Pseudo-Linear Time-Invariant Magnetless Circulators Based on Differential Spatiotemporal Modulation of Resonant Junctions

Submitted to the IEEE Transactions on Microwave Theory and Tech niques 1  Abstract — In this paper, we present voltage- and current-mode differential magnetless non-reciprocal devices obt ained by pair ing two single-ended (SE) circulators, each consisting of three first-order bandpass or bandstop LC filters, connected in either a wye or a delta topology . The reson ant poles of each SE circulat or are modulated in time with 120 de g phase-shifted periodic signa ls, resulting in synthetic angular-momentum biasing achieved through spatiotemporal modulation (STM). We tailor the two SE circulators to exhibit a consta nt 180 deg phase difference betw een their STM biases. Unlike convention al differential time-variant circuits, for which only the even or odd spurs are rejected, we show that the proposed con figuration cancels out all intermodulation (IM) products, thus making them operate alike linear time-invariant (LTI) circuits for an external observer. In turn, this property enhances all metrics of the resulting circu lator, overcoming the limitations of SE architectures, and improving insertion loss, impedance matching, bandwidth and noise figure. We show that this differential arch itecture also sign ificantly relaxes the requir ed modulation parameters, both in fre quency and amplitude. We develop a rigorous small-signal model to guid e the design of the proposed circuits and to get insights into th eir pseudo-LTI characteristics. T hen, we valid ate the theory with simulations and measurements showing remarkable performance compared to the current state of the art of magnetless non-reciprocal devices . Index Terms — Non-reciprocity, magnetless circulator, S TM bias, differential, pseudo-LTI , voltage-mode, current-mode. I. I NTRODUCTION IRCULATORS are three-port non-reciprocal com ponents, crucial to enable full -duplex comm unication [1]-[6], si nce they allow unidirectional signal transmission from the transmitter (TX) to the antenna (ANT), while isolating the receiver (RX) from se lf-interference. Co mmercial circulators Manuscript received on August 31, 2017. The authors are with the Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78712, U .S.A. A.A. is also with the Advanced Science Re search Center, City University of New York, New York, NY 10031, U.S.A. (corresponding author: A. A., +1.512 .471.5922; fax: +1.512.471.6598; e-mail: aalu@ gc.cuny.edu ). This work was supported by the Qualcomm Innovation Fellowship, the Air Force Office of Sci entific Research, the Defense Advanced R esearch Projects Agency, Lockhe ed Martin, Silicon Audio, the Simons Foundati on, and the National Science Foundation. A.A. is currently the Chief T echnology Officer of Silicon Audio RF Circulator. The term s of this arrangement have been reviewed and approved b y The University of Texas at Austin i n accordance with its policy on objectivity in research. are based on magnetic biasing of rare-earth ferrite materials [7]-[10], which are bulky, expensive and incompatible with standard integrated-circuit (IC) technologies. In order to remove the magnet, active circulators that utilize the intrinsi c non-reciprocal properties of transisters were explored, yet the se devices have never become popular since they suffer from fundamentall y poor noise figure and power handling performance [11]-[14]. Recen tly, linear p eriodically time-varying (LPTV) circuits h ave been presented as an alternative approach towards magnetless non-reciprocity , which can achieve low loss, small noise figure, watt-level power handling, and other benefits [15] -[34]. In partic ular, [29]-[34] proposed a synthetic spatiotem poral modulation (STM) angular-mom entum biasing of resonant junctions, which results in symmetric cir culators w ith decent performance in many metrics. The main challenge in th is app roach is that it suffers from strong IM product s in close proxim ity to the desired band, due to mixing between the RF input and the relatively low-frequency modulation signals. These products not only pose an interference problem to neighboring channels, but they also require large modulation parameters to achieve good perform ance – e.g., the m odul ation am plitude in [33] was larger than 10 Vpp – which prohibits their integration using submicron CMOS technologies. Furt hermore, they effecti vely reduce t he circul ator’s overall power handli ng i n an actual full-duplex system, since they could saturate the RX front-end and drive the TX power am plif ier into instability because o f load-pull effects. More important ly, these products enforce a bound on the minimum insertion loss of about 3 dB [31]-[33], which weak ens the argument of circulators compared to other reciprocal interfaces based on couplers or balanced duplexers [35], [36]. Therefore, the rejecti on of these products is pivot al, and as important as cancelling the fundame ntal harmonic of the TX signal at the RX port, to enable the use of STM circulators in commercial systems. Although filtering may sound a reasonable option, it is, in fact, far from being practical, si nce it suffers from many problems. Specifically, add ing filters at the circulator’s three ports increases the overall size and degrade s the total insertion loss. It also imposes a restriction on the minim u m m odulation frequency, in order to relax the requirements on the sharpness of these filters, which, in turn, requires large modulation amp litude to maintain sufficient isolation [33]. This not only increases power consumption and further complicates integration, but it m ay be even impossible Pseudo-Linear T ime-Invariant Magnetless Circulators Based on Dif ferential Spatiotemporal Modulation of Resonant Junctions Ahmed Kord, Graduate Student Member, IEEE , Dim itrios L. Sounas, Senior Member, IEEE , and Andrea Alù, Fellow , IEEE C Submitted to the IEEE Transactions on Microwave Theory and Tech niques 2 to achieve with practical varactors. Moreo ver, RF filters are typically non-reconfigurable, hence STM circulators’ tunability, which is an important feature in modern communication system s, is sacrificed. Ref. [34] proposed a partial solution to these problems based on combining two SE circulators with anti-phase STM bias through RF baluns. Yet, the results in [34] were based on heuristic investigations without deep understanding of the ultimate capabilities of this architectu re, no r a detailed simulat ion and experimental validation. In this paper, we address this issue by developing a rigorous theory for differential STM circulators and showing that they have far more interesting characteristics than the ones report ed in [34] . In particular, we show that the suitable combination o f two S E circulators i n a di fferential confi guration surprisingl y result s in the total cancellati on of all IM products for excitat ion at any port and at any frequency, thus making these circulators essentially pseudo-LTI circuits. This property, in turn, alleviates the trade-off between the IM products and the modulation parameters (recall that in SE circulators, a minim u m modulat ion frequency is required to keep the IM products below a certain level [32], [33]), enabling a strong reduction of both m odulation frequency and amplitude, while still achieving remarkable performance. In terestingly, this implies that the differential STM b ias synth esizes a truly continuous angular m omentum , mimicki ng mechanical m otion that does not use temporal m odulation [28], even though i t is implem ented usi ng a discret e number of resonators (three per junction). This is fully consistent with magnetic-biased circulators, in which the aligned electron dipole moments imitate a continuous rotation of the ferrite disk at a macroscopic level, while they are in fact a collecti on of quantized spins [ 7]. Furthermore, in analogy with passive mixers [37], we introduce two dual im plementati ons of differential STM circulators: (i) voltage-m ode and (ii) current-mode. In voltage-m ode circulators, which are implem ented using bandstop/delta junctions, baluns are used to subtract the IM products, since t h e y a r i s e a s common voltages at the terminals o f the constituent SE circulators [33], [34]. Conversely, in current-mode circulators, which are based on bandpass/wye junctions, IM p roducts ari se as differential currents , and they can be rejected by tying the te rminals of two SE circulators together to sum up and cancel these currents. This paper is organized as follo ws. In Section II, we briefly summ arize the SE STM circulators presented in [32], [33] and discuss their limitations. In Section III, we qualitatively ana lyze the proposed voltage- and current-mode t opologies t o explain how these circuits have emerged and to understand their differences. Then, we develop a detailed small-signal model for the voltage-mode architecture, prove t hat it is indeed IM-free, and derive analytical expressions for the S -param eters. For complete ness, simil ar analysis for the current-mode topology is provided in the appendices. In Section IV, we present simulated and measured results for the voltage-mode topology with remarkable performance nearly i n all metrics compared to the current state of the art. Finally, we draw our con clusions in Section V. II. L IMITATIONS OF S INGLE -E NDED STM C IRCULATORS The underlying physical princip le of STM circulators is the same as for m agnetic circulators , i.e. the degeneracy between two counter-rot ating m odes in a resonant junction is lifted by the applied bias. At radio frequencies, this was achieved by modulating the natural oscilla tion frequencies of bandpass or bandstop LC tanks connected in a wye or a delta topology, respectively, with 120 deg phase-shifted periodic signals having the same frequency and amplitude [32], [33]. Fig. 1 shows a simpli fied schematic for both topologies, where the resonance frequency of the n -th tank is given b y  0 cos , 1 , 2, 3 nm m n ff k V t n     (1) where n is the tank index, 0 f is the static unmodulated resonance frequency of all tanks, m V and 2 mm f    are the modulation am plitude and angular frequency, respectively, k is a constant with Hz/Volt units quantifying the effect of the modulati on voltage on the resona nce frequencies of the tanks, and   1 n n    , where 120    . Such m odulation schem e results in what we call STM angular-mom entum biasing, since it involves phase variation in space (  direction) and in time. The bias direction is, by definition, the di rection along which the phase of the modulati on signals increases. For example, (1) assumes that the phase increases in the clock-wise direction based on t he port defi nitions of Fig. 1, thus resulting in a clock-wise STM bias. This, in turn, provides a preferred sense of precession for the counter-rotating modes of the resonant 𝑎 𝑖  𝑖  𝑣  𝑣  𝑏 Fig. 1. Single-ended STM circulators: (a) Bandstop/delta topolo gy. (b) Bandpass/wye topology. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 3 junctions in Fig. 1. These m odes can be defined if we express the tank voltag es n v in Fig. 1(a) or, sim ilarly, the tank current s n i in Fig. 1(b) as a superposition of two quantities as follows     11 , jn jn n vv e v e       (2)   11 , jn jn n ii e i e       (3) where   1 jn ve    and   1 jn ie    are the aforementi oned counter-rotating m odes. Notice that the phase of these modes increases either clockwise (+) or counter clockwise (–), and it adds up to 360 deg in one cycle. In general, a common mode 0 v and 0 i whose phase is the same for all tanks may also be defined, yet i t was shown in [33] that t he cancellation of this mode is necessary to have optimal performance for STM circulators. The circuits in Fig. 1 indeed satisfy this conditi on since a non-zero common mode in either topologies would violate Kirchhoff’s laws. To avoi d confusion with the comm on and differential components deco mposition of the total signal a t the RF ports, 0 v and 0 i will b e called the in-phase mode in t he rest of this paper. The topologies in Fig. 1 achieve strong non-reciprocity without magnets, yet t hey suffer from several disadvant ages. In particular, the wye topology r equires many filters in the modulati on network, which increases the overall form factor and complicates the design. It also requires using im pedance transformers at the 50 Ohm ports to increase the load ed quality factor l Q of the resonant junction, which is necessary to achieve strong non-reciprocity [32]. Clearly, these transformer s add m ore intri nsic loss and increase the size further. Moreover , the constituent series LC tanks amplify the input RF voltage across the varactors roughly by the same order as l Q , which degrades the circulator’s linearit y and power handling. These problems can be overcome with the delta topology, whi ch however requires a large modul ation amplitude (>10 Vpp in [33]) to achieve good performance, thus increasing the dynamic power consum ption and prohibiting integrat ion in submicron CMOS technologies. Another serious problem that both topologies suffer from is the strong IM products due to mixing between RF and modulati on frequencie s. As mentioned in the introduction, these products limit the performance, particularly power handling and inserti on loss. In the next sections, we present a differen tial architecture that overcomes all these problems and results in remarkable p erformance compared to the current state of the art. III. T HEORY AND P ROPOSED C IRCUIT T OPOLOGY A. Voltage- and Current-mode Differential Topologies In o rder to understand the operation principle of differential STM circulators, consider adding a constant phase  to all modulati on signals in (1), i.e.,  1 n n     . One can prove, foll owing the analysis in [33], t hat the fundam ental component of the rotating modes, and subsequently the S -parameters, remain exactly the same, while t he IM products at m    become    0 ,, , j mm VV e             ( 4 )    0 ,, , j mm II e             (5) where   , m V     and  , m I     are the Fourier transforms of the generated IM p roducts due to an input excitation at  . The expressions of  0 , m V        and   0 , m I        as a function of the circuit elements and modulation parameters can be found in [32] and [33], respectively. This finding sugge sts that combining two SE circulators with a phase difference 180    between the modulation signals of the constituent SE circulators cancels these products entirely. Based on this observation, Fig. 2(a) and Fig. 2(b) show the proposed voltage- and current-mode differential architectures, respectively. The constituen t SE circulators in the voltage-mode topology are based on bandstop/delta junctions (see Fig. 1(a)) and are combined together using differential ports (or baluns). In contra st, the current-mode topology employs bandpass/wye junctions (see Fig. 1(b)) with their terminals directly tied together. Both topologies ensure that t he IM products of each SE circulator have opposite parities, hence they destructively interfere at all ports, while the fundamenta l components are in phase and, the refore, sum up constructively. More specifically, the voltage-mode topology yields even symme try for the IM products, th erefore resulting in an infinit e effective port im pedance for these products, as shown in Fig. 3(a). On the other hand, the fundamental components experience a virtual ground at the middle of the differential ports, leading to an effective port impedance that is half of t he original differential port impedance 0 Z , as shown in Fig. 3(b). Unlike the IM products, the finite port impedance at the fundamental frequency allows cu rrent flow and, consequently, power transfer to the ports. From duality, the current-mode topology results in opposite symmetries, i.e., odd for the IM and even for the fundamental fre quencies, as shown in Fig. 3(c) and Fig. 3(d), respectively. Fig. 3(a) and Fig. 3 (c) show that the IM products in the voltage- or current-mode topol ogies are prohibited from leaking into the external ports since they see an effective open or short circuit impedance, r espectively. Since power is conserved, these products refl ect back to the resonant junction s and they can be regarded as new excitations at m    , which mix with the modulation signal, s imilarly to the original input a t  , and generate new harmonics at m    ,  , and 2 m    . The second-order spurs at 2 m    exhibit the sam e symm etry as the fundamental component at  , hence their half-circuit models are the sam e as Fig. 3(b )-(d) and, therefore, can leak t o the external ports. In g eneral, the circulator is designed to b e matched at the fundamental h armonic but this does not guarantee matching at 2 m    , hence the IM products at these frequencies partially reflect and enter the circulator again. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 4 Similarly, this gives ri se to new harm onics a t 3 m    , which have the same symmetry as the first-order products at m    and therefore are completely r eflected back to the resonant junctions. By continuing this recursive process, we find that t he spectrum at the RF ports is expect ed to contain the even-order harmonics  , 2 m    , 4 m    , …. This qualitative analysis suggests that the perfo rm ance improvement due to the differential configuration w ould be increm e ntal, and only associated w ith a weakening of the first non-zero higher products at 2 m    . In the next section, however, we prove that both proposed circulators i n Fig. 2 do not generate any IM product, making them pseudo-LTI circuits and drastically improving the performance. This result cannot be predicted from the half-circuit m odels of Fig. 3, which are typically developed for conven tional differential ci rcuits, and it requir es a rigorous analy sis of the complete circuit. Since the analysis o f both topologies is quite similar, we focus in the rest of this paper on the voltage-mode arc hitecture, we develop its small-signal model (detailed analysis in Appendix A), and provide simulated and measured r esults with remarkable performance. For completeness, a nalysis of the current-m ode topology is provided in Appendix B. B. Linear Small-Signal Analysis of Voltage-mode Topology In this section, we provide a t heoretical analysis for the voltage-mode topology. Fig. 4(a) shows the complete circuit implem entation of the voltage -mode STM circulator where LC baluns ( rf L and rf C ) are used to realize the differential ports. Baluns ( m L and m C ) are also used to p rovide the required anti-phase STM biases of the upper and lower SE circulators from three 120 deg phase-shift ed modulation signals with an amplitude m V and frequency m f . DC biasing is com bined with these signals through the bal uns’ shunt inductance and a sufficiently large resistance B R connecting the two terminals of the baluns’ balanced port. Furthermore, the parallel LC tanks are realized using inductors 0 L and a pair of varactors (recall that each SE circulator in the voltage-mode topology is implem ented using the bandst op/delta topology shown in Fig. 1(a)). The varactors are in c ommon-cathode configuration and the common node is connected to an inductor d L so that they form together a bandpass resonance at m f , thus allowing the modulation signal to pass through while, at the same time, prohibiting the RF signal from leaking out of the delta junctio n. Under the small-signal assumpti on, the circuit in Fig. 4(a) can be simplified as shown in Fig. 4(b) where t he TX, R X and ANT ports along with the RF baluns a re all replaced with differenti al voltage sources with a total i mpeance 0 Z (output impedance of the baluns). Also, the comm on-cathode varactors and the DC/modulation network are replaced with time-variant capacitors whose capacitance, assuming weak and linear modulation, is given by   0 0 cos , 1 , 2, 3 cos , 4, 5, 6 mn n mn CC t n C CC t n               ( 6 ) where 0 C is the static capacitance of the common-cathode varactors as set by the DC bias and C  is the effective capacitance variation which is proportional to the modulation voltage m V . We also assume that the varactors’ and the inductors’ losses of each tank are combined into a dispersion-less parallel resistance 00 0 0 RQ L   , where 0 Q is the unloaded quality factor of th e tanks. Applying Kirchhoff’s laws to the n th tank in Fig. 4(b) and writing the result in a matrix form, we get 𝑎 𝑏 Fig. 2. Differential STM circulator: (a) Voltage-mode topology. ( b ) Current-m ode topology. Fig. 3. Half-circuit models for: (a) Voltage-mode fundamental harmonic. (b) Voltage-mode IM products. (c) Current-m ode fundamental harmonic . (d) Current-m ode IM products. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 5 0 00 0 0 12 1 1 , dd d c c ms c s CH v H Q v H v C H C v RZ L CH C v G v Z                ( 7 ) 0 00 11 0, cc c c d m s d Cv v v C C v C Cv RL            (8) where d dt  , 2 2 d dt   , U is the unitary matrix, H , Q , c C , and s C are m atrix operators which a re derived in Appendix A,  12 3 ,, ss s s vv v v  is the differential source excitation vector, and c v and d v are the common and differential voltage vectors, respec tively, given by , 2 ul c vv v   (9) , 2 ul d vv v   (10) where  12 3 ,, u vv v v  and  456 ,, l vv v v  are the vectors of the tank voltages in the upper and lower SE STM circulators, respectively. Equations ( 7) and (8 ) can be further simplified i f we express both d v and c v as a superposition of the j unction’s modes as expressed in (2). Appl ying this transformation and recognizing that the in-phase m ode of each SE circulator is not excited (see Appendix A) yields ,, ,, 00 /6 ,, , 00 1 /6 ,, , 000 00 00 00 22 00 23 12 3 , 3 m m m m jt jt cc m jt jt cc j dd d s j dd d vv ee CC j vv CC ee vv v e RZ v vv v RZ C L C Z C e                                                               (11) ,, , , ,, , , 00 00 0 , , 0 0 11 2 0 0 0, 2 0 m m m m jt cc c d jt cc c d jt d m jt d vv v v e C vv v v RC LC C e v e C j v C e                                                          (12) where , d v  and , c v  are the differential and common components of the counter-rotati ng modes, respectively. A circuit as in Fig. 4 with six fir st-order resonant tanks (three for each junction) would normally lead t o a sixth-order system, but the absence of the i n-phase modes reduces t he order by two, as described by (11) and (12) whi c h r e p r e s e n t a f o u r t h - o r d e r system of second-order linear differential equations. These equations can be solved by Four ier transform, which yields      /6 2 00 00 , 00 1 11 33 , j m d s m je j RC LC V ZC VD                   (13)      2 2 , 00 1 13 , 12 cm m s C jj V ZC VD             (14) where   , d V   ,   , c V   , and  s V  are the Fourier transforms of   , d vt  ,   , c vt  , and   s vt , respectively, and     2 2 22 00 0 000 00 2 00 00 23 1 23 11 . m mm RZ C Dj CR Z C L C j RC LC                               (15) 𝑎 𝑏 Fig. 4. Voltage-mode differen tia l STM circulator: (a) Complete circuit implem entation. (b) Small-signal m odel. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 6 Equations (13) and (14) show that the only frequency components existing in the circuit are the fundamental harmonics at  and the first-order IM products at m    without any additional higher-ord er products, as incorrectly predicted by the approximate analy sis of differential circuits based on half-circuit models in Sec. III.A. Also, the fundamental harmonics are exclu sively excited as differential components, allowing them t o flow to the ports while the first-order IM products are e xclusively excited as com mon components, hence they are trapped inside the resonant junction and cannot carry power t o the external ports. This result shows that, interestingly, the proposed circulator hides the intrinsic time-va riant characteristics from the external ports, making it essentially a pseudo-LTI circuit. As mentioned earlier, this IM-free characteristic not only allows to achieve very low insertion loss, increases the effective power handling , and avoids interference with n eighboring channels, but it also relaxes the required modulation parameters significantly as we show later in this paper. Finally, the S -parameters can be calculated from (13) as follows (see Appendix A)        ,, ,, 11 11 1 2, 6 3 dd dd ss VV VV j S VV               (16)        ,, ,, 21 11 1 2, 3 3 dd dd ss VV VV j S VV              (17)     ,, 31 1 12 2. 3 3 dd s VV Sj V           (18) Due to the circulator’s threefol d rotational symm etry, the rest of the S -parameters can be found by rotating the indices as       1, 2 , 3 2 , 3 , 1 3 , 1, 2  . With proper choi ce of the circuit elements and m odulation parameters, , d V  can be designed to destructively interfere at one port and sum up at the other, as required to achieve infinite isolation. More specifically, port 3 can be isolated if ,1 43 ds VV j    , which results in the following conditions on the m odulation parameters when substituted in (13):     2 22 2 0 2 00 2 2 22 2 0 00 3 , 3 rf m rf rf rf m l rf m rf m rf rf l Q Q                                   (19)  2 22 2 0 2 00 22 2 0 3 , 2 rf m rf rf l rf rf m Q C C                          (20) where 0 lr QQ Q   and  00 0 32 r QZ C   (recall that 00 0 0 QR C   is the unloaded quality f actor o f the resonant tanks). For operation at a given frequency rf  , (19) and (20) can be used to calculate the re quired modulation parameters m  and C  to achieve infinite isolation. Quite interestingly, the same conditions lead to unitary transmission at the third port, assuming a lossless circuit. A lso, from power conservation, this leads to perfect matching at the input port. Therefore, the proposed differe ntial circuits allow, in the lossless case, to realize an ideal circulator with S -matrix given by 001 100 , 010 S            (21) while SE circulators do not have this advantage, since the required modulation pa ram eters to optim ize for return loss (RL), insertion loss (IL), and isolation (IX) are not the same [33]. In order to get further insight into the operation of the circulator, Fig. 5 (a)-(c) show the S -parameters at the three ports for excitation from port 1 vers us the modulation parameters m  and C  normalized to 0  and 0 C , respectively. T hese charts were generated using (16)-(18) assuming 0 50 Z   , 0 70 Q  , and an input frequency 1 rf f  GHz. It can be seen that the required modulation pa rameters to optimize RL, IL, and IX as indicated by points 1 p , 2 p , and 3 p , respectively, are nearly the sam e as mentioned earlier (the n egligible misalignm ent is due to the finite 0 Q ). Moreover, Fig. 5(d) shows the circulator’s BW, whi ch is defined as the minim um frequency range to maintain IX more than 20 dB, RL less than 20 dB, and IL less than 3 dB. In general BW i s a more relevant metric in practical systems than optimized S -parameters at a single-frequency. Optimizing the BW, however, requires 𝑐 𝑑 𝑎 𝑏 𝑝  𝑝  𝑝  𝑝  Fig. 5. S -parameters at 1 GHz versus modulation parameters for 0 50 Z   and 0 70 Q  : (a) Return loss. (b) Insertion loss. (c) Isolation. (d) Fract ional bandwidth. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 7 different modulation parameters, as indicated by point 4 p , than those at 1 p , 2 p , and 3 p . Since the capacitance ratio at 4 p is impractical, we choose in this paper to operate at 0 0.5 CC   and 0.1 mr f ff  , which assumes the maximum realistic capacitance variation of com mercial off-the-shelf varactors, hence results i n the best possible BW, and at the sam e time is close to 2 p where IL becomes minimum. In such a case, Fig. 6(a) shows the theoretical S -parameters (based on (16)-(18)) where IL, RL, and IX at the center frequency are 0.6 dB, 27.5 dB, and 31.5 dB, respectively, and BW is 3.6% (36 MHz). We would like to stress that infinite IX at the center frequency i s still possible if the circu it is designed at point 3 p . However, in practice, it is desirable to minimize the dispersion of the S -parameters over the frequency ra nge of operation, in order to avoid distortion of realistic signals with finite bandwidth. Uniform IX, in particular, is d esirable as it simplifies the de sign of the following layers of sel f-interference cancellation in full-duplex system s. For the sake of com p arison, Fig. 6(b) shows the S -parameters of a SE circulator designed using the same modulation frequency (capacitance ratio is optimized for lowest IL) where the results are clearly much worse. Specifically, IL, RL, and IX at 1 GHz all degrade to 2.31 dB, 21.8 dB, and 10 dB, respectively, and BW becomes undefined since the minim um level of 20 dB IX is not satisfied. As mentioned earlier, the drastic improvem ent in the differential architecture is due to the cancellation of IM products. Fig. 7(a) shows that the signal spectrum at all ports is indeed IM-free. Although the incident signal was assumed to be at 1 rf f  GHz in Fig. 7(a), the same conclusion holds for any input frequency. In contrast , Fig. 7(b) shows that the I M products in the SE im p lementa tion are only –12 dBc and very close to the fundamental com ponent (only 100  MHz apart) which is the reason for the poor S -parameters in Fig. 6(b). The analysis for the current -mode topology is analogous, and provided in Appendix B. Unlike their SE counterparts, the differential current-mode topol ogy has several advantages compared to the differential voltage-mode topology . Specifically, since the baluns at the RF ports are eliminated, the overall insertion loss and noise figure can be improved, the form factor and complexity are reduced, and more immunity against random mismatches is maintained. Furthermore, the original disadvantages of the SE wye topology are eliminated, i.e., additional filters are not required in the modulation network to block the modulation signal, since the differential circuit exhibits virtual ground symmetry at the RF ports. Also, impedance transformers can be removed since the requirements on the modulation paramete rs and consequently the Q -factor of the circuit are all relaxed, thus allowing to design the current-mode circuit with 50 Ohm term ination at a few GHz. However, the voltage-mode circulator still maintains an 𝑎 𝑏 Fig. 6. Theoretical S -parameters for 0.1 mr f ff  : (a) Differential. (b) Single-ended. 𝑎 𝑏 Fig. 7. Theoretical harmonic spectrum at all ports for an incid ent tone a t 1 rf f  GHz and 0 in P  dBm: (a ) Differential. (b) Single- ended. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 8 advantage in terms of linearity and power handling, since it relies on parallel LC tanks, which do not amplify the input voltage across the varactors, as the wye/current-mode topology, but rather they amplif y the current, which does not contribute to the non-linear ity of the varactors. IV. R ESULTS AND D ISCUSSION Guided by the theoretical analy sis in Section III, a PCB prototype of the differential voltage-mode STM circulator (see Fig. 4(a) for the schem atic) was designed at 1 GHz u sing off-the-shelf discrete components as listed in Table I. In o rde r to account for all parasitics, the layout was simulated using ADS Momentum and the generated S -parameters were combined with commercially available spice models of all components to perform post-layout circuit/EM co-simulations. The design was then fabricated a nd the total area occupied by all com ponents is 2×(13m m×11m m). Notice that both the top and bottom sides of the PCB are populated, each with one SE circulator to have a symmetr ic and more compact layout as shown in Figs. 8(a)-(b). Fig. 8(c) shows the measurement setup, while Table II provides a list of the u sed equipment to take th e measurem ents. A. S-parameters Fig. 9 shows the measured S -parameters in magnitude and phase from all ports for DC 7.3 V  V, 0.8 m V  V(rms), and 100 m f  MHz. Notice that the modulation parameters, particularly the amplitude, are much lower than their SE counterparts [33], yet the differential circuit still results i n much better performance. Specifi cally, the measured I L, RL, and IX at the circulator’s cen ter frequency of 1 GHz are 1.78 dB, 23 dB, and 24 dB, respectively, and the fractional B W is 2.3% (23 MHz). For the sake of comparison, the simulated 𝑐 𝑎 𝑏 VN A Signa l ge ne rators O sc illo- scop e Spe ctr um anal y z e r Pow e r supplie s A m plif ie r DUT Fig. 8. Photograph of: ( a) Top side of the fabricated prototype . (b) Bottom sid e of the fabricated prototype. (a) Experimental setup. TABLE I T HEORETICAL D ESIGN P ARAMETERS Element Value LC tanks D ~10 pF @ VDC = 7.3 V 0 L 4.3 nH RF baluns rf L 11 nH rf C 2.2 pF Modulation baluns m L 56 nH m C 43 pF Biasing/Modulation network d L 150 nH B R 100 KOhm B C 1000 pF TABLE II L IST OF THE U SED E QUIPMENT Part Model Quantity Power supply Agilent E3631A 1 Vector network analyzer Agilent E5071C 1 Spectrum analyzer R&S FSVA40 1 Oscilloscope R&S RTO1044 1 Signal generators R&S SGS100A 4 R&S SMB100A 1 Instrument amplifier Minicircuits TVA-4W-422A+ 1 𝑎 𝑏 S 11 S 21 S 31 Fig. 9. Measured S -parameters: (a) Magnitude. (b ) Phase. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 9 S -parameters are also shown in F ig. 10 where the sim ulated IL, RL, and IX at 1 GHz are 1.7 d B, 22 dB, and 24 dB, respectively, and the fractional B W is 2.3% (23 MHz). Clearly, simulated and measured S -parameters are in excellent agreement, yet they are different than the theoretical results in Fig. 6(a). The reason is that RF baluns were not included in th e small-signal analysis of Section III.B for simplicity. These baluns contribute to insertion loss and, more importantly, resu lt in a finite imbalance in the di fferential architecture due to random component mism atches. As one may expect, this imbalance is the reason for the slight asymmetry between different ports as shown in Fig. 9. The baluns’ loss is estimat ed using circuit/EM co-simulations to be 0.5 dB each, which if de-embedded from the measured results, the actual IL of the differential circulator at 1 GHz becomes 0.78 dB which is in excellent with the theoretical results in Fig. 6 (a). It is wort h mentioning that this is the insertion loss that the circuit wou ld exhibit if it were connected directly to a differential transce iver, in which case the baluns are omitted. To the best of our knowledge, this is the lowest IL of all LPTV magnetless circulators presented to-date, w ith available room for further improvem ent in a m ore optimized design or by sacrificing the low modulation parameters a nd increasing their values. Interestingly, the current-m ode topology (see Fig. 15 and Appendix B), does not e ven require such baluns, therefore the total IL in this case can be expected to be b elow 1 dB. The circulator can also be tuned fo r operation at di fferent channel s as shown in Fig. 11 by simply controlling the DC bias of the varactors, and adjusting the m odulation voltage accordingly in order to account for the d ifferent slope o f the CV characteristics at the new quiescent point. Th e maxim um tunability range, while maintaining the same specs on S -parameters, was measured to be 60 MHz (6% of the band center frequency at 1 GHz). B. Harmonic Response Fig. 12 shows the measured and simulated harmonic spectrum at both the transm itted ( tra P ) and isolated ( iso P ) ports for an incident signal at 1 GHz and 0 dBm. Despite that the measured IM products are as small as –29 dBc for a modulation frequency of only 10%, they are s till finite, m ainly due to the non-linear CV characteristics of the var actors (recall that linear time-variation produces no IM products in the proposed differential topologies, as shown in Sec. III). Random mism atches in the components, which are inevitable in practice, also play a role as they lead to finite im balance not only in the RF baluns but more generally between the constituent SE circuits in any differential architecture. However, the impact of this im balance is insignificant as can b e deduced from the fact that the measured IM products are in excellent agreement with simulations where this issue is neglected. Compared to the SE a rchitecture in [33], the IM products are at least 17 dB smaller (=1/50) e ven though m f is reduced from 190 MHz to only 100 MHz. We would like to stress that a SE implementation at 100 MHz would have 𝑎 𝑏 Fig. 11. Measur ed S -param eters at different channels by changing the DC bias and modulation voltage: (a ) Magnitude. ( b) Phase. 𝑏 𝑎 Fig. 10. Simulated S -parameters: (a) Magnitude. (b) Phase. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 10 resulted in even larger IM products and the overall performance, particularly isolation and insertion loss, would have become m uch worse. C. Power Handling and Non-Linearities Non-linearity in STM circulators is exclusively due to the varactors which ultimately lead s to compression of both IL and IX. In [33], the maxim um power (Pmax) that a circulator can handle was defined as the input power that guarantees an IL compression less than 1 dB (P1dB) and at the same tim e maintains IX more than 20 dB (IX20dB). Fig. 13(a) shows the measured and simulated results for both the transmitted power and isolation versus the input power at 1 GHz. The measured P1dB and IX20dB both occur at +28 dBm, thus resulting in Pmax of the sam e value, which is in good agreement with the simulated value of +30 dBm. We also notice a peaking in IX before Pmax since the capacitance variation C  at such high power is effectively reduced by the higher-order terms of the varactors’ non-linear CV characteristics, hence the operation point in the design charts shown in Fig. 5 is shifted down clos er to 3 p where IX is maximum. Fig. 13(b) also shows the fundamental and third-order harm onic of t he transmitted power for two in-band tones at 1 GHz with 1 MHz separation. The measured input-referred third-or der intercept point (IIP3) is found to be +31 dBm and the simulated value is +33.8 dBm . The power handling of t he prese nted circuit is slightly less th an [33] solely because the varactor s used in this paper have a lower breakdown voltage. If the same varactor were used in both SE and differential implementations, Pmax in the differential architecture would actually be 3 dB larger, since input power is halved betw een the upper and lower SE circulators through the baluns. Despite this artifact, the resu lts of Fig. 13 are still larger than the reported values o f any oth er magnetless circulator p resented to-date. D. Noise Figure In general, different mechanis m s contribute to the NF of STM circulators [33]. This includes incoming noise from the RF ports which remains the same as in the SE architecture, assuming the typical 50 Ohm termination in both cases. 𝑎 𝑏 IX (dB) 𝑃   (dB m) Si mu l a ted Me a s ure d IM3 𝑓𝑢 𝑛𝑑 Si mula ted Me a s ure d Fig. 13. (a) M easured and simulated P1dB and IX com pression. (b ) Measured and simulated II P3. 𝑃  𝑃  𝑓  2 𝑓  𝑓  𝑓  𝑓  𝑓  𝑓  𝑓  2 𝑓  𝑃  0 dBm 𝑓  1 GHz 𝑃  𝑃  𝑃  0 dBm 𝑓  1 GHz 𝑃  𝑃  𝑓  𝑓  𝑃  𝑃  𝑎 𝑏 Fig. 12. Harmonic spectrum at tra nsmitted and isolated ports for a single tone input at 1 rf f  GHz and 0 in P  dBm : (a) Measured. (b) Simulated. Si m ul ate d Me a s ure d Fig. 14. Measured and simulated NF. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 11 However, noise generated by th e circuit itself is doubled, very similarly to conventional differential circuits. In fact, the voltage-mode topology increases it even further b ecause of additional loss in the RF baluns . Notice that the current-mode topology (Fig. 2(b) and Appe ndix B) does not have this problem. Amplitude and phase noi se in the m odulation sources a l s o a d d t o t h e t o t a l N F , y e t i f t h e a n t i - p h a s e S T M b i a s o f t he constituent SE circulators i s generated from the sam e sources, as in this p aper, then this noise becom es strongly correlated a t the terminals of the differential ports and, therefore, cancels out. Noise folding from the IM frequencies also adds to the NF, but since the proposed differential circuits reduce these products, then their contributi on becomes negligible. The reduction of the last two noise mechanism s, i.e., from the modulation sources and due to folding, i s much stronger than the increased thermal noise gener ated by the circuit. Therefore , the overall NF im p roves. In fact, since the differential circui ts are pseudo-LTI passive system s, the NF should be exactly equal to the I L [37]. Fig. 14 shows the m easured and sim ulated NF, where the simulated value at 1 GHz is 1.8 dB, which is indeed nearly equal to the sim ul at e d I L , wh i l e th e m ea s u r e d NF at 1 GHz is 2.5 dB which is about 0.7 dB larger than the measured IL due to imperfect cancellation of the IM products as discussed earlier. To the best of our knowledge, this is the smallest measured NF for magnetless circulators proposed to-date. V. C ONCLUSION We presented here a pseudo-LTI architecture for magnetless circulators, based on comb ining two bandstop/delta or bandpass/wye junctions with anti-phase STM b ias in either a voltage- or a current-mode topology, respectively. We developed a rigorous analytical m odel for the proposed circuits and explained a detailed design procedure to achieve given specifications on insertion lo ss, return loss, isolation and bandwidth. Our theory shows that all IM products are indeed cancelled, despite the fact that the order of the system is increased because of the increased number of LC tanks. Based on this theory, we designed a PCB prototype for the voltage-mode topology and measured its performance including scattering parameters , harmonic response, power handling, and noise figure, a s summ arized in Table III in comparison with previous works . Several of these metrics surpass the results of all magnetless circulators presented to-date, with room for further improvem ent, thus making the differential STM circulator a s ignificant step on the quest towards integrated full-dupl ex comm unication systems. A PPENDICES A. Detailed Analysis of th e Voltage-Mode Topology Here, we present a detailed analysis for the differential voltage-mode topology and deri ve the equations given in section III.B. Applying Kirchhoff’s laws to the n- th tank in Fig. 4(b) and following a similar analysis to [33], we get   0 00 11 , cu m s u u u CU C C v U C C v v i RL             (22)   0 00 11 , cl m s l l l CU C C v U C C v v i RL               (23) where d dt   ,   12 3 ,, u ii i i  and  456 ,, l ii i i  are the vectors of the tank currents of the upper and lower SE STM circulators, respectively,   123 ,, u vv v v  and  456 ,, l vv v v  are the vectors of the tank voltages of the upper and lower SE STM circulators, respectively, U is the unitary matrix, c C and s C are two matrices given by     cos 0 0 0c o s 0 , 00 c o s 2 m cm m t Ct t                (24)     sin 0 0 1 0s i n 0 , 00 s i n 2 m sc m m m t CC t t                    (25) TABLE III Summ ary of the M easured Results in Com parison to Previous Wor ks . Metric This work [33] [21] [28] Technology SMT/PCB SMT/PCB CMOS/65 nm SMA RF center frequency ( MHz) 1000 1000 750 100 Mod-to-RF frequency ratio (%) 10 19 100 6 DC bias (Volt) 7.3 19.6 N/A N/A Mod. am plitude (Vpp) 2.3 10.8 1.2 8 20 dB IX BW ( %) 2.3 2.4 4.3 (†) 200 IL 1 (dB) <2 (*) <3.4 <2 5 ~ 10 RL 1 (dB) >23 >9 N/A >20 P1dB 2 (dBm) +28 +29 N/A (††) N/A IIP3 2 (dBm ) +31 +33.7 +27.5 (TX/ANT) +8.7 (ANT/RX) N/A NF 2 (dB) 2.5 4.5 4 N.A First-order IM products (dBc) –29 –11.3 N/A N/A Size (mm 2 ) 2×(13×11) (**) 13×11 5×5 N/A 1 O v e r t h e B W . * B a l u n s c o n t r i b u t e 1 d B . † R e q u i r e s e x t e r n a l i m p e d a n c e t u ning. 2 At center frequency. ** 13×11 mm 2 on each side of the PCB. †† Limited to +10 dBm by the break down voltage of the used 65nm technology. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 12 and 23    . The tank currents u i and l i can be related to the source currents  12 3 ,, ss s s ii i i  , which flow through the differential ports, using KCL as follows , s ul iG i G i   (26) where 11 0 01 1 . 10 1 G          (27) Next, the tank voltages u v and l v can be expressed as a superposition of differentia l and common components by rewriting (9) and (10) as follows , uc d vv v  (28) . lc d vv v  (29) Substituting (26)-(28) into (22) and (23) yields 00 00 00 0 11 1 , dd d cc m sc s Gv Gv Gv RC LC CC GC v GC v i CC C            (30) 0 00 11 0. cc c c d m s d Cv v v C Cv C Cv RL            (31) The source currents s i can be related to the source voltages s v by applying KVL along the loops shown in green dashed lines in Fig. 4(b), which results in 00 12 , s sd Gi Gv Qv ZZ  (32) where 010 001 . 100 QUG        (33) Substituting (32) into (30) yields (7) and (8), where H is given by 12 1 11 2 . 21 1 HG G          (34) Equations (7) and (8) can be furt her sim plified if we express d v and c v as a superposition of the in-phase, clockwise and counter-clockwise modes. This mode decomposition can be expressed through the followi ng matrix transform ation: 1 1 , , dd cc vT v vT v       (35) where   ,0 , , ,, dd d d vv v v     and  ,0 , , ,, cc c c vv v v     are the vectors of the mode voltages for the differential and comm on components, respectively, and the operator T is given by 22 11 1 1. 1 jj jj Te e ee                (36) Applying this transformati on to (7) and (8) y ields 00 00 00 00 0 0 12 1 1 , dd d cc m sc c H vH Q v H v RC Z C LC CC Cv Cv G v CC Z C                             (37) 00 00 0 0 11 0, cc c c d m s d CC vv v C v C v RC LC C C                (38) where 1 /3 /3 00 0 30 0 , 00 j j HT H T e e              (39) 1 /3 /3 10 0 00 , 00 j j QT Q T e e                     (40) 1 /6 /6 /6 /6 000 1 2, 3 2 jj jj GT G e e j ee j                     (41)      1 /3 /3 /3 /3 000 3 0, 2 0 mm mm jt jt cc jt jt CT H C T e e ee                        (42)     1 /3 /3 /3 /3 00 0 3 0. 2 0 mm mm jt jt ss jt jt j CT H C T e e ee                          (43) Equations (37) and (38) can be reduced to (11) and (12) if we recognize that the in-phase m ode of each SE circulator is not Submitted to the IEEE Transactions on Microwave Theory and Tech niques 13 excited, i.e. ,0 ,0 ,0 2 ul d vv v   and ,0 ,0 ,0 2 ul c vv v   are both equal to zero, and assuming an excitation at only port 1, since a general s v can b e constructed using a linear superposition of individual port excitations. In order to find the S -parameters, (26) is also transformed into th e basis of the rotating modes, resulting in ,, 1 ,, 00 13 0 11 . 3 01 3 sd s sd iv j v iv ZZ j                   (44) Fourier transforming (44) yields     ,1 , 00 11 13 , 3 ss d IV j V ZZ      (45) where  , d V   are given by (13). Using the inverse trans- formation s s IT I   , the actual source currents   123 ,, ss s s I II I  are found as follows    1, , , ss s II I     (46)     2, , , jj sk s s Ie I e I       (47)    3, , . jj ss s Ie I e I       (48) By definition, the S -parameters are given by          01 11 1 02 21 1 03 31 1 12 , 2, 2, s s s s s s ZI S V ZI S V ZI S V             (49) which can be simplified to (16)-(18). B. Detailed Analysis of th e Current-Mode Topology For completeness, we present the analysis of the dual differential current-mode topol ogy (see Fig. 4(b)) in this appendix. The complete circ uit implementation of this topology is shown in Fig. 15(a). Following similar steps as in Appendix A and applying Kirchhoff’s laws to the n - th tank in Fig. 15(b), we get   00 0 11 , 33 uu c u i n Li R i E U C i E v C        (50)   00 0 11 , 33 ll c l i n Li Ri E U C i E v C        (51) where  2 0 1 , 1 CC    (52)  0 2 0 2 1 1, 1 C C CC         (53) 21 1 12 1 , 11 2 E               (54)   12 3 ,, u ii i i  and   456 ,, l ii i i  are the vectors of the tank currents of the upper and lower SE STM circulators, respectively,   123 ,, in vv v v  is the vector of the input voltages at the ports and i s given by (24). The tank currents u i and l i can, sim ilarly, be expressed as a superposition of differential d i and common c i components as follows 𝑎 𝑏 Fig. 15. Current-mode diff erential STM circulator: (a) Complete circuit implem entation. ( b) Sm all-signal model. Submitted to the IEEE Transactions on Microwave Theory and Tech niques 14 , uc d ii i  (55) . lc d ii i  (56) Also, in v can be related to the source voltages s v using a simple KVL which gives 0 2 in s c vv Z i  , (57) Substituting from (55)-(57) i nto (50) and (51) yields 0 00 0 0 0 0, 33 dd d c c R ii E i E C i LL C L C       (58) 00 00 0 0 0 0 32 1 . 33 3 3 cc c c d s RZ iE i E i E C i E v LL C L C L         (59) Equations (58) and (59) can be transformed into the basis of the rotating modes , d i  and , c i  using the same m atrix given by (36), which results in ,, , 0 ,, , 00 0 , , 00 0 0, 2 0 m m dd d dd d jt c jt c ii i R ii i LL C i e i LC e                                  (60) ,, , 00 ,, , 00 0 , 1 , 00 0 2 0 1 . 23 0 m m cc c cc c jt d s jt d ii i RZ ii i LL C i e v i LC L e                                    (61) Notice that the in-phase modes ,0 d i and ,0 c i are equal to zero and we also assumed  1, 0 , 0 s v  for simplicity. Applying Fourier transform to (60) and (61) yields      2 0 , 00 0 0 1 3 , mm c s R j j I LL L C VD                (62)    , 2 1 00 , 6 m d s I j VD LC           (63) where  , d I   and  , c I   are the Fourier transforms of  , d it  and  , c it  , respectively, and  D   is given by    2 2 00 00 0 00 2 0 00 0 2 2 . mm RZ Dj LC L LC R j LL C                                (64) Applying KCL at the input term inals, the source currents   123 ,, ss s s I II I  can be found as follows: 2, s ul c I II I   (65) where   123 ,, cc c c I II I  is given by      1, , , cc c II I         (66)    2, , , jj cc c Ie I e I           (67)    3, , . jj cc c Ie I e I           (68) Finally, the S -parameters are calculated using (49) which yields     0 11 , , 1 4 1, cc s Z SI I V         (69)        0 21 , , , , 1 2 3, cc cc s Z SI I j I I V               (70)        0 31 , , , , 1 2 3, cc cc s Z SI I j I I V               (71) R EFERENCES [1] J. 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