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7/23/2019 Design Considerations for Asymmetric Half Bridge Converters
http://slidepdf.com/reader/full/design-considerations-for-asymmetric-half-bridge-converters 1/16
FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
>
Lnhksodi Eld`
Nmstrnet – Ngdhk vnr`dus i`hcs df sdft-sw`tel`hk
edhvortors, nsyggotr`e Q[G lnbf-mr`cko edhvortors lnvo
crnwh nttoht`dh cuo td tlo`r s`gpb`e`ty, `hloroht zord vdbtnko
sw`tel`hk (ZVU) enpnm`b`ty, nhc f`xoc-froquohey dpornt`dh.
Sl`s pnpor prosohts nh nhnbys`s nhc rov`ows prnet`enb cos`kh
edhs`cornt`dhs fdr nh nsyggotr`e lnbf-mr`cko edhvortor. @t
`hebucos cos`kh`hk tlo trnhsfdrgor nhc soboet`hk edgpdhohts.
N stop-my-stop cos`kh prdeocuro w`tl n cos`kh oxngpbo w`bb
lobp ohk`hoors cos`kh nh nsyggotr`e lnbf-mr`cko edhvortor.
@. @HS]DC_ES@DHSlo offdrt td dmtn`h ovor-`herons`hk pdwor cohs`ty
df sw`teloc-gdco pdwor suppb`os lns mooh b`g`toc
my tlo s`zo df pnss`vo edgpdhohts. Dpornt`dh nt
l`klor froquohe`os edhs`cornmby rocueos tlo s`zo df
pnss`vo edgpdhohts, suel ns trnhsfdrgors nhc
f`btors9 ldwovor, sw`tel`hk bdssos lnvo l`hcoroc
l`kl-froquohey dpornt`dh. Sd rocueo sw`tel`hk
bdssos nhc nbbdw l`kl-froquohey dpornt`dh, rosdhnht
sw`tel`hk nhc sdft-sw`tel`hk toelh`quos lnvo mooh
covobdpocY>-?W
.
Slo rosdhnht sw`tel`hk gotldc prdeossos pdwor
`h n s`husd`cnb gnhhor my ut`b`z`hk tlo rosdhnheo
cur`hk tlo oht`ro sw`tel`hk por`dc. Kohornbby, tlo
dutput vdbtnko `s rokubntoc my vnr`nmbo froquohey
edhtrdb nhc tlo eurroht dr vdbtnko wnvofdrg `h tlo
rosdhnht hotwdri lns n s`husd`cnb slnpo.
Gonhwl`bo, sdft-sw`tel`hk toelh`quos ut`b`zo tlo
rosdhnheo dpornt`dh dhby cur`hk tlo sw`tel`hk
trnhs`t`dh td sdftoh tlo sw`tel`hk elnrnetor`st`es df
tlo cov`eos. [loh tlo sw`tel`hk trnhs`t`dh `s dvor,
tlo edhvortor rovorts td Qubso-[`ctl-Gdcubnt`dh
(Q[G) gdco. U`heo rosdhnht dpornt`dh dhby deeurs
cur`hk tlo sw`tel`hk trnhs`t`dh, tlo pnrngotors df
rosdhnht edgpdhohts nro hdt ns er`t`enb ns `h n
rosdhnht edhvortor. Gdrodvor, tlo sw`tel`hk
froquohey `s f`xoc nhc ons`by syhelrdh`zoc td tlo
dtlor pdwor stnkos td g`h`g`zo OG@.
Ngdhk vnr`dus i`hcs df sdft-sw`tel`hk
edhvortors, tlo nsyggotr`e Q[G lnbf-mr`cko
edhvortor lns crnwh nttoht`dh cuo td `ts s`gpb`e`ty
nhc `hloroht zord vdbtnko sw`tel`hk (ZVU)
enpnm`b`ty. F`kuro > nhc = sldw tlo mns`e e`reu`t nhc
typ`enb dpornt`dh wnvofdrgs df tlo nsyggotr`e
Q[G lnbf-mr`cko edhvortor. Sl`s edhvortor lns
sovornb ncvnhtnkos dvor dtlor sdft-sw`tel`hk
tdpdbdk`os: bdw GDUFOS vdbtnko nhc eurroht
stross, sgnbb dutput enpne`tdr nhc `hcuetdr,g`h`gug edgpdhoht eduht, nhc s`gpbo edhtrdb.
+
VD
-
] d
^=
(>-C)
h:> @ p
Bbip
Bg
EM @
cs=
@ g
V ̀h
+
Vc
-
@ cs>
^>
(C)
+
Vroe
-
VEM
F`k.>. Nsyggotr`e Q[G lnbf-mr`cko edhvortor.
F`k.=. Syp`enb wnvofdrgs df nh nsyggotr`e Q[G lnbf-mr`cko edhvortor.
Cos`kh Edhs`cornt`dhs fdr Nsyggotr`e
Lnbf-Mr`cko Edhvortors
7/23/2019 Design Considerations for Asymmetric Half Bridge Converters
http://slidepdf.com/reader/full/design-considerations-for-asymmetric-half-bridge-converters 2/16
FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
=
Sl`s pnpor prosohts cos`kh edhs`cornt`dhs df nh
nsyggotr`e Q[G lnbf-mr`cko edhvortor. @t `hebucos
cos`kh`hk tlo trnhsfdrgor nhc soboet`hk tlo
edgpdhohts. Slo stop-my-stop cos`kh prdeocuro
oxpbn`hoc w`tl n cos`kh oxngpbo w`bb lobp ohk`hoors
cos`kh nh nsyggotr`e Q[G lnbf-mr`cko edhvortor.
@@. @CONB@ZOC DQO]NS@DH Q]@HE@QBO DF
NUPGGOS]@E Q[G LNBF-M]@CKO EDHVO]SO]
F`kuro 5 `bbustrntos tlo s`gpb`f`oc e`reu`t c`nkrng
df nh nsyggotr`e Q[G lnbf-mr`cko edhvortor nhc
`ts `conb wnvofdrgs, wloro tlo boninko `hcuetnheo
df tlo trnhsfdrgor `s `khdroc. Slo CE mbdei`hk
enpne`tdr (EM) nets hdt dhby ns nh ohorky sdureo
fooc`hk tlo bdnc wl`bo tlo uppor sw`tel (^=) `s
edhcuet`hk, mut nbsd ns n mbdei`hk enpne`tdr td
provoht trnhsfdrgor snturnt`dh.
F`k. 5. U`gpb`f`oc e`reu`t c`nkrng df nh nsyggotr`e Q[G lnbf-mr`cko
edhvortor nhc `ts wnvofdrgs.
Nppby`hk n vdbtnko-soedhc mnbnheo oqunt`dh
nerdss tlo trnhsfdrgor pr`gnry s`co ns:
( ) (> )`h EM EMV V C V C∝ ⋏ 7 ⋏ ∝ (>)
Slo vdbtnko nerdss EM `s dmtn`hoc my:
EM `hV V C7 ⋏ (=)
wloro C `s tlo cuty eyebo rnt`d df bdw-s`co
GDUFOS (^>).
@conb`zoc dutput vdbtnko `s dmtn`hoc my nvornk`hk
tlo soedhcnry-s`co roet`f`or vdbtnko (Vroe) ns:
( )(> )`h EM EM
d
V V V V C C
h h
∝7 ⋏ + ⋏ ∝ (5)
wloro h `s tlo trnhsfdrgor turhs rnt`d (H p/Hs).
Uumst`tut`hk Oqunt`dhs = `htd 5, tlo `conb`zoc
dutput vdbtnko `h Oqunt`dh 5 `s s`gpb`f`oc ns:
=(> )`hd
V C CV
h
⋏ ∝ ⋏7 (3)
F`kuro 3 sldws tlo hdrgnb`zoc kn`h us`hk
Oqunt`dh 3. Slo dpornt`dh sldws syggotry fdr tlotwd dpornt`dh rnhkos df C frdg 1% td ?1% nhc
frdg ?1% td >11%. U`heo tlo sbdpo df tlo kn`h
eurvo `s rovorsoc nt ?1% cuty eyebo, cuty eyebo
sldubc mo o`tlor mobdw dr nmdvo ?1% fdr dutput
vdbtnko rokubnt`dh. @h kohornb, dpornt`dh rnhko frdg
1% td ?1% `s proforroc s`heo tlo edhvoht`dhnb
Q[G edhtrdbbor enh mo usoc ns `t `s. Gonhwl`bo,
dpornt`dh rnhko nmdvo n ?1% cuty eyebo roqu`ros
ncc`t`dhnb e`reu`try td n edhvoht`dhnb Q[G
edhtrdbbor cuo td tlo rovorsoc edhtrdb pdbnr`ty.
F`k. 3. Hdrgnb`zoc vdbtnko kn`h wloh `khdr`hk tlo cuty eyebo bdssos
enusoc my boninko `hcuetnheo.
F`kuro ? sldws s`gpb`f`oc pr`gnry-s`co eurroht
wnvofdrgs. Slo trnhsfdrgor pr`gnry-s`co eurroht (@ p) `s
tlo sug df tlo dutput `hcuetdr eurroht roforroc td ns tlo
pr`gnry s`co (@BD /h) nhc tlo gnkhot`z`hk eurroht (@G).
F`k. ?. Qr`gnry-s`co eurroht wnvofdrgs.
7/23/2019 Design Considerations for Asymmetric Half Bridge Converters
http://slidepdf.com/reader/full/design-considerations-for-asymmetric-half-bridge-converters 3/16
FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
5
@h stoncy-stnto dpornt`dh, tlo gnkhot`z`hk eurroht
`s nutdgnt`enbby ncjustoc sd tlnt tlo nvornko
elnrk`hk eurroht df EM dvor dho eyebo `s zord. Slo
gonh vnbuo df tlo trnhsfdrgor gnkhot`z`hk eurroht,
@G.nvk, `s dmtn`hoc frdg tlo edhc`t`dh fdr zord hot
eurroht tlrdukl tlo CE mbdei`hk enpne`tdr (EM):
. .( ) ( )(> )D D G nvk G nvk
@ @ @ C @ Ch h
+ 7 ∝ + ∝ (?)
. (> = ) D G nvk
@ @ C
h7 ∝ ⋏ (6)
wloro @d `s tlo dutput bdnc eurroht.
Ns dmsorvoc, tlo gonh vnbuo df tlo trnhsfdrgor
gnkhot`z`hk eurroht `s zord wloh cuty eyebos fdr tlo
GDUFOSs nro ovoh (?1% nhc ?1%). Gonhwl`bo,
tlo gonh vnbuo df tlo gnkhot`z`hk eurroht `heronsos
ns tlo cuty eyebo gdvos nwny frdg ?1%, wl`el
sldubc mo tlo wdrst enso fdr trnhsfdrgor cos`kh. @h
dtlor wdrcs, g`h`gug nhc gnx`gug cuty eyebos
nro tlo wdrst ensos fdr dpornt`dh rnhko mobdw ?1%
cuty eyebo nhc nmdvo ?1% cuty eyebo, rospoet`voby.
@@@. USONCP-USNSO NHNBPU@U DF NES_NB
NUPGGOS]@E Q[G LNBF-M]@CKO EDHVO]SO]
F`kuro 6 sldws tlo selognt`e df nh nsyggotr`e
Q[G lnbf-mr`cko edhvortor nhc F`kuro ; sldws `ts
typ`enb wnvofdrgs. @h F`kuro 6, Bg nhc Bbi nro tlo
gnkhot`z`hk `hcuetnheo nhc offoet`vo boninko
`hcuetnheo `h tlo trnhsfdrgor pr`gnry s`co,rospoet`voby, nhc EM `s tlo CE mbdei`hk enpne`tdr
wldso vdbtnko `s nbgdst edhstnht cur`hk dho
sw`tel`hk eyebo.
F`k. 6. Uelognt`e df nh nsyggotr`e Q[G lnbf-mr`cko edhvortor.
F`k. ;. Syp`enb wnvofdrgs df nh nsyggotr`e Q[G lnbf-mr`cko edhvortor.
@h kohornb, tlo nsyggotr`e Q[G lnbf-mr`cko
edhvortor edhs`sts df tlroo stnkos, ns sldwh `h F`kuro 6:
n squnro wnvo kohorntdr, ohorky trnhsfor hotwdri, nhc
roet`f`or hotwdri.
Slo squnro wnvo kohorntdr prdcueos n squnro wnvo
vdbtnko (V c ) my cr`v`hk sw`telos ^> nhc ^=
edgpbogohtnr`by. @h drcor wdrcs, tlo bdwor nhc
uppor GDUFOSs cuty eyebos sldubc mo C nhc >-C,
rospoet`voby. N sgnbb conc t`go `s usunbby
`htrdcueoc motwooh tlo edhsoeut`vo trnhs`t`dhs.
Slo ohorky trnhsfor hotwdri edhs`sts df n CE
mbdei`hk enpne`tdr nhc trnhsfdrgor. Slo ohorky
trnhsfor hotwdri rogdvos tlo CE dffsot df tlo
7/23/2019 Design Considerations for Asymmetric Half Bridge Converters
http://slidepdf.com/reader/full/design-considerations-for-asymmetric-half-bridge-converters 4/16
FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
3
squnro wnvo vdbtnko (Vc) us`hk tlo CE mbdei`hk
enpne`tdr (EM), tloh trnhsfors tlo puro NE
squnro wnvo vdbtnko td tlo soedhcnry s`co
tlrdukl tlo trnhsfdrgor. Slo trnhsfdrgor
pr`gnry-s`co eurroht, @ p, bnks tlo vdbtnko nppb`oc
td tlo trnhsfdrgor pr`gnry s`co cuo td tlo
boninko `hcuetnheo, wl`el nbbdws tlo GDUFOSs
td mo turhoc dh w`tl zord vdbtnko. Ns sooh `h
F`kuro ;, nftor dho df tlo sw`telos lns mooh
turhoc dff, @ p elnrkos (dr c`selnrkos) tlo
GDUFOS dutput enpne`tnheos nhc ovohtunbby
sw`hks tlo sw`tel vdbtnkos frdg dho `hput rn`b td
tlo dtlor (frdg V`h td krduhc dr frdg krduhc td
V`h). Sloh, @ p edht`huos td fbdw tlrdukl tlo nht`-
pnrnbbob mdcy c`dco. Ns bdhk ns tlo mdcy c`dco
edhcuets, tlo edrrospdhc`hk GDUFOS enh mo
turhoc dh w`tl zord vdbtnko.
Slo roet`f`or hotwdri prdcueos n CE vdbtnko myroet`fy`hk tlo NE vdbtnko w`tl roet`f`or c`dcos
nhc n bdw-pnss BE f`btor. Slo roet`f`or hotwdri
enh mo `gpbogohtoc ns n fubb-wnvo mr`cko dr
eohtor-tnppoc edhf`kurnt`dh.
@h stoncy-stnto nhnbys`s, tlo fdbbdw`hk
nssugpt`dhs nro gnco:
Slo conc t`go `s hokb`k`mbo s`heo `t `s vory sgnbb
edgpnroc td tlo sw`tel`hk eyebo.
Slo boninko `hcuetnheo `s guel sgnbbor tlnh tlognkhot`z`hk `hcuetnheo.
Slo CE mbdei`hk enpne`tdr, EM, `s bnrko ohdukl
td hokboet tlo vdbtnko r`ppbo nerdss EM.
Slo dutput f`btor `hcuetdr dporntos `h edht`hudus
edhcuet`dh gdco.
Nbb e`reu`t obogohts nro `conb nhc bdssboss.
Slo cuty eyebo fdr bdwor GDUFOS, C, `s b`g`toc
mobdw ?1%.
Slo enpne`tdrs E> nhc E= `hebuco hdt dhby tlo`htorhnb dutput enpne`tnheo df GDUFOSs, mut
nbsd tlo oxtorhnb pnrns`t`e enpne`tnheos.
N. Dpornt`dh Gdco Nhnbys`s
Slo nsyggotr`e lnbf-mr`cko edhvortor lns o`klt
dpornt`dh gdcos cur`hk dho sw`tel`hk eyebo. Slo
oqu`vnboht e`reu`t df onel gdco `s `bbustrntoc `h
F`kuro <.
(n)
(m)
(e)
(c)
(o)
7/23/2019 Design Considerations for Asymmetric Half Bridge Converters
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
?
(f)
(k)
(l)
F`k. <. Dpornt`dh gdcos df nh nsyggotr`e Q[G lnbf-mr`cko edhvortor.
Gdco @ (td~t>): Slo bdwor sw`tel (^>) `s
edhcuet`hk nhc n vdbtnko df (V`h-VEM) `s nppb`oc td
tlo trnhsfdrgor pr`gnry s`co (V pr ). Slo trnhsfdrgor
pr`gnry-s`co eurroht (@ p) `s tlo sug df tlo dutput
`hcuetdr eurroht roforroc td ns tlo pr`gnry s`co
(@BD/h) nhc tlo gnkhot`z`hk eurroht (@G). Sl`s gdco
`s s`g`bnr td tlo pdwor trnhsforr`hk gdco df n
edhvoht`dhnb lnbf-mr`cko edhvortor.
Gdco @@ (t>~t=): Slo bdwor sw`tel (^>) `s turhoc
dff nt t> nhc tlo pr`gnry s`co eurroht, @ p, elnrkos E>
nhc c`selnrkos E=. Sl`s gdco edht`huos uht`b tlo pr`gnry-s`co vdbtnko crdps td zord (uht`b E= `s
c`selnrkoc td VEM) nt t=. U`heo tlo soedhcnry s`co `s
edhhoetoc td tlo pr`gnry s`co, E= `s c`selnrkoc
us`hk tlo ohorky stdroc `h tlo boninko `hcuetnheo
nhc dutput `hcuetdr.
Gdco @@@ (t=~t5): Nt t=, tlo trnhsfdrgor pr`gnry-
s`co nhc soedhcnry-s`co vdbtnkos moedgo zord nhc
tlo soedhcnry s`co `s coedupboc frdg tlo pr`gnry
s`co. Sloh, tlo dutput `hcuetdr eurroht, @BD, mok`hs
td froowloob `h tlo soedhcnry s`co tlrdukl tlo
roet`f`ors wl`bo E= edht`huos td mo c`selnrkoc. U`heo
tlo soedhcnry-s`co `hcuetdr (Bd) `s c`sedhhoetoc
frdg tlo pr`gnry s`co, E= `s c`selnrkoc us`hk tlo
ohorky stdroc `h tlo boninko `hcuetnheo dhby. Nftor
E= `s fubby c`selnrkoc, tlo mdcy c`dco df ^
= mok`hs
edhcuet`hk mofdro t5.
Gdco @V (t5~t3): Slo mdcy c`dco df ^= `s
edhcuet`hk nhc tlo vdbtnko nerdss tlo sw`tel ^= `s
ebngpoc nt zord. My turh`hk dh ^= wl`bo tlo mdcy
c`dco `s edhcuet`hk, zord vdbtnko sw`tel`hk (ZVU)
`s nel`ovoc. Cur`hk tl`s gdco, n vdbtnko df -VEM `s
nppb`oc nerdss tlo boninko `hcuetnheo nhc tlo
pr`gnry-s`co eurroht (@ p) coeronsos. Sl`s gdco
edht`huos uht`b @ p, pbus tlo gnkhot`z`hk eurroht (@G),
moedgos -@BD/h. Cur`hk tl`s gdco, tlo trnhsfdrgor
soedhcnry-s`co vdbtnko rogn`hs zord, wl`el enusoscuty eyebo bdssos df ns guel ns:
3 5>
=d bi
B
s U `h
t t @ B C
S S h V C
∝7 7
⋏ ⋏ ⋏ (;)
wloro Ss `s tlo sw`tel`hk por`dc.
Gdco V (t3~t?): Slo uppor sw`tel (^=) `s
edhcuet`hk nhc -VEM `s nppb`oc td tlo trnhsfdrgor
pr`gnry s`co (V pr ). Slo trnhsfdrgor pr`gnry-s`co
eurroht (@ p) `s tlo sug df tlo dutput `hcuetdr eurroht
roforroc td ns tlo pr`gnry s`co (-@BD/h) nhc tlo
gnkhot`z`hk eurroht (@G). Sl`s gdco `s s`g`bnr td
tlo pdwor trnhsforr`hk gdco df n edhvoht`dhnb lnbf-
mr`cko edhvortor.
Gdco V@ (t?~t6): Slo uppor sw`tel (^=) `s turhoc
dff nt t? nhc tlo pr`gnry-s`co eurroht @ p elnrkos E=
nhc c`selnrkos E>. Sl`s gdco edht`huos uht`b tlo
pr`gnry-s`co vdbtnko moedgos zord (uht`b E> `s
c`selnrkoc td V`h-VEM) nt t6. U`heo tlo soedhcnry-
s`co `hcuetdr (Bd) `s edhhoetoc td tlo pr`gnry s`co,
E> `s c`selnrkoc us`hk tlo ohorky stdroc `h tlo
boninko `hcuetnheo nhc dutput `hcuetdr.
Gdco V@@ (t6~t;): Nt t6, tlo trnhsfdrgor pr`gnry-
s`co nhc soedhcnry-s`co vdbtnkos moedgo zord nhc
tlo soedhcnry s`co `s coedupboc frdg tlo pr`gnry
s`co. Sloh, tlo dutput `hcuetdr eurroht, @BD, mok`hs
td froowloob `h tlo soedhcnry s`co nhc E> edht`huos
td mo c`selnrkoc. U`heo tlo soedhcnry s`co `s
c`sedhhoetoc frdg tlo pr`gnry s`co, E> `s
c`selnrkoc us`hk tlo ohorky stdroc `h tlo boninko
7/23/2019 Design Considerations for Asymmetric Half Bridge Converters
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
6
`hcuetnheo dhby. Nftor E> `s fubby c`selnrkoc, tlo
mdcy c`dco df ^> mok`hs edhcuet`hk mofdro t;.
Gdco V@@@ (t;~t<): Slo mdcy c`dco df ^> `s
edhcuet`hk nhc tlo vdbtnko nerdss tlo sw`tel ^> `s
ebngpoc nt zord vdbtnko. My turh`hk dh ^> wl`bo tlo
mdcy c`dco `s edhcuet`hk, zord vdbtnko sw`tel`hk
(ZVU) `s nel`ovoc. Cur`hk tl`s gdco, n vdbtnko df
(V`h-VEM) `s nppb`oc nerdss tlo boninko `hcuetnheo
nhc tlo pr`gnry s`co eurroht (@ p) `heronsos. Sl`s
gdco edht`huos uht`b @ p g`hus tlo gnkhot`z`hk
eurroht moedgos @BD /h. Cur`hk tl`s gdco, tlo
trnhsfdrgor soedhcnry s`co vdbtnko rogn`hs zord,
wl`el enusos cuty eyebo bdssos df ns guel ns
< ;=
=
(> )
d bi B
s U `h
t t @ B C
S S h V C
∝7 7
⋏ ⋏ ⋏ ∝ (<)
M.
Zord-Vdbtnko Uw`tel`hk Edhc`t`dh
Ns c`seussoc `h tlo oxpbnhnt`dh df dpornt`dh gdco,
nftor ^> `s turhoc dff nt t>, E= `s c`selnrkoc frdg V`h
td VEM my tlo ohorky stdroc `h tlo boninko
`hcuetnheo nhc dutput `hcuetdr. Sloh E= `s fubby
c`selnrkoc my tlo ohorky stdroc `h tlo boninko
`hcuetnheo dhby. Slorofdro, tlo ZVU edhc`t`dh fdr
sw`tel ^= `s k`voh ns:
= =
= > =Y ( )W ( )Y Wbi p `h B @ t E E CV 8 + (0)
wloro Bbi `s tlo offoet`vo boninko `hcuetnheo `htlo trnhsfdrgor pr`gnry s`co nhc @ p(t=) `s tlo pr`gnry
s`co eurroht nt t=, k`voh ns:
= .
(> )( )
=
d `h U p G nvk
g
@ V C CS @ t @
h B
∝7 + + (>1)
Nftor ^= `s turhoc dff nt t?, E> `s c`selnrkoc frdg
V@H td V@H-VEM my tlo ohorky stdroc `h tlo boninko
`hcuetnheo nhc dutput `hcuetdr. Sloh E> `s fubby
c`selnrkoc my tlo ohorky stdroc `h tlo boninko
`hcuetnheo dhby. Slorofdro, tlo ZVU edhc`t`dh fdr
sw`tel ^> `s k`voh ns:
= =
6 > =Y ( )W ( )Y(> ) Wbi p `h
B @ t E E C V 8 + ∝ (>>)
wloro @ p(t6) `s tlo pr`gnry-s`co eurroht nt t6, k`voh ns:
6 .
(> )( )
=
d `h U p G nvk
g
@ V C CS @ t @
h B
∝7 ∝ + ∝ (>=)
U`heo @ p(t=) `s nbwnys bnrkor tlnh |@ p(t6)| nhc CV`h `s
nbwnys sgnbbor tlnh (>-C)V`h fdr cuty eyebos sgnbbor
tlnh ?1%, tlo ZVU edhc`t`dh df Oqunt`dh 0 `s
snt`sf`oc gdro ons`by tlnh tlnt df Oqunt`dh >>. Slus,
tlo edhc`t`dh k`voh my Oqunt`dh >> sldubc mo
snt`sf`oc td nel`ovo ZVU fdr mdtl GDUFOSs. @t `s
roqu`roc td `heronso tlo boninko `hcuetnheo (Bbi
) dr
`heronso tlo vnbuo df |@ p(t6)| td kunrnhtoo ZVU fdr
mdtl sw`telos dvor w`co bdnc rnhko. Sdd bnrko n
boninko `hcuetnheo `heronsos tlo vdbtnko dvorslddt
nhc r`hk`hk `h tlo roet`f`or c`dcos, wl`el rosubts `h
us`hk l`klor vdbtnko rntoc roet`f`ors nhc pdwor
c`ss`pnt`dh `h tlo shummor e`reu`t td ebngp tlo
vdbtnko dvorslddt. Nhdtlor wny td kunrnhtoo ZVU
dvor n w`co bdnc rnhko `s td `heronso tlo vnbuo df
|@ p(t6)| my rocue`hk tlo gnkhot`z`hk `hcuetnheo.
Ovoh tldukl tl`s gotldc `heronsos edhcuet`dh
bdssos dh tlo pr`gnry s`co, `t `s gdro offoet`vo tlnh`herons`hk tlo boninko `hcuetnheo fdr l`kl `hput
vdbtnko nppb`ent`dhs wloro tlo edhcuet`dh bdssos `h
tlo pr`gnry s`co nro hokb`k`mbo.
Vks>
Vks=
Vcs>
Vks>
Vcs=
@Q
Vpr
C >-C
V`h
V`h
V`h-V
EM VEM
Vks>
Vks>
Vks=
Vks=
Vpr V
`h-V
EM
VEM
V`h-VEM VEM
Vcs=
Vcs>
gdco @@ nhc @@@ gdco V@ nhc V@@
t>
t5
t=
t?
t;
t6
VEM
V`h-V
EM
F`k. 0. Cotn`boc wnvofdrgs cur`hk tlo sw`tel`hk trnhs`oht.
7/23/2019 Design Considerations for Asymmetric Half Bridge Converters
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
;
E. CE Elnrnetor`st`es Edhs`cor`hk Cuty Eyebo
Bdssos
@h tlo `hput td dutput vdbtnko robnt`dhsl`p df
Oqunt`dh 3, tlo cuty eyebo bdssos enusoc my boninko
`hcuetnheo `s `khdroc. @h prnet`enb cos`kh, ldwovor,
cuty eyebo bdssos tnio n edhs`cornmbo ngduht df tlo
sw`tel`hk eyebo nhc sldubc mo edhs`coroc td dmtn`htlo edrroet dutput vdbtnko oqunt`dh. Ns dmsorvoc `h
F`kuro ;, cuty eyebo bdssos tnio pbneo hdt dhby
cur`hk tlo ^> edhcuet`dh por`dc (CB>), mut nbsd
cur`hk tlo ^= edhcuet`dh por`dc (CB=). Edhs`cor`hk
tlo cuty eyebo bdssos frdg Oqunt`dhs ; nhc <, tlo
dutput vdbtnko df Oqunt`dh 3 `s gdc`f`oc ns:
= >
( )( ) (> )`h EM EM
d B B
V V V V C C C C
h h
∝7 ⋏ ∝ + ⋏ ∝ ∝ (>5)
Uumst`tut`hk Oqunt`dhs =, ;, nhc < `htd >5, tlo
dutput vdbtnko `h Oqunt`dh >5 `s s`gpb`f`oc ns:== =
(> ) ( )`h d bi d
U
V @ BV C C
h h S 7 ⋏ ∝ ∝ (>3)
Ns dmsorvoc `h Oqunt`dh >3, cuty eyebo bdssos
`heronso ns tlo boninko `hcuetnheo `heronsos. Ovoh
tldukl bnrko boninko `hcuetnheo `heronsos ZVU bdnc
rnhko nhc nbbov`ntos tlo rovorso-roedvory prdmbog
my rocue`hk c`/ct, `t rosubts `h rocueoc trnhsfdrgor
turhs rnt`d, wl`el enusos gdro vdbtnko stross dh tlo
roet`f`or c`dcos nhc `heronsoc edhcuet`dh bdssos cuotd cuty eyebo bdssos. Edhs`cor`hk tlo dvornbb
off`e`ohey, boninko `hcuetnheo rosubt`hk `h ?~>1%
cuty eyebo bdssos (CB>+CB=) `s prdpor.
@V. COU@KH Q]DEOC_]O
@h tl`s soet`dh, n cos`kh prdeocuro `s prosohtoc
us`hk tlo selognt`e df F`kuro >1 ns n roforoheo. N
eohtor tnppoc trnhsfdrgor `s usoc `h tlo roet`f`or
stnko nhc tlo `hput vdbtnko `s nssugoc td mo
suppb`oc frdg pdwor fnetdr edrroet`dh (QFE) pro-
rokubntdr. N CE-CE edhvortor w`tl n >0=[/=3Vdutput lns mooh soboetoc ns n cos`kh oxngpbo. Ns n
gnjdr cov`eo, FUFN-sor`os Fn`rel`bc Qdwor Uw`tel
(FQUSG
) `s ogpbdyoc, wl`el `s nh `htokrntoc Qubso
[`ctl Gdcubnt`dh (Q[G) edhtrdbbor nhc
GDUFOSs spoe`f`enbby cos`khoc fdr Zord Vdbtnko
Uw`tel`hk (ZVU) lnbf-mr`cko edhvortors w`tl
g`h`gnb oxtorhnb edgpdhohts. Edgpnroc w`tl n
c`seroto GDUFOS nhc Q[G edhtrdbbor sdbut`dh, tlo
FUFN-sor`os enh rocueo tdtnb edst, edgpdhoht eduht,
s`zo, nhc wo`klt, wl`bo s`gubtnhodusby `herons`hk
off`e`ohey, prdcuet`v`ty, nhc systog rob`nm`b`ty.
Slo cos`kh spoe`f`ent`dhs nro ns fdbbdws:
Hdg`hnb @hput Vdbtnko: 311VCE (dutput df QFE
stnko)
Dutput: =3V/<N (>0=[)
Ldbc-up S`go ]oqu`rogoht: =1gs (?1Lz B`ho
Froquohey)
CE B`hi Enpne`tdr df QFE Dutput: 551¿F
Uw`tel`hk Froquohey: >11iLz Gnx`gug Cuty Eyebo df Q[G Edhtrdbbor: ?1%
Edhtrdb
@E
Vee
VCBBVee
UK QK
VES]
LVee
Hs
Vd
Hp Hs
C]>
_5
IN35>
V`h
(QFE dutput)
C]=
BD
ED
EM
] sohso
E`h
]S
FM
EUFUFN=>11
F`k. >1. ]oforoheo e`reu`t df nh nsyggotr`e Q[G lnbf-mr`cko edhvortor fdr tlo cos`kh oxngpbo.
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
<
YUSOQ->W Cof`ho tlo Uystog Upoe`f`ent`dhs
Slo fdbbdw`hk spoe`f`ent`dh sldubc mo cof`hoc.
Ost`gntoc Off`e`ohey ( O ff ): Slo pdwor edhvors`dh
off`e`ohey gust mo ost`gntoc td enbeubnto tlo
gnx`gug `hput pdwor w`tl n k`voh gnx`gug
dutput pdwor. @f hd roforoheo cntn `s nvn`bnmbo, uso O ff 7 1.<<~1.0= fdr bdw-vdbtnko dutput nppb`ent`dhs
nhc O ff 7 1.0=~1.06 fdr l`kl-vdbtnko dutput
nppb`ent`dhs. [`tl tlo ost`gntoc off`e`ohey, tlo
gnx`gug `hput pdwor `s:
d
`h
ff
Q Q
O 7 (>?)
@hput Vdbtnko ]nhko (V ̀hg`h
nhc V ̀hgnx
): Slo
gnx`gug `hput vdbtnko wdubc mo tlo hdg`hnb QFE
dutput vdbtnko:
gnx.`h D QFE V V 7 (>6)
Sldukl tlo `hput vdbtnko `s rokubntoc ns n
edhstnht my tlo QFE pro-rokubntdr, `t crdps cur`hk
tlo ldbc-up t`go. Slo g`h`gug `hput vdbtnko,
edhs`cor`hk tlo ldbc-up t`go roqu`rogoht, `s k`voh
ns:
g`h =
.
=`h L_
`h D QFE
`h
Q S V V
E 7 ∝ (>;)
wloro VD.QFE `s tlo hdg`hnb QFE dutput vdbtnko,
SL_ `s tlo ldbc-up t`go, nhc E`h `s tlo CE b`hi mubienpne`tdr dh tlo `hput s`co.
(Cos`kh Oxngpbo) Nssug`hk tlo off`e`ohey `s 0=%,
>0==10
1.0=d
`h
ff
Q Q
O [ 7 7 7
gnx
. 311`h D QFE V V V 7 7
5g`h = =
. 6
= = =10 =1 >1311 56;
551 >1
`h L_ `h D QFE
`h
Q S V V V
E
∝
∝
⋏ ⋏ Û7 ∝ 7 ∝ 7
Û
YUSOQ-=W Cotorg`ho tlo Srnhsfdrgor Surhs]nt`d (h7Hp /Hs)
Ns dmsorvoc `h Oqunt`dh >3, boninko `hcuetnheo
df tlo trnhsfdrgor rosubts `h cuty eyebo bdssos nhc
nffoets tlo `hput td dutput vdbtnko rnt`d. Edhs`cor`hk
tlo fdrwnrc vdbtnko crdp `h tlo roet`f`or c`dco,
Oqunt`dh >3 `s gdc`f`oc ns:
== =(> ) ( ) d BI
d F `h
U
@ BV V V C C
h h S + 7 ∝ ∝ (><)
wloro VF `s tlo soedhcnry-s`co roet`f`or c`dco
vdbtnko crdp.
Sloro nro tlroo cos`kh pnrngotors `h Oqunt`dh ><:
cuty eyebo (C), trnhsfdrgor turhs rnt`d (h), nhc
boninko `hcuetnheo. @t `s hoeossnry td cotorg`ho tloboninko `hcuetnheo pr`dr td tlo trnhsfdrgor turhs
rnt`d nhc cuty eyebo. ]ntlor tlnh eldds`hk tlo
boninko `hcuetnheo nrm`trnr`by, `t `s ronsdhnmbo td
cotorg`ho tlo boninko `hcuetnheo my edhs`cor`hk tlo
cuty eyebo bdssos. Slo cuty eyebo bdssos nro
npprdx`gntoby:
> = gnx =
>6
( )
`h bi Bdss B B
`h U
Q B C C C
V S
⋏ ⋏7 + ≏
⋏ (>0)
@t `s typ`enb td sot tlo cuty eyebo bdssos ns ?~>1%
df tlo sw`tel`hk eyebo. Dheo tlo boninko `hcuetnheo`s cotorg`hoc, tlo trnhsfdrgor turhs rnt`d `s
enbeubntoc my sdbv`hk Oqunt`dh >< fdr h w`tl n k`voh
C ns:
=(> ) ( (> )) 3( )
( )
`h `h d F d bi s
d F
V C C V C C V V @ B f h
V V
∝ + ⋏ ⋏ ∝ ∝ +7
+(=1)
[loh enbeubnt`hk tlo trnhsfdrgor turhs rnt`d w`tl
g`h`gug `hput vdbtnko nhc gnx`gug cuty eyebo
edhc`t`dh, tloro sldubc mo sdgo gnrk`h dh tlo
gnx`gug cuty eyebo df tlo Q[G edhtrdbbor td
kunrnhtoo dutput vdbtnko rokubnt`dh.
(Cos`kh Oxngpbo) My eldds`hk cuty eyebo bdssos ns
0% df tlo sw`tel`hk eyebo, tlo boninko `hcuetnheo `s
dmtn`hoc ns:gnx =
= 6
( )
>6
1.10 311 >1 >135
>6 =10
Bdss `h U bi
`h
C V S B
Q
L ¿
∝
⋏ ⋏7
⋏
⋏ ⋏ Û7 7
⋏
Edhs`cor`hk ?% gnrk`h dh tlo gnx`gug cuty eyebo df
tlo Q[G edhtrdbbor (?1¾5%), tlo wdrst-ensognx`gug cuty eyebo td enbeubnto tlo trnhsfdrgor turhs
rnt`d `s eldsoh ns 3=%. Nssug`hk tlo fdrwnrc vdbtnko
crdp df tlo roet`f`or `s >.=V, tlo turhs rnt`d `s k`voh ns:
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
0
(Cos`kh Oxngpbo – Edht`huoc)
g`h
gnx gnx
g`h =
gnx gnx
= 6 5
(> )
( )
( (> )) 3( )
( )
56; 1.3= (> 1.3=)
(=3 >.=)
(56; 1.3= (> 1.3=)) 3(=3 >.=) < 35 >1 >11 >1
( )
6.=
`h
d F
`h d F d bi s
d F
d F
V C Ch
V V
V C C V V @ B f
V V
V V
∝
⋏ ∝7
+
⋏ ⋏ ∝ ∝ ++
+
⋏ ⋏ ∝7 +
⋏ ⋏ ∝ ∝ + ⋏ ⋏ Û ⋏ Û+
+
7
YUSOQ-5W Enbeubnto tlo Hdg`hnb Cuty Eyebo
fdr Gnx`gug @hput Vdbtnko nhc Fubb-Bdnc
Edhc`t`dh
U`heo tlo turhs rnt`d `s cotorg`hoc `h USOQ-=, tlo
hdg`hnb cuty eyebo fdr gnx`gug `hput vdbtnko nhcfubb bdnc edhc`t`dh `s dmtn`hoc my sdbv`hk:
gnx == =(> )( ) ( ) d bi
d F `h hdg hdg
U
@ BV V V C C
h h S + 7 ⋏ ∝ ∝ (=>)
Slo hdg`hnb cuty eyebo `s k`voh ns:
gnx gnx
( ) => > 3Y W
=
=
d F d bi
`h `h U
hdg
h V V @ B
V hV S C
+∝ ∝ +
7 (==)
(Cos`kh Oxngpbo)
gnx gnx
6
6
( ) => > 3Y W
=
=
6.= =?.= = < 35 >1> > 3Y W
= 311 6.= 311 >1 >11.53
=
d F d bi
`h `h U
hdg
h V V @ B
V hV S C
∝
∝
+∝ ∝ +
7
⋏ ⋏ ⋏ Û∝ ∝ +
⋏ ⋏ ⋏ Û7 7
YUSOQ-3W Dutput @hcuetdr Cos`kh
Slo eurroht r`ppbo df tlo dutput f`btor `hcuetdr `s
k`voh ns:(> )
( )=
Y W(> )
`hd F
D bi B U
d `h
V CV V
@ Bh @ CS B h V C
∝∝ ∝
∆ 7 Û ∝⋏ ∝ (=5)
wloro Bd `s tlo dutput `hcuetdr.
@t `s typ`enb td sot tlo r`ppbo ns >1~=1% df tlo
rntoc dutput eurroht.
(Cos`kh Oxngpbo) My eldds`hk tlo eurroht r`ppbo df
tlo dutput `hcuetdr ns =1% df tlo rntoc dutput eurroht,
tlo dutput `hcuetnheo `s k`voh frdg Oqunt`dh =5 ns:
66
(> )( )
=Y W
(> )
311 (> 1.53)( =?.=)
= < 35 >16.= Y1.53 >1 >1 W< 1.= 6.= 311 (> 1.53)
5=.5
`hd F
D bi d U
B `h
V CV V
@ Bh B CS @ h V C
L ¿
∝∝
∝∝ ∝
7 Û ∝∆ ⋏ ∝
⋏ ∝ ∝ ⋏ ⋏ Û7 Û ⋏ Û ∝
Û ⋏ ⋏ ∝
7
Slo dutput `hcuetdr `s sgnbbor tlnh tlo fdrwnrc
edhvortor w`tl tlo sngo eurroht r`ppbo nhc dutput
pdwor.
YUSOQ-?W Cotorg`ho tlo Gnkhot`z`hk
@hcuetnheo
Ns dmsorvoc `h Oqunt`dhs >1 nhc >=, tlognkhot`z`hk `hcuetnheo vnbuo `s ebdsoby robntoc td
tlo ZVU edhc`t`dh, s`heo `t cotorg`hos tlo poni
eurroht bovob cur`hk tlo sw`tel`hk trnhs`t`dh. Df tlo
twd ZVU edhc`t`dhs `h Oqunt`dhs 0 nhc >>,
Oqunt`dhs >> `s tlo wdrst enso fdr dpornt`dh w`tl n
cuty eyebo mobdw ?1%. Slus, Slo ZVU edhc`t`dh fdr
mdtl GDUFOSs `s k`voh ns:
=
.
=
> =
(> )Y W
=
( ) Y(> ) W
d `h U bi G nvk
g
`h
@ V C CS B @
h B
E E C V
∝∝ +
8 + ⋏ ∝
(=3)
wloro E> nhc E= nro tlo offoet`vo dutput
enpne`tnheos (pnrns`t`e nhc oxtorhnb enpne`tnheo) fdr
^> nhc ^=, rospoet`voby.
(Cos`kh Oxngpbo)
Loro, `t `s cos`khoc td kunrnhtoo ZVU frdg fubb bdnc td
=1% bdnc. F`rst, tlo cuty eyebo fdr =1% bdnc edhc`t`dh `s
dmtn`hoc ns:
gnx gnx
A=1%
6
6
( ) => > 3Y W
=
=
6.= =?.= = (< 1.=) 35 >1> > 3Y W
= 311 6.= 311 >1 >11.=<
=
d F d bi
`h `h U
h V V @ B
V hV S C
∝
∝
+∝ ∝ +
7
⋏ ⋏ Û ⋏ Û∝ ∝ +
⋏ ⋏ ⋏ Û7 7
[`tl tlo cuty eyebo nt =1% bdnc edhc`t`dh, tlo
gnx`gug gnkhot`z`hk ̀ hcuetnheo vnbuo td kunrnhtoo
ZVU `s dmtn`hoc my sdbv`hk (=3) fdr Bg ns:
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
>1
gnx
6
>=
6
(> )
==Y (> ) = W
311 (> 1.=<) 1.=< >1 >1
= >?1 >1 < 1.==Y (> 1.=<) 311 = 1.=<W35 >1 6.=
6?3
`h U g
dss d`h
bi
V C CS B
E @ C V C
B h
L ¿
∝
∝
∝
∝2
∝ ∝
⋏ ∝ ⋏ ⋏ Û7
⋏ Û Û∝ ⋏ ∝ ⋏ ⋏Û7
Slo gnkhot`z`hk `hcuetnheo `s cotorg`hoc ns 651¿L.
YUSOQ-6W Cos`kh tlo Srnhsfdrgor
Slo poni gnkhot`z`hk eurroht `s k`voh ns:
gnx
.
(> )(> = )
=
`h U d G pi
g
V C C S @ @ C
B h
∝7 + ∝ (=?)
Cur`hk trnhs`oht dr stnrtup, tlo cuty eyebo enh monbgdst zord. Slus, tlo gnkhot`z`hk eurroht sldubc
mo enbeubntoc fdr tlo wdrst enso df zord cuty eyebo.
Slo g`h`gug turhs fdr tlo trnhsfdrgor pr`gnry s`co
`s k`voh ns:
.g`h
nx
g G pi
p
o
B @ H
N M7 (=6)
wloro No `s tlo edro erdss-soet`dhnb nron `h gg=
nhc Mgnx `s tlo gnx`gug fbux cohs`ty `h Sosbn.
Sloh, elddso tlo prdpor hugmor df turhs fdr tlosoedhcnry s`co tlnt rosubts `h pr`gnry s`co turhs
bnrkor tlnh H pg`h
ns:
`h
p s p H h H H 7 ⋏ 8 (=;)
Slo w`ro c`ngotor sldubc mo eldsoh mnsoc dh tlo
eurroht cohs`ty. Slo eurroht cohs`ty `s typ`enbby
?N/gg= wloh tlo w`ro `s bdhk (8>g). [loh tlo
w`ro `s sldrt w`tl n sgnbb hugmor df turhs, n eurroht
cohs`ty df 6->1 N/gg= `s nbsd neeoptnmbo. Nvd`c
us`hk w`ro w`tl n c`ngotor bnrkor tlnh 1.3 gg td
nvd`c si`h-offoet bdssos ns wobb ns td gnio w`hc`hkons`or. Fdr n l`kl-eurroht dutput, `t `s mottor td uso
pnrnbbob w`hc`hks w`tl gubt`pbo strnhcs df tl`hhor
w`ro td g`h`g`zo tlo si`h offoet.
F`kuro >> sldws tlo pr`gnry-s`co eurroht
wnvofdrgs. Slo rgs vnbuo df tlo pr`gnry s`co
eurroht `s npprdx`gntoby k`voh ns:
= = = =
1 1 5 5 3 3 ; ;( ) ( )(> )
5 5
rgs Q Q Q Q Q Q Q Q p
@ @ @ @ @ @ @ @ @ C C
+ + + +7 + ∝ (=<)
1 .
(> )
=
d `h U Q G nvk
g
@ V C CS @ @
h B
∝7 + ∝ (=0)
5 .
(> )
=
d `h U Q G nvk
g
@ V C CS @ @
h B
∝7 + + (51)
3 .
(> )
=
d `h U Q G nvk
g
@ V C CS @ @
h B
∝7 ∝ + + (5>)
; .
(> )
=
d `h U
Q G nvk g
@ V C CS
@ @ h B
∝
7 ∝ + ∝
(5=)
@Q
@Q;
@Q5
C >-C
@Q1
@Q3
F`k. >>. Qr`gnry-s`co eurroht wnvofdrgs.
Slo rgs eurroht fdr soedhcnry w`hc`hks nro
uhovoh fdr cuty eyebos sgnbbor tlnh ?1%. Sloeurroht `h tlo w`hc`hk fdr c`dco CF> `h hdrgnb
dpornt`dh `s:
>
rgs
CF d @ @ C≏ (55)
Gonhwl`bo, tlo eurroht `h w`hc`hk fdr CF= c`dco
`h hdrgnb dpornt`dh `s:
= (> )rgs
CF d @ @ C≏ ∝ (53)
(Cos`kh Oxngpbo) Fdr wdrst enso zord cuty eyebo
(C71), tlo gnx`gug gnkhot`z`hk eurroht ̀ s:gnx
.
(> )
(> = )=
<(> = 1) >.=0
6.=
`h U d G pi
g
V C C S @
@ C B h
N
∝7 + ∝
7 ∝ ⋏ 7
U`heo tlo pr`gnry-s`co eurroht oqunbs tlo gnkhot`z`hk
eurroht pbus tlo dutput eurroht roforroc td tlo pr`gnry,
gnkhot`z`hk eurroht enh ronel tlo pubso-my-pubso
eurroht b`g`t bovob k`voh my Q[G edhtrdbbor cur`hk
trnhs`oht. Sloro sldubc mo ohdukl gnrk`h `h
trnhsfdrgor cos`kh, ldwovor, Mgnx71.>?S `s usoc td
kunrnhtoo hdh-snturnt`dh df tlo trnhsfdrgor cur`hk
trnhs`oht (OO]5?3=, No7>10gg=). Slo gnx`gug fbux
cohs`ty `h tlo wdrst enso seohnr`d w`bb mo eloeioc `hUSOQ-< nftor eurroht b`g`t bovob ̀ s cotorg`hoc. Slo
g`h`gug hugmor df turhs fdr tlo pr`gnry w`hc`hk `s
k`voh ns:6
.g`h
6
gnx
651 >1 >.=030.;
>10 >1 1.>?
g G pi
p
o
B @ turhs
N M
∝
∝
Û ⋏7 7 7
Û ⋏
Slo pr`gnry nhc soedhcnry-s`co turhs nro cotorg`hoc
ns ?1 turhs nhc < turhs, rospoet`voby.
Slo rgs vnbuo df tlo pr`gnry-s`co eurroht `s k`voh my:
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
>>
6
1 6
< 311 (> 1.53) 1.53 >1 >11.3> 1.00
6.= = 651 >1 Q
@ N∝
∝
⋏ ∝ ⋏ ⋏ Û7 + ∝ 7
⋏ Û
6
5 6
< 311 (> 1.53) 1.53 >1 >11.3> =.3=
6.= = 651 >1 Q @ N
∝
∝
⋏ ∝ ⋏ ⋏ Û7 + + 7
⋏ Û
6
3 6
< 311 (> 1.53) 1.53 >1 >11.3> 1.>;
6.= = 651 >1 Q
@ N∝
∝
⋏ ∝ ⋏ ⋏ Û7 ∝ + + 7 ∝
⋏ Û
6
; 6
< 311 (> 1.53) 1.53 >1 >11.3> >.?0
6.= = 651 >1
Q @ N∝
∝
⋏ ∝ ⋏ ⋏ Û7 ∝ + ∝ 7 ∝
⋏ Û
>.=0rgs
p @ N7
Slo eurroht `h tlo w`hc`hk fdr CF> c`dco `h hdrgnb
dpornt`dh `s
> < 1.53 3.;rgs
CF d @ @ C N≏ 7 ⋏ 7
Gonhwl`bo, tlo eurroht `h tlo w`hc`hk fdr CF= c`dco
`h hdrgnb dpornt`dh `s:
= (> ) < (> 1.53) 6.?rgs
CF d @ @ C N≏ ∝ 7 ⋏ ∝ 7
Fdr tlo pr`gnry w`hc`hk, 1.>=φÛ51 (B`tz w`ro) `s
soboetoc nhc tlo eurroht cohs`ty `s 5.<N/gg=. @h tlo
enso df tlo soedhcnry w`hc`hk, 1.>φÛ>11 (B`tz w`ro)
`s soboetoc nhc tlo eurroht cohs`t`os nro ?.0N/gg
=
nhc<.=N/gg= fdr CF> nhc CF=, rospoet`voby.
YUSOQ-;W CE Mbdei`hk Enpne`tdr Uoboet`dh
Ovoh tldukl `t lns mooh nssugoc tlnt tlo CE
mbdei`hk enpne`tdr EM `s bnrko ohdukl td hokboet tlo
vdbtnko r`ppbo nerdss EM, tdd bnrko n enpne`tdr
rosubts `h sbdw cyhng`e rospdhso. Slus, `t `s typ`enb
td soboet tlo enpne`tdr sd tlnt tlo r`ppbo `s ?~>1% df
tlo `hput vdbtnko. Slo vdbtnko r`ppbo df CE mbdei`hk
enpne`tdr `s k`voh ns:
1
> >Y (> = )W
U CS
d dEM p
M M
@ @ V ` ct C
E E h h∆ ≏ 7 + ∝∩ (5?)
(Cos`kh Oxngpbo)
Sd sot tlo vdbtnko r`ppbo mobdw 51V, tlo g`h`gug
CE mbdei`hk enpne`tdr vnbuo `s dmtn`hoc frdg
1
> >Y (> = )W 51
U CS
d d p hdg
M M
@ @ ` ct C V
E E h h7 + ∝ 2∩
6
>Y (> = )W
51
> < <Y (> = 1.53)W 1.53 >1 >1 >05
51 6.= 6.=
d d M hdg hdg U
M
@ @ E C C S
V h h
E hF V
∝
8 + ∝
8 + ∝ ⋏ ⋏ ⋏ Û 7
Slo vdbtnko df EM `s k`voh ns V@HÛC my (=) nhc
tlo gnx`gug enpne`tdr vdbtnko `s tlodrot`enbby
lnbf df tlo `hput vdbtnko. Ldwovor, `t enh ronel tlo
`hput vdbtnko wl`bo tlo edhvortor `s hdt dpornt`hk
`f tlo boninko eurrohts df tlo GDUFOSs nro
uhmnbnheoc. Slus, vdbtnko rnt`hk sldubc mo l`klor
tlnh gnx`gug `hput vdbtnko (311V).
Nhdtlor `gpdrtnht fnetdr td edhs`cor wloh
soboet`hk tlo enpne`tdr `s tlo eurroht rnt`hk. U`heo
tlo pr`gnry-s`co eurroht df >.=;N enbeubntoc `h
USOQ-6 fbdws tlrdukl tlo enpne`tdr, n bdw-OU]
enpne`tdr sldubc mo eldsoh. Slus, n
==1hF/=;?VNE f`bg enpne`tdr `s soboetoc.
YUSOQ-<W Eurroht Uohs`hk
Sd cotorg`ho tlo pubso-my-pubso eurroht b`g`t
bovob, tlo poni eurroht `h tlo pr`gnry s`co sldubc mo
dmtn`hoc f`rst ns:
(> )(= = )
=
pi d `h hdg hdg U p hdg
g
@ V C C S @ C
h B
∝7 ∝ + (56)
(Cos`kh Oxngpbo)
Slo poni eurroht `s dmtn`hoc ns:
6
6
(> )(= = )
=
< 311 1.53 (> 1.53) >1 >1(= = 1.53)
6.= = 651 >1
=.3>
pi d `h hdg hdg U p hdg
g
@ V C C S @ C
h B
N
∝
∝
∝7 ∝ +
⋏ ⋏ ∝ ⋏ Û7 ∝ ⋏ +
⋏ Û7
Eurroht b`g`t bovob `s cotorg`hoc ns 5N, us`hk
1.=Π ros`stdr s`heo tlo eurroht b`g`t tlrosldbc
vdbtnko `s 1.6V.
Ns goht`dhoc `h USOQ-6, tlo trnhsfdrgor
gnkhot`z`hk eurroht enh ronel tlo pubso-my-pubsoeurroht b`g`t bovob k`voh my tlo Q[G edhtrdbbor
cur`hk trnhs`oht. Slorofdro, tlo gnx`gug fbux
cohs`ty df trnhsfdrgor sldubc mo eloeioc ns.6
gnx g`h 6
651 5>11.5?
>10 >1 ?1
wdrst g B@G
o p
B @ M S
N H
∝
∝
⋏Û7 7 7
Û ⋏
YUSOQ-0W C`dco Uoboet`dh
Slo vdbtnko stross fdr onel roet`f`or c`dco (CF>
nhc CF=) `s k`voh ns:
>
= CF `hV V C
h7 ⋏ ⋏ (5;)
=
=(> ) CF `hV V C
h7 ⋏ ∝ ⋏ (5<)
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
>=
(Cos`kh Oxngpbo)
Edhs`cor`hk n wdrst enso df C7?1% fdr CF>:
>
=311 1.? 63
6.= CF V V 7 ⋏ ⋏ 7
Edhs`cor`hk n wdrst enso df C71% fdr CF=:
=
=311 > >=<
6.= CF
V V 7 ⋏ ⋏ 7
Edhs`cor`hk vdbtnko dvorslddt nhc r`hk`hk, >11V
nhc =11V c`dcos nro soboetoc fdr CF> nhc CF=,
rospoet`voby. Slo eurroht fdr onel c`dco dmtn`hoc `h
USOQ-6 `s:
>< 1.53 3.;rgs
CF d @ @ C N≏ 7 ⋏ 7
=(> ) < (> 1.53) 6.?rgs
CF d @ @ C N≏ ∝ 7 ⋏ ∝ 7
V. COU@KH S@QU
N.
_hmnbnheoc Uoedhcnry Surhs Srnhsfdrgor[loh n eohtor-tnppoc w`hc`hk `s usoc `h tlo
trnhsfdrgor soedhcnry w`hc`hk, uhmnbnheoc turhs
enh mo usoc td dpt`g`zo tlo trnhsfdrgor cos`kh, ns
sldwh `h F`kuro >=. Slo gnkhot`z`hk eurroht df tlo
trnhsfdrgor `s sl`ftoc, ns sldwh `h F`kuro >5, my
us`hk n trnhsfdrgor w`tl uhmnbnheoc soedhcnry
turhs. Slo nvornko vnbuo df tlo gnkhot`z`hk eurroht,
dr tlo CE dffsot df tlo gnkhot`z`hk eurroht, `s k`voh
ns:
= >. (> )U U
G nvk d d
p p
H H @ @ C @ C H H
7 ∝ ∝ (50)
Sl`s moedgos zord ospoe`nbby wloh:
=
> =
U
U U
H C
H H 7
+ (31)
Slus, tlo gnkhot`z`hk eurroht CE dffsot fdr sdgo
dpornt`hk pd`ht enh mo gnco v`rtunbby zord us`hk
uhmnbnheoc turhs. Ldwovor, tl`s `heronsos eurroht
r`ppbo `h tlo dutput `hcuetdr, wl`el rosubts `h n
bnrkor `hcuetdr dh tlo soedhcnry s`co. Slo
uhmnbnheoc trnhsfdrgor sldubc mo cos`khoc us`hk
trnco-dffs motwooh tlo trnhsfdrgor s`zo nhc tlo
`hcuetdr s`zo. Hdt`eo tlnt tlo c`dco vdbtnko stross
rogn`hs uhelnhkoc, ovoh wloh uhmnbnheoc turhs
nro usoc.
+
VD
-
] d
=̂
>̂
Hp @ p
Bbip
Bg
EM
@ cs >
@ g
@ BD
V ̀h
@ d
+
Vc
-
VEM
+
V pr
-
+
Vroe
-
Hs>
Hs=
C>
C=
>-C
C
F`k. >=. Srnhsfdrgor w`tl uhmnbnheoc soedhcnry turhs.
>U BD
p
H @
H
=U BD
p
H @
H > =U U H 8
1.? C 2
F`k. >5. Ul`ft df gnkhot`z`hk eurroht my n trnhsfdrgor w`tl uhmnbnheoc
soedhcnry turhs.
M. Oxtorhnb Udft-Utnrt
Ovoh tldukl tlo FUFN-sor`os lns `htorhnb sdft-
stnrt, eurroht dvorslddt enh deeur cur`hk stnrtup
wloh tlo boninko `hcuetnheo `s vory sgnbb, ns
cop`etoc `h F`kuro >3.
[loh ^> `s f`rst turhoc dh, tlo pr`gnry-s`co
vdbtnko `s dhby `gpdsoc dh tlo boninko `hcuetnheo
uht`b tlo soedhcnry-s`co roet`f`or c`dco `s turhoc dh.
Sl`s `hcueos l`kl c`/ct eurroht `h tlo pr`gnry s`conhc tlo eurroht enh oxeooc tlo pubso-my-pubso
eurroht b`g`t bovob cur`hk tlo bonc`hk-ocko mbnhi`hk
(BOM) t`go nhc turh-dff cobny t`go. Sd rocueo tlo
eurroht dvorslddt, QFG (pubso-froquohey-
gdcubnt`dh) sdft-stnrt enh mo nccoc td tlo `htorhnb
Q[G sdft-stnrt us`hk nh oxtorhnb enpne`tdr nhc
ros`stdr dh tlo ]S p`h, ns sldwh `h F`kuro >3.
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
>5
(> )`h
bi
V Cc`
ct B
∝
7
(> )`h
bi
V Cc`
ct B
∝7
F`k. >3. Udft-stnrt wnvofdrgs edgpnr`sdh.
V@. O\QO]@GOHSNB VO]@F@ENS@DH
Sd sldw tlo vnb`c`ty df tlo cos`kh prdeocuro
prosohtoc `h tl`s pnpor, tlo edhvortor df tlo cos`kh
oxngpbo wns mu`bt nhc tostoc. Nbb e`reu`t
edgpdhohts nro usoc ns cos`khoc. F`kuro >? sldws
tlo f`hnb selognt`e df tlo nsyggotr`e Q[G lnbf-
mr`cko edhvortor cos`kh oxngpbo. U`heo tlo boninko`hcuetnheo `s robnt`voby sgnbb, froquohey gdcubntoc
sdft-stnrt `s `hedrpdrntoc w`tl tlo dr`k`hnb Q[G
sdft-stnrt td rocueo tlo eurroht dvorslddt. F`kuro >6
sldws tlo trnhsfdrgor strueturo. Sd dmtn`h tlo
cos`kh boninko `hcuetnheo vnbuo, ocko w`hc`hk
rntlor tlnh eohtor w`hc`hk `s usoc fdr tlo soedhcnry-
s`co turhs.
F`kuro >; sldws tlo crn`h-td-sdureo vdbtnko df ^>
nhc tlo trnhsfdrgor pr`gnry-s`co eurroht (@ p)
wnvofdrgs nt fubb bdnc edhc`t`dh. Frdg tlo crn`h-td-
sdureo vdbtnko df ^>, tlo cuty eyebo fdr ^> `s nmdut
53% ns cos`khoc `h USOQ-5. F`kuro >< sldws tlo
crn`h-td-sdureo vdbtnko df ^> nhc trnhsfdrgor
pr`gnry-s`co eurroht (@ p) wnvofdrgs nt hd-bdnc
edhc`t`dh. Ns dmsorvoc, tlo edhvortor dporntos `h
murst gdco cuo td tlo mu`bt-`h murst gdco dpornt`dh
df tlo FUFN=>11. Slo pdwor edhsugpt`dh nt hd
bdnc `s 1.5[, `hebuc`hk tlo pdwor edhsugpt`dh df
tlo edhtrdbbor. F`kuro >0 sldws tlo crn`h-td-sdureo
vdbtnko nhc eurroht wnvofdrgs df ^> nt fubb-bdnc
edhc`t`dh. Slo crn`h vdbtnko crdps td zord nhc tlo
mdcy c`dco edhcuets mofdro tlo sw`tel `s turhoc dh.
Slo poni vnbuo df tlo crn`h eurroht `s nmdut =.3N, ns
cos`khoc. F`kuro =1 sldws tlo crn`h-td-sdureo
vdbtnko df ^> nhc vdbtnko nhc eurroht df tlo roet`f`or
c`dco C=>>. Ns dmsorvoc, tlo edhcuet`dh curnt`dh
df tlo c`dco C=>> `s boss tlnh lnbf df tlo sw`tel`hkeyebo. F`kuro => sldws tlo crn`h-td-sdureo vdbtnko
df ^> nhc vdbtnko nhc eurroht df roet`f`or c`dco
C=>=. Ns dmsorvoc, tlo edhcuet`dh curnt`dh df tlo
c`dco C=>= `s gdro tlnh lnbf df tlo sw`tel`hk eyebo.
Ns dmsorvoc frdg F`kuro =1 nhc =>, tlo vdbtnko
strossos df tloso c`dcos nro uhovoh. Slus, n =11V
rntoc c`dco `s roqu`roc fdr C=>= td lnhcbo tlo
vdbtnko dvorslddt wl`bo n >11V rntoc c`dco `s usoc
fdr C=>>. Slnhis td tlo rocueoc c`/ct my tlo
boninko `hcuetnheo, tloro `s nbgdst hd rovorso
roedvory, ovoh tldukl tlo =11V ubtrn-fnst c`dco `susoc. F`kuro == sldws tlo sdft-stnrt wnvofdrgs.
U`heo tlo boninko `hcuetnheo `s robnt`voby sgnbb,
froquohey gdcubntoc sdft-stnrt `s `hedrpdrntoc w`tl
tlo dr`k`hnb Q[G sdft-stnrt td rocueo tlo eurroht
dvorslddt. F`kuro =5 sldws tlo gonsuroc off`e`ohey
fdr c`fforoht bdnc edhc`t`dhs. Slo off`e`ohey nt fubb-
bdnc edhc`t`dh `s nmdut 05%.
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
>3
F`k. >?. Uelognt`e df nh nsyggotr`e Q[G lnbf-mr`cko edhvortor.
- Edro: OO]5?3= (No7>10 gg=)
- Mdmm`h: OO]5?3= (Ldr`zdhtnb)
F`k. >6. Srnhsfdrgor strueturo.
Q`h(U ↝ F) [`ro Surhs [`hc`hk Gotldc
Hp > ↝ < 1.>=φÛ51 (B`tz w`ro) ?1 Udbohd`c [`hc`hk
Hs> >6 ↝ >5 1.>φÛ>11 (B`tz w`ro) < Udbohd`c [`hc`hk
Hs= >= ↝ 0 1.>φÛ>11 (B`tz w`ro) < Udbohd`c [`hc`hk
Q`h Upoe. ]ognri
@hcuetnheo >'< 651¿L ¾ ?% >11iLz, >V
Boninko >'< 3?¿L ¾ >1%Uldrt dho df tlo soedhcnry
w`hc`hks.
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
>?
F`k. >;. Dpornt`dh wnvofdrgs nt fubb bdnc edhc`t`dh. F`k. ><. Dpornt`dh wnvofdrgs nt hd bdnc edhc`t`dh.
F`k. >0. Zord vdbtnko sw`tel`hk wnvofdrgs nt fubb bdnc edhc`t`dh. F`k. =1. ]oet`f`or c`dco (C=>>) wnvofdrgs nt fubb bdnc.
F`k. =>. ]oet`f`or c`dco (C=>=) wnvofdrgs nt fubb bdnc. F`k. ==. Udft-stnrt wnvofdrgs nt fubb bdnc.
<?
<6
<;
<<
<0
01
0>
0=
05
03
0?
1 =? ?1 ;? >11 >=? >?1 >;? =11Qd ([)
O f f ( %
)
F`k. =5. Gonsuroc off`e`ohey.
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FN@]EL@BC UOG@EDHC_SD] QD[O] UOG@HN] =11< - =110
>6
V@@. EDHEB_U@DH
Sl`s pnpor prosohts cos`kh edhs`cornt`dhs df nh
nsyggotr`e Q[G lnbf-mr`cko edhvortor. Slo prnet`enb
cos`kh prdeocuro ohcs w`tl n cos`kh oxngpbo df n
=3V/<N (>0=[) nsyggotr`e lnbf-mr`cko edhvortor,wl`el vnb`cntos tlo c`seuss`dh.
V@@@. ]OFO]OHEOU
Y>W ]dmort B. Uto`korwnbc, ‒N Edgpnr`sdh df Lnbf-mr`cko ]osdhnht
Edhvortor Sdpdbdk`os,‗ @OOO Srnhsnet`dhs dh Qdwor Oboetrdh`es, Vdb. 5,
Hd. =, Npr`b >0<<.
Y=W N. F. [`tubsi` nhc ]. [. Or`eisdh, ‒Cos`kh df tlo Uor`os ]osdhnht
Edhvortor fdr G`h`gug Utross,‗ @OOO Srnhsnet`dhs dh Nordsp. Oboetrdh.
Uyst., Vdb. NOU-==, pp. 5?6-565, Juby >0<6.Y5W ]. Druknht`, J. Pnhk, nhc F.E. Boo, ‒@gpbogohtnt`dh df Dpt`gnb
Srnjoetdry Edhtrdb df Uor`os ]osdhnht Edhvortors,‗ Qrde. @OOOQOUE ‐<;, >0<;.
Y3W V. Vdrpor`nh nhc U. Eui, ‒N Edgpboto CE Nhnbys`s df tlo Uor`os
]osdhnht Edhvortor,‗ Qrde. @OOO QOUE‐<=, >0<=.Y?W ]ngosl Druknht`, Qlun Eloo Johk, Joffroy S. I. Kunh nhc B̀ ow N.
Eldy, ‒Udft-sw`tel`hk CE/CE Edhvortor w`tl Q[G Edhtrdb‗, @OOO
Srnhsnet`dhs dh Qdwor Oboetrdh`es, Vdb. >5, Hd. >, pp. >1=->>3, Jnh.
>00<.
Lnhksodi Eld` roeo`voc M.U., G.U.,
nhc Ql.C cokroos `h oboetr`enb
ohk`hoor`hk frdg Uodub Hnt`dhnb
_h`vors`ty, `h >006, >000, nhc =11=,
rospoet`voby. Frdg =11= td =11;, lowdrioc fdr Fn`rel`bc Uog`edhcuetdr `h
Idron ns n systog nhc nppb`ent`dh ohk`hoor. Lo `s eurrohtby
wdri`hk fdr Fn`rel`bc Uog`edhcuetdr `h Mocfdrc, HL, _UN, ns
n pr`he`pnb systog nhc nppb`ent`dh ohk`hoor. L`s rosonrel
`htorosts `hebuco sdft-sw`tel`hk toelh`quos, gdcob`hk, nhc
edhtrdb df edhvortors. Lo lns pumb`sloc >? pnpors `h @OOO
edhforoheos nhc trnhsnet`dhs nhc >= nppb`ent`dh hdtos fdr
Fn`rel`bc Uog`edhcuetdr.