Frequency Independent Framework for Synthesis of Programmable Non-reciprocal Networks

Frequency Ind ependent Fram ework for Synthesi s of Programm able Non-recipro cal Netwo rks Ruochen Lu 1 , John Krol 1 , Liuqing Gao 1 , and Songbin Gong 1 1 University of Illinois at Urbana Champaign Abstract: Passive and linear n onreciprocal net works at micr ow ave frequencies h old great promises in enabling new front-end archit ectures for wireless communication systems. Their nonreciprocity has been achieved by disrupting the time -reversal symmetry using various forms of biasing schemes, but only over a limited frequency range. Here we demonstrate a fram ework for synthesizing theoretically frequency- independent multi-port nonreciprocal n etworks. The framewor k is highly expan dable, and can have an arbitrary nu mber of ports while simultaneously su stainin g balan ced performance and providing unprecedented programmability of non -reciprocity. A 4-port circulator based on such a framework is implemented and tested to produce bro adband nonreciprocal performance from 10 MHz to 9 00 MHz with a temporal switching effort at 23.8 MHz. With the combination of br oad bandwidth, low temporal eff ort, and high programmability, the framework could inspire new ways of implementing multiple input multiple output (MIMO) communication systems for 5G. Introduction: Microwave frequency nonreciprocal networks that bear non -reciprocal resp onse have long been sought after for a wide range of applications, includin g full -duplexing radios 1,2 and quantum computing 3,4,5 . Most commonly utilize d nonreciprocal multiport networks are isolator s and circulators. Conv entionall y, non - reciprocity is obtained by magnetically biasing a ferrite material with in which the electromagnetic wave propagates at different ph ase velocities in the opposite directions 6,7 . In a circular structure based on a material of such properties, constructi ve and destructive interference of the clockwise and counter -clock wise propagating waves can exist at different nodes aro und the circular resonator, thus establishing transmission and isolation through ports situated at these nodes. Motivated by attaining non-reciproc ity for mo re integrated RF and microwave application s, temporal modulations, applied to either reactive 8,9,10 or conductive 11,12 elements, have recently been explore d t o produce a momentum -biasing equiv alent to the magnetic o nes and break the reciprocity. T hese approaches all rely on wave interference or mode splitting cause d by biasing in a r esonant structure. In other words, the bandwidth over which their desirable non -re ciprocal performance can be m aintain ed are sensitive to phase delays between adjacent ports of the network. Although wide -band phase nonreciprocal gyrators 12 can be engineered to enhance the bandwidth of such ne tworks, these type of non-reciprocal devices are inherently frequency dependent. Moreover, demonstrations on temporally modulated nonreciprocal so far are prim arily two port gyrators 11,13 and three po rt circulators 14 . Conceivably, both magnetic and temporal modulation based approaches can be expand ed t o a network with more ports by exploiting established circuit topologies or simply networking several 3-port circulators. However, the possibilities of r econfiguring the non-reciprocity in thes e approaches ar e limited. For instance, only a small subset of circulation sequences through all ports are accessible among all permutations, due to the limitations arising from its topology and application of momentum biasing. We show a framework f or synthesizing a frequency independent and broadly programmable non- reciprocal network with an arbitrary number of ports ( 2N) using switches and an array of dispersionless delay lines. The generalized 2N-po rt framework can also be elegantly reduced to 3 port or 2 port device with more compact size and less sw itched delay lines than the se quentially swi tched delay line s 15 . This concept attains m ulti-port n on-reciprocity by equally multiplexing the input signal o nto N delay lines in the time domain and later aggregating the delayed signals off N delay lines consecutively at the intended port. The timing offset betwe en switches addressing each port results in only one port rec eiv ing signal a t any given time from an excitation port. Unlike the abovementioned momentum biasing approaches , the non-reciprocal performance of our network is only dependent of the t im e delay s, inste ad of phase de lays , and therefore is frequency independent. More impressively, the network has far more programmable states than any alternative reconfigurable non -reciprocity. Such pro grammability of nonreciprocity in a multi-port network will ins pire new applications in m ultiple input multiple out put (MIMO) communication systems. Results: Fig. 1. a. Sch ematic symbol o f a circulator o f 2N p orts and clockwise circulator. b. Concept of 2N -port non- reciprocal n etwork. c. Switch control waveforms app lied to the n etwork for produ cing the nonreciprocity. d. 3-port circulator derived from the 2N framework Fig. 1 shows the 2N -port framework consisting of 2N ports equally situated o n the both sides o f N identical delay lines that each has a time delay of δ . On either side of the delay lines, each port is fanned out to connections with all delay l ines through a sin gle pole sin gle throw (SPST) switch that presen ts open in the off state. Therefore, composition o f such a 2N -port network require N delay lines a nd 2N 2 SPSTs (or 2N SPNTs). The clock signal for controlling each switch is de noted as C(t,m,n) , where t is the time, m is the port number, n is the delay line number. All the clocks have a period of 2 Nδ , and a duty cycle of 1/N . Within the time range [ 0 , 2 N δ ] , t h e control signal can be represented as: ( , , ) = H [ t − ( j − 1 ) δ ] − H [ t − ( j + 1 ) δ ] for j ≠ 0 H [ t ] − H [ t − δ ] + H [ t − ( 2N − 1 ) δ ] − H [ t − 2Nδ ] for j = 0 where H is the Heaviside step function, and j is the r emainder of the modulo operation. j = mod(m + 2n − 2,2N) C(t,m,n) is designed to turn o n only one switch, among the switches connected to Port m, at any give n time so that the signal is sequentially time -m ultiplexed onto the N delay lines. On the other side of d elay lines, Port m+1 is controlled by C(t ,m+1,n), which is designed to be a time delayed version o f C(t,m,n) with a timing offset of δ so that the signal, after traversing N delay lines , will be collected and de -multiplexed into Port m+1. In the re ve rse path, signals fed into Port m+1 , after be ing time multiplexed onto a nd traversing the delay lines, are subsequently rejected by po rt m because the switching control clocks, C(t,m,n), are the time advanced version of C(t,m+1,n). In other words, all switches are turned off as signal arrive s Po rt m from Port m+1. On the other hand, switches on Port m+2 a re synchronized with the ar rival of signals from Par t m+1 to aggregate them fr om the dela y lines. The exce ption exists for Port 2N, to which the fed sign al s will be circulated to Port 1. For a 2 N-port network that consists of infinitely fast an d lossless switches and lossless dispersionless delay lines, and is addressed by ideal square wave control signals, infinite ly large isolation, zero insertion loss , and zero return loss in the c irculati o n are predicted. The perfectly synchronized time -domain m ultiplexing and de-multiplexing on opposite ends of the N delay l ines allow signal incident fr om Port m to exclusively transmit to Port m+1, w hile the energy leakage in the reverse order is completely forbidden. Note that in our generalized framework, N has to be an even number as required by the symmetr y of the network. For producing odd number of ports, a network with an even number of ports can be redu ced to have one less port by leaving one port o pen, which essentially elim inates N SPST s. As an example seen in Fig. 2a and 2b, a 4-port netwo rk is re duc ed to a 3-port circulator that is typically sought after for full - duplex radio applications. Fig. 2. a. Schematic of a 4 -po rt circulator. b. schematic of a 3 -port circulator reduced from a 4 -port circulator. c. Switch control waveforms for producing c lockwise (from 1 to 4) circulation. 4-port Broadband Circulator and Experimental Validation To e xperimentally validate our framework, we cho ose to produce a 4-port circulator based on the 2N p ort framework with a frequency span from DC to 1 GHz . Fig. 2 shows the sch ematic and control waveforms. Fig. 3 . a. Picture of the implemented 4 -port circ ul ato r consisting two switching modules and two microstrip delay lines. b. Block diagram of the constructed system . c. Simulated S -parameters of the 4- port circulator. Fig. 4. Measured S-parameter performance of the 4 -port circulator. As seen in Fig. 3 (a), the prototype is implemented with connectorizied switching and delay line modules. Two delay line modules, with each end connected to a switching mo dule, form the nonreciprocal network. We take the modular approach for experimental validation as it allows more nodes to experimentally observe performance and analyze l oss in t h e 4 -port network. Based on our design for a 4 -port switching module, only 2 single pole single throw series switches that present near open circuit to the input in the off -state are needed. In practice, open -reflective switches are not co mmo nly available with fast s witching ti me. Alternatively, 4 short -reflective s witches, Minicircuit MSW 2-20+, are arranged in a lattice configuration (see supplementary m aterials) to equivalently produce switching of two SPST o pen-reflective switches. MSW 2 -20+ has a fast switching time of 2 nS, which minimizes the insertion loss due to switching. The delay line modules are implemented using Roger Duroid 6010.2LM boards with m eandering microstrip structures to produce a total group de lay of 10.5 ns with slight dispersion that is less than 1 ns . In operation, the switches are controlled by 4 clock signals that have a period of 42 nS , and a frequency of 23.8 MHz. The slightly increased delay is caus ed by the additional e lectrical length in the control boards. The switches o n the same side of the de lay line are complimentarily driven while the switches on the opposite ends of the same delay are driven with a timing o ffset of 1 0.5 nS. The clock signals are generated by two s ync hronized dual-channel Tektronix arbitrary function generators, and fed to the control ports on the switching modules. Advanced Design Sy stem (ADS) is used for sim ulating the 4-port performance. The switches have 2 ns switching time, an on -state resistance of 3Ω, and an off -state resistance of 60 kΩ. The delay lines are represented by their S -parameter performance, w hich is modele d using ADS momentum. To extract the frequency domain response o f the network, a series of time domain simulation s with varying single tone inputs to P ort 1 are performed before Fourier transform is performed to attain scattered power out of other ports at the input frequency. As seen in Fig. 3c, t he simulation sh ows a broadband (up to 0.9 GHz) nonreciprocal performance. An IL of 3 dB at low frequencies is caused by the non -ideal switch properties and the loss in the delay lines. A great isolation over 30 dB is o bserved simultaneously. The performance degrades at higher frequencies due to the additional loss in the delay lines. The m easurement o f 4-port was do ne using a setup shown in Fig. 3 b. The no n-reciprocal network is tested with a 4 -port Keysight P NA-X network analyzer. Calibration is performed with Keysignt 85052D calibration kit to move the measurement reference planes to the connectors on the switch modules. 4 -port S- parameter is subsequently c haracterized with IF bandwidth o f 1 kHz and a measurement power level of −5 dBm. As seen in Fig. 4 , broadband non-reciprocal respons e s are obtained from 10 MHz to 0.9 GHz . A minimum IL of 5.1 dB is obtained at low frequencies . Isolations of 35 dB is measured between the adjacent ports, and 20 dB between the diagonal ports. As the frequency increases to the self -resonance in the delay lines around 0.9 G Hz, the IL and isolation perfo rmance gradually decay s to 7.6 dB and 24 dB, respectively. The measured performance slightly deviates from the simulated results due to the simplification of sw itches in the model and multi-reflections on delay lines caused by impedance mismatch at ports. Discussion  Frequency Independent Performance As discussed earlier, the frequency in dependent performance of nonreciproicity is the outcome of perfect synchronization o f t ime-domain m ultiplexing and delays in the forward path, and complete off- synchronization between them in the backward route. W e recognize that a number of causes in practice can compromise t he frequency independent performance and yield a broadband pe rform ance instead. For instance, the electromagnetic delay lines typically exhibit disp ersion, which causes the synchronization between switc hing and delay to degrade as the operating freque ncy moves off the design center frequency. To reduce size, d elay lines based on slow-wave or meandering structures often have a cut off frequency that also limits the BW of the nonreciprocal network. Ot her types of delay lines with smaller sizes, e.g. acoustic delay lines 16 , usually have passbands over w hich low insertion loss and constant group delay can be m aintained. Nonetheless, with our frequency independent framework a s the basis, the bandwidth over which non -reciprocity is enabled should be o nly limited by the compo nents chosen for implementation. It is worth noting t hat the frequency independent performance is no t dependent on the temporal e ffort applied in the system. Unlike the momentum biasing approaches where the ban dwidth of nonreciprocity is fundamentally limited by the modulation frequency used to pr oduce momentum biasing, the switching frequency in o ur frame work is only se t by the time delay length i mpo sed by the delay lines. Provided with low loss delay lines to render long group delays, the switching frequency can be reduced to a mere fraction of the non -reciprocal bandwidth ( e.g. 23.8 MHz switching frequency for m aintaining a nonreciprocal bandwidth o f 900 M Hz in our case ), consequently giving rise to simpler and lower cost clock generati on, less phase delay in cl ock sig nal fanout, an d minim ized overall temporal effort. One caveat in operating o ur framework lies in the resulting group delays between ports, wh ich is longer than those of ferrite circulators. The refore, such systems might not be a go od fit for timing -sensitive applicatio ns (e.g. radar front ends).  Network expandability without compromising performance and symmetry Expanding a momentum -biased 3-port circulator into a n N-port circulator is a non-trivial task. Simply adding more folds o f symmetry in the structure will n ot pr oduce unilateral circulation. In o ther word s, the excitation at one port will be nonreciprocally received at more than one por t . A t ypical way to attain nonreciprocal net wo rks with more ports using momentum -biased devices is to network 3 port circulat ors in various manners, such as the m ethod reported for creating macroscale topological materials 17 . W ith each added circulator in the network , the number of ports in the network can be max imall y increased by one, t hus suggesting a heavy c ost in component co unts and cl oc k feeds for co nstructing multi -port nonreciprocal networks beyond 3 ports. Additionall y, networking 3 -port circulators often breaks the network structural symmetry and creates unbalanced paths between ports. Consequently, higher insertion loss are expected for paths that require the signal to traverse more in the c omposed multi -p ort network to reach destination port s. For the even-port o peration in our time-multiplexed framework, o ne c an add 2 more ports to the network with each added delay line, which compare s fa vorably against the network expans ion via interconnecting 3 port circulators. For networks with an odd number of ports, the cost o f expansion is the same, except for adding t he last port , which require s a delay line for its ow n. More importantly and more advantageously in our framework , all transmission paths are balanced with the same insertion loss and delay regardless the number of ports . T hus, the 2N-network maintains N fo lds of symmetry in both the structural design and performance.  Programmability o f nonre ciprocity with a rich space of permutations Enabling pro grammable RF circuits has been the holy-grail problem for designing highly adaptive RF systems in the past decade, fo cusing primarily on either passive reciprocal network s, such as filters 18,19,20,21 , antenna tun ers 22 , and phase shifters 23 , or active/nonreciprocal circuits, such as amplifiers 24 . Programmability of passive non-reciprocity has rarely been visited even thoug h the current carrier aggregated communicatio n syste ms can greatly benefit fr om pr ogramm abl e non-reciprocity in front ends 25 . Te mporal modulated no n-rec iprocal systems have recently revived the hope for achieving such programmability without compromising other r elevant performance specifications. Our framework is readily programmable by first re-shuffling the clock waveforms applied to the switches on one side of the delay lines, and then adjusting the clocks on t he ot her side acco rdingly. Thro ugh this practice, any po rt on one side of t he delay lines can be co nfigured to circulate to any port on the other side of the delay lines, thus allowing for a rich space of non-reciprocal states. The accessible states for the 2N-port nonreciprocal network can be studied as S-matrix permutations with the only limitation that circulation between por ts on the same side of the delay lines cannot be established. Therefore, assuming all ports are matched, the component s in the sha ded region s o f the S-matrix, seen in Fig. 5a, are inaccessible for programming . On the other hand, assuming the network is lossles s and S -matrix is unitary, the sub-matrices outlined by the red boxes in Fig. 5a have a single complex component in each row and column. P rovided that the im plementati on is balanced with identi cal switches on b oth sides of th e identical delay lines, these complex components are identical with a magnitude of 1 , and are denoted as α s. Note that t he programming of t he network change s neither the structural no r the performance symmetry. In other words, the pro gram m ing does n ot cha nge the value of α in the S-matrix. Fig. 5. a. Acce ssible and forbidden regio ns of the S -Matrix fo r programing. b. Programmable non-reciprocal states as a function of numb er o f ports. To det ermine the number of programmable non -reciprocal states, we c an first p o pu late the top right sub- matrix, referred as sub-matrix A onward, with allowed permutations, which is ! . With each permutati on of A, we can then exam the allowed permutation s of sub-mat rix B in the lowe r l e ft quarter. Due to non- reciprocity of the netw o rk ( S i j ≠ S ji ), N components are determined as 0 in B for a given permutation of A. Consequently, the n u m b e r o f w a ys t o po p u la t e B for a g iv e n A is given b y : ( × , ) = ! − × ( − 1 ) ! + × ( − 2 ) ! − × ( − 3 ) ! + ⋯ + ( −1 ) × × ( 1! ) + ( −1) × Thus, the number of nonreciprocal states , Ω , for a 2 N-port n e tw o rk is : Ω (2 ) = ( ! ) × P( × , ) As seen in Fig. 5b, t his repr esents an exponential growth of programmable non -reciprocal st ates as the number of ports increases . Method: The loss in the system can be understood with an analytical approach focusing the switching loss, which is defined as the inse rtion loss caused the switching process. T hus, when analyzing switch loss, the delay lines are modelled as l ossless and perfectly m atched tr ansmission lines. Fundamentally, the switch loss is the result of momentarily losing signal during the switching from one delay line to the other. Such a loss is inevitable using switches with small but not zero switch on and off time. The Insert ion lo ss due to switching is determined by how much signal is lost proportionally over time, and thus related to ratio of switch ti me ( t s ) to delay time ( δ ). The switches are rep resented as time -varying re sistances ( R switch ) durin g switching on and o ff pe riods. They linearly change resi stance from an off -state resistance ( R off ) to on-state resistance ( R on ) o ver a switching period ( t s ) upon the application o f control waveforms, which are assumed to be perfect square waves of 50% dut y cycle. In a 2δ period, R switch can be d e sc rib e d as: R ( t ) = ⎩ ⎪ ⎨ ⎪ ⎧ R + ( R − R ) ∙ t t for 0 < ≤ R for t < ≤ 2 δ − R + ( R − R ) ∙ t − 2δ + t t for 2δ − < ≤ 2 δ Consider the upper line in Fig. 2a, the input signal from 0 < t < t s experiences a time v arying transmission coefficient of h(t) when transmitting through the switch controlled by L 1 . Then, t his signal is dela yed by δ , and from δ < t < δ + t s the signal experiences a t ransmission o f h(t - δ) when transmitting through the switch controlled by R 1 . Given that sw it c h i ng ti m e ( t s ) i s s maller than δ , h(t ) c an be described as: ℎ ( t ) = 2 ( ) + 2 for 0 < ≤ 2 δ The transfer function, betwee n P o r t 1 t o 2 as s e e n in F i g . 2 a , is g i v e n a s: ( ) = ( ) ∙ ∙ ( − ) = ( ) Where H(ω) is the Fouri er transform of h(t) . It is noteworthy that when t s >0, the system transfer function has components other than the DC com p one nt. It implies that the non -ideal switching produces signals at frequencies other than the input signal (e.g. the carri er frequency), which is another interpretation o f the switching loss. In additio n , in se rt io n loss is als o in t r oduced by by R o n and R off . Thus, the total insert ion loss ( IL ) between ports can b e d e sc rib e d a s: = − 2 0 l og ( / ) = −20 log ( 0 ) Based on the analytical closed form expression o f switching loss as a function of t s and δ , a 2D contour plot o f switching loss with swit ching time vary ing from 0 to 5 ns, and group delay of delay lines v aryin g from 10 to 50 ns is plotted i n Fig. 6. An R on of 6 Ω and a n R off of 120 kΩ are a ssumed for the switches used in implementation. Fig. 6. Dependence of switch ing loss on switching time and group delay. Data availability All relevant data is available upon r equest. Acknowledgements This work is p artially supported DARPA MTO signal processing at radio frequency program (SPAR) program under grant number HR0011 -17-2-0004. Author contributions R. L., J. K. and S. G. conc eived the ideas for the frequency independent nonreciprocal networks. R. L . J. K. and L. G. developed the theoretical model for predicting the performance. R. 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