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Theory of Linear and nonlinear circuits - Engberg J.

Engberg J. Theory of Linear and nonlinear circuits - Wiley & sons , 1995. - 154 p.
ISBN 0-47-94825
Download (direct link): noisetheoryoflinearandnonlinear1995.pdf
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1.3. Transformations between sets of noise parameters
63
To rn, g„, and Z,,:
G„
To F,_min, gn, and ZSoF-
Qnc =
1 - (y/Cr, / R„ +Gi - j 8^) Z, v
l+(y/Gn/Rn + G?,-jB^)Z; 1 + 1
To Ta, Tn, and T7 eJ ,f~'\
(4.87)
GJRr. + |17|2 Gn + Д„|У7Г (4.88)
GjRn \ (i;j2 (4'S9)
min — 1 + 2 ^RnGj + \J RnGn + (RnG7)2 ^ (4.90)
gn = Gn + ЛП|У7!2 ■ (4.91)
To F,
min j Qnc ■>
and Г SOF'-1
Femin = 1 + 2 ^RnG7 + yRnGn + I RriG-, )~ у (4.9
4 Re[^]j
(4.94)
1 — ( JgvJRn f G‘ — j В Zx
Гsof = ----------/ / — -ч-- (^-9о)
1 + {^jG,,j Rn + Gl - ]B~,) Z{
Tn = -A (\ZA2G^ + R„\l 4- !Z,!2|KJ2 + 2(R,G-. - Л'тДЛЛ (4.96! )Kl| \- - - l ■ - ■■ - - -j/ ■
T, =
— (;Z,l2Gn -t- R„ ' i + |/,Гг!У,|г - 2(R,<?, + -Vi 5,)j'l (4.97
Rj| V ** I 11 ‘ ' • ' 1 ■ 1/
т »>*■' = - io (: X';2 - Rf' G_. 4- R,. ! 1 - '■7,-С-\У.лг J- •> V.( V, I > 1,!3 -
’ I R\ | ' I '’J " l ........ *....... "i!
+ j2PlT0\XiGn + ЙП(Х,|У;|2 - Я,)] (4.08)
= Ri -r j A'i is the complex reference impedance at the inpiu ;>ort.
64
4. Noise parameters
From Femim Rn, and YSof
To Gn, Rn, and Y-t\
г — 1N {Femin 1)
— '-T bure mm ) . j-.
4 Rn Rn = Rn
= " 2Я,-Gsof _ j Вsof
To rn, 5„, and Z7:
_ (4 RnGsoF - F°mm + l)(-Fe7Tun 1) Г" “ 4Дп]У50/гР
Sn = KnlKsOfl2
2 min ~ 1 - 2 RnG'soF + jZRy.BsOF
7 2 Rn ITsofP
To Fe min, Qm and Zsof ■ Fs min — Fe min 9n = Rn jXyoi7!
Zsof
Ysof
\Ysof\2 To F, min, Qnc i and Г sof•
_ 4Re(Zl]Rn
Vnc
Г sof =
IrfteHr W
1 - Y$qfZi 1 + YsofZ'
To Ta, Tp, and T-,
T'
Ta = f^l'i -i- IZ.rl^OF!2) + -Rif/7;™ - 1)
- ° /? ([?, Гг •■■■-■< 1' — У.ГЛ^^Г-^
Пч " -w, -*• * — Г .' j
Тв = ш (Лп(1 + iZll2!KsOFl2) “ - 1)
т 2 Rn(R\GsoF + A'1Bsof’)'1
(4.99)
(4.100)
(4.101)
(4.102)
(4.103)
(4.104)
(4.105)
(4.106)
(4.107)
(4.108)
(4.109)
(4.110)
(4.112)
4.3. Transformations between sets of noise parameters
65
= -jj (Rn [l - |YSof\2(R\ - A'i )"j + 2RnXiBSOF)
I |
+ j^PiToRn{BsoF + AilYsofi2) (4.113)
From rn, gn, and Z7
To Gn, Rn, and Y~,:
G.
тп/дп + \Z-,\2 Rn — Tn + 3n\Z-A~
Y,,
rn! 9ix + l^j2
To Fg min ^
Rn, and Ysof:
femin = I + 2 (gnR'. + \fgnrfn + (ffn-ДЦ)2 ^
-Rjl — Tl “К j? R | ^7 i
yfrj 9n; " j ib0F i"n/?n + iz^l2
To Femint 9nr and Z$OF•
Fem,n ~ 1 + 2 (gnR+ \J9nr: n + [gnR^)2 J
9n = 9n
Zsof = \Jrn/gn + Rl ~ j*
io trmiЦ„с. ana 1 sof" F -
1 с mtn —
Q,c =
Г SOF =
1 + 2 j gnR- + \j9-n.rn + <gntl~,)-
\
4 R,efiTi: (r„ 4- yn \Z^\:)
• i‘i- - j A- -6 у
\ \/Tni3n + Ri, - i А'-, -г 2Г*
\J ^nj On“b J 7 ^ 1
. It J~n -L - n Y Л- 7Г
(4.114)
(4.115)
(4.116)
(4.117)
(4.118)
(4.119)
(4.120)
(4.121)
(4.122)
(4.123)
(4.124)
(4.125)
66
4. Noise parameters
J.O ia, 10, ana eJ ^:
Гг
Ta — | + gn R7 + i?i)2 + (X~, + A'i)2
Tfl = -Щ ('■" + Зп [(л7 - Rif + (A% + A'j)2!
= -Щ (r„ + 9n [(R:, - R\) + (Л% + A',)2]; + j 2piTog„(X.r + A'i)
From te jriijij 9m and ZsoF To Gn, Rn, and Y.f:
^ _ { 4 9nRsOF t* min 4“ 1 )(^1)
"" = -ignlZsoFl2
Rn = gn\ZsoF?
v _ Fsof ~ 1 ~ 2 gnRsOF + j2gnXsop 7 2 gn\ZsoF\~
To Fe
min j Rn, and Ysof-
Fe min Fe Trim
Rn = gn\ZsoF\2 v ZsOF*
*SOF | /~t ;o
\JJSOF\~
To тп, Qtl j and Zy
I)2
=
To ^TIC? and r^Q/T.
rn — -^50/*4 min 1)
<7n = 9n
F - 1
1 * ,Tlin L
TsOF =
F
L e m;n
4 RejZij ijn \ ZsofV
itefpr W
^50 F -7r- a. 7*
I
j) (4.126)
j) (4.127)
)
(4.128)
(4.129)
(4.130) (-1.13 i)
(4.1.32)
(4.133)
(4.134)
(4.135)
(4.136)
(4.137)
(4.138)
(4.139)
(4.140)
4.3. Transformations between sets of noise parameters
67
To Ta, Tp, and Г,е^:
T-, = (gn{\ZsoF\2 + \Z\\2) + i2i(Femi'n - 1)
To
m
- 2 gniRsofRi + XsofX i))
(;?n(|ZsoH2 + ~ R\(Femi„ ~ 1)
,(RsofRi - XsofXi))
^ (I^sof!2 - (Д? - A-,2) - 2XSOFA'!)
?Ogn №1
+ j2plr03n(>1 - Xsof)
(4.141)
(4.142)
(4.143)
From /’emin, Qnc, and Га-of
To Gn, Rn, and У-,'
Gn — Re[Zi](Fe:
1)
Wnc(i ! 1 SOt i ) ( ^етn’.n 1/
QncFsopz; + Zip
Rn = У-, =
(4.144)
(4.145)
QnA^so/Zi + Zi\2
4 Re[Zx]
2 Re[Zt] (Femin - 1) + Qr.c{\^S0F\'2Zi - Z{) + 2 jQneIm[r50FZ.] Qnc iГSOF^i + Z\ j"
(4.146)
To F,
min *
Rn, and
Rn =
Ysof =
Qnc\^;ofZl + Z\ j2 4 Re[Zi]
J- ~ TgQF L'sofZ' + Zv
To r„, gn, and Z-,:
= Re[Zi] [Femtn - 1) Qnc |1 - Tsof!2
Qnc I L “ ^50f|2
I'i-i'*')
(4.148)
(41421
(4.1о L)
68
4. Noise parameters
2Re[Zj] (Fcmin - 1) + Qnc (|Tsofj~^i - - 2jlm[lgoy ZiJ)
Qncjl- - Г50^р
To Fem;n) gn. and Zs
(In =
Zsof =
Qnc |1 - Tsof I2 4 Re[Zi]
Гдор Z* + Z\
1 - Г50Р-
To Ta, Tp, and T^e^y.
Ta = PlT0(/’emm — 1 + Gncl^SC/F^) Tp = PiT0(l - Femm -f Qnc)
T,e>^ = Pi To QnC jr^o^! e~ Jfsoy
(4.152)
(4.153)
(4.154)
(4.155)
(4.156)
(4.157)
(4.158)
From Ta, Tp, and T7 e1
To Gn, Rn, a,nd K7:
Gn
№1
To
Т*Тз - T2
Rn =
|Zi|2(Tq + Tp -f iTyCosify) - 4 A'i Ty(Xi cos y7 + Д, sin ip^) 1
(4.159)
4T0jR:l
x (j^il'l^a + To + 2T7cosi/?^) - 4 Xj T7(Лi costpy + Ri sin <^7)j (4.160) У7 =
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