Synthesis and Realizability in Design of Minimum-Simulation Inline Antenna Arrays

This article discusses a simplified design technique of antenna arrays. The technique relies on a single electromagnetic simulation of an arbitrary inline core element, for instance, a printed patch antenna or a waveguide slot, and several identical core elements are then cascaded by means of transmission lines with given length and characteristic impedance. Realizability constraints as well as extended possibilities are discussed. Design examples are presented and characterized for validation.


I. INTRODUCTION
I N THE current landscape of wireless telecommunications, phased arrays are gradually replacing some traditional single-feed antennas, such as parabolas and horns, in several applications.Architecturally, however, phased arrays demand multiple radio frequency chains, which are unpalatable where cost, consumption, and number of components must be minimized and where dynamic beam steering is not required [1].
Single-feed arrays, on the other hand, involve one RF chain feeding a number of elements via a static distribution network, responsible for amplitude and phase at each element [2], [3].Among these, inline antenna arrays (Fig. 1) merely require each element to be coupled with the neighboring ones, thus a more compact implementation than corporate distribution networks.
Typical practical examples include series-fed patch antennas, leaky wave antennas, and slotted waveguides.
The design of inline antenna arrays normally requires an equivalent model (i.e., a resonator), which can then be plugged into a circuit to determine the required interconnections: these techniques can be relatively simple yet powerful [4], [5].Still relying on a simple equivalent model, other techniques require different elements along the array [6], [7], which conversely demand several separate electromagnetic simulations to dimension each element in accordance with the synthesis output.The extension to other antenna types today involves  a modeling step (not trivial and implying several electromagnetic simulations) and the development of a suitable synthesis technique for the array, which, if lacking, must be replaced by several heavy electromagnetic optimizations.The principle described here derives from [8]: cascading sections all relying on the same "core" radiating element.This article provides an insightful description as well as realizability constraints and expanded possibilities, together with step-by-step examples.
In detail, the proposed technique: • requires in principle only one electromagnetic simulation of a core 2-port antenna without additional assumptions; • allows to design with closed formulas an inline array of core elements connected by transmission lines; • neglects mutual coupling between the radiators; crosscoupling compensation can be however applied [9]; • does not require equivalent models of the core element; • may result in a non-uniformly spaced linear array; • allows easy beam steering, inserting just a phase shifter between elements without redesigning the whole array.The procedure works at a single frequency and, hence, may yield a narrowband design, but still a reasonably good starting point for optimization in case a wider bandwidth is needed.

II. BASIC DESIGN METHOD
The main aspects of the design method are mutuated from [8], but here recalled and revised with several novel and specific insights.For simplicity, the discussion is referred to a transmitting array, while reception is obtained by reciprocity.
Remark 1: The proposed technique makes each section, at design frequency f 0 and at reference impedance Z 0 : 1) matched at its input, when its output is closed on Z 0 ; 2) radiating an illumination-dependent fraction of the input power and forwarding the rest to the following sections; 3) provide radiation in-phase with the following sections, in a considered spatial direction.In the following, the general nth section is designed fulfilling the three requirements above.All impedances will be normalized to Z 0 and denoted with lowercase letters.The synthesis philosophy is to start from the terminal section, proceeding backward toward the feed of the array.

A. Section Structure
Each section, as shown in Fig. 2, comprises an input phase shifter (normalized impedance z (n)  I and electrical length φ (n) I ), a core radiating element, and an output network, including a "mapping" phase shifter (z (n)  M , φ (n) M ) and, for all nonterminal sections, an "adjustment" phase shifter (unitary characteristic impedance, electrical length φ (n) A ).The core radiating element is a reciprocal 2-port antenna, with external behavior fully known via an electromagnetic simulation returning: 1) The scattering parameters referred to impedance Z 0 2) Radiated field in the desired spatial direction, measured at a prescribed far-field distance when fed from each of the two ports with a prescribed power P. The radiated field component is denoted as S R1 (when fed from port 1 with port 2 on a matched load) and as S R2 (when fed from port 2 with port 1 on a matched load).Only the relative phase and amplitude of S R1 and S R2 are relevant; hence, the chosen far-field distance is not influent.All the N sections use the same core element; thus, in principle, just one electromagnetic simulation is needed for the method.Under these assumptions and feasibility constraints, in fact, the array designed by the proposed procedure will perform as intended without needing any further simulation or optimization, as discussed further in Section IV.
The phase shifters have scattering parameters given in Fig. 3 (top) and are implemented by transmission lines giving the same scattering parameters, as explained in Appendixes I and II.The mapping phase shifter can, in principle, be replaced with more complicated structures (not treated here) to overcome manufacturing constraints.The output network of the terminal section is constituted just by the mapping phase shifter, open-or short-circuited, and does not expose a second port.

B. Matched Input
According to the hypotheses, each nonterminal section is closed onto another similar section, and thus, a divide-etimpera choice is to impose that each section alone is matched where (n) M is the reflection observed looking into the output network closed onto a matched load, which must, thus, fulfill For whatever choice of z (n) I and φ (n) I , S I C(n) can be computed as cascade of the input phase shifter and the known core (1), resulting in the needed (n)  M for input matching The task of transforming a matched z 0 = 1 load at port 2 of the section into the required (n)  M is demanded to the "mapping" phase shifter, as described in Section II-D.The choice of z (n) I and φ (n) I is, however, constrained by the other requirements, as explained in the following.

C. Operating Power Gain
Of the requirements set out in Remark 1, the one concerning quotas of radiated and transferred power is considered now, which is directly related to the power gain of the section.
Assuming that the core network has only radiation losses, whatever power enters the network through port 1 is either radiated or transferred on to the following stages.Several illumination distributions [10] can be used, i.e., uniform, Dolph-Tchebychev, and Orchard [11], [12], [13], [14], each with different efficiencies and side lobes.The case of uniform illumination with in-phase radiation is considered for simplicity: • The terminal section must radiate all the power it accepts.
• The last-but-one section must radiate half of the accepted power and forward the other half to the terminal section.
• In general, the nth section counted from the feed point, being the last-but-m with m = N − n has Power forwarded to the following sect.
Power accepted at its port 1 Power radiated by section n Power accepted at its port 1 Fig. 3 shows these formulas for a uniform array, confirming that the radiated power from each element is the same.When a nonuniform illumination I (n) is instead required, the power gain of each section can be computed as in Appendix II Thus, (5) or the chosen illumination distribution provides the required power gains at the beginning of the procedure for all sections n = 1, . . ., N .The operating power gain [15] of a general 2-port network (whose scattering parameters are denoted as Ŝn,m ) is a function of the load onto which the network is closed L ; hence, the notation G P ( L ) Fig. 3. Top: generic phase shifter and its scattering parameters as a function of its electrical length φ and normalized characteristic impedance z.Bottom: power gain (solid line) along an N = 10 array and its complementary radiated power fraction (dashed line) of each section for uniform illumination; actual radiated power (dotted line) from each section is proportional to the cumulative product of the power gains of the previous sections.
The overall section's operational power gain G I C M A P (0), which must equal (5), is the product of the individual gains and, due to S M A(n) being lossless is, thus, The S I C(n) network instead is lossy seen from the two physical ports, as radiation also occurs, but it is matched at its input when closed onto (n)  M of (3), and thus, where the last equality is obtained by replacing (3), itself function of the S I C(n) scattering parameters.Since all of them depend on z (n) I and φ (n) I , as well as on the core's scattering parameters, the expression can be developed to obtain of which an example is given in Fig. 4, and with with period π , as observable by inspection.For z (n) I = 1, besides, the gain becomes independent on φ (n) I and equal to the "critical" value G I C c , with (n) M = S 11 / .For the designer, the purpose is to identify the (z (n) P prescribed by the chosen illumination distribution; see (5).This can be achieved by either sweeping z (n) I and φ (n)  I , while evaluating the G I C M A(n) P (0) surface, or in closed form, as described in the following.Minimum and maximum values of the achievable operating power gain can be derived to establish the limits (see Section III).When the desired G (n)  P is within the achievable range, (9) can be manipulated to obtain φ (n) I as a second-order polynomial solution for each value of z (n) By any of the above methods (sweeping the surface or closed-form contours), one identifies the set of input phase shifter parameters, which fulfill the desired power gain.One pair (z (n)  I , φ (n) I ) must be selected among those.For any nonterminal section, only those achievable within the realizability constraints of the mapping network are acceptable, as outlined in Section II-D, whereas the terminal section does not impose further constraints on realizability.

D. Mapping Phase Shifter
A given pair (z (n) I , φ (n) I ) implies the required (n) M via (4), which must be obtained by transforming the matched load z 0 = 1 through the mapping phase shifter.
For the terminal section n = N , the required 2-port power gain is G I C M A(N ) P (0) = G N P = 0 (since there are no further loads along the array), which implies that (N ) M = 1.This can be easily obtained if the mapping phase shifter is simply Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
an open-or short-circuited stub with unitary impedance z (N ) M = 1 and electrical length φ Therefore, for any candidate pair of (z (n)  I , φ (n) I ), the required mapping phase shifter can be determined.If the obtained values are not acceptable due to manufacturing constraints, the candidate pair must be rejected or a different output network considered, for instance, a single-stub network as general solution or using Section III-C.

E. Radiated Field
The radiation from an entire nonterminal section n < N can be computed considering the ( S I C M A(n) ) network as a fictitious 3-port network where port 3 observes the radiation in the far-field point.Since port 2 is closed onto a matched load, the contribution reaching the observation point from section n is simply S I C M A(n) 31 : nothing is reflected at port 2; therefore, S I C M A(n) 32 does not intervene.The adjustment phase shifter (itself matched yet unknown) does not affect the radiation from section n in any way, and hence, the following expressions are valid also for the terminal section (n = N ).The radiated component is, in fact, computed by cascading the input phase shifter with the (n)  M -loaded core, as shown in Fig. 5

F. Adjustment Phase Shifter (Only for Nonterminal Sections)
The adjustment phase shifter must enforce that the radiated field from the following sections, which receive the forwarded (nonradiated) power quota from section n, has the desired Fig. 5. Schematization of equivalent cascades used to compute scattering terms of a nonterminal section.Top: the radiation term is obtained by loading the core section with M and then cascading with the input phase shifter.Bottom: the transmission term is the forwarded contribution toward the adjustment phase shifter, hence merely the cascade of the input, core, and mapping blocks, neglecting radiation.
phase relation with the radiated field from section n.For simplicity, we assume that radiated fields must all be in phase in broadside.Radiation from section n is known via (14).The following part of the array (sections n + 1 to N ) behaves as a matched 1-port antenna, radiating with a phase delay denoted as ∡S I C M A(n+1) 31 in the observation point, computable from the previous steps of the design.The adjustment shifter delays the forwarded quota, where the I C parameters are computed by the rules of cascade and the parameters S M(n) of the mapping shifter follow Fig. 3.The adjustment phase shifter must, therefore, satisfy For a chosen pair (z (n) I , φ (n) I ), finally, the adjustment phase delay must be determined to be positive with a suitable integer M, so that a physical delay line can implement it For in-phase recombination in a direction different from broadside or if a nonuniform-phase illumination is used (such as sidelobe suppression methods, such as [16] or using [9] to compensate for mutual coupling by altering phase and amplitude tapering), the adjustment phase delay determined via (18) must be altered according to the desired tilt angle from broadside and physical spacing between the elements.
The adjustment phase shifter completes the current section.The overall physical spacing between core sections is affected by the choice of φ I , z I , and therefore, different choices may lead to larger or tighter spacing.Additional N • 360 • can moreover be added to any of the involved tx lines section, if required due to mechanical constraints or to mitigate mutual coupling.The overall physical spacing between core sections, however, can be instead reduced, if necessary, by meandering or slow-wave lines, for instance, leveraging [17].

III. LIMIT CASES
The procedure outlined in Section II is, in principle, very general but might yield unacceptable results, for a given implementation technology, or not succeed at all.These limitations arise as a consequence of the absence of assumptions on the core element: for instance, one might try the technique on a very poor radiator, such as a very narrow microstrip patch, with extremely small radiation capabilities (S 11 = S 22 ≈ 0, S 21 ≈ 1); trying to enforce (5) on the last element (m = 0) would fail, as the corresponding (9) could hardly be equated to 0. Intuitively, a poorly radiating element is ill-suited (e.g., requires extreme impedances) to be the core of the terminal section or of the very next sections in an array.
This depiction shows one way of failure or returning an unacceptable design.A systematic discussion of other constraints and countermeasures is presented in the following, still avoiding any further electromagnetic simulation.

A. Gain Extreme Values
The surface of G I C M A P (0) as a function of z (n)  I and of φ (n) I exhibits maxima and minima, which may be incompatible with the values required by the illumination [e.g., (5)].The surface can be described by contour lines, of which a parametrization has been given in (11).Some values of G P , however, yield empty contour lines, which occur when (11) has no real solutions; the boundary between two or zero real solutions for φ I occurs when the radicand is zero where the three coefficients α, β, and γ all depend on For a given G P , up to two values of z I represent the leftand right-most extents of the contour.Such extreme values are denoted here as z E I (G P ) and are computed as follows: If there are no real values for z E I (G P ), the contour is empty, meaning that the surface never takes the G P value.This happens when the outer radicand in (20) is complex, which starts to happen when the innermost radicand is zero (β 2 = 4αγ ).The condition yields the extreme value of operating power gain achievable with the chosen core section which yields a positive real result due to S representing a strictly passive network.This limit value creates two possible ranges: G P < G P L or G P > G P L .In most cases, G I C C > 0 and G I C C < G P L ; thus, the achievable gain is determined by the first range: 0 < G P < G P L .If the illumination function requires values beyond these limits, the core element must be altered.However, bounds on the practically acceptable values of impedance for the input phase shifter might further restrict the range of attainable gain.

B. Realizability of Mapping Phase Shifter
Once a pair z (n) I , φ (n) I has been selected to provide the gain required by the current stage, the corresponding needed reflection coefficient (n)  M at port 2 of the core element is known.While other mapping structures (mentioned in Section III-C) do not impose restrictions, if this task is to be accomplished by a simple phase shifter, certain constraints apply.
Equation ( 13) introduces the formulas for computing the needed impedance z (n)  M and electrical phase delay φ (n) M , with the condition for physical realizability.The condition can be depicted by M belonging to one of two 0.25-radius circles in the complex plane.These two circular regions for M must then be converted back to a condition on z I , φ I , so that realizability boundaries can be drawn together with the constant gain contours, as done in Fig. 4. The boundaries are described by the real solutions (up to 4) for every value of z I The region (inner or outer with respect to this boundary), which contains the z I = 1 axis, is the acceptable one, if |S 11 / + 0.5| ≤ 0.5 ∨ |S 11 / − 0.5| ≤ 0.5.

C. Countermeasures
An arbitrary core element might prevent some steps of the synthesis from succeeding do to a few occurrences: 1) All candidate pairs z I , φ I fulfilling the required operating power gain would require unacceptable z M impedances.
2) The required power gain can only be obtained by unacceptable z I impedances for the input phase shifter.
3) The required operating power gain at step n is beyond the limit values of achievable gain of (21).To tackle the first case, one can simply draw the z I , φ I contour for the needed value of operating power gain according to (11) and superimpose the boundary determining physical realizability of the mapping phase shifter (22), as done in Fig. 4. Based on these limits, it is immediate to identify what pairs of z I and φ I yield a realizable mapping phase shifter and select one as appropriate.In case the whole operating-powergain contour does not cross the physically realizable-mapping boundary, then the whole contour is either acceptable or not.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.When there are no acceptable pairs, however, a possible countermeasure is to introduce a phase delay φ 2 on port 2 of the core element (S 22 ← S 22 e −2 jφ 2 , S 21 ← S 21 e − jφ 2 , and S R2 ← S R2 e − jφ 2 ).This addition amounts to inserting a z = 1 matched transmission line section (Fig. 6) to the core element and changes the boundary of ( 22) for physical realizability of the phase shifter.This can similarly be seen as presenting the same required M to the original core section, but implementing it by means of two phase shifters: one with unitary normalized characteristic impedance rotate M and one nonunitary to complete the mapping function, which always provides a solution.Other options can be similarly devised for implementing the impedance mapping with different structures, i.e., a single-stub network.
The very same approach can be considered also for tackling the other two failures.If the required operating gain contour demands unacceptable z I , an additional phase delay (unitaryimpedance transmission line section) on port 2 of the core does alter neither the operating power gain surface (9) nor its contour lines (11).Conversely, an additional phase delay φ 1 on port 1 of the core element (S 11 ← S 11 e −2 jφ 1 , S 21 ← S 21 e − jφ 1 , and S R1 ← S R1 e − jφ 1 ) does alter the operating power gain surface and, hence, the available z I , φ I contour pairs and the corresponding mapping phase shifter.This countermeasure, however, can only help limitedly, since just the B quantities in (9) are affected in the gain surface, and the gain bounds of (21) still remain.An additional delay φ 1 is not, thus, sufficient to address the third failure possibility: in these cases, a different core element must be chosen, although possibly only in the unrealizable sections of the array.

IV. PRACTICAL PROCEDURE
The design procedure, under the assumption of maximizing broadside radiation for an array of N elements, requires knowledge of the desired illumination distribution along the array G (n) P for n = 1 to N , e.g., using (5) for uniform illumination.Electromagnetic data of the 2-port antenna core element must be known, described by S 11 , S 21 , and S 22 and the radiated field in the observation point, expressed as S R1 and S R2 , whose absolute magnitude and phases are irrelevant.The procedure, sketched in Fig. 7, provides the inline antenna array as cascade of core elements with interconnecting phase shifters, i.e., transmission line sections of different characteristic impedances.Parasitic reactive effects due to abrupt impedance changes can be compensated by using equivalent circuits, available in the literature for several implementations, e.g., rectangular waveguide discontinuities and microstrip lines.As visible, no electromagnetic simulation or optimization of the whole array is needed.However, if the feasibility constraints are not fulfilled (e.g., impedances are beyond manufacturable limit), then circuit and/or electromagnetic optimization would be needed.Similarly, if additional requirements, such as bandwidth or squint limits, are introduced, the final array should be optimized, although using the design result as good initial point.When losses are not negligible, for instance, when using low-quality substrates or conductors and especially at mm-wave frequencies, each section will dissipate; therefore, a fraction of power will neither be radiated nor forwarded to the following sections.Based on this, the power gains of each section at design time should be increased slightly.For example, for uniform illumination in a two-element array where both sections loose about 1 dB, the power gain of the last-but-one should be increased to −2.6 dB, instead of the theoretical −3 dB.The precise amount of correction, however, strictly depends on which elements introduce losses and a rigorous treatment is very difficult.

V. EXAMPLES
This section presents manufactured examples, all dedicated to broadside microstrip arrays involving uniform illumination, for simplicity, and to prove the design extensibility.The first example is a step-by-step demonstration of the procedure to N = 4 elements at 21 GHz.The second instead proves the extensibility of the previous array to a longer one.The last example investigates the 31-GHz empirical prototype of [8], using the improvements presented here.

A. 4 × 4 Patch Array at 21 GHz
A simple patch has been designed according to the basic design formulas [10] applied to a Rogers 4350B substrate 10-mil thick.The by-the-book unoptimized layout is shown in Fig. 8, with a 35-µm-thick copper for the patch and two Z 0 = 50-feedlines.Reasonable impedances achievable with this technology are between 25 = 0.5Z 0 and 120 = 2.4Z 0 .
1) Simulation: The patch with the two feedlines is modeled in Ansys Electronics Desktop.Both ports are set up to have the electric field directed from ground toward the trace at the center of the feedline, and reference planes deembedded up to the recessed feedpoints.The far-field observation point is placed in far-field at 14 000λ (corresponding to 200 m) on top of the patch, to sample electric field along the polarization axis.The 3-D simulation is run with a tight convergence requirement, since the whole array will be designed based on this single simulation.The simulated behavior is shown in   2) Considerations: The considered core element creates the G P surface of Fig. 4, evaluated via (9).The operating gain for z I = 1 is G I C C = 0.8147, from (10).The achievable power gain, via (21), is bound to be within 0 and 0.884, making this core element good for N < 9 arrays (which need to radiate significantly via each section) and poor for N ≥ 10 (whose initial sections radiate very little, as visible in Fig. 3).
To verify this intuition, a few contour lines can be drawn using (11): specifically, the figure shows the contour lines corresponding to the power gains, which are needed along a uniform-illumination array, following (5), confirming the computed bounds.By further inspection via (11), a 0 power gain can be obtained with z I > 2.46 or z I < 0.17, although both values are unfeasible with the chosen technology.
3) Terminal Section: The terminal section should exhibit a null power but, as shown, this requires unacceptably high or low impedances for the input phase shifter.However, the choice z (N ) I = 2.4 and φ (N ) I = 81 • gives a gain of 0.038 (−14.2 dB), which is acceptably close to the ideal 0. With this choice of input phase shifter for the terminal section, (4) can be used, which yields the supposedly unimodular reflection coefficient (N ) M = 0.987∡29.9 • .Approximately, however, the output section can still be an open-circuited stub with z (N ) M = 1 and electrical length obtained by (12) as φ (N )  M = 165 • .The nonrealizability region for the mapping phase shifter (gray in Fig. 4) does not apply to the terminal section.The radiated field can be computed via ( 14) as S I C M A(N ) 31 = 2.58∡−79 • : values larger than unity can happen due to normalization.
Only for demonstration purposes, the obtained terminal section can be simulated (Fig. 10).Due to the series reactances created at impedance discontinuities [18], a high-impedance phase shifter enclosed between two 50-wider traces is altered.Proper compensation of such effects can be computed: applied to the 120-input line with an ideal electrical length of 81 • enclosed between 50-interfaces (port 1 of the core, and section's port 1), formulas suggest that the physical line should actually be 118 and 75 • long, as simulated in Fig. 10.
4) N − 1 Section: The last-but-one section in a uniform-illumination array must exhibit an operating power gain G N −1 P = 0.5 (−3 dB).The corresponding contour in Fig. 4 is within the achievable gain limits; however, as a nonterminal section, the realizability region of the mapping phase shifter (the shaded gray area in the figure) is here relevant.In this case, we chose z (N −1) I = 2.04 and φ (N −1) I = 47 • , which requires (N −1) M = 0.714∡0.8• .The choice results from sampling a few pairs on the contour and within the realizability region of the mapping phase shifter and selecting the one corresponding to the most realizable mapping impedance applying (13) at every sample.The obtained mapping phase shifter in the chosen point is z (N −1) M = 120 and φ (N −1) M = 85 • .Radiation from the current section is predicted by ( 14) as S I C M A(N −1) 31 = 1.85∡−60 • .The transmission through section N − 1, except for the adjustment shifter, is computed by (15) and found to be S I C M(N −1) 21 = 0.707∡19 • .Thus, the adjustment phase shifter must properly delay the forwarded portion to match the −60 • from the N − 1 section, thus fulfilling (18) to give φ (N −1) A = 0.3 • .5) N −2 and N −3 Sections: To complete the four-element array, we apply the same steps of Section V-A4 twice more.The N − 2th section must implement an operating power gain of G (N −2) P = 0.67 (−1.7 dB); thus, the contour available in Fig. 4 is even larger.For realizability of the mapping phase shifter, the input shifter is selected as far as possible from the boundary: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.11, with simulated and measured performance.Specifically, simulations are carried out for the full electromagnetic layout and also for the ideal array: the former exactly reflects the designed layout, whereas latter is obtained as computation of the ideal array factor applied to an array of terminal sections, used as a reference of the theoretical uniform illumination.
Electromagnetic simulations of the full array predict the return loss peak at 21.07 GHz, whereas the measurement shows it at 21 GHz, as intended.Such shift is anyway well within usual inaccuracies and fabrication spread and tolerance.
The overall array gain in boresight in the approximate array simulation predicts optimistically 16.2 dBi, neglecting all distribution losses (about 1 dB) and undesired couplings.The full electromagnetic simulation instead predicts 15.1 dBi at 21 GHz, while the measured peak gain is between 13.8 and 14.8 dBi.Such uncertainty, due to a spurious reflection in the anechoic chamber during the gain measurements (not present during radiation pattern cuts) unfortunately spotted when the chamber was no longer available, indicates some unexpected loss, possibly due to the uncalibrated connector and transition.
The normalized radiation patterns, measured and simulated by full electromagnetic simulation, instead agree very well in the E-plane, whereas the approximated array simulation again optimistically predicts lower and symmetrical sidelobes, neglecting the effect of the binary division tree on one side.The H-plane cut also shows a very good respondence in terms of sidelobe levels, although with minor differences in the notches angles between the full electromagnetic simulation and measurement.These can be ascribed to typical fabrication inaccuracies as well as imperfections of the simulation model, for instance, due to the simple first-order radiation boundary used.Overall, the agreement between measured and expected behavior (either as approximate array simulation or full array simulation) is rather good, especially without optimization.

B. 8 × 1 Patch Array at 21 GHz
The example of Section V-A successfully synthesized a 4 × 1 array with uniform illumination and radiating in broadside.The proposed technique, in case of uniform illumination,  allows to directly extend the array by adding more sections without redesigning the previous ones.This is possible due to the fact that the only dependence on the number of elements (N ) lies in the operating power gain, but (5) actually depends only section counted backward from the terminal one.
To prove this simplification, an N = 8 × 1 array at 21 GHz has been addressed.The four terminal sections, in fact, require exactly the same power gain as the design of Section V-A5, and thus, the whole 4 × 1 array can be reused.Furthermore, four elements n = 4 to 1, corresponding to m = 4 to 7, are appended at the input.The contours of the G P surface are the remaining ones in Fig. 4, along which the following pairs have been chosen: z The simulation confirms that the synthesis outcome is successful, with a very good matching at 21.04 GHz; measurements instead show the best return loss at 20.85 GHz and a still acceptable level even down to 20.5 GHz, whereas the simulated |S 11 | curve already reaches −7 dB at 20.75 GHz.
The H-plane radiation pattern is very smooth, although measurements show a steeper decay beyond 55 • ; the measured E-plane radiation pattern follows very well the one simulated as approximate array, although with an asymmetry due to the protruding connector and feeding cable on one side, neglected by both simulations.The sidelobe levels also coincide reasonably well with the approximate array simulation (using ideal uniform illumination from an array of decoupled patches), thus implying that the designed network indeed provides a good uniform and in-phase illumination from all sections.The simulated gain is between 13.2 and 13.4 dBi in the full electromagnetic simulation and in the approximate array one, but unfortunately no gain measurement has been possible in the allotted timeframe for the characterization setup.Again, the agreement between measurements and expected results is rather good, and the design has been carried out merely by extending the 4 × 1 array without any redesign.
For comparison, a traditional inline 8 × 1 array of resonating elements would require halving the radiation resistance of each with respect to an already-designed 4 × 1 array, thus needing further simulations or an analytical equivalent model.
C. 5 × 1 Butterfly Patch Array at 31 GHz From [8] The example provided in [8] involves a core element whose power gain surface reaches only G P = −5 dB, which is insufficient for a terminal section: that paper, thus, implements a shift of the reference plane of the core element in the terminal section.In light of the bounds described in Section III-A, that modification is now explainable: the computation of the analytical maxima and minima for gain via (21) yields 0 < G P L < 0.898 (compatible with an ideal terminal section) but achieving a very low gain requires unacceptably impedances for z (N )  I : (20) for G N P = −15 dB, in fact, returns 160 or 20.4 .The suggested countermeasure of adding a phase delay to port 1, thus, is the only choice: a φ N 1 = −30 • section could be removed here, which brings the required input impedance to an acceptable value [about 110 , confirmed by (20)].
For the other sections, instead, this additional delay is not needed, as their original contours include acceptable impedances.Instead, another issue arises due to the realizability of the mapping phase shifters: most of the required contours for the nonterminal sections require extreme z M values.The empirically proposed solution was to introduce an offset (φ (n) 2 − 30 • ) in port 2's reference plane of the core element in all the nonterminal sections.This phenomenon is correctly predicted by the boundary proposed here as (22), which follows the unrealizable (white) regions of [8, Fig. 5].

D. 4 × 1 Slotted Waveguide Array at 27.25 GHz
As further example of application of the proposed technique, a slotted waveguide example is described, designed for application in a fixed-wireless access base station.The radiating element is a 45 • -slanted slot in the narrow wall of a WR28 waveguide (7.11 × 3.56 mm).The slot itself is not 13.Core element as narrow-wall slot in WR28 waveguide and its simulated scattering parameters and radiation characteristics at 27.25 GHz.optimized: 1.8-mm wide and carved into the 0.2-mm-thick metal waveguide.With the required single electromagnetic simulation of the core element, its scattering parameters are obtained at 27.25 GHz, as reported in Fig. 13.
The scattering parameters are provided as input to the procedure in Section IV in order to design a four-element antenna array with electric field along the transversal direction, y.
The resulting cascade of phase shifters is then converted to waveguide, where the narrow-wall height is modulated according to the required impedances.Based on the reactive effects of such discontinuities, an analytical compensation is applied to embed them into equivalent phase shifters, which are then implemented by actual waveguide sections.
The final design is 50-mm long, as shown in Fig. 14, where the terminal section is closed by a short-circuited stub; the spacing between elements is about one wavelength, and slots are reversed to provide the correct constructive superposition along the desired polarization axis.The magnitude of S 11 is also very good at the design frequency, being less than −25 dB, implying that the impedance-matching objective was reached.From basic array theory, the expected gain is expected 6 dB above the core element, thus 10 dBi, which is reasonably close to the simulated value (without losses) of 10.5 dBi.
For comparison purposes, a traditional design has also been carried out to obtain radiation in broadside direction, also shown in Fig. 14, because of the available equivalent model developed ad hoc (see [19] and following works) constituted by a simple resonator.Half-wavelength sections are used to have many such resonators in parallel and radiating in phase, and thus, the resonator must be properly chosen depending on the number of elements of the array.The overall footprint of the traditional design is smaller (42 mm) because of the reduced spacing, and its peak gain is about 10.7 dBi.On the other hand, the closeby sidelobes are lower for the proposed array, while the sidelobes at 40 • are lower in the traditional design.
To verify the changes to aim in nonbroadside direction, the array visible in Fig. 14 (left) can easily be modified.Only the adjustment phase shifters are affected, and their lengthening can be computed analytically, so that the maximum radiation occurs at 10 • , as reported in Fig. 15.
The return loss of the array designed at 10 • is also better than 20 dB at the design frequency, and the squint graph confirms that the peak gain of 10.5 dBi occurs slightly above 9 • .The squint of the traditional design is limited, from −1 • to +3 • , whereas the proposed design (designed at 10 • ) has a larger squint, from +1.8 • to +12 • .No folding of the phase shifters has been leveraged in this example, which justifies the larger footprint and the increased squint in frequency.This example confirms the simple adjustment needed to steer the proposed array, whereas the traditional array design would need complete redesign, since altering the electrical delays between elements would also detrimentally affect the needed transformations for impedance matching.

VI. CONCLUSION
The method outlined here discusses a technique for the design of inline antenna arrays requiring ideally only a single electromagnetic simulation.It relies on a single core network, which must be properly simulated in its scattering and radiation characteristics.These are then manipulated by means of analytical formulae to compose the complete array by interposing phase shifters.Every section of the array is designed basically independently of the other stages.Although the synthesis works at a single frequency, the obtained array can constitute a good starting point for later wideband optimizations.Due to intrinsic limitations, which may arise due to the chosen core network, the design flow may fail at some stage, but simple modifications can overcome them.The method moreover allows to accommodate different core sections along the array, if needed, and different impedance-transformation networks, such as stubs.
Two prototypes have been designed and manufactured without any optimization, and measurements confirm the expected behavior.Specific limitations emerged in a previously manufactured sample have also been justified.An additional example in slotted waveguide has been provided, to validate the procedure and also show the easy modification necessary to aim the array not in broadside.

APPENDIX I DERIVATIONS
For easy reference, some of the relations used in this work are derived in this appendix.Scattering parameters of an arbitrary 2-port network are defined, as shown in Fig. 16, referred to an impedance Z 0 .When an arbitrary load, expressed as reflection coefficient L referred to the same impedance Z 0 , terminates port 2, the reflection coefficient observed looking into port 1 is obtained by eliminating variables a 2 and b 2 replacing L = a 2 /b 2 .The result is used, for instance, in (2).
Scattering parameters referred to impedance Z 0 corresponding to a phase shifter with phase delay φ and its own characteristic impedance Z p can be computed starting from the scattering parameters referred to Z p : S 11,Z p = S 22,Z p = 0 and S 21,Z p = S 12,Z p = e − jφ .These are then transformed by a change of reference impedance [20] The result is found in Fig. 3

(top).
When two reciprocal 2-port networks A and B are cascaded by joining port 2 of network A with port 1 of network B, their Z 0 -referred scattering matrices S A and S B can be converted to chain matrices T A and T B , according to [20] This result is used in (3), (8), and ( 16) to expand the scattering parameters S I C of the cascade of the input phase shifter and of the core section and also in (15) to cascade the mapping and adjustment sections as well ( S I C M A ).
licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 2 .
Fig. 2. Structure of section n in the array and involved scattering parameters.

Fig. 4 .
Fig. 4. Colored background: achievable operating power gain surface [(9), in linear scale] as a function of z I and φ for the set of scattering parameters used in Section V-A; solid black: contour lines corresponding to the power gain values along a uniform-illumination array; gray-shaded regions with dotted boundaries: combinations of z I and φ I corresponding to a mapping network (when treating a nonterminal section) not realizable as a simple phase shifter, described in Section III-B.

Fig. 6 .
Fig.6.Additional delays on the core element and modified parameters.

Fig. 8 .
Fig. 8. Base element for 21-GHz examples of Sections V-A and V-B.The bottom wall is modeled as infinite copper ground plane.The top view shows the reference planes where the scattering parameters are computed and the electric field mode polarity of the two ports.

Fig. 9 .
Fig. 9. Base element for 21-GHz examples: simulated scattering parameters versus frequency and gain patterns along E-and H-plane cuts at 21 GHz.Only values at boresight and at 21 GHz are used for the synthesis technique.

Fig. 9 ,
Fig. 9, although not strictly needed for the design procedure: reasonable matching and a slightly asymmetry in the H-plane cut, due to the nonoptimized design of the base element.Relevant to the array synthesis, instead, are the scattering parameters at 21 GHz and the polarized field component in the observation point when fed alternatively with 1 W from port 1 (giving 26.1 mV/m with phase 174 • , while port 2 is closed onto a 50-load) and from port 2 (26.1 mV/m with phase 354 • , while port 1 is closed onto a 50-load), normalized to the former S 11 = −0.05382+ j0.05101; S 22 = −0.05201+ j0.05056 S 21 = −0.85484+ j0.31274; S R1 = 1∡0 • ; S R2 = 1∡180 • .

Fig. 10 .
Fig. 10.Terminal section using the 21-G H z core element of Section V-A1.An additional 50-line is added as input, though entirely numerically de-embedded.Equivalent model and simulated performance are also shown.

=
42 • ; z (N −6) I = 0.77, φ (N −6) I = 49.3 • ; z (N −7) I = 0.63, and φ (N −7) I = 51 • .Again, formulas to compensate the reactive effects of width discontinuities in microstrip have been applied to create the final layout.The result is shown in Fig. 12, as modeled in Ansys Electronics Desktop HFSS and as manufactured without any optimization.

Fig. 14 . 4 ×
Fig. 14. 4 × 1 array designed for maximum radiation in broadside direction with linear polarization along y and corresponding traditional design.Simulated radiation patterns at 25.5 GHz are shown on the top right.

Fig. 15 . 4 ×
Fig. 15. 4 × 1 array designed for maximum radiation toward 10• with linear polarization along y, and peak gain-versus-frequency behavior and S 11 response versus frequency.As reference, the simulated behaviors of the traditional design aiming in broadside (visible in Fig.14) are also shown.The only elements changed with respect to the one aiming in broadside are the adjustment phase shifters, highlighted in green.

• 1 − 1 .
, from Z p to Z 0 [defining = (Z 0 − Z p )/(Z 0 + Z p )] S11 S 12 S 21 S 22 = S 11,Z p − S 12,Z p S 21,Z p S 22,Z p − S 11,Z p − S 12,Z p − S 21,Z p 1 − S 22,Z p − . The overall chain matrix is then T = T A T B , which is finally converted back to the overall Z 0 -referred scattering matrix SS = S 11A − S 11B A S 21A S 21B S 21A S 21B S 22B − S 22A B 1 1 − S 11B S 22A.