APPLICATION NOTE 3169

Abstract: An RF amplifier is an active network that increases the amplitude of weak signals, thereby allowing further processing by the receiver. Receiver amplification is distributed between RF and IF stages throughout the system, and an ideal amplifier increases the desired signal amplitude without adding distortion or noise. Unfortunately, amplifiers are known to add noise anddistortion to the desired signal. In a receiver chain, the first amplifier after the antenna contributes most to the system noise figure. Adding gain in front of a noisy network reduces the noise contribution from that network.

and:

Equations 1 and 2 show that the V

and:

In other words, V

In an alternate representation of the noisy two-port network (

and:

The values of I

and:

Other representations, besides those shown in Figures 1b and 2, can be derived for a noisy, two-port network. A convenient representation for noise analysis places the noise source at the input of the network (

Representing the noise-free, two-port network in Figure 3 by its ABCD parameters in equations 9 and 10 show:

and:

Equations 9 and 10 show that there is no simple way to evaluate V

The relationship between noise sources V

and:

Comparing equations 1 and 2 with equations 11 and 12, it follows that:

and:

Hence, solving equations 13 and 14 for V

and:

An alternate method for determining V

and:

A source connected to the noisy two-port network (

Because I

Because noise from the source and noise from the two-port network are uncorrelated:

and equation 20 reduces to:

Substituting equation 20 into equation 19 gives:

There is some correlation between external sources V

Furthermore, the relation between I

Y

Multiplying equation 26 by V

Substituting equation 26 into equation 23 produces the following expression for F:

Noise produced by the source is related to the source conductance by:

where G

and the uncorrelated noise current can be expressed in terms of an equivalent noise conductance G

Substituting equations 29, 30, and 31 into equation 28, and letting:

and:

gives:

The noise factor can be minimized by properly selecting Y

Hence, from equation 34:

The dependence of the expression in equation 34 on G

This gives:

Solving for G

The values of G

From equation 36, the minimum noise figure F

Solving equation 39 for G

Using equation 42, equation 34 can be expressed as:

Solving equation 39 for G

Equation 44 shows that F depends on Y

where m = R

y

Admittances y

Expressing y

When the noise figure is expressed as a function of a circle, it can be used with a Smith chart for optimum noise-figure matching in specific applications:

For LNA input matching, a noise circle is positioned on the Smith chart as follows:

From equations 51 and 52, one can visualize the noise performance of an LNA by plotting the noise circles on the Smith chart. This technique allows the designer to see the effect of tuning in order to estimate the practical noise performance.

The preceding section demonstrates that for each LNA (indeed, for any two-port network), there exists an optimum noise figure. LNA manufacturers often specify an optimum source resistance in the data sheet. As an alternative, data sheets for the MAX2656 and other LNAs specify an optimum source-reflection coefficient.

To design an amplifier for minimum noise figure, determine (experimentally or from the data sheet) the source resistance and bias point that produce the minimum noise figure for that device. Then force the actual source impedance to "look like" that optimum value with all stability considerations still applying. If the Rollet stability factor (K) is calculated to be less than 1 (K is defined as a figure of merit for LNA stability), then you must be careful in choosing the source and load-reflection coefficients. For an accurate depiction of the unstable regions, it is best to draw stability circles.

After providing the LNA with optimum source impedance, the next step is to determine the optimum load-reflection coefficient (Γ

where Γ

Figure 5's application employs a MAX2656 LNA operating at a PCS receiver frequency of 1960MHz and noise figure of 2dB (as requested by design). It must operate between 50Ω terminations. As described in the MAX2656 data sheet, the optimum bias resistance (R

A source impedance with noise-equivalent resistance R

A MAX2656 LNA operating at 1960MHz has the following S parameters (expressed as magnitude/angle):

- S

- S

- S

- S

The calculated stability factor (K = 2.684) indicates unconditional stability, so we can proceed with the design. Figure 5 shows design values for the inputmatching network. First, a Smith chart for input matching shows (in blue) the 2dB constant-noise circle requested by design (

For Larger Image

For convenience, we chose a source-reflection coefficient of Γ

The value of arc Γ

C

This value and the normalized load-resistance value are plotted in

For Larger Image

The arc OΓ

- Gonzalez, Guillermo; Microwave Transistor Amplifiers, Analysis & Design; Second Edition, Prentice Hall, Upper Saddle River, New Jersey 07458.
- Bowick, Chris; RF Circuit Designs; Howard W. Sams & Co. Inc., a publishing subsidiary of ITT.