Fa-'
- E1fi41le l Dec 203C $e,rn-l-larn '1.\ttl,ll .1.2 Sunuurn ol ltr:rults lir ltrctangrrlar \\arrtuirie
$eatNo:1?
Ve*ue: E5-O5-e0le:. {.ltrrrttt t t-i TE,,,,, MOdC TMr,, Mode
1l\Bl-U -1.1 \ilil|rnNri {rl 11.'illl\ li)r l}lixlltl I'lrlr \\r\r{uklt a^1Fi aJue
Qilantity 1 i \1 \i,rLir I \1. \ir,rl. I lr,, \|rlc kt il*;l- +{*W J{*;kF;A;ttt?
P-tt rl rl
2z 2n
l'.
tr tr.
2n )t
, I t' r't ll fi i
t)
a( R,lnd zkRslpqd 2k? R"l kpn,t
lJ ;i
I E: 0 ,4 sm(nxy/d1s*i0z 0 udilieleUW'rL t2 ran d Prana
Itt, 0 0 Bcos(nty/d\e-lfi, owtnmmwt ,p ,p
mftx nl(v
(jap/k)E sin(nrr-ldle*i\z E,O E- Srn rP'
I r, o 0 '....,'..._ Stn
ab -e
I t, ev,ldJe-iqz \-jfilk)Acos(rny/dte-i0t 0 mfrx nfiv
H1 cos JP'
f r.
I
Vol\dre-J\: (joe/kslAcwlnty/d1e'ifz 0 ,{ cos
i'
e- 0
lr, o o (iflkc)&sin(nry/dle-ia, -
jopnt , mfrx nry _iu-
sln ie rr -iBmr cos frfrx
_4aB nrv
'::::: sin.::!_e-Jb:
Y z Zpl = ltt / )t/ Zyy = pa/ k Z1B = k4lfl E3 --;-l
kibah
cos
klaob
-
F - iaunn A sin mrr
__:___l_ nrv
_ cde:?'Jpz -i0nn B *in mrx
--4lljl
ilrv iP:
'::::- 6s]:!-e-
Stripline kioah klb a
, WV0'0 0t jfimr mrx nr! *iB- iwnn
*1"" max
b
nrv
,ry'r -:-l stn ms rF' ' 8 sin ""'" cos':::!-e-Jfr:
kloab -e klbdh
-
j!{! n"or*n' immr mrx nrv
Hr ,in'o,}' - cos !)l):) r;rnJ)!-"-i$z
o-)fr" -#f
trbat) ktaab
J zrc=
kn
7 zw: iBn
Itr (3.176),. =3x ld dw is ttr spaed of liglu i! fre{Fs. Urhg (2-ll) trd (2.16)
allNs !s to wrile d1e cheaddistic impedarcc ofu ffiimksion line as
eherc I atrd C e tte irducr@c ard c@rib4e p.r uit ldgfl of fE tre- Th6, w From (3,30) the dielectric attenuation is
cu fird Zq if we k@' C, A! menti@ed prryisly, Laple's equatiou @a be wlved by
onfomd mapping to nnd $e capscitue per uit leagrtr ofiF bd lhc cuhing * tand (310.6)(0.001)
solutioo irvolvo ooplieied ipe.id irrciiG [6], r for prctical corsutdioB simpb ad - --; * = 0.155 Np/m.
fomhs harc bef, dewloped by cwe fitritrg to dte csct slutis [6, fl- The rcultirg
fmul, for ch@tsisric impedse is
--f
The surface resistarrce of copper at 10 6Hz is R, = 0.026 O, Then from {3.181)
the conductor attenuatioo is
(3.179a)
2.7 x 10-3 R"e,ZoA
". = = 0.122 Np/m.
30"6
whqe Fr is the elrb.rile x,,ly'/r of thc cflter cgndEior giwn by
since ,1, = 4.74. The tolal attfiuatior cofftant i$
a : ud * uc: 0.27? Np/m.
1l. I rgbr
In dB,
Thw fomulas awmc a srip with zw thickrc ard m quotfd s beiBg a$Wte to
abod t% ofthe ffi Hults. It is sm frm (3.179) tbar &e chracteristic impedance
n{dB) = 20logea * 2.41 dB/m.
dereus a: Ihe srip width R imltm.
When designhglcimitr one u*ally needs to tird the saip widrh, giq the
At l0 CHz, the wavelengh oa ttreQlis
chaBctsisti( impedmce (and h€ight, ard relBtiK peminivity €,), wtich rcquis the
iwrc of the li)rolss i, (3.179). Such fomulq hrc beo dcrived q
(l.l80a) so in terms ofwavelength the attenuetion is
o(dB) : (2.41X0.0202) = 0.O49 dB/t. I
-l (l.l80b)
, *J
AXAMPLE 3.7 MICROSTRIP LINE DESIGN
Design a microskip line on a 0.5 mm alumina substrate (6r : 9.9, tan 6 : 0.001)
Micro-strip line for a 50 Sl characteristic impedaflce. Find the length of this line required
to produce a phase delay of 27fi" at 10 GHa and compute the total loss oa this
lbe, assuming copper conductors" Compare the results obtained from the approx-
imate formulas of (3.195) -(3.199) with those {iom a microwave CAD package.
Solutian
First find W/ d fot Zs : 50 f,2, and'initially guess th^t W/ d < 2. From (3. I 97),
,ii I t.l ilr, r,''llr lrr.... ll!,t iit\a .,i,lt),' , t :il... A:2.142. Wd = 0.9654.
field lines are in the dielecric region and some are in air, the effective dielectric canstant
satisfies the relation
So the ccndition that W/d < 2 is satisfied; otherwise we would use the expression
tor W/d > 2. Then the required line width is W :0.9654d:0.483 mm. From
l<1"<er (3.195) the effective dieleetric constant is en - 6.665. The line length, €, far a
270" ptrase shift is found as
afld depends on the substrale dielectric coflstalrl, thc substrate thickness, the conduclor
width, and the frequenry. o :270" : il: JekoL
Wc rvill present approximate design formulas for the eftbctive dielectdc constant, charac-
teristic irnpedance, and atteruation ofmicnrstrip linq these resul8 are curve-tit approximations k11 : "" J : 209.4 m-t .
to rigorous quasi-static solutions [8, 9]. Then we wili discuss additional aspects ofmicrosrip c
lincs, including Aequencydqrendent effects, higher order modes, and parasitic effects. 270" (n / 180"r
t : ----+:--::8.72 mm.
Je"ko
Formulas for Effective Dielectric Canstant, CharacteriEtia
lmpedance, and Attenuatlon
The cffective dielectric constanl ofa microstrip line is given approximately by
il 1!,t,
The effective dielectric constant can be interpreted as the dielectric constant ofa homo-
geneous medium that equivalently rEplaces the air and dielectric regions ofthe rnicrostrip
line. as shown in Figure 3.26, The phase velocity and propagation constant are then givetr
by (3.193) and (3.194).
Given the dimensions oftle microstrip line, the characteristic impedance can be cal-
culated as
For a given characteristic impedance Zo and dielectric constant 6/, the Wld ratio cen be
fouad as
(3.197)
Cavity Resonators
ftund=tx. t=1.2.3. (6.18)
tr,!rilrifr,,lr.:rlr1Lr..NitrIrrr1l,,.n,rr,r,,rrnfiritL!fl:rt.rji:!,,al.n.,r.i.NglhlL'r:l
rl 1if r.\.rlrrn tl1.rn;. \r ir,,,iIi ll s\rhriirrs rr. ir,::lNi. Ii)i (,1h.r l.ir!tlF ,)! L,,l
'\ ri(nrirn..rir.irurrrlliiirI|,..li]rtgi,i.|.r:iI]rritri,.,i.titrr(i,r.
;l
Iirtr rr tr:L :.t:r , 1lr! Tl ,) \l rr$,|.in1 rltlc,, ilrr (r\rr! x,rJi.r ttrr rr
i.r,,r,' hir.,rr'rrrni[]i.iri,,r.ni,,r..iirn,'n.nnif!i,.,rfi.r,.i,rrf,l,., ,.
ii,..1x$'rrfi.t\.lr i1,.tr:,!iin, r,,rri,...iiirll',-.,,'rll1..,,,rr,.iri.!,r,rrl,.
If, < a < ./, the doflinant rson&t ftode (lowcst re$nmt frequency) will be rhc TEror
,rode, corespondiflg 10 tbe TElo dominant wareguide mode in a shoned guide oflcngrh
ls/2, atd is similar to the short-circuilcd i/2 transmision line rsonatoi The dominant
TM rsonant modc i! tlc lMlto modc.
, S-Parameters
We can fiod S11 by applying an incident waye al pon l, [ir, and measuring
the oulcoming wave pott 2, V;" . This is equivalcnt to thc tmosmission cocffi-
"t 2:
cied 1'rom port I to port
Frcnr the frct that 5l r * Sz: = 0, we know thar I/i = 0 when port 2 is t€rmiraled
in Zo = 50 Q, md tbat /r+ = 0. In this w w€ have that llf = \ ad Y; =
I/2. By applyinS a voltage ll
at port I and using voltage division twice we find
(4.40) I/t = ,a tr the voltage acmss the 50 Q load rsi$tor at port 2:
\ .Irrrllr ir -Ira!i Il lll, \crr(latillt Illlt r\ ri)ti !rc LirlrtilrrreLl lrs
v; =v2= - ("#*=)("*=) =oio1v,
(4,41) 4l.il4 = 141.8(58.56)/(141.8 + 58.56) is the re,sistffie of the parallel com-
where
bimtim of the 50 Q load md the 8.56 Q resistor wirh the 141.8 Q csistor. Tbus,
512*$21 =$,'/Ql"
In.rvords,t4.4l)saysrhatSi, isfoundbydrivingport.ywilhaniflcidentwaveofvoltage
lf the input power is 1V{12/22a, then rhe ourpur Wwr h ly;12/220=
tr',* and measuring thr'reflected wave amplitude {- coming out of port i. The incident ls2tvfl2 /2zo : ll2tf l2zolvll2 = lvl P /42o, which is onc-h&tfi -3 dB) of
the input powcr
\", rtL,n
'll,^r l), \ liir f.itl1(tirt! :|,tlt:. Lirjr l,; ,1,..1u:t,tltt.l li,,rtr ilr,: 1l rLrt ) ii
rrtrtlrtr ,rr.l irr'r'\\t:Li. i ir-,i r..rtIt.t il.,\lLilti.litilt tili rl)irt.r!lfJlrit( il t1,r,l.rrtr:i: :t , .,1
.rll tirrlilrrs.rrrrrit:rtirr,l {iili.,rf.rr:ltr,,tjri i i,rr..It.,irrl tltr|lr,itr,,Lrr.r.:tr:rtr,,l,l
matched loads. .CrllctlllLl,.rtttttili.t'.rIirctr lir:rilii.Irf:l.r'\rr']It{r!11,. i l.,,tit,;llr:ii,:t,:t.ll
,....,.,
C/
)- rull,Dlr{.4 EvALUArroNor-pARAMETERs
' rrnd ilre f parameters of the 3 dB attenuator circuit sho*n in Figure 4.8.
Solution
From (4,41), Srr can be found as the reflection cocfrcient seen at port I when
port 2 is terminated in a matched load (Zs 50 O): -
v-t z!)t - zol
t" =
tln-=o=
F(r)1''.=o =
ffil',**''
Pon \ L Potl
1 '-v2 'v- 2
SiElna,l FiowChart r Rule 1 (Series Rule). Tr.o branches, whose cormon node hm only one incoming
and one outgoing wave (branches in series). may be combined to fom a silgle
branch whose coefficient is the product ofthe coefficients olthe original branches.
'!:, t.: s:r5:z
c_+-_'tr! > _____. +> H Figure 4.16a shows the flow paphs for this rule. It$ derivation follows from the
t, i. t, r\ v3 basic relation
I.r - .!.r1 I'1 - .!::.5:i I r. (4.?s)
(a)
. [{ulc2lPlraileli{rritt. l\rohranchestiorrollrctrnnrirrrrioditoanothcrconrrrl,rr
notlc lbralcltts irr Iarallt:L) ntrr, br, ronbirtr-tj inio.r.rirtlc brarch u,hosc coelticii.nt
,tn is thc surn ol tht cotlircirntr oi the orLginrl Lrr anchcs. l-igu.e I iih shorr s thc tlol
..:1.
So +So uraphs ii:rthis rulc. Thr ilrrir;rtiorr tLrllorr. lionr tlu ohliLrris rciatiirr
=N ffi Ii - 5ri't +.\ii-l - (.1, +Si,rt'r .
Vr Y2
t4.761
r Rule-1 tSril LoLpl{Lrle).\\:hcrtrnodtlrirrrselfloollirhiiurrirtlruTbcgiitsrnri cntls
\, or) lhc:anrr'irLrile)ol coelllcicrrt.1. Llrr scli:loop rrr: be ciilinatcri L.r uruitillrjng
coclhcttlls ol llrt bumcitcs lctiliqg tltlt no.lc bli 1,ii I - .tt. irigure -:1. L6c rhotrs tlic
{b) ilur urephs lirr this ulc. rr hich cair be dcrir erl ts lirlloq r. []r orn tlrt originrl rretr Lrrl.i
r
\,.' ltar c
s:z l.-.n,1;i\::l:. {4.71a)
Jr = \r:l:. (4.77b)
s:r
-'*'Qp " =>
I"I Y2 k'3 ltt
1* 5n
y2
s3?
v-1
Elirninatirg l'2 gives
(4.78)
(c) which is seen to be the transfer function for the reduced gmph ofFigure 4.t6c.
. RBle 4 (Splilti[g Rule). A node may be split into two seprate rodes as long as the
.s,. tr4 s:r t/i s+: t'4 resulring flow graph confains, once and only once, each combination of separate
+ (not self-lcops) input and ouqrut branches that coflnect to the originai node. This
nJIe is illusirated it Figure 4.16d and lollows from the obseryatioil that
yl t, ,,\ (4.79)
in both the original ffow graph md the flow gmph widr the split node.
- E1fi41le l Dec 203C $e,rn-l-larn '1.\ttl,ll .1.2 Sunuurn ol ltr:rults lir ltrctangrrlar \\arrtuirie
$eatNo:1?
Ve*ue: E5-O5-e0le:. {.ltrrrttt t t-i TE,,,,, MOdC TMr,, Mode
1l\Bl-U -1.1 \ilil|rnNri {rl 11.'illl\ li)r l}lixlltl I'lrlr \\r\r{uklt a^1Fi aJue
Qilantity 1 i \1 \i,rLir I \1. \ir,rl. I lr,, \|rlc kt il*;l- +{*W J{*;kF;A;ttt?
P-tt rl rl
2z 2n
l'.
tr tr.
2n )t
, I t' r't ll fi i
t)
a( R,lnd zkRslpqd 2k? R"l kpn,t
lJ ;i
I E: 0 ,4 sm(nxy/d1s*i0z 0 udilieleUW'rL t2 ran d Prana
Itt, 0 0 Bcos(nty/d\e-lfi, owtnmmwt ,p ,p
mftx nl(v
(jap/k)E sin(nrr-ldle*i\z E,O E- Srn rP'
I r, o 0 '....,'..._ Stn
ab -e
I t, ev,ldJe-iqz \-jfilk)Acos(rny/dte-i0t 0 mfrx nfiv
H1 cos JP'
f r.
I
Vol\dre-J\: (joe/kslAcwlnty/d1e'ifz 0 ,{ cos
i'
e- 0
lr, o o (iflkc)&sin(nry/dle-ia, -
jopnt , mfrx nry _iu-
sln ie rr -iBmr cos frfrx
_4aB nrv
'::::: sin.::!_e-Jb:
Y z Zpl = ltt / )t/ Zyy = pa/ k Z1B = k4lfl E3 --;-l
kibah
cos
klaob
-
F - iaunn A sin mrr
__:___l_ nrv
_ cde:?'Jpz -i0nn B *in mrx
--4lljl
ilrv iP:
'::::- 6s]:!-e-
Stripline kioah klb a
, WV0'0 0t jfimr mrx nr! *iB- iwnn
*1"" max
b
nrv
,ry'r -:-l stn ms rF' ' 8 sin ""'" cos':::!-e-Jfr:
kloab -e klbdh
-
j!{! n"or*n' immr mrx nrv
Hr ,in'o,}' - cos !)l):) r;rnJ)!-"-i$z
o-)fr" -#f
trbat) ktaab
J zrc=
kn
7 zw: iBn
Itr (3.176),. =3x ld dw is ttr spaed of liglu i! fre{Fs. Urhg (2-ll) trd (2.16)
allNs !s to wrile d1e cheaddistic impedarcc ofu ffiimksion line as
eherc I atrd C e tte irducr@c ard c@rib4e p.r uit ldgfl of fE tre- Th6, w From (3,30) the dielectric attenuation is
cu fird Zq if we k@' C, A! menti@ed prryisly, Laple's equatiou @a be wlved by
onfomd mapping to nnd $e capscitue per uit leagrtr ofiF bd lhc cuhing * tand (310.6)(0.001)
solutioo irvolvo ooplieied ipe.id irrciiG [6], r for prctical corsutdioB simpb ad - --; * = 0.155 Np/m.
fomhs harc bef, dewloped by cwe fitritrg to dte csct slutis [6, fl- The rcultirg
fmul, for ch@tsisric impedse is
--f
The surface resistarrce of copper at 10 6Hz is R, = 0.026 O, Then from {3.181)
the conductor attenuatioo is
(3.179a)
2.7 x 10-3 R"e,ZoA
". = = 0.122 Np/m.
30"6
whqe Fr is the elrb.rile x,,ly'/r of thc cflter cgndEior giwn by
since ,1, = 4.74. The tolal attfiuatior cofftant i$
a : ud * uc: 0.27? Np/m.
1l. I rgbr
In dB,
Thw fomulas awmc a srip with zw thickrc ard m quotfd s beiBg a$Wte to
abod t% ofthe ffi Hults. It is sm frm (3.179) tbar &e chracteristic impedance
n{dB) = 20logea * 2.41 dB/m.
dereus a: Ihe srip width R imltm.
When designhglcimitr one u*ally needs to tird the saip widrh, giq the
At l0 CHz, the wavelengh oa ttreQlis
chaBctsisti( impedmce (and h€ight, ard relBtiK peminivity €,), wtich rcquis the
iwrc of the li)rolss i, (3.179). Such fomulq hrc beo dcrived q
(l.l80a) so in terms ofwavelength the attenuetion is
o(dB) : (2.41X0.0202) = 0.O49 dB/t. I
-l (l.l80b)
, *J
AXAMPLE 3.7 MICROSTRIP LINE DESIGN
Design a microskip line on a 0.5 mm alumina substrate (6r : 9.9, tan 6 : 0.001)
Micro-strip line for a 50 Sl characteristic impedaflce. Find the length of this line required
to produce a phase delay of 27fi" at 10 GHa and compute the total loss oa this
lbe, assuming copper conductors" Compare the results obtained from the approx-
imate formulas of (3.195) -(3.199) with those {iom a microwave CAD package.
Solutian
First find W/ d fot Zs : 50 f,2, and'initially guess th^t W/ d < 2. From (3. I 97),
,ii I t.l ilr, r,''llr lrr.... ll!,t iit\a .,i,lt),' , t :il... A:2.142. Wd = 0.9654.
field lines are in the dielecric region and some are in air, the effective dielectric canstant
satisfies the relation
So the ccndition that W/d < 2 is satisfied; otherwise we would use the expression
tor W/d > 2. Then the required line width is W :0.9654d:0.483 mm. From
l<1"<er (3.195) the effective dieleetric constant is en - 6.665. The line length, €, far a
270" ptrase shift is found as
afld depends on the substrale dielectric coflstalrl, thc substrate thickness, the conduclor
width, and the frequenry. o :270" : il: JekoL
Wc rvill present approximate design formulas for the eftbctive dielectdc constant, charac-
teristic irnpedance, and atteruation ofmicnrstrip linq these resul8 are curve-tit approximations k11 : "" J : 209.4 m-t .
to rigorous quasi-static solutions [8, 9]. Then we wili discuss additional aspects ofmicrosrip c
lincs, including Aequencydqrendent effects, higher order modes, and parasitic effects. 270" (n / 180"r
t : ----+:--::8.72 mm.
Je"ko
Formulas for Effective Dielectric Canstant, CharacteriEtia
lmpedance, and Attenuatlon
The cffective dielectric constanl ofa microstrip line is given approximately by
il 1!,t,
The effective dielectric constant can be interpreted as the dielectric constant ofa homo-
geneous medium that equivalently rEplaces the air and dielectric regions ofthe rnicrostrip
line. as shown in Figure 3.26, The phase velocity and propagation constant are then givetr
by (3.193) and (3.194).
Given the dimensions oftle microstrip line, the characteristic impedance can be cal-
culated as
For a given characteristic impedance Zo and dielectric constant 6/, the Wld ratio cen be
fouad as
(3.197)
Cavity Resonators
ftund=tx. t=1.2.3. (6.18)
tr,!rilrifr,,lr.:rlr1Lr..NitrIrrr1l,,.n,rr,r,,rrnfiritL!fl:rt.rji:!,,al.n.,r.i.NglhlL'r:l
rl 1if r.\.rlrrn tl1.rn;. \r ir,,,iIi ll s\rhriirrs rr. ir,::lNi. Ii)i (,1h.r l.ir!tlF ,)! L,,l
'\ ri(nrirn..rir.irurrrlliiirI|,..li]rtgi,i.|.r:iI]rritri,.,i.titrr(i,r.
;l
Iirtr rr tr:L :.t:r , 1lr! Tl ,) \l rr$,|.in1 rltlc,, ilrr (r\rr! x,rJi.r ttrr rr
i.r,,r,' hir.,rr'rrrni[]i.iri,,r.ni,,r..iirn,'n.nnif!i,.,rfi.r,.i,rrf,l,., ,.
ii,..1x$'rrfi.t\.lr i1,.tr:,!iin, r,,rri,...iiirll',-.,,'rll1..,,,rr,.iri.!,r,rrl,.
If, < a < ./, the doflinant rson&t ftode (lowcst re$nmt frequency) will be rhc TEror
,rode, corespondiflg 10 tbe TElo dominant wareguide mode in a shoned guide oflcngrh
ls/2, atd is similar to the short-circuilcd i/2 transmision line rsonatoi The dominant
TM rsonant modc i! tlc lMlto modc.
, S-Parameters
We can fiod S11 by applying an incident waye al pon l, [ir, and measuring
the oulcoming wave pott 2, V;" . This is equivalcnt to thc tmosmission cocffi-
"t 2:
cied 1'rom port I to port
Frcnr the frct that 5l r * Sz: = 0, we know thar I/i = 0 when port 2 is t€rmiraled
in Zo = 50 Q, md tbat /r+ = 0. In this w w€ have that llf = \ ad Y; =
I/2. By applyinS a voltage ll
at port I and using voltage division twice we find
(4.40) I/t = ,a tr the voltage acmss the 50 Q load rsi$tor at port 2:
\ .Irrrllr ir -Ira!i Il lll, \crr(latillt Illlt r\ ri)ti !rc LirlrtilrrreLl lrs
v; =v2= - ("#*=)("*=) =oio1v,
(4,41) 4l.il4 = 141.8(58.56)/(141.8 + 58.56) is the re,sistffie of the parallel com-
where
bimtim of the 50 Q load md the 8.56 Q resistor wirh the 141.8 Q csistor. Tbus,
512*$21 =$,'/Ql"
In.rvords,t4.4l)saysrhatSi, isfoundbydrivingport.ywilhaniflcidentwaveofvoltage
lf the input power is 1V{12/22a, then rhe ourpur Wwr h ly;12/220=
tr',* and measuring thr'reflected wave amplitude {- coming out of port i. The incident ls2tvfl2 /2zo : ll2tf l2zolvll2 = lvl P /42o, which is onc-h&tfi -3 dB) of
the input powcr
\", rtL,n
'll,^r l), \ liir f.itl1(tirt! :|,tlt:. Lirjr l,; ,1,..1u:t,tltt.l li,,rtr ilr,: 1l rLrt ) ii
rrtrtlrtr ,rr.l irr'r'\\t:Li. i ir-,i r..rtIt.t il.,\lLilti.litilt tili rl)irt.r!lfJlrit( il t1,r,l.rrtr:i: :t , .,1
.rll tirrlilrrs.rrrrrit:rtirr,l {iili.,rf.rr:ltr,,tjri i i,rr..It.,irrl tltr|lr,itr,,Lrr.r.:tr:rtr,,l,l
matched loads. .CrllctlllLl,.rtttttili.t'.rIirctr lir:rilii.Irf:l.r'\rr']It{r!11,. i l.,,tit,;llr:ii,:t,:t.ll
,....,.,
C/
)- rull,Dlr{.4 EvALUArroNor-pARAMETERs
' rrnd ilre f parameters of the 3 dB attenuator circuit sho*n in Figure 4.8.
Solution
From (4,41), Srr can be found as the reflection cocfrcient seen at port I when
port 2 is terminated in a matched load (Zs 50 O): -
v-t z!)t - zol
t" =
tln-=o=
F(r)1''.=o =
ffil',**''
Pon \ L Potl
1 '-v2 'v- 2
SiElna,l FiowChart r Rule 1 (Series Rule). Tr.o branches, whose cormon node hm only one incoming
and one outgoing wave (branches in series). may be combined to fom a silgle
branch whose coefficient is the product ofthe coefficients olthe original branches.
'!:, t.: s:r5:z
c_+-_'tr! > _____. +> H Figure 4.16a shows the flow paphs for this rule. It$ derivation follows from the
t, i. t, r\ v3 basic relation
I.r - .!.r1 I'1 - .!::.5:i I r. (4.?s)
(a)
. [{ulc2lPlraileli{rritt. l\rohranchestiorrollrctrnnrirrrrioditoanothcrconrrrl,rr
notlc lbralcltts irr Iarallt:L) ntrr, br, ronbirtr-tj inio.r.rirtlc brarch u,hosc coelticii.nt
,tn is thc surn ol tht cotlircirntr oi the orLginrl Lrr anchcs. l-igu.e I iih shorr s thc tlol
..:1.
So +So uraphs ii:rthis rulc. Thr ilrrir;rtiorr tLrllorr. lionr tlu ohliLrris rciatiirr
=N ffi Ii - 5ri't +.\ii-l - (.1, +Si,rt'r .
Vr Y2
t4.761
r Rule-1 tSril LoLpl{Lrle).\\:hcrtrnodtlrirrrselfloollirhiiurrirtlruTbcgiitsrnri cntls
\, or) lhc:anrr'irLrile)ol coelllcicrrt.1. Llrr scli:loop rrr: be ciilinatcri L.r uruitillrjng
coclhcttlls ol llrt bumcitcs lctiliqg tltlt no.lc bli 1,ii I - .tt. irigure -:1. L6c rhotrs tlic
{b) ilur urephs lirr this ulc. rr hich cair be dcrir erl ts lirlloq r. []r orn tlrt originrl rretr Lrrl.i
r
\,.' ltar c
s:z l.-.n,1;i\::l:. {4.71a)
Jr = \r:l:. (4.77b)
s:r
-'*'Qp " =>
I"I Y2 k'3 ltt
1* 5n
y2
s3?
v-1
Elirninatirg l'2 gives
(4.78)
(c) which is seen to be the transfer function for the reduced gmph ofFigure 4.t6c.
. RBle 4 (Splilti[g Rule). A node may be split into two seprate rodes as long as the
.s,. tr4 s:r t/i s+: t'4 resulring flow graph confains, once and only once, each combination of separate
+ (not self-lcops) input and ouqrut branches that coflnect to the originai node. This
nJIe is illusirated it Figure 4.16d and lollows from the obseryatioil that
yl t, ,,\ (4.79)
in both the original ffow graph md the flow gmph widr the split node.
