Kinetic inductance of coupled superconducting microstrip lines

PHYSICA E Physica C 282-287

Kinetic inductance

(1997) 2529-2530

of coupled superconducting

microstrip lines

J. F. KangKb,R. Q. Hana, G. C. Xiongb, X. Y. Liu”, Y. Y. Wang” %rstitute of Microelectronics, Peking University, Beijing 100871, P.R.China t’Department of Physics,

Peking University,

Beijing 10087 1,

P.R.China

Using the conformal mapping technique, simple analytical expressions for the kinetic inductance of coupled superconducting microstrip lines in odd-mode and even-mode have been derived. Based on the formulae, the geometrical effects of kinetic inductance are discussed.

1. INTRODUCTION Understanding dependence of kinetic inductance on geometric parameters is necessary in the analysis and design of superconducting transmission lines. .From the view point of design, simple expressions are convenient. However, no simple expressions have been given for the coupled superconducting microstrip lines up to this point. In this paper, we present two expressions in odd- and even-mode by using the conformal mapping transformation technique. 2. ANALYTICAL SOLUTIONS

London equations must also be satisfied in the superconducting regions. As an approximation, we assume that the normal magnetic field along the dielectric-superconducting boundary is zero, and the vector potential Az=l at the striplines surface or Az=O at the ground-plane surface. Based on the symmetry of the structure, usually only a half of the coupled microstrip lines is considered. Because the symmetric plane is either a magnetic wall in the even mode case, or an electric wall in the odd mode case, the y axis is the constant potential line with Az=O in the odd-mode, and is the constant flux line in the 84 _ --0. By using the ax conformal mapping technique, the Z-plane can be transformed to the Z-plane. The required transformations are as follows: even-mode with

The geometry of the coupled superconducting microstrip lines is shown in Fig. 1. For simplicity, we assume that the thickness t of the superconducting microstrip lines and ground plane is much less than the magnetic penetration depth h of superconductors in the following derivation. The field solutions of the coupled superconducting microstrip structures must satisfy Maxwell’s equations. In addition, the

(1)

w’= c,s-1 O

F

(the even-mode)

(2b) (3)

(l-W*)(l-k%2)

We relate the z component of the vector potential A=(O,O,Az) as Az=Re(Z’). In the complex variable theory, the x component of the

X

z

The geometry of the Fig. 1 superconducting microstrip lines.

(2a)

da

Z’$_ Y

(the odd-mode)

coupled

0921-4534/97/%17.00 0 Elsevier Science B.V. All tights reserved. PI1 SO921-4534(97)01333-6

magnetic field H,” and H”, in the boundary

JR Kang et al./Physica C 282-287 (1997) 2529-2530

2530

(i.e. in the surface of microstriplines) is obtained,

c, ;sh($z)

e=

~o’+l)[(k-1)C~+(k+l)(o’+l)][(k+l)c*++(k-l)(o’+l)]

By using the formulae given by Chang [l] and expressions of ZY,”and g , the odd-mode and even-mode kinetic inductance are obtained. Lkin _ &A 0

I

Lkin _ poh e I

wherea

o

a ae

_ -- ,I2

(odd-mode)

(4)

(even-mode)

(5)

w+; s

w++ ae=fj

H,Odx’

s

H:&’

z

1

1=2K(k) and K(k) is the complete integral.

W=Zm

elliptic 2

4

3. RESULTS AND DISCUSSIONS

kinetic

inductance

,$

and

e

.i

lb

sld

Fig. 2 (a) shows the plots of odd- and evenmode kinetic inductance L,,“” and Leki” with respect with s/W for different values of d. The plots indicate that the differences between Lokin and L,“’ become larger with increasing s and increasing d. Fig. 2 (b) presents the comparisons between the

d=lm

magnetic

o,e resulting from the magnetic inductance Lex field energy stored in the dielectric region. The results show that for small s, d, and W, the kinetic inductance is the dominant in the total inductance, but for large s, d and W, the magnetic inductance is dominant.

4. CONCLUSION We have derived two analytical formulae for odd-mode and even-mode kinetic inductance of coupled superconducting microstrip lines by

(a) Odd- and even-mode kinetic inductance L,“” and L,“” with respect with s/W for different values of d. (b) The comparisons between

the

kinetic

inductance

L$

and

magnetic inductance Lze

using the conformal mapping transformation technique. Based on them, the characteristics of kinetic inductance for coupled superconducting microstrip lines can be analyzed easily. Such results means that, in smaller sizes, kinetic inductance is dominant in the total inductance, but in larger sizes, the magnetic inductance is dominant. REFERENCES: 1. W.H.Chang, J. Appl. Phys. 50(12), (1979) 8129