Modelling nonlinearity in superconducting split ring resonator and its effects on metamaterial structures

Physica C: Superconductivity and its applications 540 (2017) 26–31

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Physica C: Superconductivity and its applications journal homepage: www.elsevier.com/locate/physc

Modelling nonlinearity in superconducting split ring resonator and its effects on metamaterial structures Behnam Mazdouri a, S. Mohammad Hassan Javadzadeh a,b,∗ a b

Department of Electrical Engineering, Shahed University, Tehran, Iran Institute of Modern Information and Communications Technologies (IMICT), Shahed University, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 15 April 2017 Revised 27 June 2017 Accepted 14 July 2017 Available online 18 July 2017 Keywords: Superconducting split ring resonator Nonlinear behavior Superconducting metamaterial Circuit modelling Nonlinear modelling

a b s t r a c t Superconducting materials are intrinsically nonlinear, because of nonlinear Meissner effect (NLME). Considering nonlinear behaviors, such as harmonic generation and intermodulation distortion (IMD) in superconducting structures, are very important. In this paper, we proposed distributed nonlinear circuit model for superconducting split ring resonators (SSRRs). This model can be analyzed by using Harmonic Balance method (HB) as a nonlinear solver. Thereafter, we considered a superconducting metamaterial filter which was based on split ring resonators and we calculated fundamental and third-order IMD signals. There are good agreement between nonlinear results from proposed model and measured ones. Additionally, based on the proposed nonlinear model and by using a novel method, we considered nonlinear effects on main parameters in the superconducting metamaterial structures such as phase constant (β ) and attenuation factor (α ). © 2017 Elsevier B.V. All rights reserved.

1. Introduction In recent years, researchers have paid great attentions to the field of metamaterial structures, because of their unique electromagnetic properties that not found in the natural materials [1,2]. Researches and developments in metamaterial structures lead to different applications such as invisibility cloak [3], perfect absorbers [4], imaging devices [5], quantum material [6,7], and etc. Regarding to rapid developments in communications devices that consist of metamaterial structures such as multi-band filters [8] and transmission lines [9], one of important challenges is their relatively high losses. In practice, applying metamaterial structures with high losses in communications devices can lead to difficulty in observation of exotic electromagnetic properties of these modules [10]. On the other hand, superconducting materials have low microwave surface resistance and can be used in metamaterial structures to reach small sizes with low losses and high quality factors. Therefore superconducting materials are good candidate to apply in metamaterial structures. Some of the designed and fabricated superconducting metamaterial devices reported in [11]. In addition to the property of low losses in superconducting materials, they have ALSO unique properties such as flux quan∗ Corresponding author at: Persian Gulf HW Faculty of Engineering Shahed University Tehran 3319118651 Iran. E-mail address: [email protected] (S. Mohammad Hassan Javadzadeh).

http://dx.doi.org/10.1016/j.physc.2017.07.005 0921-4534/© 2017 Elsevier B.V. All rights reserved.

tization, Josephson effect, full diamagnetization and nonlinear behavior. Therefore using superconducting materials in metamaterial structures can lead to new opportunities for nonlinearity, tuning and switching behaviors and so on [11]. Nonlinearity in superconducting structures is because of dependence of the superfluid density on the current distribution. These nonlinear behaviors are also because of nonlinear Meissner effect (NLME) that reported in [12–14]. Nonlinear behaviors of superconducting materials such as harmonic generation and intermodulation distortion (specially third-order IMD) can cause limitations for using them in communication devices. Therefore, analyzing and modelling of nonlinearity in superconducting structures are very important to predict and realize their unusual nonlinear behaviors [15]. Several models for superconducting microstrip transmission lines (SMTLs) proposed in [16,17]. In the other model, accurate distributed nonlinear circuit model presented for SMTLs based on both quadratic and modulus dependence of phenomenological on current [18]. Similarly nonlinear models proposed for discontinuities such as open-end and gap [19], step-in-width [20] in SMTLs. In the other studies distributed nonlinear circuit models presented for superconducting microstrip straight bends [21], CPW rectangular-spiral resonator [22], symmetric and asymmetric parallel-coupled microstrip lines [23]. In this paper, we are going to present nonlinear model for superconducting split ring resonators (SSRRs) which are shown in Fig. 1. Our proposed model can be analyzed nonlinearly with Harmonic Balance method (HB) using ADS Software. This model can be used to predict nonlinearity in superconducting structures

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L2

L3

L1

g (a) L1 Fig. 2. Proposed equivalent circuit model for rectangular SSRRs.

2. Linear equivalence circuit model

L2 W

(b)

In this section we are going to propose linear equivalent circuit model for SSRRs that shown in Fig. 1. In this paper we consider only Fig. 1(a) and our proposed equivalent circuit model is shown in Fig. 2. In similar way, distributed equivalent circuit model for Fig. 1(b) can be obtained. As shown in Fig. 2, coupled lines shown in Fig. 1(a) are symmetric with widths different from attached transmission lines in above and bottom. In our proposed model the distributed circuit models for coupled lines and transmission lines were used. All components in the proposed circuit model can be obtained from [23,24]. 3. Distributed nonlinear circuit model

Fig. 1. General configuration of SSRRs. (a) Rectangular split ring resonator (b) Circular split ring resonator.

Superconducting materials are intrinsically nonlinear due to dependence of the superfluid density (ns ) on the current density (j) which is called nonlinear Meissner effect (NLME). As mentioned in [16], relation between variation of ns and the current density j can be defined by the following nonlinearity function f(T, j):

consist of SSRRs such as superconducting metamaterial filters. To validate the accuracy of proposed nonlinear model, this model used to predict nonlinear behaviors in a superconducting metamaterial filter consist of SSRR that fabricated in [8]. The results obtained from the proposed model are in good agreement with the measured ones from [8]. Thereafter, based on new method, we analyzed nonlinearity effects on main parameters of the superconducting metamaterial structures such as phase constant (β ), attenuation factor (α ). This paper is organized as follows. In Section 2, linear equivalent circuit model of SSRR is introduced. In Section 3 we considered theory of nonlinearity in superconducting materials and the equivalent distributed nonlinear circuit model for SSRR is proposed. Section 4 presents modelling nonlinearity in a superconducting metamaterial filter consist of SSRR. Also based on proposed model, we predicted nonlinearity effects on variation of phase constant (β ) and attenuation factor (α ). Finally conclusions are drawn in Section 5.

f (T , j ) =

ns ( T , 0 ) − ns ( T , j ) ns ( T , 0 )

(1)

In relatively small current levels, j/jc , the nonlinearity function f(T, j) has quadratic phenomenological dependence on current density and for the nonlinearity function we have:

 2

f ( T , j ) = bθ

j jc

(2)

For the case of relatively high current levels (j/jc ), the nonlinearity function f(T, j) has modulus phenomenological dependence on current density. So we have:

     c

j f (T , j ) = bθ  j

(3)

Where bθ and bθ defined in [18]. Now we are going to present nonlinear equivalent circuit model for SSRR that shown in Fig. 1(a). For this purpose, we used

28

B. Mazdouri, S. Mohammad Hassan Javadzadeh / Physica C: Superconductivity and its applications 540 (2017) 26–31

L1

R1 = R0S + RS ( T , i )

(5)

L2 = L0C + LC (T , i )

(6)

R2 = R0C + RC (T , i )

(7)

50 ohm

L2

(4)

L3

L1 = L0S + LS ( T , i )

L5

proposed equivalent circuit model that shown in Fig. 2 and then we considered linear series resistances (R1 and R2 ) and series inductances (L1 and L2 ) in nonlinear regime. So for L1 , L2 , R1 and R2 we can define:

Where L0S , R0S , L0C and R0C are linear parts of inductive and resistive components in formulas (4) to (7) and i is current flow in the equivalent circuit model. From [18] based on considering both quadratic and modulus nonlinearity dependence on current density, the nonlinear components of transmission lines can be written as follows:

LS (T , i ) = {L2S (T )IK2 + L3S (T )|IK3 | + L4S (T )IK4 + L5S (T )|IK5 | + L 6 S ( T )

IK6

}

W2 50 ohm

L4

W3 W4

Fig. 3. The superconducting metamaterial filter based on SSRR.

(8) The parameters q (T) and m (T) in Eqs. (10), (11), (15) and (16) are respectively the quadratic and modulus geometrical nonlinear factors (GNFs) and can be written as Eqs. (17) and (18):

Where

L 2 S ( T ) = L q ( T ) L3S (T ) = 0.5Lm (T )α (t ) L4S (T ) = −0.5Lq (T )α (t ) L5S (T ) = 0.125Lm (T )α 2 (t ) L6S (T ) = −0.125Lq (T )α 2 (t )





(9)

In these equations Lm and Lq are respectively modulus and quadratic inductances and can be calculated from (10) and (11).

L m =

μ0 λ2 (T , 0 ) m (T ) jm ( T )

(10)

L q =

μ0 λ2 (T , 0 ) q (T ) jq2 (T )

(11)

Where

jc

jc jq =  , jm =  b bθ ( T ) θ (T )

(12)

And α (T) can be calculate form [18]. For nonlinear resistive components we have:

RS (T , i ) = {R2S (T )IK2 + R3S (T )|IK3 | + R4S (T )IK4 + R5S (T )|IK5 | + R6S (T )IK6 }

j4 ds

q (T ) =  ( j d s )4

(13)

j3 ds

m (T ) =  ( j d s )3

(17) (18)

The integrations in (17) and (18) must be taken over transmission lines cross section. As well as, IK is current flow in Kth section of distributed equivalent circuit model for transmission lines that shown in Fig. 2. From [23] based on considering quadratic nonlinearity dependence, for nonlinear components of coupled lines we have:

LC (T , i ) = LC (T )In2

(19)

RC (T , i ) = RC (T )In2

(20)

Where LC , RC are nonlinear inductive and resistive components of coupled lines respectively and can be calculated in similar way for quadratic nonlinear components in (11) and (16). In is current flow in nth section of distributed equivalent circuit model for coupled lines that shown in Fig. 2. To predict nonlinear behaviors of SSRR, the proposed distributed nonlinear model shown in Fig. 2 can be analyzed using Harmonic Balance method (HB) with ADS software.

Where

R 2 S ( T ) = R q ( T ) R3S (T ) = 0.5Rm (T )α (t ) R4S (T ) = −0.5Rq (T )α (t ) R5S (T ) = 0.125Rm (T )α 2 (t ) R6S (T ) = −0.125Rq (T )α 2 (t )

4. Nonlinear results and discussion

(14)

In these equations Rm and Rq are respectively modulus and quadratic resistances and can be calculated from following equations [18]:

2 + a (T ) q (T ) jq2 (T )

(15)

2 + a (T ) m (T ) jm ( T )

(16)

Rm = σ1 (T , 0 )ω2 μ20 λ4 (T , 0 ) Rq = σ1 (T , 0 )ω2 μ20 λ4 (T , 0 )

To validate the accuracy of proposed linear and nonlinear circuit models that shown in Fig. 2, we considered quad-band superconducting metamaterial filter based on split ring resonator that designed and fabricated in [8]. Configuration of this filter is shown in Fig. 3. This filter is fabricated on 0.5 mm thick MgO wafer (the relative dielectric constant of substrate is 9.78) with double-sided YBa2 Cu3 Oy (YBCO) films. Introduced filter designed to operate at four frequencies 1.57 GHz, 2.1 GHz, 4.6 GHz and 5.1 GHz and designing parameters of this filter are: L1 =5.4, L2 =6.15, L3 =13.35, L4 =0.2, W1 =0.5, W2 =5.6, W3 =0.1, W4=0.1 (units: mm). To predict nonlinearity in superconducting structures such as superconducting metamaterial filters we need an accurate linear circuit model and then we define nonlinear components which

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Fig. 4. The proposed distributed nonlinear circuit model for superconducting metamaterial filter in Fig. 3 based on the proposed model for SSRRs.

L p = L0 p + Lnp (T , i )

f or p = 3, 4

R p = R0 p + Rnp (T , i )

f or p = 3, 4

f or u = 3, 4

f or u = 3, 4

-30 -40 -50 -60 -70

(23)

Rnu (T , i ) = R2u (T )I2j + R3u (T )|I3j | + R4u (T )I4j + R5u (T )|I5j | + R6u (T )I6j

-20

Measured S11 Measured S21 Modeled S11 Modeled S21

(22)

Lnu (T , i ) = L2u (T )I2j + L3u (T )|I3j | + L4u (T )I4j + L5u (T )|I5j | + L 6 u ( T )

-10

(21)

Where L0p and R0p are linear parts of series inductive and resistive components shown in Fig. 4. From [18] for nonlinear parts of Lp and Rp we have:

I6j

0

Magnitude (dB)

added to linear circuit model. We propose equivalent circuit model shown in Fig. 4 for introduced superconducting metamaterial filter based on SSRR that shown in Fig. 2. As shown in Fig. 3, we have two attached transmission lines with widths of W2 and W4 which located above and bottom of SSRR. In similar way for modelling of SSRR, we can define series resistances (R3 , R4 ) and series inductances (L3 , L4 ) in distributed circuit model for attached transmission lines. Additionally, we defined (C3 , C4 ) as capacitive components. We analyzed proposed model linearly using ADS software and scattering parameters results presented in Fig. 5. As shown in Fig. 5, there are good agreement between results from our proposed model and measured ones. Now based on our proposed linear model, we are going to consider nonlinear behaviors in introduced filter shown in Fig. 3. For this purpose we must consider proposed equivalent circuit model that shown in Fig. 4 in nonlinear regime. We can define nonlinear resistive (R3 , R4 ) and inductive components (L3 , L4 ) as follows:

-80 -90

1

2

3 4 Frequency (GHz)

5

6

Fig. 5. S-parameter results from equivalence circuit model analyzed by ADS software is compared with measurement results from [8].

(24)

Where L2u ,...,L6u and R2u ,...,R6u can be calculated in similar way explained in (9) and (14) respectively. Other nonlinear components in Fig. 4 defined in Eqs. (4) to (20). We analyzed our proposed nonlinear model that shown in Fig. 4 nonlinearly, (all inductive and resistive components considered in nonlinear regimes), with Harmonic Balance method using

ADS software. Fig. 6 shows fundamental and third-order IMD at f= 1.57 and f= 2.1GHz from proposed nonlinear model compared with measured ones from [8]. The results obtained from the proposed nonlinear model are in good agreement with measured ones.

30

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20

0

Power Out (dBm)

-20

-40

-60 Measured Fundamental f=1.57GHz Measured Fundamental f=2.1GHz Measured IMD f=1.57GHz Measured IMD f=2.1GHz Modeled Fundamental f=1.57GHz Modeled Fundamental f=2.1GHz Modeled IMD f=1.57GHz Modeled IMD f=2.1GHz

-80

-100

-120 -5

0

5 10 Power in (dBm)

15

20

Fig. 6. Input-power dependence of the amplitude of the fundamental signal and the IMD signal at T= 77 K for superconducting metamaterial filter at f= 1.57 GHz and f=2.1 GHz. Calculated IMD is compared with measured results from [8].

Based on proposed linear and nonlinear models, we are going to consider nonlinearity effects on main parameters in introduced superconducting metamaterial filter. In first step, we computed phase constant (β ) and attenuation factor (α ) for introduced filter using following equation:



γ = p(α + β ) = cos−1

A+D 2



(25)

Where A and D are ABCD matrix elements and p is the total length of introduced superconducting metamaterial filter. Fig. 7 shows variation of attenuation factor (α ) and phase constant (β ) versus different frequencies from proposed model compared with commercial software. There are good agreement between results from proposed model and the results from commercial software. The results presented in Fig. 7 show that our presented linear model is enough accurate to predict attenuation factor (α ) and phase constant (β ) in linear regime. So based on our proposed linear model and defined nonlinear components we can consider nonlinearity effects on attenuation factor (α ) and phase constant (β ). To consider nonlinearity effects on attenuation factor (α ) and phase constant (β ), we computed electric currents vectors in all branches (Ik , In , Ij ) in proposed model shown in Fig. 4 using Harmonic Balance method (HB) in nonlinear regime. These computation have done for two input powers pin = 12, 20 dBm at different frequencies from 0.1 GHz to 6 GHz. So in specific input powers we can calculate all nonlinear components shown in Fig. 4 . Then we can calculate α and β from formula (14) in nonlinear regime as follows:

γ (T , i ) = p(α (T , i ) + β (T , i ))   −1 A (T , i ) + D (T , i ) = cos

2

Fig. 7. Linear variation of (a) attenuation factor (α ), (b) phase constant (β ) of introduced superconducting metamaterial filter from proposed model compared with commercial software.

and 20 dBm at T= 77 K respectively. It can be found from Fig. 8(a) that nonlinearity can increase attenuation factor in pass-band frequencies in introduced superconducting metamaterial filter and it increases more in higher input power. Furthermore, from Fig. 8(b) it can be found that nonlinearity can change left/right hand frequency ranges and it can create left hand regions at new frequency ranges. 5. Conclusion

(26)

As you can see in formula (26), α and β were defined as functions of currents flow in branches. Fig. 8(a) and (b) show variation of α and β in nonlinear regime in two different input powers 12

In this paper, we proposed a distributed nonlinear model for superconducting split ring resonators (SSRRs) based on considering both quadratic and modulus dependence of phenomenological on the current density. This model can be used to predict nonlinearity in SSRRs and other superconducting structures consist of SSRRs. To

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circuit model in two input powers pin = 12, 20 dBm and at different frequencies from 0.1 GHz to 6 GHz at T= 77 K. Thereafter, we calculated α and β based on all components values in proposed models in linear and nonlinear regimes. Our results show that nonlinearity can increase attenuation factor in pass-band frequencies in introduced superconducting metamaterial filter and it increases by increasing in input power. Finally, it has been shown that nonlinearity can change left/right hand frequency ranges and it can create left hand regions at new frequency ranges. References

Fig. 8. Nonlinear variation of (a) attenuation factor (α ), (b) phase constant (β ) of introduced superconducting metamaterial filter at two input power Pin= 12, 20 dBm at T= 77 K compared with linear regime.

validate the accuracy of our proposed linear and nonlinear models, we considered designed and a fabricated superconducting metamaterial filter based on SSRR from [8]. There are good agreement between linear and nonlinear results from the proposed model and measured ones. Additionally, we considered nonlinear effects on main parameters, phase constant (β ) and attenuation factor (α ), in introduced superconducting metamaterial filter. For this purpose, we computed the current flow in all branches of proposed equivalent

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