A First-Order Noise Analysis of the GBT L-Band Receiver Front-End

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1 A First-Order Noise Aalysis of the GB L-Bad eceiver Frot-Ed ichard F. Bradley Jauary 30, 003 Itroductio ecet ivestigatios of GB baselies associated with observatios of cotiuum radio sources have revealed the presece of several frequecy depedet aomalies i the spectra produced by traditioal calibratio ad ormalizatio practices [1,, 3]. he (sig - ref / ref formulatio, where sig is the sigal produced at the receiver s output whe the telescope is poitig o source ad ref is the correspodig sigal whe the telescope is poitig off source, idicates overall large-scale frequecy structure as well as periodic, small-scale frequecy structure. hese structures are preset i the L-, C-, ad X-bad receivers that were evaluated durig the commissioig process, ad it may well be preset at other bads. hrough a series of tests performed by. Fisher ad. Norrod, the cause of this spectral aomaly withi the GB istrumetatio has bee localized to beig ahead of the first mixer i the receiver s F compoet chai. It was agreed that first-order modelig ad simulatio of this cascaded microwave etwork was ecessary to begi to uderstad the origi of this pheomeo so that corrective actio ca be take. L-bad was chose for the simulatio sice a well-uderstood equivalet circuit model is available for the low-oise amplifier used i this bad. A block diagram of the F frot-ed compoets of the L-bad receiver are show i Fig. 1. he F compoets at a physical temperature of 300 K iclude the feed, itercoectig waveguide, dewar vacuum seal, ad a sectio of the thermal gap. At 15 K, there is aother sectio of the thermal gap, the ortho-mode trasitio (OM, calibratio coupler, ad low oise amplifier. he system oise temperature of this receiver o the GB was measured to be approximately 18.5 K. A rough estimate of the oise budget [4] assigs 5 K for atmosphere ad CMB, 5 K for spillover, 4-5 K for the feed, widow, OM, ad coupler cascade, ad 3-4 K for the low osie amplifier. he feed legth is about 3.3 meters [4] yieldig a total electrical legth from amplifier to the feed aperture of about 4 meters [4]. A typical composite source / sys spectra from MHz for a cotiuum radio source take with the GB [3] is show i Fig.. hese spectra are illustrative of the typical frequecy structure ecoutered i GB baselies. here is a overall slope across this bad together with a siusoidal-like variatio. It is these structures that will be addressed with the simulatio ad modelig described here. I additio, the spectra shows some very small-scale structure that is believed to be caused by the telescope s ics [5] alog with kow egative-goig iterferece spikes [5]. Both of these structures reside outside the scope of this study. 1

2 Noise heory I order to obtai a rough estimate of the various oise cotributios withi a microwave etwork, it is commo practice to model the loss as merely a perfectly matched atteuator that geerates oise proportioal to its physical temperature, as described by equatio. 7-55a o p. 91 of Kraus [6]. his approach yields a quatity that is actually the miimum equivalet oise temperature, mi, of the matched lossy compoet. Implicit i this model is the assumptio that the oise waves emaatig from the two ports of the atteuator are statistically idepedet. his is usually ot the case i practice. he complex two-port oise model, which is defied by four oise parameters, is a more complete model of the oise properties of a F compoet [7, 8]. A derivatio of this oise model is give i Appedix A, but it ca be summarized as EFG = MIN + Z Z, (1 where mi is the miimum oise temperature of the etwork, G is the oise coductace, Z=+jX is the imum oise impedace, ad Zs = s+jxs is the impedace preseted to the iput of the atteuator. ref = 90 K. his equatio shows that the amout of oise power as a fuctio of frequecy that flows out of the output port of the lossy two-port etwork will deped o the complex circuit impedace that termiates the compoet s iput port. Note that the secod term is a positive goig quadratic fuctio of the differece betwee the iput impedace ad the imum oise impedace. his fuctio is cetered about the imum oise impedace. Compoet Models he approach aded here for modelig the frequecy behavior of the L-bad frot-ed is to first develop a suitable equivalet circuit ad oise model for each of the critical compoets i the F sigal path that are thought to be cotributig to the varyig baselie pheomeo. A circuit simulator, amely Agilet s Advaced Desig ystem (AD, was used to determie the F etwork behavior ad calculate (sig - ref / ref i order to emulate GB observatios. hree critical etwork elemet have bee idetified: 1 basic circuit loss, the cascade comprised of the feed, dewar waveguide trasitio, ad OM, ad 3 the low oise amplifier. Details of each compoet are give below. Basic Circuit Loss he oisy elemets withi the etwork are assumed to be geerated by discrete losses that ca be modeled by a pi-type atteuator pad made from idividual AD resistor models. A careful study of the behavior of the overall circuit model uder various coditios cofirmed that it was ideed fuctioig as a complex two-port oise model. wo atteuators were chose to illustrate the differeces betwee the perfectly matched model ad the full complex two-port oise model. I both cases, the atteuators were desiged for 0.1 db power loss usig the hyperbolic formulae

3 give i [9] ad are at a physical temperature of 300 K. Atteuator #1 was desiged to perfectly match the embeddig etwork impedace of 50 ohms. Atteuator # was desiged for 100 ohms but used i the same 50 ohm embeddig etwork as atteuator #1. his situatio could represet a loss elemet whose imum oise resistace differs sigificatly from the etwork impedace. he results of the AD simulatios are summarized i able 1. able 1 AD simulatio results for the two atteuators described i the text. Zatt [ohms] 1 [db] 11 [db] [K] mi [K] [ohms] X [ohms] G [m] Note that if 1 is used to predict the oise cotributio we would calculate 6.9 K for atteuator #1 ad 8.76 K for atteuator # which is quite close to. However, we completely miss the fact that is ot equal to mi for atteuator #. he extra 1.73 K is actually comig from the secod term i Eq. 1. I order to gauge the effect of the secod term i Eq. 1, a AD simulatio was carried out o the two atteuators for the coditio of a varyig iput impedace, with s ragig from ohms ad Xs ragig from -5 to +5 ohms. he results are plotted i Fig. 3. he oise temperature of both atteuators is a relatively weak fuctio of the iput reactace sice both have X = 0 ohms. he -5 to +5 ohms variatio about the imum reactace produced oly a 0.5% or less icrease i the oise temperature. However, the effect of varyig the iput resistace is more proouced. For Atteuator #1, the ± 5 ohm variatio aroud 50 ohms (which is also i this case produced oly a 0.5% icrease i above mi. O the other had, Atteuator #, whose imum oise resistace is 100 ohms, produced a ± 6% variatio i about the value at 50 ohms. Feed, dewar trasitio, ad OM cascade eflectrometry measuremets of the feed, dewar trasitio, ad OM cascade were made by. rikath [10] ad are reproduced i Fig. 4. hese measuremets idicate reflectios at the -5 to -30 db level across the mid portio of the bad. It was decided that a detailed model of this etwork was uecessary at this time i favor of a simplified etwork cosistig of a cascade of several 50 ohm trasmissio lies with small-value capacitors placed i shut at each of the juctio poits, as illustrated i Fig. 5. he lies were modeled as lossless compoets so that a give amout of circuit loss could be iserted at various locatios alog the lie to study the effects. A was used to calculate the level ad periodicity of these reflectios as a fuctio of shut capacitace ad lie legth. hese parameters were adjusted util a first-order fit at midbad was obtaied betwee the model ad the measured data. he calculated 11 ad termial impedaces are show i Fig. 6 3

4 his lossless etwork was the placed ahead of the a atteuator as show i Fig. 7. he atteuator was modeled as described i the previous sectio with both types beig cosidered. A plot of the oise temperature versus frequecy for the cascaded etwork is show i Fig. 8. Note that i both cases the impedace variatio preseted to the iput of the atteuator traslates to oise temperature fluctuatios for the atteuator. hese fluctuatio are much more proouced for atteuator # (Z = 100 ohms which is cosistet with the oise model. Low oise amplifier he amplifier used i the GB L-bad receiver is a three-stage, sigle-eded desig created by Gallego ad Pospieszalski [11]. he three HFE s are type FH0X that were maufactured by Fujitsu. A equivalet circuit model for the HFE is show i Fig. 9. A block diagram of the amplifier is give i Fig. 10. Basic capacitor models were used i place of the more complete equivalet circuit models for the chip capacitors for this simulatio. AD was used to simulate the performace of this amplifier with iput ad output ports termiated i 50 + j0 ohms. he calculated -parameters are show i Fig. 11. Calculated oise parameters are give i Fig. 1. he iput impedace preseted to the amplifier was systematically varied over the rage 45 < i < 55 ohms ad -5 < Xi < +5 ohms ad the oise temperature at 1.4 ad 1.7 GHz was calculated. he results are preseted i Fig. 13. Frot-Ed Cofiguratios ad imulatio esults everal AD simulatios were performed to gai a uderstadig of the frequecy structure observed i [o - off] / off for cotiuum radio sources. I all simulatios, the characteristic impedace of the etwork was 50 ohms ad a AD two-port oise source model, perfectly matched to 50 ohms, was used to emulate a frequecy idepedet atea temperature for both o ad off source coditios. wo idetical etworks, oe with a(o ad oe with a(off were aalyzed simultaeously so that the [o - off] / off could be easily calculated. Details of the idividual simulatios together with their results are preseted below. Note that receiver gai drifts are ot modeled here ad that a perfectly stable receiver is assumed. Large cale, No-Periodic Frequecy tructure he first simulatio was desiged to examie the origis of the large-scale frequecy structure. It was hypothesized that this structure was a cosequece of amplifier oise. he etwork, a block diagram of which is show i Fig. 14, cosisted of the atea temperature oise source followed by the amplifier model. a(off was 5 K while a(o was varied from 0-10 K i steps of K. he results are show i Fig. 15. hese data are preseted graphically i two colums. he left had colum shows the o-source oise temperature (o, off-source oise temperature (off, [(o - (off], ad [(o - (off] / (off for the etwork operatig over the GHz bad. he right had colum shows the same sequece for a badwidth of 40 MHz ear 1.4 GHz. hese plots clearly show the frequecy structure of the amplifier oise is carried through to the [(o - (off] / (off due to the use of this frequecy depedet ormalizig factor, (off. Its 4

5 effect is a fuctio of atea temperature. I additio, the arrow bad formulatio idicates just how such broad bad structure could have bee easily hidde i past observatios. his effect is ot idicative of a iheret receiver problem, but a calibratio ad ormalizatio issue that will affect observig techique. Note that receiver gai drifts are ot modeled here ad that a perfectly stable receiver is assumed. mall cale, Periodic Frequecy tructure he secod simulatio was desiged to examie the origis of the small-scale, periodic frequecy structure. he etwork cosisted o the atea temperature oise source, followed by the shut capacitor loaded trasmissio lies, ad the amplifier. rasmissio lie legth Le was icreased to 1000 cm to ehace the frequecy variatio for visualizatio purposes oly. he results are preseted i Fig. 16. he left had colum shows the o-source oise temperature (o, off-source oise temperature (off, [(o - (off], ad [(o - (off] / (off for the etwork operatig over the GHz bad. he right had colum shows osource oise temperature (o, off-source oise temperature (off, ad [(o - (off] / (off for the same etwork operatig over a 40 MHz badwidth ear 1.4 GHz. I additio, 11 for the trasmissio lie sectio oly is show. Note that i this case, 9.5 K had to be icluded ito a for both o ad off source to make [(o - (off] / (off comply with GB observatios. I additio, the variatio i [(o - (off] / (off is much smaller i amplitude tha observed o the GB but is i agreemet with that expected from the amplifier due to impedace variatio of the loaded trasmissio lie. he amplitude variatio is about the same if we make Le3=1000 cm ad Le=10 cm. hese simulatios suggest that although the variatio i amplifier oise is certaily oe compoet of the oise, there remais aother oise mechaism that domiates. A clue to the other mechaism comes from the 9.5 K eeded to artificially raise the system temperature to that of the GB L-bad receiver. he previous simulatio suggests that this oise is probably ot due to spillover sice this compoet would simply raise the overall oise temperature of the receiver without ehacig the variatio we have observed o the GB. herefore, the extra oise was removed from a, modeled as a atteuator, ad placed betwee the loaded trasmissio lie ad the amplifier. wo atteuators were ivestigated: oe desiged for Z = 50 ohms ad the other with Z=150 ohms. he physical temperature of the atteuator was set to 300 K sice a atteuator at a cryogeic temperature of 15 K would eed to have a atteuatio of over db to produce 9.5 K of oise, a situatio that does t represet the loss measuremets. he simulatio results are give i Fig. 17 for a 0.1 db, Z = 50 ohms atteuator (Le = 1000 cm ad Fig. 18 for a 0.06 db, Z = 150 ohms atteuator (Le = 1000 cm. Fig. 19 shows the results for the 0.06 db, Z = 150 ohms atteuator with Le = 135 cm. o first order, the results i Fig. 19 are i agreemet with the GB measuremets preseted i Fig.. 5

6 Coclusios As a word of cautio, the simulatios results ad coclusios based o them as preseted i this report should be take as pure speculatio util they ca be verified through careful experimetatio either o the bech or via GB observatios. With that said, there are several coclusios that may be draw from the simulatios: he frequecy structure observed i the (sig - ref / ref is ot preset i (sig ref but occurs as a result of the frequecy depedece iheret i the ref sigal. his effect is ot idicative of a iheret receiver problem, but a calibratio ad ormalizatio issue that will affect observig techique for wide badwidth applicatios. he large scale, o-periodic structure is probably due to the frequecy depedet oise temperature of the low oise amplifier. he small-scale, periodic frequecy structure is primarily due to oise geerated by a ohmic loss at a physical temperature of 300 K located ear the OM ed of the microwave etwork cosistig of the feed ad the vacuum widow / thermal gap. his oise is effected by reflectios at the -5 to -30 db level withi the structure. he simulatios idicate that the imum oise impedace for this source differs sigificatly from the waveguide impedace which causes oise fluctuatios with frequecy that are o the same scale as that of the foud i the GB baselies. Hece, this mechaism appears tho be the root cause for the frequecy structure observed. he reflectios that occur withi the feed, dewar widow, thermal gap, ad OM will cause frequecy depedet variatios i the oise temperature of the low oise amplifier. However, the amplitude of these variatios appear to be too small to accout for the size of the variatios observed i the GB baselies. Based o bech measuremets ad GB baselie data, the reflectios withi the feed ad vacuum widow / thermal gap structure appear to be a result of two or more reflectios, with oe pair spaced about 135 cm apart. he use of a isolator or a balaced amplifier will ot affect the primary source of the baselie structure. However, such devices will reduce the gai variatios i the amplifier which are a direct result of the multiple reflectios. moothig the gai variatio with frequecy should help to improve receiver stability by reducig the receiver s sesitivity to the physical characteristics of the frot-ed etwork. 6

7 efereces [1] Fisher, J.. ample of Baselie tability of the -3 GHz GB eceiver, Project A_B_0119, November 19, 00, NAO, Dec. 3, 00. [] Fisher, J.. ecod Baselie est of the -3 GHz GB eceiver, Project BAEJF010, December, 00, NAO, Dec. 3, 00. [3] Fisher, J.. Further Ivestigatio of L-Bad Baselies o Cotiuum ources, Project BAEDB0117, December 17, 00, NAO, Dec. 3, 001. [4] Norrod,., Private Commuicatio, Ja [5] Fisher, J.. Private Commuicatio, Ja [6] Kraus, J.D. adio Astroomy, McGraw-Hill, New York, 1966, p. 91. [7] Pefield P. ad. P. afuse, Varactor Applicatios, M.I.. Press, Cambridge, MA. 196, pp [8] Colli,. E. Foudatios for Microwave Egieerig, d Ed. McGraw-Hill, New York, 199, pp [9] I, eferece Data for adio Egieers, 6 th Ed., Howard W. ams, Idiaapolis, IN, 1975, Ch. 11. [10] rikath,. Private Commuicatio, Ja, 003. [11] Gallego, J. D. ad M. W. Pospieszalski, Desig ad Performace of Cryogeically- Coolable, Ultra Low-Noise, L-Bad Amplifier, NAO Electroic Divisio Iteral eport, No. 86, Mar

8 Appedix A Aded from lecture otes - Bradley & Fisher, A 535 Itroductio to adio Astroomy Istrumetatio, Uiversity of Virgiia, prig 001. A.1 he Noise emperature of a wo-port Network as a Fuctio of its ource Impedace Cosider the oisy two-port etwork havig a complex source impedace, Z, at the iput as show i Fig. A.1a. Let s assume for the momet that this impedace has o oise associated with it. Fig. A.1b shows the same etwork with all of the oise geerated withi the etwork represeted by its equivalet oise temperature, N, which is ow associated with the source impedace, Z. here is a value of Z which miimizes the total oise out of a two-port etwork. We refer to this imum value of Z as Z. Figure A.1 Noisy etwork with source impedace Z (a Noisy etwork. (b Equivalet oise temperature represetatio. Let us compute N ad Z i terms of the parameters of a equivalet oise model for the etwork. We choose the othe-dahlke Noise Model, which represets the etwork of Fig. A.1 by the equivalet circuit show i Fig. A.a. his circuit cosists of two oise sources: a voltage source, v, ad a curret source, i. 8

9 v = 4k EF v = 4k N i = 4k G EF Figure A. (a Noise model of etwork. (b hevei equivalet circuit. wo parameters, a resistace ad a coductace, associated with these sources may be defied as = v 4k EF (A.1 ad G = i 4k EF, (A. where EF = 90 K. We ca treat the oise sources as radom variables, ad recall that the variace of a radom variable is Var[ X ] = E[( X m ] = ( x m f ( x dx. + X X X imilarly, we ca defie the covariace of two radom variables, i this case v ad i, as [ ] cov[ v, i ] = E ( v m ( i m v i (A.3 (A.4 9

10 ice we ca iterchage the expected value ad summatio, ad otig that our radom variables have zero mea, we ca expad Eq. A.4 as [ ] [ ] [ ] [ ] E ( v m ( i m = E v i m E i m E v + m m, v i v i v i [ v i ] cov[ v, i ] = E. (A.5 Give two radom variables, the ratio of the covariace to the product of the stadard deviatios is the correlatio coefficiet, ñ, such that ρ [ v i ] E[ vi ] v σ i E[ v ] E[ i ] = cov, σ = ( { }, (A.6 where we have made use of the fact that both v ad i have zero mea. Note that the correlatio coefficiet is complex: ñ = ñ +jñ I. he objective of the calculatios that follow is to fid as a fuctio of, G, ñ, ad Z that gives idetical mea square voltage from both equivalet circuits, v = v + i Z = v + i Z + e i v Z. (A.7 We ow substitute the expressios for the voltage ad curret ito Eq. A.7 ad solve for / EF givig EF G ( [ ] X G = ρ ρ I X. (A.8 he fuctio / EF (, X is real-valued ad the miimum poit o the surface defied by this fuctio ca be foud by settig the gradiet equal to zero, EF = (, X (, X ( X EF,, X = EF r. 0 (A.9 First, we begi by takig the partial derivative of Eq. A.8 with respect to X givig (, X EF X G ρ X G I = = 0. (A.10 10

11 olvig for X = X yieldig the equatio for X as X = ρ I. (A.11 G Next, we take the partial derivative of Eq. A.8 with respect to givig X ρ EF I G G X (, G X = + + = 0 (A.1 olvig for = gives = + X G G X ρ I. (A.13 ubstitutig X = X from Eq. A.11 gives = [ 1 ρ I ]. (A.14 G We ca ow determie Z from Eqs. A.14 ad A.11 as Z = + X = [ 1 ρ I ] + ρ I, G G Z =. G (A.15 he miimum oise temperature, mi, ca be foud by substitutig ad X ito Eq. A8 givig MIN EF G G = + ( + X + ( ρ ρ I X. (A.16 11

12 With a little algebra we ca simplify this equatio to the followig form: [ ] = G 1 ρ + ρ. (A.17 MIN I EF ice the fuctio (Z is aalytic aroud the miimum, we ca fid a compact expressio for ear MIN by aylor series expasio of about the poit MIN, ( Z = ( Z Z = Z d ( Z + Z d Z Z = Z Z (A.18 1 d ( Z + Z d Z Z = Z Z. he first term is simply MIN. he secod term is zero sice the expasio is about the miimum poit. he third term yields, d ( Z d( Z d ( Z d Z EFG Z = + EFG =, G EF 1 d ( Z EFG Z Z = Z Z d Z Z = Z ρ,, (A.19 or EFG = MIN + Z Z. (A.0 If we defie the parameter N = G the, = + MIN N Z Z EF. (A.1 1

13 here are a couple of importat poits to make regardig Eq. A.1. We ca ow defie four parameters associated which the oise geerated withi ay two-port etwork: MIN,, X, ad N. Whe the source impedace is equal to the imum oise impedace, the oise temperature of the etwork is equal to the miimum oise temperature, MIN. However, if there is a mismatch betwee Z ad Z, the the oise temperature will always be higher the MIN ad related to the square of the differece betwee these impedaces. 13

ANALYSIS OF EXPERIMENTAL ERRORS

ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder