Effects and Models of Offset-Via in Electromagnetic Band Gap Structure

Via is an essential component in the design of an Electromagnetic Band Gap (EBG) structure. This paper investigates the effects of the via in a mushroom-like EBG structure and proposes an equivalent circuit model of the via. It is revealed that when the via in the mushroom-like structure is offset, the element reflection phase will vary within 720 degrees instead of 360 degrees phase range in a conventional design. This phenomenon can be explained by a new circuit model that introduces a mutual inductance, and the corresponding electromagnetic properties of mushroom-like EBG structure can be quantitatively analyzed by this model. In addition, the applicability and error analysis of the model are discussed in this paper. The theoretical modeling and analysis enable an efficient and in-depth exploration of the via functions in similar EBG structures, and also provide more theoretical guidance and assistance for the design of reflection and transmission units based on EBG structure.

Connecting via is an essential component in a mushroomlike EBG structure.So a theoretical model is necessary to understand the coupling characteristics between the via and other components.There are several related studies about the model establishment and the theoretical analysis of the via in electromagnetic structures [15], [16], [17].Relevant calculation models of the via inductance have been also proposed in [18], [19].However, due to the lack of an analytical solution for via inductance and its susceptibility to other structures, a widely applicable model is yet to be established.In the case of the classic mushroomlike structure featuring a central via, several studies have examined its analysis and modeling [3], [20], [21], [22].There are also analytical models that employed the nonlocal homogenization models [23], [24], [25].Whereas, for the modified mushroom-like structure with offset vias, there is no accurate theoretical analysis and modeling yet.Direct application of the relevant formula in [3], [26] results in significant errors.Hence, it is necessary to develop an accurate model for the mushroom-like EBG structure with offset vias.
In a mushroom-like EBG structure, the offset distance of the via will bring great difference to the properties of the structure.The original symmetrical mushroom-like structure has only one resonant frequency, whereas the structure with a single offset via will introduces two resonant frequencies.Moreover, the range of the reflection phase has also been expanded from 360 degrees to 720 degrees.As mentioned in the previous paragraph, there is a lack of comprehensive research dedicated to the qualitative and quantitative analysis of this phenomenon.
In this paper, an equivalent circuit model of the offset-via is proposed to explain the double resonance phenomenon in the EBG structure and calculate the reflection phase.Then, an improved method in which only one simulation of the EBG unit without vias is required to obtain phase curves for the unit at all bias distances is proposed to enhance accuracy.
This article is organized as follows: Section II provides an overview of the structure configuration and the phenomenon under study.The model of the central-via mushroom-like structure is discussed in Section III, with details of the parameter analysis and error analysis.Next, an innovative circuit model of the offset-via mushroom-like structure and comprehensive discussion are presented in Section IV.The comparison between the full-wave simulation and the theoretical calculation of various EBG structures are illustrated in Section V. Finally, conclusions are drawn in Section VI.

II. CONFIGURATION AND PHENOMENON
The aim of this work is to understand and analyze the function of the offset via in the EBG structure, as shown in Fig. 1(a).The symbols of main parameters are labeled in Fig. 1(b).Furthermore, ε 1 is the dielectric constant of the medium, and ε 0 is the dielectric constant in the vacuum.r is the outer radius of the via.In a reference design, W = 6mm, h = 2mm, g = 1mm, r = 0.2mm, ε 1 = ε r ε 0 = 2.2ε 0 .All the simulations are carried out by commercial electromagnetic software Ansys HFSS.
When the incident wave is x-polarized, the solid line in Fig. 2 is a phase curve of the reflection coefficient when the via is in the center, that is, t = 0.The dashed lines are phase curves of the reflection coefficients when the offset distance is 0.5mm, 1.5mm, and 2.5mm, respectively.Once the via is offset, the phase curve changes from single resonance to double resonance.Hence, the range of the reflection phase is 720 degrees.Furthermore, a mutation frequency of 7.8 GHz can be defined in these reflection phase curves, where the reflection phase passes through 180 degrees.Although the phase curve changes with the offset distance, the mutation frequency hardly changes.
To explain the phenomenon that appear in the offset-via structure above, we first review the existing central-via model in previous research [3].We will discuss the applicability of the model, identify scenarios that result in noticeable phase errors, and then provide a correction method to enhance the accuracy of the model.Next, a new circuit model of the offset-via mushroom-like structure is proposed, which is used to quantitatively analyze the dual resonance of the reflection phase, and explain why the mutation frequency remains unchanged with the change of the t.

III. CIRCUIT MODEL FOR THE CENTRAL-VIA EBG A. CIRCUIT MODEL
The reflection analysis of this periodic EBG structure can be equivalent to a transmission line with a characteristic impedance of Z 0 and a load impedance of Z S .This equivalent model has been widely used in previous studies and has been proved to be a reasonable approximation [3], [21], [27].The relationship between reflection coefficient and impedance is shown in (1).
In the case of normal incidence of an electromagnetic wave in air or vacuum, it is commonly approximated that Z 0 = 377 ohms.When Z S is an imaginary number, the amplitude of the reflection coefficient is 1.When Z S = 0, it can be inferred that the reflection phase is −180 degrees or +180 degrees (depending on Z S → 0 + or Z S → 0 − ).When Z S → +∞ or Z S → −∞, the reflection phase is 0 degrees, which is denoted as the resonance point.For the Z S analysis of a mushroom-like EBG structure with a central via, Sievenpiper et al. proposed a calculation method [3], as illustrated in Fig. 3.The current flowing in this mushroom-like structure is divided into two parts: i c and i 1 .i c is the current flowing through the equivalent capacitance C 0 originated from the upper patches between two adjacent units.i 1 is the current mainly flowing through equivalent inductor L 0 caused by the upper metal patch and metal ground.In fact, the effect of this equivalent inductance can also be regarded as the effect of combination of a transmission line of length h connected a short-circuit load impedance.When value of h is much smaller than the wavelength, the results of two calculation method are basically consistent.Then equations ( 2) and ( 3) can be listed as the current and voltage equations in the S-domain: The equivalent circuit of Z S in a single central-via mushroom-like structure is shown in Fig. 3(b).So, we can calculate the equivalent surface impedance, as in (4): Next, we will show the formulation of the L 0 and the C 0 in detail.The C 0 is given by the capacitance between coplanar metal patches in adjacent units, which can be derived by the conformal mapping.The inductance calculation is based on the formula for calculating the inductance of the solenoid [3].The current line density on the upper and lower patches is approximately i/(W+g), and the area inside the coil is (W+g) *h.According to the above relationship, the value of the L 0 and the C 0 can be calculated in ( 5) and ( 6): ) As mentioned earlier in this section, the effect of this inductance L 0 can also be regarded as the effect of combination of a transmission line of length h connected a short-circuit load impedance.So, according to the transmission theory, the equivalent impedance of this combined structure is given by: c represents the speed of light.
When the h λ g , the impedance Z L in ( 7) is equal to the jωL 0 in (6).The thickness of the array used in the current research generally satisfies this condition, thus validating the consistency between the model that equates the medium to an inductance and the model that equates the medium to a transmission line.

B. DISCUSSION AND UPDATE
The EBG unit is simulated in HFSS, with periodic boundary conditions and Floquet port excitation applied.The simulation results are compared with the circuit model results to verify the accuracy of the circuit model.
Firstly, we take W = 6mm, g = 1mm, r = 0.2mm, ε r = 2.2, but h varies from 0.5mm to 2.5mm and 5.0 mm.The comparison of the reflection phase calculated by the circuit model and the reflection phase simulated by the HFSS is shown in Fig. 4. It is noticed that when h, the thickness of the dielectric substrate, is relatively thick, the result of the circuit model is similar with that of the full-wave simulation.With h decreases, the difference between the results of the simulation and the model becomes larger.Nevertheless, the circuit model can still capture the overall trend of the phase curve change.Next, to further understand the error of the model, we fit the simulated phase curve to obtain an updated value of C 0 (updated method), and compare with those calculated by the circuit model (original method) and simulated by HFSS in Fig. 5. Equation ( 5) suggested the capacitor remain constant when the h change.Nevertheless, according to the value obtained by the updated method, the simulated C 0 will vary with h, and the error range is within 20% when the h is relatively large.This discrepancy arises because when the distance of the metal patch and the metal ground is close, the coupling effect between them will be stronger.When calculating C 0 by (5), only the coupling of metal patches between adjacent units was considered.Therefore, when the coupling effect between the metal patch and the metal ground strengthens, the error between the calculated capacitance and the actual capacitance will increase, resulting in a larger deviation of the phase curve.
Next a simulation model is designed to verify the effect of dielectric thickness h on C 0 and confirm the accuracy of the value by updated method, as depicted in Fig. 5(a).To obtain the capacitance C 0 value, we extracted the patch between two adjacent units from the original mushroom-like structure, and set the surrounding medium and air distribution according to the original structure.
The Fig. 5(b) presents the comparison of the capacitance obtained by original method, the capacitance obtained by the updated method, and the capacitance obtained by directed capacitance simulation.The results clearly confirm our previous conclusion.When h is relatively small, the value of C 0 tends to be larger.Additionally, when h is greater than a certain value, C 0 hardly changes with the value of h.
It is worthwhile to point out that the discrepancy in C 0 among the different value of h will become a crucial factor contributing to the accuracy of this model.Therefore, it is recommended that the value of C 0 in (5) needs to be updated by the value fitted in the simulated curve to minimize the error caused by calculation value in (5).

IV. CIRCUIT MODEL FOR THE OFFSET-VIA EBG A. CIRCUIT MODEL
As mentioned before, this type of periodic structure can be equivalent to a transmission line with a characteristic impedance of Z 0 and a load impedance of Z S .However, unlike the previous model, the mushroom-like structure with an offset-via exhibits a double resonance phenomenon.Therefore, the circuit model of Z S will also be different in this case.
A new circuit model is proposed, including a mutual inductance to calculate the value of Z S , as shown in Fig. 6(a).The current flowing in the overall mushroom-like structure is divided into three parts: i 1 , i 2 , i C .i C is the current flowing through the equivalent capacitance C 0 originated from the upper patches between two adjacent units.i 1 is the current mainly flowing through equivalent inductor L 0 caused by the upper metal patch and metal ground.i 2 is the loop current flowing through the connecting via and the equivalent capacitance C p formed by the upper patch and the metal grounded.L v represents the equivalent inductance when the current flow through connecting via.We can list the (8), ( 9) and (3) as the current and voltage equations: The equivalent circuit of Z S in the offset-via case is illustrated in Fig. 6(b).Regarding these parameters with t, we will explain their derivation in detail later.It is worth mentioning that although the via L v is physically connected to C 0 , due to the periodicity of the EBG structure, the positive charge and negative charge in C 0 cancel out within the same unit, and the current i 2 flowing through the connecting L v does not enter C 0 , but all enters C p , forming a displacement current between the center of the patch and center of the ground.Therefore, in the equivalent circuit, the via L v can be disconnected from the capacitor C 0 .Ultimately, the sum of i 1 and i C represents the current flowing into the unit cell.According to the definition of the current and voltage at both ends of the transmission line, the equivalent surface impedance can be calculated in (10): 1) L 0 AND M The L 0 is still the value in (6).From a similar derivation in the value of L 0 , the magnetic flux generated by i 2 in the i 1 loop is t W+g L 0 i 2 .And the magnetic flux generated by i 1 in the i 2 loop is t W+g L 0 i 1 .Hence, The magnetic flux generated by i in the i 2 loop is 2) C P The C p is the equivalent parallel metal plate of the upper patch and the lower ground, so the value is shown in (12): It is worth noting that W>>g is required here.If this condition is not satisfied, because the area of the upper patch is W 2 and the corresponding effective area of the metal ground is (W+g) 2 , there will be a small deviation between the capacitance in ( 12) and the actual capacitance.
3) L V Due to absence of analytical solutions for calculating the inductance of the via, various numerical solutions are proposed in previous literature, such as given in [18], [19].However, after theoretical analysis and parameter fitting, it is found that the formula mentioned in these articles is not suitable for mushroom-like structure, probably because of the higher frequency band and the periodicity of unit.So, a new circuit model for the via in mushroom-like is proposed, based on the abstraction of the physical model and parameter fitting.
The inductance generated by the via is abstracted as electromagnetic characteristic of an infinitely long metal cylinder.The magnetic flux flowing through the via to the edge of the patch is μhI 2π ln W+g 2r when the W>>r.The estimated value of L v is shown in (13):

B. DISCUSSION AND UPDATE
Then the comparison of the simulated phase curve and calculated phase curve by ( 13) is illustrated in Fig. 7.We take W = 6/7*P, g = 1/7*P, r = 0.2mm, ε r = 2.2, but P, which represents the periodicity of the unit, varies from 4.5mm to 11mm.Double resonance phenomena are observed in both the simulation results and calculation results.It can qualitatively predict the phase variation pattern, and provide the corresponding phase values at each frequency.However, this method has a certain margin of error, around 10% in operating frequency.The numerator term in (10) shows the relationship between the mutation frequency and the parameters of the unit, as shown in (14): With the value of C p calculated in (13) and the simulated mutation frequency in (14), the value of L v corresponding to the specific parameter can be calculated.Then using the relationship between the inductance and h, W, g, r in (13), we can further fit the L v and other parameters to obtain an updated via inductance which is more accurate, as shown in (15): The relationship between the inductance of via and other unit parameters in is similar to the relationship derived from electromagnetic theory in (13), except for a little difference in two coefficients.The coefficients of determination in the derivation and fitness of (15), which can evaluate the goodness of fit, exceeds 0.99 when W is between 4 mm and 20mm, R is between 0.1mm and 3mm, h is between 1mm and 6mm.
Although this formula is obtained through curve fitting, it can be employed as an empirical formula for predicting performance of EBG structures in commonly used microwave frequency bands.It demonstrates good general applicability.
Similar to the previous section, we designed a simulation model with periodicity to verify the effect of W on L v and confirm the accuracy of the value obtained by the updated model.Fig. 8 shows the comparison of the inductance value calculated by ( 13) and (15), and the inductance obtained by simulating via in a validation model.It can be observed that the curve calculated by (15) and the curve simulated by the validation model are relatively consistent across different values of h.
It is worthwhile to point out that the model can also provide an explanation for the mutation frequency that does not change with the variation of t and h.By substituting equations ( 12) and ( 15) into ( 14), we obtain (16): t and h are not included in (16).This means that changes in t and h do not affect mutation frequency.

V. COMPARISON OF CIRCUIT MODEL AND NUMERICAL SIMULATION A. PROCEDURE
In the previous section, a pure analytical method using equivalent circuit model is proposed to analyze the mushroom-like EBG structures with offset via.Several examples are presented in this section to verify the accuracy of the proposed model.When aiming for an accurate reflection phase curve, special attention should be paid upon the C 0 value and L v value.Hence, a procedure is summarized as follows: 1. Obtain a simulated phase curve of a centralvia mushroom-like structure with the EBG parameters (W, g, h, ε 1 ).
2. Perform parameter fitting on the simulated phase curve to determine the updated values of C 0 .
3.Replace the values in ( 5) with the updated C 0 values obtained above, and replace the value in (13) with the value in (15).
4.Substitute these updated values into (10) to calculate the reflect phase curve.
After the correction of C 0 and L v , the consistency between the calculated phase curve and the simulated phase curve will be greatly improved.
When using this method to predict phase curves, we only need one simulation process for the unit without via to obtain phase curves for all bias distances.

B. EXAMPLES
The improved method has been proposed to be compared with the original method and the simulation for the same structural parameter, as shown in Fig. 7. Relative to the original purely analytical method, the accuracy of the curves Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.was significantly enhanced.The error of the operating frequency decrease from nearly 10% to 2%.Then, more examples with different parameters will be demonstrated to further validate the effectiveness and accuracy of this updated method.
Firstly, W = 6mm, h = 5mm, g = 1mm, r = 0.2mm, ε r = 2.2 are taken as the original parameter.Then, offset distance t changes from 0.5mm to 2.5mm to compare the simulation results and theoretical model results in Fig. 9(a).The EBG unit is simulated in HFSS, employing periodic boundary conditions and Floquet port excitation.Then, the examples with different h, W and ε r are compared in Fig. 9(b)(c)(d).
Overall, the calculated phase curves exhibit good agreement with the simulated phase curves, as depicted in Fig. 9(a).Even under varying conditions of h, W and ε 1 values, the simulated phase values and the calculated phase values continue to exhibit remarkable consistency, as shown in Fig. 9(b)(c)(d).
Dual resonance phenomena with 720 degrees phase range are observed in these cases.In particular, two resonant frequencies can be identified when the reflection phase equals to 0 degree, and one mutation frequency can also be identified when the reflection phase equals to 180 degrees.According to (1), at the resonant frequency, Z S = ∞; at the mutation frequency, Z S = 0.
The dual resonance phenomena can be understood from (10).When Zs=0, the numerator needs to be 0. Since Zs in (10) has only one zero-point, so the phase curve has only one mutation frequency.When Zs= ∞, the denominator needs to be 0. Since Zs in (10) has two poles, so the phase curve has two resonant frequencies.
The parametric effect on the mutation frequency is studied first.It is noteworthy that, for different values of t and h while keeping other parameters constant, it can be observed that the mutation frequency remains almost unchanged, as depicted in Fig. 9(a)(b).Increasing W and ε 1 will reduce the mutation frequency, as depicted in Fig. 9(c)(d).The reason for the above phenomenon can be understood from ( 12), ( 14), (15).Changing t does not affect L v and C p , thus keeping the mutation frequency unchanged.On the other hand, increasing W and ε r results in an increase in C p, while the increase of W also causes an increase in L v , both resulting in a decrease in the mutation frequency.Additionally, increasing h causes C p to decrease and L v to increase, but these effects counterbalance each other, ultimately maintaining the mutation frequency unchanged.
Next, this curve also has two resonance points, that is, the 0-phase point.This resonance points corresponds to the two zero points in (10).Increasing t and h will also increase the resonance frequency of the high resonance point and decrease the resonance frequency of the low resonance point.Increasing W and ε 1 will reduce the resonance frequency of the two resonance points simultaneously.This phenomenon can also be understood by analyzing the variations in parameters as described in (10), similar to the earlier analysis of how parameters affect the mutation frequency.

C. SUMMARY
This paper primarily proposes three methods to obtain the phase curve of the single-offset mushroom-like EBG structure.The characteristics of each method are summarized and emphasized as follow: Method 1: Method 1 is a purely simulation-based approach.It is relatively accurate, but the simulation process is time-consuming.
Method 2: Method 2 is a purely analytical approach.It utilizes the equivalent circuit model from Section IV, along with (5) and (13).Its computational time is negligible compared to simulation, approximately 1.1 seconds, but there is approximately a 10% error in the operating frequencies.The comparison between Method 1 and Method 2 is illustrated in Fig. 7.
Method 3: Method 3 is a semi-analytical approach that utilizes a small amount of simulation.It leverages the model of the structure without vias to fit C 0 and calculates L v using (15).This method is accurate and significantly reduces the computation time compared to the pure simulation method (Method 1), as it only requires one simulation to determine the phase curves for all bias distances.The comparison between Method 1 and Method 3 is illustrated in Fig. 9.
Finally, the time consumption for obtaining phase curves using method 1 and method 3 is recorded.The frequency range considered is from 3 to 15 GHz, with t varying from 0.4mm to 2.7mm at an interval of 0.1mm.The method 1 is solely relying on full-wave simulation, requiring 173 minutes.The method 2 requires 1.1 sec in this example, The method 3 requires 4 minutes plus 2.17 seconds.
Hence, the analysis of the mushroom-like structure by this circuit model can help us to design an EBG structure efficiently.The reduction in time consumption primarily stems from a decrease in the number of simulations.In method 2 and method 3, we can not only provide a qualitative explanation but also accurately obtain quantitative reflection coefficient curves in just one simulation, which would typically require multiple simulations in method 1.Therefore, the time required is significantly decreased.

VI. CONCLUSION
This paper investigates the characteristic of the via in the mushroom-like EBG structure.An equivalent circuit model of the EBG structure, including a mutual inductance accounting for the offset via, is proposed to understand and analyze the double resonance phenomenon.The systematic theoretical analysis and modeling conducted in this study allow for a more efficient and in-depth exploration of the via functions in EBG structures.Moreover, it exhibits good generalizability.For example, extending similar analysis methods to the case of oblique incidence: by using a similar analytical approach, we can also equivalently represent the EBG unit under oblique incidence as a similar circuit model.In the future, this basic model can be extended to analyze EBG cases of multiple offset vias, and enable to design reflectarray antenna and transmitarray antenna with vias.

FIGURE 1 .
FIGURE 1. Configuration of a mushroom-like EBG structure with an offset via.(a) 3D view of the EBG structure; (b) side view of the EBG structure.

FIGURE 2 .
FIGURE 2. Reflection phases of a mushroom-like EBG structure varying with different offset distance t in x-direction.

FIGURE 3 .
FIGURE 3. Model of a mushroom-like EBG with a central-via.(a) Schematic diagram of current distribution (b) Diagram of equivalent circuit.

FIGURE 4 .
FIGURE 4. Comparison of the phase simulated in HFSS and the phase calculated by circuit model of single central-via mushroom-like structure.

FIGURE 5 .
FIGURE 5. Configuration of the Verification Simulation model C0.(a) Adjacent patch capacitor model.(b) Comparison of C0 calculated by model (original method), C0 extracted from reflection phase (updated method) and C0 obtained by the direct capacitor simulation.

FIGURE 6 .
FIGURE 6. Model of a mushroom-like EBG with an offset-via.(a) Schematic diagram of 2D current distribution (b) Diagram of equivalent circuit.

7 .
Comparison of the simulated phase curve and original method calculated phase curve of the offset-via mushroom-like structure.

FIGURE 8 .
FIGURE 8. Configuration of a simulation model Lv.(a) An inductor model of the via in a mushroom structure (b) Comparison of Lv calculated by model (original method), Lv calculated by our empirical formula (updated method) and Lv obtained by the direct inductance simulation.

FIGURE 9 .
FIGURE 9. Comparison of the simulated results in HFSS and the calculated results by the updated equivalent circuit model of the offset-via mushroom-like EBG structure.(a) Reflection phases under the original parameters: W = 6mm, h = 5mm, g = 1mm, r = 0.2mm, εr = 2.2.(b) Reflection phases when h = 2mm.(c) Reflection phases when W = 9mm.(d) Reflection phases when εr = 3.5.