cpw_calculations#

For calculations of CPW parameters. Referenced primarily as a tool for some components.

@author: Thomas McConkey, as part of https://uwspace.uwaterloo.ca/handle/10012/13464

Key References:

D. Schuster, Ph.D. Thesis, Yale University (2007) https://rsl.yale.edu/sites/default/files/files/RSL_Theses/SchusterThesis.pdf

Goppl et al., Journal of Applied Physics 104, 1139044 (2008) https://aip.scitation.org/doi/abs/10.1063/1.3010859?journalCode=jap

Mohebbi and Majedi, Superconducting Science and Technology 22, 125028 (2009) https://iopscience.iop.org/article/10.1088/0953-2048/22/12/125028/meta

Functions

`effective_dielectric_constant`(freq, s, w, h, ...)	Returns the film and substrate thickness dependent effective dielectric constant for a planar CPW transmission line.
`elliptic_int_constants`(s, w, h)	Calculates the complete elliptic integral of the first kind for CPW lumped element equivalent circuit calculations.
`guided_wavelength`(freq, line_width, ...[, ...])	A simple calculator to determine the guided wavelength of a planar CPW transmission line.
`lumped_cpw`(freq, line_width, line_gap, ...)	A simple calculator to determine the lumped element equivalent of a CPW transmission line.

effective_dielectric_constant(freq, s, w, h, t, q, Kk0, Kk01, eRD=11.45)#

Returns the film and substrate thickness dependent effective dielectric constant for a planar CPW transmission line. Assumes package ground can be ignored.

Parameters:

freq (float) – The frequency of interest (eg. 5*10**9)
s (float) – The width of the CPW trace (center) line, in meters (eg. 10*10**-6).
w (float) – The width of the CPW gap (dielectric space), in meters (eg. 6*10**-6).
h (float) – Thickness of the dielectric substrate, in meters (eg. 760*10**-6).
t (float) – Thickness of the thin film, in meters (eg. 200*10**-9).
q (float) – Filling factor of the CPW in question
Kk0 (float) – The complete elliptic integral for k0
Kk01 (float) – The complete elliptic integral for k01
eRD (float, optional) – The relative permittivity of the substrate. Defaults to 11.45.

Returns:

etfSqrt is the effective permittivity for a CPW transmission line, considering film and substrate thickness.

Return type:

float

elliptic_int_constants(s, w, h)#

Calculates the complete elliptic integral of the first kind for CPW lumped element equivalent circuit calculations.

Parameters:

s (float) – The width of the CPW trace (center) line, in meters (eg. 10*10**-6).
w (float) – The width of the CPW gap (dielectric space), in meters (eg. 6*10**-6).
h (float) – Thickness of the dielectric substrate, in meters (eg. 760*10**-6).

Returns:

Contents outlined below

Return type:

tuple

Tuple contents:

ellipk(k0) (float): The complete elliptic integral for k0
ellipk(k01) (float): The complete elliptic integral for k01
ellipk(k1) (float): The complete elliptic integral for k1
ellipk(k11) (float): The complete elliptic integral for k11

guided_wavelength(freq, line_width, line_gap, substrate_thickness, film_thickness, dielectric_constant=11.45)#

A simple calculator to determine the guided wavelength of a planar CPW transmission line. Assumes the substrate has relative permeability of 1. Assumes package grounds are far away.

Parameters:

freq (float) – The frequency of interest, in Hz (eg. 5*10**9).
line_width (float) – The width of the CPW trace (center) line, in meters (eg. 10*10**-6).
line_gap (float) – The width of the CPW gap (dielectric space), in meters (eg. 6*10**-6).
substrate_thickness (float) – Thickness of the dielectric substrate, in meters (eg. 760*10**-6).
film_thickness (float) – Thickness of the thin film, in meters (eg. 200*10**-9).
dielectric_constant (float) – The relative permittivity of the substrate. Defaults to 11.45, the value for Silicon at cryogenic temperatures.

Returns:

Contents outlined below

Return type:

tuple

Tuple contents:

lambdaG: The guided wavelength of the CPW based on the input parameters, in meters. This value is for a full wavelength. Divide by 2 for a lambda/2 resonator, 4 for a lambda/4.
etfSqrt: Effective dielectric constant (accounting for film thickness)
q: Filling factor

lumped_cpw(freq, line_width, line_gap, substrate_thickness, film_thickness, dielectric_constant=11.45, loss_tangent=1e-05, london_penetration_depth=3.0000000000000004e-08)#

A simple calculator to determine the lumped element equivalent of a CPW transmission line. Assumes a lossless superconductor. The internal geometric series inductance is ignored.

Parameters:

freq (float) – The frequency of interest, in Hz (eg. 5*10**9).
line_width (float) – The width of the CPW trace (center) line, in meters (eg. 10*10**-6).
line_gap (float) – The width of the CPW gap (dielectric space), in meters (eg. 6*10**-6).
substrate_thickness (float) – Thickness of the dielectric substrate, in meters (eg. 760*10**-6).
film_thickness (float) – Thickness of the thin film, in meters (eg. 200*10**-9).
dielectric_constant (float, optional) – The relative permittivity of the substrate. Defaults to 11.45, the value for silicon at cryogenic temperatures.
loss_tangent (float, optional) – The loss tangent of the dielectric. Defaults to 10**-6, reasonable quality silicon.
london_penetration_depth (float, optional) – The superconducting london penetration depth, in meters. It is advised to use the temperature and film thickness dependent value. If circuit geometries are on the scale of the Pearl Length, the kinetic inductance formulas breakdown. Defaults to 30*10**-9, for Niobium.

Returns:

Contents outlined below

Return type:

tuple

Tuple contents:

Lk (float): The series kinetic inductance, in Henries.
Lext (float): The series geometric external inductance, in Henries.
C (float): The shunt capacitance, in Farads.
G (float): The shunt admittance, in Siemens. #NOTE:double check if right units
Z0 (float): sqrt(L / C)
etfSqrt**2: Effective Dielectric Constant
Cstar: External Inductance

-----Lext + Lk--+---+---
                |   |
                C   G
                |   |
----------------+---+---