Noise Figure Minimization of RC Polyphase Filters

Transcription

1 Noie Figure Mimization of olyphae Filter Jáno advánzky Abtract - ideband uppreion of polyphae filter i dependent of the ource and load impedance. Thi property i valid for any number of tage and any detung between the tage. onequently, noie figure mimization can be done dependently of the adjutment of ideband uppreion. A formula for noie figure of paive two-port i obtaed, and noie figure of polyphae filter i mimized. The noie figure i trongly dependent on the ource reitance and capacitance. A formula for the lower bound of the noie figure of n tage polyphae filter i given. For a two-tage filter providg better than -5 ideband uppreion over two octave, a mimum noie figure of 0.8 wa found. INTODUTION polyphae filter (F are widely ued communication ytem []. Their typical application are modulation and demodulation of gle ideband ignal. Due to their implicity, they are popular IF tage of application-pecific tegrated circuit. Noie F i qualitatively analyzed [], where the relation between reitor value of different tage i dicued for reducg noie. A pioneerg paper on F i [3]. Derivation of the tructure and it ideband cancellation property are explaed there detail. None of the mentioned publication conider termation of F for mimum noie figure. However, a F uually follow the put mixer of a receiver thu it contribution to the ytem noie i ignificant. In thi paper we how that the ideband uppreion of an F i dependent of the impedance of the ource and the load under very general condition (ection. A formula for noie figure of paive two-port i obtaed (ection 3 and applied for our cae (ection 4. A an example, noie figure of a two-tage F, deigned for ideband uppreion lower than -5 over two octave, i mimized to 0.8 (ection 5. Experimental verification i preented ection 6. erhard chulte* TANFE FUNTION OF AN F The one-tage F i an 8-node tructure (Fig. [3] Figure : A one-tage F In Fig., all reitor and capacitor are identical. In a multitage filter, node are connected to node of the next tage, repectively. eitor and capacitor value of different tage are not necearily identical. In thi paper, we conider the cae when all capacitor are identical and only reitor of different tage may differ from each other. Input and put port are configured a hown Fig., node are parenthee: (,3, (,4, 3 (5,7 and 4 (6,8. Thu our firt goal i to obta voltage ga from port to port 3 and 4, repectively. V V 4 3 ort ort 3 olyphae filter ort ort 4 Figure : Excitation and loadg of the F autriamicroytem A, A-84 Unterpremtätten, chlo remtatten, Autria, phone: , Jano.advanzky@autriamicroytem.com

2 eneral formula for voltage ga of arbitrary number of tage i not known. For le than 30 ideband uppreion over ome frequency band, one or two tage are ufficient, thu we obta voltage ga for one and two tage filter: = ( V 3, jω + + V 4, = jω ( jω ω V 3, = (3 D jω( + V 4, = D (4 D = jω( (5 ω where Eq. (-( how voltage ga for one tage, Eq. (3-(5 for two tage, j i the imagary unit, ω i the angular frequency and i the load impedance. V3, i the ga from port to port 3 for tage, etc. eitor value the firt and econd tage are denoted by and, repectively. The ratio of the voltage ga i V 4, = jω (6 V 3, jω( V 4, = (7 V 3, + ω + where Eq. (6 i the ratio for one tage and Eq. (7 for two tage. The ratio of the voltage ga can alo be expreed by the amplitude and phae match factor m and ϕ: V 4 π = m exp j + ϕ (8 V 3 where m and ϕ are real quantitie, m 0. For perfect match, m= and ϕ=±kπ (k=0,,. By Eq. (6-(8, amplitude and phae match factor are dependent of the ource and the load impedance. The ideband uppreion can be expreed term of the amplitude and phae match factor: = 0log [ m co( ϕ ] + [ m ( ϕ ] [ + m co( ϕ ] + [ m ( ϕ ] (9 A a conequence, the ideband uppreion i alo dependent of the ource and load impedance. Thi property will appear important noie mimization. ideband uppreion uually depend on the difference between the two ource impedance and that of the load impedance, thu we have to take care for the ymmetry of termation realization. 3 NOIE OF INEA AIVE TWO-OT Our goal i to mimize the noie figure of a F. Due to the required ymmetry for termatg impedance, it i ufficient to conider noie figure for one put and one put, the other beg properly termated (Fig. 3. Our model i equivalent to the impedance matrix model. The circuit Fig. 3 i conidered to be thermal equilibrium, havg common noie temperature T for all noie ource. Thermal noie the two-port i modelled by the noie of. The two-port i excited by a ource with noiy ource impedance and loaded by a noiele load impedance. v n =4kT B V V V 3 7 Noiy two-port Figure 3: and are aumed to produce thermal noie. The two-port i modelled by, and the relation between V and V. ort and 4 of Fig. are termated by and, repectively The noie figure i defed a N F = (0 N where the ignal to noie ratio are N = N = ( n 5 n and we denoted the ignal and the noie with ubcript and n, repectively. The ignal power at the put i, aumg uoidal ignal with amplitude V : V = ( The noie power at the put i the noie power comg from the impedance : 4kTB n = (3 + Noie from port Fig. i cancelled due to equipotentiality. In Eq. (3, k i the Boltzmann contant (.3807*0-3 Joule/ K, T i the noie temperature and B i the noie bandwidth. The ignal power at the put i and the noie power i V = (4

3 kTB n = (5 + Noie figure of a two-tage polyphae filter Aplac 7.6 autriamicroytem A Apr n F/ df/ n n n 00.0k.0M 0.0M 00.0M.0 f/hz F_AA F_formula deltaf Fig. 4. omparion of Eq. (6 to the analyi reult By combg the equation above, the noie figure i obtaed: F + = (6 V + In Eq. (6, V = V / V i the loaded voltage ga. In Fig. 5 we compare the noie figure value obtaed by the circuit analyi program AA [4], to thoe of Eq. (6. Agreement i good. 4 NOIE FIUE MINIMIATION In thi ection, we ue Eq. (6 for mimizg of the noie figure of a F. The noie mimum i found with repect to. Followg thi tep, noie figure for the next tage (ummg buffer can be mimized with varyg. In the lat two tep, the variance of the ideband uppreion with repect to and i exploited. et u conider firt the one-tage filter. Analytic expreion for V,, and are = (7 V, jω + +,, jω + + = (8 jω + jω + jω + + = (9 jω + jω + With thee expreion ubtituted to Eq. (6, the noie figure at ω = / i: + + X X + F = (0 where = + jx. Noie figure at ω i dependent of a expected. The noie figure ha a mimum at = X = ( However, I realization, ductive termation i not allowed. With X = 0, the noie figure ha a mimum at It value i = ( ( F m = + (3 that i, Now let u contue with a two-tage filter. ubtitutg the expreion for V,,, and, to Eq. (6, the noie figure at center frequency ω = / i obtaed a follow, obervg that ω correpond to the local maximum of the ideband uppreion:

4 ( ( (3 + F = 4 (4 where all termation are conidered a reitive. The noie figure i dependent of the load a before. The noie figure ha a mimum at (3 + = (5 + 3 We note that cae of =, Eq. (5 reduce to Eq. (, and m ( F = (6 that i, Thi i the lower bound of the noie figure of two-tage F. By comparion of Eq. (3 and (6, it i eay to fd the general formula for lower bound of the noie figure of n tage: m n ( F ( = (7 n + Thi noie figure occur at the frequency of perfect amplitude match when all reitor of different tage are identical. We checked the validity of Eq. (7 by computer imulation for n=, 5 uccefully. noie matchg, the ource reitance for mimum noie figure i =.39 kω, and the noie figure i F =0.8. Noie figure at the center frequency a a function of the ource reitance and capacitance are given Fig. 5 and 6, repectively. It can be hown that f c and max determe filter bandwidth. Therefore the capacitance value a an put variable cannot be replaced by an equation for bandwidth. noie figure optimization of a polyphae Aplac filter7.6 autriamicroytem A Mar f 00.0f.0p 0.0p 00.0p noiefigure Fig. 6. Noie figure a a function of ource capacitance, when ource reitance i tuned to mimum noie figure at zero ource capacitance 5 EXAME: A TWO-TAE DETUNED OYHAE FITE A an example, the noie figure of a two-tage detuned F i mimized. Input data for the deign i the capacitance value =8 pf (due to the etimated ilicon area, the center frequency f c=0 MHz and the ideband uppreion at center frequency max=-5. The followg equation are olved for and : f = (8 π + = max 0 log (9 + + where Eq. (9 follow from Eq. (7-(9 and the condition for local maximum of. noie figure optimization of a polyphae Aplac filter7.6 autriamicroytem A Mar k 3.0k 0.0k30.0k 00.0k noiefigure Fig. 5. Noie figure a a function of the ource reitance, when ource capacitance i zero Eq. (8-(9 have two pair of olution, one of them i =.7 kω, =3.6 kω. Applyg Eq. (5 and (4 for 6 MEAUEMENT Thi i a verification of the theoretical ideband uppreion curve of a F and alo a verification of the tatement that the ideband uppreion i dependent of the ource reitance (Fig. 7. ideband uppreion of a polyphae Aplac filter 7.70 autriamicroytem A ep k 00.0k 400.0k f/hz _k _3k Fig. 7. Theoretical ( and meaured (_k, _3k ideband uppreion of a two-tage F Fig. 7 ha been obtaed ug a realized two-tage detuned F with circuit element value = 490 Ω, =9 Ω and =000 pf. Node wa excited with an opamp put erie with a reitor. Node 3 wa grounded. Node and 4 were termated alo by. Output port were ymmetrically termated by differential amplifier. All opamp are M38. The voltage ga have been meaured by a H3575A a-hae Meter. From the voltage ga ideband uppreion ha been calculated ug Eq. (9. A hown Fig. 7, = kω and =3 kω wa applied. ocidence between theoretical and meaured

5 curve i good, alo demontratg that ideband uppreion i really dependent of the ource reitance. Thu the bai of the ma tatement of thi paper i verified. 7 ONUION ome reult on noie figure mimization of F have been obtaed. In ection voltage tranfer function are given. It i hown that the ideband uppreion i dependent of the ource (and load termation, thu noie figure can be mimized with degradation of the ideband uppreion. In ection 3 a formula for noie figure of paive two-port i hown. In ection 4 noie figure of F are mimized. In ection 5 we apply our reult deigng and noie matchg of a two-tage detuned F. In ection 6 the dependence of the ideband uppreion of the ource reitance i verified experimentally. The dependence of the ideband uppreion from ource and load i proved here for one- and two tage. Thi property wa oberved a generally valid for arbitrary number of tage, however. Acknowledgment The author would like to acknowledge the leaderhip of the autriamicroytem A for providg excellent condition for thi reearch. Kd help from Mr.. Huzka and Dr. H. Novak readg the manucript and from Mr. I. Nagy performg the meaurement of the ideband uppreion, are greatly acknowledged. The author are grateful to the econd anonymou reviewer for hi valuable comment. eference [] J. rol, M. teyaert: MO Wirele Tranceiver Deign, Kluwer Academic ubliher, 997 [] F. Behbahani, Y. Kihigami, J. eete and A. A. Abidi: MO Mixer and olyphae Filter for arge Image ejection, IEEE Journal on olid-tate ircuit, Vol. 36, No. 6, June 00, pp [3] M. J. gell: gle ideband Modulation ug equence Aymmetric olyphae Network, Electrical ommunication, Vol. 48, No. -, pp. -5, 973 [4] AA 7.6, 00 AA olution orporation,