APPENDIX 4B Andis n th Amanyg CD TANSFE FUNCTIONS IN CONTINUOUS CONDUCTION MODE (CCM In this st, w will driv th transfr funt v / d fr th thr nvrtrs ratg CCM 4B- Buk Cnvrtrs Frm Fig. 4-7, th small signal diagram fr a Buk nvrtr is shwn Fig. 4B-. Dfg th utut stag imdan s as th aralll mbat f th filtr aaitr and th lad rsistan, s r ( sc = = ( r s( r C sc (4B- In any ratial nvrtr, r, and thrfr, r. Makg us f this assumt Eq. 4B-, s (4B- sc s d sc r v Fig. 4B- Equivalnt iruit f avrag buk nvrtr. Dfg ff as th sum f th filtr dutr imdan s and th utut stag imdan s, s ff = r C s s C C (4B-3 Thrfr, th Fig. 4B- by vltag divis Andis CD 4-
v s = = d ff r C s s C C (4B-4 4B- Bst Cnvrtr Frm Fig. 4-7, th small signal diagram f a Bst nvrtr is shwn Fig. 4B-a. In this iruit, th d stady stat ratg t valus an b alulatd as fllws: I = (4B-5 Equatg th ut and th utut wr, I = I (4B-6 Substitutg Eq. 4B-5 t Eq. 4B-6, I = I I = = (4B-7 In Fig. 4B-a, th sub-iruit lft f th markd trmals an b rlad by its Nrtn quivalnt, as shwn Fig. Fig. 4B-b. Th sub-iruit lft f th transfrmr FIg. 4B-b an b transfrmd t th right, as shwn Fig. 4B-, whr = ( D s (4B-8 d di s v : D (a Andis CD 4-
d s s di v : D s (b s ( D d ( D s di s v ( s d ( D s s v (d Fig. 4B- Equivalnt iruit f avrag bst nvrtr. Th tw urrnt surs Fig. 4B- an b mbd and usg th Thvn s quivalnt, th quivalnt vltag Fig. 4B-d is v q s = d ( D (4B-9 Usg th quivalnt vltag Eq. 4B-9 and alyg th vltag divis th iruit f Fig. 4B-d, v s = d ( D r C s s C C (4B-0 4B-3 Buk-Bst Cnvrtr Frm Fig. 4-7, th small signal diagram f a Buk-Bst nvrtr is shwn Fig. 4B- 3a. First, w will alulat th valus f th ndd quantitis at th d stady stat ratg t. Andis CD 4-3
In a Buk-Bst nvrtr, I = (4B- D = (4B- D Equatg th ut and th utut wr, I and hn, I = I (4B-3 = (4B-4 S, I = I I, I = D ( D (4B-5 Cnsidrg th sub-iruit t th lft f th markd trmals Fig. 4B-3a and drawn Fig. 4B-3b, whr, i i = i (4B-6 = Di (4B-7 Eqs. 4B-6 and 4B-7 ar valid gnral nly if i = i = 0. Thrfr Fig. 4B-3b, v = d (4B-8 ( D Shrtg th trmals as shwn Fig. 4B-3, s In Fig. 4B-3, = = ( (4B-9 i i Di D i i = d (4B-0 ( D s Substitutg Eq. 4B-0 t Eq. 4B-9, d s s = (4B- i Andis CD 4-4
Frm Figs. 4B-3b and 4B-3, and Eqs. 4B-8 and 4B-, th Thvn imdan t th lft f th markd trmals Fig. 4B-3a is v Th = = s whr, (4B- is = ( D (4B-3 v Th = d (4B-4 ( D s d ( di v :D ut (a i i Dv d D s v : D (b i i s d D i s :D ( Fig. 4B-3 Equivalnt iruit f avrag buk-bst nvrtr. Andis CD 4-5
With this Thvn quivalnt, th iruit f Fig. 4B-3a, an b drawn as shwn Fig. 4B-4a. s d ( D di s v Fig. 4B-4a Equivalnt iruit f avrag buk-bst nvrtr (ntd. Th sub-iruit t th lft f th markd trmals an b rrsntd by its Nrtn quivalnt, as shwn Fig. 4B-4b. s d s di v s Fig. 4B-4b Equivalnt iruit f avrag buk-bst nvrtr (ntd. Cmbg th urrnt surs and rrsntg th sub-iruit Fig. 4B-4b by its Thvn quivalnt as shwn Fig. 4B-4, ( D (4B-5 vq = d sd s d sd ( D s v Fig. 4B-4 Equivalnt iruit f avrag buk-bst nvrtr (ntd. Hn, v sd d = ( D r C s s C C (4B-6 Andis CD 4-6
APPENDIX 4C DEIATION OF PAAMETES OF THE CONTOE TANSFE FUNCTIONS 4C- CONTOE TANSFE FUNCTION WITH ONE POE-EO PAI Th ntrllr transfr funt givn blw nsists f a l at th rig and a lr air t rvid has bst k s / G ( s = s s/ (4C-a T analy this transfr funt, th l at th rig an b mittd s w knw that 0 it trdus a has f 90, by dfg anthr transfr funt as fllws: ' s / G( s = k s / (4C-b whr φ = G ( s = tan tan (4C- ' 4C-- Frquny at whih φ bst Ours Th maximum angl φ bst rvidd by th ntrllr urs at th gmtri man f th r and l frqunis, as shwn blw. (This gmtri man frquny is mad t id with = whr is th rss vr frquny. T fd th frquny at whih φ bst urs, w will st th drivativ f th has angl t r: Thrfr, r, d 0 d φ = = Frm Eq. 4C-5, = 0 ( ( ( ( (4C-3 (4C-4 = 0 (4C-5 = (4C-6 Andis CD 4-7
whih shws that th has angl f th ntrllr transfr funt rahs its maximum at th gmtri-man frquny. 4C-- Drivg th r and Pl Frqunis Substitutg Eq. 4C-6 t Eq. 4C-, r, φ φ bst = tan tan (4C-7 bst = tan tan (4C-8 Nt that tan x = t x and π tan y t y =. Thrfr, Eq. 4C-8 φ π π bst = tan tan = tan (4C-9 W will df an trmdiat variabl, alld th K-fatr, as K bst = (4C-0 Slvg Eqs. 4C-9 and 4C-0 r, K bst K bst φbst π = tan 4 φbst = tan 45 (4C- (4C- 4C-- alig th Cntrllr Transfr Funt with a Sgl O-Am Th ntrllr transfr funt Eq. 4C- an b ralid by a sgl -am iruit as shwn blw. Andis CD 4-8
C * ( v v C v Figur 4-C Cntrllr imlmntat f G ( s, usg Eq. 4-C(a, by an -am. In Fig. 4C-, btag th ut-utut rlatshi and marg it with th transfr funt f Eq. 4C-, C k = = = C ( C C C CC (4C-3 Frm Eq. 4C-3, trms f C = C = C / = /( C (4C-4 ( k 4C- CONTOE TANSFE FUNCTION WITH TWO POE-EO PAIS Th ntrllr transfr funt givn blw nsists f a l at th rig and tw lr airs t rvid has bst k G ( s = s ( s / ( s / (4C-5 T analy this transfr funt, th l at th rig an b mittd s w knw that 0 it trdus a has f 90, by dfg anthr transfr funt as fllws: G ( s = k ' ( s / ( s / (4C-6 whr φ = G ( s = tan tan (4C-7 ' A drivat similar t st 4C- shws that th has aks at a frquny f that is th gmtri man f th l and r frqunis, similar t that st 4C-: Andis CD 4-9
= (4C-8 Nxt, w will us th trignmtri idntity that x y tan x tan y tan = xy (4C-9 and frm Eqs. 4C-7 and 4C-8, at frquny, th has bst is φ bst = G ( s = tan ' (4C-0 φ ( bst tan = (4C- Dfg K bst as K f bst = (4C- f and usg Eqs. 4C-0 and 4C-, 0 K bst = tan 45 φ 4 bst (4C-3 Th ntrllr transfr funt Eq. 4C-5 an b ralid by a sgl -am iruit as shwn blw. C 3 C 3 C ( v rr v Figur 4-C Cntrllr imlmntat f G ( s, usg Eq. 4-C5, by an -am. In Fig. 4C-, btag th ut-utut rlatshi and marg it with th transfr funt f Eq. 4C-5, trms f Andis CD 4-0
C = /( k C 3 3 3 ( / = C = /( C = /( / C = /( (4C-4 Andis CD 4-