Magnetless Circulators Based on Spatiotemporal Modulation of Bandstop Filters in a Delta Topology

AP1507-1042.R1 1  Abstract — In this paper, we discuss the design rationale and guidelines to build magnet-less circulators based on spatio-temporal modulation of resonant junctions consisting of first-order bandstop filters connected in a delta topology. Without modulation, the junction does not allow transmission between its ports, however, when the natural oscillation f requencies of the constituent LC filters are modulated in time with a suitable phase pattern, a synthetic angular-momentum bias can be effectively imparted to the junction and a transmission window opens at one of the output ports, thus realizing a circulator. We develop a rigorous small-signal linear model and fi nd analytical expressions for the harmonic S -parameters of the proposed circuit, which significantly facilitate the design process. We validate the theory with simulations and further discuss the large signal response, including power handling and non-lin earity, and the noise performance. Finally, we present measured results with unprecedented performance in all metrics for a PCB prototype using a Rogers board and off-the-shelf discrete components. Index Terms — Full-duplex, non-reciprocity, magnet-less circulator, bandstop filters, s patio-temporal modulation. I. I NTRODUCTION IRELESS comm unications hav e significantly advanced since th e first generation of cellular services was launched in Japan in 1979. Yet, all deployed syst ems up to date are half-duplex, employing either frequency or time division diplexing for bi-directional comm unication, therefore limi ting the maximum transmission rate allowed by available resources. In a full-duplex system, both the transm itter (TX) and the receiver (RX) operate simultaneously at the same frequency, Manuscript received on March 31, 2017. This work was supported by the Q ualcomm Innovation Fellowship, the Air Force Office of Scientific Resear ch, the Defense Advanced Resea rch Projects Agency, Silicon Audio, the Sim ons Foundation, and the National Science Foundation. The authors are with the Department of Electrical a nd Computer Engineering, University of Texas at Austin, Austin, TX 78712, U SA. A. A. is also the Chief Technology Officer of Silicon Audio RF Circulato r. (corresponding author: A. A., +1.512.471.5922; fax: +1.512.471. 6598; e-mail: alu@mail.utexas.edu ). which, in principle, doubles the capacity of wireless channels [1], [2]. The key challenge in full-duplexing is to have sufficient isolation between the TX and RX nodes (typically 100  dB) to avoid self-interference, i.e., leakage of the strong TX signal into the RX path. Several works from both academ ic and industrial groups have recen tly proposed a combination of radio-frequency (RF) and baseband digital signal processing (DSP) techniques [2]-[4] to achieve this goal. RF cancellation is an absolute necessity in full-dupl ex system s to avoid saturation of the analog-to-digital converters (ADC) and it can be classified into t hree categories: ( i) antenna-based [1]-[3], (ii) circulator-based [3]-[5] and (iii) mixed-signal approaches [6]-[9]. Antenna-based techniques require at least two antennas and are sensitive to their p lacement, thus weakening the argument for full-duplex as compared to conventional MIMO systems, which can also doubl e the throughput using multiple antennas. Altern atively, mixed sig nal ap proaches exploit the fact that the TX signal is already known, and therefore, aim at subtracting it at the RX node. However, what the transceiver actually knows is the clean baseband digital TX signal which becomes very different after it goes through t he noisy and non-linear RF chain to be up-converted to the carrier frequency. Therefore, mixed signa l approaches, if not carefully designed, may end up adding more interference at t he RX node. The stringent 100 dB specification on self-interference cancellation (SIC) in full-duplex systems can be obtained while using a single antenna by combining DSP and mixed-signal techniques with circulators as depicted in Fig. 1. Nevertheless , circulators come with their own challe nge, which is the necessity of breaking reci proc ity. In general, this can be achieved using: (i) magnetic-biased anisotropic materials [5], (ii) active devices [10]-[15], (iii) non-linearities [16], [17] , or (iv) linear periodically-time-varying (LPTV) circuits [18]-[30] . For decades, non-reciprocity h as been almost exclusively achieved through m agnetic biasi ng of ferrite m aterials, l eading to bulky devices, which are incompatible with conventional integrated circuit (IC) technologies. To get rid of magnets, active approaches have been pur sued over the years, but they suffer from a fundamentally poor noise f igure, lim ited power handling and small dynamic range, resulting in devices that cannot be deployed in commercial systems. Also nonlinear Magnetless Circulators Based on Spatiotemporal Modulation of Bandstop Filters in a Delta T opology Ahmed Kord, Graduate Student Member, IEEE , Dim itrios L. Sounas, Member, IEEE , and Andrea Alù, Fellow , IEEE W AP1507-1042.R1 2 elements have been explored to realize magnet-free non-reciprocal devices, but they i nherently lead t o signal distortions, and operate onl y over a limited range of input intensities. Recently, LPTV circuits, based on modulated varactors or banks of switched capacitors [18]-[30], have been presented as a p otential alternative to wards non-reciprocity, without the drawbacks of the previous approaches. In particular, [18], [19] present ed the idea of parametrically modulating a transmission line (TL) by loading it with varactors and injecting a modulat ion signal at one port. Such a line allows signal propagation in one direction (the direction opposite to propagation of the modul ation signal) as a conventional TL, while in the opposite direction it mixes the injected m odulation with the R F signal, upconverting the lat ter to a different frequency. Such an approach necessitates the TL length to be larger than the wavelength and, more importantly, it requires the us e of a diplexer to isolate the two counter-propagating RF signals in the frequency spectrum, thus making it n ot suitable for integr ation and less attractive when compared to high performance m agnetic-biased circulators. In [24], [25], a fully int egrated CMOS m agnet-less circulator was presented, where staggered commut ation using N -path filters was shown to be equivalent to a highly m iniaturized non-reciprocal phase shifter, which, when embedded in a loop of reciprocal phase shifters, can lead to an asymmetric circulator. In this work, the asymm etry of the circuit improved power handling of the TX/ANT p ath as compared to the ANT/RX path, but it also resulted in sensitivity to impedance mism atches at the RF ports, thus requiring the use of reconfigurable impedance tuners and leading to asymmetric S -parameters. Furthermore, the commutator circuit requires a modulati on frequency equal to t he RF band’s center frequency, which leads to challenges in power consumption in the modulation path, especially in the RF and mm-wave bands, and it also complicates the rejection of any modulation leakage at the RF ports due to switch parasitics. A related approach for the realization of magnet-less circulators follows the so-called angular-mom entum biasing, and it is based on a loop of three resonators modulated in tim e with 120  phase difference between each other, so that a synthetic angular-momentum is effectively im parted to the circuit [21]-[23]. This approach results in symmetric circulators, which are insensitive to random mismatches, and require a small modulation freque ncy in t he order of 20-30% of the RF band’s center frequency, hence reducing power consumption along the modulation path. Furtherm ore, it allows for easy filtering of the modulati on leakage at the RF ports an d the opportunity of CMOS implem entation via less-scaled and high-voltage CMOS technologi es, which can handle high power. The first circulator based on this approach was presented in [22] by connecting three ladder LC resonators in a delta topology. However, such a connection is non-optimal, as it supports a non-zero common mode that leads to poor matchi ng and an insert ion loss in the order of 25 dB , despite t he fact that isolatio n can still be in the order of 40 dB. In [23], it was shown that the comm on m ode problem can be overcome if series LC resonators are connected in a wye topology. One problem with thi s approach i s that it requires a large number o f filters in order to prohibit the modulation signal from leaking into the RF ports and vice versa, t hus significantly com plicati ng its im plement ation and increasing insertion loss. Moreover, in this topology, the LC resonators are connected in series with the 50 Ohm port impedance, thus lim iting the loaded Q -factor that can be achieved with realistic inductors and in turn the circulator’s p erform ance. The circuit reported in [23] showed a measured insertion l oss of 9 dB at 130 MHz. Here, we present a new implementat ion of angular- mome ntum circulators that overcomes the mentioned problems of [22], [23]. In particular, we p ropose connecting parallel LC tanks in a delta topology, which can be considered t he dual cas e of the wye topology in [23]. Interestingly, this circuit does n ot allow transmission to any port without modulati on, however, when angular-momentum is imparted, the degenerate poles of the loop split allowing non-reciprocal transmission with a far more superior performance in all metrics compared to [22], [23]. To the best of our knowledge, this is the first time a bandstop junction is used as the building block to r ealize circulators, based on any form of biasing, including DC magnets. Like the wye topology in [23], the delta topology presented here does not support a com mon mode, thus enabling good matching. At the same time, it requires less filters and provides a greater control over the loaded Q -factor wi thout the requirement of using impedance t r ansformers at the ports, thus reducing the complexity of t he circuit and ma king it more suitable for integration. In addition , it exhibits b etter power-handling, since, in contrast to [23], the voltage across the varactors is not amplified by the circuit’s resonance (instead, there is an amplification of the current , but this ha s no effect on the ci rcuit’s non-linearity). In this paper, we show and experimental ly validate that the proposed circuit results in magnet-less circulators with unprecedented performance, satisfying n early all metrics of practical systems, e.g., RFID, hence making this work an important step towards the comm ercialization of full-duplex communications in the near future. The paper is organized as follo ws. In Section II, we present the new circuit and develop an analytical model for its linear small-signal response, including the harmonic S -parameters, and describe the design proce dure based on thi s theory. In Fig. 1. Circulato r in a full-duplex transceiver. AP1507-1042.R1 3 Section III, we provide numerical simulations and investigate the circuit’s large-signal response, including maxim um power handling and non-linearity, followed by a discussion of the noise figure and its associated m echanisms. In Section IV, we provide measured results and discuss the experimental setups for the fabricated prototype in detail. Finally, we draw our conclusions and an outlook on th e next steps in this exciting line of research in Section V. II. T HEORY AND P ROPOSED C IRCUIT A. Proposed Circuit Topology Fig. 2 shows the complete imple mentati on of the proposed circuit which consists of three identical parallel LC tanks connected in a l oop. These tanks represent first-order bandstop filters with a center frequency given b y 1 ,1 , 2 , 3 2 n n fn LC   (1) where n is the tank index, and L and n C a r e t h e t o t a l inductance and capacitance of the n th tank, respectively. The capacitance n C is realized via varactors to allow a continuous modulati on of the tanks’ resona nce frequencies, as required in the angular-momentum approach. The modulation signals have the sam e frequency m f and am plitude m V , and phase difference 120   between different tanks . Varactors are also stacked in pairs to i mprove power handli ng and to increase the circuit’s 1dB compression point (P1dB). They are also connected in a common-cathode configuration to improve the input-referred third-order intercept point (IIP3) [32]. Modulation is applied to the varactors through m atching networks ( L m and C m ), which also act as bandstop filters for th e RF s ignal, thus prohibiting its l eakage i nto the modulati on ports. An important feature of the proposed circuit is that the modulati on si gnals see a virtual ground at the RF ports because of symm etry, therefore alleviating the necessity of using three additional filters as co mpared to [23] and simplifying the modulati on network significant ly (Appendix A). Furtherm ore, the matchi ng networks at the modulation-signal path amplify the modulation voltage, t hus rel axing the thus relaxing the requirements on out put voltage from the modulation signal generators (Appendix A). Finally, DC b iasi ng is combined with the modulati on signals through sufficient ly large resistors B R . B. Linear Small-Signal RF Analysis At the RF frequency, the com plete circuit in Fig. 2(a) can be simplifi ed as shown in Fig. 2(b), in which the varactors and their DC/modulation network are replaced with time variant capacitors given by 0 1 k nk n k CC a v     , (2) where 0 C is the static capacitance of the varactors set by the DC bias D C V , k a are the coefficients of a poly nomial that models the non-linear CV characteristics of the v aractors around their quiescent point, and mod ,1 , 2 , 3 rf nn n vv v n   (3) is the total AC voltage across the n th LC tank. S -parameters are, by definition, calculated under t he small-signal assum ption mod rf nn vv  , so (2) yiel ds the following expression for the effective capacitance s een by the RF signal: 2 01 2 () mod mod nn n CC a v a v     . (4) Assuming weak and linear modulation, so that we can keep terms up to first order, (4) becom es   0 cos nm n CC C t      , (5) where 1 m Ca V   is the effective modulation capacitance, m V and m  are the modulati on vo ltage and frequency, respectively, and   1 n n    , where 120   is the constant phase difference between different modulation signals. W e also assume that the varactors’ and inductor’s losses of each tank are combined into a disper sion-less parallel resistance R , through which we can define a total “unloaded” quality factor 0 / QR L   . This assumption is acceptable for narrow bandwidths, as is the case in this paper. Equation (5) embeds both the DC bias and the modulation signal into a time-variant capacitance, therefore the tank AC voltages in Fig. 2(b) should Fig. 2. Proposed magnet-less circulator: (a) Complete schematic . (b) Linea r small-signal model at rf f . The circuit is connected to sources with a real impedance 0 Z and voltages s n v , where 1, 2 , 3 n  is the port index. (c) Simplified m odel at m f . AP1507-1042.R1 4 be interpreted as the RF signal rf nn vv  . Also, the RF ports are assumed to have a real impedance 0 Z , which is typically equal to 50 Ohm. Applying Kirchhoff’s laws to the n th tank in Fig. 2(b) and writing the result in a matrix form, we get 0 3 s Z A vB v v G v L      , (6) where d dt  ,   12 3 ,, vv v v  is the tank voltages vector,   123 ,, ss s s vv v v  is the input excita tion vector, and A , B , and G are matrices which depend on the circuit elements and modulati on param eters as deri ved in Appendix A. Equation (6) can be written in an even simpler form if, in analo gy with magnetic circulators, we express n v as a superposition of three modes, i.e.,   11 cm jn jn n vv v e v e        , where cm v ,  1 jn ve    , and  1 jn ve    are defined as the common, clockwise and counter-clockwise modes, respectively . The comm on mode here refers to the in-phase signal component of t he voltages n v . The clockwise m ode refers to the signal component whose phase i ncreases in a clockwise direction by 120   , sim ilar to a TL wave propagating in that direction. Similarly, the counter-clockwise mode refers to a propagating wave in the opposite direction, i.e., a signal whose phase increases counter-clockwise. As an example, an excitation at port 1 as shown in Fig. 3(a), where 0 V is a constant amplitude, can be decomposed using superposition into these predefined modes as shown in Fig. 3(b)-(d), while noticing that 2/ 3 4/ 3 10 jj ee    . More generally, for excitation at all por ts, this can be expressed through the m atrix transformat ion 1 vT v    , (7) where   cm ,, vv v v     is the vector of the defined mode voltages and the operator T is given by 22 11 1 1 1 jj jj Te e ee                . (8) Applying this transformation and considering the case when only port 1 is excited, as in Fig. 3, i.e.,   1, 0 , 0 s v  , we get 0 0 0 cm cm cm 3 30 , Z ZC v v v L     (9)  0 00 0 0 01 33 31 2 33 1 33 , 26 m m jt jt ms Z ZC v Z C e v v R jZ Z Ce v v j v L                        (10)  0 00 0 0 01 33 31 2 33 1 33 . 26 m m jt jt ms Z ZC v Z C e v v R jZ Z Ce v v j v L                      (11) Notice that a general s v can be constructed using a linear superposition of individual port excitations and that excitatio n from the other ports can be inferred from excitation at port 1 based on the circuit’s symmet ry. Equation (9) has only the trivial so lution cm 0 v  . This means that the common mode is not excited and that the input power is split between the counter-propagating modes v  and v  , which are given by the coupled differential equations ( 10) and (11). These equations can be solved by applying the Fourier transform, yielding (Appendix B)     1 1 1 , ks m k VH V k         (12) where      2 0 00 0 0 3 3 33 6 , 3 1 m m Z j CZ j L H Z D j R                       (13)     2 0 1 33 4 , m j jC Z H D           (14)     2 0 1 33 4 , m j jC Z H D          (15)     11 0. HH     (16) The denominator functi on  D   is given by Fig. 3. (a) Excitation at port 1 resulting in the tank voltages n v . (b)-(d) Decomposition of Fig. 5(a) u sing the superposition of three mod es: (b) Comm on mode, (c) Clock-wise m ode, and (d) Counter clock-wise m o de. AP1507-1042.R1 5     2 22 2 0 2 00 00 2 00 00 9 4 33 31 33 31 . m mm DC Z ZZ CZ j LR ZZ CZ j LR                                 (17) Equation (12) shows that the volta ge at any frequency  depends on the input voltages at different frequencies, in particular the frequencies  , m    and m    , a s a r es ul t o f the modulation. A graphical representation of the frequency mixing mechanism is provided i n Fig. 4(a). If the input si gnal is monochrom atic with frequency  , (12) and Fi g. 4(a) show that the voltages have three components at frequencies  , m    and m    , as shown graphically in Fig. 4(b), wit h ampl itudes       2 0 00 0 1 3 3 33 6 3 1 m s m Z j CZ j V L Z VD j R                       , (18)      2 0 1 33 4 m m s j jC Z V VD           , (19)   0 m V    . (20) Next, in order to find the S -parameters, (18)-(20) are transformed back to the tank voltages   123 ,, VV V V  . In particular, using the inverse transformation of (7) and recognizing that cm 0 V  , we find       1 kk k VV V     , (21)    2/ 3 2/ 3 2 jj kk k Ve Ve V       , (22)    2/ 3 2/ 3 3 jj kk k Ve Ve V       , (23) where km k     and 1, 0 , 1 k  is the harmonic index. Using Kirchhoff’s laws, the tank currents   nk I  can be calculated from the voltages  nk V  (see Appendix B), then t he harmonic S -parameters for excitation at port 1 can be found as follows:             01 3 11 1 02 3 21 1 01 2 31 1 ,2 , ,2 , ,2 . kk kk l s kk k s kk k s ZI I S V ZI I S V ZI I S V                      (24) If 0 k  , k    and (24) give the conventional S -parameters defined for linear time-invariant (LTI) systems which relate the input and output powers at t he same frequency, while 0 k  gives the transformation of the input signal at  to t he intermodul ation products at km     . I n t h e n e x t sections, the S -parameters refer to the conventional ones, unless stated otherwise. Also, due t o the circulator’s threefold 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  , resulting in 11 22 33 21 32 13 31 12 23 SS S SSS SSS    , (25) Simil arly, the harmonic input impedance at port 1 is found by        1 1 0 11 3 ,. p s in k pk k k V V Z Z II I         (26) Like for the S -parameters, 0 k  yi elds the conventional input impedance, which relates voltages and currents having the same frequency. Generally, if 0 k  ,  , in k Z   corresponds to a trans-impedance which relates the input voltage at  to the excited current at k  . It is also worth mentioning that     11 0 11 1, , 1, k in k k S ZZ S         still holds for all k . Equations (24), (25) give the S -parameters as a function of the circuit elements L , 0 C , Q (recall that Q is the unloaded quality factor of the LC resonators, incorporating both inductor’s and varactor’s losses) and the modulati on parameters m f and C  . These parameters should be chosen for operation around a given frequency rf f while achieving certain specifications on insertion loss (IL), return loss (RL) , Fig. 4. (a) Frequency mixing map for wideband signals. (b) Freq uency mixing map for narrowband signals. AP1507-1042.R1 6 isolation (IX) and BW . Fig. 5 shows the S -parameters and the fractional bandwidth versus the normalized modul ation frequency mr f f f and normalized m odulation amplitude 0 CC  for 1 rf f  GHz, 0 7.67 C  pF, 3.4 L  nH and 70 Q  , which i s a reasonable value for a PCB desi gn. The values of L and 0 C were selected by assuming an initial value for one of them and finding the other such that the resonance frequency   0 0 12 fL C   occurs at the design frequency rf f . The circulator’s bandwidth is defined as   IL IX BW mi n , ff     , (27) where IL f    and IX f    are the frequency ranges where insertion loss (IL) is less than  dB and isolation (IX) is m ore than  dB, respectively. For the results in Fig. 5(d), we choose 4   dB and 20   dB, in which case, one can find that IL 4 d B f   is always greater than IX 2 0 dB f   , i.e., the ban dwidth is determined by the minimum isolation specification. Fig. 5 allows us to find the modulation parameters for which any of the S -Parameters or t he bandwidth become optimum (m inimum 11 S , ma ximu m 21 S , mi nimu m 31 S an d ma ximu m BW ). These quantities do not n ecessarily become optimum under the same modulati on paramete rs, showing that we may need to trade off one or several of them when we select the modulation parameters. In this paper, we give priority to isolation and we choose the circuit to operate at point 3 p , where isolation at the design frequency is maxim um. The various circuit parameters for this operation point are summarized in Table I. Other choices that give priority to other metri cs, or a combination o f them, depending on the design specificat ions, are equall y valid . For example, if m aximizing isolation at the center frequency is not i mportant , one could select point 4 p as the operation point, where the 20 dB isola tion bandwidth is maxim um. Notice that point 4 p is also cl ose to point 2 p , where insertion loss is minim um. In general, if the generated charts do not provide a solution that meets the given specifications, one can try different values of L and 0 C that satisfy 0 rf f f  until reaching the desired solution. Fig. 6(a) shows the S -parameters in dB using (18)-(25) and the values g iven in Table I. Insertion loss, return loss and isolation at the center frequency of 1 GHz are 2.9 dB, –10.8 dB and 56 dB, respectively, and the fractional BW based on the definition of (27) is 2.7% (27 MHz). Input impedance is also shown in Fig. 6(b) which is almost real, but not equal to 50 Ohm (the port im pedance), at the design frequency rf f . This is related to the fact that the circuit was not chosen to operate at point 1 p in Fig. 5(a), where the return loss becomes very small. Impedance matc hing can be improved by using simple LC matchi ng networks at the RF ports at the expense of increasing the form factor and the overall resistive loss [31 ]. Fig. 6(c) and Fig. 6(d) also show the phase response and the transmissi on group delay 21 g dS d     , respectively. Clearly, g  is almost flat in the 2.7% operational band, thus im posing minima l dispersion to the transmitted si gnal through the circulator and allowing its use wi th any coherent or non-coherent digital modulation schem e, as in practical communicat ion systems. Fig. 6. Theoretical results: (a) S -parameters magnitude. (b) S -parameters p hase. (c) Input i mpedance. (d) Transm ission group delay. TABLE I T HEORETICAL D ESIGN P ARAMETERS Element Value Q 70 L 3.4 nH 0 C 7.67 pF 0 / CC  0.46 m f 190 MHz Fig. 5. S -parameters at 1 f  GHz versus modulation parameters f o r 1 rf f  GHz, 0 7.67 C  pF, 3.4 L  nH and 70 Q  . (a) Return loss. (b) Insertion loss. (c) Isolation . (d) Fractional bandwidth. AP1507-1042.R1 7 Fig. 7 shows the circulator’s harmonic response at the transmitted and isolated p orts, i.e., t P and iso P , respectively, for a monochromatic input at 1 rf f  GHz and 0 in P  dBm. Notice that the presented circulato r is rotationally symm etric as mentioned earlier, hence Fig. 7 results apply for excitation at any of the RF ports where, for example, if the input is inciden t from port 1, then t P and iso P become 2 P and 3 P , respectively. The IM products are only –10 dBc, where dBc is a norm alized unit with respect to the carrier (center) frequency, which pose s an interference problem to neighboring channels and may saturate the R X front-end, especially at high TX power, even i f the fundamental harmonic is sufficiently attenuated. It can be shown ei ther in simulat ions or from the analysis in Section II. B that these products are reduced w hen using a la rger m odulation frequency, yet at the expense of increasing the power consumption in the modulation path. Further reduction is possible using channel or ba nd pre-selection filters t hough, again, this would put a restriction on the m inimum m odulation frequency, h ence on the overall power consumption, to relax the requirements on the sharpness of these filters. III. S IMULATION R ESULTS Based on the theoretical results presented in the previous section, a PCB circulator operating at 1 GHz was designed using off-the-shelf discrete components, as listed in Table II. These components were chosen based on the design param eters in Table I. The layout on a Rogers board was simulated using ADS Momentum and the generated S -parameters were combined with the rest of the circuit components in ADS to perform p ost-layout circuit/EM co-simulations. By followin g such approach, we take into acc ount all parasitics, e ither due to the finite l ength of t he interconnecting transmission lines or the pads of the components’ footprints. Commercia lly available measured S -parameters of the passive RLC elements and the full non-linear spice model with package parasitics of the varactors were also used. A. S-parameters Fig. 8 shows the simulated S -parameters with and without modulati on for DC 21.6 V  V. Without modul ation ( 0 m V  ), the circuit is, clearly, reciprocal with the same transfer function seen between the input port, e.g. port 1, and any of the output ports, i.e., 21 31 SS  . As mentioned before, the proposed circulator without modul ation is a b andstop filter, i.e., at th e resonance frequency transmission is not allowed towards any of the output ports, wi th the input power mostly reflected back and residual transmission is merely due to the finite qu ality factor of the constitu ent LC tanks. Fig. 8 shows that indeed without modulation, transmission is 20  dB at the unmodulated center frequency of 1 .06 GHz. When spatio-temporal modulati on is applied with 190 m f  MHz and 3.62 m V  V, the S -parameters become non-recipr ocal: for ex citation from p ort 1, 2 and 3 the power is mostly transmitted to ports 2, 3, 1, respectively. We also notice that t he center frequency is shift ed down to 1 GHz, because o f the varactors’ second-order non-linearity, as will be explained later. The achieved inserti on loss, return loss and isolati on at 1 GHz are 2.8 dB, –11.34 dB and 55 dB, respectively, which are all in agreement with the linear small-signal analytical res ults of Sect ion II. On the other hand, the fractional BW decreases to 1.8% (18 MHz), because of all the parasitics which were neglected in the theo retical analysis. The simulated phase response and group delay are also shown in Fig. 9. Fig. 10(a) presents the S -parameters for different DC voltages, showing that the circulator can be reconfigured to operate at different frequencie s spanning a range of 100 MHz, which is about six tim es larger than the instantaneous bandwidth, by sim ply cont rolling the DC bias and adjusting the modulati on voltage to maintain the same S -parameters. This Fig. 12. S im ulated r econfigur able S -param eters. Fig. 7. Sim ulated harmonic spectrum at the transmitted and isol ated ports for a single tone input at 1 rf f  GHz and 0 in P  dBm. Fig. 8. Simulated S -parameters: without modulation (dashed lines) and with modulation (solid lines). TABLE II L IST OF THE D ISCRETE C OMPONENTS U SED IN O UR D ESIGN Element Value D ~8 pF @ VDC=21.6V L 3.3 nH m L 72 nH m C 24 pF B R 100 KOhm B C 1000 pF AP1507-1042.R1 8 range can be extended further if the modulation frequency is also controlled, e.g. using voltage co ntrolled oscillators (VCO ) to generate the modulat ion signals. B. Power Handling and Non-Linearities Non-linearity in the circuit is p redominantly due to the use of varactors, related to the non-linear CV characteristics as given by (2) and the finite forward-conduction and breakdown voltages, f V and B V , respectively. In order to maximize the power handling of a given varactor, the bia sing should be such that the circulator operates in the middle channel of the tunability ban d at DC 2 f B VV V   , which is approximate ly equal to 20 V for the varactors used in this paper. Clearly, as the D C bias deviates from t his optimal value, e.g. as we t une the circulator for operation at a different channel, the maxim um power that the varactor can handl e decreases. This r equires to set a back-off limit from both f V and B V in order to maintain, approximately, the same linearity performance in all channels. In order to investigate the second factor that contributes to the non-linear characteristics of the circuit, i.e., the non-linear CV curve of the varactors around the quiescent point, we invoke (2) and substitute for     cos cos nm m r f r f vV t V t   , which yields     0 1 cos cos cos km kr f lk m rf k l bk t d k t Cb ek l t                     , (28) where k b , k d and lk e are polynomial coefficients, which up to the first-order term are given by  22 00 2 3 11 3 3 11 3 1 , 2 3 , 4 3 . 4 mr f mm rf rf bC a V V ba V a V da V a V           (29) Notice th at for simplicity, the subscript n was dropped and the sinusoidal phases were assumed zero since they are irrelevant to this analysis. The DC term 0 b represents the effective static capacitance in the presence of l arge modul ation and RF signals while taking into account the varactors’ non-linearities. We notice a shift in su ch capacitan ce with respect to the static value 0 C due to the second order non-linearity. Generally, all even order terms in (2) would resu lt in a similar shift b ut the seco nd order one is predominant. For a convex CV curve, as for the majority of commercial varactors, 2 a is positive, therefore the resonance frequency is shifted “down” in agreement with the result in Fig. 8 . For rf m VV  , this shift is constant and independent of the RF signal, however, when rf V becomes sufficiently large, 0 b changes with the i nput power, or in other words, the varactor is compresse d. Clearly, when this occurs, both IL and IX at the fixed design frequency, e.g. 1 GHz, will also compress. The rest of the terms in (28) correspond to harmonic variation of the varactor’s cap acitance at different frequencie s. For example, 1 b corresponds to the effective capacitance variation at the imposed modulation frequency m  which, in contrast to (5), is a non-linear function of the modul ation amplitude m V due to the non-linear characteristics of the varactors’ CV curve. The term 1 d corresponds to a harmonic capacitance variation at the RF frequency rf  , effectively corresponding to modulation of the circuit with a modulati on frequency rf  . Since such a modulation frequency is very far from the optimum modulation frequency m  , this term has a negligible effect on the circuit’s response. Simil arly, the oth er Fig. 9. (a) Sim ulated phase response. (b) Simulated group delay . Fig. 10. Sim ulated reconfigurable S -parameters. AP1507-1042.R1 9 terms in (28) correspond to modulati on of the circuit at other individual or mixed harm onics of m  or rf  and for the same reason as for 1 d , they have a neglig ible effect on the circuit’s response. Fig. 11 shows th e o utput power at th e transmitted port and the isolation versus the input power, assuming that the input frequency is equal to the desi gn frequency of 1 GHz. We can see that both the out put power and the isol ation compress after a certain input power, due to the non-linear characteristics of the varactors, as explained before. The maximum allowable input power level is defined as   max IX min , in in X PP P    , (30) where in X P is the input X dB compression point and IX in P   is the maximum input power to mainta in  dB isolation (IX) at the center frequency. For the typical values: 1 X  dB and 20   dB, Fig. 11 shows that the achieved max P is 26.7 dBm. Such a large value exceeds by sev eral orders of magnitude what conventional active approaches can achieve either in PCB or IC platforms [10]-[14]. It is also worth mentioning that this larg e compression point is partially due to the stacking of the varactors in pairs, as in Fig. 2(a). Such an approach h alves th e RF voltage on each v aractor, thus increasing the maximum allowable input power by 3 dB, approximately. In general, stacking N varactors can improve the compression point by  10 ~2 0 l o g N , but it also complicates the modulation network and, more important ly, increases the insertion loss, due to the parasitic loss of the varactors. Finally, the circuit is also t ested with two in-band input tones at 1000 0.5  MHz, as shown in Fig. 12 where IIP3 is found to be 33.8 dBm. Notice that the difference between IIP3 and P1dB is not necessarily 9.6 dB as expected in third-order time-invariant system s [33], due to the time-varying characteristics of the circuit. C. Noise Figure Noise performance of the ANT/RX path is another critical metric for the performance of the circulator, since it comes at the forefront of the receiving path superseding the low noise amplifier (LNA); therefo re, it is highly desirable that the circulator adds minimal noise in order not to limit the overall signal-to-noise r atio (SNR) of th e transceiver. In the follo win g discussion, we detail the different noise mechanism s in the proposed circuit and their impact at the circulator’s performance. A lthough the analysis focuses on the RX port, the results also apply to the ANT and TX ports, due to the symmetry of the circuit. Total noise at the RX port can be decomposed into three components: (i) incoming noise from the ANT and TX ports, (ii) noise added by the circuit itself, a nd (iii) noise resulting from random variations in the modulation signals, including am plitude and phase noise. The TX and ANT noise are both transmitted to the RX p ort through the circulator’s harmonic S -parameters. Notice that since the circuit is tim e-v ariant, o utput noise at the RX port at a particular frequency not onl y comes from the ANT or TX noise at the sam e frequency, but it also folds from the IM frequencie s as shown in Fig. 13. Therefore, the RX noise due to the ANT and TX noise is given by        ______ 2 2 ______ 2 ANT 21 ______ RX ,1 2 2 TX 31 , , N mm kN m vk S k v vS k                       , (31) where N is the total number of the IM products and  ______ 2 TX v  and  ______ 2 ANT v  are the power spectral d e nsities (PSDs) of the TX and ANT incoming noise, respectively. Notice that   22 31 21 ,, SS     over the circulator’s BW; therefore, TX’s contribution at 0 k  in (31) can be neglected. Also, as mentioned earlier, only the second-order products at m f f  are excited under the linear small-signal assumption. Hence, (31) simplifies to         ______ ______ 1 2 22 RX ,1 A NT 21 1 ______ 1 2 2 TX 31 0 1 , , mm k mm k k vv k S k vk S k                  , (32) and the harmonic S -param eters in such a case are given by (18) -(25). One can argue that the circulator should not be penali ze d by the input noi se folding, since there is no desired si gnal at the IM frequencies. A channel pre-se lection bandpass filter, which is similar to image r eject filters in heterodyne transceivers [ 31], Fig. 11. Sim ulated compression point and m aximum power handling . Fig. 12. Sim ulated two-tone test and achieved IIP3. AP1507-1042.R1 10 can thus be added at the antenna port in order to knock down this out-of-band noise as shown in Fig. 13. Unlike DSB m ixers, however, where the conversion gain seen by the desired signal and its image is the same, IM harmonic transfer functi ons in th e circulator’s case are, in fact, much smaller than insertion los s, i.e.,     21,31 21 0 ,, m k Sk S        , therefore, noise folding from IM frequencies is already very small compared to the in-band noise in N and can be neglected e ven without filtering. As for the noise added by the circuit itself, it is attributed to the thermal noise of the biasing resistors, the inductors’ fini te quality factors, the varactors and the output impedance of the modulation sources. Fig. 14 shows these noise sources where noise of the biasing resistors B R is neglected compared to the input noise from the ports, since 0 B RZ  . Also fo r simplicity, total noise of the k -th LC tank, due to the varactors and the finite quality factor of L , is lumped into a sin gle parallel current source ____ 2 , tk i . One can split each ____ 2 , tk i into two fully correlated shunt current sources with opposite currents at the two termi nals of t he corresponding tank, therefore allowing t he calculation of their contribution at the RX port using the harmonic S -parameters. Furthermore, incoming noise from the modulati on/DC ports is injected into the comm on-cathode node of the corresponding tank through a voltage source ______ 2 , mk v in series with a noiseless complex impedance m Z where ______ 2 , mk v and m Z can be found using Thevenin-equivalence looking back into the modulation ports. One can interpret ______ 2 , mk v as an amplit ude variation of t he modulation signal applied to the varactors, which is equivalent to an effective random capacitance variation   n Ct  in (5). Simil arly, phase noise of the modulation signals can be represented as a random phase   n t  , hence (5) is rewritten as follows:     0 cos nn m n n CC C C t t t            . (33) Equation (33) shows that in a reali stic scenario, the modulatio n signal is not a pure sinusoidal tone with frequency m f , but a random signal with a finite bandwidth whi ch increases the folded noise into the circulator’s instantaneous BW and further degrades its NF. More im portantly, noise in (33) at m f in particular is indistinguishable from the desired modulat ion signals at the same frequency, therefore, it would result in a random variation of the harmonic S -parameters, both magnitude and phase, hence the RF signal would incur undesired amplitude and phase modulation around the RF frequency. Clearly, this would lead to a fuzzy conste llation in a practical comm unication system a nd increase the bit error rate, yet the significance of this effect depends on the comm unication scheme itself and is beyond the scope of this paper. Analytical calculation of the exact total output noise at the RX port can be very t edious; however, sim ulation tools such as ADS harmonic balance significantly simplify the task. Also, since the presented circulator i s a passive LPTV circuit, one should expect an overall noise figure (NF) close to the inserti on loss [31] but slightly higher due to noise folding and modulation phase noise. Fig. 15(a) shows that the simulat ed NF is 2.9 dB at the center frequency, which is only 0.1 dB higher t h a n I L , a n d l e s s t h a n 3 . 1 d B o v e r the circulator’s instantaneo us BW. In obtaining these results, we assum ed that the modulation sources are uncorrelated a nd have the phase noise characteristics shown in Fig. 15(b) in order to mim ic the experimental setup we used in Section III.B. IV. M EASURED R ESULTS The circuit, as described in Section III (Fig. 2(a) for the schematic and Table II for the components), was fabricated on a PCB with a total form factor of 25mm ×23mm and an area occupied by the core p art (without the SMAs) of only 13mm ×11mm . Fig. 16(a) and Fig. 16(b) show a photograph of the experimental setup and the fabricated prototype, respectively, w hile Table III provides a list of the used equipment. Fig. 15. (a) Simulated NF vs frequency. (b) Phase noise of the modulation sources. Fig. 13. Channel pre-selec tion filter to reject out-of- b and input noise folding from I M frequencies. Fig. 14. Therm al noise sources generated by the circuit. AP1507-1042.R1 11 A. S-parameters and Output Spectrum Fig. 17 shows a block diagram for the experimental setup used to measure the S -param eters. The modulation signals were generated using three phase-l ocked RF sources, which were controlled through a laptop. With the help of an oscilloscope, the sources were configured to generate three signals with the same amplitude and phase difference of 120 deg. Next, two of the RF ports were connected to the VNA and the third port was terminated with a 50 Ohm load. Ideally, a 4 port VNA would give the entire S -matrix with one m easurement, while a 2 port VNA requires performing three different measurements in order to construct the 33  matrix. Finally, the DC port is connected to a power supply. Fig. 18 shows the measured S -parameters, with and without modulati on, for three different channels correspondi ng to different DC bias and modulation voltages. The modulation parameters (at the signal generators) in this case were chosen to get maximum isolation at the center frequency of each channel resulting in IL, RL and IX of 3.3 dB, –10.8, 55 dB, respectively, and 2.4% instantaneous BW (based on the definition of (27)). We also notice that matching is non-optimal , i.e., 11 S is not minimum at the circulator’s center frequency. This is mainly due to the varactors’ package uncertainities, which resulted in a self-resonance frequency (SRF) closer to 1 GHz than expected, thus leading to an asymm etric resonance of the unmodulated junction, as can be seen from 11 S in Fig. 18(a). It is worth menti oning that even with this asymmetry, the circulator is threefold rotationally symmetric from its po rts, since the same S -parameters were measured between differen t ports, thus still preserving the insensitivity to por t mi smatches as compared to [24]. Fig. 19 sho ws the o utput spectrum at both the transmitted an d isolated ports for a monochrom atic input at 1 rf f  GHz and 0 in P  dBm. The output power t P is about –3.3 dBm which is indeed equal to IL in P  as expected, while 48 iso P   dBm is 7 dB larger than IX in P  simply because the isolation null is not perfectly aligned with 1 GHz. We also see the second-order IM products at rf m f f  and rf m f f  which are –15 dBc and –11.3 dBc, respectively, in a fair agreement with the theoretical results. Furthermore, two additional tones exist at 5 m f and 6 m f , which are basically higher-order harmonics of the modulati on signal due to the varactors’ non-linear CV characteristics. It is therefore desirable not to have / rf m f f as an integer number, in order to avoid any of these h arm onics falling on top of the desired signal. Nevertheless, these Fig. 16. Photograph of: ( a) Experim ental setup. (b) Fabricated prototype. TABLE III L IST OF U SED E QUIPMENT Instrum e nt M odel Q uantity Power supply Agilent E 3631A 1 Vector network analyzer Agilent E 5071C 1 Spectrum analyzer R &S FSVA40 1 Oscilloscope R & S RTO1044 1 Signal gener a tor s R& S SGS100A 4 R& S SM B100A 1 In stru m e n t am p lif ier M i nicir c uits T VA- 4W - 422A+ 1 TABLE III L IST OF U SED E QUIPMENT Instrument 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 Fig. 17. Experim ental setup of S -parameters m easurements. Fig. 18. M easured S -parameters: (a)-(b) Without modulation. (c)-(d) With modulation. AP1507-1042.R1 12 harmonics are very small, e.g. the fifth harmonic is less than 40  dBc, and therefore can be neglected compared to the rf m f f  IM products. Notice that these harmonics are independent of the RF si gnal a nd do not change when the input power is varied, therefore, the y can be entirely cancelled out using simple DSP al gorithms. B. Power Handling and Non-Linearities In the measurement of P1dB, a monochromatic tone at 1 rf f  GHz is generated via a signa l generator and applied to port 1 of the circuit through an amplifie r, as shown in Fig. 20(a). The amplifier was used in order to be able to tune the supplied power in P to the circuit at higher levels than the maxim um output power of the generator. In this case, in P is restricted by the output compression point of the amplifi er which i s +34 dBm. The other two ports of the circuit are interchangeably connected to a spectrum analyzer and a 50 Ohm load in order to measure both transmission and isolation. The output powers TX P and IX P at the fundamental frequency of 1 GHz are measured using the spectrum analyzer and the measurement is repeated for different values of in P where the results are shown in Fig. 21. Both P1dB and max P (based on the definition of (30)) are found equal to 29 dBm whi ch is a remarkable number for magnet-less circulators with such form factor. IIP3 is measured by com bining two tones at 1 1000 0.5 f   MHz and 2 1000 0.5 f  MHz with the same power in P and feeding them to the circulator’s input port, as shown in Fig. 20(b). Then the spectrum analyzer is used to measure the output power of the fundame ntal component at 1 f and the third-order IM product at 12 2 f f  . Fig. 22 shows the results where IIP3 was found t o be 33.7 dBm. To the best of the authors’ knowledge, these are the largest reported P1dB and IIP3 of all magnet -less circulators proposed to date . C. Noise Figure Fig. 23 shows a block diagram for the experimental setup used to measure the circulator’s NF based on the Y factor method. A calibrated 6 dB excess noise rati o (ENR) source was connected to the circulat or’s input port and biased with a 28 V DC signal, the transm it port was connected to a spectrum analyzer through a high-pass filter (HPF) with a cut-off frequency larger than m f but much smaller than rf f , and the Fig. 21. Measured P 1dB and ma ximum power handling. Fig. 19. Measured harmonic spect rum at the transmitted and isol ated ports fo r a single tone input at 1 rf f  GHz and 0 in P  dBm. Fig. 20. Experim ental setup of: (a) P1dB and (b) IIP3 m easureme nts. Fig. 22. Measured I IP3. AP1507-1042.R1 13 isolated port was terminated to 50 Ohm. The HPF was added to reject any residual leakage from the m odulation signals which, despite being small, may still require further attenuation to g et below the noise floor in order not to overload the spectrum analyzer’s RF port while measuring the NF. Notice that this filtering is only possible because m f is much smaller than rf f in the angular-mom entum circulator, while if both frequencies were equal, e.g. as [24] requires, more complicated techniques must be utilized to reject t he m odulation leakage, which may, i n turn, degrade the circulator’s power handli ng. Fig. 24 shows the measured results when the circulator is configured to operate in the center channel of Fig. 18. In such a case, NF is 4.5 d B at 1 GHz and less than 4.7 dB over the circulator’s instantaneous BW. These values were slightly larger than predicted in simulations mainly because the measured insertion loss was 0.5 dB higher. It is also worth mentioni ng that the phase noise contributi on to the NF can be reduced by using one source and generating the three modulati on signals on-board using phase shifters such that they become perfectly correlated, hence the random phase vari ations of the generator’s output signal equally appear in the three modulati on signals. V. C ONCLUSION We presented here the concept, design principles and experimental verification of magnet-less circulators based on spatio-temporally m odulated fir st-order bandstop filters connected in a delta topology. We developed a rigorous analytical model, from which we extracted the optimal modulation parameters of the circuit in order to achieve given specifications on insertion loss, return loss, isolation and bandwidth. B ased on this model , we desi gned a PCB protot ype and measured its performance in cluding scattering paramet ers, harmonic response, power handling, and noise figure as summ arized together with the theoretical and simulated results in Table IV in com parison with previous works. These results show that the proposed circuit is very promising for the realization of tunable and IC compatible magnet-less circulators with small form-factor, lo w lo ss, large isolation, high power handling, and l ow NF, as required in practical full-duplex systems, and has definite advantages compared to other recently proposed approaches to magnet-free circulators. TABLE IV S UMMARY OF R ESULTS AND C OMPARISON TO P REVIOUS W ORK Metric This work [22] Meas. [23] Meas. Theo. Sim. Meas. RF frequency (MHz) 1000 1000 1000 170 130 DC bias (Volt) N.A. 21.6 19.6 2 1.7 Mod. frequency (MHz) 190 190 190 15 40 Mod. source voltage (rm s) N.A. 2.5 3.8 0.42 0.9 DC power consumption (mW) 0 0 0 0 0 Isolation 1 (dB) 56 55 55 40 50 Insertion loss 1 (dB) 2.9 2.8 3.3 25 9 Return loss 1 (dB) –10.8 –11.3 –10.8 N.A. 4 Instantaneous BW 2 (%) 2.7 1.8 2.4 N.A. N.A. Tunability BW (%) 10 10 6 30 (*) N.A. P1dB (dBm) N.A. 28 29 N.A. N.A. (**) IIP3 (dBm ) N.A. 33.8 33.7 N.A. N.A. (***) Pmax 3 (dBm) N.A. 26.7 29 N.A. N.A. Max. IM product (dBc) –10 –10 –11.3 N. A. N.A. Modulation leakage (dBc) N.A. –60 –60 N.A. N.A. NF 1 (dB) N.A. 2.9 4.5 N.A. N.A. Size (mm 2 ) N.A. 13 11  13 11  25 25  22 17  1 At center frequency 2 Defined in (27) 3 Defined in (30) *Applies only for IX **P1dB(sim) = +3.2 dBm ***IIP3(sim ) = +5.7 dBm Fig. 23. Experim ental setup of NF measurem ent. Fig. 24. Measured NF (in blue) and fitting curve (in red). AP1507-1042.R1 14 Several of these m etrics, incl uding BW, i nsertion loss and unwanted intermodulation products, can be further improved using higher-order modulated filters or a differential architecture, which will be the sub ject of future investigation s. A PPENDICES A. Modulation Network and DC Biasing As mentioned in Section II.B, the modulation signals see a virtual ground at the RF ports b ecause of symmetry. To prove this fact, we con sider mr f f f  , as is the case in this paper. In such a case, the impedance of the RF inductors L at m f is small enough that they can be replaced with a short circuit. Therefore, each pair of common cathode varactors in Fig. 2(a) appear in parallel and can be effectively replaced with a singl e varactor as shown in Fig. 2(c). Similarl y, the internal impedance 0 Z of the three RF ports als o appear in parallel and are lumped into a single resistance 0 /3 eq RZ  . Because o f this resistance, the modulation voltage that is applied across the varactors and enters in (4) for the effective capacitance seen by the RF signal is generally not equal to the m odulation voltages mk v after the modulation filters, but p mk vv  , where p v is the voltage at the modulati on frequency across eq R . However, due to the 120 deg sym metry of the m odulation signals at the three varactors, the current flowing through eq R is zero, making the node P a virtual ground for the modulation signals and the modulation si gnals across the varactors equal to mk v . The virtual ground feature simplifie s the design of the modulation network since the circuit in Fig. 2(c) can now be split into three identical and in dependent circuits as sho wn in Fig. 25 for the k th branch. Such a circuit can be designed to amplify the volt age at the varactor compared to the source voltage, thus relaxing the requirements on the modulation signal generators s rc V . By using simple circuit analysis, the voltage gain m G can be found as 1 2 2 1 1 1 mme q m ms r c m sr c m m k LC V GR VL C             , (34) where s rc R is the finite output impedance of the modulation sources, m L and m C are the matching network’s series inductance and shunt capacitance, respectively, and mk eq mk CC C CC   . The condition to get the maxi mum g ain is given by 2 2 1 1 1 mme q mm k s r c m LC LC R           . (35) Also, in o rder to prohibit the RF signal from leaking into th e modulation ports, the following condition m ust be satisfied: 20 m LL  , (36) such that rf m L  is large enough to be considered an open circuit for the RF signal. Equations (35) and (36) can now be used to find the values of m L and m C for optimal operation of the circulator. Finally, the DC bias can be combined with the modulati on signals through a bias tee or a sufficiently large resistance, as B R in Fig. 2(a), since there is no DC current flow in the circuit . Furthermore, for proper bi asing of the varactors, the RF ports are also DC grounded again through a bias tee or a large resistance. B. Detailed Small-Signal Analysis Here, we p resent a detailed anal ysis for the small-signal model of Fig. 2(b) and derive the equations given in section II.B. Applying Kirchhoff’s laws to the n th tank in Fig. 2(b), we get , n L n L vi   , (37) , n L nn n n v iC v R i    , (38) where d dt  , , L n i is the current in the inductor of the n -th tank, n i is the total curren t in the n -th branch (see Fig. 2(b)) and 1, 2 , 3 n  . Taking the deriv ative of (38) and sub stituting it into (37), we get 11 n nn n n n Cv C v v R L i           . (39) Substituting (5) in to (39), we get   0 cos 11 sin mn n mm n nn n CC t v Ct v v R L i                    . (40) The vol tages n v can also be related to the source voltages 1, 2 , 3 s v at the RF ports as follows    11 0 1 3 2 0 2 1 ss v v Zi i v Zi i             , (41)    22 0 2 1 3 0 3 2 ss v v Zi i v Zi i             , (42)     33 0 3 2 1 0 1 3 ss v v Zi i v Zi i             . (43) Substituting (41)-(43) into (40) and using 123 0 vv v   from Kirchhoff’s voltage law, we get t hree differential equations fo r n v , which were co m pactly w ritten in t he matrix form (6) where Fig. 25. L -section m atching network of the k -th m odulation branch. AP1507-1042.R1 15  00 3 c A ZC U C C   , 0 1 ms B UZ H C C R        , U is the unitary m atrix, and 11 0 01 1 10 1 G          , (44) 21 1 12 1 , 11 2 H             (45)                 2 cos cos cos 2 cos 2 cos cos 2 , cos c os 2 cos 2 mm m cm m m mm m tt t Ct t t tt t                             (46)                1 2s i n s i n s i n 2 sin 2 sin s in 2 . sin sin 2 sin 2 sc m mm m mm m mm m CC tt t tt t tt t                                (47) Applying the m atrix transform ation (7),(8) to (6) yields 0 3 s Z A vB v v G v L            , (48) where 1 0 00 0 00 3, 22 22 mm mm jt jt jt jt AT A T C CC Ze C e CC ee C                    (49) 1 0 00 0 00 100 33 3 1, 22 33 3 1 22 m m mm jt jt mm jt jt mm BT B T jZ j ZC e ZC e R jj Z ZC e ZC e R                          (50)     1 00 0 11 33 3 3 . 66 3 11 33 3 3 66 3 j GT G j j j jj                   (51) For   1, 0 , 0 s v  , (48) can b e decomposed into the three equations (9)-(11). The solution of (9) is cm 0 v  . On th e o t h er hand, in order to find the solutions of (10), (11), we apply th e Fourier transform, yielding  22 00 0 0 0 0 1 )) )) )) 3 3( ( ( ) 2 33 1( ( ( ) 2 3 (3 3 ( , 6 mm m m s m ZC Z C V Z jV Z C R Z V V V j L V j                             (52) 22 00 0 0 0 0 1 3 ( ) () () 2 33 1( ) ( ) ( ) 2 3 () ( 3 3 ) () , 6 3 mm mm m s VZ C V Z jV Z CV R Zj Vj V L ZC                               (53) where   V   and   V   are the Fourier transforms of   vt  and   vt  , respectively. Solving (52) and (53) for   V   and   V   leads to (12)-(17). The current flowing in each branch can also be found by transforming (39) t o frequency domain, result ing in         0 1 1 11 , 2 nn jn mn m jn mn m Ij C V Rj L j eV C eV                           (54)       1 0 2 11 , jn nm n mn m m j IC e V jC V Rj L                    (55)       1 0 2 11 . jn nm n mn m m j IC e V jC V Rj L                    (56) Finally, the harmonic S -parameters are calculated using (24) and (25). R EFERENCES [1] J. I. Ch oi, et al. “Achieving single channel, full duplex wireless comm unication,” Proc. of the sixteenth annual int. conf. on Mobile computing and networking , ACM, 2010. [2] M. Jain, et al. “Practical, r eal-time, f ull duplex wireless,” Proc. of the 17th annual int. conf. on Mobile computing and networking , ACM, 2011. [3] D. Bharadia, E. McMilin, and S. Katti, “Full duplex radios,” ACM SIGCOMM Computer Comm. Rev . , vol. 43, no. 4, pp. 375-386, 2013. [4] D. W . Bliss, P . A. Parker , and A. R. Ma rgetts, “Simultaneous tr ansmission and reception f or improved wire less network performa nce,” IEEE/SP 14th W orkshop on S tatistical Signal Processing , IEEE , 2007. [5] D. M. Pozar, Micr owave engineering. John W iley & Sons, 2009. [6] M. Duarte and A. Sabharwal, “Full-duplex wireless comm unication s using off-the-shelf radios: Feasibility and first results,” Confer ence Recor d of the Forty Fourth Asilomar Conference on Signals, Systems and Computers , IEEE , 2010. [7] M. Duarte, C. Dick, and A. Sabharwal, “Experiment-driven characterization of f ull-duplex wireless systems,” IEEE T ra ns. on W ireless Comm. , vol. 1 1, no. 12, pp. 4296-4307, 2012. AP1507-1042.R1 16 [8] A. Goel, B. Analui, and H. Hashemi, “Tunable duplexer with pass ive feed-forward cancellation to im p rove the RX- TX isolation,” IEEE T rans. on Cir cuits and Systems I: Regular Papers , vol. 62, no. 2, pp. 536-544, 2015. [9] J. Zhou, et al. “Integrated wideband self-interference cancellation in the RF domain for FDD and full-duplex wireless,” IEEE Journ al of Solid-S tate Cir cuits , vol. 50, no. 12, pp. 3015-3031, 2015. [10] S. T anaka, N. Shimomura, and K. O htake, “Active circulators: Th e realization of circulators using transistors,” Pr oc. IEEE , vol. 53, no. 3, pp. 260-267, 1965. [11] Y . A yasli, “Field effect transistor circulators,” IEEE T rans. on Magnetics , vol. 25, no. 5, pp. 3243-3247, 1989. [12] G . Carchon and B. Nanwelaers, “Power and noise limitations of a ctive circulators,” IEEE T rans. on Microw . Theory and T echn. , vol. 48, no. 2, pp. 316-319, 2000. [13] C.-H. Chang, Y .-T . Lo, and J.-F . Kiang, “A 30 GHz Active Quasi-Circulator with Current-Reuse T echnique in CMOS T echnolog y, ” IEEE Micr ow . Components Lett. , vol. 20, no. 12, pp. 693-695, 2010. [14] T . Kodera, D. L . Sounas, and C. Caloz, “Artificial Fara day rota tion using a ring metam aterial structure without static magnetic field,” Appl. Phys. Lett. , vol. 99, p. 031 14, July 201 1. [15] T . Kodera, D. L. Sounas, and C. C aloz, “Magnetless Nonreciproca l Metamaterial (MNM) technology: Application to microwave components,” IEEE Trans. Micr ow . Theory and T echn. , vol. 61, no. 3, pp. 1030-1042, Mar . 2013. [16] K. Gallo and G . Assanto, “All-optical diode in a periodically p oled lithium niobate waveguide,” Appl. Phys. Lett. , vol. 79, no. 3, pp. 314-316, July 2001. [17] L. Fan, J. W an g, L. T . V arghese, H. She n, B. Niu, Y . Xuan, A. M . W einer, and Q. Minghao, “An all-silic on passive optical diode,” Science , vol. 335, pp. 447-450, Jan. 2012. [18] S. Qin, Q. Xu, and Y . E. W ang, “Nonreciprocal components with distributedly modulated capacitors,” IEEE T rans. Micr ow . Theory and T echn. , vol. 62, no. 10, pp. 2260- 2272, Oct. 2014. [19] S. Qin and Y . E. W ang, “Broadband parametric circulator with ba lanced monolithic integrated distributedly modulated capacitors (DMC), ” Pr oc. IEEE MTT -S In t. Micr ow . Symp. Dig. , San Francisco, C A, USA, May 2016. [20] R. F leury , D. L. Sounas, C. F . Sieck, M. R. Haberman, and A. A l ù, “Sound isolation and giant linear nonreciprocity in a compact acoustic c irculator ,” Science , vol. 343, pp. 516-519, Jan. 2014. [21] D. L. Sounas, C. Caloz, and A. Alù, “Giant non-reciprocity at t he subwavelength scale using angular m omentum -biased metamater ials ,” Nat. Commun. , vol. 4, p. 2407, Sept. 2013. [22] N. A. Estep, et al. “Magnetic-free non-reciprocity and isolation based on parametr ically modulated coupled-resonator loops,” Natur e Physics , vol. 10, no. 12, pp. 923-927, 2014. [23] N. A. Estep, D. L. Sounas, and A. Alù, “Magnetless Microwave Circulators Based on Spatiotem porally Modulated Rings of Couple d Resonators,” IEEE Trans. Micr ow . Theory T ech n. , vol. 64, no. 2, pp. 502-518, 2016. [24] N. Reiskarimian and H. Krishnaswamy , “ Magnetic-free non-recipro city based on staggered com mutation,” Natur e comm., vol. 7, no. 4, 2016. [25] J. Zhou, N. Reiskarimian, and H. Krishnaswamy , “ Receiver with integrated magnetic-free N-path -filter-based non-reciprocal cir culator and baseband self-interference cancellation for full-duplex wireles s,” IEEE ISSCC Dig. T echn. Papers , pp. 178-180, Jan. 2016. [26] M. M. Biedka, R. Zhu, Q. M. Xu, and Y . E. W ang , “Ultra-Wide B an d Non-reciprocity through Sequentia lly-Switched Delay Lines,” Sci. Rep. , vol. 7, p. 40014, Jan. 2017. [27] J. Kerckhoff, K. Lalumière, B. J. Chapm an, A. Blais, and K. W . Lehnert, “On-Chip Superconducting Microwav e Circulator from Synthetic Rotation,” Physical Review Applied, vol. 4, no. 3, Sep. 2015. [28] F . Lecocq et al., “Nonreciprocal Microwave Signal Processing wi th a Field-Program mable Josephson Amplifier ,” Phys. Rev . Appl. , vol. 7, no. 2, Feb. 2017. [29] K. M. Sliwa et al., “Reconfigurable Josephson Circulator/Direct ional Amplifier ,” Phys. Rev . X , vol. 5, no. 4, Nov . 2015. [30] H. Lira, Z. Y u, S. Fan, and M. Lipson, “ Electrically Driven Non reciprocity Induced by Interband Photonic T ransition on a Silicon Chip,” Phys. Rev . Lett. , vol. 109, no. 3, Jul. 2012. [31] E. Schwartz, “Broadband Mathcing of Resonant Circuits and Circulators,” IEEE T rans. Microw . Theory T echn. , vol. 16, no. 3, pp. 158–165, 1968. [32] K. Buisman, L. C. N. de V reede, L. E. Larson, M. Spirito, A. Ak hnoukh, T . L. M. Scholtes, and L. K. Nanver , “ Distortion-free varactor diode topologies for RF adaptivity ,” Proc. IEEE MTT -S In t. Micr ow . Symp. Dig. , Long Beach, CA, USA, June, 2005. [33] B. Razavi, RF microelectr onics . V ol. 2. New Jersey: Prentice Hall, 1998.