
Keywords: MAX2655,MAX2656,lna,optimizing noise figures,lownoise amplifier
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LowNoise Amplifier (LNA) Matching Techniques for Optimizing Noise Figures
© Jul 08, 2004, Maxim Integrated Products, Inc.


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 and
distortion 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, twoport network with internal noise sources (
Figure 1a), the effects of those sources are represented by the external noisevoltage 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 noisefree, twoport 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 opencircuit measurements in the noisy twoport 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
opencircuit voltages.
Figure 1. A noisy twoport network (a) can be modeled by a noisefree twoport network (b) with external noise voltage sources V_{n1} and V_{n2}.
In an alternate representation of the noisy twoport network (
Figure 2), the external sources are the current noise sources I
_{n1} and I
_{n2}. Equations 5 and 6 represent the noisefree twoport network:
and:
The values of I
_{n1} and I
_{n2} in Figure 2 follow from short circuit measurements taken in the noisy twoport network, as shown in equations 7 and 8:
Figure 2. A noisy twoport network can also be represented by a noisefree, twoport network with external noisecurrent sources I_{n1} and I_{n2}.
and:
Other representations, besides those shown in Figures 1b and 2, can be derived for a noisy, twoport network. A
convenient representation for noise analysis places the noise source at the input of the network (
Figure 3).
Figure 3. Again, a noisy, twoport network can be represented as a noisefree, twoport network with external noise sources V_{n} and I_{n} at the input.
Representing the noisefree, twoport 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 shortcircuit 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 opencircuit 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 noisefree, twoport 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 twoport 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 twoport network. Thus,
noise power is proportional to the mean square of the shortcircuit current (denoted by activelow I
_{sc}²) at the input port of the noisefree amplifier; and noise power due to the source alone is proportional to the mean square of the source current (activelow I
_{s}²). Hence, the noise figure F is given by:
Figure 4. This noise model lets you calculate the amplifier noise
figure.
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 twoport
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 noisefigure 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 twoport network, the noise figure measures the
amount of noise added to a signal transmitted through the
network. For any practical circuit, the signaltonoise ratio
(SNR) at its output is worse (smaller) than at its input. In
most circuit designs, however, the noise contribution of
each twoport 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 twoport 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 sourcereflection 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 loadreflection
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
loadreflection coefficient (Γ
_{L}) needed to properly
terminate the LNA's output:
where Γ
_{S} is the sourcereflection 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 thirdorder adjustable intercept point (IP3). Designed for PCS phone applications with gain
selected by logic control (14.5dB in highgain mode and 0.8dB in lowgain 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. This typical operating circuit for the MAX2656 LNA shows design values for the inputmatching network.
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 sourcereflection coefficient Γ
_{OPT} for minimum noise figure in a 1960MHz application (F
_{MIN} = 1.79dB) is:
A source impedance with noiseequivalent 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 constantnoise circle requested by
design (
Figure 6). For comparison, note the dottedline
depiction of constantnoise circles corresponding to the
noise figures of 2.5dB, 3dB, and 3.5dB.
For Larger Image
Figure 6. The solid circle on this Smith chart depicts the desired (optimum) 2dB noise figure for a MAX2656 PCS LNA with input matching.
For convenience, we chose a sourcereflection coefficient
of Γ
_{S} = 0.3/150° on the 2dB constantnoise 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/(2
πf) =
15/[2
π 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 2
πf) = 1/[55.55 x 2
π x (1.96 x 10
^{9})] = 1.46pF,
rounded to 1.5pF.
C
_{1} is simply a highvalued DCisolation capacitor and does not interfere with the input matching. The chosen Γ
_{S} provides the loadreflection coefficient needed to properly terminate the LNA:
This value and the normalized loadresistance 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.
For Larger Image
Figure 7. The MAX2656 PCS LNA has output matching for a desired (optimum) 2dB noise figure.
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 2
πf) = 1/[22.5
x 2
π 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 lowcost 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.
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APP 3169: Jul 08, 2004
APPLICATION NOTE 3169,
AN3169,
AN 3169,
APP3169,
Appnote3169,
Appnote 3169
