08.Noise

154 NOISE

Fmin D minimum noise figure

Rn D equivalent noise resistance (usually device data are given in terms of a normalized resistance, rn D Rn/50

YG D GG C jBG D the excitation source admittance

Yopt D Gopt C jBopt D optimum source admittance where the minimum noise figure occurs

While a designer can choose YG to minimize the noise figure, such a choice will usually reduce the gain somewhat. Sometimes the noise figure is expressed in terms of reflection coefficients, where

where Y0 and Z0 are the characteristic admittance and impedance, respectively. Then the noise figure is given as follows:

The noise figure expression (8.40) and its equivalent (8.42) are the basic expressions used to optimize transistor amplifiers for noise figure. The derivation of Eq. (8.40) is the subject of the following section. Readers not wishing to pursue these details at this point may proceed to Section 8.9 without loss of continuity.

8.8TWO-PORT NOISE FIGURE DERIVATION

The work described here is based on the IRE standards published between 1956 and 1960 [8,9]. A noisy resistor can be modeled as a noiseless resistor in series with a voltage noise source. In similar fashion a two-port can be represented as a noiseless two-port and two noise sources. These two noise sources are represented in Fig. 8.6a as a voltage vn and a current in. The two-port circuit can be described in terms of its ABCD parameters and internal noise sources as

v1 D Av2 C Bi2 C vn

8.43

i1 D Cv2 C Di2 C in

or, as shown in Fig. 8.6, as a noiseless circuit and the noise sources referred to the input side. If the input termination, YG, produces a noise current, iG, then the circuit is completed. The polarity markings on the symbols for these sources merely point out the distinction between voltage and current sources. Being noise sources, the polarities are actually random. The Thevenin´ equivalent circuit in

156 NOISE

The product of the noise voltage and the uncorrelated current in Eq. (8.48) can be expressed by substitution of Eq. (8.49) into Eq. (8.48):

Because hvniŁui D 0 from Eq. (8.48), the product of the noise voltage and the correlated current can be found using Eq. (8.47):

The resistance, Rn, is the equivalent noise resistance for hv2ni, and Gu is the equivalent noise conductance for the uncorrelated part of the noise current, hi2ui. The total noise current is the sum of the uncorrelated current and the remaining correlated current:

Now the expression for the short circuit current in Eq. (8.45) can be modified by Eq. (8.51):

Furthermore hi2ni can be replaced by Eq. (8.55):

hi2sci D hi2Gi C hi2ui C jYcj2hv2ni C jYGj2hv2ni C YŁGYchv2ni C YGYŁc hv2ni 8.58

The noise figure, given by Eq. (8.46), can now be put in more convenient form:

The noise figure, F, is a function of the input termination admittance, YG, and reaches a minimum when the source admittance is optimum. In particular, the optimum susceptance is BG D Bopt D Bc. The value for Fmin is found by setting the derivative of F with respect to GG to zero and setting BG D Bc. This will determine the a value for GG D Gopt in terms of Gu, Ru, and Gc:

Substituting this into Eq. (8.61) provides the minimum noise figure, Fmin:

The correlation conductance, Gc, can be replaced from the total noise figure expression in Eq. (8.61) by Eq. (8.64) to give the following expression for F:

In these expressions, f is the operating frequency in GHz, the capacitance is in pF, and the transconductance in Siemens. The constants k1, k2, k3, and k4 are empirically based fitting factors. The expression for Ropt in Eq. (8.72) differs from that originally given by Fukui, as modified by Golio [11]. The circuit elements of the equivalent FET model in Fig. 8.7 can be extracted at a particular bias level. The resistance, Ri, is often difficult to obtain, but for the purpose of the noise estimation, it may be incorporated with the Rg. The empirically derived fitting factors should be independent of frequency. They are not quite constant, but over a range of 2 to 18 GHz average values for these are shown below [11]:

k1 D 0.0259 k2 D 2.966 k3 D 14.51 k4 D 162.6

These values can be used for approximate estimates of noise figure for both metal semiconductor field effect transistors (MESFETs) as well as high-electron mobility transistors (HEMTs).

The transistor itself can be modified to provide either improved noise characteristics or improved power-handling capability by adjusting the gate width, W. The drain current, Ids, increases with the base width W. Consequently those equivalent circuit parameters determined by derivatives of Ids will also be proportional to W. Also the capacitance between the gate electrode and the source electrode or between the gate electrode and the drain electrode will be also proportional to W. This is readily seen from the layout of a FET shown in Fig. 8.8. The gate resistance, Rg, scales differently, since the gate current flows in the direction of the width. Also the number of gate fingers, N, will reduce the effective gate resistance. The gate resistance is then proportional to W/N. These relationships may be summarized as follows:

gm / W

1

Rds /

W

Cgs / W

Cgd / W

W

Rg /

N

The Fukui equations, (8.70) to (8.73), for the newly scaled equivalent circuit parameters are given as follows:

Reference should be made to [11] for a much fuller treatment of modeling MESFETs and HEMTs.

The bipolar transistor has a much different variation of noise with frequency than does the FET type of device. An approximate value for Fmin for the bipolar transistor at high frequencies is [12]

In this equation, Ic is the dc collector current, rb is the base resistance, and ωT is the frequency where the short circuit current gain is 1. Values for Yopt and Rn are also given in [12] but are rather lengthy. A somewhat more accurate expression is given in [13].

Comparison of Eq. (8.84) with the corresponding expression for FETs, Eq. (8.70), indicates that the bipolar transistor minimum noise figure increases with f2, while that for the FET increases only as f. Consequently designs of low-noise amplifiers at RF and microwave frequencies would tend to favor use of FETs.

8.10PROPERTIES OF CASCADED AMPLIFIERS

Ideally amplifiers are completely unilateral so that there is no feedback signal returning to the input side. Under this condition analysis of cascaded amplifiers

162 NOISE

results in some interesting properties related to noise figure and efficiency. The results obtained will be approximately valid for almost unilateral amplifiers, even if some of the “amplifiers” are attenuators. The following two sections deal with the total noise figure and total efficiency respectively of a cascade of unilateral amplifiers.

8.10.1Friis Noise Formula

The most critical part for achieving low noise in a receiver is the noise figure and gain of the first stage. This is intuitively clear, since the magnitude of the noise in the first stage will be a much larger percentage of the incoming signal than it will be in subsequent stages where the signal amplitude is much larger. For a receiver with n unilateral stages, the total noise figure for all n stages is [14]

where NTn is the total noise power delivered to the load. This can be expressed in terms of the sum of the noise added by the last stage, Nn, and that of all the previous stages multiplied by the gain of the last stage:

If the nth stage were removed, and its noise figure measured alone, then its noise figure would be

By substituting Eq. (8.87) into Eq. (8.86) an expression for the noise figure is obtained that separates the contributions of the noise coming from the last stage only from the previous n 1 stages:

The second term in Eq. (8.91) is the same as Eq. (8.86) except that n has been reduced to n 1. This process is repeated n times giving what is known as the

Friis formula for the noise figure for a cascade of unilateral gain stages:

Clearly the noise figure of the first stage is the most important contributor to the overall noise figure of the system. If the first stage has reasonable gain, the subsequent stages can have much higher noise figure without affecting the overall noise figure of the receiver.

8.10.2Multistage Amplifier Efficiency

For a multistage amplifier, the overall power efficiency can be found that will correspond in some way with the overall noise figure expression. Unlike the noise figure, however, the efficiency of the last stage will be found to be most important. Again, this would appear logical since the last amplifying stage handles the greatest amount of power so that poor efficiency here would waste the most amount of power. For the kth stage of an n stage amplifier chain, the power added efficiency is

where

Pok D output power of the kth stage Pik D input power to the kth stage

Pdk D source of power which is typically the dc bias for the kth stage

When Eq. (8.94) and Eq. (8.95) are substituted into the efficiency equation, Eq. (8.93), the power from the power source can be found:

The efficiency for the whole amplifier chain is clearly given by the following:

Pdk

iD1