A general analysis of error in noise measurement systems

A GENERAL ANALYSIS OF ERRORS IN NOISE MEASUREMENT SYSTEMS # L. P adell, A. Come on, A. Rami ez* ABSTRACT Gene al and app oxima ed exp essions a e p esen ed which allow he iden i ica ion and quan i ica ion o e o s in noise igu e measu emen s ha use he Y- ac o me hod. This o mula ion is a gene aliza ion o ha in /2/ o all measu emen si ua ions (in pa icula i is no necessa y o include an inpu isola o ). The o mulas a e u ilized o de e mine he e o limi s on he noise igu e measu emen o a mic owa e low-noise ampli ie . INTRODUCTION Causes o e o in noise igu e measu emen pe sis e en in mode n, mi c op ocesso -con olled measu emen sys ems. E ec o e o s is especially impo an in he cha ac e iza ion o low-noise de ices, whe e depa u es o indica ed noise igu e alues om ac ual ones a e mo e signi ican . Analysis o such e o s has gene a ed a conside able- amoun o li e a- u e. In 1975 Mamola and Sannino p esen ed a gene alized analysis o sou - ce misma ch e ec s on noise pa ame e measu emen s/l/. The exp essions de- i ed he e allow he de e mina ion o he co ec alue o he e ec i e inpu noise empe a u e o he wo-po om he expe imen al da a. In 1981 S id s udied he special bu impo an case in which an ideal (s12=0) in- pu isola o is used in he measu emen /2/. In 1984 Kuhn s udied he in lu ence o he gain a ia ion wi h inpu impedance on measu emen e o s /3/. In he p esen wo k gene al and app oxima ed exp essions a e de i ed which allow he iden i ica ion and quan i ica ion o e o s in noise igu e measu emen s. This o mula ion gene alizes he one in /2/ o all measu emen si- ua ions (namely, i is no necessa y o include an inpu isola o ) and al- so i akes in o accoun he second s age co ec ion. Non con e ing wo- po s a e assumed. As a p ac ical example, he o mulas a e u ilized o de e mine he e o limi s on he noise igu e measu emen o a mic owa e low noise ampli ie . NOISE MEASUREMENTS IN REAL CONDITIONS Figu e I shows he se -up and quan i ies in ol ed in a noise igu e measu emen o a wo-po de ice using a wo-s a es noise sou ce (**) Wi hou loss o gene ali y a 'cold' sou ce empe a u e T =T is assumed c o h oughou he pape . The noise 'igu e shown by he ins umen (Fdmeas) di- e s om he igh Fd alue ( he alue unde nominal condi ions sO) because se e al depa u es om an 'ideal' measu emen si ua ion occu : - The noise sou ce e lec ion coe icien is no he nominal. In addi i- on, i changes be ween he wo s a es ('cold' and 'ho '). This causes *E.T.S.E.Telecomunicacio. Dep . o Teo ia del Senyal i Comunicacions A.M.R. G oup. Ap.30002, 08080 Ba celona - Spain +This wo k has been suppo ed by he CAICYT (Resea ch P ojec 3346/83 and TELEFONICA (Con ac e . U.P.C. 619-TC/368) (**) See also Glossa y o e ms below 924 G m, Ted, Tem , Te , G d also O change. - The e exis unce ain ies in he knowledge o sou ce ho empe a u e (Thd s. Th). - The noise me e also con ibu es o he o al ead noise powe s. To ac coun o his e ec a second s age co ec ion using he F iis' o mu la is commonly done using calib a ion da a. - The e is ins umen a ion unce ain y. Ins umen a ion unce ain y is no conside ed in he ollowing de i a- ion bu i is added in he inal calcula ions. Fo each noise sou ce s a e (1 and 2) he ollowing powe s a e ead by he ins umen du ing measu e- men (N1,N2) and du ing calib a ion (Nlc,N2c): N1 k(T + T (P )*B.G (P ) I c e SC SC N2 k(Th + T (P ))B13G (P 2h~a + e ( shV .G (PsQ Ni IC k(T em sc' msc1'~, N (k(Tc + Te(Fs)) .B.G m(sh) 2c ha em sh m sh Noise igu es (Fmeas) a e calcula ed by he ins umen using he exp e ssion /5/,/6/: Y(I - T ITO) + (T h/T - 1) meas y - I whe e Y is he Y- ac o ac ually measu ed by he ins umen , de ined by: Y = N2/N1 (du ing measu emen ) (2a) Y = N2c/N 1c (du ing calib a ion) (2b) Subs i u ing (2a) in (1) yields F (P SC ) m (T h-T ) meas M (T -T )+ T (N F (P1 ) -HMF (P (3)(* h ha o o h sh c SC This undamen al exp ession ela es he displayed (measu ed) noise i gu e o a chain ( o med by he wo-po unde measu emen and hemeasu e- men sys em), o pa ame e s o he measu emen se -up and also iden i ies he causes o e o . Fmmeas can be ob ained in a simila manne by subs i u ing (2b) in (1). The ins umen calcula es he de ice gain om /5/: N - N N2 1I Gdm~eas N -N (4) 2c Ic and hen i deembeds Fdmeas using he F iis' o mula: F -1 F F - mmeas (5) dmeas meas Gdmeas EXPRESSION OF NOISE MEASUREMENT ERROR Subs i u ing (1), (2), (3) and (4) in (5) he ollowing exp ession e la ing he displayed noise igu e o he wo-po unde measu emen o (*) See Glossa y o e ms below 925 he pa ame e s o he measu emen sys em is de i ed: M h(T -T)+T M mF (h -m( sc) mc h F ( )m (ThT ) + mh ha o mhm s SCch0Gad (O)N/M1 Fdmeas Mh(T - T') + T'(hF ( sh) - M cF ( (6) (S) We a e in e es ed in he expec ed limi s o he e o AF (*) Exp e- 11 d~* ssions o he e o limi s can be ob ained by di e en ia ing he F iis' o mula F ( ) -i F =0 - 0 mo (*) d(O) F (O) Gd (0) yielding = 3F @FI+Fm oo Fd = - G (O) iFm + G (0) Gad (7) ad ~~~~ad Subs i u ing o F meas, Gdmeas in (7) we ob ain, wi h he lowes o de o app oxima ion (which is su icien i e o s a e no exceedingly high) F ( sc )Mc(ThTo) Fm( P )Mmc (Th To) F M N F SC ____F ______ Mc h meas + h _0 imeas-I (8) d F (F -)V(Th-T ) meas F ( )M h(Th-T) G + M G kOJl'$ThaTo) measFm 00 m h dmeas Mmh o dmeas o which he addi ional app oxima ions mh (T ha. 0) + TO(&hF (sh) McF (Psc)) 14h(Tha To) M (T -T )+T (M F (P ) -M F (P )) 14 (T - T ) mh ha o o mhm sh mc m sc mh ha o ha a e ul illed in mos p ac ical cases (noise sou ce ENR > 5dB), ha e been made. Fo low noise ampli ie s, designed o ha e a noise igu e nea he mi- nimun, i is easonable o assume ha sligh de ia ions o he sou ce im- pedance om he nominal alue do no p oduce la ge a ia ions o he ac- ual noise igu e /3/, so ha exp ession (8) can be app oxima ed by AF N(M (Th-TO) -OIF in(Mc ThTO) Fmmeas_ I4h N F na-I1 (*) d Mh(Tha-TO) meas(Mm (Tha°TO) Gdmeas Mh N Gdmeas Exp ession (9) indica es ha o low-noise ampli ie measu emen he p e alen causes o e o a e he sou ce impedance a ia ion due o he noise sou ce swi ching and he ENR unce ain y, and i is hough o be use ul in a R&D labo a o y en i onmen . I also shows ha including a well cha ac e ized isola o a he noise sou ce ou pu esul s in lowe e o limi s, in ag eemen wi h /4/. Exp ession (9) should no be used o wo- po s ha ing an inpu isola o ; in ha case F ( SC)/F (0) is known /2/ F ( ) I1 - s7 ji2 F (0) I - SC12 (assuming ideal isola o , s21=O) (*)See Glossa y o e ms below 926 and (8) leads o less pessimis ic limi s han (9), acco ding o /2/. No e ha (9) makes only use o F meas, FxmeasesGdmeas whjchcan be di ec ly ead du ing calib a ion and measu emen . PRACTICAL EXAMPLE Noise- igu e measu emen o a low-noise na owband mic owa e ampli 'ie a e conside ed. The ampli ie is used in he 2nd IF sec ion (2.62 GHz) o 20-30 GHz ea h s a ions ecei e s and i was designed o mee e y low noi se eque imen s (a ound 1 dB NF). I has wo s ages wi h NEC's NE75083 MES- FET's. The measu emen sys em is based upon a HP 8970 B noise igu e me e and HP 346B noise sou ces. Fou measu emen s we e made. The 'i s wo used a noise sou ce wi h ENR = 15 dB. The hi d and ou h used a noise sou ce wi h ENR = 5 dB. The second and ou h added an inpu isola o . Expe imen- al esul s o Fdmeas we e: (a) - Fdmeas = 1.17 dB (ENR 15 dB, no inpu isola o ) (b) - Fdmeas = 1.28 dB (ENR 15 dB, wi h inpu isola o ) (c) - Fdmeas = 1.07 dB (ENR 5 dB, no inpu isola o ) (d) - Fdmeas = 1.35 dB (ENR 5 dB, wi h inpu isola o ) The e o limi s we e hen calcula ed using (9) o (a) and (c) and(8) o (b) and (d) in he ollowing manne (see Table I)- I'scI,dT'shl and ENR unce ain y we e known om noise sou ce manu ac u e s da a. Ilns umen a- ion unce ain y we e also known om noise igu e me e manu ac u e 's da a. I'i| was measu ed. Thus i was possible o calcula e Mc/Mh unce ain y. In a simila way, Mmc/Mmh, Mh/Mmh and N/Mo could ha e been calcula ed om mj (manu ac u e 's da a) and j ol (measu ed). Howe e , as Gdmeas is high (29 dB), he con ibu ion o he second and hi d e ms in (9) is negligi- ble. Combining Mc/Mh unce ain y wi h ENR and ins umen a ion unce ain ies (in bo h wo s case and RSS manne ) he e o limi s we e ob ained. Figu e 2 shows he measu ed noise 'igu es and he compu ed e o limi s o Fd. Using ansis o manu ac u e 's da a, he app oxima ion F (Tsc)/F (0) Z 1 was calcula ed o a ec he es ima ion o he e o limi s by less han ± 0.07 dB in case (a) and less han ± 0.02 in case (c). CONCLUSIONS A gene al exp ession ela ing he noise igu e o a wo-po displayed by a noise measu emen sys em ha uses a Y- ac o me hod, o he a ious pa ame e s appea ing in a eal measu emen si ua ion in p esen ed. Simpli- ied exp essions a e de i ed o es ima e e o limi s in a noise igu e measu emen o low-noise ampli ie s e en hose no including isola o s a hei inpu s. These exp essions only equi e measu ed da a o Fm, Gd,| iJi and j oj and manu ac u e 's da a o he es o he pa ame e s. Calcula- ions show ha e o s can be as high as ± 0.74 dB when measu ing a I dB NF ampli ie wi h no inpu isola o and ha app osima ion F ( sc)/F (O) % I is jus i ied o ampli ie s wi h noise igu e nea he minimun. In his case, he measu emen e o s a e domina ed by misma ches. 927 GLOSSARY OF TERMS Fdmeas Measu ed Noise Figu e o he De ice Fmmeas Measu ed Noise Figu e o he Noise Me e F meas: Measu ed Noise Figu e o he Chain Gdmeas: Measu ed Gain o he De ice AFd = Fdmeas - Fd(O) Fd De ice Noise Figu e Fm Noise Figu e o he Noise Me e F Noise Figu e o he Chain Gad : De ice A ailable Gain G : Chain T ansduce Gain G m : Me e T ansduce Gain Te : E ec i e Inpu Noise Tempe a u e o he Chain Tem : E ec i e Inpu Noise Tempe a u e o he Me e Th : Noise Sou ce Tempe a u e (Calib a ion Value) Tha : Noise Sou ce Tempe a u e (Ac ual Value) sc,h : Noise Sou ce Re lec ion Coe icien (Cold/Ho ) Fm : Inpu Re lec ion Coa icien o he Noise Me e Fi : Two-Po Inpu Re lec ion Coe icien (du ing Measu emen ) 110 : Two-Po Ou pu Re lec ion Coe 'icien (du ing Measu emen ) '00 - Io when s'=0 (1 - I1 ( 12)- II 212) ( s' Ii) = Misma ch Coe icien ji - I'1 2j2 Mc = ( sc, i) Mh = (Fsh-, I;) Mmc = ( sc i m) Mmh = ( sh, m) N = ( ooI Fm) MO = (O,I'i) k Bol zmann's cons an B Sys em Bandwi h (much na owe han ha o he Two-Po ) REFERENCE S /1/ G. Mamola, M. Sannino "Sou ce misma ch e ec s on measu emen s o linea wo-po noise em- pe a u es" IEEE ans. on I.M., ol IM-24 n.3, sep . 1975, pp. 239-242 /2/ E. S id "Noise measu emen o low-noise GaAs FET ampli ie s" Mic owa e Sys ems News, no . 1981, pp. 62-70 /3/ N.J. Kuhn ''Cu ing a sub le bu signi ican cause o noise igu e e o "' Mic owa e Jou nal, june 1984, pp. 85-98 /4/ E. S id ''No'ise measu emen checklis elimina es cos ly e o s' Mic owa e Sys ems News, dec. 1981, pp. 88-107 /5/ "Noise igu e measu emen s. P inciples and applica ions" Hewle -Packa d semina , 1982 /6/ "Fundamen als o RF and Mic owa e Noise Figu e Measu emen s" Hewle -Packa d Applica ion No e 57-1, July 1983 928 il measu ed a,c b,d Mc/Mh unce ain y(dB) a,c b,d (.M! (W.C.) 15 dB 0.03 0.04 ±0.34 0+.02 0.56 0.061 0.26 5 dB 0.01 0.01 ±0.11 ±0.006 ENR unce ain y: ±0.3 dB (W.C.), ±0.1 dB (RSS) Ins umen a ion unce ain y: ±0.1 dB TABLE I - Pa ame e s in e eining in AFd Mm( s F m) CALIBRATION G m( s, m) TemC s) Gpd( i , m) L _ _ _ , _ F ( s) I +Te ( s) /To Fmc s)u1+Tem( s)/To G ( 3, i) FIGURE 1 Te ( s) - Block Diag. 1 . q:: 0) -o 03 . _- , C-,- I a CD 0) :D - 1a 5 4 AMPLIFIER INCLUDINC 3 AN INPUT ISOLATOR B (IslI =-0.061) (EXPRESSION (3)) I 9 ~ ~~~~~Ii LIMITS DUE TO i 1 TRUMNTATION ANDD- . ERUNCERTAINTY: ----WORST CASE 1-T--------I 1- '- '1-RSS -1 MEASUREMENT WITH A ' 15 DB ENR NOISE SOURCE WORST CASE MEASUREMENT WITH BB ': 5 DB EIIR IIOISE SOURCE RSS - BjA/ MEASUREMENT WITH 15 DB ENR NOISE SORCE WORST CASE MEASUREMENT WITH EBBS /5 DB ECR NOISE SOURCE , j- -7------- AMPLIFIER NOT INCLUDING INPUT ISOLATOR (i ,1FJ=0.56 . B' /: . (EXPRESSION (4)) . | | I A .5 I Unce ain y (dB) FIGURE 2 - E o Limi s 929 Noise Sou ce ENR(dB) I scI I shl Nl c, N2 C Ni ,N2 - - - i - - s I I I I I I I I_ - a