# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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6.5. Transformer coupled feedback

133

Y. =

Yn Yu]

у V [О.ЧЭ)

Г 21 >'22 j

where Z\ = r\ + jutL\, Zi — r2 + jw I? and Zn = j w M. corresponding to the impedances of the two inductances and the mutual inductance. The noise currents and voltages are all uncorielated and the equivalent noise two-port of the amplifier

is shown. The noise currents and voltages are determined from the noise parameters

by Equations (2-7) and (2.9).

As the noise sources are uncorrelaten, the noise factor equals the sum of the output noise powers originating from the noise sources and from the source at standard noise temperature divided by the output noise power from the source at standard

noise temperature. There are seven noise sources and each of them gives a contri-

bution to the output noise power. Eight (numbered) equations can be written as follows where the right sides of seven of the equations are either zero or contain a noise source:

— Vi + Z2I2 + ZqI3 + Z0I4 = j (6.119)

— Vn 4- ZJl + Ztl2 + Zgli — I (6.120)

I ~ebl

— + Z0h + Zoh + Z-iIi = I (6.121)

l ~еь3

— V4 4- Zi) 1-2 "r Z1 f-f- 7.1 f ] = I (6.122)

Yu Yu Vi

Y21 Y22

Yu Yl3 ' V,

t'21 Y22 V3

- 1 0 0 - 1

L 0

0

_ 1

h

h

h

L/3

[»’ t". or 0 (6.123)

’ 0 ' -eJYy - Yu) 1

0 а ^21

(6.121)

(6.125)

Vi

и

(6.126)

The noise sources are taken one at a time and in the other equations the zeros are used. These equations can be written in matrix form as shown:

134

6. Noise of embedded networks

— "i 0 0 0 0 z2 Zo Zo V'-i

0 -1 0 0 Zi Zi 0 Zo V2

0 0 -1 0 Zo Zo 0 z2 Уз

0 0 0 -1 0 z0 Zi Zx Vi

I'll 0 Y13 0 -1 0 0 0 h

I21 0 Y22 0 0 0 -1 0 h

Ys 0 0 1 1 0 0 h

0 0 1 1 0 0 0 0 h _

Here tor each noise contribution a is equal to one of the seven vectors below containing only one noise source each:

0 0 0 ~ec] 0 0 0

0 “e6i 0 0 0 0 0

0 0 ~eb2 0 0 0 0

0 0 0 0 -ec2 0 0

0 0 0 0 0 -eJY, - Yu) —

0 0 0 0 •0 ea^21 0

-ics 0 0 0 0 0 0

0 L 0 0 0 0 0 0

Solving Equation (6.127) seven times for /, 7i = h + /4 where m = 0 1, ..., 6 it

follows from Cramer’s rule [13] that Im is of the form

=

S “b bmJSm

D

where So = ios. = еь1, s2 = e>,2. 53 = eci, s4 = eC2, s5 = ea, s6 = ia and D

is the determinant. am and bm are dependent on the matrix Y and Zo, Zi and Z2. It is not necessary to compute D as it is not used in the following, but it is also of the form a-Ys + b~. By use of the algebraic computer program Maple V [12] the quantities я.0 — Яд and bn - bg are computed as:

«0 = 0

4,i = ^ 11 Z,,/: — Vи\i\Z\ + 1 22ZnZ. + 2 Z:)

П, j “ IhZqZt — Y\2Z0 — V 21 Z2 T У 22 ^^Z2 4" 2 Zn

6[ = 0

«2 = —Y\\Z\Z2 + Y12Z0Zl + y-2iZoZ2 - i'22^0 - Z\ ~ z2 b2 — — 1 — У11 Z2 -h У12 Za

6.5. Transformer coupled feedback

135

аз — AyZoZiZ2 — AyZg + VnZoZ] - V^Zg

+ У2,ZxZi - 2 Y21Z0 + V%2+ 2 20

^3 = 'XyZqZi + У'п^о + YnZ2

а4 — — i\yZiZ^ ~ ^11 ZyZi 4 Y12Z0Z2 "b V21 Zq/,2

+ Y-nZl - Y-iZxZ, YnZ'i - - Jfc

bA = -1 - AyZf - YUZ2 - Y22Z2

dj — AyZg ■— X у ZoZ\ Zj — V; 1 ZijZ\ + V, 1 Z\]Zi + 2 V 21Z"

-Y2lZxZ2 - Y2lZ\ + V.ZoZi - yTZ0Z2 -- V., V >1 Zn Z, — V-, V'j 1 Zj Z-J — V, У 22 Z;^ + } . V j j Z, i Z; Zj

65 - — A}' Z,Z, — y„Z0 — У21 Z2 Г V-, Z;i — 21 Zo + V- j Zn Zj

ae = >2iZ02Z2 — У21 Zj Z2 — Y22Zq + Y22Z0ZlZ2 -f ZqZi — ZqZ2

6(5 = — y2iZ2 + \n1Zr\Z1 4- Zq

As |/mj2 is proportional to the mth contribution to the output noise power it follows that

where Gs — Re[y,’]. As Gs is proportional to |^01“ - and 4 к To Л/ can be cancelled in numerator and denominator - the identity reduces to

E6m=1 ji^.Ys + b„).= c;, + K + y;,|2 (6 l2g)

jao + <>0\-

Introducing the two vectors

with the scalar product (u, тг) ~ V. -i; r‘ and norm ]|u;| = ■/ u. u) and as an = 0.

(6.128)

V

(6.130)

(Oi.si . h-

Iк Vs + ^oll2 im2

G'n 4- R'JYs + y.:i2

(6.132)

Equation (6.132) leads to

1.36

6. Noise of embedded networks

К = S- > о (6.133)

2Re[yi’(i;1,r0)] = 2 Kelly}',;'jjv,!!'] (6.134)

which gives for Ys = 1:

and for Ys = j-

1ш[(г-0, и,)] = 1т[У^ ||V’21

Finally

GUM2 = IWP-lkllW

r, Ц-foll2 il^il!2 - |К,«г)|2 ^ ,rpn

6^- - -----------n4kR^------------ - 0 (6Л36}

ana the expressions for the noise parameters R'n. G'n and YI, are found.

6.6 Mixed input

In uiiiei lu obtain a low noise factor and simultaneously a good input power match, it is sometimes the practice to ground the input transistor somewhere between the base and the emitter and thus apply the input signal to both base and emitter by means of a lossless transformer as shown in Figure 6.I6.

For such a circuit the noise param-t.prs ran be calculated by expressing the output noise power density with the source at standard noise temperature, and from this an expression can be made for Fe. This expression is identically equal to the standard expression for Fe (Equation (4.11)) tor all source admittances and thus the following equations are found:

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