A Modified Transmission Line Approach to The Analysis of Superconducting Microwave Resonators
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Air
Su
Z Pezoelec, fic Lay
Im
y x Figure 1: HTSC - AIN Resonator Equivalent Circuit Diagram THEORY Determining Equivalent Inductance For A Section Of Sunerconductor The intent of the inductor analysis is to find an equivalent circuit element that may be used in the transmission line analysis. Using a section of material with dimensions of dx, dy, and dz, a current is introduced and assumed to flow in only the x - direction. There is assumed to be no magnetic field variation in the y - direction, or aHy/•y = 0. The magnetic energy stored in a section of material is given in equation 1 as
WM
ffv.n.Wdv jJvv
2 .dv .
(1)
Setting equation 1 equal to the energy stored in an inductor we obtain
LI2
=
fJv lt°H 2 " dv
(2)
where B = ýtfi, I = sfJ.- ds, and ýt = gto = 47t x 10-7 H/m in HTSC. Along with Ampere's Law in integral form, we now have two constitutive relations between the magnetic fields and the current passing through the sample. Analyzing the differential current traveling in the xdirection, and approximating the integrals as differentials while assuming that aHy/ly = 0, Ampere's law can be approximated as
(z) ally az 324
(3)
Substituting equation 3 into equation 2 we can now eliminate the current from the previous equations:
LJffs---•z
.dy.dz
ývJH 2 .dx dy dz.
2
(4)
Assuming a solution of the form [I = H.Ioe-', we can solve equation 4 for the equivalent inductance in the circuit. Expanding the exponentials and simplifying, the resulting inductance is LX
(5)
oaX
Lx-&AyAz
where LX is the incremental inductance of a current flowing in the x - direction. A similar calculation was performed for L,. These results are for a sample that is entirely superconducting. In this analysis, the case when the sample is not entirely superconducting will be investigated, since for most HTSC there remains a small finite resistance that will be accounted for in the transmission line analysis. The effective area of the superconductor will be treated as being decreased by a percentage equal to the ratio of superconducting to non-superconducting regions, i.e. equation 5 becomes
(6)
LSC-2°tAx 2
x-
KAyAz
where K is the ratio of superconducting to non-superconducting regions. Similarly, the nonsuperconducting region will have an inductive component and a resistive component to its impedance given by LNo NSC
2
AR
a (l- K)AyAz'
xNSC =
pAx
(1- K)AyAz
(7)
where p is the resistivity of the superconductor just before the sample passes through the transition and becomes superconducting. [11 Determining The Equivalent Resistance For The Air-Superconductor Interface The equivalent resistance as shown in Figure 1 at the free space boundary of the sample simulates the power lost into the half space that may be used in the transmission line analysis. Assuming an area AxAy, we will consider the power in free space at the boundary. Starting with the Poynting vector in the z direction, P, = E, x HY, we then use plane wave relations to relate the electric and magnetic fields in free space, Hy = Er/n, where il = Vt
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