Coupling Matrix Representation of Nonreciprocal Filters Based on Time Modulated Resonators
JJOURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 1
Coupling Matrix Representation of NonreciprocalFilters Based on Time Modulated Resonators
Alejandro Alvarez-Melcon,
Senior Member, IEEE,
Xiaohu Wu,
Member, IEEE,
Jiawei Zang,Xiaoguang Liu,
Senior Member, IEEE, and J. Sebastian Gomez-Diaz,
Senior Member, IEEE
Abstract —This paper addresses the analysis and design ofnon-reciprocal filters based on time modulated resonators. Weanalytically show that time modulating a resonator leads to a setof harmonic resonators composed of the unmodulated lumpedelements plus a frequency invariant element that accounts fordifferences in the resonant frequencies. We then demonstrate thatharmonic resonators of different order are coupled through non-reciprocal admittance inverters whereas harmonic resonators ofthe same order couple with the admittance inverter comingfrom the unmodulated filter network. This coupling topologyprovides useful insights to understand and quickly design non-reciprocal filters and permits their characterization using anasynchronously tuned coupled resonators network together withthe coupling matrix formalism. Two designed filters, of ordersthree and four, are experimentally demonstrated using quarterwavelength resonators implemented in microstrip technology andterminated by a varactor on one side. The varactors are biasedusing coplanar waveguides integrated in the ground plane of thedevice. Measured results are found to be in good agreement withnumerical results, validating the proposed theory.
Index Terms —Coupling matrix, microwave filters, non reci-procity, spatio-temporal modulation, time modulated capacitors.
I. I
NTRODUCTION N ON-RECIPROCAL components are of key importancein many electronic systems, such as radar or mobileand satellite communications [1]. Traditionally, such com-ponents have relied on magnetic materials, such as ferrites,under strong biasing fields. Increasingly stringent technologi-cal demands, in constant pursuit of integration, affordability,and miniaturization, have triggered the recent emergence ofmagnetless non-reciprocal approaches to break the Lorentzreciprocity principle [2] and the subsequent development ofdevices such as circulators [3]–[13], isolators [14]–[20], and
Manuscript received Month DD, YYYY; revised Month DD, YYYY;accepted Month DD, YYYY.This work is supported in part by the National Science Foundation withCAREER Grant No. ECCS-1749177 and by the grants PRX18/00092 andTEC2016-75934-C4-4-R of MECD, Spain.A. A. Melcon was on sabbatical in the Department of Electrical and Com-puter Engineering at UC Davis. He is now with the Department of Informationand Communication Technologies, Technical University Cartagena, 30202,Spain. e-mail: [email protected]. Wu is with the Department of Electrical and Computer Engineering,University of California, Davis, USA. e-mail: [email protected], [email protected]. Zang was a visiting student in the Department of Electrical and ComputerEngineering at UC Davis. He is now with the School of Information andElectronics, Beijing Institute of Technology, Beijing 100081, China.J. S. Gomez-Diaz and X. Liu are with the Department of Electricaland Computer Engineering, University of California, Davis, USA. e-mail: { jsgomez, lxgliu } @ucdavis.edu even non-reciprocal leaky-wave antennas [21]–[23] operatingin the absence of magneto-optical effects.In this context, the concept of non-reciprocal filters basedon time-modulated resonators have recently been put for-ward [24]. The operation principle behind this type of filtersrelies on tailoring the non-reciprocal power transfer amongthe RF and intermodulation frequencies to create construc-tive/destructive interferences at the input/output ports. Thefilters were analyzed in [24] through a dedicated spectraldomain method combined with ( ABCD ) parameters, anduseful guidelines on how to optimize the frequency, amplitude,and phase delay of the signals that modulate the resonatorswere given. A practical prototype was also experimentallydemonstrated using varactors and lumped inductors.Here we should remark that the combination of time mod-ulated resonators with sinusoidal modulation signals will en-hance the generation of the two first intermodulation products(the so called +1 and − harmonics [16]). The minimizationof higher order intermodulation products may be interesting,since it will help to keep under control the power conversionbetween harmonics, and to simplify the tailoring processneeded to create constructive/destructive interferences at theinput/output ports.In this paper, we develop a coupling matrix representationof non-reciprocal filters based on time modulated resonators.Starting from the initial unmodulated equivalent circuit, amulti-harmonic equivalent network is rigorously derived, tak-ing into account the nonlinear harmonics (also known asintermodulation products) that are internally excited. By in-troducing the concept of harmonic resonators, the resultingstructure is represented with a simple network based on aspecific coupled resonator topology. It is analytically shownthat resonators of identical harmonic orders are coupled withthe admittance inverters found in the original unmodulatednetwork while resonators of different harmonic orders are cou-pled through non-reciprocal admittance inverters. In addition,analytic formulas are derived to represent the new harmonicresonators with Frequency Invariant Susceptances [25], [26]that accounts for differences in the resonant frequencies. In thisway, all the resonators of the resulting network are expressedin terms of the original unmodulated resonators.It is important to emphasize that the analytic calculationof the non-reciprocal admittance inverters and the frequencyinvariant susceptances for harmonic resonators, together withthe derived coupling topology, permits to analyze and designnon-reciprocal filters using an asynchronously tuned coupledresonator network and the classical coupling matrix formalism a r X i v : . [ ee ss . SP ] M a y OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 2 [26]. Here the term asynchronously tuned is used to refer tocoupling topologies having resonators tuned at different reso-nant frequencies. The formalism permits to easily considerfilters of any order with an arbitrary number of nonlinearharmonics. As detailed below, this approach also sheds lighton the underlying mechanisms that enable non-reciprocalresponses in time-modulated filters. Besides filters operating atidentical input/output frequencies, this technique can also beapplied to analyze devices that exhibit non-reciprocity betweenthe fundamental frequency and desired nonlinear harmonics.After a review of the equivalent network for coupled res-onators filters in Section II, we introduce in Section III the cou-pling matrix formalism for time-modulated filters. Numericalstudies are first presented, including the convergence behaviorof the scattering parameters with the number of harmonics.To demonstrate the usefulness of the proposed approach, inSection IV we present the design of two non-reciprocal filtersof third and fourth orders. The filters are then experimentallydemonstrated in Section V using quarter wavelength resonatorsimplemented in microstrip technology. Coupled microstriplines are terminated with varactors on one side to build timemodulated resonators. A compact structure is obtained byintegrating the feeding network of the modulating signal in thesame board as the filter. This is achieved by using a coplanarwaveguide feeding network in the ground plane of the device.Numerical results obtained with the theory presented in thispaper show good agreement with respect to measurementsobtained from the manufactured prototypes.II. E
QUIVALENT N ETWORK OF C OUPLED R ESONATORS F ILTERS
Let us start from the basic ideal equivalent network ofa lossless in-line filter represented by lumped elements andadmittance inverters as shown in Fig. 1. Fig. 1(a) shows thenormalized lowpass filter prototype with all capacitors normal-ized to ( F) and the source and load impedances normalizedto ( ). For the sake of clarity, but without loss of generality,we consider a network composed of three resonators (networkof order N = 3 ). The ( N + 2 ) coupling matrix can be used tocharacterize this network [26], leading to M = M P M P M M M M M
00 0 M M M P M P . (1)Here we have used the notation ( P ) and ( P ) to refer to thesource ( S ) and load ( L ) terminations. This notation will bemore convenient when investigating the non-reciprocal behav-ior of the network in the next section. Note that in this matrixthe diagonal elements ( M u,u with u = 1 , , · · · , N ) representthe frequency invariant susceptances shown in Fig. 1(a), whilethe off-diagonal elements ( M u,u +1 ) represent the values ofthe admittance inverters of the network. Frequency invariantsusceptances are used in Fig. 1 to account for asynchronouslytuned topologies [26]. Fig. 1. Equivalent circuit of an ideal lossless filter based on lumped elementsand admittance inverters. (a) Normalized lowpass prototype with all elementshaving unitary values. (b) Lowpass prototype scaled to arbitrary capacitancevalues ( C ) and port impedances ( R P , R P ). (c) Bandpass network resultingfrom a standard lowpass to bandpass transformation. This coupling matrix relates the currents and nodal voltagesin the normalized network shown in Fig. 1(a). The Kirchhoff’scurrent law in this network can be written in matrix form as I = (cid:104) G + j ω C + j M (cid:105) · V , (2)where the whole admittance matrix has been expressed as thesum of three simpler matrices. In this expression ( C ) is amatrix containing the capacitors of the network C = C C C
00 0 0 0 0 (3)and ( G ) is the so called conductance matrix, which containsthe port admittances as G = G P G P , (4)and for the network in Fig. 1(a): C = 1 F, G P = 1 /R P =1 Ω − , G P = 1 /R P = 1 Ω − . In this system of equations( I ) represents the current excitation vector and ( V ) containsthe unknown nodal voltages [see Fig. 1(a)], as I = I P , V = V P V V V V P . (5)From this normalized network, a scaled lowpass circuitas shown in Fig. 1(b) can be obtained. Capacitors and portimpedances are scaled to arbitrary values ( C ), and ( R P = OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 3
Fig. 2. Coupling topology of the in-line filter shown in Fig. 1. /G P , R P = 1 /G P ), respectively. Note that during theproduction of a particular filter, the transformation ratios ( C )are calculated with the information of the practical technologythat will be used during the filter implementation. In any case,the response of the scaled network is the same as the originalnetwork if the values of the admittance inverters ( J u,u +1 ) andfrequency invariant susceptances ( B u ) are conveniently scaledas J P = M P (cid:112) G P C, J u,u +1 = M u,u +1 C, (6a) J NP = M NP (cid:112) C G P , B u = M u,u C, (6b)If a standard lowpass to bandpass transformation is appliedto the network of Fig. 1(b), the capacitors are transformed intoresonators, thus obtaining the traditional bandpass networkshown in Fig. 1(c). In this network all resonators are equaland take the values C p = Cω F B , L p = F B ω C = 1 ω C p , (7a) F B = ω c − ω c ω , ω = 2 π f , (7b)where f is the center frequency of the passband and ω c and ω c are the lower and upper angular equi-ripple cut-offfrequencies of the passband, respectively.The network shown in Fig. 1(c) represents a bandpass filterwith the so called in-line coupling topology, as illustrated inFig. 2. In this figure, white circles represent the resonators ofthe structure ( r u ), while dashed circles represent the terminalports with reference impedances ( R P = 1 /G P , R P =1 /G P ). Also, solid lines connecting the circles representthe ideal admittance inverters of the network ( J P , J u,u +1 , J NP ).If Kirchhoff’s current law is applied to the nodes of thebandpass network shown in Fig. 1(c), the following linearsystem of equations is obtained I P = G P jJ P jJ P Y (1) p jJ jJ Y (2) p jJ
00 0 jJ Y (3) p jJ P jJ P G P · V P V V V V P (8)where Y ( u ) p is the admittance of the resonators, calculated as Y ( u ) p = j ω C p + 1 j ω L p + j B u . (9)Similarly as before, it is now convenient to express the matrixof the system as the sum of three matrices as I = (cid:104) G + Y inv + Y p (cid:105) · V . (10) The first matrix is again the conductance matrix defined in (4).The second matrix is symmetric and contains the values of theadmittance inverters of the network Y inv = j J P J P J J J
00 0 J J P J P . (11)Finally, the third matrix represents the admittances of theresonators Y p = Y (1) p Y (2) p Y (3) p
00 0 0 0 0 . (12)Note that the size of all these matrices is the same as thatof the regular coupling matrix with ports, namely ( N + 2) × ( N +2) . Also, we want to remark that the admittance invertersare located in the off diagonal elements of (11), and that theinformation of the resonators appears in the diagonal entriesof (12). We stress that all matrices involved in the formulationare symmetric, therefore assuring that the considered networkis completely reciprocal.III. N ETWORK WITH T IME M ODULATED R ESONATORS
Applying time-varying signals to modulate the capacitors ofthe bandpass network shown in Fig. 1 makes the system non-linear [27], [28]. In this work, we will consider that the valuesof the capacitors are modulated in time with the followingsinusoidal variation C ( u ) p ( t ) = C p (cid:104) m cos( ω m t + ϕ u ) (cid:105) , (13)where ω m is the angular frequency of the modulating signal, ϕ u is the initial phase, and ∆ m is the modulation index. Eventhough we will use the same modulation frequency and mod-ulation index to modulate all capacitors, their initial phasesmay be different along the network, i.e., ϕ u = ( u −
1) ∆ ϕ with u = 1 , , · · · , N . It will be shown later in this paperthat this phase difference is the key mechanism that enablesnon-reciprocal responses [24].In this scenario, a number of nonlinear harmonics (orintermodulation products) N har are generated in each res-onator, resulting into the equivalent network shown in Fig. 3.These nonlinear harmonics are coupled by the time modulatedcapacitors. For simplicity, the figure only shows N har = 3 harmonics (i.e., k = · · · , − , , , · · · with k denoting theorder of a given nonlinear harmonic).The application of Kirchhoff’s current law on the networkshown in Fig. 3 leads to a linear system with a structure verysimilar to the previous one given in (10). However, each entryin the matrix system becomes now a submatrix of size N har × OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 4
Fig. 3. Equivalent circuit of the ideal filter shown in Fig. 1(c) when a timedomain signal is used to modulate the value of the capacitors. Color boxesrepresent the admittance coupling matrix between generated harmonics. N har due to the generated nonlinear harmonics. In this way,the vector containing the nodal voltages becomes V = V P V V V V P , V P = V P , − V P , − V P , V P , +1 V P , +2 , (14a) V u = V u, − V u, − V u, V u, +1 V u, +2 , V P = V P , − V P , − V P , V P , +1 V P , +2 , (14b)where the number of harmonics considered is five ( N har = 5 , k = · · · , − , − , , , , · · · ) and the total number of un-knowns in the system of linear equations becomes ( N +2) N har . We recall that in our notation ( u ) is an integersweeping the physical resonators ( u = 1 , , · · · N ). Then ( V u )of (14b) are simply the to N + 1 entries of ( V ) shownin (14a). Then, following the same strategy as before, theconductance matrix is written as G = G P G P , (15)where ( ) denotes the zero matrix. The other sub-matricesare diagonal and represent the loads to the new generatedharmonics as G P = G P U and G P = G P U , with U beingthe identity matrix. In addition, the matrix of the admittanceinverters can now be written as Y inv = j J P J P J J J
00 0 J J P J P , (16)where the submatrices J u,u +1 = J u,u +1 U are also diagonaland represent the couplings of same order harmonics betweenthe different resonators. Here we should remark that with the equivalent network employed, which uses ideal frequencyindependent inverters, the couplings of same order harmonicsbetween different resonators are all identically affected bythe original inverters. This is a narrowband approximation,usually introduced in the theory of coupling matrices [26].In real implementations, harmonics will be affected by theinverters in a slightly different way, due to their intrinsicdispersive nature. These dispersive effects maybe importantfor wideband responses, and special techniques may be neededto preserve accuracy [29], [30]. However, for narrowbandresponses (fractional bandwidths typically less than 10%), thenarrowband approximation usually gives good results [26].Finally, the matrix that contains the resonator admittancesbecomes Y p = Y (1) p Y (2) p Y (3) p
00 0 0 0 0 . (17)Each admittance submatrix represents the coupling among thedifferent nonlinear harmonics generated in a resonator witha time-modulated capacitor. In this paper we have used thetheory reported in [31], [32] to model this non-linear behavior.Note that this theory is based on considering ideal capacitors.Applying the theory reported in [31], [32] permits to expresseach of these submatrices as Y ( u ) p = Y b + j ω n N ( u ) c + j B u U , (18)where ω n is a diagonal matrix containing the angular frequen-cies of the nonlinear harmonics (spectral matrix), namely ω n = ω − ω m ω − ω m ω ω + ω m
00 0 0 0 ω + 2 ω m . (19)The matrix Y b includes the presence of the inductors in themodulated resonators and can be expressed as Y b = 1 j L p ω n − . (20)Finally, N ( u ) c models how the nonlinear harmonics are exciteddue to the modulated capacitors and it can be written as N ( u ) c = C p D ( u ) E ( u ) C p D ( u ) E ( u ) C p D ( u )
00 0 E ( u ) C p D ( u ) E ( u ) C p . (21)The new elements of this matrix depend on the modulationindex and on the phases of the modulating signal as D ( u ) = ∆ m C p e − j ϕ u , E ( u ) = ∆ m C p e + j ϕ u . (22)By doing straightforward operations with these matrices, thefinal admittance submatrix in (18) can be written as shown OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 5 Y ( u ) p = Y ( − r + jB u jD ( u ) ( ω − ω m ) 0 0 0 jE ( u ) ( ω − ω m ) Y ( − r + jB u jD ( u ) ( ω − ω m ) 0 00 jE ( u ) ω Y (0) r + jB u jD ( u ) ω
00 0 jE ( u ) ( ω + ω m ) Y (+1) r + jB u jD ( u ) ( ω + ω m )0 0 0 jE ( u ) ( ω + 2 ω m ) Y (+2) r + jB u (23) in (23) (top of the page). In this last expression, we haveemployed the following auxiliary admittance Y ( k ) r = j C p (cid:16) ω + k ω m (cid:17) + 1 j L p (cid:16) ω + k ω m (cid:17) . (24)The form of the matrix shown in (23) admits an interestinginterpretation of the non-linear phenomenon in terms of cou-pled network resonators. Following the coupling matrix for-malism, the elements in the diagonal represent new resonatorsdue to the generated nonlinear harmonics (we will call themharmonic resonators). Therefore, each physically modulatedresonator gives rise to N har new harmonic resonators yieldingto a network of order N har N . These resonators have differentresonant frequencies, transforming the original structure intoan asynchronously tuned coupled resonators network.The resonant frequencies of the new harmonic resonatorscan be obtained by equating the diagonal elements of thematrix shown in (23) to zero. However, following the couplingmatrix formalism, it would be convenient to formulate allresonators to be equal, with additional frequency invariantsusceptances to account for differences in the resonant fre-quencies. This can be accomplished by first writing (24) as Y ( k ) r = j ω C p + j C p k ω m + 1 j ω L p (cid:16) k ω m /ω (cid:17) , (25)and then applying the following Taylor expansion
11 + x ≈ − x + · · · , x < (26)to the third term to obtain Y ( k ) r ≈ j ω C p + 1 j ω L p + j (cid:32) C p k ω m + k ω m ω L p (cid:33) . (27)Note that this Taylor expansion can be used in this contextsince, in general, we will assume: ω m << ω . This assumptionis again related to the narrowband approximation assumedthroughout the paper, and to the fact that to achieve goodpower conversion between non-linear harmonics, the modu-lation frequency should lay within the passband of the filter[24].The comparison of this expression with (9) shows thatthe harmonic resonators can be made all equal to the staticresonators in the unmodulated network. The differences inresonant frequencies can be modeled with additional frequencyinvariant susceptances, defined as ˆ B k = C p k ω m + k ω m ω L p , (28)where, in order to make the frequency invariant susceptancestruly independent on frequency, the center frequency of the passband ω has been used in the last definition. The ap-proximation will remain valid for narrowband filters. Thesefrequency invariant susceptances can also be formulated interms of the initial lowpass capacitors as ˆ B k = 2 k ω m Cω F B . (29)It can be observed that the frequency invariant susceptancesassociated to harmonic resonators depend on the order of thenonlinear harmonic itself k , on the modulation frequency ω m and on the passband bandwidth. This expression is also veryuseful, since it will directly translate into the diagonal elementsof the coupling matrix for the non-reciprocal filter by settingthe lowpass capacitor to unity, i.e., C = 1 .It is illustrative to compare the structure of the matricesshown in (8) and in (23). Specifically, the off diagonal el-ements of the matrix (23) indicate that the new harmonicresonators are coupled following an in-line coupling topologyamong them. However, it can be observed that the matrix is notsymmetric. This indicates that these harmonic resonators arecoupled through non-reciprocal admittance inverters. Follow-ing this idea, we define a non-reciprocal admittance inverter torepresent the coupling between two different harmonics k − and k , belonging to a specific physical resonator u , as (cid:40) J ( k,k − u = D ( u ) (cid:2) ω + k ω m (cid:3) , Low to up. J ( k − ,k ) u = E ( u ) (cid:2) ω + ( k − ω m (cid:3) , Up to low. (30)so a coupling from a lower order harmonic to an upper orderharmonic will use the top formula of (30), while a couplingfrom an upper order harmonic to a lower order harmonic willinvolve the bottom formula. An explicit expression for thisnon-reciprocal inverter can be obtained in the lowpass domainas J ( k,k − u = ∆ m Cω F B e − jϕ u (cid:2) ω + k ω m (cid:3) J ( k − ,k ) u = ∆ m Cω F B e + jϕ u (cid:2) ω + ( k − ω m (cid:3) (31)where the center angular frequency of the passband has beenused to define frequency invariant inverters.Here we remark that these admittance inverters are differentfrom those shown in (16). Admittance inverters in (16) comefrom the unmodulated network, and they couple same orderharmonics between different physical resonators. On the con-trary, these new admittance inverters play an important role inthe non-linear process occurring within each time modulatedresonator. As a consequence, the new admittance invertersin (31) model the couplings between the different harmonics,generated, due to the non-linear process, within the samephysical resonator. OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 6
Fig. 4. Coupling topology of the in-line filter shown in Fig. 1(c) when thecapacitors of the resonators are modulated with a time varying signal.
These last expressions indicate that the coupling betweenadjacent harmonic resonators belonging to a specific physicalmodulated capacitor can be controlled with the modulationfrequency ω m , modulation index ∆ m , and initial phase of themodulation signal ϕ u . Moreover, the degree of non-reciprocityof the coupling depends on both, the initial phase of the modu-lating signal and the modulation frequency. These expressionsrepresent the values along the off-diagonal elements of thecoupling matrix for the final non-reciprocal filter, once thevalue of the lowpass capacitor is set to unity ( C = 1 ).The analysis presented above permits an insightful interpre-tation of non-reciprocal filters in terms of an asynchronouslytuned coupled resonators network. As already indicated, the or-der of the equivalent network is N N har . Its coupling topologyis further shown in Fig. 4. In this figure, harmonic resonatorsare identified with white circles as r ( k ) u . These harmonicresonators are defined with the same inductors ( L p ), capacitors( C p ) and frequency invariant susceptances ( B u ) as the orig-inal static resonators. However, the new frequency invariantsusceptances ( ˆ B k ) given in (29) must be added to correctlyrepresent their resonant frequencies. Furthermore, solid linesrepresent regular inverters modeling the couplings of sameorder harmonics between different physical resonators, as de-fined in (6). Finally, lines terminated in arrows represent non-reciprocal inverters modeling the couplings between differentorder harmonic resonators belonging to the same physicalresonator, as defined in (30) or (31). It is also interestingto note that this coupled resonator network can easily becharacterized with the traditional coupling matrix formalism[26], using the results obtained in this Section. In this case thesize of the coupling matrix is ( N + 2) N har × ( N + 2) N har .It is interesting to note that according to the admittance in-verters expressed in (31), the coupling increases with the orderof the harmonics. This implies that higher order harmonics willundergo very high couplings, which is a somewhat counter-intuitive scenario. The situation, however, can be explainedwith the coupling topology shown in Fig. 4. This topologyexplicitly states that couplings to higher harmonics can onlyoccur from contiguous harmonics. Therefore, the power cannotbe coupled from the fundamental frequency to harmonics of (a) (b)(c) (d)Fig. 5. Different paths that can be followed by electromagnetic waves totravel from port to port (top row) and from port to port (bottom row)in the coupling topology described in Fig. 4. very high orders, with a very strong coupling.The topology shown in Fig. 4 explicitly shows that the non-reciprocal response in time-modulated filters originates dueto the non-reciprocal coupling [see (31)] between adjacentnonlinear harmonics that appear in time-modulated resonators.Following this scheme, the underlying non-reciprocal mecha-nism can be intuitively understood as follows. Electromagneticwaves propagating from port can reach port and keep thesame oscillation frequency by (i) going through the admittanceinverters that link the different resonators at the fundamentalfrequency, as in regular in-line filters (see Fig. 2 and Fig. 5a);and (ii) going through an ideally infinite number of routes(assuming an infinite number of nonlinear harmonics) thatappear in the topology due to the presence of harmonicresonators. One specific example of these routes, illustrated inFig. 5b, involves the harmonic admittance inverters J ( − , , J ( − , − , J ( − , − , and J (0 , − that impart a total phase of + ϕ + ϕ − ϕ to the waves propagating therein. The outputat port is then conformed by the interference of the wavescoming from all possible routes. Let us now consider thedual case, i.e., waves coming from port and propagatingtowards port . As in our previous analysis, propagating wavescan follow the path of common in-line filters (see Fig. 5c)plus potentially any of the ideally infinite routes enabled byharmonic resonators. The former leads to reciprocal contri-butions whereas any of the paths that encompasses nonlinearharmonics introduces non-reciprocity due to the non-reciprocalresponse of the impedance inverters. For instance, Fig. 5dshows the route previously analyzed but considering now theopposite propagation direction of the waves. This specific pathinvolves the same harmonic impedance inverters as before, buttraversed in the opposite direction, thus providing a total phaseof − ϕ − ϕ + 2 ϕ to the waves (negative with respect to theprevious scenario). For instance, assuming ∆ ϕ = 45 ◦ , the totalphase difference between forward and backward paths in this OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 7 example is of ◦ . It is thus evident that an adequate controlof the phase imparted by each time-modulated resonator is keyto control the response of this type of filters. At port , wavescoming from all routes interfere to construct the output signal.Strong non-reciprocity at the same frequency arises due to thedifferent wave interference that appears in ports and .The design of time-modulated non-reciprocal filters can becarried out following the guidelines shown in [24]. In suchdesign, the goal is to optimize the modulation frequency andindex as well as the initial phase of the modulation signalapplied to each resonator to (i) independently manipulatethe interference of all waves that merge at ports and to boost non-reciprocity; (ii) maximize the energy coupledto nonlinear harmonics; and (iii) ensure that most energyis transferred back to the operation frequency at the deviceports to minimize loss. It is important to remark that it isrequired to modulate at least two physical resonators to enablenon-reciprocal responses [24]. If one modulates just a singleresonator, the incoming energy will simply be distributedamong various nonlinear harmonics that will then propagatethrough the network. Finally, note that we have focused onnon-reciprocal responses at the same frequency. It is indeedpossible to design devices based on time-modulated resonatorsthat exhibit non-reciprocal responses between the fundamentalfrequency and any desired nonlinear harmonic. These deviceswill be governed by the topology shown in Fig. 4 and willfollow the theory developed here.IV. N UMERICAL R ESULTS
Using the coupling matrix formalism derived above, asoftware tool for the analysis of non-reciprocal in-line filtershas been developed. In this Section, we will investigate theconvergence of the numerical algorithm as a function of thenumber of harmonics N har included in the calculations.The first example is a filter of order three whose unmodu-lated response has equal ripple return losses of RL = 13 dB.The filter coupling matrix yields M = . . . . . . . . . (32)This coupling matrix gives the response of the normalizedlowpass prototype. The bandpass response is adjusted to havea bandwidth of 47 MHz, with a center frequency of f =975 MHz ( F B = 4 . ). By using the procedure shown in[24], the modulation parameters were optimized, leading tothe following values: f m = 22 . MHz, ∆ m = 0 . , and ∆ ϕ = 35 ◦ .Here we should remark that the design of this filter is notyet completely determined by synthesis techniques. Rather,the coupling matrix shown in (32) gives the initial responseof the unmodulated filter. Once this response is established,the parameters of the modulation signals are optimized toobtain the desired non-reciprocal response, using the procedurereported in [24]. Frequency (GHz) -40-30-20-100 M agn i t ude o f S ( d B ) N har =3N har =5N har =7 (a) Frequency (GHz) -40-30-20-100 M agn i t ude o f S ( d B ) N har =3N har =5N har =7 (b) Frequency (GHz) -40-30-20-100 M agn i t ude o f S ( d B ) N har =3N har =5N har =7 (c) Frequency (GHz) -40-30-20-100 M agn i t ude o f S ( d B ) N har =3N har =5N har =7 (d)Fig. 6. Scattering parameters of the third order non-reciprocal filter designedin Section IV. Results are computed with the coupling matrix approachintroduced in this work using an increasing number of harmonics in thenumerical method. In general, the design of this filter fully from synthesistechniques is very complex, and will involve (i) the calculationof suitable reflection and transmission polynomials to properlyrepresent the desired (non-reciprocal) transfer functions, (ii)the extraction from these polynomials of a suitable couplingmatrix and (iii) the transformation of the obtained couplingmatrix into a form that represents the coupling topology shownin Fig. 4.Fig. 6 shows the scattering parameters at the fundamentalfrequency, obtained for this filter with increasing numberof harmonics N har = 3 , , . Here numerical results wereobtained from the responses of the coupling matrices forthe time modulated network. The coupling matrix is easilycalculated starting with the coupling matrix given in (32) forthe unmodulated network, and with the selected parametersfor the modulation signal ( f m , ∆ m and ∆ ϕ ). Then, usingthe coupling topology shown in Fig. 4, the coupling matrixentries for the time modulated network are computed with (29)and (31) (with C = 1 ). It can be observed that the resultsare in general very stable, showing only small differences asthe number of harmonics is increased. Note that the algorithmconverges using just N har = 5 harmonics and increasing fur-ther the number of harmonics leads to negligible changes in thesimulated response. Results show that the filter has a passbandwhich is quite flat in the forward direction with a bandwidth of MHz measured at the return loss level of dB. It shouldbe stressed that, even though the network is non-reciprocal it issymmetric and thus return losses from both ports are identical.Insertion losses within the passband in the forward directionare . dB. Since the network is lossless, these losses are infact due to power that is converted to nonlinear harmonicsand is not converted back to the fundamental frequency. Avery strong non-reciprocity is obtained at the center of the OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 8 passband, being the insertion loss of about dB. Overall,the insertion losses in the backward direction are greater that dB within the whole useful bandwidth. We observed inthis case that fairly good isolation can be obtained at thecenter of the passband. However, the isolation deterioratesat the edges of the useful bandwidth. As explained in theprevious section, the non-reciprocity is obtained by provokingenergy conversion from the fundamental frequency to thegenerated non-linear harmonics. Although these conversioneffects are non-reciprocal in magnitude and phase, the mainmechanism that allows to obtain high non-reciprocity is thedifference in phase between the forward and backward paths.Therefore, high isolation is obtained by adjusting the phasesamong the resonators to produce phase cancellation effects inthe backward direction. With a small number of resonators(three in this example), these cancellation effects can bemade efficient in a narrow bandwidth. Moreover, as it willbe discussed in our next example, there is a trade-off betweenthe isolation level and the bandwidth where this isolation isachieved. In general, larger isolation values can be obtainedbut only over a narrower bandwidth.If we define the directivity between the forward and back-ward directions as D = | S | | S | , (33)then a directivity of D = 14 . dB is obtained at centerfrequency. Moreover, the directivity within the useful passbandis always better than D = 5 . dB.To demonstrate the convergence of the algorithm when theorder of the network is increased, we have also designed afourth order non-reciprocal filter. For this second example thereturn losses of the unmodulated filter are RL = 18 . dB,leading to the following coupling matrix M = .
997 0 0 0 00 .
997 0 0 .
873 0 0 00 0 .
873 0 0 .
68 0 00 0 0 .
68 0 0 .
873 00 0 0 0 .
873 0 0 . .
997 0 . (34)This time the bandpass response is adjusted to have a band-width of 58 MHz at a center frequency f = 890 MHz, givena fractional bandwidth of F B = 6 . . After optimization, theparameters of the modulated capacitors are f m = 19 MHz, ∆ m = 0 . , and ∆ ϕ = 48 ◦ .Fig. 7 shows the simulated scattering parameters withincreasing number of nonlinear harmonics N har = 3 , , .It is evident that the response is inaccurate if only threeharmonics are included in the calculations. After increasingfurther the number of harmonics, the differences among thedifferent simulations reduce considerably, especially for thereflection characteristic and the forward transmission coeffi-cient. We have verified that including additional harmonics inthe simulations leads to negligible variations in the simulatedresponse, which indicates that good convergence is obtainedwith nine harmonics. As expected, this study shows that more Frequency (GHz) -50-40-30-20-100 M agn i t ude o f S ( d B ) N har =3N har =7N har =9 (a) Frequency (GHz) -50-40-30-20-100 M agn i t ude o f S ( d B ) N har =3N har =7N har =9 (b) Frequency (GHz) -50-40-30-20-100 M agn i t ude o f S ( d B ) N har =3N har =7N har =9 (c) Frequency (GHz) -50-40-30-20-100 M agn i t ude o f S ( d B ) N har =3N har =7N har =9 (d)Fig. 7. Scattering parameters of the forth order non-reciprocal filter designedin Section IV. Results are computed with the coupling matrix approachintroduced in this work using an increasing number of harmonics in thenumerical method. harmonics needs to be used in the numerical simulations whenthe order of the network increases.Moreover, it has been previously shown [16], that in thistype of modulated resonators only the two first higher orderharmonics are important in the non-linear process. Conse-quently, the minimum number of harmonics that need to beconsidered in the numerical simulations should grow, with thenumber of resonators in the network, according to the rule: N har = 2 ( N −
1) + 1 . Note that the convergence resultspresented for the third and fourth order filters, shown in Fig. 6and Fig. 7, are in agreement with this rule.The filter shows an almost flat response for the transmissioncoefficient in the forward direction, having a bandwidth of MHz measured at a return loss of RL = 12 dB. Theinsertion losses in the forward direction are smaller than IL = 3 . dB within the useful passband. Again, theselosses correspond to power converted from the fundamentalfrequency into nonlinear harmonics that is not convertedback into the fundamental frequency. The response of thefilter shows a strong non-reciprocal behavior in the backwarddirection. Around the center frequency, the directivity is betterthan D = 13 . dB in a bandwidth of MHz. In the wholeuseful passband, the directivity is shown to be better than D = 9 dB.At this point it is interesting to observe that the optimummodulation frequency ( f m = 19 MHz) is slightly smallerthan the bandwidth of the filter. This condition assures thatthe two first intermodulation products can be strongly excited,while the generation of higher order intermodulation productsare minimized. Also, we emphasize that the response of thefilter was optimized to achieve a good trade-off between theisolation level, and the bandwidth where it is achieved. Otheroptimization criteria are possible, for instance by increasing
OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 9
Frequency (GHz) -50-40-30-20-100 S - pa r a m e t e r s ( d B ) S RigorousS CMS ADSS RigorousS CMS ADS (a)
Frequency (GHz) -50-40-30-20-100 S - pa r a m e t e r s ( d B ) S RigorousS CMS ADSS RigorousS CMS ADS (b)Fig. 8. Scattering parameters of the fourth order non-reciprocal filter designedin Section IV computed using the commercial tool ADS (cross symbols), thecoupling matrix approach introduced in this work with approximations (CM;dashed line) and without approximations (solid lines). Approximations involvethe use of (27)-(29) and (31). In both calculations the number of harmonicshas been fixed to N har = 9 . further the isolation level, at the expense of reducing thebandwidth where this isolation is achieved. For instance wehave verified that by decreasing the frequency of the mod-ulation signal to f m = 18 MHz, the directivity increasesto D = 33 . dB, although in a narrow bandwidth of only . MHz. In any case, this example shows that the proposedsystem offers high flexibility in the characteristics that can beachieved, that could be adapted to many different scenarios.As validation of the theory presented in this paper, weemploy this last filter design to compare our results withthose obtained with the commercial tool ADS [33]. Herewe remark that the ADS results were obtained using idealbuilt-in models to implement the time modulated capacitorsthrough (13), combined with the large signal scattering pa-rameters analysis module. In addition, we also check whatis the impact of the approximations introduced in order toformulate the frequency independent elements required by thecoupling matrix formalism. Essentially, the approximationsinvolve (i) the representation of the harmonic resonators withthe frequency invariant susceptances of (29), instead of usingthe rigorous admittances given in (24); and (ii) the use offrequency independent admittance inverters of (31), instead ofthe rigorous expressions shown in (30). Fig. 8 compares thefilter response using these two different approaches, and usingthe commercial tool ADS. It can be observed that our theory(Rigorous) agrees perfectly with the results obtained withADS. Small differences can be observed between these tworesults (ADS, Rigorous), and the results obtained introducingthe approximations (CM). This indicates that the impact ofthe approximations introduced is indeed small, especially fornarrowband filters.V. P
RACTICAL R EALIZATION
In this Section we present the fabrication and measure-ment of the two previously designed non-reciprocal filters,implemented in microstrip technology. Fig. 9 and Fig. 10show the details of the filters together with pictures of themanufactured prototypes. It can be observed that the topmetalization layer contains the input/output RF feeding lines (a) (b)(c) (d)Fig. 9. Geometry of the third order filter designed in microstrip technology.(a) Detail of the top metalization layer. (b) Detail of the bottom metalizationlayer. Panels (c)-(d) show a picture of the top and bottom metalization layersof the fabricated prototype, respectively.TABLE ID
IMENSIONS ( IN MILLIMETERS ) OF THE FABRICATED RD - ORDER FILTER ( SEE F IG . 9). W W W S S S h l l
50 3.44 3 2.66 0.36 0.22 11.3 153 72 l l l l l l l Φ Φ IMENSIONS ( IN MILLIMETERS ) OF THE FABRICATED TH - ORDER FILTER ( SEE F IG . 10). W S S S h l l l
70 4.56 2.21 0.21 9.3 160 73 27.8 l l l l l l l l and that the resonators are realized using quarter wavelengthtransmission lines terminated on one side with a via-holeconnected to a varactor. On the bottom metalization layer,the ground plane of the microstrip line is modified to feed thevarious varactors (from Skyworks, model SMV1234) with thecorresponding modulating signals using coplanar waveguides.In the figures we also show the positions where the varactorsare soldered in the board. Note that a choke lumped inductorof value nH is incorporated to increase the isolationbetween the signals oscillating at f and f m . It should beemphasized that this implementation enforces that the RF andmodulating signals are supported on different planes of thesubstrates which significantly increases the isolation betweenthem ( > dB). The substrate material used for the fabricationis Rogers RT/duroid 6035 HTC with a relative dielectricconstant (cid:15) r = 3 . and a thickness of . mm. The finaldimensions of the fabricated prototypes are collected in Table Iand Table II for the third and fourth order filters, respectively.Fig. 11a shows the measured results for the third order filterin the absence of any modulation and compares them withthe simulated response using the coupling matrix formalism. OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 10 (a) (b)(c) (d)Fig. 10. Geometry of the fourth order filter designed in microstrip technology.(a) Detail of the top metalization layer. (b) Detail of the bottom metallizationlayer. Panels (c)-(d) show a picture of the top and bottom metalization layersof the fabricated prototype, respectively.
Here we should remark that the simulated responses of thefilters are all obtained with the theoretical analysis presentedin Section III. In addition, it can be observed in the measuredresponse some deviations with respect to the response ofthe designed prototype shown in Fig. 6. The differencesare mainly due to the insertion losses within the passband,which are around IL = 2 . dB, and to some parasitic crosscouplings that were not taken into account during the initialdesign. These two factors have been included in the simulatedresponses obtained with the coupling matrix formalism derivedin this work, shown in Fig. 11. Losses in the resonators aremodeled with an additional resistor connected in parallel. Theresponse show in Fig. 11a is used to extract the unloadedquality factors of the resonators, giving Q U = 114 . Thisunloaded quality factor is small, but it is not uncommon ofplanar technology [34], and especially when using microstripline printed resonators. In addition, we have found that thedrop of selectivity in the lower side of the passband is mainlydue to a non negligible cross coupling between the ports andthe second resonator, giving normalized coupling factors or M P = M P = 0 . . Although of much weaker value,there is also a small parasitic coupling between the first andthird resonator, which is modeled with a normalized couplingfactor of M = 0 . . It can be observed that the agreementbetween measured and simulated results are reasonably good,once losses and parasitic couplings are included in the derivedcoupling matrix formalism.Fig. 11b presents the measured versus simulated resultswhen the modulating signal is applied to the varactors and thefilter is excited from the first port. It can be observed that thefilter behaves as in the unmodulated case, with increased lossesof around IL = 4 . dB that account for both dissipation effectsand the power converted into nonlinear harmonics. The usefulbandwidth measured at a return loss level of RL = 11 dBis MHz. Fig. 11c shows the response of the prototypewhen it is excited from the second port. The filtering responseis suppressed and instead the device behaves as an isolatorthat attenuates all incoming power. Maximum non-reciprocity
TABLE IIIB
ASIC ELECTRICAL PERFORMANCES OBTAINED FOR THE TWOMANUFACTURED FILTERS . IL (dB) RL (dB) D (dB) F B (%)Third order 4.5 11 13.8 4.6Fourth order 4.4 11 13.6 6.4 is achieved at the center of the passband with a directivityof D = 13 . dB. It can be observed that when losses andparasitic cross couplings are included in the coupling matrixmodel, a good agreement is maintained between measured dataand numerical simulations.Measurements corresponding to the fourth order filter areshown in Fig. 12. Fig. 12a plots the response of the filterbefore introducing the modulating signal and compares itwith respect to the response of the ideal circuit. Again thebandwidth and the ripple level obtained within the passbandare very similar. Measured results exhibit a perfectly constantequi-ripple response, since the resonant frequencies of theresonators are slightly tuned with constant voltages appliedto the varactors. The insertion losses due to dissipation effectsin the resonators and in the varactors are slightly larger than inthe previous filter, obtaining a minimum level of IL = 3 . dBthat slowly increases towards the end of the passband. Theinsertion losses measured in the unmodulated case (Fig. 12a)were used again to extract the unloaded quality factor ofthe resonators, obtaining essentially the same value as in theprevious example. This is something to be expected, since thesame resonators as before were used in this second prototype,and the same technology was used for manufacturing. Inany case, this also shows high repeatability of the employedmanufacturing process.Measured results again show a drop in selectivity as com-pared to the designed response of Fig. 7, especially in thelower side of the passband. Once more we found that thisis due to parasitic cross couplings not taken into accountduring the initial design process. In the comparison shownin Fig. 12, we can observe good agreement between measuredand simulated responses when losses and parasitic couplingsare included in the derived model. Again, we found that thestrongest parasitic couplings occur between the ports and theclosest non contiguous resonators: M P = M P = 0 . and M P = M P = 0 . . However, non negligible parasiticcouplings have also been found between internal resonators: M = M = 0 . and M = 0 . .Fig. 12b presents the measured results obtained from themanufactured prototype when the modulating signal is appliedto the varactors and the filter is excited from the first port.The fabricated prototype behaves as a filter with a usefulbandwidth of MHz measured at a return loss level of RL = 11 dB. With respect to the unmodulated case, theinsertion losses in the forward direction have increased to IL = 4 . dB. As in the previous case, the extra lossesare due to power converted into nonlinear harmonics that isnot converted back to the fundamental frequency. Excitingthe device from the second port significantly attenuates the OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 11
Frequency (GHz) S - pa r a m e t e r s ( d B ) S -SimulatedS -MeasuredS -SimulatedS -Measured (a) Frequency (GHz) S - pa r a m e t e r s ( d B ) S -SimulatedS -MeasuredS -SimulatedS -Measured (b) Frequency (GHz) S - pa r a m e t e r s ( d B ) S -SimulatedS -MeasuredS -SimulatedS -Measured (c)Fig. 11. Measured response of the manufactured third order non-reciprocal filter and comparison with respect to the numerical results obtained with theproposed technique (losses and parasitic cross couplings have been included in the coupling matrix approach). (a) Unmodulated response (note that in thiscase S = S and S = S ). (b)-(c) Response obtained when the modulating signal is applied to the varactors and the filter is excited (shown in theinset using a magenta arrow) from the first (b) and the second (c) port. Frequency (GHz) S - pa r a m e t e r s ( d B ) S -SimulatedS -MeasuredS -SimulatedS -Measured (a) Frequency (GHz) S - pa r a m e t e r s ( d B ) S -SimulatedS -MeasuredS -SimulatedS -Measured (b) Frequency (GHz) S - pa r a m e t e r s ( d B ) S -SimulatedS -MeasuredS -SimulatedS -Measured (c)Fig. 12. Measured response of the manufactured fourth order non-reciprocal filter and comparison with respect to the numerical results obtained with theproposed technique (losses and parasitic cross couplings have been included in the coupling matrix approach). (a) Unmodulated response (note that in thiscase S = S and S = S ). (b)-(c) Response obtained when the modulating signal is applied to the varactors and the filter is excited (shown in theinset using a magenta arrow) from the first (b) and the second (c) port. propagating energy. The strong non-reciprocity predicted bythe initial simulations is confirmed by the measurements.Around the center frequency of the passband, the directivityis better than D = 13 . dB in a bandwidth of MHz.Across the entire passband, the directivity is always better than D = 7 . dB. In general, good agreement between measuredand simulated responses are obtained when losses and parasiticcouplings are included in the derived coupling matrix model.For reference, the basic performances for both manufacturedfilters are collected in Table III.Another important characteristic of these devices for manyapplications is the power handling levels. The hardware builtcould not be tested under high power signals. Primarily, thepower handling will be limited by the technology used to builda similar unmodulated circuit [35]. However, an interestingfuture research topic will be the assessment on how theadditional circuitry needed by modulation signals and thepresence of varactors affect the power handling levels, andwhich arrangements are more appropriate to reduce theseundesired effects. VI. C ONCLUSION
We have presented the analysis of non-reciprocal filtersbased on time modulated capacitors using a coupling ma-trix formalism. From the initial topology of the filter, anovel coupling topology using harmonic resonators is firstderived. Closed form analytic expressions have been obtainedto represent the harmonic resonators with frequency invariantsusceptances, thus obtaining the diagonal elements of thetraditional coupling matrix. Also, non-reciprocal admittanceinverters have been analytically computed to account for thecouplings between harmonic resonators, thus obtaining the off-diagonal elements of the coupling matrix. The derived analysismethod has been validated with the design and fabricationof third and fourth order filters implemented in microstriptechnology. Measured results on the fabricated prototypes, andresults obtained with a commercial tool are found to agree wellwith respect to numerical calculations obtained using the newcoupling matrix formulation, thus fully validating the theorypresented.
OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, AUGUST 20XX 12 A CKNOWLEDGMENT
Authors are grateful to Rogers Corporation for the generousdonation of the dielectrics employed in this work.R
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