Measurement and uncertainty analysis of a cryogenic low-noise amplifier with noise temperature below 2 K


  • Dazhen Gu,

    1. Electromagnetics Division, National Institute of Standards and Technology, Boulder, Colorado, USA
    2. Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, Colorado, USA
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  • James Randa,

    1. Electromagnetics Division, National Institute of Standards and Technology, Boulder, Colorado, USA
    2. Department of Physics, University of Colorado, Boulder, Colorado USA
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  • Robert Billinger,

    1. Electromagnetics Division, National Institute of Standards and Technology, Boulder, Colorado, USA
    2. R. Billinger is deceased 17 February 2013.
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  • David K. Walker

    1. Electromagnetics Division, National Institute of Standards and Technology, Boulder, Colorado, USA
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Corresponding author: D. Gu, Electromagnetics Division, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA. (


[1] We report measurements and uncertainty analysis on a cryogenic low-noise amplifier (LNA) with a very low noise temperature (NT), among the lowest noise performances reported at microwave frequencies. The LNA consists of three stages of InP high electron mobility transistors with a gate length of 130 nm. It exhibits about 44 dB gain and less than 2 K average NT in the operational band of 4 GHz to 8 GHz. A detailed uncertainty analysis is outlined to evaluate a variety of error sources in the measurement. The calculated uncertainty shows as low as 0.1 dB on the measured gain of about 44 dB and 0.18 K on the measured NT of 1.65 K, indicating excellent measurement accuracy. A breakdown of the uncertainty components helps identify the major causes of the overall uncertainty and enlightens us about how to further improve accuracy. It is important to know the actual physical temperature of the passive termination that is used as a cryogenic noise source in experiments. Due to its large temperature gradients, the commercial matched load is replaced by a custom-made attenuator that is isothermal and consequently provides reliable NT measurements of the LNA. The precision measurement technique developed at the National Institute of Standards and Technology is independent from the manufacturers' characterization method. This study marks the first time that such a low NT from a cryogenic LNA is verified independently with such a low uncertainty.

1 Introduction

[2] Cryogenic low-noise amplifiers (LNAs) are indispensable devices in ultra-sensitive radio receiver systems. The applications of these systems can be found in radio astronomy [Webber and Pospieszalski, 2002] and emerging millimeter-wave and terahertz imaging [Rodriguez-Morales et al., 2006]. Cryogenic LNAs produce very small noise contributions while providing appreciable gain to the detection signal. The best noise performance to date is achieved with InP high electron mobility transistors (HEMTs) operating at cryogenic temperatures. Progress on modeling [Malmkvist et al., 2008] and fabrication techniques [Ye et al., 2005] has allowed the continuous advance of cryogenic HEMT performance. Despite the fast development and the great demands from various applications, there are a limited number of reports on accurate characterization of noise properties of cryogenic amplifiers [Laskar et al., 1996; Hu and Weinreb, 2004; Randa et al., 2006; Cano et al., 2010; Russell and Weinreb, 2012].

[3] The underlying difficulties of accurate measurements on cryogenic LNAs are twofold. First, these LNAs operate in a cryostat, cooled by either an open-flow method or a closed-cycle cryocooler. The input port and the output port of the LNAs are not directly accessible to a user. Extraction procedures are required to calibrate the noise properties of the cables that guide the signal from the outside of the cryogenic systems at room temperature to the LNAs inside the cryogenic systems at low temperatures. Second, cryogenic LNAs often exhibit ultra-low noise temperature (NT) below 10 K, and some of them are approaching 1 K (for example, 1.2 K at 6 GHz was reported in a recent publication) [Schleeh et al., 2012]. This imposes a significant accuracy requirement on the extraction procedures to precisely characterize any external components in the system. Furthermore, a thorough analysis of the uncertainty associated with the NT is required in order to make measurements credible.

[4] Several years ago, we reported a method of precisely measuring the gain and the NT of cryogenic LNAs at the National Institute of Standards and Technology (NIST) in Boulder, Colorado [Randa et al., 2006]. The method showed very good accuracy in a broad frequency bandwidth. NT below 2 K was recently reported on newly developed LNAs from our collaborators [Schleeh et al., 2012], and a measurement comparison was then made at NIST. In this paper, we report a very low NT measured by us. A high-gain (> 40 dB) and low-noise (∼ 2 K) cryogenic amplifier was characterized by a radiometric system. The gain of the cryogenic LNAs has been verified independently from vector network analyzer (VNA) measurements. The uncertainty analysis shows excellent measurement accuracy and indicates consistency with the manufacturer's own measurements.

[5] The measured NT of the particular LNA is among the lowest noise performance reported. Other measurements of very low amplifier NTs include the following: Our previous publication [Randa et al., 2006] showed a result of 2.3 ± 0.3 K at 7 GHz for a cryogenic LNA operating over the 1–12 GHz range; a minimum Te of 1.4 K was reported in the 4–8 GHz band in 2003 [Wadefalk et al., 2003], and a recent publication showed a 1.2 K minimum Te with an average Te of 1.6 K in the same frequency band [Schleeh et al., 2012], both made by InP HEMTS; a two-stage InP HEMT LNA with 1.5 K NT in the band of 2–4 GHz was reported by Mellberg [Mellberg et al., 2004]; in 2009, Bardin presented a SiGe heterojunction bipolar transistor with 1.75 K at 2.9 GHz in his thesis [Bardin, 2009]; more recently, a large number of InP HEMT made LNAs with an average of 3.5 K to 4.5 K in 1–4 GHz band along with other cryogenic LNAs covering higher frequency bands were deployed in a radio astronomy system in New Mexico, USA [Pospieszalski, 2013]. Aside from the NIST report, most of these results lack a rigorous uncertainty analysis. Our recent result of below 2 K NT for a cryogenic LNA is substantiated by a reliable uncertainty calculation with an accuracy as good as 0.18 K.

[6] This paper is organized as follows. We present a description of measurement procedures, mainly including the analytic formulation to derive measured gain and NT, in section 2. A detailed uncertainty analysis is given in section 3. The uncertainty analysis along with appendices of the mathematical derivation is more comprehensive than our previous publication [Randa et al., 2006]. In section 4, experimental results are reported with some improvements in terms of lower NT and smaller measurement uncertainty in comparison to our previous work [Randa et al., 2006]. We discuss the importance of the thermal equilibrium of the passive termination (PT) used in the NT measurements as well as detailing individual items of the measurement uncertainty in section 5. The isothermal issue of the PT is newly revealed, which actually explains some discrepancies in an earlier conference paper [Gu et al., 2012]. In the end, we conclude with the measurement results, emphasize the uncertainty analysis, and propose a possible improvement on the measurement apparatus in section 6.

2 Procedures and Theory

[7] The device under test (DUT) was housed in an open-cycle liquid helium cryostat at about 4 K. The only accessible ports are the two 7 mm connectors at room temperature, as shown in Figure 1a. Precision adapters are used to translate the 2.9 mm precision connector (PC) on the cryostat to a PC-7 connector interface for measurements on the NIST coaxial radiometric measurement system. The adapter noise characteristics are combined together with the overall cable noise characteristics in the analysis. To de-embed the NT and the gain of the cryogenic DUT, the properties of the cables have to be known first. Specific measurements are designed to take care of the gain extraction and the noise temperature extraction independently. The two-port configuration in Figure 2a is used for measurements to calculate the gain, whereas the one-port configurations in Figures 2b and 2c are used for measurements to determine the NT through the input path and the output path, respectively. In the one-port configuration, an additional thermal sensor (marked as “TS2” in Figure 1b) is used to closely monitor the physical temperature of the PT.

Figure 1.

(a) A disassembled view of the cryostat. The operation space is inside the 4 K shield. (b) A view of the inside of the 4 K shield. The setup as shown is the one-port configuration that includes a passive termination, an attenuator in this case, at the input port of the LNA. Both the attenuator and the LNA are seated on top of a copper block that is attached to the 4 K base plate. “TS1” and “TS2” indicate two thermal sensors.

Figure 2.

Illustration of radiometric measurements on a cryogenic DUT. (a) Measurement setup for gain extraction. (b and c) Internal PT for measurements of NT through the input cable and the output cable.

2.1 Extraction of Cable Properties

[8] The cryostat contains two cryogenic vessels, which are filled with liquid nitrogen and liquid helium, respectively. As a result, the cables possess nonuniform temperature distributions from about 4 K to 296 K. Besides the nonnegligible loss they present, these cables contribute noise in addition to the noise from the DUTs. The most feasible way to characterize their loss and their added NT is through a series of radiometric measurements [Randa et al., 2006]. We name the gain of the cables as α1 and α2 (both smaller than one), corresponding to the input cable “I-1” and the output cable “2-O”, respectively. Also, the NTs contributed from “I-1” and “2-O” are denoted by ΔT1 and ΔT2. We replace the DUT with a short thru section and connect two known noise sources (NSs), one hot NS with NT of Th and one cold NS with NT of Tc, sequentially to the input port of Figure 2a. Next, we measure the output NT with a radiometer system.

[9] The total gain of the cables is given by

display math(1)

where math formula and math formula represent the measured NT on port “O” for the two-port configuration (with a short thru section instead of the DUT in Figure 2a) when the hot NS and the cold NS, respectively, are connected to port “I.”

[10] Radiometric measurements on port “I” and port “O” allow us to determine ΔT1 and ΔT2 when an internal PT terminates the input cable and the output cable, respectively, as shown in Figures 2b and 2c with the DUT replaced by a short thru. The internal PT functions as a cryogenic NS providing a NT near 4 K to the thru connector. ΔT1 and ΔT2 can be obtained from

display math(2a)
display math(2b)

where math formula and math formula are the measured NTs on port “I” (Figure 2b) and port “O” (Figure 2c), and THe is the liquid helium temperature. The values of α1 and α2 can be taken approximately as math formula with negligible influence on the LNA NT calculation. THe is measured by a calibrated thermal sensor that is directly attached to the PT, as shown in Figure 1b.

2.2 Measurement on LNA

[11] After the losses and the NTs of the cables are extracted, we perform a set of measurements with the LNA connected. The measurements of the LNA are broken down to the similar steps as we have done for the thru configuration in section 2.1 to determine the gain from Figure 2a and determine the NT from Figures 2b and 2c. The mathematical derivations for gain and NT calculations from measurements can be found in the previous publication [Randa et al., 2006]. Here we list only the basic formulas, as follows.

[12] For the overall gain, we have

display math(3)

where math formula and math formula are the measured NTs on port “O” when the hot NS and the cold NS are connected to port “I,” respectively, as shown in Figure 2a. Γout is the measured reflection coefficient on port “O” for the Figure 2a configuration. The last fractional term in (3) represents the correction to the modified gain of the cable “2-O” due to the nonnegligible mismatch on the LNA output.

[13] We obtain the effective NT Te from

display math(4)

where math formula is the measured NT on port “I” referring to Figure 2b and math formula is the measured NT on port “O” referring to Figure 2c. math formula and math formula are the modified gains of cable “1-I” and “2-O” (Figures 2b and 2c) due to reflection at the LNA output. Tv is the NT corresponding to the vacuum fluctuation (Tvhf/(2kB), h is the Planckconstant, f is the frequency, and kB is the Boltzmann constant).

3 Uncertainty Analysis

[14] For measurements of such low-noise temperatures, the uncertainty analysis is crucial. Radiometric measurement uncertainty arises from a number of sources, such as the primary noise standards, the mismatch effect between the DUT and the measurement port, the different losses presented in the measurement ports, and nonlinearity [Grosvenor et al., 2000]. Besides all these sources that eventually propagate to the overall uncertainty of G and Te, we need to go beyond to take account of the variation of the quantity (G or Te) between the ideal matched case (ΓS=0) and the actual measurement conditions (ΓS≠0). This is proven to be critical in our study, since G and Te of most cryogenic LNAs are referred to the gain and the effective NT with an ideally matched input, although a perfectly matched termination never exists in practice. Different measurement laboratories may use a nearly matched termination with different reflection coefficients to measure a DUT. Hence, this uncertainty should be included in the reports. To our knowledge, this concept has not yet been adopted in most publications of NT measurements on cryogenic LNAs.

3.1 Uncertainty of G

[15] The gain is determined by combining (1) and (3), from which we write its uncertainty as

display math(5)

The first two terms under the square root follow directly from the usual formula for propagation of uncertainties [Taylor and Kuyatt, 1994]. The imperfectly matched load (ML) error term uML(G) is the uncertainty due to our assumption that all the input terminations in the measurements are reflectionless. The first term in (5) is evaluated from the expression for α1α2 in (1). To evaluate uML(G), we take the difference between the full expression for the available gain and the form that it assumes for ΓS=0. Algebraic details are outlined in section A; the result is

display math(6)

where scattering (S-) parameters are those measured between ports “I” and “O” with the amplifier present, as in Figure 2a. |ΓPT| is the magnitude of the reflection coefficient of the PT. The remaining contribution to u(G) is from measurement of α1α2G, the gain between ports “I” and “O.” From (3) and considerations similar to those above, we can write

display math(7)

where math formula, ΔTin=ThTc, and uML(α1α2G) again refers to the uncertainty due to imperfect matching conditions. From section A,

display math(8)

where |ΓNS| is the magnitude of the reflection coefficient of the NS.

[16] Equations (6) and (8) deal with different conditions. Referring to Figure 2a, equation (8) accounts for the imperfect NS reflection in the real experiments, where the cables and the LNA are viewed collectively as a consolidated component. Equation (6) represents an embedded condition, where the LNA is viewed alone. Although it is not obvious, uML(G) implicitly contributes to the measurements in Figures 2b and 2c.

3.2 Uncertainty of Te

[17] For the uncertainty in the effective input noise temperature of the amplifier, we return to (4) and write

display math(9)

where uML(Te) accounts for the variation of Te when the LNA input is connected to an imperfectly matched termination. (Note that there is a typographical error in (16) and (17) of Randa et al. [2006] and an algebraic error in (17) there, with minor consequences.) Most of the terms in (9) are straightforward to evaluate. The uML(Te) term requires some algebra and approximations (see section B), which result in

display math(10)

[18] Equation (10) clearly indicates that uML(Te) will present an uncertainty of more than 3%(=10−30/20) of the value of Te even for a PT with a 30 dB reflection loss and a perfectly matched LNA input with math formula. As we will show in section 5.2, uML(Te) plays a nonnegligible, sometimes even a dominant, role in the overall uncertainty of Te.

4 Experimental Results

[19] The cryogenic LNA under test in this paper is a three-stage InP HEMT with more than 40 dB gain and about 2 K NT. Its operation is specified between 4 GHz and 8 GHz with a dissipation power of a few milliwatts at cryogenic temperatures. These HEMTs were fabricated by use of microstrip monolithic microwave integrated circuit process. The HEMT gate length was 130 nm, and its gate width was 2 × 100 μm. The optimized gate recess of these HEMTs made possible the low gate current and the high transconductance at the low drain current, which in turn enabled the outstanding low-noise performance [Schleeh et al., 2012].

[20] A noise diode with about 1000 K absolute NT (excess noise ratio of about 4 dB) was used as the hot NS, and a room temperature ML functioned as the cold NS. In addition, a liquid-nitrogen cooled ML was used as another NS for a consistency check. However, only the radiometric measurements with the noise diode and the room temperature ML were included here. The NIST noise-figure radiometer (NFRad) was used to measure the NT [Grosvenor et al., 2000]. NFRad measurements were set at integer frequencies from 4 GHz to 8 GHz. A custom-made attenuator with a loss of about 20 dB was used as the PT in the one-port configuration measurements: Figures 2b and 2c. Unlike the earlier measurements [Randa et al., 2006], there were issues of thermal equilibrium for the PTs, which are discussed below in section 5.1. Every measurement was repeated twice to check the repeatability. The difference between the two measurements was included as the Type-A uncertainty to account for the random variation.

[21] The properties of the cryostat cables and the LNA were successfully extracted from measurements following the procedures mentioned in sections 2.1 and 2.2. Figure 3 shows the measured results of the cables and the LNA. We also checked the reciprocity of the cable by flipping the NSs to port “O” and measuring the NT on port “I,” noted as the backward measurement. Two-port VNA measurements were performed on port “I” and “O” when the input and output cables were connected by the short thru and by the LNA, respectively. The difference of the two was considered to be the gain of the LNA by VNA measurements. The extracted gain from radiometric measurements was compared to VNA measurements, showing very good agreement.

Figure 3.

Experimental results of cables and the LNA. (a) Cable gain and NT. Both the forward measurement and the backward measurement agreed with VNA measurements. (b) LNA gain. NIST measurements showed agreement, but they are slightly lower than values from the manufacturer. (c) LNA NT. NIST data and manufacturer's data mostly agree.

[22] The input and output cables overall produced about 3 dB loss in the frequency range of 4 GHz to 8 GHz, as shown in Figure 3a. The measurement uncertainty of α1α2 represented less than 1% error of the value. The cables' effective NT was in the range of 39 K to 49 K, with a maximum of 1.3 K uncertainty. In Figure 3b, the extracted gain value of the LNA from radiometric measurements was consistent with |S21|2from the VNA measurements at NIST. However, the manufacturer-specified gain was slightly larger, with a difference as much as 0.3 dB compared to NIST measurements. In Figure 3c, measured NTs by NIST showed slightly higher values than the manufacturer's data, although they overlapped within the uncertainty.

[23] Measured values of the gain and the NT along with the uncertainty in comparison to the manufacturer's own measurements are listed in Table 1. We achieved as small as 0.1 dB uncertainty in the gain measurement and as small as 0.18 K uncertainty in the NT measurement. The breakdown of contributions to the measurement uncertainty is discussed in section 5.2.

Table 1. LNA Gain and NT Measurement Comparisona
FrequencyNIST GainManufacturer GainNIST NTManufacturer NT
  1. a

    Uncertainty from manufacturer excludes uML(Te).

444.15 ± 0.1544.462.08 ± 0.261.87 ± 0.65
544.20 ± 0.1244.521.65 ± 0.181.41 ± 0.65
643.78 ± 0.1043.981.90 ± 0.201.74 ± 0.65
743.16 ± 0.1143.412.00 ± 0.191.70 ± 0.65
843.80 ± 0.2143.822.33 ± 0.332.32 ± 0.65

5 Discussion

5.1 Thermal Equilibrium of PT at Cryogenic Temperature

[24] It is evident from (4) that the reading of THe has a direct impact on the accuracy of the effective noise temperature. In the preliminary investigation of this work [Gu et al., 2012], we used a commercial ML as the PT for the cryogenic input NS for measurements as described in Figures 2b and 2c. Because of the small footprint of the commercial ML, we were unable to attach a thermal sensor directly on the ML body. The determination of THe relied on the thermal sensor (“TS1”) that was located in the vicinity of the ML. It turned out that the package of the ML was actually slightly warmer than the reading from “TS1,” and more importantly, the resistive element inside its package was much warmer (by more than 1 K) than the outside due to a poor thermal conduction between the package and the resistive element. As a result, the calculated Te of the LNA by using the commercial ML was about 1.5 K higher.

[25] In order to overcome this problem, we replaced the commercial ML with a custom-made attenuator that provided negligible temperature gradient between the metallic package and the attenuator chip [Cano et al., 2010]. Note that different from Cano's original approach of using it as a two-port attenuation component, we used the attenuator as a one-port PT in our experiments. The replacement by the attenuator proved to be crucial. Radiometric measurements showed a lower NT from the configurations of Figures 2b or 2c when the attenuator was used as a PT, as opposed to the same configuration when the commercial ML was used as a PT. This clearly indicated that the reading of “TS2” provided a reliable physical temperature of the attenuator chip, while the use of “TS1” to estimate the ML temperature was unreliable. In the experiments, we observed “TS2” readings were slightly higher than “TS1” readings by 0.15 K, which was much higher than u(Te) mentioned in section 5.2, indicating the importance of the direct attachment of a thermal sensor on the PT. This issue did not arise in Randa et al. [2006], because we were able to mount the LNA and the commercial ML directly on the base plate of the cryostat. As a result, the thermal conduction was much greater and that in turn allowed the commercial ML to reach closer to liquid helium temperature. The LNA in this study has a different packaging configuration and requires an extra copper block for suspension from the base plate, as shown in Figure 1b. Such a setup incurs reduced thermal conduction and results in a nonisothermal commercial ML.

5.2 Uncertainty Assessment

[26] The VNA measurement uncertainty is estimated to be 0.5 dB for the high-gain LNA measurement (> 40 dB), which is obtained from the uncertainty calculator provided by the VNA manufacturer [Agilent Technologies, 2012]. Therefore, all the gain values from radiometric measurement are within the VNA measurement uncertainty in Figure 3b. The major contributions to u(G) come from the NT measurement of Th-O and the mismatched condition.

[27] The uncertainty analysis of Te shows three major components: the contribution from the α1α2G, the mismatched condition, and THe. The separate values are listed in Table 2. α1α2G represents the overall gain of the signal path, and its accuracy certainly plays a major role in the uncertainty of Te. The contribution from uML(Te) varies greatly and becomes dominant when the input of the LNA is poorly matched (at 8 GHz in this case). uML(Te) is evidently a nonnegligible factor that accounts for 20% to 78% of the overall uncertainty. The omission of its contribution consequently causes an underestimate of the measurement uncertainty, and it should be therefore included in the analysis.

Table 2. Breakdown of u(Te)
Frequencyu(Te)math formulauML(Te)u(THe)

[28] The contribution from u(THe) is estimated from the temperature variation in the experiments. The typical accuracy of our thermal sensor is about 5 mK, and its long-term stability is about 25 mK. The monitored temperature of the PT varies somewhat from one experiment to another, especially when the LNA is flipped 180° as shown in Figures 2b and 2c, and the location of the heat dissipation by the LNA, in turn, changes. The standard deviation of the mean THe is about 40 mK. We therefore estimate u(THe) to be 50 mK.

[29] The uncertainty breakdown shown in Table 2 indicates that math formula and uML are equally important. We further investigate the uncertainty components of math formula according to (6) and list them in Table 3. The uTO) term is evidently the dominant factor, which arises from the uncertainty of radiometric measurements of NT. This term is directly related to the instrument accuracy of the NFRad, and its improvement will be limited. Therefore, uML(Te) is the only remaining term that can be possibly reduced to improve the accuracy in the future. Referring to (10), the further reduction of the uncertainty due to the imperfectly matched condition relies on the minimization of |ΓPT| and math formula. The S-parameters of a LNA are intrinsic. Thus, the only avenue for improvement is to lower the reflection magnitude of the PT. The custom-made attenuator used in this study exhibits about −18 dB to −26 dB reflection in the frequency band of 4 GHz to 8 GHz at about 4 K. Future improvements to achieve an uncertainty smaller than 0.2 K across the band may require a PT with a broadband reflection below −30 dB at cryogenic temperatures, especially at the frequency points where math formula is somewhat large.

Table 3. Breakdown of u(α1α2G)/(α1α2G)
Frequency (GHz)math formulamath formulamath formulamath formula

[30] The accuracy of the gain and NT on the LNA was comparable to that of the previous results. Note the previous results contain some underestimation of uML(Te) due to algebraic error in (17) in Randa et al. [2006]. Therefore, the current result shows some improvements, which are attributed to a better matched LNA input that resulted in a smaller contribution in the mismatched condition.

6 Conclusion and Future Work

[31] In conclusion, we successfully measured a cryogenic amplifier with 44 dB gain and a minimum 1.65 K NT in the frequency range of 4 GHz to 8 GHz by radiometric measurements. The measurement accuracy was as good as 0.1 dB for the gain and 0.18 K for the NT. For those who are more accustomed to noise figure (NF) in dB, defined by NF≡10log10[(290+Te)/290], we measured NF as low as 0.024 dB, with an uncertainty of 0.003 dB.

[32] The measurement uncertainty in this study is based on a rigorous uncertainty analysis. We expand the regular error propagation to include uncertainty originating from imperfectly matched terminations. Our analysis indicates that this uncertainty source is nonnegligible, sometimes dominant, when computing the overall uncertainty. The lowest measurement uncertainty on both the gain and the NT reflects some improvements in comparison to the result in Randa et al. [2006] (from 0.2 dB to 0.1 dB for gain uncertainty and from 0.3 K to 0.2 K for NT uncertainty). One possible enhancement to further reduce u(Te) requires a PT with less reflection magnitude over the frequency band of interest in the cryogenic environment.

[33] The precise measurement of the physical temperature of the PT is emphasized in the paper. A custom-designed PT, such as a ML or an attenuator, instead of off-the-shelf components, is advised. The PT must have a high thermal conductivity between its package and its inner element. A thermal sensor integrated inside the PT package is optimal to monitor its temperature.

[34] In the current study, the PT placed in the cryostat is equivalent to a NS at the fixed temperature of about 4 K connected to the input of the DUT. A local heating technique to warm the cryogenic components to variable temperatures is reported in Russell and Weinreb [2012]. Such a technique can be adapted in our system to heat the PT to temperatures other than 4 K. This capability would enable us to produce a cryogenic NS with a set of NTs directly to the DUT, which is similar to the approach of a hot and an ambient NS sequentially connected to the port “I” to extract the total gain, as shown in Figure 2a. The additional advantage is that the new apparatus would allow us to decompose the input and output cable properties and provide two extra independent checks of the LNA Te from individual one-port configurations.

Appendix A: Derivation of uML(G)

[35] For the mismatch contribution to the uncertainty in the available gain, we first look into the full expression for the available gain (GA),

display math(A1)

where ΓS is the reflection coefficient of the input termination (the PT or the NSs), Γout is the reflection coefficient looking into the output of the component: Γout=S22+(ΓSS21S12)/(1−ΓSS11).

[36] uML(GA) can be solved from the root mean square (RMS) of the difference between GAS=0) and GAS≠0), given by

display math(A2)

where GA0=GAS=0).

[37] In the following content, we provide a convenient analytic formula of uML(GA). The other alternative is to plug the magnitude of all S-parameters and Γ from measurements in (A2) and solve the RMS average from numerical integration. Our study shows that these two methods produce a negligible difference.

[38] The quantity in brackets of (A2) is expanded using the approximation (1−x)−1≈1+x, valid for small |x|. This leads to

display math(A3)

where R{·} denotes the real part of a complex number and (·)is the conjugate of a complex number. We have kept only the lowest order in small quantities. Since we do not know the relative phases of the scattering parameters and of ΓS, we take the RMS over those phases, leading to

display math(A4)

[39] When we view the cables and the LNA as a consolidated element, (A4) simply reproduces (8) by replacing GA and ΓS with α1α2G and ΓNS. On the other hand, when we consider the LNA itself, we just need to replace ΓS with ΓPT in (A4) and calculate the S-parameters of the LNA to obtain (6). We do not measure the S-parameters of the amplifier between planes “1” and “2,” but we can estimate their magnitudes from the measured S-parameters between planes “I” and “O” with the following relationships;

display math(A5)
display math(A6)
display math(A7)

We substitute all these relations into (A4) and retrieve (6):

display math(A8)

Appendix B: Derivation of uML(Te)

[40] For the mismatch contribution to the uncertainty in the effective noise temperature, uML(Te), we again estimate it by the magnitude of the terms that are neglected by assuming that ΓS=0. The effective NT of an amplifier is a function of ΓS due to its intrinsic noise properties,

display math(B1)

where Tmin is the minimum effective NT when the input of the LNA is connected to a termination that shows the optimal reflection coefficient ΓSopt, and t is a factor that characterizes how much Te varies as different terminations with ΓS are connected. For simplifying algebra, we find it convenient to work with noise matrix elements (X1, X2, and X12) in the wave representation. Our notation, based on Wedge and Rutledge [1992], can be found in Randa [2002]. The X-parameters are elements of the intrinsic noise matrix, referred to the amplifier input plane. In particular, X2 is the effective NT when a perfect ML terminates the LNA: X2=TeS=0).

[41] In terms of the X-parameters, uML(Te) can be calculated from TeS≠0)−TeS=0), given by

display math(B2)

[42] By expanding the denominator and omitting higher order terms of ΓS, we end up with

display math(B3)

[43] We then take RMS of (B3) over unknown phases and reach

display math(B4)

[44] We make the approximation X2Te and estimate the magnitudes of X1 and X12. In our experience, X1X2. For X12, it is reasonable to take the RMS of the forward and backward noise waves and write math formula. We then reduce (B4) to

display math(B5)

which is the formula used in (10) for the variation of Te when an imperfect PT terminates the LNA input.


[45] We thank N. Wadefalk at the Chalmers University of Technology in Sweden for providing test components and helpful discussions.

[46] The work of this publication is supported by NIST, an agency of the U.S. government, and is not subject to U.S. copyright.