UNIVERSIDAD DE CANTABRIA TESIS DOCTORAL

Transcription

1 UNIVERSIDAD DE CANABRIA Depatamento de Ingenieía de Comunicaciones ESIS DOCORAL Cyogenic echnology in the Micowave Engineeing: Application to MIC and MMIC Vey Low Noise Amplifie Design Juan Luis Cano de Diego Santande, Mayo 010

2 Chapte III Measuement echniques fo Cyogenics he design of any cicuit stats with the modelling of its diffeent components and finishes with the chaacteization of the cicuit itself. Both fo taking measuements, fom whee the equied models ae extacted, and fo chaacteizing the final design, diffeent measuements have to be pefomed. In the case of low noise amplifies Scatteing paametes (S-paametes) and noise measuements ae usually enough fo modelling and final chaacteization. In cyogenics these measuements ae even moe challenging than at oom tempeatue. Fo the S-paametes measuement the desied calibation planes ae not diectly accessible and theefoe some modifications to the classical measuement techniques have to be applied. hese modifications usually compomise the measuement accuacy. Fo the noise, not only the calibation planes ae not accessible but also the noise souce powes involved in the R techniques ae lage compaed with the measued noise values, which affects noise accuacy. his chapte gives an oveview of some calibation methods to measue cyogenic S-paametes focusing in a technique to avoid dift eos as well as systematic eos in the RF ange. hen, the cold-attenuato technique, which is one of the most widespead methods fo measuing noise in cyogenics, is explained in detail. Moeove, a compehensive study of the uncetainty in this technique using Monte-Calo analysis is pesented. Finally, it is shown a new design of a chip attenuato that impoves accuacy in noise measuements with the cold-attenuato technique. 55

3 Chapte III Measuement echniques fo Cyogenics 3.1. Oveview of Calibation echniques fo Cyogenic S-Paametes Measuements he calibation is a pocess though which diffeent pecisely known devices, called standads, ae measued using a netwok analyze in ode to obtain the diffeences between measued and actual values. Fom these measuements, a set of paametes, called eo tems, descibing the test set ae calculated, and the measuement efeence plane is moved to the DU pots. he above mentioned eo tems chaacteize the systematic eos intoduced by the netwok analyze and the diffeent hadwae used to connect this equipment to the DU (cables, connectos, adaptes ). hese systematic eos ae time invaiant and theefoe can be chaacteized and eliminated though pope calibation [3.1]. As well as these, andom eos and dift eos may be pesent duing the measuements; the fists ae andomly time vaiant so they can not be eliminated, while the late ae poduced by pefomance vaiations of the diffeent components once the calibation has been made. hese vaiations ae mainly poduced by tempeatue changes and can be eliminated with successive calibations. At oom tempeatue ( = 96 K) seveal calibation techniques can be applied to eliminate systematic eos [3.]-[3.4]. he connection of the diffeent standads at the desied calibation plane is staightfowad and it is not time consuming, which avoids the appeaance of dift eos. At cyogenic tempeatue the DU is enclosed by the Dewa so that its pots ae not diectly accessible. heefoe, calibation techniques applied at oom tempeatue need to be conveniently modified. At this point it is needed to define the concept of themal cycle, which is the peiod of time that includes a cyostat cooling down to the test tempeatue, a measuement, and a cyostat heating up to oom tempeatue. A themal cycle usually takes seveal hous, anging fom five hous to one day depending on the available cyogenic system. Accoding with this definition, the diffeent measuement techniques can be divided into two main goups: those which need only one themal cycle fo the calibation and measuement, and those which need one themal cycle fo each standad chaacteization duing calibation and one last themal cycle fo the final measuement. Fom the pevious classification it is clea that measuement techniques included in the second goup may be affected by dift eos that need to be emoved fo pecise calibation Measuement techniques involving one themal cycle In these techniques all the standads ae intoduced at the same time togethe with de DU in the Dewa. Once the cyostat is cooled, the calibation is pefomed and the 56

4 3.1. Oveview of Calibation echniques fo Cyogenic S-Paametes Measuements measuement is taken. he complete calibation pocess is caied out just befoe the measuement, thus avoiding the appeaance of dift eos. Any of the calibation techniques developed fo oom tempeatue can be used with these measuements just intoducing the suitable numbe of standads. he poblem aises with the method followed to exchange fom one standad to othe at the desied calibation plane. Depending on the selected method the measuement accuacy can be seiously affected. Fo components o devices (not connectoized) the chaacteization can be made fast and accuate if a cyogenic pobe station is available [3.5]-[3.7]. In this case a calibation substate including all the standads is cooled close to the DU and the measuement is made in the same way as at oom tempeatue. Modification of standads electical chaacteistics with tempeatue may affect the measuement pecision. If a cyogenic pobe station is not available o the components and systems ae connectoized then diffeent electical paths ae used fo each element. he diffeences between paths, which ae assumed to be equal, poduce a degadation of measuement accuacy, limiting the maximum fequency ange. o change between these diffeent paths it is necessay a switching method eithe inside o outside the Dewa. An electomechanical switch can be used inside of the Dewa to exchange between the diffeent standads and the DU. he advantage of this option is that only one RF feedthough is needed fo all the elements and theefoe the diffeences between paths ae educed. he main dawback is that a switch coveing the full band and woking at cyogenics with small o negligible diffeences between its ways may be vey difficult o impossible to find. A sketch of a calibation and measuement using this pocedue is shown in Fig DC CONROL CRYOSA DEWAR Calibation Plane DC CONROL DU RF FEEDROUGH SWICH SHOR HRU SWICH RF FEEDROUGH LINE Figue 3.1. Example of calibation and measuement in one themal cycle using switches inside the Dewa. A hu-reflect-line (RL) calibation technique is applied in this example though a home-made RL test-fixtue designed fo non-insetable devices. he calibation plane is moved fom netwok analyze to DU pots. 57

5 Chapte III Measuement echniques fo Cyogenics An altenative to the pevious method is to use a feedthough fo each element in the cyostat [3.8]. he switch between diffeent feedthoughs can be made manually outside the Dewa. he advantage of this technique is the absence of switches woking at cyogenic tempeatues which eliminates a geat souce of inaccuacy; on the othe hand, the main dawback of this method is the need of electically simila feedthoughs and connectos. he moe diffeent they ae the lowe maximum fequency they can be used. Figue 3. shows this altenative technique. Fig. 3.. Example of calibation and measuement in one themal cycle using diffeent feedthoughs fo the standads and DU. A hu-reflect-line (RL) calibation technique is applied in this example though a home-made RL test-fixtue designed fo non-insetable devices. he calibation plane is moved fom netwok analyze to DU pots Measuement techniques involving seveal themal cycles In these techniques moe than one themal cycle is necessay depending on the calibation method and the selected stategy. he main advantage of these techniques is that all the elements duing the calibation and measuement have the same, o almost, electical path; but, as a consequence of the vey time consuming calibation pocedue, dift eos may be pesent in the measuement. Following with the example pesented in Fig. 3., a measuement applying the hu-reflect-line (RL) calibation could be caied out in fou themal cycles if only two feedthoughs ae used. In each themal cycle one standad is connected to the feedthoughs while the othe elements ae left open. Between themal cycles, the element to be measued must be changed manually. In this technique diffeent connectos ae used fo each standad and DU. Since these connectos ae consideed to be electically equal duing the calibation then the measuement accuacy is affected. o ovecome the poblem of diffeent connectos in non-insetable devices special test-fixtues can be developed [3.9]. his technique impoves calibation accuacy and can be used at highe fequencies than the othe techniques. he poblem is to design a test-fixtue eadily configuable fo diffeent devices and standads that woks popely 58

6 3.1. Oveview of Calibation echniques fo Cyogenic S-Paametes Measuements at cyogenic tempeatues with a minimum size. hee ae some commecial altenatives fo oom tempeatue test-fixtues [3.10] but they ae too big to be installed in the Dewa and thei pefomances unde cyogenic tempeatues have not been checked. Finally, a hybid technique is applied in [3.11] to measue S-paametes of a connectoized system. In this technique the main calibation is made at oom tempeatue at the desied efeence plane using an electonic calibation kit. hen it is supposed that small vaiations ae poduced upon cooling and some modifications ae applied to the fist calibation to take into consideation these vaiations. hese vaiations ae obtained fom the measuement of diffeent standads in diffeent themal cycles. he advantage of this method is that the chaacteization of the above mentioned vaiations ae pefomed once fo the available setup and only the oom tempeatue calibation is needed fo the subsequent measuements. he dawback of this technique is that it is only advantageous if the setup in the Dewa is always the same, i.e. the efeence is always set at the same plane, othewise this technique has the same advantages and disadvantages than the othes pesented in this section Accuate hu-reflect-line calibation avoiding dift eos Excluding on-wafe measuements with a cyogenic pobe station, among the measuement techniques pesented in pevious sections those which need seveal themal cycles ae moe accuate since they use the same electical path fo all the elements. he RL calibation is one of the most accuate techniques within the oom tempeatue calibation methods using a minimum numbe of standads: one hu, one Reflect (eithe Shot o Open) and a Line (λ/4 longe than hu standad at cental fequency). On the othe hand, it has a limited bandwidth (fequency span/stat fequency atio of 8:1) that can be extended using additional Line standads [3.1]. Since the RL calibation uses minimum numbe of standads the calibation pocess is faste than othe techniques such us Shot-Open-Load-hu (SOL) which equies one themal cycle moe. heefoe the RL method has been selected in this thesis to be adapted to cyogenic measuements. he basic setup fo applying RL calibation is shown in Fig

7 Chapte III Measuement echniques fo Cyogenics Fig Measuement setup applying RL calibation at cyogenic tempeatue. In the geneal calibation pocedue at cyogenics the netwok analyze emains uncalibated until all the standads ae measued which usually takes many hous o even days. Moeove, once the netwok analyze is calibated, the DU measuement is delayed one additional themal cycle. If the measuement equipment is pone to vay its pefomance with time then dift eos will be pesent in the final measuement. As stated in Section 3.1, dift eos can be eliminated with successive calibations, theefoe a modification of the geneal calibation pocedue to enable calibation of the netwok analyze befoe each measuement is poposed in this thesis. he main idea behind this modification is to pefom the calibation calculations (at efeence plane B in Fig. 3.3) using a compute, while the netwok analyze is used only to obtain the diffeent measuements duing the time consuming calibation. Since it is not necessay to keep the standads measuements in the analyze memoy duing the calibation, then this equipment can be calibated (at efeence plane A in Fig. 3.3) befoe each standad is measued, thus eliminating its dift eos. Once the systematic and dift eos of the netwok analyze have been eliminated, the DU S-paametes at efeence plane B need to be obtained using the diffeent measuements taken at efeence plane A. he calculations to eliminate de systematic eos in the setup between efeence planes A and B ae pesented in the following subsection. 60

8 3.1. Oveview of Calibation echniques fo Cyogenic S-Paametes Measuements heoetic backgound of RL calibation In geneal, systematic eos associated with a two-pot system can be chaacteized with a set of twelve eo tems [3.13]. (a) (b) Fig wo-pot 1-tem eo model; (a) 6-tems fowad, (b) 6-tems evese. he meaning of the diffeent tems is the following: - e 00 : Fowad diectivity - e 11 : Pot-1 match in the fowad diection - e : Pot- match in the fowad diection - e 10 e 01 : Fowad eflection tacking - e 10 e 3 : Fowad tansmission tacking - e 30 : Fowad isolation - e 33 : Revese diectivity - e : Pot- match in the evese diection - e 11 : Pot-1 match in the evese diection - e 3 e 3 : Revese eflection tacking - e 3 e 10 : Revese tansmission tacking - e 03 : Revese isolation In the RL technique the calculation of these tems stats with the sepaation of the measuement system in two diffeent pats containing the tems at the input and output of the DU. In ode to do this, it is necessay to eliminate the isolation eo tems [3.13]. he measuement of these tems is caied out though the use of matched loads at both pots and measuing both paametes S 1 and S 1, thus: 61

9 Chapte III Measuement echniques fo Cyogenics m e S1 30 = (3.1) e ' = S 1 m 03 (3.) Whee supescipt m denotes that the paametes ae measued. If the insetion loss of the DU is not high (same ode of the test-fixtue o equipment isolation), then these tems can be estimated as zeo and theefoe they can be omitted in the calibation pocess [3.14]. Afte sepaating the eo set into two blocks the new flow gaph epesenting the system is as shown in Fig m S O S X { S DU SY a 1 e 10 S DU1 e 3 b e 00 e 11 S DU 11 S DU e e33 b e a 1 01 S e 3 DU1 Fig tem eo model. Now the poblem is educed to find eight eo tems. his is the model associated to old netwok analyzes, such as 870C fom Hewlett-Packad, which can not distinguish between the intenal matching of pot 1 and pot, and theefoe they assume that the impedance of the intenal switch is the same in both diections [3.15]. Newe netwok analyzes, such as model 8510C fom Hewlett-Packad, can pefom measuements to eliminate the eos intoduced by the switch at its diffeent states. Accoding with the calibation pocedue poposed in this thesis, once the standads and the DU have been measued at calibation plane A thee is a set of fou S-paamete files, which ae conveniently manipulated to obtain fist the eight eo tems in Fig. 3.5 and finally the DU S-paametes at calibation plane B. hese eo tems epesent the test-fixtue between calibation planes A and B. he DU S-paametes at calibation plane B ae obtained with equations (3.3)- (3.6). S S DU m m ( e S )( e S ΔS ) m m eso1so O11 O Y 11 = (3.3) D e S S + m m ( e S )( e S ΔS ) m m 11 DU = O1 O 1 33 O 11 O11 D X (3.4) 6

10 3.1. Oveview of Calibation echniques fo Cyogenic S-Paametes Measuements S S e10e3s D m = O1 DU1 e01e3s D m = O 1 DU 1 (3.5) (3.6) D ΔS Whee, m m ( e S ΔS )( e S ΔS ) m m = e11 eso 1SO 1 11 O 11 X O Y (3.7) X = e e e e ΔSY = ee33 e3e3 (3.9) A compehensive deivation of the pevious expessions can be found in Annex II. hese calculations ae based in the infomation given in [3.3], [3.13], [3.16] and [3.17]. (3.8) Expeimental veification of the poposed technique he expeimental veification of the technique pesented in the pevious section is caied out at oom tempeatue since a suitable DU cyogenic measuement is not available to compae with. Howeve the calibation pocedue is caied out in the same way as it was fo a cyogenic measuement. A tansisto, model AF-551M4 fom Avago echnologies, is mounted in a suitable test-fixtue designed fo measuing it applying the RL calibation technique in the 0.4 to 3. GHz band (8:1). his test-fixtue, shown in Fig. 3.6a, also includes the hu, Reflect (shot-cicuit) and Line standads. he test-fixtue emulates the measuement setup of Fig. 3.3 whee the calibation plane A would be the coaxial connectos efeence plane in Fig. 3.6a, and calibation plane B would be the tansisto efeence plane. Hence, the contibution of the diffeent coaxial connectos and the netwok analyze dift eos is coected with this technique in this case. Fist, the netwok analyze is calibated at calibation plane A (coaxial calibation SOL) and then one of the standads is measued fom this calibation plane. Aftewads, the analyze is etuned to its oiginal state (PRESE) to simulate the loss of calibation accuacy poduced by time consuming measuements (dift eos). his pocedue is epeated fo the othe standads and fo the tansisto biased at the desied bias point, in this case V ds = V and I ds = 10 ma. Second, these fou measuement files ae intoduced in the developed softwae to calculate the tansisto S-paametes at calibation plane B (3.3)-(3.6). Finally, the same tansisto is measued once again but in this case using the nomal RL technique at calibation plane B to compae with the calculated esult. his compaison is pesented in Fig. 3.6 showing almost identical cuves between the calculated (ed cuves) and measued (blue cuves) S-paametes. 63

11 Chapte III Measuement echniques fo Cyogenics (a) S(,1) S(1,)*80 S(1,1) and S(,) S(,) S(,1) and S(1,) S(1,1) feq (400.0MHz to 3.00GHz) (b) feq (400.0MHz to 3.00GHz) (c) Fig Compaison of the RL calibation technique between the nomal pocedue not suitable fo cyogenic application (blue) and the poposed pocedue adapted fo cyogenic application (ed); (a) RL test-fixtue; (b) measued and calculated S 11 and S ; (c) measued and calculated S 1 and S Noise Measuement at Cyogenics: Cold-Attenuato echnique Intoduction to the noise measuement he noise measuement is pobably the most impotant measuement fo micowave enginees woking at cyogenics since the main objective behind cooling the systems is the eduction of its themal noise. Most of the noise measuement techniques, both at oom and cyogenic tempeatues, ae based on the noise lineaity of two-pot linea systems. he noise powe at the system output is linealy dependant on the noise at the input [3.]. If one point and the slope, o two points, of the staight line that defines the linea system ae known, then the noise powe at the output, without noise at the input, i.e. the noise added by the DU, N a, can be obtained. Fom this value it is easy to calculate the noise facto, F, o the effective input noise tempeatue, e, though (3.10) and (3.11). heefoe, it would be enough a noise souce poviding two noise powes ( c and h ), and the measuement of the output powes, to be able to tace the staight line as pesented in Fig his technique is known as Y-Facto method. 64

12 3.. Noise Measuement at Cyogenics: Cold-Attenuato echnique N a + k0 BG F = (3.10) k BG 0 a e = N kgb (3.11) Whee k is the Boltzmann s constant (k = J/K), B is the bandwidth, and 0 is an IEEE standad tempeatue defined as 0 = 90 K. Fig Relationship between souce tempeatue and noise output powe in linea two-pot systems. If the souce tempeatue is zeo then the output powe is the noise added by the system, N a, which is elated with its effective input noise tempeatue, e. he effective input noise tempeatue is a paamete much moe useful fo low noise devices such as found in cyogenic applications. e is pefeed fo low noise devices because it is a moe sensitive indicato when N a is small. Fom Fig. 3.7 a moe convenient equation fo calculating e can be obtained (3.1). his equation shows that e has not a efeence tempeatue and uses the Y-Facto, which is the atio of the output noise powes measued at the eceive when two diffeent input noise powes ae pesented to the DU. h Yc e = (3.1) Y 1 o obtain a pecise noise measuement it is advisable that the measued value of e to be between c and h o, at least, to be as close to c as possible. In cyogenics e has vey low values compaed with the noise powes deliveed fo common noise souces. Due to the andom natue of these signals a small eo in thei measued values in the eceive may poduce lage eos in the calculation of e. o ovecome this poblem thee ae seveal techniques developed in cyogenics to educe the noise powes at the input of the DU. Some of them use altenative noise souces poviding adequate powe values, while othes ty to educe the noise powes aiving to the DU 65

13 Chapte III Measuement echniques fo Cyogenics intoducing some element with high insetion loss befoe the DU. All these techniques, biefly descibed in [3.3], ae commented in next section fo completeness Oveview of noise measuement techniques in cyogenics Hot and cold loads In this technique, two independent loads ae placed outside the cyostat poviding suitable input noise powes. hese two loads ae connected to the input line though a switch avoiding the need of connect and disconnect altenatively the loads and thus speeding up the pocess. A sketch of this technique is shown in Fig Hot Load CRYOSA h c D.U. P ON Noise Figue Mete LN e, G P OFF Cold Load Fig Diagam of the hot and cold loads noise measuement technique. Usually, the hot load is a esisto at oom tempeatue which povides a h of aound 300 K while the cold load is a esisto immesed in liquid nitogen (LN ) which poduces a c aound 77 K. his technique has some poblems elated with its accuacy: - he accuacy of noise measuement is geatly dependent on the diffeence between the impedances pesented by the loads to the DU, which usually diffe significantly fom each othe. - Long-tem epeatability of insetion loss and impedance match of the switch is cucial fo accuate esults. - Commecial cold loads ae bandwidth limited and custom equipment needs to be developed fo high fequency measuements. - Even though the noise measuement can be automatized the test speed is slow hindeing the optimization pocess when seveal measuements ae equied. - Finally, the noise powes at DU input may be still fa fom the measued e value, leading to inaccuacy poblems Noise figue mete with noise diode his is the typical technique used fo noise measuements at oom tempeatue. In this method a solid-state noise diode is used as a noise souce poducing two diffeent noise powes when it is switched fom one state to othe. A block diagam of this setup is shown in Fig

14 3.. Noise Measuement at Cyogenics: Cold-Attenuato echnique Fig Diagam of the noise measuement technique with noise figue mete and noise diode. he noise tempeatues at the noise souce output ae vey diffeent in this technique anging fom aound 300 K, when the diode is in its OFF state, and thousands of kelvins, depending on the diode Excess Noise Ratio (ENR), when the diode is in ON state. heefoe, this method is suitable fo devices with effective input noise tempeatues included in the pevious ange. But, at cyogenics this method pesents two main poblems: - he effective input noise tempeatue of cyogenic amplifies is well below 50 K in the fequency ange whee these noise diodes ae used and then the eo poduced by this appoach is noticeable. - he noise souce output impedance changes significantly fom one state to othe poducing an impotant ipple in the noise measuement. Moeove, the DU s input noise tempeatue is also aised due to the impedance changes. Noise souces with low ENR ae pefeable fo this technique since they ae made placing a suitable attenuato at the souce output, thus educing the ENR value and the diffeences between output eflection coefficients Hot and cold load his technique should not be confused with the fist one pesented in Section Now thee is only one load placed at the DU s input inside of the Dewa. his load is cooled by the cyostat and heated up with a esisto attached to it in ode to poduce the two diffeent noise powes needed fo the Y-Facto method. he sketch of this technique is shown in Fig emp. Senso Hot-Cold Load Heate CRYOSA h c D.U. e, G P ON P OFF Noise Figue Mete Fig Diagam of the hot and cold load noise measuement technique. 67

15 Chapte III Measuement echniques fo Cyogenics his technique ovecomes many of the dawbacks of pevious methods. Fist, the diffeences between load eflection coefficients in both states, cold and hot, ae likely small which educes the uncetainty in noise measuements; and second, when the heate is OFF the load is at the same physical tempeatue as the DU, thus poducing a c value close to the measued e value. On the othe hand, two seious poblems still emain in this technique: - he cooling and heating pocesses ae vey slow and theefoe the test speed is educed consideably, making this technique inadvisable fo boadband measuements if anothe method is available. - In ode to apply this technique special cae must be taken to avoid DU to be heated when the heate is ON, othewise the noise esults will be affected by this tempeatue incement. A good themal isolation between the load and the DU could be vey difficult to achieve Cold sky and ambient apetue load In this technique a calibated test feedhon, with a vey low side-lobe patten to minimize ambient noise pickup fom the gound, is placed at oom tempeatue at the input of the DU. he cold tempeatue is just the sky tempeatue which has a value of a few kelvins, while the hot tempeatue is accomplished though the use of an apetue load at ambient tempeatue. A adiomete is used to measue the powe atio between the sky and the ambient apetue load as shown in Fig { Fig Diagam of the cold sky and ambient load noise measuement technique. his method has the advantage of achieving high pecision fo fequencies below the millimete-wave ange since the atmospheic weathe has small effects in the efeence sky tempeatue. Futhemoe, the noise measuement is made with the DU installed in an antenna, which epoduces its final application and theefoe the measuement conditions ae moe ealistic. On the othe hand, this technique has some dawbacks: 68

16 3.. Noise Measuement at Cyogenics: Cold-Attenuato echnique - he calibated feed-hon may be difficult to design depending on the fequency and bandwidth. - he measuement setup needs to be placed in a suitable location with stable and chaacteized atmospheic conditions. he noise measuement accuacy is geatly affected by the cold tempeatue efeence. Radio obsevatoies ae usually the only place whee this technique is applied. - he test speed is low due to manual switching between loads Cold-attenuato technique In Section 3... it was commented that the use of low ENR noise souces was advisable due to the eduction of diffeences between the output eflection coefficients in both souce states. he decease of ENR in a noise souce is achieved intoducing a suitable attenuato afte the noise diode and thus the diffeences between the above mentioned impedances ae educed twice the attenuation value [3.4]. he cold-attenuato technique is a modification of this method in the way that an attenuato is placed and cooled befoe the DU, inside of the Dewa as shown in Fig With this appoach, as well as an impovement in the eflection coefficients vaiation, thee ae thee moe impotant advantages: - he noise powes ae attenuated befoe the DU, thus poviding suitable powe values fo the Y-Facto method in a cyogenic envionment. When the noise souce is OFF then the c value is the physical tempeatue of the attenuato plus few kelvin esulting fom the OFF ( OFF ~ 300 K) value attenuated by the coldattenuato; when the noise souce is ON the high ON value (typically ON > 9000 K) is attenuated obtaining a suitable h value aound 100 K o less. - he tempeatue and insetion loss pofile in the input line is impotant fo accuate noise measuements. In this technique the insetion loss of the input line can be made negligible egading the attenuato insetion loss and theefoe this method is less sensitive to eos in the tempeatue and insetion loss chaacteization of the input line. - Finally, no mechanical switches ae needed and thus the test speed is impoved, enabling fast and boadband noise measuements. 69

17 Chapte III Measuement echniques fo Cyogenics { Fig Diagam of the cold-attenuato noise measuement technique. Assuming that the input line has a constant tempeatue distibution and a constant popagation loss the tempeatues at the DU input can be calculated with (3.13) [3.5]. If the insetion loss of the input line is not negligible compaed with the attenuato then the pevious simplification may lead to noticeable eos and then a linea tempeatue distibution has to be consideed in this input line. Expessions fo calculating the input tempeatues in this last case can be also found in [3.5]. h c = ON, OFF + 1 p 1 L pad Lpad, (3.13) Whee p is the physical tempeatue of the attenuato and L pad the total insetion loss of the input line including the attenuato. If the input path is fully chaacteized in the measuement band then the calculation of the input tempeatues is staightfowad and the calculation of e is simple using (3.1). hee ae many factos which influence the accuacy of the noise measuement with the cold-attenuato technique, but in many cases the eo in the value of the cold efeence becomes dominant [3.4], [3.6]. A high value of the insetion loss in the attenuato will help in poviding a well defined cold efeence; howeve this value should be kept within adequate limits in ode to impove the noise measuement accuacy. It is clea that small attenuation values should not be used in ode to avoid poblems commented peviously. On the othe hand, lage attenuation values poduce small diffeences between the hot and cold tempeatues at the DU input. he andom natue of noise signals poduce fluctuations in the powe measued by the noise figue mete. hese small fluctuations may geneate a lage measuement eo when calculating the Y-Facto if the diffeences between hot and cold tempeatues ae small [3.]. heefoe, thee should be an optimum value of the insetion loss of the attenuato which may be dependent on the details of the measuement setup. In the following calculation, a simple setup is consideed whee only the fluctuations in eceived noise powes, due to limited eceive bandwidth and integation time, ae taken into account. 70

18 3.. Noise Measuement at Cyogenics: Cold-Attenuato echnique Fom (3.1), to obtain the optimum insetion loss fo the attenuato it is needed to calculate the minimum e uncetainty due to the uncetainty in Y, poduced by the fluctuations in noise powe measuements. Equation (3.14) gives the standad deviation of e, Δ e, in tems of standad deviation of Y, ΔY. e Δ e = ΔY (3.14) Y Fo this simple appoach, it can be consideed that Y is function of two uncoelated andom vaiables, noise powes P ON and P OFF, and theefoe ΔY can be calculated fom (3.15). ΔY Y = ΔP P ON ON ΔP + P OFF OFF (3.15) Fluctuations in the measued powes ae dependent on the noise figue mete bandwidth, B, and integation time, τ, as stated in (3.16). ΔP ON PON, OFF ( h, c + e ) kgb, OFF = = (3.16) B τ B τ Whee k is the Boltzmann s constant and G the DU tansduce gain. Intoducing (3.16) and (3.15) in (3.14) the standad deviation of e is obtained as a function of h and c (3.17), which ae dependent on the attenuato insetion loss (3.13). Δ e = ( h B τ + )( ( e h c ) c + ) e (3.17) Combining (3.17) and (3.13) the optimum value fo the attenuato insetion loss, fo a given setup, can be obtained. In Fig. 3.13, Δ e as a function of total path attenuation is plotted fo the following setup: B = 4 MHz, τ = 1 sec, p = 15 K, e = 0 K and OFF = 96 K. 71

19 Chapte III Measuement echniques fo Cyogenics Δe (K) Attenuation (db) Fig Δ e (coveage facto of 3 [3.7]) vesus total input path attenuation fo two diffeent values of noise souce Excess Noise Ratio (ENR): ENR = 0 db in solid line and ENR = 1 db in dashed line. he excess noise atio of noise souces changes with fequency. In the boadband noise souce available in ou laboatoy, model 346CK01 fom Agilent echnologies 1, the ENR anges fom almost 0 db at 1 GHz to less than 1 db at 40 GHz. Accoding with Fig. 3.13, the optimum attenuation value is elated with the ENR of the noise souce. heefoe the best esults should be found with a 15 db attenuato at high fequencies and a 0 db attenuato at low fequencies. As well as the eceive uncetainty investigated hee to obtain the optimum attenuato insetion loss thee ae many othe eo souces that need to be examined in ode to make a complete estimation of the measuement uncetainty. hese noise souces ae investigated in next section, togethe with the calculation of the total noise measuement uncetainty using Monte Calo analysis Estimation of Uncetainty in Noise Measuements (based on [3.6]) Intoduction Duing the measuement pocess the noise figue mete (NFM) applies the Fiis equation (3.18) to obtain the DU noise since this noise is measued togethe with the NFM own noise. sysmeas meas meas = DU + meas DU = meas sys (3.18) meas G G DUmeas DUmeas 1 Agilent echnologies, Inc., Santa Claa, CA USA. 7

20 3.3. Estimation of Uncetainty in Noise Measuements Whee the system noise tempeatue, sys, is measued at the DU noise measuement pocess, and the eceive noise tempeatue,, is measued at the eceive calibation pocess. he DU gain can be obtained fom pevious measuements. In the following analysis a geneal noise measuement configuation is consideed, i.e. it can be applied to diffeent setups and measuement techniques, not only the cold-attenuato technique at cyogenics. Futhemoe, in ode to extent the geneality of this analysis diffeent noise souces ae consideed fo the calibation and measuement pocesses. he configuations fo the calibation and measuement ae pesented in Fig (a) NS cal hcal ccal (Receive) Noise Figue Mete N Y cal = cal N 1cal meas (b) Fig Configuations and definition of vaiables fo the analysis; (a) measuement; (b) calibation. Whee N i ae the measued noise powes in the eceive duing calibation and measuement fo the two diffeent noise souce powes. he calculation of the noise tempeatues of the eceive and the system is staightfowad using (3.19) and (3.0). sysmeas meas = = hmeas hcal Y Y Y Y cal meas cal 1 meas cmeas 1 ccal (3.19) (3.0) he eceived noise powes can be obtained fom (3.1)-(3.4). N = kb( + ) (3.1) 1cal c cal meas N = kb( + cal h cal meas N = kbg ( + + 1meas DU meas c meas DU meas meas N = kbg ( + + meas DU meas h meas DU meas meas ) ) ) (3.) (3.3) (3.4) And finally, the DU gain is calculated as (3.5). 73

21 Chapte III Measuement echniques fo Cyogenics G DU meas = N meas N cal N N 1meas 1cal h h cal meas c c cal meas (3.5) In ode to calculate the uncetainty in the DU noise tempeatue and gain the vaiables in (3.19), (3.0) and (3.5) ae consideed to have a andom natue. hus, a lage numbe of measuement iteations can be simulated and the statistics can be extacted fom these esults. his pocess is known as Monte Calo analysis. o obtain the diffeent noise powes at the eceive it is consideed the andom natue of all the vaiables involved in thei calculation. As the hot and cold noise tempeatues of the calibation and measuement appea both in equations (3.19), (3.0) and (3.5) and in the definition of the eceived noise powes (3.1) (3.4), then thei andom natue is only consideed in this last case; othewise the andom behavio will be consideed twice and theefoe the calculation will be wong. heefoe the noise tempeatues ae going to be used as deteministic o andom independently, denoting with supescipt the andom natue. he deteministic hot and cold tempeatues ae calculated with (3.6)-(3.9) c cal = amb (3.6) hcal = ccal ENR cal (db) 10 (3.7) = amb 1 + c meas p 1 (3.8) Lattn Lattn hmeas = cmeas ENRmeas ( db) Lattn ( db) 10 (3.9) Whee amb is the ambient tempeatue fo the noise souce (typically amb = 96 K), ENR cal is the calibation noise souce excess noise atio, L attn is the total insetion loss of the input path between the noise souce and the DU, ENR meas is the measuement noise souce excess noise atio, and p is the physical tempeatue of the elements in the input path. In (3.7) and (3.9) the hot tempeatues ae defined fom thei cold tempeatues, i.e. with the constant diffeence h c. his is because the ambient tempeatue is consideed to be diffeent fom the standad tempeatue 0, and the noise tempeatues in both states of the noise souce ae not independent. he noise geneated in the intenal attenuato of solid-state noise souces is added to both states [3.]. 74

22 3.3. Estimation of Uncetainty in Noise Measuements Calculation of the eceived andom noise powes he noise powe measued at the eceive with the DU and the noise souce in ON state is given by (3.30), whee the Boltzmann s constant and eceive bandwidth ae omitted since they ae common to all expessions and they do not have a andom natue. Subscipts denotes andom behavio of the vaiables. ( ) N = ΔGc G meas tduon h + meas (3.30) syson he gain in (3.30) efes to tansduce gain, which is the atio between the powe deliveed to a load and the powe available fom the souce. Since it is necessay to obtain the available gain to calculate the system noise tempeatue, then the tansduce gain is deived as a function of the available gain (3.31). G tduon 1 Γout 1 Γ ON ec = G avdu (3.31) ON 1 Γ Γ outon ec he diffeent eflection coefficients in (3.31) ae defined in Fig and (3.3). { Fig Vaiables definition fo the noise powe calculation at DU measuement. Γ out ON = S S1S + 1 S 1 11 Γ Γ meas meas ON ON (3.3) Whee the S ij ae the DU Scatteing paametes at the desied fequency. he available gain is given by (3.33). avdu S 1 1 ΓmeasON = 1 Γ S11 meas 1 Γ ON out G (3.33) ON ON 75

23 Chapte III Measuement echniques fo Cyogenics On the othe hand, the system noise tempeatue is calculated fom Fiis fomula (3.34). sys ON ( Γ, Γ ) R out ec = ON DU ON + (3.34) GavDU ON In this analysis it is consideed that the DU noise tempeatue is affected by the noise souce eflection coefficient. In ode to account fo this effect the DU noise paametes ae used to calculate the DU noise tempeatue in (3.34) though equations (3.35) and (3.36). DUON g = min + 0 Z g Z ON opt (3.35) R n g Z g ON 1+ Γ = Z0 1 Γ meason meason (3.36) Whee Z g is the impedance of the noise souce, Z 0 is the chaacteistic impedance, 50 Ω, and min, g n, and Z opt ae the noise paametes of the DU. Finally in (3.34) thee is a function, R, which is the eceive noise tempeatue (with an isolato at the input) and it depends on the DU output eflection coefficient and the eceive input eflection coefficient as given by (3.37) [3.6]. Γa Γb Γ a, Γb ) = ec + ( ec + iso ) (3.37) (1 Γ )(1 Γ ) R ( a b Hee, ec is the minimum eceive noise tempeatue, i.e. the value measued when the eceive is pefectly matched, and iso is the physical tempeatue of the isolato placed at eceive input. In (3.30) the tem ΔG c was intoduced to take into account the uncetainty when the insetion loss of the intenal attenuatos of the NFM ae measued duing the calibation pocess. his infomation can be found in the NFM manuals as a value expessed in db, theefoe a andom vaiable is geneated following a nomal distibution (NORM) with a mean value of zeo and a standad deviation given by the manual (3.38). When this vaiable is conveted fom db to linea scale then it is intoduced in (3.30) as a multiplying facto instead of a sum. ΔGc ΔGc ( db) = NORM 0, (3.38) kgc 76

24 3.3. Estimation of Uncetainty in Noise Measuements ΔGc = 10 Δ Gc (db) 10 (3.39) he paametes k i appeaing along this analysis ae the coveage factos fo the andom vaiables and ae used to give the expanded uncetainty, i.e. the ange whee it is believed that the given value is found with a cetain pobability. Geneally a facto of k i = defines an inteval with a level of confidence of 90%, a facto of k i = implies a level of confidence of 95%, and a k i = 3 defines a confidence geate than 99% [3.7]. Since in the nomal distibution the standad deviation value is needed then the uncetainty value given by the equipment manufactue is divided by its coveage facto, which sometimes is also given by the manufactue. Once the mean value of the eceived noise powe has been calculated using (3.30) it is necessay to obtain its uncetainty, which is poduced by the limited bandwidth and integation time of the eceive (3.40). ΔN = NORM, meas N meas Bτ 0 (3.40) heefoe, the new value of the eceived noise powe is given by (3.41). N N N meas + Δ meas meas = (3.41) Finally, anothe uncetainty is included in this analysis: the non-lineaity of the NFM which may be pesent between measuements in ON and OFF states. his effect can be included in the analysis as an uncetainty that only is added in the ON state. Once again this data, when available, ae given in db, theefoe the nomal distibution geneated fo this effect is added as a multiplying facto in the final noise powe (3.4)- (3.44). Δ GN meas ΔG ( db) = NORM 0, kg (3.4) ΔG N meas = 10 Δ G (db) N meas 10 (3.43) N N meas Δ meas = G (3.44) N meas his concludes the calculation of the eceived noise powe with the DU and the noise souce in ON state, including all the consideed uncetainties. An analogous pocess can be followed to obtain the eceived noise powe with the DU and the noise 77

25 Chapte III Measuement echniques fo Cyogenics souce in OFF state, N 1meas, just changing ON fo OFF in the subscipts and h fo c in the tempeatues. he NFM non-lineaity uncetainty is not consideed in this case. Fo the calibation, the calculation of the eceived noise powes is caied out in a simila way than fo the DU measuement. he ae only two diffeences: the tem that includes the uncetainty in the insetion loss measuement of the intenal attenuatos in the NFM, ΔG c, is not consideed, and the available gain of the DU is 1. Again, in the calculation of the powe with the noise souce in ON state, N cal, is consideed the effect of the non-lineaity of the eceive, ΔG, which is not included in the calculation with the noise souce in OFF state, N 1cal. he equations fo the calculation with the noise souce in ON state ae given in (3.45)-(3.51). N cal ton ( + ( Γ, Γ )) = G (3.45) hcal R calon ec Gt ON 1 ΓcalON = 1 Γ 1 Γ calon Γ ec ec (3.46) Δ = NORM 0 N, cal N N N cal + Δ cal cal N cal Bτ (3.47) = (3.48) Δ GN cal ΔG ( db) = NORM 0, kg (3.49) ΔG N cal = 10 Δ G (db) N cal 10 (3.50) N = N ΔG (3.51) cal cal N cal Definition of andom vaiables Most of the andom vaiables used in the pevious sections have not been defined yet and they need to be geneated fom data obtained fom manufactues o measuements. All the andom vaiables ae consideed to follow a nomal distibution (NORM) o a unifom distibution (UNIF); in geneal, magnitudes will follow the nomal distibution wheeas phases will follow the unifom distibution in the ange (0, π). 78

26 3.3. Estimation of Uncetainty in Noise Measuements In ode to geneate andom data following a nomal distibution with mean value μ and standad deviation σ it is used the Box-Mulle tansfom (3.5) [3.8]. NORM ( μ, σ ) μ + σ lnu cos(πv ) = (3.5) Whee U and V ae two andom values geneated following a unifom distibution in the ange (0, 1). o geneate the data following the unifom distibution in the ange (a, b) (3.53) is used. UNIF ( a, b) a + ( b a) U = (3.53) In the following, the eflection coefficients ae defined as andom vaiables with a magnitude equal to the wost value of the available data and its phase unifomly distibuted in (0, π). he eflection coefficient of the calibation noise souce in ON state is given by (3.54). Γ calon = Γ j calon cal e φ max (3.54) Γ Γ cal (db) max 0 cal max = 10 φ = UNIF(0,π ) cal ON (3.55) (3.56) he noise souce eflection coefficients in both states ae not independent and hence the coefficient in the OFF state is geneated fom the one in the ON state though the use of the diffeence of eflection coefficients (3.57)-(3.59). Γ caldiff = 10 Γcal diff ( db) 0 (3.57) φ UNIF( 0,π ) cal diff = (3.58) cal OFF cal ON cal diff jφ caldiff Γ = Γ + Γ e (3.59) About the andom noise tempeatues geneated by the noise souce they ae calculated fom the ENR data of the souce and the ambient tempeatue measued by an additional themomete. Both tempeatues, cold and hot, ae not consideed as independent. cal = (3.60) c amb 79

27 Chapte III Measuement echniques fo Cyogenics h = c cal cal ENRcal (db) 10 (3.61) ENR db) = NORM ENR ΔENR ( db), k ( db) cal cal ( cal (3.6) ENR cal Δamb = amb NORM amb, (3.63) k amb Fo the measuement noise souce the calculations of the eflection coefficients and noise tempeatues ae simila to those caied out fo the calibation noise souce. he only diffeence is elated with the pesence of an input path fom the noise souce to the DU which is chaacteized by its insetion loss (including the attenuato o not) and its physical tempeatue. his diffeence is accounted fo using (3.64)-(3.69). Γ measdiff = 10 Γmeas diff ( db) Lattn ( db) 0 (3.64) = amb + 1 c meas p 1 Lattn L attn (3.65) L attn = 10 L attn (db) 10 (3.66) L db) = NORM L ΔL ( db), k ( db) attn attn ( attn (3.67) L attn Δ p = p NORM p, (3.68) k P h = c meas meas ENRmeas ( db) Lattn ( db) 10 (3.69) he input eflection coefficient of the eceive is defined in a simila way than the othes. Γ Γec ( db) max 0 ec = 10 max φ UNIF( 0,π ) ec = (3.70) (3.71) Γ ec = Γ ec max e jφ ec (3.7) And finally, the S-paametes of the DU ae defined by (3.73)-(3.75). 80

28 3.3. Estimation of Uncetainty in Noise Measuements S ij = Sijmax e jφs ij (3.73) S ij = 10 max Sij (db) max 0 (3.74) 0, φ UNIF( π ) S ij = (3.75) his concludes the mathematical analysis of the noise measuement uncetainty using the Monte Calo method, which can be adapted to diffeent measuement setups and stategies. Softwae pogams have been developed fo these calculations unde Matlab and Mathcad platfoms [3.4]. hese pogams can be downloaded fom [3.9] Uncetainty estimation in the cold-attenuato technique he pevious uncetainty analysis is applied in this section to the study of the cold-attenuato technique and the influence of some paametes in the final uncetainty. he setup fo this measuement was pesented in Fig he noise souce used fo eceive calibation is the model N4000A wheeas fo the DU measuement the noise souce model N400, both fom Agilent echnologies, is used. he eason fo using two noise souces with diffeent ENR is to educe measuement ipples and thus impove its accuacy [3.6]. he input path is compised of the input coaxial line and the cold-attenuato. he eceive used fo this analysis is the NFM model N8975A fom Agilent echnologies with an isolato placed at its input in ode to impove the eceive matching. Finally, the data fom a LNA designed at Cento Astonómico de Yebes, Guadalajaa, Spain, fo the IF of the band 9 of ALMA poject [3.30] ae used fo the DU. In able 3.1 the values of all paametes at the cente fequency of 8 GHz used fo this analysis ae pesented. Paamete Definition Value Unit Calibation Noise Souce: N4000A ENR Excess noise atio of the calibation noise souce 5. db ΔENR Uncetainty of the ENR 0.14 db k ENR Coveage facto of ΔENR Γ max Wost value of the souce eflection coefficient in ON -9 db Γ diff Wost value of the diffeence in coefficients between ON and OFF -48 db amb Measued ambient tempeatue 97 K Δ amb Uncetainty of the ambient tempeatue themomete 1 K k amb Coveage facto of Δ amb Measuement Noise Souce: N400A ENR Excess noise atio of the measuement noise souce 14.1 db ΔENR Uncetainty of the ENR 0.13 db k ENR Coveage facto of ΔENR Γ max Wost value of the souce eflection coefficient in ON -4 db Γ diff Wost value of the diffeence in coefficients between ON and OFF -4 db 81

29 Chapte III Measuement echniques fo Cyogenics p Measued physical tempeatue of the attenuato 1.5 K Δ p Uncetainty of the attenuato tempeatue senso 1 K k p Coveage facto of Δ p L att otal attenuation of the input path 15 db ΔL att Uncetainty of the attenuation measuement 0.15 db k Latt Coveage facto of ΔL att Receive: N8975A ec Minimum noise tempeatue of the eceive 1500 K iso Measued physical tempeatue of the isolato 97 K Γ ec_max Wost eflection coefficient value at eceive input -0 db B Receive bandwidth 4 MHz τ Receive integation time 1 sec ΔG c Uncetainty in intenal attenuatos insetion loss measuement 0.17 db k Gc Coveage facto of ΔG c ΔG Uncetainty poduced by eceive non-lineaity 0.05 db k G Coveage facto of ΔG DU: LNA 4 1 GHz Band 9 of ALMA S 11 Input eflection coefficient -3.5 db S 1 ansmission coefficient 33.5 db S 1 Revese tansmission coefficient -47 db S Output eflection coefficient -13 db min Minimum noise tempeatue (noise paamete) 3.74 K g n Noise paamete 6.74e-5 S Re(Z opt ) Real pat of optimum noise impedance (noise paamete) 77.9 Ω Im(Z opt ) Imaginay pat of optimum noise impedance (noise paamete) 71.1 Ω Monte Calo Paametes Iteations Numbe of iteations in the Monte Calo analysis k Coveage facto fo the Monte Calo esults able 3.1. Data fo the Monte Calo analysis in the cold-attenuato technique. Fo this DU it is expected a nominal gain of 33.5 db and a nominal noise tempeatue of 6 K at the fequency of 8 GHz when cooled down to 1.5 K. With the paametes shown in able 3.1 the Monte Calo analysis gives the following esults extacted fom the developed pogam with a coveage facto of. 8 (a) Fig Results of the Monte Calo analysis fo the ALMA LNA using the cold-attenuato technique; (a) gain esults; (b) effective input noise tempeatue esults. (b)

30 3.3. Estimation of Uncetainty in Noise Measuements heefoe, a total uncetainty of aound 1.7 K in the noise tempeatue fo the pesented setup is found. his value agees well with othe studies made with othe DUs, setups and mathematical appoaches [3.31]. aking advantage of the developed pogam it is easy to vay some paamete values in ode to check the change of the total uncetainty, thus extacting conclusions that may help duing the measuement pocess. Some plots ae pesented in the following figue to show these vaiations. Δe (K) Δe (K) Δe (K) ΔENRm (db) (a) Δp (K) (c) Γmax (db) (e) Δe (K) Δe (K) Δe (K) ΔLatt (db) (b) Latt (db) (d) ΔG (db) (f) 83