[For a copy of this article in PDF format, which displays figures and equations, click

here

. Requires Adobe Acrobat Reader,

free download.

]

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 add noise and distortion to the desired signal.

In a receiver chain, the first amplifier after the antenna contributes most to the system noise figure (assuming low loss in front of the amplifier). Adding gain in front of a noisy network reduces the noise contribution from that network.

The first part of this article is a refresher addressing the theoretical noise figure for a two-port RF network, and the optimum reflection coefficient for the minimum noise figure.

The rest deals with using scattering parameters (S parameters) as a design tool to match impedances for minimum noise figure. The analysis considers optimum noise matching for an SiGe low-noise amplifier.

Two methods are available for analyzing the effect of noise in electronic devices and circuits. The first method substitutes equivalent noise sources at appropriate physical locations in the small-signal model for the device. As an example, consider the noise produced by two resistors in series (figure 1a). The noise model of a resistor (figure 1b) produces an open-circuit voltage whose mean-square value is:

Because V_n1 and V_n2 are statistically independent (uncorrelated), the mean value of the product term in equation 1 is zero. Thus:

When noise sources are uncorrelated, the results show that superposition can be used to calculate the total mean-square noise voltage.

The second method for analyzing the effect of circuit noise models the noisy circuit as a noiseless circuit plus external noise sources. For a noisy two-port network with internal noise sources (figure 2a), the effect of those sources can be represented by the external noise-voltage sources V_n1 and V_n2, placed in series with the input and output terminals, respectively (figure 2b). Those sources must produce the same noise voltage at the circuit terminals as do the internal noise sources. The values of V_n1 and V_n2 are calculated as follows: Representing the noise-free two-port network in figure 2b by its Z parameters, we can write:

Equations 2 and 3 show that the Vn₁ and Vn₂ 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 (I₁ = I₂ = 0),

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 3), the external sources are the current-noise sources I_n1 and I_n2. Representing the noise-free two-port network, we can write:

The values of I_n1 and I_n2 in figure 3 follow from short-circuit measurements taken in the noisy two-port network. That is:

Besides that of figures 2b and 3, other representations can be derived for a noisy two-port. A representation convenient for noise analysis places the noise source at the input of the network (figure 4). Representing the noise-free two-port network in figure 4 by its ABCD parameters, it can be written:

The previous equations show there is no simple way to evaluate V_n and I_n in figure 4 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 the noise voltages V_n1 and V_n2 in figure 2b (which require only open-circuit measurements). The relationship between noise sources V_n and I_n in figure 4 and noise sources V_n1 and V_n2 in figure 2b is derived from the following.

Using Z parameters to represent the noise-free two-port network in figure 4, it can be written:

and

Comparing (2) and (3) with (4) and (5), it follows that:

Hence, solving (6) and (7) for V_n and I_n gives:

An alternate method for determining V_n and I_n relates them to the noise sources I_n1 and I_n2 in figure 3. In this case the relationships are:

and

A source connected to the noisy two-port network (see figure 5) 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 I²_SC) 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 (I²_S). Hence, the noise figure F is given by:

Because I_sc = -I_s + I_n + V_nY_s, it follows that the mean square of I_sc is given by:

And, because noise from the source and noise from the two-port network are uncorrelated, we have:

Equation 11 reduces to:

Substituting (12) into (10) gives:

There is some correlation between the 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_un) and one correlated to V_n (I_nc). Thus:

Furthermore, defining the relation between I_nc and V_n in terms of a correlation admittance, Y_c, gives:

However, Y_C is not an actual admittance in the circuit. It is defined by (15) and calculated as follows:

From (14):

Multiplying (16) by V*_n, taking the mean, and observing that:

we obtain:

Substituting (16) into (13) produces the following expression for F:

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

where G_s = R_e[Y_s]. The noise voltage can be expressed in terms of an equivalent noise resistance R_n as:

The uncorrelated noise current can be expressed in terms of an equivalent noise conductance, G_u, namely:

Substituting (18), (19), and (20) into (17), and letting Y_c = G_c + jB_c and Y_s = G_s + jB_s gives:

The noise factor can be minimized by properly selecting Y_S. From (21), F is decreased by selecting B_S = -B_C. (22)

Hence, from (21):

The dependence of the expression in (23) on G_s can be minimized by setting:

Which gives:

Solving for G_S, we obtain:

The values of G_S and B_S in (24) and (22) 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. This is given as:

From (23), The minimum noise figure, F_min, is:

Solving (24) for G_U/G_opt and substituting into (26) gives:

Using (27), we can write (21) as:

Solving Equation 24 for G_U and substituting into (28), the expression for F can be simplified to:

Equation 29 shows that F depends on Y_opt = G_opt + jB_opt, and on F_min. When these quantities are specified, the value of F can be determined for any source admittance, Y_s. This equation can also be expressed as:

where r_n = R_n/Z₀ is the normalized noise resistance. And y_s = Y_SZ₀ is the normalized source admittance. Hence:

and 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 help formulate the noise figure (NF) of (30) 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 Г_opt versus frequency, as:

When the noise figure {F in (31)} is expressed as a function of a circle, it can be used with a Smith chart for optimum noise-figure matching in specific applications as show in the following set of equations.

For LNA input matching, a noise circle is positioned on the Smith chart's center and radius.

Center:

Radius:

For any two-port network, the noise figure gives a measure of the amount of noise added to a signal transmitted through the network. For any practical circuit, the signal-to-noise ratio at its output will be worse (smaller) than at its input. In most circuit designs, however, you can minimize the noise contribution of each two-port network 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 on the data sheet.

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. Next, force the actual source impedance to “look like” that optimum value. All stability considerations still apply, of course. If the calculated Rollet stability factor (K) is less than 1 (K is defined in the literature as a figure of merit for LNA stability), then the source and load-reflection coefficients must be carefully chosen. For an accurate graphical depiction of the unstable regions, it is best in that case to draw stability circles.

After providing the LNA with optimum source impedance, the next step is to determine the optimum load-reflection coefficient (ГL) needed to properly terminate the LNA's output:

where Г_S is the source-reflection coefficient necessary for minimum noise figure. (The asterisk in the above equation indicates the conjugate of the complex quantity Г_L.)

For practical examples to support the theory of optimum noise matching for LNAs, we examine an LNA (figure 6) with high third-order adjustable intercept point (IP3). This particular design is for personal communications system (PCS) phone applications with gain selected by logic control (14.5 dB in high-gain mode and 0.8 dB in low-gain mode), the amplifier exhibits an optimum noise figure of 1.9 dB (depending on the value of bias resistor, R_bias).

The figure 6 application employs an LNA operating at a PCS receiver frequency of 1960 MHz and noise figure of 2 dB. It must operate between 50Ω terminations. For this particular device, the optimum R_bias for minimum noise figure is 715Ω. The optimum source-reflection coefficient Г_OPT for minimum noise figure in a 1960 MHz application (F_MIN = 1.79 dB) is:

A source impedance with noise-equivalent resistance R_N = 43.2336Ω yields the minimum noise figure.

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

S₁₁ = 0.588/-118.67°

S₂₁ = 4.12/149.05°

S₁₂ = 0.03/-167.86°

S₂₂ = 0.275/-66.353°

The calculated stability factor (K = 2.684) indicates unconditional stability, so we can proceed with the design. Figure 6 (a typical operating circuit) shows design values for the input matching network. First, a Smith chart for input matching shows (in blue) the 2 dB constant-noise circle requested by design (figure 7). For comparison, note the dotted-line depiction of constant-noise circles corresponding to noise figures of 2.5 dB, 3 dB, and 3.5 dB.

For convenience, we choose a source-reflection coefficient of ГS = 0.3/150° on the 2 dB constant-noise circle. The normalized 50Ω source resistance is transformed to ГS using three components: The arc Г_SA (clockwise in the impedance chart) gives the value of series inductance L₁. Arc B_O (clockwise in the admittance chart) gives the value of shunt capacitor C₁.

The value of arc Г_SA measured on the plot is 0.3 units, so Z = 50(0.3) = 15Ω. Thus, L₁ = 15/ω = 15/(2πf) = 15/(2π)(1.96 × 10⁹) = 1.218 nH, rounded to 1.2 nH. Value of the arc B_O measured on the plot is 0.9 units, so 1/Y = Z = 50/0.9 = 55.55Ω. Thus, C₂ = 1/(55.55(ω)) = 1/(55.55(2pf)) = 1/(55.55)(2π)(1.96 × 10⁹) = 1.46 pF, rounded to 1.5 pF.

Capacitor C₁ is just a high-valued DC isolation capacitor, and does not interfere with the input matching. The chosen Г_S provides the load-reflection coefficient needed to properly terminate the LNA:

This value and the normalized load-resistance value are plotted in Figure 8, which also shows a possible method for transforming the 50Ω load into Г_L. For this example, note that a single series capacitor provides the necessary impedance transformation.

The arc OГ_L (counterclockwise in the impedance chart) gives the value for series capacitor C₃. The value of arc OГ_L measured on the plot is 0.45 units, so Z = 50(0.45) = 22.5Ω. Thus, C₃ = 1/(22.5(ω)) = 1/(22.5)(2πf)) = 1/(22.5)(2π)(1.96 × 10⁹) = 3.608 pF, rounded to 3.6pF.

1. Guillermo Gonzalez, Microwave Transistor Amplifiers, Analysis & Design, 2nd ed. (Upper Saddle River, New Jersey: Prentice Hall, 1996).

2. Christopher Bowick, RF Circuit Design, (Newnes, 1997).

Alphonse Harter is the corporate field application engineer for Maxim Integrated Products Inc. (www.maxim-ic.com) in the France office. He covers the southern Europe region for the wireless products group. He may be reached at alphonse_harter@maximhq.com.

RSS    Save to Del.icio.us  Digg This