# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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From these results various parameter curves are drawn with the values of the lossless feedback elements as coordinates. Such a graph is shown in Figure 6.4, where the real and imaginary parts of the load admittance and the transducer gain for that load admittance are the parameters. The graph is to be read as follows.

Each point in the graph assigns a load admittance {Gl , Bl) to a feedback pair (В/ , Xj) such that input matching and minimum noise measure are attained simulta-

108

6. Noise of embedded networks

neously. The corresponding transducer gain can also be read from Figure 6.4.

It should be noted that simultaneous input power match and minimum noise measure without feedback can only be obtained with a transducer gain less than one. With only series feedback (X/ = 12 П) the optimization is obtained with a transducer gain of about 11.5 dB.

Lehmann and Heston [4-] have constructed an integrated low noise amplifier on X-band using this technique.

2 + jW0.810-9n ■ Intrinsic chiP i 2±J^10-9fi --------------_______f- ' 1 '

Gate

jw0.610'12 S

o.i 1---------'

Si_________1

j и 0.6 10-12 s

Drain

I 2 4- j^lO_9n -----------------j :--------------

•Source

Figure 6.5: Transistor with lumped encapsulation.

Example 6.3 Consider an encapsulated transistor as shown in Figure 6.5. Data for the encapsulated transistor are

iu = 10 4- 32.1 mS

У12 = 0.5 - j 0.866 mS

j'-’i = 19.5 - j 29 mS

yJ2 = 1 4- j 3 mS

Rn = 25 a

On = 4.8 mS

— 24-77.0 mS

and the data for the encapsulation are seen in Figure 6.5, By adding the negative admittance of the two capacitors in parallel with the input and output terminals they disappear. This is done by applying Equations (6.1) - (6.7). Using Equations (6.22) -(6.28) the Z parameters appear. Then the resistances and inductances in series with the intrinsic transistor are removed by adding the negative amount in Equations .’6.3)

- (6.14-). Finally conversion to the Y parameters is performed by the Equations !n 15)

6.2, Reference plane transformation of noise parameters

109

(6.21). The results at 1 GHz - computed with a Fortran program1 - are as follows'.

Уп = 14.41 + j 0.3113 mS

K-12 = 0.787 - j 0.7576 mS

У21 = -36.23 — j 31.19 mS

Y22 = 1.948 — 7I.1I4 mS

Rn - 23.37 а

Gn - 4.269 mS

Ky = 1.260 + 7 6.552 mS

6.2 Reference plane transformation of noise parameters

When working with noise parameters it is often convenient to know them at a reference plane different from that where the noise parameters are actually measured. This is a kind of embedding as a piece of transmission line in front of a transistor can be considered as an embedding element. The result leads to the noise parameters of a transmission line as a two-port. In the next section this two-port can be an embedding element in parallel or in cascade with other two-ports. In this section, formulae are presented for transformation of noise parameters along a transmission line with known characteristic impedance and known attenuation and phase constants. It is assumed, however, that the temperature of the transmission line is the standard noise temperature of 290 К [5j.

When calculating the noise performance of an active element embedded in passive circuit elements, it is necessary, at microwave frequencies, to include distributed elements as well as lumped elements. The most significant distributed element with respect to noise performance is the transmission line leading to the first active element. This section considers the noise parameters of a network consisting of a transmission line with known constants preceding an active, two-port with known noise parameters. The result can be used to transform the noise parameters measured in one relerence plane to another along the known transmission line. The transformation works both ways; it adds the additional noise contribution when the transmission line is made longer and subtracts noise when the reference plar.p is moved closer to the active two-poTf. This feature is useful when correcting measured noise data of an active two-port and is carried out by adding a “negative" length of transmission hne.

\_____1:— Гч

СС i^pcuuu \j.

110

6’. Noise of embedded networks

With trazs^iision line constants Zo (real and positive) and 7 = a + j в and with / the electrical length of the transmission line in front of the known two-port with noiie parameters Rn. Gn and Ky the new (primed) noise parameters are presented below as

Zo f,:ai

\Z£ (G; + В

|e:ai _ e-2ci + ZoG^ rp:a, + e_2al _ .2cos2/3;j Rn

Zo

- 2 Z0Gy (e2al - e-2a,j - 4 Z0By sin 2pi

-e-a' + e~2ai -2 cos 2,3/]} (6.29)

G\ = 7^7- {e2“‘ + e~2ai - 2 + Z0Gn |e-al - (T2*'1

T- Rn [Zo {G\ + B2 + Z0-2) (e3al - e-2“')

- 2Gy (e2*' - e-'2a')]} (6.30)

К = :Tjf {zoGn sin 231 + Rn [z0 (С2 + B2 - Zsin 2pi

- 2B-, cos 23/|} (6.31)

G'n = ~ je2ai - e~2al + Z0Cn [e2al + e”20' + 2 cos 2pi]

'-0

Rn

- Y [z0 [Gi + Bl j [e-al + + 2 cos 2/j/J

-2ZeGyU2aI - e+ 420Д., sin 2/3/

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