Distributed Nonlinearities in Microwave
Superconducting Devices
M. M. Assefzadeh
assefzadeh@ee.sharif.edu
Main Reference:
Analysis and Simulation of the Effects of Distributed
Nonlinearities in Microwave Superconducting Devices; C. Collado, J.
Mateu, and J. M. O’Callaghan; IEEE Transactions on Applied
Superconductivity, March 2005
Seminar for the Course:
Principles of Applied Superconductivity
Professor: Dr. Fardmanesh
June 2010
Outline
Introduction
High Tc Superconductive Devices
Nonlinearity Drawbacks
Nonlinearities in Superconductors
Analytical and Phenomenological
Superconductor Films
IMD and Third

Harmonic Generation in
SDevices
Nonlinear Transmission Line
Line Resonators
Superconductor Characterization with IMD Measurements
Harmonic Balance for Simulation of Superconducting Devices
Conclusion
2
Introduction
3
Motivations to use SC (esp. HTS) in electronics:
Low Surface Resistance
Reaching to High Currents
in Devices
Reduced Power Dissipation
and Delay
Unique Quantum Accuracy
Low Noise from Cryogenic
Operation
Taken from HTS Microwave Devices Lecture, Colorado University
Large

Area Double

Sided
YBCO Thin Films
Taken from The Web Page of
Semiconductor Physics Group of Leipzig
University
Introduction
4
HTS Microwave Devices such as:
Planar Filters
Resonators
Microstrip and
Stripline
Transmission lines
Benefits of SC Filters:
Low volume
Reduced insertion losses
High selectivity
Drawbacks
of High Powers:
Nonlinearities
Recognize Them
Simulate Them
Predict Them!
Taken from HTS Microwave Devices Colorado; Northrop Grumman
Nonlinearities in Superconductors
5
Intrinsic Concept of Nonlinearity in Superconductors
Assumptions
Microwave Frequencies (Low Surface Resistances)
Two Fluids Model (Nonlinearly)
Studying Intrinsic Nonlinearity
1. Nonlinear Conductance and Penetration Depth
Intrinsically:
Less Cooper Pairs when we have applied current.
Phenomenological
: Experimental works claiming the current
dependent penetration depth
2. Dependence of Electric Field on Surface Current
Nonlinear Conductance and
Penetration Depth
6
Analytical Approach
Basis:
Nonlinearity characterization function
Taylor expansion:
Approaching to:
Phenomenological Approach
Measuring current dependent
penetration depth and after fitting
Data:
Small Signal values (J~0):
Linear Conductance
Linear Penetration Depth
No Dependence On J
Large Current Magnitudes:
Increased
note that this is the resistive
conductance in the two fluid model
Increased
Time Domain Equation Between
E
and
J
s
:
assuming q
uasi exponential decay of
the electromagnetic fields:
Nonlinear inductive equation:
Assuming E
in two linear and NL components:
Deriving nonlinear parts of surface resistance and inductance:
Superconducting Films
From Nonlinear Electric Field to Nonlinear Surface Impedance
7
Talking
about
J
o
Talking
about
J
S
Nonlinear Distributed Parameters in
Transmission Lines
8
Intrinsic Nonlinearities in SC affecting Parameters in Transmission
Lines:
These nonlinearities follow the same nonlinear rules as the nonlinearity
function f(T,J).
Quadratic Nonlinearities:
Modulus Nonlinearities:
Nonlinear equivalent
circuit of a
superconducting
transmission line segment
with length
dz
; Taken from
the main reference.
IMD & Third Harmonic Generation
Derivation from intrinsic nonlinearities
9
Definitions:
Third Harmonic
: An effect of nonlinear devices creating freq. of
3f
.
IMD
: The unwanted amplitude modulation of signals containing different
frequencies.
In our work, we consider the products
f
12
= (2f
1
–
f
2
)
&
f
3
= 3f
1
for a
signal with two frequencies f
1
&
f
2
.
From Wikipedia
The spectrum
of an RF signal
containing two
fundamental
frequencies
IMD and 3rd Harmonic Generation in
Nonlinear Transmission Lines
10
Matched Transmission Line
Experimental use of
Transmission Lines:
1) Quadratic or modulus
nonlinearity?
Spurious powers against
sources powers slope:
3:1
and 2:1 for Quadratic and
Modulus
2) Resistive or inductive
nonlinearities?
From The Main Reference
IMD and 3rd Harmonic Generation in
Nonlinear Resonators
11
The Same Theory Analysis Applies for
Line Resonators
Disk Resonators and Cavities
Hairpin Resonator,
measurements fit theory (dots
are measured)
Quantitative results:
Taken From
The Main
Reference
In the Order of
J
c
Results of Harmonic Balance Simulation
12
Harmonic Balance:
A high performance method to simulate
nonlinear circuits
Linear part in Freq. domain
NL part in Time domain
Current Distribution
Along a SC Matched Line
Simulation (Dots) Versus Theory
Taken From the Main Reference
Results of Harmonic Balance Simulation
13
Matched SC Line: Powers
Delivered to the output
Dashed lines from the
theory, solid lines simulated
Inset chart: The error
between calculations and
simulations
10% Error for Input Power of
45dBm (=33W); The Effect of
Higher Order Nonlinearities.
Taken From the Main Reference
At the
fundamental
frequency
At the IMD 12
frequency
Conclusion
14
Nonlinearities due to high powers
Theory and phenomenological approaches
SC thin film devices: Theoretical solutions
Resulting intermodulation distortion and 3rd
harmonic generation
Simulations reveal the effects of extra high
powers; Higher order nonlinearities
References
15
[1]
Carlos
Collado
, J. M. (MARCH 2005). Analysis and Simulation of the Effects of
Distributed Nonlinearities in Microwave Superconducting Devices.
IEEE TRANSACTIONS ON
APPLIED SUPERCONDUCTIVITY
, 26

39.
[2]
T.
Dahm
and D.
Scalapino
, “Theory of intermodulation in superconducting
microstrip
resonator,” J. Appl. Phys. , vol. 81, no. 4, pp. 2002
–
2002, 1997.
[3]
T.
Dahm
, D.
Scalapino
, and B.
Willemsen
, “Phenomenological theory of intermodulation
in HTS resonators and filters,” J.
Supercond
., vol. 12, pp. 339
–
339, 1999.
[4]
B. A.
Willemsen
, T.
Dahm
, and D. J.
Scalapino
, “Microwave intermodulation in thin film
high

Tc superconducting
microstrip
hairpin resonators: Experiment and theory,” Appl. Phys.
Lett
., vol. 71, no. 29, pp. 3898
–
3898, 1997.
[5]
HTS Materials and Devices.
(
n.d
.). Retrieved from Colorado; Northrop Grumman:
http://boulder.research.yale.edu/Boulder

2000/transparencies/talvacchio

lecture1/colorado

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