ' .'50 .j 8SA2

ta f1 -:-~ 2m ...' ... .. deg

1 1 i , . 326,56;'4 5 , 0 . ~. 75~9-6 ;~ .. .

t : a n : : -1 4~(!);' .. qeg. .'" 21 .S '

45,.0' 63 ,43

',1 .~ " .; 8~ i:8 7'. 86.42

88:.56'89".71

- - ' 1 1 . . ' ,I "tan-1 02$cl. : i t I : f 'LG(jlCil) " J .' p.Gi r i t S . . i : . i 11. de~r:'. . " i . . ' , ' ' , ' l"hase plot.'82.818 7 . 1 1 3 . . '89;42

,_ .- " ._:11.9:l~:~2 . . 'f. . ~21 .:94~ ..;.22 " 9 .. . ._ :_25.53~;:_26 ....: h

1 .. 4 3 ,3 . : 57..1

2 . 6 . 5 6 . '. ,45. .06 8 . 1 '9 . ' : . - .

. ~33:47 #:_3:3..' 4 " 8 " 5 '5 ~ - - - - 4 '9 ' . 'j__ '._ '.. Tr-r" .... _", L'k '

. I.. ',-_'';::6~~62~-70

. _ :S5 .7 f ;" '-86On t h , ~ '~amese.~lldgg~ap~~treet:ch6osea scaleb i ~an i i : ::10o:0nY~iSb~iijlerig:h\tside aUlle s e r n i l o g . :g l i rph $ h e e t ~Mart : the c a lc u ra te d p h a s e angile o n th e m8iph s b i e e t , Joill t hepo i i n t s b y : a i smoO~~GUive.T h e .ma g f1 l~ h id ea n d p h a s e p lo ts are

s ho w n ln fig 3,.6~l," . . ..3.7 .POLAR PLOT : ,

. . The.polarp.Fbtofa 's innsoidalt ramsfe1i fu~ction GUj(o)isa:pl.ot of the m-agrHtude'ofG(JOJ)vetsus the'phase 'a ng le .o f G(ji ll) oripom~:coor(linate~ as '~I . is -va ried f rom zero 1 Q : iU f ini ty . Thus: the polar p Ib t isth elocus of ve cto rs IG (j"m ) 1LG (jm )" a s ..00'is varied from zero t o in'finitJ. :ThtqJOlair plot is ,a lso called Nyquist ,.

+90n'-.2700

, ..'. ..The polar plot is usually plotted on a polar graph sh ee t. .The 'polar graphsheet has concentric circles and radial-lines. The 'circles "ra


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F or m inimum phase transfer fun ctio n with only p oles" typ e-mmrb er o f th e.system determ in es .th equadrant at-which the polar plot starts and the order of the- system determinesthe quadrant atwhich the

plot ends. The minimum phase systems are _systems with all poles and zeros ..on left halfof s~ .lane.The strut and end o f polar plot of all poleminimum phase system are shown ~n fig 3.21'& 3.22 .espectively. Sometypical sketches of polar plot are' shown -in table-S, 1. - .

The~'chan,g:ehl shape of polar plotcan be predicted due to addition ofapole or zero. ,--, l. .W ben a p ole- is' ad ded to a system .the polar plot end .point w ill sh ift by -90.2. When a zero i s - ad~d to a system the polar plot endpoint wm shift by +900

' . S ta I'1 t oftype-2-'.system'

. .Start of type.:3,1 . _.system . t . Endof$rrIo r d e r syste ,m . E : n d o f' 4 thorder:system_.........Start o f t yp e ...Qsys tem

Start o f -type-1t. .system.F ig 3.11.~S ta rt o f'p o ia r P lo t.o f a ll poleminimum phase system . . . . .

~LE:-3.1 : Typical Sketcbes of Po~ Pl~t '

,End of 2nc'(f. j

. ~ t ". . . '. 6 -.As w .~ 0, GU (O )~ a)L - 9ftAs 0 0 , ~ .a), G(jre) -+ OL -ISO"

-".' ... ,As co ~ oo~. GUm ) -+ OL-180 ----~~------~------~------~~--------------

'~~ ---- - .._ ....


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'G U ID ) _.' ..: -- ~" .J~. '- . - -. .- -.: .. ": -: (I-'iijro1D ",(1+ J r o T 2}(l+ jtoT3) ..' ',. __ ../.\~

1 . ....- , e 0 . ' f ; \. . . . -. ". .,. '.~. . '. ... ". .~t. . ; . J l - ~ @ i i i ~ ~ : 1 ~ T r - - . J l : ~ i f1 ~ ~ ~ ; ~ t2 l~ M ; i i i~ ; ~ i~ r 3 ' .. ~90o6 l ' ~ ". . .. . 1 ." . . ~1 - ....1 T '~'l'--=. - .-~.. - .:~ . _:- .~o_ :- 0 _ . _ " L( ....an ro T1 - ta tI. O "H 1 ; --" tan o)T~J'. " ( 1 + ~ 2 t lH l~ ' r o .~ { H t - i " ( o ~ ) . . ' . . . .

As. m 40" 'O(jt) ~ lZ::O~~.. A ~ s r o -4 06 , .. . ; < . ? : { J ~ t : ? _ z ~~?~~... .. .. . . .. ."._". .." " _. . . __.. _" ,

__ -


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I.. G(5)=--S- . ... : ..... . 1. 1 1."G{JW) = ~ = .. ' . , ,= ~L- 90. . .' Jco o)L90 ill:As.~ .40, '. G(~}~ 9 '. J L - 90As ffi - - + 00; G{jro) ~ O L . - : - ,9 00

.' S(jJ'?'O_90'

~.... : l+sT-':'. G(s)~~ - ..-' . sT:.". . 'G.- . '(,'.""-),,' = = . . 1 +jro.T '1 - I 1. 1 I L 90'" I1 1 . . . . . . ~+ .. = .- . . . +, =- -,1+

v ,.. jsr .OJT ". roTL90" . roT.As IJ) .~ 0,' G(jro} - - + . ooL - 900c +1 .

. ~ .A s 0) . - - - ? , 00, . G Om ), ~ DL - 90'" +1

1 .

;;'27fl .: 1 ..G(s) :; s m :OO! 0)c'm('Il",0.".. :.....'0C

. .' .G O O ) = jO ) = m .L 9{J " .:.....~As co ~ '0, G(jru) ~ OLge~As 00.---+ 00" . G(joo) ~ o o . L 9 , O " -1800 . m=O . 80"0.-900

.G(s)_:: l+sT .:~.O(joo) = 1 + jciJ T=1 + ,mT L90"

As OJ -40, G(5 oo) ~ 1 +0L 90~ -.AsO) --)-co; G O o o ) . ~ 1+ c O ; L 9 0 " - r o : ; ; ; 0

. - 1

E T E R M U I A I I O N O f G A I 'N M ,A R G I I N A N D P H A S E M A ,R G I N F R O M , P O L A R P L O T . . .. '.' .r _ . ._ ..The gainmargiD isdefined.as the inverse of the magnitude ofG(jro) at pha se c ro sso ve r fre qu en cy .he pbase"cros:sover freq'.eRcy is. the frequency at whichthephase ~ofG(jm)is 180. -Let the polar plot'~ut the 1800ax is at pointB and the.~agnitude ,circl'H>assingt'hrough the pointB

e 0B ' N ow -the O-aiI1margi~, Kg = ifGB,. If the point B lies within un ity circle, 'th en th~ Gain margin. isotherwise negative. (If the polar plot is drawninordinary graph sheet using rectangular coordinatesthe.point B .is the cutting point ofGOm) l o c u s with negative real axis and 'K g " " " 1 1 1 G B !where Os is tbe .agnitudecorresponding to point .B):. . '.. . - . . .' . . .. . ' t h e plta~e margin isdefined as, p.hasemargi~ T " 180'+ +where'$ is the phase angle ofar tgain crossover frequency.The.gaililcrossover:frequency isth~fr,equen6;at which the magni~ude:.fG(jro) is unity. . , .

- 4 1 ~ _ . . .


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Let the polar plot cut th e '\ffiitycircle at point A asshown in fig 3,23 and 3 ;24:. N 6~ f u . e ' : p n a s e r ;~art4'1?yis givenby L~O~' Le , if LAO]? is below'~ 1800 a x i s tbe:n thephasemasgin is positive andjfit_'~'I S above ....180 axis then t he phas .emargin is negative, - . ' . . '. , .". r-

-180...;....-..~+=..,._~,......~~-----r--O()p

:t.' :- '~

... ". . 1 r c - g o ? , Ga l ri . .m arg ln , .K~~ ~.,. P I1asemargin, 1= 1 1 , 0 0 + % t

'. .~'.

. ; '",.. ' 1Gain r 'r 't :; Ir 'O l I lI . k" =-.. "",+,,:::1 ."'g . ~. .

I! _900OI;.,~~~ " 100.h. . ; ,l la ; :H : ;rna rg l n ,Y = ...,' :+' t' g :: .

Fig 123 : Polarplo: showtngpo:sitille gainmargin and phase margln:, .G A IN AD JU S TM ~ NT USING POLA!R PL OT .'T o [Determine Kfor SpedJjed GM

Fig 3.24: P olar p lo{sJ iow ,lng negative gain . .. m a r g i n andphase matgin:" ....,

. Draw G O m j : locuswithK =L Let it cut the -"1800' axis, atpoint B. correspondingto a gain of.9a' Let the. specified gain: ~ 1 i 3 0 o _ ' . _ .,,-A+-+-"''f'"''-~~T-~-""""?_.~oQmargin be x d b . : .Por this gain margin, the G U - c o ) locus win cut . 1 . ,'-180 atpoint.Awhosemagnimde is GA, .,. . '.' 1 .' "' "1 x. Now, 20og-', = x - : : : :: ; :. log-' =-GA . GA .2 0

, I .. . 9 'A = lOi.j2(J - I , , ~

-90" .Fig :$ .2 5 : Pol~rplotfor diffe~rli- values ofK ., '.,', . " ,'.. . .', .'. G'Now-the value of K IS g iv en bY~K=_ 1 k.. . ". Os.- . - . .. .

,l(~ K >1~ then the' system' gain should be increased,.If,K " I, then the' syst~im gain should be reduced.. . -,To :D ete rm in e K r o t Spec i f ied P M, .Draw 'G(jw) locus with K ~-1;:'Let it cus the unity e i r e - m e a tpointB. (The gainat po.int~Bis GB' and e q u a l to .un ity) . . L e t the, ..-sp ecified p hase margin bex" . . , '..'For a phase margin o f x", ;let ~gp; be the phase' angleof': .

GO!))) at gain crossover frequency,:.x~= 1 800 + , J , ; .:::}.'J- ." .= x" . . . : .180. .: .'t'gcx 'fs=

" I~ the polarplot, the radial line corresponding to q . g C X ~iIf .cut the locus' of G-Gro).with . K = l at point A and the magnitudec('l:~,esponding to that point ,be GA

1-270" ~ ..

,',' - '. '-. .s

, - ~ ~;:":.laOo--t---t--:r---.--~-.----.:--.-----+-~. s:.",c r ,

" >F r g 3.26: GainadJitsiinentfor '.",:.:~~.. requiredphase .mafgiit' '-.. "'.


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3 42. ( G - -1)"'B- ,

.

. .

'. ' . . T h e o p en JO~l-ltranSfeffLilldiotlofa.unity f e edba c k s y s t em i s . given by' G{s ) = 1/5(1 +s) (1'+25). Sk e t c tr ih e p o l a l'll l lo t a n de n 1 : li :~ e . i l i e -g i a il i l m a r g in a nd phase m a rgin .. ' .

SOLUTlo.N .' ... Given that, G ( s ) = l_1s(l-t;$} ( 1 1 , . + 2 5 )

Puts =,j(i).._ .

'_r- 1 .. .. .G(jQ)} =.. ( 1 ' ) ' C - I + ' " i?..,",. ' . ' , J m - + J m - I. ' : J " ' l ! J" . .. .. T heoo rner f r e q : u e r i c i i e s . a r e - I D ci .:; 112 ~{).5 'r a d l s e c ; a n d [(Q'c;2 = 1 r a d / s e c . The magnitJde a n d phase. ~mg , l e o fGa i o ) Bfecome r f r equenc l e s , ( j I n d f o . f f requenc ies a ro un d c o rn e r fnlquenGreS ;qnd tabu la ted . Hl tab l .e-1. Us i i ng pdfarto .t h e p o . 1a r c o o r d jr ia ~ $ i r S : t e d ;,n1 : a b l ' e - - 1 are ,converted to~ n g u l a r coordinateS a n d ta b u la te d tn t ab f~ '

plot us ing_p(il~rcoordina.t~'is.s~etehoo ona polar grap'~'Slh~etas shown in fig 3,. ; , 1. ' ,Th'e polar plotusi~gis ske~hed on[;~,"ordmatygraph sheet as s h own in fi g .3:.7.2. .' . . '.'..' . - . ..'., . . 1 - .1' ...G(jffi). = O m ) (l~ j~) (1~ j2~) = toL9G6 " ' h ~ m 2 'Ltan-1o o 4 1 + 4 m 2 L t a n - 1 1 2 r o

1 . '. 1 1= = . . . . . ' . - L - $OO' : " ; ' t an - 00 ,_ ta n - 2m;.[ID~(1+:m~)(1"'f~2). '.:"',' .. '. '. ':"" .. 1 .'.' 1 ".' 1 ..'~IG(J~)l; :.-'~., .;:;- ' - - - - : ~ " " " " ' " " " = = = = = =" ". [. . .ro~(1+:'ffi2H1+:4ro2)"; oi41+ '4m2+ 0)2+ 4 r . . o " . QHb+ 5(rl + 4 < 0 4

. , . . .

Maghitude and pbase~ , o f 000) at .. ari~us frequencies.. ra dJ se a

. I:. ' . I G ( } : o J I0.35

1 . 8 " .0.7 . . 0 . . 3,2

. .0.45 0 . 6 t . O0.5. .

.1.2' 0.9 '- 1 7 1~)

- . ! ': - deg.- , 1 4 4 I . _ .... ' : 1 5 { ) ' - ; 1 ' . ; . 1 5 6 . - -162 498 . 1

'1 " . .0 0' r ad / sec eL35 :0..4 ...0.45 . 0 . 5 0 . ! 6 - 0.7 to.

-1.14 ~.89 , -0.7' ,-0 ..9,J'-0.37 , ;Q~14 0 0.09-1.29

, '-1.78 .: -1.56 '.' ~1 . 3 [ 7. ."-0 '-9 .': .-0 . .61

TG a In m a r gin , K g ,= = 1.4286P h a s e ma t g iit l; t = . +120


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300~330

2 0 < ). , . ,340 1 0 -350 {le.

3 , S O ' "'~10 340- 2 00

3.43. . , ..30 .

. '.:t20'" ."-400: .. ::

310, .'.. . .,.5 0" .. !j.-,

(}Q .'

{ J"

0.0

"l~.~ . ' ; , 1 : 1- , , " 1'no~' ::l:140~' .';i

L-.:::"",::,,~~~_,;:;;,,"";';:""'_:;:;::;....~_~-.:...~=~_";:~_--=~=::'_...l,~:""'''';:;;~~~''::'~~'''';;;:4;:::....1 - .~--210" -200 -190" -18{)0 -170O,tg-160 . -150,0' '.'~. 150" 160" 110" .l80" 1900 .-. 2000 210'" .!. I I B .


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3{1"~330

2 0 ". , . , 340

10-350" {l".3$0''~10 340"- 2 00

3.43. . ,. .. 3 0 .

. : . : 2 0 ' " ,' . . : . . 4 0 . . - :

I

310,', .'.. . .,.5 0" .. !j.-,

OO .

{ J"


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'".._r"

. . . . . . 11.9.~~- +Ie,8I.....

~ J .,-..,8-!"-:',- -0


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345 :-, .. .EXAMP.LE 3.8..

. . _' ..SOLUTION ..

. '. (~iVenIiat;,G(s) :::11s2(1+~}:(1+~~) '.. .', ' .. ", :.> '. 1 1 ' ". " Put: s = j d : J , : . . : '. G (J ro ):;;; '., :'2 .' '." . .' .. ' ' . { j I G 1 1 : I ) (1+Jro) {1 + J2m} "

.. . -.'The c;:omerft~q~enci~!1ir ie ' f i l lcl =0.0 rad /sec and IDa.=1 ' rad($e"C~,The 'm:agnitud~' and ' P h a ~ e 'al!'gf:e:.9 , f' , < 3 ( ] < O ) a re ::'.c a lc u la te d fu rth e c om e r c ~ u enQ ie s a n cH l"ie q ue n c ie s a r ound comerffequenci~s,~l'reabu l a t e d in tab le- - ' fHs lng th-e'polar~(Y:: r eaangu l a rconve rs ian ; l bepo la t r oo rd ina tes n S t e d t l 1 't a b l e -1a fe oon Ve ~ to ;racta'ngular cOord ina tes "and ' t = l b u l a t e t l l , n , t a b l ' E r : : : ;2, the poJariplot us ing;pola l r ,~coord lnates r fSs.ketCIJ~Oi ena pOI'ar.g~h $ h e e ( a : $ ' :~hQwn ! i r i fig 3,8.'1-.The'poiar . :p lot us. ing : :redan ;gu la r c oo r d i n a t e s is ske i tched :on -an p : rd i na ry gr:ap l i ls"he:e t .asshawn 'in ftg 3.8.2., .' ..,.... . '.:.;G(to) = 2 t .', t'.. . (jel). (1+j'm:}.(1+j2@,l,',; 1-- . '" . -"ID2~1 '80~1+(jJ1L:tan-1r;lJ 41+4m2 'L 't an~'~2 ID . , .(~(jw);;;; . , .1. :.' .. ' -?(~ '1 ;80 '~ tan~Joo , . . ta l i !~ '12M). . . ' C D 2 J 1 + (iJ2'~ l+ 4002 ,' " , . . :. : ' . '.lG (j(D '}I= ,", '1 , ....,: '. . . , ::}~.~ .' . o . : - ' . .. ' .. 0).2'J1+cr/Jl+,"4m,2 "ro2:~(1+Q}2) f '1 l - :+ 4 m 2 ) ' . " :1.~ ----:----;======;::===:m 2 J 1 - + : o s r o 2 + % 4' .

. . L G(jm )=.- 1 : ~ o o : : - , t a : ~ : _ 1 m:....tan-1 2 , m .

. " :/

.. . __ :- ". .,

. ,,~!., ..

- ..'TABLE - t:. M ; ! g m r u d e :~ d " : p h : a s e plot' ofG(jml)' ~tvan0l!ISf ieqti~ndes .

. ,

, c

'0.55 0,61.9' ,1.5

0.65 , . 0.7'. 'I 'O}5 1.01 .1

. 'I

0 . 3LG (1n). . ' I ' -deg' " . ~246 ,I--. ~2s1 ,-:256 " ':-261 '-265' '~269' .'r '-27300 .. ",'

.: .l"a dIs ec .' 0..45 . 0;5 .'0.09. i

0:.55 '1 .0, !0.29. I : "9~) 3.01' .. 2.36 . 1 : ..84 . .1 . 1..48 ',1',2,". to 0..8

RESULTGa i n marglin,~,~ . o ,P h a s e . maJYin,y= " : 9 1 ) 0 . '

" . " 1


47/99

"3 .46

", '

3 ,0 0-,330Q :

2 00.-,3400' , 10": _ 3 ; S O Q 35'0'"-io- 340 '' ~20< i " , J JOO-30'0

iQ " '

Q O ' .

,,-,.,8N"0'T' 1 " " " : " " " 1'--'- - -8" . : p, . . . . ..__,. . . .- -sv- - -" " " " ' "

O < > ,

-0 .2'30' ' : ' '130

, ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ _ ==~~~~~~~~~~~~,.:,'22,0"-140"

160" 200"


48/99

..47 :" -.

. ,


49/99

EXAMPlE3.9'"Iha openlooptransferfu~:diOn of'a u r )i :n y te e d bac k systemis given , by,

" (1 :+ 'O.2S} ( 1 + " ( ) , Q r 2 5 : s )G(s) ~ .' . '.. .. ,.. .",' .", ., ' ~ S3 {1 + n .0 05 s } j( 1+~O .OO t s )'.. .', . .

Sketch the polar plotand determine th e phase margin .SOLUTION! .'

.' . (1+ jl12ro)f1 + , m o . ( J . 2 S ( ! ] } . . G ( J ffi ) = = . . . . - " . . . ' . . - - , . , . . .. '. . {}m)3 (1+ -jO .OO:5m) '(1+jO.OOb) .

_ . , ~ 1 +o . , 2m )'2~ta~~10,2ro , J r - ~ + - : ,( - O " " " " " ' : ~ " " " ' 2 ; - 5 r n " " ' , ) - ; 2 ' Ltan~1o:025m.ro3'L27QQ '41+(o.:605m)2 L t a n - 1 ,O.00500 Jl+(o.ci~1ro)2 Ltan 10 . , 0 0 1 m -. ' . \ ~ . . J ~ + (O ; 2 ~ -) ~ :' : J 1 "O . ' O ? 5 ~ / ~ . .IG(Jm)~-, '" .' ',' '." ,-

.. '. .a13 J~+ {O.005IDt~~1+ (O.OOb.}2LG(jm)= tan-102m +~n-1.0.'OI2~ffi,_2700.,.. .tan-\O.O,o-Si- tan10:001 w .

. The magnitiJd-s'and ptia~s,e,8n,gle 'of G(jro) are. calculated for var ious frequ,encjes and' listed, in,Wb le - 1 ., U s i n g tJ 1e p o : l a r t O rectangular c o n v e r $ i o n , i h e po la r coo r d in a te s , i is ted in t a l b i : e : - 1 are conve r ted to reCtangular CQOrdiliat~and ta :b :u l~ ted i n 't ab l& ' -2 . The ,PD ' le r p !l ot u s ,i n' : p e ' l a - l i i ' Doardiiii1a~s'iss k e r o n e ( i ( j n S : po[arg: r ;aph. s h ee t a s s h o w n in fig 3 .9 '.1 ', Thep ola r I110tius i ingl rectangular coo rd lna t e s i.s,k em h e d on ,an Elrd inary,gfaph 'sneet i3$ s h ow n in fig!3 :92 . .

,\.

0.9 0.950 ' Hl .. ,"1.1 . 1.2~,~.~~-----+~--~----_", lG(to)lI1.4 1.2".1.0 0.8," 0.6.LG(fro), .deg

'. '1 ;." 70,2

~ 1 .4-0.4

I . -259' . ! . -258 ;_257'~256 I : -255 -24~,. '

r r o ~ . , ~ r _ a i l l_ s e _ c ~ ' ' _ '_ _ - r ~ O ~ ' . ._ _ , _ _ O . _ 9 _ 5 _ . ~ ' ~ 1 ~ . O ~ ; ~ _ . ~ 1 ~ . 1 . ~ ~_1 : _ . 2_ " ~ ~ - ~ _ 1 _ A . _ ~ ~ ' , J L _ _ 1 . _ . 1_ ~~(jm r . ~O,'27 ,~O.25, ~O:22 -0.1'9 .-0.16 :~O,:1:2 L:O:07 ~1 1 i - G , - ' 1 i u - " - ) ~--+-j -.1.-37-"1--. . c . . ; . . . 1 . - 1 7 - F -O : - 9 7 - ~ . - . " + - 1 - 0 . - , 7 - 8 - ' -i--015='8-~'I~O-.38.. I . ' O;.~

RESULTP h a s e m a rgill " jI = . . . . .rr


50/99

0

. ':. ,"_ ..

....j-20", ' . . .. .. .; 4 0 "I.

"

e, .

0" .0

.'2200.-1400

-210''''1 5 _ f l " _

-200"1600

.' -"!90"170"

-18 0"180"

.-170 .' .1900 ~r6l:l"2 0 0 " -lSOQ2 1 0 ~


51/99

.... ,~- . . . . '. ,

. 1 . . .

'.~.

", ., ,. .

I ,

: ;.'..: .~

. . . I. . . I

,.'Ii. . . . . . . . 1 1 .

I.' Im i ' > :

I II",:.. .. .. .' , I ., ,

:~,:~:,M.

. . . .'1 '.. '.: . . : " .. .. . 1 1 1 . '..' .. . .. . ..' . .II...... . . ", ...... , .._ .,. .; ; .i~ r '." _ :;: < ;::~ .. ::," ::. '#.:~.:..,.:.:.

I '. ,

.~'

' l 1 l i " . : : .. ,' .. ,... . ,.. .. .

. .. , ._... ..' .., ..

: . . . . .. , .. . . I . ' .

. . . ~'. !~ .

.....'. ;' . ' . , . .. .~~

. .~'. . ..L .. .. . .. -: 1 1 , ~ . : . . .

. ~. ~;-!:- 0,- '_P

. . ,... ,

" .:I':.. .

~.~~

! . . . '~ . ,

," 5Q\,j.i . . .~.~ ; ."N.,'r.)

. ~". . .,~, . ,-:z .."-'8 ..-

.~'.0oo . , : , , ", " , ," >+'.......-I'-"~8'~'.0.oo.. '"1-P ' " " " i..,___",.'",-;,'-sy:'--..8, g ' - " ~ . ..0

'& ' .

. . . . . . . . . . .'-"Ii.-,S ':.0.. 0


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3_L?}_ .__.. 'EXAMPLE 3.110

, . 1f1heo p e n loop transfer fiunctfon 'of a unity.feedback'system [s:giv,en b y G ($~)~ 1Is ( 1 +-s)2. Sketch th e poi l~~ plot and .:d f~ ten t \ i nEHl1e ga i lnand p h a s e m a' l li l r l.SOLUTION

Given that, 'G (s) "" 1 1 / $(1+ s ) 2 _ .. . .Puts=jco, ... . ',. .GOro}~ .1 = ' :'1. .'. ':' .': j~ { 1 + _ j P J Y " jn (1+jm)(1+'jo)

: . ~ . Th e com etfreq ,ue tic, IS (9C1 = .1 fad f se c .Th~ magn i t u d e and phas-e:aJ1lgt~ofG.( j ID)a re , ca l cu l a ted for COrner irequ e rlC Y ,and. f r equenc ies . a r ound G O m ~ r f F e q u e n c Y a nd ! t a l b u J a t e d i if i l1iatile:-1..US1:f tg.pgl !ar i tO-rectangularconvers ion th e p o la rc oo ifd i:rn a te s '.: I i :s~edm ' tS 'b fe-< are c onv e r t e d t on :!dtangu la rcoord~n ta teS 'and tabufated In ' t a b l J " ; 2 , . T h e , p o la r plol ~ s jr ig p o la r coordlnstes is 'sketched on apolar ' g taph sheet a s s h own I n fig 3 . .1O~1. The p o '( ar plo t u s:in g iF~ilgular coordin.ates a re ske t ched on a n .' o l T d i n a l r Y 9 1 l f C J p h l s t l e e t a s s h ow n in fig ,3 , ,102 . ": - ' .

- . L9'O'~" C--;-1'2 "ta- --1. ' J 1 + ' " 2L'ta -1 .()) ... "1+0)- ,,;... 11 IT ! ' < r o , . n m1 i .---'-'-:"'.........~ ...~ Lf-90- 2 ta n - 1 iO O ~ .. - ~ {~1+~2 r~

' I G ( - . ) 1 1.'.. 1" .. ~(i) =. . ~'.'. . .' m(1+ro2} W +.(1)3.. L J 3 (fm } :: = -90 ~ 2t8n-1m

.'

fAB'~ ..: .M ~ itO d e aU d .p .hras:e..o f (;(jm). a t v :a :'r iou s reqoe, nc ie s - . I .

. " . . . _ ' . . - . .r a . . ,. . . . : /~ s e c ~ : . _ _ : . . ; . . . . . .. . . .. . . . .. .. .; -_O_A~ ! F - O _ ~ . 5 _ -. . . . . . ._O __ , 6 ~ '1'. ~ O _ .7 _ . . - + 1 ~O " " " , . ; , 8 " " " ~ o + J_ ' . ; - _ O _ . 9 ~ . : - - 1 _ . 0 = =--If-1_'.1~---I. - . I " ~, ~)I" 2,2, .1..6 1..2 1 . ' (1 .8- .OJf 0;5 0.4 .." _ .

LGtjo) -_ . I, f -. d eg ~1M : I - - 1 4 3 . j ": '15,1 -159':"'1:'67 -.: '. -174- 1 1 . :"'180I ' . _. . . I 1 . . . . ~ '-185

'Gj(;o) .' J-1 .5B J ~O.96 ,4).58--{J1.36 _-~(J:181 0'.06 ' . . 1'O A 0 . 6I 0 . 7 , 0 . 8 '0.9 1.0 1 1 . 1 1I~.5 -0.40 . 0 3 J - . . -0 I. ' . ' ~ W -1..53 . . . , 1 _ .2 8 . -1.05 -fi.93 ~n ,8 : ., . . _:_O ~-6

RESULT. 'Galn margiril,K";2-.. . . g-. .. . P ha se m a rgin , i'= .21" . . . . .


53/99


54/99

3,53

~ ..

. . . .

~~-~'--.:'"... -" '.", .. ". -~.'O~.:~":"'~. ".,

, ::+;---! .

~ .. .j_ L~~' ~.

- .... - -.. ,'.'., - _',

..~" ,- 'r ..:......" .:.-.J..:._'~~':-.- ~

~'t'_ 1-

.. :~-:. . ;, "

' ; ' : ~ + - - ~ ~ - - :

,.,.I~

','';'_,''

~.,'~~, " " ' "_ - -.~ .. _, .. . ,"".-.:.

Fig LIO.-1.; , Polar plotof dUro) _::_/ [ j _ r o (I+jroi] ( u s x . - n g rectangular coordinates)


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3 . 54

O o ns id e r ,a 'u nityfe ed b ac ks ys te m h a vin g a n o pe n lo o p tra ns fe r fun c t io n O (s ) = = ' _ K '.. . . . . ':.- ... ,s{1 + O.?s)(1+ O.05s}Sk~ t c h t h e .p a l la r p l o t- a n d d~terminetbe value O f ' k s o that.( i) Ga i n margin is 1 8 d b '{i T) : P h a s e m a r g i n is 6 '1 Y ;

. - . . .-U ' T ' J O : t . '... I :._ i .. , " f : i " ' t : - _ . . . . ..Giventhat,- G(s);;; ... ' .. ,K: .., .'.....The po l a r plotls sketched byta~dingK'= 1 - . ',. . 5 (HO~.2S) (1+'0.055) , .'. ..'.. ':' ..:.purK = 1 . a n d s = J o o .in G(s).. ..: GUm}=. ,.. .. ,. 1 1 . . . , '. ' . : ' . '. ' . ' . . J m . ( 1 1+ J O . 2 o o ) (.1+ JO.05m)

. .. Th e co rn e r fre quen c ie s a re roc;= 110.2 = S'radlsec a n d (i)c2 =1/0.05 = 2D radlSec_.The.magnitude a n d phase ang l e ofr o ) a :r e q a lc u la te d fr ir va r io lJ s 'IT e q lJ E )h c ie s an d ~ a b l; llla \e d i n ta lb le -1 . USil1lgpo! 'arto rec tangu f ,~ rconve rs ! o r i the po l a r c oo r d l n a t e ss t e d in mb Je ..t a re CO I IV f ,lr ta dt o r e c ta ng i. Jla r c o a r d li i[) a ~eSa n d ' t a bU !~a tedin t a ble -2 . T h e : p o la r plb t us ing . pol a r c o ord ina t e s ' isk ~ t 6 h e d o ' r i: a p o l a . r g ra p ' h 5 h o o t a , s . , :s h O : W n i n r f i : g j; . 1 4 : 1. P 'o ia r p b t us ,H i~ r e c t anQ iU la i e oo rd i n a te s : i s 's ' k e to hea -on 'an : c it d iM ryg r a p i i s t r e e fa s s h o W n f n - f i g 3 :; 1 1 " . . 2 ;" :. . . ; . _ . . ' . . . . . ' . . . ' . . . . " . . '_ . . . , ~ ' . . , . . . . . . " , .- ( . , 1 .G _ i r e ) =. (1' '0 ' 2 )(1 'poSm)JM , +1 r o . +J : ' " . , ' , '

1 1_.----IF~~==~~--~~======--~~---;:oL90b J1+ ( O . 2 ( o i ~ L : t a n - : - ~ : O L 2 r o J 1 . + ( O . 0 5 w } 2 Ltan-1 Q : O ! ) w '.' 1 . , . ,,''. . "- = . " . . : ' . . . : L(....:90 -tan-lO . 2 o T h _ ta n, .. 1 0. 05m) .m J l+ (0.2m}2 4 1 . + (O .05m)2 .. . ' . .

: . IG( jm }[=' .. .1. . . ... .. and LG(jco)~ - '9(J- tan;'1 O.2m.; 18n-1 uo5i l l '. ' . ' - . Q J ~ l+ ( o . . ~ ( i ) f 4 1 + (0 .0&,0 )2

"~--~ro~'--~'~---~---r~~~'------'~--2"'---.I . - , - - - - - r - - - - - .rad/sec 3' 4'

~l01~~ . .' 0 .5 . - 0 .3 . O .2 -~ ; 'L~)'deg.. ~! ....98:

.t _' - ' 1 1 7 . 5 < . '..:.129.4" 0 : - 1 " 4 0 ,

. . , . m~d/sec . 5

.0.146 7 9 1 0 : 1 " 1j'

0 . 1 0.:07 0.05 . 1 0.04, '. 0 . 0 - 3 0 OJJ2LG(}:o) . deg - ' : " '1 >5 7 . ~1&t-.' -176-149 . - 1 8 0 - . - : - 1 - 8 4 -1'95

:BLE-2 : R~w and: Imaginary Parts ofG(jm) at Va n o u s ~'Jrequ.ettCies . . .rored/sec

-0.97

I '1I 2 3-0.23 I - .0.190..6 .0.. 1 I . . '-4.15...

4 .'.-0.23 . -0.23 . -D.24

-1.21 -0. .44


56/99

. G , I ( )n ) -0 .-120 .=_O~092' ~J)67 "-,0 .050 : --0,04 ~O.030 -0.019-~ o _ ; - , 5 9

,55 'illrad/sec 6 . -. -:. . : . : - . ~. "11, .: I ' . 14q( jm) -0.072 -{).Q39 ~.019 1 ' - 0 . _ 0 0 3 4 : , O 0.002 0.005'',lnttu;l'polarpl~sllowri in,fig 3.1:1.1 aad 3.11.2th~rearetwopIOts,. ma~keda'scurve~l and rurv ,e- IL 'Th:esetwo. loc i i ' ~ re

ketchedlrMthdJnerentscalestodiearlytlete:rminethegainmarginafl~phasemarg~n. ' , ' , ,Fr~m~hepOlarp i a - t " wi th , K::::1, . - . ., . . _ G a i l 1 m a l i : g im , K i l l '= , 1 : 10 .04 : :: ;2 5 : , , .Gaill margin in db= 20 lu!)25 ='28 db.Ph~1Sem:~rgi i l l1,1~ 76"'. '

(i) . ~ . ".. W U n " K : : : 1 1 : ;I~tGam) c u t U l ~ 1 8 0 ' ! " llxi~ ,~tpoint,B ~nqgf.l~AQQr r eSpo~ i ng> t o that pok.. tb e ,G IB , From t he p ,o l.~ t:p IQ t '= 0 .0 4. T h ,e g i;lin m a r g in o f2 8~ d b wm n K = 1lhas;,toberedecedto l8,db a n d so IKha~tobei~creasedto av,~ lue:9r~rman. ,

\. ; L e t G A beth~:ga[na~...80"for a gain mfu;9in of 1 1 - 8d b . '

l 1 18ogl 'n " ;=; 20' '. = > , ,,A

' ... ,', ... ' ,',. ',.. . GA' (1 1 2:5 - . ..The value of K IS gwen by, K ~ -- """~ .., 3 ..125. ' .. , Gs,O.04 -,"[ii) ,., .. ".-'" j ,W ith K : 1 , th e! ph as e margin is 7 6 . T h ' r S ' has ' t obe reducedt ( ) 60"\ HenCE!ga in h as to be ' i n c r e a s ed .- .' :tell ' ( I ~be lhe phase atG(jrul fo i: ' aph~se'rl'largi;n of 60 0

. , :. ,. 60 "'= :180=++ga .+ g c ; Z = 60 -1 80" ::;::-1 2Q o

. , , " ~ " . ,I n t fn e ,p o ll a r p lo t i le , ., ..120 " li:ne,~t'1tbe IOaJ Iso f GO( i !l )a t pointCand uuttn e un ity c ird e a t po in t D .Let; Go = : Magnitude o f G ym ) at p , a in tC .

G~ = = Magnltude of G O m J _ a t point O ~ F ro m th e' polariplot, G c; = 0.425 an d Go = = 1, .

... Go 1,. 3'", N !o~'K'''' -, '- 2 " '5 3', ' ' ' , ' ~Gc- 0.425- ' ..

,i'(a~,When K:~1(3ain mBi l ig in , K 'I I .. 2$

Giai in margin in d b , : . ,= '28db,.'

(b ) When K ;:;:1!Phase margin. y ;;;.tBe , " ' ,(c) , Fora,gainmargioof1-8db,K = = 3~12.5( d ) F o r a p h as e ,m a rgin ilf6il~.K = 2.353


57/99

3,.56

::20"-40"

I

13 t O O"-500

290'' .,..70"

3000 .'. ":"6 0 ..

, 2800-:-80"

.. ..- .8;.n.Q,o.;-'. . . . . . .'--'r"""'.9C 'io.~,......j- -9~. . . . . .6 0 < >

...100"

-210'"15n

-20W'160e -19W'170" -1801800.'"'-170"1'900 -1:60

0

20 00 -15:02100. _ " _ ' , . , ' "


58/99

, "'; .

; ;

" , - I I

, ,

. ':I ,IL'"",~

.f:, ' ' .1 '

" .6, " ',15, I, ! ,'f{l2

,11" 'A_I _ . -Z '~ .

I~~~t' L 3

.< 1_ : : l < \' . . ,

, I,~, " " ,.,i( ,',

, '

II. , .:: - : ', ,

I~,_ ,r ," ,

",' ,S E ', ~ . .

:~

,,'II,:, ,gil}'

"~ :.. ~

, ', ,

" '. , " ," .

i:! : .I,

" ,

. .. ,.I''I'

I.'.

-Fig 3 : .11 .2 : Po la r p lo t 0 / OOro),= 1 / j O ) (1+ jO . 2ro ) (1 +jO.05~ID), ,(usingrectangular'coordinates)


59/99

3.58

. . . . . . I i


60/99

3 0 ' : >~330"

Hl~.-351)" .

: 3 : 5 : 0 -10'(>' 3 4 0 < >-20" 330"-30"

7+.0"~40c

.~:~.- - -:280" 'd'-.. +~I- ,gO" ,t.. ~~8."270? 0H-++H-~~-+-r+++-i-+-!O++++++-:I-++++H-I -90" '.p- -. . . . . . .

4 0Q ' ;

0'" .0. I

.'2J~O" ." ' :" '13.0"

0~ .

~21O" '~200" ~ 1 9 { ) 1 O ~ 1 !WC -noG -;'60'" -150l50" 1600 170" 180" i.90" 20D" 210"..


61/99

- . .Fig ,3 .12 .2 : Po la r pia/of G{ jro ) = 1/U0 (1+jO.Sro)'(I+j4ro)], (usingtectangular'coo,.dinat~s).- ' ,


62/99

3.61Case (i)'. W ith K = 1, Ile tG(jro}cuttlhe-1i -SO"axis afpoiRt: B and gain correspond.i:ngto that: point be Gs. From the polar p l ' O ! t ,GB "" OA4 . The g a m margin ofT12 db with K = 1 has to b e ,mcreased to 20 d b and $0K has to E ) e decreased to a value-~ than

- .' J

LetGA be-:tbeg:aillat-180"fura gainmargin of20,~b.~I- . 1 .. J ,fOW , 20 log - = 20. . .' GA.._ l . 20 ..logl -- =-.-'=1GA 2{}' 1 - .~"" 10'=0 10, . ~G A , -

. 1!''G ;;;;;~""O.1 . 'A, ' 1 1 0 '

."

T he va ,iiue o f K ls given by, K '~ G !'I "" ._ .O J = 0.227. . Gs 0,44 1Case (Ii)"

. . . . .With K "" 1 , th e p ha se m a rgin Is 1S ". T his h as tob e ' increased to 3 0" H en ce th e gain'has to bedecreased,Le t ~gc2' b e th e p ha se o f G (j( \) fO f a ph as e ma rg i n of 30. . . .

. .: 30" =180"+


63/99

3,~'62In ' another method, first the Bodeplot of GU m ) is sketched. From t h e Bode plot the magnitude and ~

s :e 'for va rious va lues of frequen.cy, (0 a re rio te d a n d tabulsted, U sing these 'values .th e N iche ls plot issketched as expla ined earlier.' --. \D ~ I E ~ , M I N A n O N O F GAIN M A I R G , I I N AND P' IHAsE'MARGIN Ffm"M i N I C H O L S .P IL O T

. The' gain margin in db is given by the negative of db magnitude of GO m) at the. phase .crossoverfrequency, (0 The Q) is the frequency at w hich phase-of G (jru) is -1 800 If the db magnitude of Gum). . ". pc pc . . . .' '. . . . . .' .at rope. is .negative then gain margin is .positive and vice versa. .

. . Lete be -the phase angle of G t h o ) . , at aain cro ss o ver frequency co . The (b. . 'isth e - fre quency at.'fga . -, !J. 0- .' " '"""1, ge gc .w hich th e db m agnitude ofG (jro) iszemNow th e phase m argin , 1 is given by y =l~O'+

EXAMPLE 3.13. C '_; , f' d'"'~ k . h" " f - ~ , , ' 'G 'C) K(1+10s). sk ~ " ~ 1 > ; . th N" h I._ O IilS lu er 'a u nity e e .1 X:IC .s ys te m i a V ln g a n open Ic op tra ns e r,i,un c tio n .. S . ::::; 2 : . . . . . . " C~ . ' ;e Ie 0 s:-' . . _. -. .:'. S (1+s ) { 1 1 + ' 2 :s ) . - . .

. - ,p le t a n d de teml . ine : the va lue of K ~Q th a t(:i}G ~ in m a r gin is 1 0d b , (I i) P h a s e margi l1l ' 1510"., .SOl , .UTI ,ON

. ,. K(ft 10s)Given tha~ G{s) = . c_' __ .' .. _ '.. .. . S20+ s}(I+ 2s}


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3.63

I .~"'f\Oh.l~~ ..b ! - K(Jt Il~ti)) __ .Flg_3,,13.1.: N ic ho ls p lo t 0/ G(;ro) =(1m y (1 +jm)(1 +j2roj .


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. T he s in us oid a l tra n sfe r fu l '1 u ; t i o n G U m } ls ob ta ined b y le W r n g t6s = J & ~A: l soputK = 1~

',' '. ,[". ~1+100a)

..! .ro I 'a d /s e c '. 0 . ; , 2 0.,4

,.. 14 . 3 . to . 1.4 ". .....5,.3. f -15.2 ~.22,5~)I. db . 34:1. 1 ,6 0.8 ,to 1,5 3 . 0 . 4.0,' .

L~)deg25.4 t~t3

-150 ~164 . -181 .:...1 94 '-2 ,04 , . I~222 . ..,.232 ' i -244." ~250' F r om~he Nichols P lo t o i t hega inmargin and p h a s e m a rgin o f the sySrel' l1 'W h en i K = 18ra,

.Gain marg:ini= ,~i : :5db. Ptlase'margin=~45~"

Gain adjltlstm@nt fu r , r~ q tiife d g ain m .a rgiin .. ... F o r a ga in m argin -o f '1i0 d b., th e m a g nb ud e o f GUm ) s ho ulc l b e -10 clb , w h en th e p ha se ' is ~180o) . When K.=1; itlie .m a g nitu de o f Guru) is +1:9.5d1bco r r e s p ond i ng to p h a se a n gle o f -1i80" ..Hen c e ifw e add-29 ..5 d b to e ve ry potnt o f Gum ) , ll M . e n

downwa rd s a n d n 'W i U c ro ss -1 iB O "a xis a t a .m ag n it U lq eo f ~ 1Oqb. T h e 'mag n I t u d e obrrect i0n is ' i n d , ; epend~n to ffre q!je nc ya nd s o t t T j s g a i n : ' C a t J be eoot r ibutedbythe t em1 lK . let.tJj'JliS;a llie o f K b e K1;Devalu e' ofK , is ca lcu la ted by equa t ing2t'1 1 1 . U '29 5'~b' .. . . . . . ." . . .VlOQr..lto- ., u .' . " .' : .' . ..' -., .'1" K' -2aSog, = . . 20 ~.

- , ,29.5K i'= 1,0. .21} . = " 1 0 . 0 3 3 5. - . .Gain adjustment for requiTed phase ma'rrgin .

!Jet;4gc2.: l :phase'OfGQ~')i a t g a in c r oSs t i'v e r fr e que r iCY 'fo r ~ j p h ase ma:rg in o f 10 0' .' . .. .. Ph : a sem a rg in , 't2;; 1;80 +t~,..','.$gel ::.y2 : . , . . 1 8 0 0 : . = 1 0 ' ~ -.180~:::;:_ ; , l " l O ~

, ,.., '. . '., " . . '.,W h en K '!= '1,1:tiJem agliJir ude o f GUm ) is + :2 3 d b co !:r e SpOm : fr in gto a phase of -110"'; Butfufaphasemargi~ o f 10 ' ;,t h is ga ins h ou ld b e m a d e ' z e ro . l i enee ifwe a d d ~23db t oeve r i po i r n t o fG{ j r o } [ locus t hen ihe p lo t shU tS i dovmwa rd sand i twm c r o s s "'-170''-ax is at magn i t ude O f 0 db . The magn i tUde co r reC t l on i s i ndepende l l t : o ffr eque l1CY a 'Mdso , th r s gain c a n b e conb ibu t ed , by t he terrilK.l~etthisvalu~rofKbe~. TtJ,evalUeof~Jscalculated.by eq lUa~ng .2,eJQ{f~ to ~23db, : .

-:23, ._. K . 0 " " 1 0 ' 20 .= O ' 0 ' 7 2 ' .. '."RESULT'

( a ) 'WhenK= 1, ., ..G a l lt in a r gin . - ":"19 ,5db

(b)(c )

PhBsemargil1'- .- 45 .. .F or a ga in :ma rg in o f lO db , ,K = K l ~ . 0.0335

Foraphase'marginof10",K= ~ = 0.07


66/99

s.e:The c l o s ed loop transfer function of the system is given by,

C(s) G(s}---= _ = = M(s)R(s) . [+0(3) H(s} .The sinusoidaltraesfer function is obtained by replacing. s by jru~ .

. ' .. . G U ru ,).. M(jOJ)=- . '. .. '. .' I+G(jm) HGro)Let, M(jm) =~a

where, M= Magnitude of dosed loop transfer function.a ";,,hase of closed loop transfer function. .

-' \ '. The magnitude, and phase of closed loop system a r e functions of frequency, m . . T h e sketch ofmag nitu de" and p hase of dosed loop systemwith respect to m is dosed loop f requency . response p lo t, T hemagnitude and phase of dosed loopsystem for various values of frequencycan be evaluatedanalytically .or grap hically. T he analytical m eth od o f'd eterm im ng the frequ ency resp on se in volves ted io us calculations. '-Two graphical m ethods are availableto determine th e . clo sed loop frequency response from open loopfrequency response, They are,:. , .

1. M and N circles2. Nichols chart

3.10'M AND N.CU: iCLESThe magnitude of dosed loop transfer function with unity-feedback can be shown to' be in the form

ofcircle for every value of M o _ These

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Cs Unit III Full