### 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.

### Amplifier Noise Figure

To analyze the effect of circuit noise, one can model the noisy circuit as a noiseless circuit plus external noise
sources. For a noisy, two-port network with internal noise sources (Figure 1a), the effects of those sources are represented by the external noise-voltage sources V_{n1} and V_{n2}, placed in series with the input and output terminals, respectively (Figure 1b). Those sources must produce the same noise voltage at the circuit terminals as the internal noise sources. The values of V_{n1} and V_{n2} are calculated in equations 1 and 2. Representing the noise-free, two-port network in Figure 1b by its Z parameters:

and:

Equations 1 and 2 show that the V_{n1} and V_{n2} values can be determined from open-circuit measurements in the noisy two-port network. It follows from these equations that when the input and output terminals are open (I_{1} = I_{2} = 0) (equations 3 and 4):

and:

In other words, V_{n1} and V_{n2} equal the corresponding
open-circuit voltages.

In an alternate representation of the noisy two-port network (Figure 2), the external sources are the current noise sources I_{n1} and I_{n2}. Equations 5 and 6 represent the noise-free two-port network:

and:

The values of I_{n1} and I_{n2} in Figure 2 follow from short circuit measurements taken in the noisy two-port network, as shown in equations 7 and 8:

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 (**Figure 3**).

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_{n} and I_{n} in Figure 3, using open- and short-circuit measurements. From a practical point of view, those values (V_{n} and I_{n}) can be expressed in terms of noise voltages V_{n1} and V_{n2} in Figure 1b (which require
only open-circuit measurements).

The relationship between noise sources V_{n} and I_{n} in Figure 3 and noise sources V_{n1} and V_{n2} in Figure 1b is derived as follows. Using Z parameters to represent the noise-free, two-port network in Figure 3:

and:

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

and:

Hence, solving equations 13 and 14 for V_{n} and I_{n} gives:

and:

An alternate method for determining V_{n} and I_{n} relates them to noise sources I_{n1} and I_{n2} in Figure 2. It is easy to show that the relations in this case are:

and:

A source connected to the noisy two-port network (Figure 4) is represented by a current source with admittance Y_{s}. It is assumed that noise from the source is uncorrelated with noise from the two-port network. Thus,
noise power is proportional to the mean square of the short-circuit current (denoted by active-low I_{sc}²) at the input port of the noise-free amplifier; and noise power due to the source alone is proportional to the mean square of the source current (active-low I_{s}²). Hence, the noise figure F is given by:

Because I_{sc} = -I_{s} + I_{n} +V_{n}Y_{s}, it follows that the mean square of I_{sc} is given by equation 20:

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_{n} and I_{n}. Hence, I_{n} can be written as the sum of two
terms—one uncorrelated to V_{n} (I_{nu}) and one correlated to V_{n} (I_{nc}). Thus:

Furthermore, the relation between I_{nc} and V_{n} in terms of a correlation admittance Y_{c} is defined as:

Y_{c} is not an actual admittance in the circuit; it is defined by
equation 25 and calculated as follows. From equation 24:

Multiplying equation 26 by V_{n}*, taking the mean, and
observing that :

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_{s} = Re[Y_{s}]. The noise voltage can be expressed in terms of an equivalent noise resistance R_{n} as:

and the uncorrelated noise current can be expressed in terms of an equivalent noise conductance G_{u}:

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

and:

gives:

The noise factor can be minimized by properly selecting Y_{s}. From equation 34, F is decreased by selecting:

Hence, from equation 34:

The dependence of the expression in equation 34 on G_{s} can be minimized by setting:

This gives:

Solving for G_{s}:

The values of G_{s} and B_{s} in equations 39 and 35 give the
source admittance, which results in the minimum (optimum) noise figure. This optimum value of the source admittance is commonly denoted by Y_{opt} = G_{opt} + jB_{opt}; that is:

From equation 36, the minimum noise figure F_{min} is:

Solving equation 39 for G_{u}/G_{opt} and substituting into equation 41 gives:

Using equation 42, equation 34 can be expressed as:

Solving equation 39 for G_{u} and substituting into equation 43, the expression for F can be simplified:

Equation 44 shows that F depends on Y_{opt} = G_{opt} + jB_{opt}, and on F_{min}. When these quantities are specified, the value of noise figure F can be determined for any source admittance
Y_{s}. This equation can also be expressed as:

where m = R_{n}/Z_{0} is the normalized noise resistance and y_{s} = Y_{s}Z_{0} is the normalized source admittance:

y_{opt} is the normalized value of the optimum source admittance:

Admittances y_{s} and y_{opt} can be expressed in terms of reflection coefficients:

Expressing y_{s} and y_{opt} in terms of reflection coefficients helps formulate the noise figure (equation 45) as a function of those coefficients. This formulation is more convenient for industrial LNA applications because in most data sheets, the LNA characteristics are expressed as a table of S parameters and the optimum reflection coefficient G_{opt} vs. frequency:

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.

### Designing for Optimum Noise Figure

For any two-port network, the noise figure measures the amount of noise added to a signal transmitted through the network. For any practical circuit, the signal-to-noise ratio (SNR) at its output is worse (smaller) than at its input. In most circuit designs, however, the noise contribution of each two-port network can be minimized through a judicious choice of operating point and source resistance.

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_{L}) needed to properly
terminate the LNA's output:

where Γ_{S} is the source-reflection coefficient necessary for
minimum noise figure. (The asterisk in the above equation indicates the conjugate of the complex quantity Γ_{L}.)

### Applications

A practical example to illustrate the theory of optimum noise matching for LNAs is the MAX2656 - an LNA (Figure 5) with high third-order adjustable intercept point (IP3). Designed for PCS phone applications with gain
selected by logic control (14.5dB in high-gain mode and 0.8dB in low-gain mode), the amplifier exhibits an optimum noise figure of 1.9dB (depending on the value of bias resistor R_{BIAS}). The MAX2655/MAX2656 IP3 is
adjusted with a single external bias resistor (R_{BIAS}), which lets you optimize the supply current for specific applications.

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_{BIAS}) for minimum noise figure is 715Ω. The optimum source-reflection coefficient Γ_{OPT} for minimum noise figure in a 1960MHz application (F_{MIN} = 1.79dB) is:

A source impedance with noise-equivalent resistance R_{N} = 43.2336Ω yields the minimum noise figure.

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

- S_{11} = 0.588/-118.67°

- S_{21} = 4.12/149.05°

- S_{12} = 0.03/167.86°

- S_{22} = 0.275/-66.353°

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 (Figure 6). For comparison, note the dotted-line depiction of constant-noise circles corresponding to the noise figures of 2.5dB, 3dB, and 3.5dB.

For convenience, we chose a source-reflection coefficient
of Γ_{S} = 0.3/150° on the 2dB constant-noise circle. The
normalized 50Ω source resistance is transformed to Γ_{S}
using two components: the arc Γ_{S}A (clockwise in the
impedance chart) gives the value of series inductance L_{1}.
Arc BO (clockwise in the admittance chart) gives the
value of shunt capacitor C_{1}.

The value of arc Γ_{S}A measured on the plot is 0.3 units,
so Z = 50 x 0.3 = 15Ω. Thus, L1 = 15/ω = 15/(2πf) =
15/[2π x (1.96 x10^{9})] = 1.218nH, rounded to 1.2nH. The
value of the arc BO measured on the plot is 0.9 units, so
1/Y = Z = 50/0.9 = 55.55Ω. Thus, C_{2} = 1/(55.55 x ω) =
1/(55.55 x 2πf) = 1/[55.55 x 2π x (1.96 x 10^{9})] = 1.46pF,
rounded to 1.5pF.

C_{1} is simply a high-valued DC-isolation capacitor and does not interfere with the input matching. The chosen Γ_{S} provides the load-reflection coefficient needed to properly terminate the LNA:

This value and the normalized load-resistance value are plotted in Figure 7, which also shows a possible method for transforming the 50Ω load into Γ_{L}. For this example, note that a single series capacitor provides the necessary impedance transformation.

The arc OΓ_{L} (counterclockwise in the impedance chart)
gives the value for series capacitor C_{3}. The value of arc
OΓ_{L} measured on the plot is 0.45 units, so Z = 50 x 0.45 =
22.5Ω. Thus, C_{3} = 1/(22.5 x ω) = 1/(22.5 x 2πf) = 1/[22.5
x 2π x (1.96 x 10^{9})] = 3.608pF, rounded to 3.6pF.

### Conclusion

These calculations have determined the matching components required for optimum noise performance in the LNA of Figure 5. Of course, in low-cost applications where optimum noise performance is not mandatory, C_{3}
can be omitted, and the MAX2656 can be connected directly to a 50Ω system.

### References

- 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.