Keysight Technologies Keysight EEsof EDA Overview on Noise in Ring Topology Mixers

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1 Keysight Techologies Keysight EEsof EDA Overview o Noise i Rig Topology Mixers Article Reprit This documet is owed by Keysight Techologies, Ic. but is o loger kept curret ad may cotai obsolete or iaccurate refereces. We regret ay icoveiece this may cause. For the latest iformatio o Keysight s lie of Keysight EEsof electroic desig automatio (EDA) products ad services, please go to:

2 Itroductio Noise i Rig Topology Mixers Rick Poore Agilet EEsof EDA For a classical double-balaced rig-diode mixer, it is expected that the coversio loss (4 5 db) should match the oise figure. This documet explores the formal defiitio of oise figure ad shows the A oise figure defiitio. It is also show that the oise figure for a mixer does ot have to exceed its coversio loss, ad that loss i the mixer balus is a importat source of oise that must be cosidered. Throughout this discussio we assume a low-side ijectio, dow-covertig mixer, although the issues discussed are equally applicable to ay mixer. RF is the iput sigal, is the local oscillator, ad IF is the output sigal. I terms of frequecies, RF+IF ad there exists a iput image IM IF that ca also mix dow to the same IF as the RF. We are ot cocered with the output image sigal +RF. Throughout this paper we will shift omeclature back ad forth betwee oise voltage ( v ), oise power (N) ad equivalet oise temperature (T). They are all iterrelated by the equatio: N v R kt () L Noise temperature is used preferetially as it has reasoably sized umbers: 447 pv of oise ito a 5Ω load is the same as 4 W is the same as 9K. We try to use the rigorous defiitios of oise factor ad oise figure. The oise factor F is a algebraic quatity greater tha or equal to oe. The oise figure NF is the oise factor expressed i decibels ad is greater tha or equal to zero. Rig-Diode Double-Balaced Mixer The rig diode double balaced mixer show to the right cosists of a rig of four diodes ad two ideal ceter tapped trasformers (balus). Oe trasformer applies the sigal across the rig ad aother trasformer applies the RF sigal across the rig. The large sigal voltage (typically drive at +7 dbm) alterately switches oe pair of diodes (D ad D3) ad the the other pair (D4 Copyright, 3 Agilet Techologies

3 Noise i Rig Topology Mixers ad D) o. This switchig actio causes mixig of the RF sigal applied to the diodes by the other trasformer. This is a broadbad mixer, limited mostly by the physical costructio of the trasformers. Typical commercial mixers such as the MiiCircuits JMS- will operate for ad RF sigals from to 5 MHz.. Sigal Performace The spectrum to the right shows the simulated IF output from the mixer. The is MHz at +7 dbm ad the RF is 5 MHz at -5 dbm. Due to the double balaced ature of the mixer, the ad RF sigals are suppressed from the output. There are two primary output sigals at IFRF (5 MHz) ad the output image at RF+ (5 MHz). The output does show eve harmoics of the ad eve order mixig products from the RF ad odd harmoics of the (e.g. 75 MHz 3* MHz 5 MHz, 4th order).. Mixer ais to IF Just as the mixer shows a umber of output toes at differet frequecies, the mixer is also capable of covertig a umber of differet iput frequecies to the IF at 5 MHz. This balaced mixer will covert ay iput frequecy ( k f ± 5MHz ) to IF, where k is odd. If there is ay imbalace, the it will also covert from those frequecies where k is eve. The primary coversios will be from the RF frequecy at 5 MHz ad the iput image at 75 MHz. The simulated coversio gais from all possible iput sidebads are show to the right. The iput frequecy is LS f + Sidebad f IF (LS* MHz + Sidebad*5 MHz). The coversio gai from the RF frequecy, 5 MHz, is 4.84 db (.337) ad the coversio from the image iput frequecy, 75 MHz, is db (.38). The sum of the coversio gais from all of the higher order harmoic iput images is 3.47 db (.45). The sum of all coversio gais is.5 db (.744). These higher order gai terms will be importat later whe we discuss the measuremet of the mixer oise figure. Agilet EEsof EDA 7 Jue 3

4 Noise i Rig Topology Mixers 3.3 Ideal Switchig Mixer The output spectrum of a ideal switchig mixer ca be show to have the form [6]: V IF cos π 3 + cos 5 ( π ( f RF + f ) t) + cos( π ( f RF f ) t) cos( π ( f + 3 f ) t) cos( π ( f 3 f ) t) RF ( π ( f + 5 f ) t) + cos( π ( f 5 f ) t) + K RF The coefficiets i terms of the order are reproduced i the table below. order mag(v) mag(v ) db(v ) total.. The last colum shows the coversio gais from a sigle iput to each of the output frequecies k f ± f RF. A ideal switchig mixer will have a 3.9 db coversio loss for the primary coversio f ± f RF. The total ormalized power i the output spectrum is.; power is coserved as there are o losses i the ideal switchig mixer. By reciprocity, these terms also represet the coversio gais from the odd order terms k f ± f IF to f IF. 3 Noise Figure 3. A Bad Defiitio of Noise Figure for a Mixer A popular book o mixer desig [] gives a iterpretatio of the mixer oise factor: F T 3 5 RF RF () + SSB (3) where Tssb is the total iput-referred oise added by the mixer. The cotext of this equatio explicitly states that oise from the iput image is ot cosidered (but as will be show later, should be). This does match the defiitio of oise figure used for liear circuits such as amplifiers, but as will be show, should ot be applied to mixers. Rewritte i simulatio terms, F T T v RL + k T (4) k T where v is the oise voltage preset at the IF port whe the RF ad IF termiatios are oiseless; this is the output-referred oise added by the mixer. The + term represets the iput termiatio oise that mixes dow from the RF frequecy to the IF frequecy ( k T ). This oise factor defiitio is the same as that used for liear two ports. I a liear circuit there is o frequecy traslatio ad thus oly oe gai exists. Noise factor ca also be thought of as the sum of the oise geerated by the mixer ad the oise geerated at the iput processed through Agilet EEsof EDA 7 Jue 3

5 Noise i Rig Topology Mixers 4 the mixer divided by the oise geerated at the iput sigal processed through the mixer. Example What umbers does this geerate for a mixer? Simulatig gives v66. pv ad.38 (-4.84 db), so: F (v /k R L +T ) / (T ) ( ) / (.5 db) A mixer with a 4.84 db coversio loss gives a oise figure of.5 db. Shockig! It is assumed that the output termiatio is oiseless (questioable for the rig diode mixer; see below) ad that oly oise at the iput termiatio at the RF frequecy is mixed to the output at the IF frequecy. This secod assumptio is icorrect, ad is caused by a icorrect readig of the IEEE defiitio of oise figure, or by usig the oise figure defiitio for etworks without frequecy traslatio. The author of [] does poit out that this igores iput termiatio oise from the iput image frequecy, ad does iclude that term i a alterative SSB oise figure defiitio: Example For the same mixer: F T T T + SSB (5) F (v /k R L + T ) / (T ) (4.+9.3)/ (3.84 db) The oise figure ow rises to 3.84 db. However, this igores other sources of oise from iput harmoic frequecies that ca mix to the IF frequecy. While small, these are preset i the rig diode mixer, ad ideed, most mixers. From a detailed readig of the oise factor defiitio, we shall see that all of these terms must be icluded i the defiitio. 3. IEEE Defiitio of Noise Factor The IRE, predecessor to the IEEE, origially defied oise factor for liear twoports i 96 []. This defiitio was clarified ad exteded to mixers i 963 [3]. The 963 defiitio reads: At a specified iput frequecy the ratio of to ) the total oise power per uit badwidth at a correspodig output frequecy available at the output port whe the oise temperature of its iput termiatio is stadard (9K) at all frequecies ) that portio of ) egedered at the iput frequecy by the iput termiatio at the stadard oise temperature (9K). Note : For heterodye systems there will be, i priciple, more tha oe output frequecy correspodig to a sigle iput frequecy ad vice versa; for each pair of correspodig frequecies a oise factor is defied. ) icludes oly that oise from the iput termiatio which appears i the output via the pricipal-frequecy trasformatio of the system, i.e. via the sigal-frequecy trasformatio(s), ad does ot iclude spurious cotributios such as those from a uused image-frequecy or a uused idlerfrequecy trasformatio. Agilet EEsof EDA 7 Jue 3

6 Noise i Rig Topology Mixers 5 Note : The phrase "available at the output port" may be replaced by "delivered by system ito a output termiatio". Note 3: To characterize a system by a oise factor is meaigful oly whe the admittace (or impedace) of the iput termiatio is specified. ) represets the umerator of the oise factor fractio ad ) represets the deomiator. Clearly ) is the total oise output power at the IF port ad frequecy, ad says othig about ot cotaiig oise due to the iput termiatio at the image frequecy (or ay other frequecy). Ideed, the text i the paper makes it clear that ) icludes all sources of oise, icludig ay ad all trasformatios of oise from the iput termiatio to the IF port ad frequecy, all oise geerated by the mixer itself, ad oise arisig from the output termiatio. This represets two departures from the defiitio of oise factor for o-frequecy traslatig etworks. First, the umerator should iclude all trasformatios of iput termiatio oise to the IF port ad frequecy. The traslatio of this oise to IF is uavoidable ad must be cosidered. I discussig ), the origial paper divides the output oise power ito two parts: A) The cotributios to the output oise power due to oise power available from the impedace termiatios that are coected to all accessible ports except the output port uder operatig coditios. These cotributios ca be described by assigig appropriate oise temperatures to the iput termiatios at all of the various resposes. For the sake of modelig, the origial paper splits the iput termiatios ito a umber of separate termiatios, where each termiatio ca represet a sigle source frequecy that ca be traslated by the mixer to the specified IF frequecy ("a respose"). The appropriate oise temperatures for the iput termiatios are all the same temperature, the IEEE stadard oise temperature (9K). B) All other cotributios to the output oise power. These iclude oise geerated withi the receiver compoets as well as oise resultig from ay frequecy coversios iteral to the receivig system. Noise geerated i the load ad reflected at the receiver output may also cotribute to this output oise power. [italics added] I the framework of [3] the oise geerated withi the mixer NN is characterized by a sigle umber, T e : N e ( ) N k T K+ 3 (6) where is the primary coversio gai from RF to IF, is the coversio gai from the iput image to IF, ad 3 ad higher represet the other possible higher- order trasformatios to IF, icludig the possible leakage of iput termiatio oise at the IF frequecy to the output termiatio. A sigle umber is used because this is oise that is commo to all iput resposes. For most mixers, ad are approximately the same ad are much larger tha 3 K. Ufortuately, most texts ad equatios assume ad completely igore 3 K by assumig they are zero. While this may be true for a well-desiged mixer with good filterig, for a broadbad mixer such as the rig-diode double balaced mixer they are ot zero. For example,.33,. 38 ad 3 + K Everyoe s iterpretatio of ) is correct, i that it represets oly the trasformatio of oise at the RF port ad frequecy to the IF port ad frequecy (the pricipal trasformatio ). By Agilet EEsof EDA 7 Jue 3

7 Noise i Rig Topology Mixers 6 dividig by oly ) gives the sigle sidebad oise figure, as it is assumed that oly a sigal at oe iput sidebad is of iterest. There has bee some cocer as to what type of gai should be used i these equatios: available or trasducer. Available gai measures the power that could be delivered to a load that is cojugately matched to the output port to that available from the iput source. Trasducer gai measures the power that is delivered to the actual load, matched or ot, to that available from the iput source. While the defiitio of oise figure icludes the word available ( the total oise power... available at the output port ), Note states that available at the output port may be replaced by delivered by the system ito a output termiatio, which souds like trasducer gai. I the text of the paper, the gai is clearly defied as a trasducer gai: the ratio of the sigal power delivered... ito the output circuit to the sigal power available at the iput. The text also clearly uses the phrase trasducer gai at several poits. 3.3 A Implemetatio of Mixer Noise Figure So let s apply our kowledge from reiterpretig the defiitio of oise figure ad see what it gives to our mixer. v F ( + + K ) RL + k T + k T The umerator cosists of two terms. The first term is the oise cotributed by the mixer itself ad has ot chaged from the old defiitio (4). The secod term is the oise due to oise at the iput termiatio at all frequecies that mix to the IF frequecy, icludig the possible direct coversio of iput termiatio oise at the IF frequecy to the output termiatio at the same frequecy. I the origial defiitio this term was represeted oly the oise at the RF frequecy k T ). The IF termiatio is still treated as oiseless. ( Example Ruig the umbers:.38,.337, F ( ) / (4. db) The oise figure is ow 4. db by icludig all of the possible coversios of iput oise. 3.4 Iclusio of Output Termiatio Noise [3] specifically states about the total oise at the output B): Here, NL is a term that takes ito accout the cotributio of oise geerated i the load ad reflected at the output of the receiver, sice the effective iput oise temperature defiitio does ot iclude such a cotributio. I order to evaluate NL oe must kow the impedace ad the equivalet oise temperature of the load, ad the output impedace of the receiver. I most cases, NL will be egligible compared to the other terms i this expressio. The cotributio NL may be of importace i systems with isufficiet gai... Further: A few words of explaatio may be i order for the reaso why Te [the iput referred mixer geerated oise] excludes the load oise cotributio ad Top [the total output (7) Agilet EEsof EDA 7 Jue 3

8 Noise i Rig Topology Mixers 7 oise] icludes it. I excludig the load oise from the defiitio of Te the effective iput oise temperature of a sigle respose two-port is brought ito direct correspodece with its oise figure F accordig to the formula Te9(F-). The oise figure cascadig formula of sigle respose two-ports may thus be adapted to Te. The cascadig formula facilitates the evaluatio of Te of a amplifier chai. The operatig oise temperature Top is iteded to be a measure of the oise of a receivig system. The iclusio i Top of the cotributio of the load-oise gives a more realistic measure of system oise performace i those cases i which the load oise is appreciable. Sadly the ext paragraph reads: Heceforth, we shall disregard NL, because i most cases of iterest it is egligible. ad everyoe who has published aythig further o mixer oise has igored it. This eglect of NL is cosistet with the stadard view of oise factor for liear etworks. Yet as we ca see from simulatio, or cosider from commo sese, a low oise mixer with a coversio gai less tha uity will atteuate ay iput oise ad sigal ad add little oise of its ow, so ay oise geerated i the output termiatio will be o-egligible ad should play a part i the sigal to oise ratio at the output Termiatio Noise at the IF Frequecy If S is the output reflectio coefficiet ad the termiatio is at the stadard oise temperature (9K) the the amout of oise preset at the output termiatio due to oise geerated by the output termiatio at the IF frequecy is: N k T S L + (8) But if the termiatio is matched ( S ) the oly a oise power of k T will be see. To compesate for the matched case, we should reduce N L by k T to cosider oly the excess oise from the load mismatch: ( + ) N L k T S (9) Addig this to the umerator of the oise factor defiitio gives: v F ( + + K+ ) RL + k T + k T Example Simulatio of the oise from the IF port at the IF freq gives 54 pv, which correspods to 4.4 K. Subtractig To9 gives: F ( ) / (5.98 db) By icludig oise geerated by the mismatched output termiatio, this brigs the oise figure up to 5.98 db Termiatio Noise at Frequecies Other Tha IF There is aother iterestig pheomeo that takes place i the rig-diode double-balaced mixer at the output termiatio: self mixig. The output termiatio geerates broadbad oise. The oise at some frequecies ca eter the mixer at the output, get processed by the N L () Agilet EEsof EDA 7 Jue 3

9 Noise i Rig Topology Mixers 8 oliearity ad reappear at the IF frequecy of iterest at the output termiatio. This adds a o-egligible oise source that should be icluded i the total output oise. Let us defie H i as the coversio reflectio gai at the output port from frequecy i to the IF frequecy. We will defie H ad higher, cosiderig that H (coversio from IF frequecy to IF frequecy) is really what we are computig above i N L. The additioal oise due to the output termiatio is the: N ( H + H + K ) SM k T 3 H () Note that the frequecies at the output that traslate to IF are ot ecessarily the same oes that traslate to IF from the iput. Addig this to the umerator of the oise factor defiitio gives: v F ( + + K+ ) + k T ( + S ) + k T ( H + H + K H ) RL + k T 3 + k T Example Simulatio of the self-mixig oise is demostrated i tb_selfmix. Here oise is ijected at the IF port at every frequecy except IF, ad the amout of oise comig back to the IF is measured. For this mixer.9 pv of oise was observed, correspodig to.53k. F ( ) / (6. db) The oise figure is ow 6. db by icludig all of the output self-mixig oise. This effect is egligible i active mixers built with bipolar ad field-effect trasistors, icludig FET-rig mixers. This is because the three termial oliear device provides some isolatio betwee its termials ad each sigal (RF, IF, ) ca be associated with a termial. But i diode mixers, the three sigals must be shared amog two termials, reducig the isolatio. (We use the term isolatio here i a maer differet tha ormally used i mixers to discuss - IF, RF-IF isolatio). 3.5 Summary of Noise Figure Cotributios Let s ow itemize all of the terms i the umerator of the oise figure equatio ad their cotributio to the sigle sidebad oise figure. Name Value (K) Partial Noise Figure k To Diodes 4..5 k To k To (3+...+) A NFssb value IF mismatch IF self-mixig The partial oise figure is computed for each term by summig from the top to each lie the total oise to form the umerator ad dividig by the first term: F terms T ). ( () Agilet EEsof EDA 7 Jue 3

10 Noise i Rig Topology Mixers 9 4 Measuremet of Noise Factor 4. Measuremet Equipmet Noise figure is measured with a oise figure meter such as the Agilet 897B. A oise figure meter works via the hot-cold method. The iput termiatio is replaced by a oise source that is switched betwee a cold temperature (room temperature or 9K) ad a high temperature (,K for a +5.5 db excess oise ratio (ENR),,K for a +5.5 db ENR). By measurig the output oise i both the cold ad hot states ad kowig the ENR, the oise factor ca be computed. T F hot 9 9 N hot N cold N The hot-cold techique is oly capable of measurig the all sidebad oise figure, ormally called the double sidebad oise figure (which eglects the higher order dow coversio products). The above equatio is equal to: F ASB v RL + kt cold k T ( + + K+ ) ( + + K+ ) What most texts ad oise figure meters do whe covertig to a sigle sidebad oise figure is drop the higher order coversio terms 3 K ad assume whe derivig their versio of (4). This erroeously gives: This is coverted to sigle sidebad by dividig by : F F B SSB (3) (4) v RL + k T (5) k T v RL + k T (6) k T But for a mixer with ay measurable coversio from the higher frequecies, this will be i slightly error. The correct equatio is: F SSB v ad the error i the faulty approximatio is: df R SSB L + k T k T ( + + K+ ) + + K+ The 897B reports to the user the all sidebad coversio gai + + K+ ad oise figure. To correct these to sigle sidebad, the factor ( + +K ), i decibels, should be added to both values. F SSB F K+ (7) (8) (9) Agilet EEsof EDA 7 Jue 3

11 Noise i Rig Topology Mixers The 897B ca make its ow +3 db correctio ( + ) whe reportig the results, but we have show that this ca be i error by up to.5 db for a typical mixer due to higher order coversios that are icluded i the measuremet. Example For our example mixer, should be replaced by (+...+)/.6. The error i the oise figure for igorig this is.6/ or.65 db; the actual oise figure is.65 db higher tha the meter-corrected SSB oise figure. The table below shows the uderestimated error for several differet mixer circuits that would be made if a oise figure meter coverted the measured double sidebad oise figure to sigle sidebad by dividig by two. Name Descriptio (+...+)/ /(+...+), db mixa rig diode mixer.4.3 jmixer JFET rig mixer..4 mix4 ilbert cell with amps.7.5 e6 ilbert cell.5. mix3 NE6 ubalaced BJT mix A Mixer NF6.. repa sigle diode.5.46 mix3 Lage coupler, diodes.5.4 mix3 BJT sigle balaced mix5 Ratrace, diodes mix pager_mixer, BJT.6.3 ideal switchig mixer Simulatig a Hot-Cold Measuremet A adds two output variables: NFssb ad NFdsb. NFssb is the IEEE stadard sigle sidebad oise figure defied i (7) or (7), ad NFdsb is the all (double) sidebad oise figure defied i (4). ad is equivalet to what is measured by a oise figure meter usig the hot-cold techique. We ca also perform a hot-cold measuremet maually i A to simulate what the oise figure meter would measure. The picture below shows the simula-tio setup required. The iput termiatio is replaced with a resistor whose temperature is swept betwee 9K ad K. The output termiatio is replaced by a oiseless resistor. The oise voltage at the output termiatio is measured ad used to compute the all-sidebad oise figure i the data display server usig the followig equatios: TcHB.Tiput[] ThHB.Tiput[] Ncvif.oise[,]**/(5*boltzma) Nhvif.oise[,]**/(5*boltzma) Fasb(Th-Tc)/Tc*Nc/(Nh-Nc) Agilet EEsof EDA 7 Jue 3

12 Noise i Rig Topology Mixers NFasb*log(Fasb) asb(nh-nc)/(th-tc) Iasb*log(asb) Example Simulatio gives Ncold48K ad Nhot755K. This gives: Fasb. (.79 db) asb.74 (-.47 db) Fssb Fasb * (+...+)/.55 (4.7 db) ssb asb * /(+...+).336 (-4.74 db) which matches the computatio doe above. 4.3 Filterig for SSB Most refereces o measurig oise figure state that sigle sidebad oise figure ca be measured simply by puttig a arrow badpass filter i frot of the RF iput port that oly passes RF, so that the mixer has o coversio gais from frequecies other tha RF. Note that this ca dramatically chage the oise figure. A badpass filter will have S ad S i the passbad ad S ± ad S everywhere else. This dramatically chages the iput load that the mixer sees ad chages the coversio gai ad the geeratio of oise. The siglesidebad measuremet ca be made to work with a filter if the filter ca have S everywhere, but that is ot typical of a physical badpass filter. Example Cosider the mixer we have bee examiig so far. Without a iput filter it has a coversio gai of db. A Chebychev badpass filter (order3) is placed at the RF iput cetered at RF (5 MHz). The mixer oise ow has a coversio gai of db. If we add a ideal filter that passes oly RF while maitaiig a broadbad S, the coversio gai remais at db. The simulated oise figures (properly adjusted to SSB) are: Filter Cgai NF NF Noe Chebychev Ideal Idetical results are obtaied though both the hot-cold simulatio ad the stadard techique discussed above. Whe the filter is added, the SSB correctio factor chages from (+...+)/.6 to uity, because the filter chages... to zero. 4.4 Measured Data A MiiCircuits JMS- mixer was measured o a Agilet 897B oise figure meter. The was 55 MHz ad the IF frequecy was 4 MHz. The measured all-sidebad coversio loss ad oise figure are show below as a fuctio of the drive power. Below +5 dbm, the oise figure is lower tha the coversio loss. So how ca the oise figure be less tha the loss? If Agilet EEsof EDA 7 Jue 3

13 Noise i Rig Topology Mixers we look at the oise figure defiitio for a mixer that geerates o oise: k T F ( + + K+ ) k T K The coversio loss is. For a ideal switchig mixer, the sum of the coversio gais is, givig a oise figure of, idetical to the coversio loss. But for a o-ideal mixer, there are additioal losses due to the diodes ad trasformers, so the sum of the gais is less tha, givig a oise figure of less tha. If the mixer has egligible higher order coversio gais, the F ( + ) (3 db), idepedet of the coversio gai. 5 Ifluece of Balu Loss o Noise The mixer simulated earlier was chaged to iclude models of real balus. Mixers were simulated usig measured data for three differet balus. Surface mout ceter tapped trasformers were purchased from MiiCircuits ad their S parameters were measured from khz to 5 Hz. The measured S parameters were the used i the simulatio of the mixer. Agilet HSMS-8 Schottky diode pairs were also used i the mixers. The mixers were simulated with f RF 95 MHz, f 55 MHz, P +7 dbm ad f IF 4 MHz. Model Impedace db Ratio Badwidth TCM-T : 3-3 MHz TCM3-T 3: 5-3 MHz TCM4-6T 4: 3-35 MHz The simulated oise results are show below. The coversio gai ad SSB oise figure are give. The equivalet oise temperature for the balus ad diodes are also listed. It ca be see that a sigificat amout of oise is geerated by the balus, similar to the amout of oise geerated by the diodes. I the mixer that has bee cosidered elsewhere i this paper, the balus were ideal: lossless ad oiseless. TCM TCM3 TCM4 Simulated coversio loss, SSB Simulated ew oise figure, ew SSB Noise figure, ew SSB eglectig balus Noise temperature (K), RF-IF balu Noise temperature (K), balu Noise temperature (K), all 4 diodes It ca be see that eglectig the balu oise will chage the oise figure by up to several db. 6 Coclusio Held ad Kerr explaied away may of the miscoceptios about mixer oise figure startig i 978 [4,5]. The misuderstadig seems to have arise with the assumptio, by Messeger ad McCoy, that a mixer ad a passive atteuator have similar oise properties ad that the () Agilet EEsof EDA 7 Jue 3

14 Noise i Rig Topology Mixers 3 temperature of this equivalet atteuator is the time average of the static oise characteristic of the diode. This led to a widely held belief that for a Schottky diode mixer, the oise-temperature ratio t... Of the early authors writig about mixer oise figure, those that wrote F N MIX + k T (3) were justified i writig + because they explicitly stated that the iput termiatio was either a ope or a short circuit at the iput image frequecy. Higher order mixig products were either igored or termiated i opes or shorts. Appedix A: Ifluece of Order ad OverSample o Simulatio Results Variatio i the coversio gai ad oise figure is observed as the order is varied. This effect is especially strog i the rig diode mixer as the diode curret is a expoetial of half a sie wave, requirig a large Fourier series to represet it accurately. Differeces i the coversio gai are oticed whe compared to a trasiet aalysis, at least util Order9 is used. The graph shows coversio gai ad oise figure as a fuctio of Order. The results did ot vary as OverSample was swept from to 3. Appedix B: FET Rig Mixers Rig mixers ca also be built with FETs []. The FETs are used as resistive switches ad are operated i the liear regio with Vds. The is applied to the gate with a DC bias ad the RF ad IF are applied to source ad drais. As log as the RF drive stays low, the Vds ca be cosidered to be zero. Ay type of FET ca be used, icludig aasfets, MOSFETs or JFETs. A extra trasformer is required i this desig to extract the IF. Cosider the basic equatio for FET drai curret i the liear regio: Agilet EEsof EDA 7 Jue 3

15 Noise i Rig Topology Mixers 4 I [ ( V V ) V ]( λv ) β V + (4) S There is oe problem i the FET oise models. The stadard FET chael oise model is i 8k T 3. But for these FETs with V, g : g m g I m V VS T m ( λv ) β + (5) ad therefore the FETs are oiseless. I this case the FET chael should be modeled as a resistace based o the small sigal resistace g which is valid i the liear regio: g d I V d V β ( V V ) Noise geerated by this resistace ca be modelig by icludig a suitable oise source i parallel with the FET so that i 4k T g. d Let s ow itemize all of the terms i the umerator of the oise figure equatio ad their cotributio to the oise figure. This mixer has a coversio gai of db ad the old A oise figure was.9 db. With the ew equatio, the oise figure is 4.54 db. Name Value (K) Partial Noise Figure k To 8.. JFETs JFET chael k To k To (3+...+) A IF mismatch IF self-mixig Note that for this type of mixer, the IF self-mixig term is egligible. This may be attributed to the uidirectioal ature of the FETs: oise comig ito the source-drai does ot traslate back to the gate/ side ad the remix to the IF. Ackowledgemets Thaks to Roger Braso of Nokia ad Prof. Stephe Log of UCSB for first discoverig this problem. Refereces [] S. Maas, Microwave Mixers, d ed., Artech House, p. 68, 993. [] IRE Subcommittee 7.9 o Noise, IRE Stadard o Methods of Measurig Noise i Liear Twoports, 959, Proceedigs of the IRE, vol. 48, pp. 6 68, 96. [3] IRE Subcommittee 7.9 o Noise, Descriptio of the Noise Performace of Amplifiers ad Receivig Systems, Proceedigs of the IEEE, vol. 5, pp , 963. [4] D. N. Held ad A. R. Kerr, Coversio Loss ad Noise of Microwave ad Millimeter Wave Mixers: Part Theory, IEEE Trasactios o Microwave Theory ad Techiques, vol. 6, S T (6) Agilet EEsof EDA 7 Jue 3

16 Noise i Rig Topology Mixers 5 pp , Feb [5] A. R. Kerr, Shot Noise i Resistive Diode Mixers ad the Atteuator Noise Model, IEEE Trasactios o Microwave Theory ad Techiques, vol. 7, pp. 35 4, Feb [6] B. ilbert, Desig Cosideratios for BJT Active Mixers, lecture otes from Mead Microelectroics short course RFIC Desig for Wireless Commuicatios Systems, Sata Clara, April 995. Agilet EEsof EDA 7 Jue 3

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