Synthesis of Symmetric, Two-Element Biomimetic Antenna Arrays Using Singular Value Decomposition

We present a method for determining the four-port S-matrix of the external coupling network (ECN) for symmetric, two-element biomimetic antenna arrays (BMAAs) by employing singular value decomposition (SVD). The presented approach greatly facilitates the synthesis procedure of symmetric, two-element BMAAs and allows all possible scattering matrices of a four-port network realizing the ECN to be easily determined. Unlike a previously-reported method that relies on solving a set of nonlinear equations to determine the S-parameters of the ECN, the present method uses only linear-algebraic equations that can be easily solved. We show that an infinite number of S-matrices exist for the ECN of a two-element BMAA given a fixed symmetric two-element antenna array. We also demonstrate that these S-matrices are different from each other only by two arbitrary phase shift matrices that represent free design parameters. In any given BMAA synthesis problem, these free parameters can be used to optimize other attributes of the response of the BMAA. We illustrated this by showing how the overall impedance-matching bandwidth of the BMAA varies as a function of a free design parameter. Two BMAAs corresponding to two different bandwidth values were fabricated and characterized to experimentally validate the proposed synthesis method.


I. INTRODUCTION
I N ANTENNA arrays or multi-antenna systems comprising closely-spaced antenna elements, significant inter-element coupling effects occur that can deteriorate the performance of the antenna system.It is often desirable to have a good impedance matching and low port-to-port coupling between the different antenna elements [1], [2], [3].Various multiport external coupling networks (ECNs) have been proposed to perform the tasks of impedance matching and decoupling in such systems [4], [5].ECNs have also been exploited to achieve additional functionalities, beyond matching and decoupling, in antenna arrays with closely spaced elements.For example, in two-element biomimetic antenna arrays (BMAAs), the ECN is used to extract the maximum available power from an incoming wave and to enhance the phase difference at the outputs of the array compared to that at the antenna terminals [6], [7].This enhanced phase difference can be useful in direction-finding applications relying on phase interferometry in small-aperture systems [8], [9], [10], [11].
In [12], a comprehensive analytic method was presented for synthesizing symmetric, two-element BMAAs.This method provides a general framework for (a) determining the elements of the scattering matrix (S-matrix) of a four-port lossless, reciprocal, and symmetric ECN and for (b) synthesizing the ECN from the calculated S-matrix.It was demonstrated in [12] that the conditions required to design a two-element BMAA resulted in an under-determined system with an infinite number of solutions for the four-port ECN's S-matrix that would satisfy these conditions.To determine the S-matrix of the four-port ECN of the BMAA, the approach presented in [12] relied on solving a system of nonlinear equations that are in general cumbersome and difficult to solve.Additionally, these nonlinear equations did not provide any intuitive understanding about the degrees of freedom available in the design of the ECN or about the additional constraints that could be applied to this underdetermined system to come up with a unique solution for the S-matrix of the ECN.These are some of the major drawbacks of the analytic method that was presented in [12].
In this paper, we extend the work presented in [12] and provide a new method for determining the S-matrix of the ECN of a two-element BMAA.We point out that two of the conditions needed to design a two-element BMAA, are also necessary and sufficient for designing impedance matching and decoupling networks for a symmetric, two-element antenna array. 1 Thus, the presented method can also be used to synthesize impedance matching and decoupling networks for any symmetric, two-element antenna array [13].To determine the S-matrix of the ECN, we use a singular value decomposition (SVD)-based analysis technique that is analogous to the one presented in [14].In [14], Wallace and Jensen used SVD to represent the S-parameter matrix for a lossless matching network, and noted that additional constraints would apply if the network is also reciprocal.In the current work, we extend SVD analysis to lossless, reciprocal, and symmetric matching networks.Then we consider the additional constraint for realizing the phase enhancement feature of a symmetric two-element BMAA.The proposed analysis framework alleviates the need to solve the difficult set of nonlinear equations proposed in [12].Indeed, only linear-algebraic equations need to be solved to determine the S-matrix of the ECN.Owing to its simplicity and linear-algebraic nature, this new formulation is especially amenable for use in numerical optimization processes aiming to improve other attributes of the ECN response (e.g., enhancing bandwidth) [15].Similar to the method presented in [12], this new formulation also shows that an infinite number of S-matrices exists that can be used to design a two-element BMAA given a fixed set of antenna elements.However, unlike the work in [12], the new formulation shows that these S-matrices differ from each other in terms of two phase shift matrices that represent the free design variables of the proposed synthesis procedure.These free design parameters may be used to improve other attributes of the performance of a two-element BMAA or an impedance matching and decoupling network used in a two-element antenna array.As an example, we show how the impedancematching bandwidth of the BMAA changes when these two phase shift values are varied.Two different S-matrices were selected for the ECN, corresponding to two different 1.Note that the ECN of a two-element BMAA designed using the approach presented in [12] serves as an impedance matching and decoupling network as well.However, the conditions needed to design only an impedance matching and decoupling network are less restrictive than those needed to design a BMAA.set of the free design parameters, to implement two versions of BMAAs.Fabricated prototypes of these BMAA versions were experimentally characterized.Measurement results show good agreement with the simulations, verifying the proposed synthesis technique.

II. SINGULAR VALUE DECOMPOSITION APPROACH FOR S-MATRIX CALCULATION
Fig. 1(a) shows the block diagram of a BMAA that includes two identical antenna elements connected to two input ports of the ECN, the outputs of which are terminated with two 50 loads.The network model of the BMAA is presented in Fig. 1(b).Interfaces 1 and 2 refer to the input and output ports of the ECN, respectively.Using this notation, we represent the original 4×4 S-matrix of the ECN in Fig. 1(b) by a simpler, partitioned 2 × 2 S-matrix.The symmetric, twoelement antenna array is represented by its 2 × 2 S-matrix as S AA .S AA is assumed known and can be obtained from fullwave simulations or measurement.In addition, the S-matrix of the ECN is given by: where S 11 , S 12 , S 21 , and S 22 are 2 × 2 block matrices.We assume that the network is lossless, reciprocal, and symmetric.These characteristics imply that: where {•} H and {•} T denote the Hermitian (conjugate transpose) and transpose of a matrix, respectively.Equation (3) requires that S 11 = S T 11 , S 22 = S T 22 , and S 12 = S T 21 .Due to symmetry of the ECN, S 12 = S 21 .It was proved in [14] that to achieve matching and decoupling, we must have The condition (4) leads to zero values for all four elements of the output reflection matrix R out .The same condition also ensures that the two-element array is impedance matched in both the common and the differential modes of excitation.R out is given by: Diagonal elements of R out , i.e., R (1,1) out and R (2,2) out are the reflection coefficients seen into the external coupling network from the 50 ports.R (1,2)  out and R (2,1)  out correspond to the coupling between the two 50 ports.While the S 11 partition of S ECN for a BMAA is determined by (4), S 12 and S 22 must also be found to determine S ECN .S 11 is represented in terms of the SVD as: where Considering that (U T 11 V 11 ) * = U H 11 V * 11 , it can be easily seen that for the equality of (10) to be valid, U 11 and V 11 must satisfy: where 11 is a diagonal phase shift matrix whose diagonal elements are complex and have a unity magnitude.Therefore, * 11 11 = I.The relation (12) and properties of unitary matrices imply: Equations ( 6) and ( 15) lead to the following expression for S 11 : Furthermore, the lossless condition of the network implies that: Equations ( 17) and (18) result in: Meanwhile, using a similar SVD analysis to that presented above, S 22 is expressed as: where 22 = U T 22 V 22 and 22 is a diagonal phase shift matrix with complex elements having unity magnitudes.Equations ( 6), (21), and (22) lead to the equality: Equation ( 23) implies that 22 = 11 and V 22 = V 11 22 where 22 is an arbitrary diagonal phase shift matrix with complex elements having unity magnitudes.Using ( 22) and ( 23), S 22 is expressed as: where 22 = 22 ( * 22 ) 2 .Thus, 22 is an arbitrary diagonal phase shift matrix with complex elements having unity magnitudes.S 12 can be also determined based on (17) and using similar SVD analysis as: where 12 is another arbitrary diagonal phase shift matrix with complex elements having unity magnitudes.A specific set of S 11 , S 12 , and S 22 matrices has to satisfy the conditions ( 19) and (20).Therefore, a constraint on the three phase shift matrices 11 , 12 , and 22 , is imposed as: * Conditions ( 4), ( 16), ( 24), (25), and (26) lead to perfect matching and decoupling, which are necessary but are not sufficient for realizing a symmetric, two-element BMAA.To design a two-element BMAA, we must also maximize the phase difference at the outputs of the ECN compared to that at its inputs.In [6], it was shown that for achieving maximum phase enhancement in a two-element BMAA, without sacrificing the output power level, the phase difference between the common-and differential-mode output signals (V comm out and V diff out ) of the ECN has to be equal to 90 It can be shown that ψ can be represented in terms of the element values of the block matrix S 12 in the following format:

S
(1,1) Here τ is a function of the intrinsic electrical characteristics of the two-element antenna array and is expressed as follows: where Z AA is the impedance matrix of the symmetric, twoelement antenna array and Z 0 is the reference impedance of the system (Z 0 = 50 ).Then, to determine ψ, S (1,1) 12 and S (1,2)   12 can be expressed using (25) as: Here α and β are the arbitrary phase shifts introduced by the phase shift matrix 12 as while σ 1 and σ 2 are the diagonal elements of (I − 2 11 ) 1/2 matrix as The other parameters in (30) and (31) are the element values of V 11 .In determining the S-matrix of the BMAA's ECN, α and β can be chosen to achieve ψ = 90 • .However, as can be observed from equations ( 28), ( 30) and (31), there are infinite sets of α and β values that result in the desired condition.Therefore, as discussed in [12], there are an infinite number of S-matrices that will provide the desired BMAA operation.
While we focused on the design of two-element BMAAs in this work, the theoretical framework presented in this section can be applied to derive the S-matrix for an ECN or an impedance matching and decoupling network for any symmetric array with an arbitrary number of elements.Assuming the number of antenna elements is N, the block matrices S 11 , S 12 , S 21 , and S 22 would be N × N. Equations ( 1)-( 26) are the same for any value of N and can be used to design a symmetric 2N-port ECN that satisfies the impedance matching and inter-port decoupling conditions.However, a new formulation of ( 27) is necessary to realize the output phase enhancement feature of an N-port BMAA.We envision using N linearly independent orthogonal excitation modes, which can be similar to the basis modes used in [16] for an N-port BMAA or to the eigen modes of an arbitrary N-port antenna array presented in [17], to express the phase enhancement conditions.The computational complexity for calculating the ECN's S-matrix is roughly estimated to be in the order of O(4N 2 + 8N 2 log(N) + 2N 3 ).

III. VERIFICATION OF THE SVD-BASED BMAA DESIGN APPROACH
The BMAA synthesis method presented in [12] was verified using two design examples in which two different Smatrices for BMAA's ECN were calculated and the BMAAs using these ECNs were implemented and characterized at approximately 625 MHz.Here, we choose one of those prototypes and demonstrate that the same S-matrix can be obtained by using the proposed SVD-based approach.The BMAA in [12], included two identical monopole antennas made of solid copper tubes having a circular cross section with the radius of 3 mm and a length of 10.3 cm.The distance between two antennas was 5 cm and they were placed above a 30 cm × 30 cm metal ground plane.The two-port impedance matrix of this antenna array was In [12], the set of nonlinear equations that would determine the S-matrix of the ECN had infinite solutions.Thus, to solve these equations, an additional independent constraint had to be imposed on the system.In one of the verification examples presented in [12], this independent condition was to create a 90 • phase difference between the output and input voltages of the ECN in the common mode.This condition does not have any physical significance and was only used in [12] to make the system of equations solvable.In this section, we use the same condition to demonstrate that the same S-matrix can be obtained using the proposed SVD-based approach without needing to solve nonlinear equations.For ease of comparison, the S-matrix of the BMAA's ECN reported in [12] is provided in ( 35)-(37) below: In finding the S 12 based on SVD of S 11 and according to (25), α and β in the phase shift matrix 12 can be chosen freely.However, it is essential to meet two constraints.First, α and β have to satisfy the maximal phase enhancement condition (27), based on (28)-(33).Second, they also have to satisfy the aforementioned design constraint used in [12].

IV. IMPACT OF THE ARBITRARY PHASE SHIFT MATRICES ON THE SYSTEM RESPONSE
The phase shift matrices, 12 and 22 , identified in ( 24) and ( 25) are free design parameters.Note that one of these two matrices can be chosen arbitrarily and the other will be calculated using (26).The ECN S-matrices obtained using different selections of these phase shift matrices all provide the desired BMAA response if (26) and ( 27) are satisfied.However, other attributes of the BMAA response such as bandwidth, complexity of implementation, design sensitivity, etc. may vary depending on the specific choice of phase shift matrices used.Therefore, the phase shift matrices may be used to control and optimize other aspects of the BMAA's response.To illustrate this promising opportunity, we synthesized symmetric, two-element BMAAs for an antenna array similar to the one examined in Section III, operating in the This measured S AA was then used to calculate the ECN's S-matrix.We gradually varied α in 12 (see (32)) from 0 • to 360 • with 1 • steps.For each value of α, we determined the value of β to achieve a 90 • phase difference between the common-and differential-mode output signals in (27) to satisfy the maximum phase enhancement condition for the BMAA.After solving the phase shift matrix 12 , we can calculate 22 and derive the four-port Smatrix of the ECN at 625 MHz following the proposed SVD-based approach.In the next step, we chose a possible fixed circuit topology for the ECN, which is shown in Fig. 2(a), and calculated the values of lumped inductors and capacitors at 625 MHz.While the ECN is based on lumped capacitors and inductors, its practical implementation on a printed circuit board (PCB) requires placing short metallic traces between the lumped components to facilitate manual placement and soldering.These metal traces are backed by a ground plane to minimize the radiation leakage, which turns them into short transmission lines (TLs).These short TLs (even a few millimeters), if not accounted for in the design process, may cause a drastic change in the response of the ECN and, therefore, severely deteriorate the performance of the fabricated BMAA.Therefore, such TLs should be taken into account when calculating the lumped element values of the ECN.To illustrate the impact of these interconnecting TLs on ECN's lumped element values, we synthesized the ECN under two scenarios: without TLs and with TLs.

A. CALCULATING LUMPED ELEMENTS OF ECN WITHOUT TLS
For the first scenario, we assumed the ECN does not have any TLs interconnecting the lumped elements and we applied the synthesis process presented in [12].Once a network topology is chosen, the reactance values of the lumped elements can be easily solved to produce the exact ECN's S-matrix derived in the previous design step.This is done by matching the two-port transmission matrices of the equivalent circuits of the lumped ECN network in the common and differential modes with those derived from the four-port S-matrix of the ECN.The inductance and capacitance values are then converted from the computed reactance values.We analytically calculated the 10 dB return loss bandwidths of each synthesized network in the common and differential modes of the excitation and identified the overlap of these two bandwidths as the BMAA's overall impedance match bandwidth. 2 In bandwidth calculations, we represented the antenna array as a frequency-dependant impedance element connected to the synthesized ECN's input.In the common and differential modes of excitation, its value was equal to Z (1,1) AA , respectively, where Z AA is frequency dependent and was experimentally acquired.

B. CALCULATING LUMPED ELEMENTS OF ECN WITH TLS
For the second scenario, we assumed there are short sections of transmission lines connecting the lumped elements together in the ECN.The ECN was simply modified by replacing each lumped element of the ECN shown in Fig. 2(a) by a lumped element connected to two identical sections of microstrip lines (with equal lengths and widths) at both ends, as illustrated in Fig. 2(b).Each TL section is modeled as a pi-network of C-L-C as shown in Fig. 2(b).We determined the values of these equivalent inductors and capacitors by matching the transmission matrix of the equivalent circuit to that extracted from full-wave simulation of the transmission line.In the design examples presented in this paper, we added identical sections of microstrip lines with a length of 3 mm and a width of 1 mm to both ends of each lumped element.Any two elements connected to each other (e.g., 2 and 3 in Fig. 2(a)) will have a 6-mm microstrip line connection between them.The values of inductors and capacitors for the equivalent C-L-C circuit of each 3-mmlong microstrip line are calculated as L = 0.84 nH and C = 0.15 pF at 625 MHz.Subsequently, the ECN's topology was modified by inserting such C-L-C circuits on both sides of each lumped component.Following a similar synthesis procedure used for the first scenario (i.e., ECN without TLs), the ECN's lumped element values can be solved from its predertermined S-matrix.Applying the same bandwidth analysis process as ECN without TLs, for each α, we 2. Recall that a BMAA must be impedance matched at both the common and the differential modes of excitation.For closely-spaced antenna elements, the bandwidth of the differential mode is almost always narrower than that of the common mode.derived an ECN's implementation and calculated its overall bandwidth.

C. NUMERICAL RESULTS
Figure 3 shows how the impedance-match bandwidth varies as a function of α for the two scenarios with and without TLs.The maximum bandwidths are recorded as 16.3 MHz at α = 359 • for the case without TLs and 15.8 MHz at α = 358 • for the other case.Note that for the scenario with TLs, there were some values of α that did not yield a solution for the ECN's lumped elements.This did not happen for the scenario without the TLs, where we could always find a solution for all lumped elements in the ECN with every value of α.Nevertheless, adding 3-mm-long TLs to the ECN does not seem to impact the overall bandwidth of the BMAA as long as the presence of the TLs is taken into account in calculating the lumped element values.Additionally, there are regions in Fig. 3 where the bandwidth changes quickly with respect to α (e.g., around the peaks and dips).This suggests that the bandwidth is sensitive to a minor change of α in these regions, which can be easily incurred by small tolerances of lumped components and fabrication.On the other hand, picking a point in flatter regions of the curves in Fig. 3 likely provides a realized circuit with a more stable response with respect to such tolerances.For this reason, we chose two representative values of α from relatively flat regions of the curves in Fig. 3 that also correspond to different bandwidths for experimental demonstration in the next section.Table 1 presents the types and values of reactive elements necessary to implement the ECNs corresponding to two representative values of α (α = 50 • and α = 120 • ) in the two scenarios of with and without TLs.For these two values of α, the values of β calculated from (27) are 100 • and −10 • , respectively.When the effect of TLs is taken into account, the values of the lumped capacitors and inductors change significantly compared to the scenario where the ECN does not have any TLs.This indicates that if the effect of TLs is not included in calculation, a substantial tuning of the element values would be required to achieve the desirable BMAA performance for fabricated prototypes.   1.
The common-and differential-mode output reflection coefficients of four different BMAA versions (with TLs and without TLs for α = 50 • and α = 120 • ) are plotted in Figs.4(a) and 4(b), respectively.Observe that all four synthesized ECNs provide excellent impedance matching for the antenna array in both modes of excitation at 625 MHz.It is illustrated that the differential-mode impedance-match bandwidth is significantly narrower than the common-mode value.Therefore, the overall impedance-match bandwidth of an BMAA is mainly limited by that of the differential mode of excitation.The overall impedance-match bandwidths of the BMAAs are 6.9 MHz (for α = 50 • ) and 3.2 MHz (for α = 120 • ) when TLs are considered in the calculation.
We calculated the phase difference between the output voltages for the two synthesized BMAAs (with TLs and calculated lumped element values) as the angle of incidence changes from −90 • to 90 • .The results are compared with the corresponding output phase difference of a regular array (the same antenna array without the ECN) to verify the phase enhancement characteristic of the BMAAs.The phase enhancement factor, which can be calculated using the formula provided in [6], is maximized under the condition of maximum power extraction if ( 27) is satisfied along with impedance matching in both the common and differential modes.Therefore, the two BMAA versions should reach the same maximum phase enhancement factor (due to using the same antenna array) despite different ECN implementations.This is illustrated in Fig. 5, where BMAA1 refers to the case with α = 50 • and BMAA2 refers to the case with α = 120 • (both with TLs).Both BMAAs have the same phase enhancement factor of 9.72 when we used the calculated lumped components as shown in Table 1.

V. EXPERIMENTAL RESULTS
We fabricated two versions of the ECN corresponding to two values of α = 50 • and α = 120 • to illustrate the difference in the overall impedance-match bandwidths of these two solutions.Note that we did not choose a value of α corresponding to the maximum bandwidth shown in Fig. 3 because such an implementation is very sensitive to even slight changes in the lumped component values.Since we have to employ commercially available capacitors and inductors to implement the prototypes, small deviations from the calculated values are inevitable in most cases.Table 2 shows the values of lumped components used in the fabricated prototypes that are closest to the calculated values listed in Table 1.

TABLE 2. Realistic values of the reactive elements used to implement two ECN versions corresponding to two values of α.
ports were connected to a vector network analyzer (Rohde & Schwarz ZVB20) to measure the BMAA's scattering parameters.
Figs. 7(a) and 7(b) show the magnitudes of the measured and simulated output reflection coefficients in the common and differential modes of the two BMAAs, plotted along their corresponding simulated values.Note that the simulation results shown in this figure were generated using the commercially available capacitor and inductor values listed in Table 2.The measurement and simulation results show reasonable agreement with only minor discrepancies for both prototypes.For the first BMAA prototype (see Fig. 7(a)), the center frequency of the differential-mode impedancematching bandwidth is shifted down by about 5 MHz.This frequency shift in the differential mode along with the widening of the common-mode bandwidth are most likely caused by the non-ideal behaviors (e.g., ohmic loss and parasitic effects) of the inductors and capacitors used to fabricate the ECN.These non-ideal effects were not taken into account in the simulations.However, the impact of this factor seems to vary between different ECN designs, as the second BMAA prototype did not exhibit such frequency shift (see Fig.  Nevertheless, these minor discrepancies do not impact the overall impedance-match bandwidths of both BMAA prototypes.Specifically, the measured overall impedance-match bandwidths are 6.9 MHz (for α = 50 • ) and 3.2 MHz (for α = 120 • ), which agree very well with the predicted values.

VI. CONCLUSION
An SVD-based approach for determining the scattering parameters of the four-port external coupling network of twoelement, symmetric BMAAs was presented in this paper.The presented technique offers a simple, linear-algebraicbased approach for determining the S-parameters of the ECN.Unlike an earlier method presented in [12], the proposed method does not rely on solving a system of under-determined, nonlinear equations.This facilitates the solution process and optimization of other attributes of the BMAA response.Additionally, the presented method can also be used to design and synthesize impedance matching and decoupling networks for any symmetric, two-element antenna array.We demonstrated that an infinite number of ECN S-matrices exist that satisfy the BMAA design conditions for a two-element antenna array.However, these S-matrices differ from each other only by two shift matrices, which represent the free design parameters for designing two-element BMAAs.These free design parameters can be used to optimize other attributes of the performances of the BMAAs.As an example, we showed how sweeping one of the phase shift values impacts the overall impedance-matching bandwidth of the BMAA.Two different versions of the BMAA corresponding to two different bandwidth values were fabricated and experimentally characterized.The measured bandwidths agree well with the calculated values, confirming the effectiveness of the design procedure as well as the validity of the proposed approach.

FIGURE 1 .
FIGURE 1.(a) Block diagram of a two-element antenna array with an external coupling network (ECN).The ECN acts as an impedance matching and decoupling network and can be designed to provide the maximum phase enhancement condition needed to implement two-element BMAAs as discussed in [12].(b) Network model for a coupled antenna array connected to the loads through the ECN.

FIGURE 2 .
FIGURE 2. (a) One possible circuit topology composed of six distinct reactive elements (minimum number of reactive elements required for realizing SECN) that is used for implementing the physical four-port ECN in a symmetric, two-element BMAA in Sections IV and V.For PCB implementation of the ECN, each reactive element is connected to two short transmission lines at its two ends.(b) Equivalent circuit model for each of the short transmission lines connecting to the lumped elements of the ECN.

FIGURE 3 .
FIGURE 3. The dependency of the 10 dB return loss bandwidth of the synthesized BMAA on the phase shift parameter α in the vicinity of 625 MHz.The bandwidth values were calculated for the two cases where the ECN is with and without the TLs.

TABLE 1 .
Types and values of the reactive elements of the ECNs (shown in Fig.2) for two cases of α.

FIGURE 4 .
FIGURE 4. Simulated output reflection coefficient in the (a) common and (b) differential modes of excitation for two different BMAAs using the same circuit topology shown in Fig. 2. The values were calculated for both scenarios with and without TLs.Types and values of the reactive elements used to realize each ECN are presented in Table1.

FIGURE 5 .
FIGURE 5. Simulated output phase responses of the two BMAA versions (for α = 50 • and α = 120 • ) when the ECNs consist of TLs and lumped elements with the calculated values shown in Table 1.The response for a regular array (e.g., the same antenna array without an ECN) is shown to illustrate the phase enhancement feature of the BMAAs.

Figs. 6
(a) and 6(b) show the fabricated prototypes of the antenna array and one representative ECN, respectively.The ECNs were fabricated on Rogers RO4003C substrates with a dielectric constant of r = 3.55, loss tangent of 0.027, and thickness of 0.508 mm.The values of the lumped elements used in two versions of ECNs are listed in Table2.Each BMAA was assembled by connecting the monopoles to the two input ports of the corresponding ECN whose two output

FIGURE 7 .
FIGURE 7. Measured and simulated output reflection coefficients of the two fabricated BMAAs in the common and differential modes when the ECN was synthesized for (a) α = 50 • and (b) α = 120 • .