Perfect matching of reactive loads through complex frequencies: from circuital analysis to experiments

The experimental evidence of purely reactive loads impedance matching is here provided by exploiting the special scattering response under complex excitations. The study starts with a theoretical analysis of the reflection properties of an arbitrary reactive load and identifies the proper excitation able to transform the purely reactive load into a virtual resistive load during the time the signal is applied. To minimize reflections between the load and the transmission line, the excitation must have a complex frequency, leading to a propagating signal with a tailored temporal envelope. The aim of this work is to design and, for the first time, experimentally demonstrate this anomalous scattering behavior in microwave circuits, showing that the time-modulated signals can be exploited as a new degree of freedom for achieving impedance matching without introducing neither a matching network nor resistive elements, that are typically used for ensuring power dissipation and, thus, zero reflection. The proposed matching strategy does not alter the reactive load that is still lossless, enabling an anomalous termination condition where the energy is not dissipated nor reflected, but indefinitely accumulated in the reactive load. The stored energy leaks out the load as soon as the applied signal changes or stops.


I. INTRODUCTION
MPEDANCE MATCHING is one of the fundamental concepts in microwave circuit theory, allowing the maximum energy transfer to a load by cancelling reflections at its terminals [1], [2]. Impedance matching is invoked when the impedances of the feeding transmission line 0 Z and of the load L Z do not satisfy the condition 0 L ZZ = , provided that the generator internal impedance equals to the input impedance of the feeding transmission line connected to the load. Without impedance matching, reflections occur inevitably at the terminals where the impedance discontinuity takes place for ensuring the continuity of the physical quantities involved in the signal propagation, i.e., voltages and currents, along the transmission line. To avoid such undesired reflections, in the last century, several impedance matching techniques have been developed, which are based on the concept of impedance transformation, i.e., adding a properly designed microwave network composed by lumped and distributed elements between the line and the load ( Fig.1(a)). A number of design techniques of matching networks [3]- [6] have been proposed in the state of the art to reach the theoretical limits imposed by Bode-Fano theorem [1] to achieve the best impedance matching conditions for antennas [7]- [10], power amplifiers [11], [12] and energy transfer applications [13]- [15], spacing from adapting matching [16]- [18] to genetic algorithms techniques [19], [20]. Indeed, considering a frequency-dependent complex load ( ) ( ) ( ) fed by a lossless transmission line with a purely resistive characteristic impedance 00 ZR = , the matching network is designed such that the load reactance vanishes and the load resistance transforms into 0 R at the desired frequency 0  . Therefore, conventional impedance matching techniques act on the network topology and the involved circuital elements, i.e., capacitance, inductance, and resistance ( Fig.1(a)).
Recently, a new degree of freedom has been introduced for achieving perfect matching: the temporal modulation of the constituting elements of the complex load [21]. Despite the earlier studies dated back to the 1960s, time-varying elements in electronic circuits [22], [23], as well as space-time-varying material properties for engineering wave propagation [24]- [31], have only recently attracted the attention of researchers, thanks to the possibility to achieve magnet-less nonreciprocity and frequency conversion [32]- [34]. In [21], it has been numerically demonstrated that a purely reactive load, i.e. an inductor or a capacitor, that is mismatched with an ideal feeding transmission line, can be transformed into a resistance R when a proper temporal modulation profile is applied to the inductance () Lt , or capacitance () Ct , of the load ( Fig.1(b)). If the temporal modulation is present, the energy is accumulated within the reactive load without generating any reflection towards the source, leading to a perfect matching. Such an approach requires extremely fast modulation of the reactive components and, therefore, the implementation 0018-926X (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. complexity prevents its realization in realistic scenarios, despite some interesting attempts have been provided [35].
In this work, we reply to the following intriguing question: can the temporal profile of the propagating signal enable a special impedance transformation, such that impedance matching of a purely reactive load terminating an ideal transmission line, as shown in Fig.1(c), is achieved? A decade ago, the use of temporal profile of incoming signal has been exploited in [36]- [38] for enabling efficient coupling in single atoms and optical resonators. More recently, the same concept has been moved in optics, for enabling energy virtual absorption in lossless systems, making them appear as matched to free-space if complex zeros of its scattering eigenmodes are excited [39]- [45], reporting also elastodynamic experimental proof in mechanical systems [46]. In particular, the authors have investigated in [45] the possibility to enable virtual absorption effects in metasurfacebounded cavities. The analysis of this problem led to the derivation of the complex zeros, and corresponding limit values of the surface impedances of the metasurface. Even though the concepts of virtual absorption [39] and virtual critical coupling [44] have already been investigated in cavities and optical applications, here, we exploit it for achieving a similarly anomalous response also in microwave circuits composed by purely reactive lossless lumped elements, leading to a virtual perfect matching, and opening the door to a new impedance matching strategy, as preliminarily shown in our recent conference work [47], [48].
This contribution aims at shedding light for the first time on this phenomenon from both the physical and the electrical point of view, through a detailed theoretical analysis, numerical and experimental results, deriving the fundamental limits of the impedance matching concept based on timemodulated excitation signals. We analyse all possible configurations of reactive loads reported in Fig. 2: single inductance ( Fig.2(a)) and capacitance ( Fig.2(b)), and series/parallel connection of those elements as reported in Fig.2(c)-(d), respectively. The parallel connection of reactive loads has been already discussed in our recent work [45], where fundamental limits of virtual perfect absorption have been analysed. Nevertheless, this study included preliminary analytic expressions, and it was related to free-space propagation, focusing on energy accumulation of the structure. This paper reveals that purely reactive loads can produce no reflection when connected to a lossless transmission line exhibiting a resistive characteristic impedance under special excitation conditions. The specific novelties introduced by this work in the state of the art can be summarized in the following four contributions: Contribution 1) We develop and discuss in detail a general theoretical analysis of all element configurations available as reactive loads, deriving the operative regions for which the phenomenon takes place as a function of the natural frequency of the circuit and the circuit time constant.

Contribution 2)
We provide a set of proper numerical simulations of all presented load cases, describing the physical insights and providing a comparison among them.

Contribution 3)
We demonstrate that the analyzed phenomenon is not related to a delay along the line, setting a numerical simulation based on an electrically long microstrip line.

Contribution 4)
We report the experimental results of perfect matching condition for the series and parallel configurations.
According to the aforementioned list, the paper is organized as follows: in sections II the theoretical analysis of the mismatching condition under complex excitation is reported and discussed (Contribution 1), considering all the different load configurations (single reactive loads in Section II.A and series and parallel connections in Section II.B). Section III is devoted to the theoretical results discussion and to the constraints and limits evaluation on the load values for achieving the virtual perfect matching (Contribution 2). In Section IV, we report the numerical simulations for purely imaginary and complex excitations, and the demonstration that the phenomenon is not related to a delay along the line (Contribution 2 and 3). Finally, Section V presents the experimental results, demonstrating that the phenomenon can be observed in lumped microwave networks (Contribution 4). In Section VI we draw the conclusions of this investigation.

II. REACTIVE LOAD UNDER TIME MODULATED EXCITATION: CIRCUITAL ANALYSIS
Let us consider a circuit where a lossless transmission line with resistive characteristic impedance 0 R is loaded by a generic reactive network. From basic transmission line theory, we can write reflection coefficient seen at the load as: Regardless of the complexity of the network, the load impedance can be always represented as 0018-926X (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
In general, this lossless transmission line is always mismatched to the load, since L X and 0 R will always have different values in the real frequency spectrum. Thus, the imaginary quantity () L jX  should be converted into a real one and equated to the real characteristic impedance 0 R of the feeding transmission line. Exploiting the temporal modulation of the incoming signal, this can be done by introducing a complex value of the excitation frequency  , i.e. ri j    =+ [41]. A complex frequency excitation is a harmonic signal of frequency r  , whose amplitude temporally varies following an exponential growing (or decaying) profile proportional to t i e  . The reflection coefficient in eq. (2) can be represented as a function of the frequency in the complex frequency plane, where it is possible to identify scattering singularities, i.e. zeros and poles, of the amplitude of the reflection coefficient function for specific combinations of real and imaginary frequency. Hereafter, we state this analysis for a single reactance case and a combination of LC in series and parallel connection loads. Inductors and capacitors are linear components, in accordance with Ohm's law even when the applied voltage signals increase due to the exponential growing in amplitude dictated by the complex frequency. Non-linearity in the circuit components may represent a further degree of freedoms for these phenomena, not considered in this work.

A. Single load cases
Now, we consider a load consisting of a single inductor L or capacitor C, whose impedance is . Starting from eq. (2), we can estimate the reflection coefficients at the load terminals for the two considered cases: where superscripts "L" and "C" identify the cases reported in In circuit theory, the quantities L  and C  are the well-known circuit time constants for a transient step response for RL and RC circuits, and their value determines how fast the time response goes to zero.
We now consider the complex frequency excitation of the two circuits in Fig.2 Fig.3(a) reports the amplitude of the reflection coefficient in the complex frequency plane, that exhibits two singularities: a pole ( Fig.3(a), red/yellow positive peak) is in the positive half-space of the imaginary frequency, whereas a zero ( Fig.3(a), blue negative peak) is in its negative halfspace. The reflection coefficient pole excitation leads to an anomalous condition for which the reflected energy is higher than the impinging one. In passive systems, this is as a special scattering condition for which the scattered field decays slower than the excitation field, giving rise to a virtual gain effect [49]. On the other side, exciting the zero implies a zeroreflection condition: the load behaves as an indefinite accumulator for the illuminating signal, without dissipating its energy but rather storing it within itself. The zero-scattering condition is satisfied as long as the complex frequency is such to engage the scattering zero, thus leading to a zero-reflection coefficient. Modifying the signal excitation frequency, the reflection coefficient changes and assumes a non-zero value, energy then leaks out from the reactive load.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. The reflection coefficient vanishes at purely imaginary frequencies , 0 LC  , as expected from Fig.3(a). An excitation signal with an imaginary frequency is not oscillating, but simply growing over time with a steepness given by the exponential factor , 0 LC e  − . This is fully consistent with circuit theory where the charging and discharging behavior of a RC (RL) circuit is described by the same exponential curve with a factor t e  − for voltage (current). It is clear, therefore, that the circuit time constant  plays a fundamental role in the zero and pole positions in the complex frequency plane. In a reciprocal lossless system, zeros and poles are always complex conjugate; increasing the time constant  , both zero and pole move inside the complex frequency spectrum, reducing the required steepness of the time-growing signal for being excited (Fig.3(b)), always conserving complex conjugation.

B. Series and parallel load cases
Let us now consider the load as a reactive network modelled as a series/parallel connection of an inductor and capacitor ( Fig.2(c)-(d)). In these two scenarios, the load reactance can be found as: where the superscripts "-" and "//" identify the series and parallel connection, respectively, and 1 res LC  = is the natural resonant frequency of the LC circuit for both configurations. Substituting eq. (6) into eq. (2), we obtain: ) are for series (parallel) configuration. By forcing eqs. (7)-(8) to vanish, we obtain the following perfect matched complex frequencies: The frequencies granting zero-reflection are two due to "  " sign in eq. (9) and seem to be purely imaginary, as in the case of a single reactive load, however, this is true only if the argument of the square root in eq. (9) is positive, i.e., 22 4 0 res  −. When this condition is not fulfilled, that is 22 4 < 0 res  − , eq. (9) assumes a complex value, that allows achieving a complex frequency with non-zero real part, i.e. harmonic signals at frequency  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Case 1) consists of a LC circuit whose 2 res   . In this scenario, the system exhibits two separate frequencies for which zero-reflection can be achieved ( Fig.4(a)-(d)). The reflection coefficient has two zero-pole pairs symmetrically distributed along the imaginary frequency tends to the frequency plane origin. On the contrary, when  decreases (Fig.4(d)), both external and internal zero-pole pair move towards the imaginary counterpart of the natural frequency, res j  , which involves special responses from the system as discussed in next cases.
In case 2), when 2 res  = , the two zero-pole pairs of previous case degenerate to the imaginary counterpart of the natural resonant frequency res j  (Fig.4(b)). The degeneration of the two zero-pole pair into two imaginary frequencies This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  for the series and parallel case, respectively. From the physical point of view, all the energy provided to the load under exponentially growing signal is stored inside reactive components (with opposite sign), which are both gradually charged in time. The energy will be accumulated progressively, and, at the same time, the voltagecurrent ratio at the load terminals always equals the transmission line resistance, thus forbidding reflections back to the source.
To provide further validation to our theoretical investigation, we can see that the proposed theoretical analysis resembles the damping behavior of RLC circuits, where the circuit behaves as overdamped, critically damped or under damped according to the roots (complex or real) of the circuit harmonic oscillator equation [50], [51]. In circuit theory, damping is caused by the energy dissipation inside the resistance of an RLC network. We stress that, in our study, the resistance of the transmission line 0 R does not dissipate energy leading to circuit damped response. Here, instead, we exploit complex excitation (having the role of resistance dissipation) to balance the corresponding circuit reflections, that are zero in the transient response.
We now illustrate the behavior of the real and imaginary part of zeros for both LC series and parallel connections, thus ,// 0  − as expressed in eq. (9), with respect to the two parameters  and res  .  (Fig.5(a)), is zero when 2 res   , (Region 1) meaning that zeros are purely imaginary, as predicted by eq. (9). In the same Region 1, the imaginary part in Fig.5(b) exhibits very high frequency values, meaning that a sharper steepness is needed to engage the desired scattering zero. This region is not of interest for seeking virtual absorption with modulated signals. On the contrary, when 2 res   (Region 2), the real part is non-zero and, for a given  , it is directly proportional to the LC resonant frequency res  . The imaginary part is, indeed, much smaller than the real part, meaning that the steepness is reduced, leading to an exponential slowly growing signal.

IV. SIMULATIONS RESULTS
The impedance matching achieved through the anomalous transient behavior of reactive loads under complex excitation has been numerically verified through CST Studio Suite [52]. We consider a lossless 50-Ohm transmission line with different loads terminating it. In circuit schematics, the impedance transmission line coincides with the generator internal resistance. In next subsections, we will focus on the circuital schematic simulations of single and combined loads that show perfect matching under purely imaginary and complex frequency excitation cases in Sections IV-A and IV-B, respectively. Finally, in Section IV-C, full-wave simulations over a designed microstrip line are exposed.  signals are recorded at the load terminals. As expected, the impinging signals are engaging the reflection coefficient zeros and no reflection takes place. Despite the entire circuit is passive, energy cannot be dissipated since the load is purely reactive, allowing only energy storing in the reactive load. As soon as the signal stops, at kick-off time 0 t , the zero-reflection condition is not satisfied anymore, forcing the reactive load to release the stored energy ( Fig.6(a), orange and light blue lines). We can furthermore observe that both zeros satisfy the circuit, and the zeros present different growing envelope according to the 0

A. Imaginary excitation cases
To explore single reactance and critical ( 2 res  = for series/parallel connection) cases, we choose inductors and capacitors values as described in Table I (Fig.6(b), grey line). The circuits transient response verifies perfect matching condition.
Reflected output signals are zero until kick-off instant 0 t , but after it the accumulated energy is released. Each output shows a different behavior according to the circuit to which it is referred to, and corresponds to RC, RL, RLC series/parallel circuits where the charging and discharging behavior is described by the same exponential curve with a factor t e  − for voltage and current, respectively.  Fig.6(c), blue signal) Perfect matching is achieved also for complex valued frequencies, showing zero reflections up to kick-off instant (Fig.6(c), orange line).

C. Full-wave microstrip verification
To give further validate the observed phenomenon of perfect matching under complex excitation, we have considered a realistic microstrip line printed on a 0.5mm-thick Referring to eq. (9), a complex zero is required, working at the microstrip operative real frequency 0 f . The exponential envelope needed to excite the reflection coefficient complex zero corresponds to the imaginary frequency 9 5 10 rad s i  = −  . In Fig.7(b) we report the recorded signals at the port terminals over time. The blue curve represents the incident signal, and the orange curve the released signal. It is clear that the phenomenon is preserved, since 1) the envelope of the released signal is opposite with respect to the incident one, whereas in case of reflection it would not, and 2) the presence of the time delay between the two signals is due to their propagation along the line. Since the signals are detected at waveguide port, the observed delay exactly corresponds to two times the line length over the phase velocity in microstrip line, that is: where eff  is the effective permittivity of the microstrip line and c is light speed. The line delay naturally occurs during This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  Fig. 9. (a) Schematic of the system for LC shunt load. Normalized simulated (b) and combined measured (c) input signal (blue curve), output signal at port (CH1) reflected towards the AFG port (red curve) and output signal detected at the oscilloscope probe (CH2) (red curve). propagation over the microstrip, and it is independent from the perfect matching phenomenon. These numerical results clearly show that the signal is not only delayed because of the line length, but it is independently stored thanks to perfect matching due to the excitation of the reflection coefficient zeros in the complex frequency plane.

V. EXPERIMENTAL VERIFICATION
In the following, we set up an experiment for validating our analysis and demonstrate the perfect matching by using a reactive load consisting of an LC parallel and an LC series connection, as detailed in subsections V.A and V.B, respectively. In both configurations we use the same components: a ceramic disc capacitor of As shown in Fig. 9a, the load is connected through a lumped-element Wilkinson Power Divider (WPD) designed at r f to the AFG and to the oscilloscope. The signal is generated by the AFG and is split in two signals, the former, CH1, is directly connected to the oscilloscope, and CH2 is connected to the WPD and combined into the line leading to the load. Here, the signal is accumulated and, later, is divided into the two ports that carry the signal back to the AFG and to the oscilloscope. Both the output impedance of AFG and oscilloscope are set at 0 50 R = , also the oscilloscope probe lead measures voltage over a 50 load. In Fig.9(b)-(c), the simulated response of the circuit detected at the probe port and the measured response detected at AFG port are reported. As expected, the experiment clearly returns perfect matching phenomenon since the signal detected at the probe is zero up to kick-off instant 0 t , that is, reflections vanish. After 0 t instant, as soon as the temporal modulated excitation stops, the signal is released and is equally split by the power divider into CH1 and CH2 signals, being the perfect matching condition not satisfied anymore. A negligible difference between simulated and measured released signals can be observed comparing the yellow curves in Fig. 9(b)-(c), due to the very small resistive behavior introduced by wires and connectors in the experiment that leads to a minor attenuation of the signals.

B. Series load
In addition to the shunt LC load examination, a second analysis for a LC series connection has been carried. Here, the considered load is composed by a series connection of eq L and eq C elements. The series configuration of these two elements provides 2 res  −  , in eq. (9). Consequently, perfect matching is obtained through two purely imaginary zeros, that are  , using the same notation of Section II. These zeros lay along the imaginary frequency axis as shown in Fig.4(a)-(d). We have arbitrarily chosen int  zero to excite the circuit. Since now the excitation int  required to get perfect matching is a static growing signal, the WPD element is not needed anymore, as shown in the experimental setup in Fig.10(a). As for the shunt connection case, CH1 carries the port signal, that comes from the BNC splitter and directly connects to the oscilloscope. CH2 carries the probe signal, detected by the probe on the breadboard. In this case, port and probe signals coincide in simulations, since retards and small dissipation are not taken into account. However, since some delays and dissipations in measures naturally occur, we show both measured port and probe signals. In Fig. 10(b) and (c), we respectively show simulated normalized signals and the combined measured signals. As for series case presented in Fig.6(b) (pink curve), perfect matching behaves similarly, in both simulated and measured results. As for the measured signals in Fig. 10(c), signals at the probe exhibit an extremely small delay with respect to the ones at the port. However, it is enough to hide the negative peak at the probe that is indeed present in the simulated signals. The accuracy and sampling speed of measuring instruments have been optimally set for detecting any variation of the signal amplitude at the measuring points. Indeed, the extremely narrow negative peak curve, presented by the output signal at the port (orange dashed signal in Fig.10(c)), is correctly detected. The absence of the negative peak on the measured signal at the probe is, thus, due to the intrinsic charging and discharging time required by reactive circuital components, that in this case is longer than the duration of such a peak.

VI. CONCLUSION
To conclude, we have presented the first experimental validation and full theoretical derivation of perfect matching of purely reactive loads terminating a lossless transmission line by exploiting the concept of impedance transformation through time-varying complex signals. In particular, the work sheds light on the anomalous phenomenon of virtual perfect matching, bringing several novelties to the current state of the art of complex signals propagation in electromagnetic systems modelled through terminated transmission lines, and, for the first time, proposing an experimental implementation. We have developed and discussed in detail a general theoretical analysis of all element configurations available as reactive loads, deriving the operative regions for which the phenomenon takes place as a function of the circuit natural frequency and time constant. The theoretical analysis has been verified through a set of proper numerical simulations of all presented load cases, describing all physical insights and providing a comparison among them. Moreover, we have clarified, by setting a numerical simulation based on an electrically long microstrip line, that the analyzed phenomenon is not related to the delay along the line. Finally, we have reported the first experimental results on virtual effects and, in particular, perfect matching condition for the series and parallel configurations. Although the required increasing envelope of the temporal profile is still a limit for making this concept exploitable in practical scenarios, the experimental verification provided in this paper may provide a base for paving the way to a new impedance matching strategy in microwave circuits, that can be exploited together with the already proposed applications based on virtual effects such as wireless energy transfer and energy harvesting [53]- [55]. that respond to the need for environment and human health protection. His research activities are focused on three fields: metamaterials and unconventional materials, in collaboration with Professor A. Alù's group at The University of Texas at Austin, USA, research and development of electromagnetic cloaking devices and their applications (First place winner of the Leonardo Group Innovation Award for the research project entitled: 'Metamaterials and electromagnetic invisibility') and the research and manufacturing of innovative antenna systems and miniaturized components (first place winner of the Leonardo Group Innovation Award for the research project entitled: "Use of metamaterials for miniaturization of components" -MiniMETRIS).
He is the author of more than one hundred publications in international journals indexed ISI or Scopus; of these on a worldwide scale, three are in the first 0.1 percentile, five in the first 1 percentile and twenty-five in the first 5 percentile in terms of number of quotations and journal quality. His main research contributions are in the analysis and design of microwave antennas and arrays, analytical modelling of artificial electromagnetic materials, metamaterials, and metasurfaces, including their applications at both microwave and optical frequencies. In the last ten years, Filiberto Bilotti's main research interests have been focused on the analysis and design of cloaking metasurfaces for antenna systems, on the modelling and applications of (space and) time-varying metasurfaces, on the topological-based design of antennas supporting structured field, on the modelling, design, and implementation of non-linear and reconfigurable metasurfaces, on the concept of meta-gratings and related applications in optics and at microwaves, on the modelling and applications of optical metasurfaces. The research activities developed in the last 20 years (1999-2019) has resulted in more than 500 papers in international journals, conference proceedings, book chapters, and 3 patents.
Prof. Bilotti has been serving the scientific community, by playing leading roles in the management of scientific societies, in the editorial board of international journals, and in the organization of conferences and courses.