Translational Circuits

Transcription

1 Circui for ll Seasons ehzad Razavi Translaional Circuis TTranslaional circuis are periodically driven, ime-varian sysems and encompass opologies such as commuaed neworks and N -pah filers. These sysems cause frequency ranslaion (shif) of impedances and ransfer funcions, a useful propery ha has led o myriad new and ineresing conceps in RF design. In his aricle, we sudy he operaion principles of hese circuis. E i RI Capaciors Roaing rushes R v o v o /E i 0 f f Frequency 3f Early Hisory The use of ime-varian sysems (e.g., mixers) for he ranslaion of ransfer funcions daes back o he lae 1940s and early 1950s. In heir 1948 paper [1], usignies and Dishal propose he arrangemen shown in Figure 1, where a mechanical wheel holding a number of capaciors roaes a a rae of f urns per second, allowing each capacior o charge o he inpu and, half a cycle laer, discharge a he oupu. The inpuoupu ransfer funcion hus appears as in Figure 1, revealing a comb filer response wih band-pass peaks locaed a he harmonics of f. In 1953, Le Page e al. [] presened he implemenaion depiced in Figure, which employs swiches and o alernaely connec each capacior o he inpu and o he oupu. lso in 1953, Smih [3] provided a comprehensive and inuiive reamen of hese circuis. Shown in Figure 3 are Smih s original drawings, suggesing ha a firs-order low-pass RC response can be ranslaed o a cener frequency of fr if he capacior Digial Objec Idenifier /MSSC Dae of publicaion: 1 January 016 Figure 1: 1948 elecromechanical sysem and is ransfer funcion. e o () C p e p (k) Figure : 1953 elecromechanical sysem wih ranslaional behavior. is replaced by an array of N commuaed capaciors ha connec o he oupu node one a a ime a a rae R e() πf r πf r ( mt i ) e (k) C 0 e o (k) of fr. (This acion is realized oday by N swiches driven by nonoverlapping phases of a clock.) 8 WINTER 016 IEEE SOLID-STTE CIRCUITS MGZINE

2 Noably, Smih also prediced he 3-d bandwidh of he resuling bandpass response o be 1/( r NRS C), where RS denoes he source resisance. This equaion proves useful for esimaing he bandwidh in erms of he oal commuaed capaciance. Smih exended he idea o a firs-order high-pass response as well, arriving a he noch filer illusraed in Figure 3. The foregoing opologies exemplify ime-varian circuis, lending hemselves o a general model proposed by Franks and Sandberg in 1960 [4] and shown in Figure 4. Each pah mixes he inpu signal wih differen phases of he local oscillaor (), ranslaes he specrum o baseband, subjecs he downconvered signals o a desired ransfer funcion, and mixes he resuls wih he phases again so as o reurn he (shaped) specrum o is original cener frequency. The erm N-pah filers was evidenly coined by Franks and Sandberg o refer o hese circuis. n imporan difference beween his absracion and Smih s circuis is ha he former assumes unilaeral sages whereas he laer enail ransparency beween he inpu and he oupu, a useful propery for impedance ranslaion. Figure 3: Smih s band-pass and noch filer implemenaions. Inpu x 1 ( ) y 1 ( ) v 1 ( ) Oupu h( ) u( ), U(s) v( ), V(s) p [ ] q [ ] x n ( ) y n ( ) h( ) v n ( ) p [ (n1)t ] q [ (n1)t ] x N ( ) y N ( ) v N ( ) h( ) p [ (N1)T ] q [ (N1)T ] Figure 4: general model of ranslaional circuis proposed by Franks and Sandberg. Impedance Translaion by Parial Commuaion Translaional circuis can shif an impedance o a well-defined cener frequency. For example, he impedance of a capacior, 1 /( j~ C), can be ranslaed o a cener frequency of ~, aking on he form 1/[ j( ~ - ~ ) C]. In oher words, he impedance of he new nework goes o infiniy a ~ = ~ raher han a ~ = 0. In his secion, we invesigae how he ranslaion occurs. Due o swiching aciviies, commuaed neworks can produce a nonsinusoidal volage in response o a sinusoidal curren. To find he impedance, herefore, we mus compue he firs harmonic of he volage. Consider he parallel RC branch shown in Figure 5, assuming Iin () = I0sin ~ in and R1C1 & Tin = ( r / ~ in). In his case, mos of he inpu curren prefers o flow hrough C1, generaing V. ( 1/ C1) # Iin( ) d. Now, le us swich C1 periodically and sudy a special case where he inpu frequency The use of imevarian sysems (e.g., mixers) for he ranslaion of ransfer funcions daes back o he lae 1940s and early 1950s. is equal o he frequency [Figure 5]. We wish o sudy he impedance seen beween and, Z. Wih he phase relaionship shown here, we observe ha he posiive half cycles of Iin flow primarily hrough C1, and he negaive half cycles enirely hrough R1. Tha is, C1 receives a half-wave recified version of Iin, hereby accumulaing posiive charge. The circui reaches he seady sage when, during each posiive half cycle, he charge delivered by Iin o C1 is equal o he charge drawn by R1 from C1. The key poin here is ha he oupu volage swing can become arbirarily large if R1 has an arbirarily high value even hough C1 periodically swiches ino he circui. This analysis culminaes in he following observaion: a sinusoidal curren of frequency fin = ~ in/( r) = f can produce a very large volage swing a fin beween nodes and in Figure 5, revealing ha he equivalen impedance of he swiched IEEE SOLID-STTE CIRCUITS MGZINE WINTER 016 9

3 (c) (d) Figure 5: simple RC circui driven by a sinusoidal curren, waveforms when C1 is commuaed, (c) waveforms when fin! f, (d) a simplified circui if fin - f is relaively large, and (e) he magniude of impedance versus frequency. C S 1 Magniude (d) S 1 C Frequency (GHz) (e) Figure 6: n RC circui using wo commuaed capaciors and he magniude of impedance versus frequency. Magniude (d) Frequency (GHz) capacior is very high a his frequency. This siuaion sands in conras o ha in Figure 5, where C1 yields a low impedance a fin. We herefore conclude ha he impedance of he capacior is ranslaed (upconvered) o a cener frequency of f, emerging as 1/[ j( ~ - ~) C1]. Of course, his propery arises from he bilaeral naure of he swich: he volage across he capacior is mixed wih he and up-convered as i manifess iself in V. We call his circui s operaion parial commuaion o emphasize ha Iin does no see a capaciance for half of he period. Wha happens if fin in Figure 5 depars from f? The synchroniciy beween Iin and he no longer holds, C1 receives boh posiive and negaive charge from Iin, and he capacior volage grows o a lesser exen [Figure 5(c)]. If fin - f is large enough, we can say ha, when S1 is on, he swing in V is approximaely proporional o 1/( C1~ in) (by virue of inegraion), and when S1 is off, he swing is proporional o R1, a much greaer value because R1C1 & Tin. We can hus approximae C1 by a shor circui and reduce he circui o ha in Figure 5(d). Here, V () is simply equal o Iin () S (), where S () denoes a square wave oggling beween zero and one. Of ineres o us is he inpu impedance in he viciniy of fin ; so we mus seek he componen of V a his frequency. Since S () has a dc value of 0.5, he ampliude of V a fin is given by 05. I0R1. We conclude ha he inpu impedance falls from (approximaely) R1 a fin = f o 05. R1 for fin somewha far from f. The simulaed plo in Figure 5(e), where f = GHz, confirms his resul, revealing very lile seleciviy in his inpu impedance. Impedance Translaion by Full Commuaion How can we improve he seleciviy of he impedance ploed in Figure 5(e)? s noed above, he componen resuling from Iin R1 overwhelms he 10 WINTER 016 IEEE SOLID-STTE CIRCUITS MGZINE

4 up-convered capaciance as fin subsanially depars from f. This means ha we mus no permi much of Iin o flow hrough R1 even when C1 is swiched ou. This is accomplished by adding one more capaciive branch ha is conrolled by [Figure 6], hus creaing full commuaion. ssuming ha C1 = C = C, R1C1 & T, and fin = f, we observe ha V is equal o V for one half cycle and o V for he oher half, growing in ampliude in boh posiive and negaive direcions. Tha is, V is defined by capaciive dynamics on op and boom, wih boh is posiive and negaive ampliudes decreasing as fin - f increases. Ploed in Figure 6, he impedance coninues o fall, exhibiing greaer seleciviy han ha seen in Figure 5(e). We should make wo remarks. Firs, he condiion R1C & T is necessary for significan up-conversion of he capaciive impedance. Each ime a swich urns on, he corresponding capacior impresses is volage upon he oupu; he longer his volage lass, he more pronounced is he effec of he capacior. Ploed in Figure 7 is he oupu volage for he exreme cases R1C & T and R1C % T, revealing a specrally rich signal in he former ha sems from subsanial up-conversion of he capacior volages. In he laer, on he oher hand, each capacior rapidly loses is charge when i is swiched in, allowing V o assume he shape of Iin. The key poin here is ha he commuaed nework need no be driven by a high source resisance so long as R1C & T. Second, he ranslaion of he capaciive impedance in Figure 6 also occurs a he higher harmonics of he. For example, an inpu of he form Iin = I0 sin( 3~ ) leads o accumulaion of charge on C1 and C and a all waveform for V. Translaion of Transfer Funcions Equipped wih he commuaed capaciors of Figure 6, we can now explore mehods of ranslaing ransfer funcions. We surmise ha (V) (V) Time (ns) Time (ns) Figure 7: The volage across a commuaed nework wih R & Tin and R % Tin. Figure 8: Low-pass o band-pass and high-pass o noch ransformaions. 0 f 0 0 f C N C N f f 0 f f IEEE SOLID-STTE CIRCUITS MGZINE WINTER

5 Figure 10: Seady-sae waveforms in a commuaed circui. T C V 1 R sw W C N T R sw Ideal Nework Figure 11: The overlap ime beween and, he equivalen resisance for differenial discharge, and (c) a single-ended equivalen. W W Figure 9: The effec of swich resisance on he ransfer funcion. 0 f f 3f f R eq R eq (c) C C R eq replacing coninuous-ime capaciive branches in a nework wih heir commuaed counerpars permis such a ranslaion. pplying his procedure o a firs-order RC filer yields he circui in Figure 8 and a band-pass response around f (and is harmonics). Similarly, as depiced in Figure 8, a high-pass secion can be convered o a noch filer cenered a f. In boh cases, RS C mus be much greaer han T. Recall from Figure 3 ha Smih predics he bandwidh of he response in Figure 8 o be 1/( Nr RS C). We can hus idenify hree criical aribues of his circui: 1) a band-pass response wih an arbirarily narrow bandwidh cenered around an arbirarily high frequency, and his aribue represens an RF filer wih an arbirarily high qualiy facor Q ; ) a response wih a well-defined, precise cener frequency, f ; and 3) a precisely programmable cener frequency. bsen in ime-invarian RF filers, his unique combinaion enables accurae channel selecion in RF receivers. Smih s expression, 1/( Nr RS C), is inriguing in ha i implies a reducion in he bandwidh as he number of commuaed capaciors increases even hough only one capacior is ied o he oupu a a given poin in ime. This puzzle can be solved if we recognize in Figure 8 ha he dwell ime of he capaciors, i.e., he pulsewidh of he phases, decreases as N increases. Consequenly, each capacior has less ime o inerac wih Iin and RS and experiences a smaller volage change as if is value were larger. Second-Order Effecs number of phenomena degrade he performance of commuaed neworks and require aenion. These include he on-resisance of he swiches, Rsw, he ranslaions of he source impedance, and he finie rise and fall imes of he. The on-resisance of he swiches in Figure 8 can be facored ou as illusraed in Figure 9, suggesing 1 WINTER 016 IEEE SOLID-STTE CIRCUITS MGZINE

6 ha he rejecion a relaively large values of fin - f is limied o Rsw/( RS Rsw) because he ideal commuaed nework iself exhibis a negligibly small impedance. To minimize Rsw, he widh of he swiches mus be increased, demanding a higher power consumpion in he pah. The magniude of Z a fin = f in Figure 6 is abou 7 d lower han he source resisance. Why does his happen? Le us plo he seadysae oupu volage as shown in Figure 10, noing ha he volage on, say, C1 begins and ends a he same value, V1, in every oher half of he cycle. If Iin = I0 sin ~ and he discharge of C1 hrough R1 is expressed as V1exp[ - /( R1C1)], one can prove ha V1. R1I0/ r provided ha R1C1 & T. Tha is, V () can be approximaed by a square wave having a peak-o-peak ampliude of 4R1I0/ r. I is remarkable ha he circui generaes a square-wave volage in response o a sinusoidal curren as if i conained harmonically scaled resonaors a f, 3f, ec. Since he firs harmonic of V has a peak ampliude of ( R1I0/ r)( 4/ r) = ( 8/ r ) R1I0, we conclude ha he impedance a fin = f is equal o ( 8/ r ) R R1, abou 1.8 d lower han R1. The oher 5-d reducion in he peak impedance in Figure 6 arises from he overlap beween he phases. Creaing a emporary resisive pah beween he op erminals of C1 and C, he overlap may simply occur because and have finie rise and fall imes, keeping he wo swiches on simulaneously wice per period [Figure 11]. The resuling differenial discharge of C1 and C can be aribued o an equivalen resisance [Figure 11] given by R eq T R 1 = sw, (1) DT where Rsw is proraed according o he fracion of he period during which he swiches are on, and he facor of Figure 1: T-coil nework a inpu. 1/ accouns for wo such evens per cycle. s illusraed in Figure 11(c), his effec ranslaes o a resisance of Req / in parallel wih each capacior and hence parallel wih Z. Quesions for he Reader 1) The commuaed capaciors of Figure 6 are placed a he anenna por of a GSM receiver so as o aenuae by 0 d a 0-dm blocker a 0-MHz offse. Wha issues does such a circui face? ) Does V0 in Figure 10 change if he circui conains four capaciive branches ha are driven by 5% duy-cycle phases? nswers o Las Issue s Quesions 1) Use a power dissipaion argumen o deermine he ransfer funcion of he circui shown in Figure 1. Z in Translaional circuis can shif an impedance o a well-defined cener frequency. We know ha Zin = RT a all frequencies. Tha is, he power delivered by Vin o he circui is equal o Vin, rms / RT. For a lossless T-coil nework, all of his power is delivered o RT, generaing an oupu equal o he inpu. The ransfer funcion is hus equal o one. C L 1 L C ESD M 1 C R T C L s 1 V DD R D C L Series Peaking Figure 13: T-coil nework a oupu. ) In Figure 13, how should LS be chosen if he damping facor of he series peaking nework mus remain around /? Since he resisance seen by LS on he righ is equal o RD, we reduce he circui o he series combinaion of C1, LS, and RD. For his nework o have a damping facor equal o /, we have g = ( RD/ ) C1/ LS = / and hence LS = RD C1/. L L 1 References [1] H. usignies and M. Dishal, Some relaions beween speed of indicaion, bandwidh, and signal-o-random-noise raio in radio navigaion and direcion finding, Proc. IRE, pp , May [] W. R. LePage e al., nalysis of a comb filer using synchronously commuaed capaciors, SEE Proc., pp. 6368, May [3]. D. Smih, nalysis of commuaed neworks, IRE Trans. Prof. Group eronau., pp. 16, Dec [4] L. E. Franks and I. W. Sandberg, n alernaive approach o he realizaion of nework ransfer funcions: The N-pah filer, ell Sys. Tech. J., pp , Sep IEEE SOLID-STTE CIRCUITS MGZINE WINTER