Modal Filter Based on a Microstrip Line With Two Side Conductors Grounded at Both Ends

This article presents a modal filter (MF) based on a microstrip line with two side conductors grounded at both ends. New analytical expressions for calculating the time response in such a structure are given. For their validation, an MF prototype was made and the transmission coefficient was measured in the frequency range from 0 to 12 GHz. In addition, the full-wave electrodynamic simulation and the analysis based on the transmission line theory were carried out. The MF frequency responses were converted into time domain responses in the Advanced Design System. At the MF output, the responses to an ultra-wideband pulse, a high-frequency damped sinusoid, and a sinusoid with Gaussian modulation were analyzed. The results for the analytical model are in good agreement with the results of measurements and simulations by the two approaches. It was experimentally proven that the structure under study provides the protection of electrical circuits from different types of interference by separating them in time and decreasing their effect on the components. Finally, signal integrity was evaluated for two data rates. It was found that the MF under investigation may be used in signal circuits up to 1 Gb/s.


I. INTRODUCTION
M ICROSTRIP lines (MSLs) are widely used to transmit electrical signals and power to various elements of electrical circuits [1].Designers are continuously modifying MSLs to improve their characteristics [2], [3].Indeed, a structure consisting of a reference conductor in the form of a conductive layer, a dielectric substrate on the reference conductor, and a signal conductor in the form of a strip on the substrate is widely used [4].However, the passband of the MSL [5] is often much wider than the bandwidth of the useful signal.This contributes to the propagation of high-frequency interference through it and requires the installation of filters to attenuate The authors are with the Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia (e-mail: indira_sagieva@mail.ru; zhechev@tu.tusur.ru;zarina.kenzhegulova@mail.ru;surovtsevrs@tu.tusur.ru;talgat@tu.tusur.ru).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TEMC.2023.3275388.
Digital Object Identifier 10.1109/TEMC.2023.3275388conducted emissions.Meanwhile, a conventional MSL with two additional conductors on top can have the properties of a modal filter (MF) [6].Such a device can protect a vulnerable circuit against ultra-wideband (UWB) pulses and microwave pulses [7] by means of modal distortions [8].Changing the reference conductor (polygon of the circuit ground) is the main disadvantage of the MF from [8].In spite of good attenuation of UWB interference in such a structure, this disadvantage may increase the return current path of neighboring components.To ensure the interference immunity of electrical circuits without changing the ground polygon, the structure from Sagiyeva and Gazizov [9] may be used.Unfortunately, such an MF is difficult to implement, since the side conductors are in the air.The purpose of this article is to present a novel MF based on an MSL with two side conductors grounded at both ends.The relevance of this study is determined by the fact that such an MF can easily be implemented on a printed circuit board (PCB) and can have an acceptable attenuation of a UWB pulse.

II. STRUCTURE UNDER STUDY
The authors propose a new MF that is a modification of a conventional MSL by adding two side conductors connected at both ends to the reference conductor.Fig. 1 demonstrates a cross section of the proposed MF.Three fundamental modes may propagate in this structure.To determine the optimal structure parameters that reduce the maximum output voltage V max and increase the differences of per-unit-length delays Δτ , we performed several steps.First, we calculated the matrix of the capacitance (C) and inductance (L) coefficients.Then, time responses on excitation of the UWB pulse were obtained (due to symmetry of the structures cross section, only two instead of three pulses were observed at the MF output).At last, amplitudes of the two pulses were equalized changing MF parameters.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.s = 0.9 mm (taken to equalize pulse amplitudes), and ε r = 4.5 (typical FR-4 substrate).This design allows one to obtain a decomposition of a UWB pulse acting between the signal and reference conductors of the MSL into two pulses of equal amplitudes by employing widely used material.The values of w and t allow the MF to be used in circuits with a dc current of at least 0.5 A. This decomposition is achieved because of different mode propagation velocities in such an MF.Fig. 2(a) shows a schematic diagram of the MF.The possibility of UWB decomposition is shown on the example of a trapezoidal pulse with an e.m.f. of 5 V and rise, fall, and flat top durations of 50 ps each [see Fig. 2(b)].The excitation was chosen because its waveform is easily analyzed and allows one to accurately determine the pulse arrival time.The circuit has the following parameters: the line length (l) of 1 m and the internal resistances of the UWB pulse source and the load (R S = R L ) of 50 Ω.The side conductors are connected to the reference one at its near and far ends.

III. PROPOSED ANALYTICAL MODEL
To compare and quickly evaluate time responses, it is advisable to use an analytical approach.Therefore, we employed an approach based on analytical representation of a signal at the nodes of a transmission line segment (using telegraphic equations) in the form of incident and reflected waves for each of the modes [10].According to this approach, first, we need to set the voltage waveform of the excitation source through vector V and the parameters of the segment terminations through matrices R S and R L .Since in the circuit [see Fig. 2(a)], all ends of the side conductors are connected to the ground plane (short-circuit case), and the structure in Fig. 1 has symmetrical cross section, R S and R L matrices for the general case have the form (for partiсular case of equality of terminations Then, based on the per-unit-length coefficient matrices L and C of the segment, the transformation matrices of modes T V and T I were calculated.They are needed to switch to the vector of modal voltage sources V m and the matrices of modal impedances R Sm and R Lm .Then, we used them to find the vector of sources of initially incident modes V 0m and reflection coefficient matrices at the near (Г S ) and far (Г L ) ends of the segment.Finally, the time response at the ends of a single segment was defined as a combination of the components of the initially incident modes and the coefficients of the reflection matrices that correspond to inhomogeneities at the junctions between the complete line segment and the terminations.On the basis of this approach, we obtained general and particular analytical models for calculating the voltage waveform at the MF output and presented them below.
Analytical models of modal voltages at the end of the line for a transmitted wave (without taking into account subsequent reflections) have the form where T a = lτ a , T b = lτ b , and T c = lτ c are modal delays; V 0a , V 0b , and V 0c are the entries of vector V 0m ; Г aa , Г ab , Г ac , Г ba , Г bb , Г bc and Г ca , Г cb , Г cc are the entries of the reflection coefficient matrix at the far end Г L (the lower L index is omitted because R Sm and R Lm matrices are the same at both ends).
Taking into account models ( 2)-( 4), the response at the MF output (node V2) took the form where T Vaa , T Van , and T Vac are the entries of matrix T V .Due to the symmetry of the MF cross section in combination with (1), the reflection coefficient leads to the zero values of the coefficients Г ab = Г ba = Г bc = Г cb = 0 of the reflection matrix.In addition, with the symmetry of the two side conductors in the MF cross section, T Vab = 0, which eliminates the second term in (5).Then, expression (5) takes the form The novelty of model ( 6) consists in the fact that it was first proposed for a new MF protective structure based on a modified MSL with two side conductors grounded at both ends.The features of this structure made it possible to significantly simplify and shorten the general models from Park et al. [10].Model (6) allows the time response of the MF to an arbitrary excitation to be obtained only on the basis of the calculated L and C matrices and a number of algebraic transformations with them.However, the model is not without drawbacks.The main model limitation is that it takes into account only signal propagation from the beginning to the end of the MF.Neither does the model ( 6) consider the losses in conductors and dielectrics.Therefore,  compared to simulation by numerical methods, the model may lead to lower resulting accuracy.On the other hand, it is also less computationally expensive, since it allows performing analysis without calculating the time response.
The set of cross-sectional parameters determines the matrices of per-unit-length parameters (see Table I).They are calculated by the method of moments implemented in the TALGAT software.
The square root of the eigenvalues of the product of these matrices determines the values of the per-unit-length mode delays: τ a = 5.796 ns/m, τ b = 5.889 ns/m, and τ c = 6.089 ns/m.However, due to the symmetry of the two side conductors, the pulse amplitude of mode b is equal to zero.As a result, only the pulses of modes a and c remain.The time interval between them is l(τ c -τ a ) = 0.293 ns.Fig. 3 presents the output voltage waveforms to the idealized trapezoidal UWB excitation, which are calculated using the TL analysis (without taking into account losses) and the proposed analytical model (6) (in Mathcad).The TL analysis is based on the method of moments to calculate L and C, the modified nodal potential method in the frequency domain, and the fast Fourier transform to calculate the time responses [11].The calculation of the time response based on the model ( 6) also requires the calculation of L and C matrices using MoM.It can be seen that V2(t) represents two pulses with an interval of 0.293 ns between them and equal amplitudes of 1.23 V each, and the responses obtained by the two approaches completely coincide.

IV. SIMULATION AND EXPERIMENTAL TECHNIQUES
To validate the proposed analytical models, as well as to experimentally confirm the modal filtration effect in the structure under study, we made an MF prototype.The value of parameter d takes into account the length of dielectric boundary.Parameters s, w, and w 1 were optimized by a heuristic search to decompose the exciting pulse into two pulses at the MF output and align their voltage magnitudes.The bandwidth was limited to the value of  4 shows a photograph of the top and bottom layers of the PCB.To reduce the PCB length, the structure was bent into a meander.The coupling between the half-turns is very weak because the distance between them is 10 mm.To connect the MF to the measuring equipment, we used coaxial-microstrip junctions.Their insertion loss does not exceed 0.5 dB in the frequency range from 0 to 12 GHz.Fig. 5 shows an experimental setup used to analyze the MF frequency response.The MF prototype was connected to a vector network analyzer (VNA) with high-frequency cables.An E50771 C (Agilent Technologies, Santa Clara, CA, USA) was used as the VNA.The frequency characteristics were analyzed in the range from 10 MHz to 12 GHz.Before measurements, a two-port SOLT calibration was performed.
After the MF was defined in the frequency domain, its characteristics were converted into time domain using Advanced Design System (ADS).A Gaussian pulse with a duration of 67 ps by 0.5 level was excited to the MF input (node V1) (see Fig. 6).This excitation corresponds to the definition of a UWB pulse from the EMC standard [12].
To assess the possibility of applying the MF structure to protect against other interference waveforms, we also considered a damped sinusoid and a sinusoid with Gaussian modulation [13].The dumped sinusoid excitation (see Fig. 7) is given in accordance with the MIL-STD-461F standard as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where E is the peak e.m.f.value, β = 1/τ c is the attenuation coefficient (at τ c = 1 ns, it is the time during which the amplitude decreases by e = 2.72 times), and f is the frequency (assumed The sinusoid with Gaussian modulation (see Fig. 8) is given by the IEC 61000-1-5-2017 standard as where E is the peak e.m.f.value, t 0 = (1/f) is the period of the carrier frequency (assumed f = 1 GHz), α = 10t 0 is the effective width of the Gaussian waveform (at the level of 1/e), and t s = 2α is the signal time shift.
To confirm the measurement results, we performed the previous numerical analysis and electrodynamic simulation of the MF in the frequency range from 0 to 12 GHz in ADS using the method of moments.In this case, fiberglass FR-4 was specified as the substrate material, and copper as the conductor.The

TABLE II N-NORMS DESCRIPTIONS AND APPLICATIONS
Svensson/Djordjevic model [14] was used to take into account the influence of the frequency dependence of losses in the dielectric.The discrepancy between the results obtained through the TL analysis and analytical models can be attributed solely to the losses in dielectrics and conductors.To assess the danger of the MF output voltage waveform, N-norms were used [15].Table II presents their descriptions and applications.
Finally, a pseudorandom binary sequence (PRBS) source of 50 000 b and data rates of 480 and 1000 Mb/s (separately) were used to analyze signal integrity.These data rates were conditioned by the bandwidth of the MF structure.The rise time (t r ) and the fall time (t f ) were 210 and 100 ps; the durations of the unit intervals (T) were 2.1 and 1 ns.At the first stage, we obtained the frequency characteristics of the structure using the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

VNA. Next, ADS was used to obtain time domain
To analyze signal integrity, the response to a single PRBS pulse was obtained.

A. Frequency Characteristics
Fig. 9 shows the frequency dependences of |S 21 | obtained from measurements, as well as from the TL analysis with losses in conductors and dielectric and electrodynamic simulation.
It can be seen that the MF under consideration is a low-pass filter.The bandwidth (defined as −3 dB) was 0.521 GHz for the TL analysis, 0.756 GHz for electrodynamic simulation, and 0.741 GHz for measurements.Note that increasing or decreasing the MF length will decrease or increase the bandwidth and frequency of the first minimum.The results obtained using the approaches are in good agreement.The maximum deviation was observed at 10.84 GHz between the TL analysis and measurements.This might have been caused by the difference between the parameter values for the real PCB and the one taken in the simulation.It could also be explained by the fact that the TL analysis did not take into account higher types of waves and radiation losses.Measurement results were influenced by connectors.

B. UWB Pulse Decomposition
First, we obtained the output voltage waveforms (see Fig. 10) for the UWB pulse excitation using different approaches: numerical models, measurements, and the developed analytical model.The last one was only employed to obtain lossless results.It can be seen that the UWB pulse is divided into two pulses of smaller amplitudes (increasing/decreasing the length of the designed MF will increase/decrease time interval between pulses).There is a qualitative agreement between the results of simulation, measurements, and the proposed analytical model.A good quantitative agreement is observed between the responses calculated by the model ( 6) and as a result of the TL analysis, as well as between those calculated by the electrodynamic approach and measured.The difference between the results of analytical and quasi-static response calculations is due to the fact that the model does not allow taking into account losses in conductors, dielectric, and radiation.The electrodynamic simulation results are in the best agreement with the measurements.In addition, we note that the difference between the measurement results and the rest might have been affected by losses of coaxial-microstrip junctions, which were not taken into account in the model ( 6) and during the simulation.Table III shows the calculated norms.
The results show a decrease in each norm compared to the input excitation.The decrease in N 1 was caused by the decomposition of the input excitation into 2 pulses of more than 3 times smaller amplitudes.Additional attenuation was provided by losses in conductors and dielectric.N 2 values significantly decreased.First of all, this decrease was caused by a decrease in the voltage rise rate caused by the decomposition of the UWB pulse.In addition, N 2 was strongly affected by the smoothing of the voltage waveform because of dispersion caused by losses.In the case of N 3 and N 4 , no significant attenuation was observed, since there was a broadening of the pulses for all cases, except for the analytical model where this was not the case.Separation of the decomposed pulses in time reduced N 5 by at least a factor of 2.

C. Damped Sinusoid Propagation
Next, we obtained the output voltage waveforms (see Fig. 11) for damped sinusoid excitation using the same approaches.It can be seen that the signal is not decomposed into components, because the excitation and even its half period have a long duration relative to the UWB pulse.Meanwhile, such decomposition will occur with an increase in the MF length.There is a good qualitative agreement of the results obtained by different approaches.There is almost complete coincidence of the responses (waveforms, delays, and peak amplitudes of half-waves) obtained as a result of electrodynamic simulation and measurements.Differences begin to appear when the reflections from the MF input arrive with a double delay at its output.There is also a good agreement between the results of the TL analysis and analytical calculations.Since the analytical model is quite simple, it does not take into account the arrival

TABLE IV N-NORMS CALCULATED FOR DAMPED SINUSOID EXCITATION
of reflections from the input, which can be seen in the response.We also calculated the norms (see Table IV).
The results demonstrate a decrease in each norm, but this decrease may not be significant, which is caused by the damped nature of the excitation.It can be seen that the average value of norm N 1 decreased only by 1.65 times.Since the excitation was not decomposed into components because of the insufficient MF length, the decrease in the norm was caused only by the propagation of the excitation through the MF and its attenuation due to losses.A significant decrease in norm N 2 was caused by a decrease in the voltage rise rate due to the influence of losses on the smoothing waveform of the sinusoid half-wave form.The comparison of the values of norms N 3 and N 4 illustrate a significant attenuation of the excitation only for the analytical model.This is explained by the fact that the model did not take into account losses, and that the reflections from the input were superimposed on the main signal waveform.The average value of the decrease in norm N 5 is 1.45 times, which indicates a decrease in the probability of component burnout in the circuit after MF.

D. Propagation of the Sinusoid With Gaussian Modulation
Finally, we obtained the output voltage waveforms (see Fig. 12) for wide radio pulse excitation using the same approaches.
The results demonstrate that due to the increase in the total duration of the excitation, the pulse was not decomposed into components.By comparing the voltage waveforms, we can see that the results obtained by different approaches agree well qualitatively (by the character of propagation and waveform).However, quantitative differences appeared for the first halfwaves and increased as half-waves arrived at the MF output.Quantitative differences in the responses obtained analytically  and by TL analysis were caused by the arrival and superposition of multiple reflections on the MF output, which were not taken into account in the analytical model.The difference in the responses obtained as a result of measurements and electrodynamic simulation appeared only in the amplitude of half-waves and were caused by the smoothing of its rise and fall caused by greater influence of losses and dispersion.Table V summarizes the calculated norms.
The results show more significant decrease in each norm (compared to a damped sinusoid), which might have been caused by the periodic nature of the excitation and its modulation.In addition, the values of norms N 1 and N 2 practically coincide with each other, which can be explained by the nature of the excitation voltage waveform.We observe their decrease by 1.8 and 2 times as a result of electrodynamic simulation and measurements, respectively.It can also be seen that the probability of dielectric breakdown (norm N 3 ) is significantly reduced.According to the measurement results, the reduction is up to five times.It is explained by the significant influence of losses and dispersion on the high-frequency component of the excitation (compared with the analytical model and TL analysis).Similar results (a significant decrease in the norm values) are observed for norms N 4 and N 5 , which can be explained by a uniform distribution and a decrease in energy due to the influence of losses and dispersion.This indicates that the possibility of burnout of protected components and damage to equipment is reduced.Fig. 14 shows the phase-frequency characteristics of |S 21 |.We can see that the phase distortion of the signal is minimal.The measurement results also agree well with the simulation results.

E. Signal Integrity
Fig. 15 shows the eye diagrams obtained when the PRBS was propagating through the MF structure.We can see that signal integrity is ensured in both cases.Thus, for the 480-Mb/s bitrate, the data-dependent jitter was 15.6 ps.For the 1000-Mb/s bitrate, it was 35 ps.In all investigated cases, the eye is still open; therefore, the likelihood of bit errors will be low.The high-frequency MF mismatch has the greatest effect on the amplitude noise.As a result, for the 480-Mb/s bitrate, the signal-to-noise ratio was 12.21.For the 1000-Mb/s bitrate, the minimal value of this parameter was 5.05.The eye diagrams show that the investigated structure can be used to transmit a useful signal even at 1000 Mb/s.

VI. CONCLUSION
This article presents the results of a comprehensive analysis of an MF based on an MSL with two side conductors grounded at both ends.An analytical model has been obtained for calculating the time response at the output of such a structure.The results of two-approach simulations, measurements, and the proposed analytical models are in good agreement.The frequency responses showed that the investigated MF is a low-pass filter.At the same time, in the time domain, the UWB interference pulse was decomposed into two pulses of smaller amplitudes.The analysis of N-norms showed a significant decrease in the influence of different types of interferences.Thus, it can be concluded that the implementation of the proposed MF transforms a widely used MSL into a protective device at no great cost.The proposed analytical models allow the time responses to be analyzed without employing computationally demanding numerical methods.In addition, by equating the terms expressing the pulse amplitudes, the analytical conditions for the equality of the amplitudes could be further obtained.Finally, signal integrity was evaluated for two data rates and proved that the proposed MF can be used in signal circuits up to 1 Gb/s.

Fig. 1 .
Fig. 1.Cross section of the MSL with two side conductors.

Fig. 4 .
Fig. 4. (a) Top and (b) bottom layers of photomask and photographs of (c) top and (d) bottom layers of the MF prototype.

Fig. 13
Fig.13shows the frequency dependences of |S 11 | and |S 21 | of the MF obtained in the measurements and electrodynamic simulations.For the MF prototype, the bandwidth is 750 MHz.We can see that the measurement results agree with the simulation ones.The selection of the data rate should be conditioned

Fig. 15 .
Fig. 15.Eye diagrams for bitrates of (a) 480 and (b) 1000 Mb/s obtained at the MF output.

a
temperature view of the three-dimensional eye diagrams.The eyes are automatically centered and displayed over 2T on the horizontal axis.The color indicates the number of crossings of a segment in the time-voltage plane.Green areas are cold, whereas red indicates comparatively more crossings.During simulation, we computed contours of constant error probability and added them to the eye diagrams as BERContour.This method allows a comparative analysis of simulation and experiment results.

TABLE I CALCULATED
PER-UNIT-LENGTH PARAMETERS

TABLE III N
-NORMS CALCULATED FOR UWB PULSE EXCITATION

TABLE V N
-NORMS CALCULATED FOR SINUSOID WITH GAUSSIAN MODULATION