Nonlinearity in superconducting CPW open-endand gap discontinuities

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Journal of Electromagnetic Waves and Applications

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Nonlinearity in superconducting CPW open-end and gap discontinuities

Abolfazl Mostaani & S. Mohammed Hassan Javadzadeh

To cite this article: Abolfazl Mostaani & S. Mohammed Hassan Javadzadeh (2017): Nonlinearity in superconducting CPW open-end and gap discontinuities, Journal of Electromagnetic Waves and Applications, DOI: 10.1080/09205071.2017.1359683

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Nonlinearity in superconducting CPW open-end and gap discontinuities

Abolfazl Mostaani^aand S. Mohammed Hassan Javadzadeh^a,b

aDepartment of Electrical Engineering, Shahed University, Tehran, Iran;^bInstitute of Modern Information and Communications Technologies (IMICT), Shahed University, Tehran, Iran

ABSTRACT

In this paper, non-linear circuit models of the superconducting CPW (SCOW) open-ends and gaps are presented. The model can be used to create a fast and accurate analysis of the nonlinearity in SCOW applications such as end-coupled filters, which have such gap and open-end discontinuities. The non-linear circuit is solved using harmonic balance (HOB) method to compute the third-order intermodulation (IMD3) distortion and fundamental outputs. As an example, we analyze the nonlinearity of an end-coupled SCOW band pass filter (BP) to validate the model. The model can be used to minimize the nonlinearity in such superconducting structures via optimizing the resonators of superconducting filters. The main difference between the proposed model and the one in the literatures is that there is no clear discussion or method about nonlinearity in discontinuous SCOW structures because of absence of closed- form formula for some equivalent circuit model of discontinuities.

As an example, for determining the value of the equivalent gap capacitance of a SCOW structure, we can’t find any easy-use and direct equation. But in this paper, we also propose a closed-form equation for calculating the such capacitances.

ARTICLE HISTORY Received 6 April 2017 Accepted 13 June 2017

KEYWORDS Superconducting CPW;

superconducting filter;

harmonic balance;

nonlinearity;

intermodulation

1. Introduction

Non-linear phenomenon in superconducting materials arises from the dependence of the surface impedance on the applied field [1–3] and limits the power control capability of superconducting devices. Nonetheless, the superconducting materials are so exciting for the engineers because of their low loss at temperature below the critical temperature (T_c).

Nowadays, the superconducting materials have a wide usage in the microwave engineering and play an important role in many devises such as superconducting filters [4,5], resonators [6], and magnetometers [7,8]. The usage of such materials will be more and more, if the engineers have the fast prediction ability of the nonlinearity in the superconducting materials. Several models have been presented in the literatures, which help to predict the non-linearities in superconducting devices such as transmission lines [9–12], bends [13], antennas [14], and coupled lines [15], but there still are several types of superconducting microwave devices which need to be modeled. One of the most attractive types of men-

CONTACT S. Mohammed Hassan Javadzadeh [email protected]

Downloaded by [Eastern Michigan University] at 09:25 23 August 2017

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tioned devices is microstrip transmission line with open-end and gap discontinuities, which is very applicable in microwave filters [16–18].

In this paper, non-linear model of open-end and gap in superconducting CPW (SCOW) structures is presented. The proposed model uses the equivalent circuit model of the superconducting open-end/gap to predict the non-linearities in such discontinuities. Based on the method of moment (MOM), the current density in all region of the structure is obtained and used to circuit model creation and finally, the circuit is solved using harmonic balance (HOB) method by a commercial software advanced designed system. This model can predict and optimize the non-linear behavior of SCOW open-end/gap and minimizing their non- linearities. As an example, a SCOW band pass filter (BP) is analyzed through the model and its third-order intermodulation (IMD3) and fundamental outputs are observed to validate the proposed model. Although, such non-linearities which are generated by open-end or gap discontinuities are not dominant in simple superconducting microwave structures, but it must be considered for the accurate prediction of the nonlinearity. As the current density around open-end or gap discontinuities is lower than on uniform transmission line, therefore, it will show lower non-linear effect. However, without considering these models, prediction of the nonlinearity could not be accurate enough. Because without this model, transmission line near to open-end or gap discontinuity is also considered as a transmission line and the result of the non-linear prediction will be more than reality. It must be mentioned that there is a big difference between the proposed model and others.

For example, the model proposed in [19] is totally about microstrip transmission lines, because the equivalent circuit model’s basic equations are completely different from SCOW structures. Also, there is a few discussion about nonlinearity in discontinuous SCPWs in the literatures, because no clear and easy-use equation is accessible to model the discontinuous section to its equivalent circuit model. For example, no closed-form equation can be found for calculating the value of equivalent gap capacitances of SCOW structures so, the engineers must use the time consuming methods. But in this paper, we also proposed a closed-form equation for direct determining the mentioned capacitance. The equation is made using the curve fitting method applied to full-wave results and can be an easy mathematical solve for all hard technical methods.

2. Nonlinearity in SCPWs

The superconductors are famous for their low impedance at finite temperatures below the critical temperature (T_C). They are also known to produce nonlinearity due to the dependence of its surface impedance (Z_S) and superfluid density (n_S) on the current density (j) at the same temperature range. The general configuration of the SCOW and its equivalent circuit model are shown in Figures1and2, respectively. For the circuit model, we have

L(T,i)=L₀(T)+L(T,i), (1) R(T,i)=R0(T)+R(T,i), (2) where,T is the temperature,iis the current distribution at the cross-section of SCOW and L0andR0are the linear terms andLandRare the non-linear terms of the transmission line. According to (1) and (2), the series elements of the circuit include two parts, where the second part denotes the non-linear effect of the superconducting material because of

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Figure 1.The general configuration of SCOW structures.W,S,t, andhare the superconducting line widths, spacing, Thickness, and substrate height with dielectric constant ofǫr, respectively.

Figure 2.Equivalent circuit model of SCOW structure with length ofdxat temperature ofT.

its dependence on the current distribution. In other word, the non-linear behavior of the SCOW is embedded into the inductance and resistance per unit length of the circuit model.

In quadratic case of nonlinearity occurred at small current levels, the non-linear elements can be written as [20]:

L(T,i)=L_q(T).i², (3)

R(T,i)=R_q(T).i², (4) where

Lq(T)= µ₀λ²(T, 0)

j²_IMD (T), (5)

and

R_q(T)=σ₁(T, 0)ω²µ²₀λ⁴(T, 0)2+a(T)

j_IMD² (T) (T), (6) in which,

j_IMD(T)= j_C

√bθ(T), (7) where, j_C is the critical current level of the superconductor. Also, a(T) and bθ(T) are positive non-linear terms, which are completely discussed in [16]. The parameter (T) is the geometrical non-linear factor (GNF), which is as follows.

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Figure 3.The general configuration of SCOW gap structures.Gis the length of the gap discontinuity.

Figure 4.Equivalent circuit model of SCOW gap discontinuity structure.

(T)=

j⁴dS (

jdS)⁴, (8)

where, dSis the differential element of the transmission line cross-section.

2.1. Non-linear circuit model of gaps

Figures3and4, show the general configuration of SCOW gap discontinuity and its equivalent circuit model which is extended from the linear model of the gap [19]. According to the model for the quadratic state we can write,

L_gap(T,i)=L_gap_q(T).i², (9) R_gap(T,i)=R_gap_q(T).i², (10) in which

Lgapq(T)=Wµ₀λ²(T, 0)

2j_IMD² (T) _gap_q(T,W), (11)

R_gap_q(T)=σ₁(T, 0)ω²µ²₀λ⁴(T, 0)W2+a(T)

2j²_IMD _gap_q(T,W), (12)

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2 4 6 8 10 12 14

−45

−40

−35

−30

−25

−20

−15

−10

−5 0

Frequency (GHz)

S−Parameters (dB)

S11 − MOM S12 − MOM S11 − FEM S12 − FEM S11 − MODEL S12 − MODEL

Figure 5.Results comparison between the model vs. MOM and FEM for S-parameters.

2 4 6 8 10 12 14

0 20 40 60 80 100 120 140 160

Frequency (GHz)

Z−Parameters (Magnitude)

Z11 − MOM Z12 − MOM Z11 − FEM Z12 − FEM Z11 − MODEL Z12 − MODEL

Figure 6.Results comparison between the model vs. MOM and FEM for Z-parameters.

where, the parameter_gap_qis the gap GNF, which is defined by:

_gap_q(T,W)=

j⁴_gapdS (

j_gapdS)⁴ −(T,W), (13) where,j_gapis the current distribution at the gap cross-section. The Equations (5) and (6) give the quadratic non-linear inductance and resistance per meter of the superconducting strip, respectively. So, the values of the mentioned items must be multiplied by the length of the superconducting strip in the simulation. Similarly, the non-linear inductance and resistance of the discontinuous section of the structure must be multiplied by the length of the discontinuity, which is equal to the width of the line. In fact, the non-linear items come

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Figure 7.General configuration of open-end discontinuity.

Figure 8.Equivalent circuit model of the open-end SCOW.

from Equations (11) and (12) can be used in the simulation without any changes (such as multiplying by the length). Now, the question “Why is the discontinuity length equal to the line width?” could be asked. If we assume the length of discontinuity along the line, the value of length becomes zero, result in the value of non-linear inductance and resistance will be zero. Moreover, the length of discontinuity must be on the superconducting line, for example, we can’t assume the length of gap as the length of discontinuity. Because, the non-linear items are related to the superconducting materials and must just be considered where the superconducting materials are. In this paper, the current distribution is computed by a commercial microwave EM simulator based on MOM. According to π network in Figure4, bothCpandCgcan be defined as follows,

C_g= − 10¹⁵ 2πfIm ₁

Y21

(fF), (14)

where,f andYare the frequency and admitance matrix (Y-matrix), respectively. We can use the values ofC_gand Y-matrix to obtain the amount ofC_pas given in the following,

C_p+C_g||C_p=Y11, (15)

For calculation of bothCpandCg, there have been proposed a few closed-form equations in the literatures, which are mostly useful for a special SCOW gap geometries [19,21]. In this paper, we suggest another closed-form equation, which is more generalized than the

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Figure 9.Geometry of the designed SCOW BP made of 400 nm thick YBCO on 1mm thickLaAlO₃at temperature of 77 K.

others. The suggested equation is based on the Equation (14), which is computed using a commercial EM simulator (SONNET) and the results are used to prepare the following equation forC_g;

C_g=y₁(W,G,ǫ_r)S²+y₂(W,G,ǫ_r)S+y₃(W,G,ǫ_r) (pF), (16) where,S,W,G, andǫ_rare the amounts of spacing between the central line and ground planes, line width, gap length, and relative dielectric permittivity of the SCOW structure, respectively andyi(W,G,ǫ_r)can be determined as follows,

y₁(W,G,ǫ_r)=p₁₁(W,ǫ_r)G²+p₂₁(W,ǫ_r)G+p₃₁(W,ǫ_r), (17) y₂(W,G,ǫ_r)=p₁₂(W,ǫ_r)G²+p₂₂(W,ǫ_r)G+p₃₂(W,ǫ_r), (18) y₃(W,G,ǫ_r)=a(W,ǫ_r)exp(b(W,ǫ_r)G)+c(W,ǫ_r)exp(d(W,ǫ_r)G), (19) in which,p_ij(W,ǫ_r)anda,b,c, andd(W,ǫ_r)are the coefficients, which are explained in Appendix1. Equation (16), gives the exact result forǫ_r≥10,W ≥200µm, andG≥100µm.

AboutC_p, we just use the general solution of formula (15). Furthermore, all of the model elements can be analytically obtained. A simple SCOW gap structure similar to Figure3, is also simulated and its S- and Z-parameters are analyzed using the model and MOM and FEM. The structure is made of 400 nm thick YBCO on 0.635 mm thickness of MgO substrate with gap width (G) of 100µm at temperature of 77 K. All of the results are compared in Figures5and6to show the accuracy of the suggested model.

2.2. Non-linear circuit model of open-ends

Figures7and8, display the general structure of a SCOW open-end and its equivalent circuit model, respectively. All of the model elements can be calculated similar to the gap case by replacing_gapwith_oc. The value of the shunt capacitance (Coc) is given in [22]

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10 10.2 10.4 10.6 10.8 11

−40

−35

−30

−25

−20

−15

−10

−5 0

Frequency (GHz)

S−Parameters (dB) S11 − MOM

S12 − MOM S11 − MODEL S12 − MODEL

Figure 10.S-parameter comparison between the model and MOM.

Figure 11.Fundamental signal response of the filter forP_in=30 dBm and its H3 generation output.

Coc =2ǫ₀ǫ_eff π

(S+W)

ln(η+ 1+η²)

η +ln1+ 1+η² η

−1 3

1 1+

1+η² + 1 η+

1+η² −

W +2

3S , (20)

and

η= G

S+W, (21)

where,Gis the gap width of open-end discontinuity.

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Evaluation of the model is performed by analysis of a SCOW BP, which is made of 400 nm thick YBCO on 1mm thickness of LaAlO₃as a substrate (Figure9). The designed BP contains four gaps, which are modeled using the introduced approach and solved using HOB method to observe the non-linear behavior of the device. C_g are obtained to be 23fF and 60fF using Equation (16) for wide and narrow gaps, respectively andC_pis obtained to be 40fF, empirically. Figure10, shows the frequency response of the BP using MOM vs. result of the proposed model forP_in=30 dBm atT =77 K. The fundamental signal and H3 generation of the SCOW BP are also determined forf =500 kHz and displayed in Figure11. According to [13,15], the H3 values must have a peak at the pass band, which the results of the model show as well.

4. Conclusion

In this paper, an accurate model for nonlinearity in open-ends and gaps structures is proposed for superconducting co-planar wave guides. The model is based on distributed circuit model of the superconducting structure can be solved fast and accurately by HOB method instead of full-wave analysis of the original structure. Through the proposed model, fundamental and third-order IMD outputs are extracted to observe the non-linear behavior of such SCOW structures. A simple SCOW gap structure is analyzed using the model and its S- and Z-parameters are compared with MOM and FEM to show the accuracy of the model.

Also, A closed-form equation for series capacitance of the SCOW gap structures is proposed.

Finally, validation of the model is performed by simulation of a SCOW BP made of YBCO on LaAlO₃substrate and observation of its nonlinearity.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

S. Mohammed Hassan Javadzadeh http://orcid.org/0000-0002-7385-3384

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Appendix 1.

For Equations (17)–(19),p_ij(W,ǫ_r)is as follows.

p₁₁=p₁W³+p₂W²+p₃W+p₄, (A1) where

p₁=2.487×10⁻²⁶ǫ³_r −1.114×10⁻²⁴ǫ_r²+1.696×10⁻²³ǫ_r−7.575×10⁻²³, (A2) p₂= −5.584×10⁻²³ǫ_r³+2.567×10⁻²¹ǫ_r²−4.041×10⁻²⁰ǫ_r+1.728×10⁻¹⁹, (A3) p₃=4.489×10⁻²⁰ǫ_r³−2.071×10⁻¹⁸ǫ_r²+3.412×10⁻¹⁷ǫ_r−1.354×10⁻¹⁶, (A4) p₄= −1.133×10⁻¹⁷ǫ_r³+5.114×10⁻¹⁶ǫ_r²−9.54×10⁻¹⁵ǫ_r+3.242×10⁻¹⁴. (A5) And

p₂₁=p₁W³+p₂W²+p₃W+p₄, (A6)

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p1= −1.667×10⁻²⁵exp(0.4767ǫ_r)+7.193×10⁻²¹exp(0.05389ǫ_r), (A7) p₂=8.858×10⁻²³exp(0.5347ǫr)−1.201×10⁻¹⁷exp(0.0689ǫr), (A8) p₃= −5.466×10⁻¹⁹exp(0.4702ǫ_r)+4.345×10⁻¹⁶exp(0.2043ǫ_r), (A9) p₄=1.179×10⁻¹⁴ǫ³_r −5.3×10⁻¹³ǫ_r²+1.059×10⁻¹¹ǫ_r−3.276×10⁻¹¹. (A10) And

p₃₁=p₁W³+p₂W²+p₃W+p₄, (A11) where

p₁=6.44×10⁻²¹ǫ_r³−2.976×10⁻¹⁹ǫ_r²+3.685×10⁻¹⁸ǫ_r−2.404×10⁻¹⁷, (A12) p₂= −1.545×10⁻¹⁷ǫ³_r +7.072×10⁻¹⁶ǫ_r²−8.248×10⁻¹⁵ǫ_r+5.811×10⁻¹⁴, (A13) p₃=1.129×10⁻¹⁴ǫ_r³−5.114×10⁻¹³ǫ_r²+4.465×10⁻¹²ǫ_r−4.351×10⁻¹¹, (A14) p₄= −2.224×10⁻¹²ǫ_r³+1.115×10⁻¹⁰ǫ_r²−2.256×10⁻⁹ǫ_r+6.98×10⁻⁹. (A15) And

p₁₂=p₁W³+p₂W²+p₃W+p₄, (A16) where

p₁= −1.641×10⁻²³ǫ_r³+7.02×10⁻²²ǫ_r²−1.114×10⁻²⁰ǫ_r+4.158×10⁻²⁰, (A17) p₂=3.141×10⁻²⁰ǫ³_r −1.461×10⁻¹⁸ǫ_r²+2.602×10⁻¹⁷ǫ_r−8.53×10⁻¹⁷, (A18) p₃= −1.458×10⁻¹⁶ǫ_r²−3.55×10⁻¹⁵ǫ_r−3.756×10⁻¹⁴, (A19) p₄=1.83×10⁻¹⁴ǫ_r²+2.647×10⁻¹²ǫ_r+7.549×10⁻¹². (A20) And

p₂₂=p₁W³+p₂W²+p₃W+p₄, (A21) where

p₁=5.51×10⁻²⁰exp(0.2692ǫ_r)−9.215×10⁻¹⁸exp(0.07579ǫ_r), (A22) p₂= −4.149×10⁻¹⁷ǫ³_r +1.964×10⁻¹⁵ǫ_r²−2.499×10⁻¹⁴ǫr+1.336×10⁻¹³, (A23) p₃=2.41×10⁻¹⁴ǫ_r³−1.148×10⁻¹²ǫ_r²+1.768×10⁻¹¹ǫ_r−7.854×10⁻¹¹, (A24) p₄= −2.09×10⁻¹¹ǫ²_r −4.347×10⁻⁹ǫr−9.819×10⁻⁹. (A25) And

p₃₂=p₁W³+p₂W²+p₃W+p₄, (A26) where

p₁= −5.6×10⁻¹⁸ǫ_r³+2.52×10⁻¹⁶ǫ_r²−2.076×10⁻¹⁵ǫ_r+2.386×10⁻¹⁴, (A27) p₂=6.96×10⁻¹⁴ǫ_r²−7.53×10⁻¹²ǫ_r−5.132×10⁻¹², (A28) p₃=2×10⁻¹²ǫ²_r +9.198×10⁻⁹ǫ_r+1.076×10⁻⁸, (A29) p₄= −3.9×10⁻⁹ǫ_r²+1.921×10⁻⁶ǫ_r+1.635×10⁻⁶. (A30) And, fora,b,candd(W,ǫ_r)we have,

a=p₁W³+p₂W²+p₃W+p₄, (A31)

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where

p₁=2.004×10⁻¹³ǫ_r²+3.568×10⁻¹²ǫ_r+2.024×10⁻¹¹, (A32) p₂= −5.49×10⁻¹⁰ǫ_r²−6.643×10⁻⁹ǫ_r−6.434×10⁻⁸, (A33) p₃=4.54×10⁻⁷ǫ²_r +7.278×10⁻⁶ǫ_r+6.411×10⁻⁵, (A34) p₄= −8.17×10⁻⁵ǫ_r²−0.0009159ǫr−0.01254. (A35) And

b=p₁exp(p₂W)+p₃exp(p₄W), (A36) where

p₁=1.475×10⁻¹³exp(0.8572ǫr)−0.006444 exp(0.0001566ǫr), (A37) p₂= −2.045×10⁻²³ǫ_r²+4×10⁻⁹ǫ_r+1.151×10⁻⁵, (A38) p₃=2.177×10¹¹exp(−2.628ǫr)+3.571 exp(−0.2393ǫr), (A39) p₄=8.616×10⁻⁶ǫ_r³−0.0005439ǫ²_r +0.01194ǫ_r−0.1003. (A40) And

c=p₁W³+p₂W²+p₃W+p₄, (A41) where

p₁= −2.12×10⁻¹³ǫ_r²−4.364×10⁻¹²ǫ_r−1.838×10⁻¹¹, (A42) p₂=5.85×10⁻¹⁰ǫ_r²+1.038×10⁻⁸ǫr+6.134×10⁻⁸, (A43) p₃= −4.8×10⁻⁷ǫ_r²−3.04×10⁻⁷ǫ_r−5.573×10⁻⁵, (A44) p₄=8.23×10⁻⁵ǫ²_r +0.0007005ǫr+0.01149. (A45) And

d=p₁exp(p₂W)+p₃exp(p₄W), (A46) where

p₁= −2.14×10⁻⁵ǫ_r²−0.0002706ǫ_r−0.01983, (A47) p₂=6.8×10⁻⁷ǫ_r²−0.0001756ǫ_r−0.00728, (A48) p₃=9.287×10⁻⁶exp(0.1575ǫ_r)−0.002951 exp(0.004102ǫ_r), (A49) p₄= −2.88×10⁻⁸ǫ_r³+1.366×10⁻⁶ǫ_r²−2.459×10⁻⁵ǫ_r−0.0004572. (A50) Therefore, all of coefficients of Equation (16) can be obtained.