III Z-Plane Analysis

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Arnold McBride
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1 III Z-Pl Aly opc to covrd. Itroducto. Ipul plg d dt hold 3. Otg th Z trfor y covoluto 4. Sgl rcotructo 5. h pul trfr fucto 6. Dgtl cotrollr d fltr III. Itroducto h dvtg of th trfor thod tht t l th gr to pply covtol cotuou-t yt dg thod to dcrt-t yt. Not: I th cour, w u th gl pl rt d th plg prod cott.

2 III. Ipul plg d dt hold Ipul plg: Not: put cotuou t gl t. Output quc of pul, wth th trgth of ch pul qul to th gtud of t t th corrpodg tt of t. Spl gl t t t 3. t t t Df tr of ut pul t, or t t t c trtd th odultg gl, d t c trtd th crrr.

3 Nt fd th Lplc trfor of t t t dt t dt t If w df or l h 3. co: dt t ) 3.3 l Dt hold crcut: Dt hold grt cotuou-t gl ht fro dcrt-t quc h t c pprotd y polyol, whr, lo h W c hv: h 3.4 If th dt-hold crcut th ordr polyol trpoltor, t clld thordr hold. Not: ghr th ordr, logr th dly. Zro ordr hold wh =. Whch h 3. 3

4 o ot th thtcl odl of th rl plr d ro-ordr crcut: h t u t ut c ut h h 3.7 Fro w hv h rfr fucto of frt ordr hold. h h h 4

5 h t t u t u t h ) Rpt th procdur of Not: ) pul plg u th plg durto qul to ro ) y cotuou t tchqu c ppld to dcrt t yt 5

6 III.3 Otg th Z trfor y covoluto t t t t L t L t L t t Lplc trfor of th product of two t fucto f t d g t C gv y: c j t Lf t g t f t g t dt Fp pdp j, c j whr Fp th LpLc trfor of f t, d p th trfor of g t So w c dtr j c j L t t p dp c j p ot: h tgrto l fro c j to c j d th l prlll to th gry th p pl d prt th pol of pfro tho of p. 6

7 L t t p p j 3.8 j c j c j j p dp p p p Whr th crcl of ft rdu th lft or rght hlf p pl. C : Evluto of th covoluto tgrl th lft hlf pl. q u, whr P hghr ordr dgr th q d dp dp p r polyol. W lo u p of q, whch tht l quto 3.8 wll hv o t lft, th othr o rducd to ro. j p p dp 3.9 qul to th u of th rdu of p th clod cotour. 3.9 p t pol of p p rdu of 3. y uttutg for, w hv rdu of p p t pol of p y chgg th otto fro p to, w ot rdu of t pol of Au h pl pol,,,, corrpodg K j K j l j j f pol t j ultpl pol of ordr, th th rdu K j K j d l! j d j 7

8 8 Epl 3. ot th trfor of C : vluto of th covoluto tgrl th rght hlf pl. h clod cotour clo ll pol of p, ut t do t clod y pol of p. R dp p j dp p j p p 3. C : h dotor two or or dgr hghr th th urtor. l R dp p p

9 j hu p dp jw thu jw p l A-3-7) 3. C : h dotor o dgr hghr th th urtor. R p dp p jw 3.3 Otg th trfor of fucto volvg th tr / ) ro hold: 3.4 Z ) frt ordr hold: 3.5 Z Epl 3. ot th trfor of 9

10 III.4 Sgl Rcotructo Splg thor: Au th gl t h frqucy pctru how low. h gl do t cot y frqucy copot ov w rd pr cod. If whr th plg prod, grtr th rcotructd copltly fro th pld gl w or w w t w, dfd, th th gl t c Rcll 3.,3. jw or jw l wo c prtd th plot. w w d w w Idl Low p fltr: uty ovr th frqucy rg: frqucy rg. w w w d ro outd th

11 Followg fgur how th frqucy pctr of th gl for d ftr dl fltrg. It how tht wh pld gl t. w w w w t c rcotructd copltly fro th Idl Low-p fltr ot phyclly rll. I lwhr w w w w t / w g t 3.6 Not: w t / ) 3.6 gv th ut-pul rpo of th dl fltr. t. It prctcl to hv th gl for t. ) I coucto yt, ph lg h ddd o rcovr th gl. It ot drl cotrol yt, ph lg y th yt utl. 3) It prctclly pol to rcotruct ctly cotuou t gl oc t pld.

12 Frqucy rpo chrctrtc of th ro-ordr hold: h jw jw w / / jw jw h w / Mgtud: Bco ro t th frqucy qul to th plg frqucy d tgrl ultpl of th plg frqucy. w / h jw w / Not: Sc th gtud ot cott, dtorto of th frqucy pctr occur th yt. Ph: jw h w / / jw w / w / w Mgtud d ph plot of th ro ordr hold Followg fgur copr th dl low p fltr wth th ro-ordr hold.

13 Foldg: h phoo of th ovrlp th frqucy pctr ow foldg. w clld th foldg frqucy or Nyqut frqucy wn w / Alg: Codr th tuto w w. Codr rtrry frqucy pot w tht fll th rgo of th ovrlp of th frqucy pctr. h frqucy pctru t w w copr two copot, jw d jw w. h lttr copot co fro th frqucy pctru ctrd t w w It ot pol to dtguh th frqucy pctru t w w fro tht t w w w. h phoo tht th frqucy copot w w w how up t frqucy w wh th gl pld clld lg. Not: to vod lg, w thr hv w w, or w hv prfltr to hd to rhp th frqucy pctru of th gl for th gl pld. dd Ocllto: It otd tht, f th cotuou t gl t volv frqucy copot qul to tgr) t th plg frqucy pld gl. w, th tht copot y ot ppr th 3

14 4 followg fgur how tht t t 3t pld t t,, w 3rd / c ) 3 3 3t dppr, th clld hdd ocllto. Not: ro: pl frqucy 3t, d th pl pot t 3t Oly wy to dtct th through crd plg rt or vrl plg rt. 4

15 III.5 h pul trfr fucto h pul trfr fucto rlt th trfor of th output t th plg tt to tht of th pld put. y y trfor of Z 3.7 y y y trfor of y c lo gv y 3.7 t t t t h h y h g h g h h g Z y y y h h h g h g h g h h h h 5

16 , th rto of th puld output ) d th puld put ) clld th pul trfr fucto of th dcrt-t yt. ) ) Strrd Lplc trfor of th gl volvg oth ordry d trrd Lplc trfor. y ) ) ) W how tht ) y t gt proof: Z t t g g t t d gt y y y h h h g h g h g h h h d 3.8 c prd ) h 3.8 Not: g th Lplc trfor of product of trfor, whr o r ordry Lplc trfor d othr r trrd Lplc trfor, th fucto lrdy trrd trfor c fctord out of th trrd Lplc trfor oprto. 6

17 rl procdur of otg pul trfr fucto: y y ) ) ) ) ) Epl 3.3 ot th trfor of 7

18 Pul rfr fucto of Ccdd Elt: u u y y ) U U U U ) ) hu y y, whr ) ) hu, ot : 8

19 Epl 3.4 Ot th trfor of followg two cofgurto: u u y y ) y ) y for ) for ) Pul rfr fucto of clod loop yt: R E + - E C) E R C C ) E C ) E hc E R C R E 9

20 th tg th trd Lplc trfor: R R E E E C ) E thu: R C ) R C ) R Fv cofgurto Clod loop pul trfr fucto of dgtl cotrol yt

21 R + - Splr Dgtl cotrollr ro ordr hold plt C ) R + E - E M ) U) D ) P ) C) ) lt df P ) th: C C C R D ) E D ) E D ) R C D ) ) D 3.9 Pul trfr fucto of dgtl PID cotrollr Alog PID cotrollr gv y t t d K t t dt d, 3. dt o whr th put to th cotrollr d th output of th cotrollr. Corrpodg dgtl cotrollr : E K I M ) K P K D 3.

22 whr: K K P K K K I Kd K D hu M ) K I D K P K D E 3. Not: q. 3. rfrrd to th potol for of th PID cotrol ch. h vlocty for of PID cotrollr gv follow: C K C R M ) K PC K I D 3.3 Not: dvtg of th vlocty for tht tlto ot cry wh th oprto wtchd fro ul to utotc.

23 Epl 3.5 Dg th dgtl cotrollr for followg yt wth P ) R + E - E M ) U) D ) P ) C) ) Au th plg t c: 3

24 ut tp rpo c) Rr: PID prtr wll dtrd prtlly. Splg prod h to cho proprly. ypclly -3 c for tprtur cotrol, -5c for prur cotrol, - c for lqud-lvl cotrol. 4

25 5 III.6 Dgtl cotrollr d fltr, ) ) ) 3.4 rprt dly of o t ut. ) -) Drct Progrg: ) ) Not: th totl dly lt +

26 6 Stdrd Progrg: ) ) ) ) ) ) )

27 Itrcocto ) ) + ) ) ) Prlll trcocto ) ) Sr cocto ) ) + + ) ) _ ) Fdc trcocto Not: For dgtl cotrollr, t portt to th dly lt, to th ug pot, to rduc th ordr of th pul trfr fucto. Epl 3.6: Codr cul LI yt fucto: 7 ) ) 4 ) 4 8 ) Drw drct for loc dgr for ) ) Drw ccd for loc dgr for ) c) Drw prlll for loc dgr for ) Soluto: 7

28 8 ) ) ) ) ) ) ) )

29 9 c) 4 4 / 3 5/ ) ) ) Lddr Progrg: to vod th coffct tvty prol to plt lddr tructur, to pd th ) to followg cotud frcto for: A B A B A B A )

30 3 Lt u df:,,, ) B B B A A B + - B A ) ),,, ) A A A B B A + - ) ) B A B A B ), A A ) h A B )

31 Not: Dgtl fltr d o lddr progrg hv dvtg wth rpct to coffct tvty d ccurcy. Lddr logc ot uqu. It y tructur cotud frcto po for roud th org tr of 3

32 Epl 3.7: Codr cul LI yt fucto: 7 ) ) 4 ) 4 8 3

33 33 Ift pul rpo fltr d ft pul rpo fltr ) ) ) ) y y y y 3.5 For quto 3.5, f ot ll r ro, th th pul rpo h ft ur of oro pl. h fltr clld Ift pul rpo fltr. If ll r ro, th th pul rpo h ft ur of oro pl. h fltr clld ft pul rpo fltr. Whch followg: ) d y 3.6 Rlto of ft pul rpo fltr: ) Z - ) Z - Z - Not: ) Ft pul rpo fltr FIR) orcurv. ccuulto rror c vodd. ) o fdc, drct progrg d tdrd progrg r 3) Pol of pul trfr fucto of FIR fltr r t th org. tl. 4) If th put gl volv hgh frqucy copot, th th dly lt dd th ft pul rpo fltr cr. ddvtg)

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