Resonant Impedance Tuners: Theory, Design, Power Handling, and Repeatability

The theory of coupled-resonator-based impedance tuners is presented for the first time, establishing coverage limitations for two- and three-resonator designs. Besides the advantages of resonant impedance tuners, practical considerations including resonator loss and limited tuning range are included in the analysis. From this theoretical background, a rigorous design procedure is presented for creating optimal resonant impedance tuners, designed for maximum Smith chart coverage, for both the two- and three-pole structures. A proof-of-concept three-pole impedance tuner is created following this procedure. This state-of-the-art tuner achieves at least 90% Smith chart coverage from 4 to 8 GHz while maintaining minimum transducer losses of 0.4 dB. The designed tuner is implemented on printed circuit board (PCB) technology and uses electronically controllable linear actuators for tuning. With these design choices, a repeatability of 48 dB is achieved, while maintaining tuning times of less than 100 ms, and is capable of operating at 100 W. This represents performance competitive with the highest repeatability tuners available today. Moreover, it simultaneously achieves $\boldsymbol {10\times }$ faster response time and $\boldsymbol {> \!\! 10\times }$ smaller footprint than state-of-the-art highly repeatable tuners.

Several technologies have been reported in the literature for implementing impedance tuners.Some of these technologies include varactors and p-i-n diodes placed along transmission lines [14], [15], [16], [17], microelectromechanical systems (MEMS) [18], [19], and ferromagnetic tuners [20].Another popular implementation is the coaxial slug impedance tuner [21], [22], [23].Despite the wide variety of presented technologies, there has been very little work on rigorous designs that result in maximum impedance coverage or optimal loss.There has been some work toward basic impedance tuner topologies, such as two-element LC networks and the slightly more complex T-and Pi-networks [24], [25], [26].
A different class of implementation is the resonant tuner.These are created using coupled resonators, which are independently tunable.Tunable resonators can be implemented with lumped components or transmission lines using varactors for tuning [27], or tunable cavity resonators [28], [29].While this class of tuners has significant potential in outperforming the above-mentioned tuners within a specific bandwidth, there exists no published analyses or design processes for achieving optimal results.Consequently, no design processes or optimal structures are known today.
This article investigates, for the first time, the theoretical coverage limits for resonator-based impedance tuners for topologies involving two and three coupled resonators.Moreover, we present the fundamental coverage boundaries even when infinite resonator tuning range is assumed.Through this analysis, the key design parameters are established and a design procedure for maximum coverage is presented.In addition, the impact of practical limitations, including finite tuning range and resonator quality factor (Q), is shown alongside the theoretical results, demonstrating the performance impact of nonideal resonators.
As a demonstration of these analytical results and design procedure, a high-power and electronically adjustable resonant impedance tuner is created and experimentally investigated.The proposed tuner is composed of three coupled high-Q evanescent-mode (EVA) cavity resonators, which are individually controlled using high-precision external actuators.EVA cavity-based structures have been successfully used mainly to realize tunable high-Q filters [30], [31], [32], [33], [34], but also in realizing passive sensors [35], diplexers [36], and frequency synthesizers [37].Such designs have been fabricated by conventional machining for low-frequency applications [32], through silicon microfabrication for high-frequency miniaturized cases [38], [39], and in printed circuit board (PCB) technology using substrate-integrated waveguides (SIWs) for 1-10-GHz applications [40].
Using coupled EVA resonators implemented on a PCB as an impedance tuner was first introduced by Semnani et al. [28] and then improved through the use of external linear actuators in [29].However, this is the first work that: 1) explores fundamental coverage limits of resonant impedance tuners; 2) presents optimal designs; and 3) demonstrates high repeatability (48 dB) that simultaneously achieves 10× faster tuning speed and 10× smaller footprint than highly repeatable tuners.
The organization of this article is as follows.Section II lays out the circuit model along with the problem definition and assumptions for the resonant impedance tuners.These circuit models are analyzed and design equations are presented for the fundamental tuning limits of two-and three-resonator structures.Following the procedure discussed in Section II, Section III details the design process for a high-power, threepole impedance tuner with optimal coverage from 4 to 8 GHz.Section IV describes the fabrication, assembly, and testing process for this tuner and summarizes the measured results.

II. THEORETICAL ANALYSIS OF COUPLED RESONATOR IMPEDANCE TUNERS
This section presents the first theoretical analysis of coupled-resonator-based impedance tuners.The key objectives of this analysis are to: 1) identify the theoretical boundaries of the forbidden regions on the Smith chart, i.e., impedances the tuner cannot match to, assuming resonators with infinite tuning range for the two-and three-pole impedance tuners at a single frequency; 2) determine the maximum impedance coverage calculated as the percent of the Smith chart coverage within the allowed regions for practical resonators with realistic (finite) tuning range and quality factor; and 3) develop a rigorous design procedure for creating optimal resonant impedance tuners for both the two-and three-pole structures.In Sections III and IV, the theoretical analysis presented here will be validated experimentally with a state-of-the-art three-pole impedance tuner implementation, simultaneously exhibiting low loss, high tuning range, wide bandwidth, and high power handling.

A. Circuit Model and Assumptions
Fig. 1 shows the general circuit topology of a coupledresonator-based impedance tuner.As the circuit model is general, the following analysis is not specifically tied to a particular technology or implementation method and is therefore useful for any circuit that can be modeled in such a way.The two transformers located at the input and output of the circuit represent the external coupling mechanisms, with coupling strength given by their transformer ratio n.The inter-resonator couplings are represented by admittance inverters, with coupling strength defined by the variable J .The shunt capacitor, inductor, and resistor between each coupling represent a parallel resonator whose lumped element parameters are chosen to accurately reflect those of the resonator technology being used for a particular implementation.
The admittance of each parallel resonator can be modeled explicitly, with Y i being the admittance of the ith resonator, and Y i = G i + j B i .The conductance of each resonator represents the associated loss and is expressed as R cond , and R die representing the radiated, conductor, and dielectric losses, respectively, and C i the equivalent resonator capacitance.The three loss terms along with the capacitance are combined into a single term for the unloaded quality factor of the resonator, Q i .The susceptance of each resonator is given by B i = ωC i − 1/(ωL i ), with L i being the equivalent inductance.
In the discussions to follow, transducer loss is presented alongside the impedance tuner's Smith chart coverage, since S 21 does not take into account the impedance transformation.Similar to unilateral transducer gain for an amplifier, the loss associated with the impedance tuner is therefore calculated as the following: (1) To simplify the theoretical analysis, a few key assumptions are made, which are listed in Table I and will now be discussed.The impact of removing particular assumptions will be addressed later in Section II-D.Starting with the first assumption, it is assumed there are no source or load transmission lines, or the electrical length of all the transmission line sections is taken to be zero.Next, it is assumed that the impedance tuners are symmetric.For the third assumption, all the transformer ratios, inverter values, and losses are treated as constant, despite them actually varying with frequency and tuning.These first three assumptions help simplify the equations and the corresponding results without loss of generality.
The fourth assumption assumes that each resonator can be independently tuned, which provides significantly more impedance coverage than requiring all the resonators to be tuned identically.The resonators are also treated as infinitely tunable, which conforms with the objective to establish fundamental impedance coverage limits for the relevant circuit topology.While real impedance tuners may not achieve these results, they still provide insight into the key design parameters and tradeoffs.The next assumption treats all the analyses at a single frequency.Reconfigurable impedance tuners inherently have narrow instantaneous bandwidths for a given match.The specifics of this bandwidth will vary based on several factors, such as impedance being matched, Q of the resonators, and coupling strengths.Operating bandwidths will refer to the range of frequencies at which the tuner provides above a specified threshold of Smith chart coverage.
For the last assumption, it is assumed that the performance of either a two-or three-pole impedance tuner will meet the requirements for most real-world applications.Although it is possible to create a single-pole impedance tuner, its performance is limited.The input admittance of a single-pole tuner is given by It can be seen to trace a circle of constant conductance on the Smith chart based on the ratio of external couplings n1 and n2.The addition of transmission lines at the input and output can rotate the constant conductance circle around the Smith chart, but it will always have limited matching performance.For higher order tuners, the additional losses and increase in physical size offset the small gains in Smith chart coverage compared with the two-and three-pole designs.This is discussed in more detail in Section II-D.
In Sections II-B and II-C, the maximum Smith chart coverage limit and an optimal design strategy will be derived for the two-resonator case, followed by the three-resonator case.

B. Two-Pole Impedance Tuner
In Fig. 2(a), the circuit model of a two-pole resonantbased impedance tuner is shown.The input admittance for the two-pole circuit is the following: A sample of the Smith chart coverage of the two-pole tuner is illustrated in Fig. 2(b).There are two regions bounded by constant conductance circles which are not covered by this tuner.The first, bounded by G max on the left side of the Smith chart, is due to circuit topology.Even considering lossless resonators with infinite tuning range, complete coverage of this region will never be possible.However, specific design choices can move the boundary, which will be discussed later.The second region, bounded by G min and the outer edge of the Smith chart, is primarily due to losses in the resonator, although its size is also impacted by the external coupling strength, to be discussed later.With lossless resonators, G min extends to the outer edge of the Smith chart, and the second region is covered.Furthermore, between the two regions bounded by G max and G min , any impedance point can be matched by the tuner.The spacing shown between points plotted on the Smith chart within the two regions is due to the simulation of a finite number of resonator combinations, and not the inability of the tuner to match to these impedances.
The first step in determining the Smith chart coverage limits of the two-pole tuner is to calculate G max .Since the desired result is a maximum of the input admittance, it can be found by taking the real part of the derivative of (3) with respect to the resonators' susceptance, setting the result equal to 0, and solving for B i from the following: To simplify the calculation, lossless resonators are assumed, i.e., G i = 0 for i = 1, 2. For high-Q resonators, this is a reasonable assumption with minimal impact on the final result, as discussed later.By inspection of the circuit model in Fig. 2(a), varying the susceptance of the first resonator only impacts the imaginary component of the input admittance.Thus, the partial derivative in (4) is taken with respect to the second resonator only and is solved from the following: The details of this calculation can be found in Appendix A, but this results in the following: The solution of (6) yields the simple result: B 2 = 0. Specifically, G max is reached when the second resonator is at resonance.Substituting this result into (3) yields the following: The above result is derived for infinite Q.While not realistic, it provides beneficial insight on the key design parameters that become hidden when the loss terms are included.For finite Q, the same procedure is followed, which results in a correction factor: The ideal case is still a reasonable first-order approximation.For the circuit values used within the coverage plots of this work, the ideal result has an 11.7% error in the size of G max boundary at a quality factor of 10.However, the error decreases to 2.6% at a quality factor of 500.These errors correspond to errors of 3.5% down to 0.02% of the total Smith chart coverage.
The next step is to calculate G min .Since practical resonators will always have loss, G min will never be zero and therefore must be accounted for.This is limited by the minimum conductance achievable when accounting for resonators with finite Q.One can solve for G min by setting |B 1 | and |B 2 | to ±∞.The solution simplifies to the following, with details contained in Appendix A: It can be seen from the results of G max and G min that they are both directly proportional to a positive power of n.This presents an optimization problem, since one of these values should be maximized while the other is minimized.Therefore, an intermediate value for the external coupling coefficient must be found to achieve an optimized tuner design.
To find the optimized value for n, an equation for the coverage area is derived using the boundaries of G max and G min .To convert each boundary into an area, their corresponding reflection coefficients are first calculated from the following: Since these circles are tangent to the left side of the Smith chart, the diameter of each circle is + 1, and the coverage area is given by To calculate the percentage of Smith chart coverage, ( 12) is divided by the area of the unit circle in the complex plane (equal to π).Substituting the solutions for G max and G min , respectively, and assuming Y L = Y 0 , the percentage of Smith chart coverage simplifies to This equation can then be used to optimize the value of n for maximum Smith chart coverage given a specified quality factor based on a particular choice of resonator.
As a demonstration of the above analysis, the Smith chart coverage for two-pole impedance tuners under several different conditions is now examined.Referring to the corresponding circuit model in Fig. 2(a), the inter-resonator coupling coefficient is given a value that is both strong and physically realizable.Typical values for J at low GHz frequencies are between 0.01 and 0.03, and therefore, 0.02 is used for this analysis.The external coupling is optimized based on the remainder of the circuit parameters using (13), constrained to values of n between 1 and 3. Fig. 3 shows plots of the results of this optimization, showing the transformer ratio that yields maximum Smith chart coverage for a given quality factor.
In Fig. 4, the impedance coverage for a two-pole tuner is plotted at several different quality factors.From the theoretical calculations for G max presented above, lossless resonators were assumed.It is now evident that this is a reasonable assumption, with G max showing very little change especially at quality factors of 50 or greater.
In Fig. 5, the impact of resonators with finite tuning range on the coverage performance of the two-pole tuner is demonstrated.Similar to finite quality factor, this change has minimal impact on G max , which is bounded when the second resonator is at resonance, the center of the tuning range.However, the impact on G min is more pronounced.The theoretical G min is approached when the susceptance of the two resonators goes to ∞.With finite tuning range, the resonators cannot achieve this condition, and thus, the overall size of the region bounded by G min is reduced, along with coverage within the region.For small tuning ratios, such as the 2:1 ratio presented in Fig. 5(a) and (b), there is portion of space within the bounded region that is not covered.This is a practical limitation for systems with finite tuning range.While the boundaries G max and G min represent the largest and smallest achievable conductances, respectively, when practical resonators are considered, there is no guarantee that all the impedances within the regions can be matched.From the above analysis and discussion, a design procedure is generated for an optimal two-pole impedance tuner as follows.
1) Design the resonator for maximum unloaded quality factor while meeting the other design requirements to achieve minimal losses while reducing G min .2) Design the inter-resonator coupling as strong as practically possible to maximize the value of G max .3) Extract the value of J corresponding to the above inter-resonator coupling strength.4) Solve for the optimal external coupling strength using (13).
Using this design procedure, Table II presents the minimum and maximum conductances, percent of Smith chart coverage, and the associated transducer loss, of a two-pole impedance tuner designed for optimal Smith chart coverage at several unloaded quality factors.The transducer loss reported in this table is the loss when source and load impedances are 50 .The actual spread of transducer losses is reported by the color and associated scale on the Smith chart coverage plots.

C. Three-Pole Impedance Tuner
The analysis for the three-pole impedance tuner follows the same procedure as for the two-pole case.The circuit topology and sample Smith chart impedance coverage are shown in Fig. 6(a) and (b), respectively.The input admittance for this circuit topology is the following: The same two coverage boundaries G max and G min for the two-resonator case exist for the three-resonator case.However, G max is much larger for the three-resonator case, resulting in a much smaller bounded region of no coverage.As with the two-resonator case, any point between these two bounds can be covered.Open space between data points is simply a result of simulating a finite number of resonator combinations.
Similar to the two-resonator case, the first step in determining the Smith chart coverage limits of the three-resonator tuner is to calculate G max .The resonators are again assumed to be lossless to help simplify the analysis, while the first resonator can also be neglected since its varying susceptance only affects the imaginary component.However, the susceptance of both the second and third resonators must be calculated using the partial derivatives in the following equation.The full details of this derivation are provided in Appendix B, but simplify to the system of equations These two equations are used to solve for B 2 and B 3 , and when simplified lead to the result: B 2 = 0 and B 3 → ∞.Substituting these values back into (14) and simplifying yield Thus, the maximum conductance occurs when the second resonator is tuned to resonance and the third resonator is at its highest possible frequency.Although this result shows any impedance is possible with lossless resonators and infinite tuning, any realistic implementation will be limited in its maximum achievable conductance.Unlike the two-resonator case, this equation for G max cannot be evaluated solely based on circuit design parameters.Instead, a numerical simulation is needed to calculate the maximum conductance based on the quality factor and tuning range of the resonators.
The next step is to calculate G min for the three-pole case.This is done by setting |B 1 |, |B 2 |, and |B 3 | to ±∞.The details are contained in Appendix B, which leads to the following result: Since only G min is directly proportional to a positive power of n, then the optimal design strategy is to make n as small as possible, i.e., external coupling as strong as possible, instead of requiring the same optimization procedure as in the twopole case.However, the same equation can be used to find the Comparing the three-pole results for G max and G min to the two-pole case, the limit for G min is the same.However, the maximum conductance of a three-pole tuner can be much higher than that of a two-pole tuner.This can easily be seen when comparing the coverage plots for the two-and three-pole tuners, e.g., Fig. 2 To demonstrate the above analysis, the Smith chart coverage of the three-pole tuner with varying conditions is now examined.Similar ranges for n and J from the two-pole case are used, with J equal to 0.02, a strong but realizable inter-resonator coupling value, and n of 1.1 is used for a very strong external coupling.
In Fig. 7, the impedance coverage for the three-pole tuner is shown at several different quality factors.Furthermore, in the calculation for G max , it is assumed that all the resonators are lossless.It can be seen in Fig. 7 that this assumption is not accurate for very low quality factors, as G max is much smaller for the case where Q is equal to 10 compared with the others.However, for quality factors of 50 and above, this assumption is reasonable.
The impact of resonators with finite tuning range is demonstrated in Fig. 8. Unlike the two-pole case, it can be seen that limiting the tuning range has an impact on both the boundaries of G max and G min .Regarding G min , its theoretical limit is approached when the admittance of each of the three resonators goes to ∞.With finite tuning range, the resonators do not reach this result, thus limiting the size of G min .However, with three resonators, the magnitude of their product approaches ∞ quicker than just two resonators.Therefore, the three-pole tuner is able to get closer to the theoretical limit of G min compared with a two-pole tuner with the same tuning range.Regarding G max , one of the conditions to reach this boundary requires tuning of the second resonator to resonance, with the third tuned to as low of a frequency as possible.With limited tuning of the second resonator, coverage up to the theoretical boundary of G max is reduced.
A design procedure for an optimal three-pole impedance tuner is now generated based on the above analysis.
1) Design the resonator for maximum unloaded quality factor given the other design requirements to achieve minimal losses while reducing G min .2) Design the inter-resonator coupling J to be reasonably strong.3) Design the external coupling n to be as strong as possible, with careful consideration toward minimizing the overall loss of the system.
Since the strength of the inter-resonator coupling has minimal impact on Smith chart coverage, and with no external coupling tradeoff to manage, the design procedure of the three-pole tuner is more straightforward than for the two-pole tuner.Using this procedure, Table III presents the minimum and maximum conductances of a three-pole impedance tuner designed for optimal Smith chart coverage at several unloaded quality factors.The transducer loss reported in this table is the loss when source and load impedances are 50 .The  actual spread of transducer losses is reported by the color and associated scale on the Smith chart coverage plots.

D. Performance Comparison and Analysis
From the above analysis of both the two-and three-pole impedance tuners, it was found that their minimum achievable conductances G min were exactly the same.Thus, for a numerical comparison of Smith chart coverage, the ratio of G max,3−pole to G max,2−pole is used.The value of G max in the three-resonator case will be much larger compared with the two-resonator case, all other parameters held constant, resulting in greater Smith chart coverage.
In Fig. 9, a comparison of percent coverage versus unloaded quality factor of a two-pole impedance tuner and a three-pole impedance tuner is shown, with identical circuit and coupling parameters.Demonstrating how these percentages translate into coverage, Fig. 10 presents Smith chart coverage plots comparing the two-and three-pole tuners with unloaded quality factors of 100 and both 2:1 and 4:1 tuning ratios.The important performance metrics for these designs are summarized in Table IV, directly comparing the coverage and transducer loss for the otherwise identical designs.The transducer losses were determined through circuit simulation.There is not an analytical formula to determine transducer loss from the quality factor of the resonators.If this could be derived, then it would be possible to relate the transducer loss to the coverage directly.
While the three-pole impedance tuner clearly achieves greater Smith chart coverage, there are practical considerations which must be taken into account.First, the physical size of

TABLE IV PERFORMANCE COMPARISON OF TWO-POLE VERSUS THREE-POLE TUNERS
the three resonator tuner will be larger than the two-resonator tuner, given the same choice of resonator technology.Second, as practical resonators will always have finite Q, the addition of a third resonator will increase the overall losses of the system.This is shown in Table IV, where the transducer loss of the three-pole tuner is always greater than the two-pole tuner, regardless of Q.Once again, the actual spread of transducer loss is always reported by the color and associated scale on the Smith chart coverage plots.As such, the choice between a two-and a three-pole design will be entirely dependent on the specific requirements of the application.One important conclusion drawn from the above results for resonant-based tuners is the following: there is no compelling reason to go beyond a three-pole design.Nearly 100% Smith chart coverage can be obtained with three resonators, and the minimal increase in coverage with the addition of a fourth resonator would certainly be offset by the increase in transducer loss.The decision between a two-and a three-pole design is primarily down to size restrictions and required Smith chart coverage.As an additional consideration in practical systems, total Smith chart coverage is not always required.For applications including amplifiers or antennas, even considering tuning across fairly wide frequency ranges, optimal impedances are often localized to specific regions of the Smith chart.Therefore, end-use of the impedance tuner also has a significant role in design choice.
To bring further insight into the design of resonant-based tuners, the effects of removing particular assumptions outlined in Section II-A are now considered.First, the effect of including transmission line sections at the input and output of the tuner is straightforward.As shown in Fig. 11, a rotation is applied to a given impedance point around the Smith chart in a clockwise manner, depending on the length of the transmission line.
The analysis up to now has been at a single frequency.However, in many circumstances, impedance tuners are required to function across a larger bandwidth of frequencies.While most of the concepts presented still hold, the assumption of constant coupling coefficients will no longer apply.Fig. 12 shows the impact of removing this assumption for a two-pole impedance tuner, where Fig. 12(a) is the original plot at f 0 , with an external coupling coefficient n equal to 1.5, and an inter-resonator coupling coefficient J equal to 0.02.The plot in Fig. 12(b) has the same circuit parameters but plotted at 1.2 f 0 , while Fig. 12(c) is also plotted at 1.2 f 0 , but with the inter-resonator and external coupling values varying by the following equations, with f being the frequency: By visual inspection of Fig. 12, the effect of varying the coupling coefficients is clear.Thus, when designing resonant-based tuners to operate over wide frequency ranges, it is important to also model how the coupling values change versus frequency.The tuner can be designed first for optimal performance at the center of the frequency range of interest, and then using the coupling model, an approximation of how the coverage will change with frequency can be evaluated similar to the example above.
Finally, the most interesting assumption to examine is the symmetric circuit.From the analysis for both the two-and three-pole tuners, the equation for maximum conductance was proportional to n 4 .Taking into account an asymmetric circuit, the maximum conductance becomes proportional to n 2  1 n 2 2 , while the minimum conductance becomes proportional to n 2  1 , the external coupling on the input side.With the input Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.and output external couplings separated, instead of finding an optimal tradeoff for a single value, n 1 can be made as small as possible to minimize G min , while n 2 is made as large as possible to compensate and increase G max .Therefore, an asymmetric tuner has potential to provide improved Smith chart coverage compared with a symmetric one.However, the improvement in coverage comes at the cost of increased transducer loss.In Fig. 13, the Smith chart coverage is plotted for a pair of two-pole tuners, both with a quality factor of 200 and identical circuit parameters except for their external couplings.Fig. 13(a) has identical external coupling structures, while Fig. 13(b) has a stronger external coupling on the input side and a relatively weaker coupling on the output.The symmetric tuner has a transducer loss of only 0.24 dB while still achieving 94.3% coverage.This coverage is increased to 98.9% for the asymmetric tuner, but the transducer loss increases to 0.50 dB.

III. DESIGN AND SIMULATION A. Impedance Tuner Design
As a proof of concept of the preceding theoretical analysis, a three-pole impedance tuner is designed for maximum possible Smith chart coverage from 4 to 8 GHz.The intended design targets for the tuner will be a resonator quality factor greater than 50 and tuning range greater than 10:1, an interresonator coupling strength J equal to 0.02, the same value used for the analysis in Section II, and an external coupling n equal to 1, the optimal value for maximum coverage.
To meet the above design targets, the three-pole impedance tuner is implemented using contactless, EVA resonators.Introduced in [41], this particular variation on previously reported EVA resonators uses a floating tuning disk rather than one attached to the walls of the cavity, allowing for much larger tuning ranges than the sealed resonator.The cross section  of a contactless EVA resonator is shown in Fig. 14.In this structure, the magnetic field wraps around the central post, effectively creating an inductor.The electric field concentrates between the central patch and the outer copper both directly across the ring and up to the floating disk and back down to the other side of the ring.Tuning of the resonator is accomplished by moving the disk either toward the cavity to increase its effective capacitance or away from the cavity to decrease the capacitance.
From Section II, the first step in creating the impedance tuner is to design resonators with minimal losses.For optimizing the unloaded quality factor of the resonators, an iterative procedure is presented in [41], which factors in design requirements including tuning range and power handling.This process is used to begin the design and to establish the resonator dimensions.Rogers TMM3 (ϵ r = 3.27 and tanδ = 0.002 at 10 GHz) is chosen as the substrate material due to its low loss, with a thickness of 1.91 mm found to be the optimal height for high quality factor while simultaneously meeting the required tuning range.Following this procedure, resonators with an unloaded quality factor of approximately 100 are created.The critical dimensions are listed below in Table V.
An input power handling of 250 W is desired for the impedance tuner design.The power handling for an EVA cavity is limited by gas breakdown within the gap between the patch and the tuning disk.The electric field in this region is strongest when the gap is at its minimum, setting a lower bound to the range of possible gap heights.A method is presented in [41] for simulating the power handling of an EVA cavity.Following this analysis, a minimum gap height of 70 µm is established to achieve the desired 250-W power  handling.To achieve the frequency tuning range requirement, the gap needs to extend vertically to over 700 µm, a condition easily met using a contactless tuning disk.
The next step in the process is designing the inter-resonator coupling.As mentioned above, the target for this design parameter is 0.02 at the tuner's center frequency.The two design parameters which control the inter-resonator coupling strength are the resonator spacing d, the distance center-tocenter between two adjacent resonators, and the iris width w, the perpendicular dimension to the resonator spacing, as shown  in Fig. 15.Decreasing the spacing between resonators or increasing the iris width leads to stronger inter-resonator coupling.These two dimensions are tuned until an inter-resonator coupling of 0.02 at 6 GHz is achieved.The circuit is simulated with only two poles and very weak external coupling, so the two peaks in S 21 are below −20 dB.An example of the result from this simulation is shown in Fig. 16, where the peaks of f 1 and f 2 are at 4.95 and 7.25 GHz, respectively.The inter-resonator coupling is then extracted using the following formula [42]: The dimensions for the final optimized resonator spacing d and iris width w are also included in Table V.The last step of the design process is to create an external coupling aperture to achieve an equivalent transformer ratio n equal to 1.The physical design parameters to tune the external coupling strength are the size of the aperture and its radial placement within the resonator, that is, its distance from the center of the resonator.While creating a larger aperture will increase the external coupling strength, there comes a tipping point where making it larger begins to increase losses and the coupling strength.This is demonstrated in Fig. 17.
Designing for high power handling and wide tuning range requires significant vertical displacement of the contactless tuning disk.To satisfy this requirement, a New Scale Technologies M3-L micro linear actuator is used as the tuning mechanism.These actuators have been used in the past for impedance tuners and filters alike [29], [39].They provide up to 6 mm of travel with 0.5 µm resolution, making them ideal for applications where large travel range is necessary.
The final 3-D model of the designed impedance tuner is shown in Fig. 15.Due to the choice in tuning mechanism and the final resonator dimensions after optimizing for quality factor and inter-resonator coupling, the tuning ring cannot be on the same side for all three resonators.Instead, the middle ring is placed on the opposite side of the board on the same side as the input and output coupling apertures.However, this does not affect the performance of the device.

B. Simulated Results
Throughout the tuner modeling process, an iterative procedure is used to create the equivalent circuit schematic.The circuit schematic then guides design changes to the 3-D full-wave electromagnetic simulation model within Ansys high-frequency structure simulator (HFSS), which is then updated.The result at the end of this procedure is an equivalent circuit schematic which properly represents the 3-D model's performance.
The equivalent circuit schematic is then used to evaluate the 3-D structure over a large number of capacitor values in a very short amount of time, significantly reducing the overall simulation time compared with performing 3-D fullwave electromagnetic simulations.The primary design metrics under consideration are the impedance tuner's Smith chart coverage and transducer loss across the 4-8-GHz frequency range.The results of these simulations at 4, 6, and 8 GHz are shown in Fig. 18.The data points assume a resonator quality factor of 100.The colors represent the transducer loss in dB, corresponding to the color scale located to the right of the Smith chart.The performance metrics are summarized in Table VI.

A. Fabrication and Assembly
The impedance tuner is implemented on a Rogers TMM3 substrate of 1.91-mm (75 mil) thickness.This board material has a dielectric constant of 3.27 and has positive qualities such as low loss and high power handling.The impedance tuner is fabricated using typical PCB fabrication processes of milling the pattern and electroplating the vias.SMA connectors are soldered to the input and output on each side of the PCB.Fig. 19 In Fig. 20, pictures of the final assembly of the three-pole impedance tuner are shown.As mentioned previously, the tuning for this device is managed through the use of M3-L linear actuators, seen in Fig. 20(a).Tuning disks are milled out of a thinner Rogers TMM3 substrate than that used for the resonators, only 1.27-mm (50 mil) thick, and attached to the M3-L with superglue.A fixture is designed and then created using 3-D printing.The M3-Ls are screwed directly into this plastic piece, and then the two fixtures are placed on the top and bottom of the fabricated PCB and held together with other screws.

B. Measured Results and Comparison
The first test of the device consists of performing coverage measurements over the frequencies of interest.This is accomplished using a Python script to control the M3-L actuators and interface with the performance network analyzer (PNA) to collect data automatically, described in Section IV-C below.The test is performed with each M3-L placed at 29 unique positions for a total of 24 389 data points.A logarithmic distribution is used to obtain more measurements at the smaller gap heights where the tuning is most sensitive.The total time for this measurement is about 3 h.Full scattering parameter measurements are taken at each data point, and transducer loss is extracted using (1) from Section II-A.
Fig. 21 shows plots of the measured Smith chart coverage of the fabricated three-pole impedance tuner using the abovementioned procedure.The color bar along the side of each Smith chart displays the transducer loss in dB.The minimum transducer loss is 0.4, 0.6, and 0.7 dB at 4, 6, and 8 GHz, respectively.The measured results are de-embedded up to the resonator plane.This removes the small amount of loss from the connector and transmission line, while also rotating the data so the small uncovered region is on the short-circuit side of the Smith chart, aligning with the majority of the plots above for ease of comparison.Using approximations for G min and G max , and taking into account the rotation from the transmission lines' sections, the Smith chart coverage is estimated to be 90%.
The other key experiment for evaluating the performance of the fabricated three-pole impedance tuner is a measurement of its power handling.Due to limitations in the laboratory equipment available at the time, the power is only tested up to 100 W. A block diagram of the high-power test setup is shown in Fig. 22.
The measured results for the high-power test are presented in Fig. 23.Since it was not possible to measure the input and output matching directly in the high power setup, a full set of S-parameters was taken at low power, and then the same gap combinations were retested at high powers.Fig. 23(a) shows S 21 for five separate gap combinations, at both low (solid line) and high (dashed line) powers.The trace color indicates the approximate input impedance of that tuner configuration, stated in the legend of the plot.Since the low-and high-power measurements are within 0.5 dB across the range of powers,  it is assumed that their input matching and output matching are well correlated.Thus, Fig. 23(b) shows the transducer loss for each of these configurations under high powers, using the measured S 11 and S 22 from the low-power results.
Table VII below compares this work to other impedance tuners both in the literature and commercially available.It can be seen that compared with other impedance tuners with small physical size, the presented device has by far the best Smith chart coverage and power handling.To get comparable performance in those departments, a much larger and heavier coaxial-slug-based impedance tuner must be used.Since these are much slower and certainly too large to integrate into any real system, they are simply forced to be used as laboratory equipment.

C. Repeatability and Speed Improvements
In previous work on impedance tuners or filters which use the M3-L for the actuation, [29], there is little effort on the control and programming side to improve speed and repeatability of the devices.First, the default error tolerance of the M3-L is set to ±1 µm and is never adjusted away from this, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.creating a limit in the repeatability of these circuits.Second, a MATLAB script handles all the control signals, which are sent to M3-L over a serial (I 2 C) interface.The process for setting a circuit configuration and performing a measurement follows a specific routine every time.A movement command is sent to M3-L to adjust it to a specific position, followed by a delay of a few tenths of a second.If there are multiple M3-Ls which need to be adjusted, then this cycle of sending a command and waiting is repeated for each actuator.However, there are occasional issues along the line of communication, and every so often M3-L would not reconfigure to the intended position.To combat this, the process of sending the movement command and pausing for 0.5 s to allow for movement is repeated two more times, for a total of three identical commands sent over the course of a couple seconds.

TABLE VII COMPARISON OF THIS WORK TO STATE-OF-THE-ART IMPEDANCE TUNERS
In this work, both the timing and error tolerance are improved.The error tolerance is reduced, even to an error tolerance as low as of 0 µm, either through the Pathway software that comes with M3-L or by sending a control configuration command over a serial interface.In practice, continuously maintaining zero position error is impossible.However, the internal feedback system of M3-L will always be working to keep it in place, and thus, M3-L is never more than ±0.5 µm out of position and, even then, not for very long.This error checking is integrated into the process for sending movement commands to M3-L.
A Python script is created to control multiple M3-Ls, checking the position of each M3-L in real-time.A movement command is sent to each M3-L, followed by a position check.If the position is not correct, the movement command is sent again.If all M3-L positions are correct, then a measurement is taken immediately.This real-time communication between the laptop and the M3-L greatly increases the measurement speed, since there is no artificial delay of 1.5 s to increase the probability that the command was received and executed properly.The communication delay has been reduced as much as possible with this improvement to the code, and now the speed is limited by the speed at which the actuators move.In Fig. 24, the difference in repeatability between the previous control and the current control methods is shown in Fig. 24(a) and Fig. 24(b), respectively.In addition, Table VIII shows the average wait time for a given error tolerance using the current control scheme.As a note, the M3-L movement speed is limited to 5 mm/s.The times listed in Table VIII do not count this movement time, simply the total time between measurements minus the supposed movement time to get to the correct position.The quantifiable repeatability of this measurement, known as the vector repeatability, is calculated using the following equation [21], [23]: Further investigations into independent harmonic tuning, possibly through the use of a separate parallel resonant structure, could improve repeatability and timing even more.

V. CONCLUSION
The theory, design, and measurement of coupled-resonatorbased impedance tuners were presented.Coverage limitations for the two-and three-resonator designs were established, with practical considerations for real-world tuners with regard to resonator loss and finite tuning range included in the analysis.Rigorous design procedures for optimal performance and maximum Smith chart coverage for both the two-and three-pole structures were detailed.A state-of-the-art threepole impedance tuner prototype in PCB technology was manufactured and tested, achieving approximately 90% Smith chart coverage from 4 to 8 GHz while simultaneously demonstrating minimum transducer losses as low as 0.4 dB and power handling up to 100 W. Furthermore, a state-of-theart repeatability of 48 dB was achieved, with tuning times less than 100 ms, representing repeatability competitive with the most repeatable tuners currently available today.Compared with other impedance tuners with small physical sizes conducive to integration, the presented tuner significantly outperforms in terms of both coverage and power handling.And while impedance tuners using coaxial slugs may achieve slightly better coverage, their tuning times are significantly slower, and their large physical size limits their integration into any system other than a laboratory bench.

APPENDIX A
This appendix contains the derivation for the input impedance, Y in , and coverage limits, G min and G max , for a two-pole resonator tuner.The circuit model is given in Fig. 25.Using this circuit model, the input admittance will be derived in a step-by-step process using the reference planes, indicated by the dashed lines in the figure corresponding to (3) in Section II-B above.Now that the input admittance of the two-pole tuner has been derived, the coverage limits can be calculated, starting with To simplify the solution, assume lossless resonators and let Then get a proper fraction by multiplying this numerator and denominator by the denominator's complex conjugate Taking the real part of the numerator now of ∂Y in For the lossless case (i.e., G i = 0 for all i), the boundary is corresponding to (7) in Section II-B above.
The minimum conductance can be determined by taking the limit of Y in as |B 1 | and |B 2 | go to ∞ corresponding to (10) in Section II-B above.

APPENDIX B
This appendix contains the derivation for the input impedance, Y in , and coverage limits, G min and G max , for a three-pole resonator tuner.The circuit model is given in Fig. 26.Using this circuit model, the input admittance can be determined.By observation, Y 2,plane of the three-pole model is equivalent to the admittance at Y of the two-pole model with appropriate changes in resonator subscripts.Thus, this will be the starting point for the derivation of input admittance for the three-pole tuner corresponding to (14) in Section II-C above.Now that the input admittance of the two-pole tuner has been derived, the coverage limits can be calculated, starting with G max To simplify and solve, assume lossless resonators and let Multiply numerator and denominator by the complex conjugate of the denominator, then only take the products from the numerator that will result in real values Re numerator Set this equal to zero and remove leading coefficients corresponding to (16) in Section II-C above.This is the first of two equations required to determine conditions for maximum

Fig. 1 .
Fig. 1.General circuit model of a resonant-based impedance tuner, consisting of an external coupling mechanism (transformers), inter-resonator couplings (admittance inverters), and shunt RLC resonators whose reactive component is variable.

Fig. 2 .
Fig. 2. (a) Circuit model and (b) sample Smith chart coverage with labeled regions that are not covered for a two-resonator impedance tuner with finite quality factor.

Fig. 3 .
Fig. 3. Sample plot using numerical optimization on the coverage equation for a two-pole tuner, showing the transformer ratio that yields maximum coverage for a given quality factor.

Fig. 4 .
Fig. 4. Coverage of a two-pole impedance tuner is plotted for resonator quality factors of (a) 10, (b) 50, (c) 200, and (d) infinity.It is evident that the quality factor has a significant effect on coverage.The transformer ratio for each plot is optimized for maximum coverage at the specified quality factor.The color and associated scale for all the Smith chart coverage plots represent transducer loss.

Fig. 5 .
Fig. 5. Finite tuning range impedance coverage for two-pole impedance tuners with quality factors of 50 (left) and 200 (right) is plotted for capacitor tuning ratios of (a) and (b) 2:1, (c) and (d) 4:1, and (e) and (f) 10:1.The impact of lower tuning ranges on G max is small, while the impact on G min is more appreciable.

Fig. 6 .
Fig. 6.(a) Circuit model and (b) sample Smith chart coverage with labeled regions that are not covered for a three-resonator impedance tuner with finite quality factor.

Fig. 7 .
Fig. 7. Coverage of a three-pole impedance tuner is plotted for resonator quality factors of (a) 10, (b) 50, (c) 200, and (d) infinity.It is evident that quality factor has significant effect on G min , yet minimal effect on G max .

Fig. 8 .
Fig. 8. Finite tuning range impedance coverage for the three-pole impedance tuners with quality factors of 50 (left) and 200 (right) is plotted for capacitor tuning ratios of (a) and (b) 2:1, (c) and (d) 4:1, and (e) and (f) 10:1.Unlike the two-pole case, lower tuning range impacts both G max and G min .

Fig. 9 .
Fig. 9. Plot demonstrating the difference in coverage between a two-pole and a three-pole impedance tuner for otherwise identical circuit parameters.

Fig. 11 .
Fig. 11.Coverage plot for a two-pole impedance tuner (a) without and (b) with 20 • transmission line sections added to both the source and load sides of the circuit model.Regions 1 and 2 are rotated around the Smith chart, along with the sample coverage points, while their size is unaffected.

Fig. 12 .
Fig. 12. Demonstration of the impact of both external and inter-resonator coupling strengths varying with frequency.The coverage for a two-pole impedance tuner at f 0 is plotted in (a), while the impedance coverage at 1.2 f 0 for constant and varying coupling coefficients is plotted in (b) and (c), respectively.The changing coefficients assume the coupling values vary linearly with frequency.

Fig. 13 .
Fig. 13.Coverage plots for a two-pole impedance tuner with a quality factor of 200 and with (a) symmetric and (b) asymmetric external coupling structures.The asymmetrical tuner allows for a larger percentage of Smith chart coverage.However, transducer loss is increased, and the coverage points near the middle are more sensitive to resonator values.

Fig. 15 .
Fig. 15.Three-dimensional model showing (a) top and (b) bottom of the designed three-pole impedance tuner.Due to space constraints, the middle resonator's ring was moved to the top side.However, this does not impact the performance of the device.

Fig. 16 .
Fig. 16.Example plot showing the simulated S 21 versus frequency of two coupled resonators to extract the inter-resonator coupling strength, with f 1 at 4.95 GHz and f 2 at 7.25 GHz.
Fig. 17.Relationship between transducer loss and extracted external coupling strength is presented, showing that losses increase above and below a certain threshold for coupling strength.

Fig. 19 .
Fig. 19.Pictures of (a) top and (b) bottom of the fabricated three-pole impedance tuner.The outer dimensions of the fabricated PCB are 48 × 27 mm 2 .
(a) and (b) shows the top and bottom of the fabricated three-pole impedance tuner, respectively.The outer dimensions of the fabricated PCB are 48 × 27 mm 2 .

Fig. 20 .
Fig. 20.Pictures of the final assembled three-pole impedance tuner.(a) M3-L with tuning disk attached.(b) Side view of the assembled structure.(c) Top view of the assembled structure.(d) Bottom view of the assembly.

Fig. 21 .
Fig. 21.Measured data for the Smith chart coverage at (a) 4 GHz, (b) 6 GHz, and (c) 8 GHz, including the transducer loss of each point.

Fig. 22 .
Fig. 22. Block diagram of the high-power test setup used to measure the power handling of the fabricated three-pole impedance tuner.

Fig.
Fig. Measured (a) S 21 under both low-(solid) and high-(dashed) power conditions, with corresponding input impedance shown in the legend and (b) transducer loss under high power for the same gap combinations, assuming identical input and output matching as the low-power measurements.

Fig. 24 .
Fig. 24.Two Smith charts showing the improvement in repeatability through changes in the control code from (a) previous works versus (b) this work.

∂Y 2
Re numerator∂Y in ∂ B 2 = − jn 6 J 2 − j2n 2 Y L B 2 ⇒ −2n 8 J 2 Y L B 2 = 0 (26)corresponding to(6) in Section II-B above.By simple observation, this equation is solved when B 2 = 0, which means it is resonant at the given frequency.To find Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.themaximum conductance, substitute the solution into the equation for Y in

TABLE I LIST
OF ASSUMPTIONS

TABLE II TWO
-POLE IMPEDANCE TUNER PERFORMANCE VERSUS RESONATOR UNLOADED QUALITY FACTOR

TABLE III THREE
-POLE IMPEDANCE TUNER PERFORMANCE VERSUS RESONATOR UNLOADED QUALITY FACTOR

TABLE V CRITICAL
RESONATOR DIMENSIONS Fig.17

TABLE VIII ERROR
TOLERANCE VERSUS AVERAGE WAIT TIME