"""
DefinedAEpTandZ0 (:mod:`skrf.media.definedAEpTandZ0`)
======================================================
Transmission line medium defined by A, Ep, Tand and Z0.
This medium is defined by attenuation A, relative permittivity Ep_r,
loss angle tand and characteristic impedance Z0.
Djirdjevic/Svennson dispersion model is provided for dielectric, default
behaviour is frequency invariant.
.. autosummary::
:toctree: generated/
DefinedAEpTandZ0
"""
from __future__ import annotations
import warnings
from collections.abc import Sequence
from numpy import imag, log, ndarray, ones, pi, real, sqrt
from scipy.constants import c
from ..constants import NumberLike
from ..frequency import Frequency
from .media import Media
[docs]
class DefinedAEpTandZ0(Media):
r"""
Transmission line medium defined by A, Ep, Tand and Z0.
This medium is defined by attenuation `A`, relative permittivity `Ep_r`,
loss angle `tand` and nominal impedance `Z0`.
Djirdjevic [#Djordjevic]_ / Svennson [#Svensson]_ dispersion model
is provided for dielectric. Default behaviour is frequency invariant.
Parameters
----------
frequency : :class:`~skrf.frequency.Frequency` object
Frequency band of this transmission line medium
z0_port : number, array-like, or None
`z0_port` is the port impedance for networks generated by the media.
If `z0_port` is not None, the networks generated by the media are
renormalized (or in other words embedded) from the characteristic
impedance z0 of the media to `z0_port`.
Else if `z0_port` is None, the networks port impedances will be the raw
characteristic impedance z0 of the media.
(Default is None)
A : number, array-like, default 0.0
Attenuation due to conductor loss in dB/m/sqrt(Hz)
The attenuation :math:`A(f)` at frequency :math:`f` is:
.. math::
A(f) = A\sqrt{\frac{f}{f_A}}
f_A : number, default 1.0
Frequency scaling in Hz for the attenuation. See A.
ep_r : number, array-like, default 1.0
Real part of the relative permittivity of the dielectric:
:math:`\epsilon_r'=\Re[\epsilon]`.
If `model='frequencyinvariant'`, the complex relative permittivity is:
.. math::
\epsilon_r(f) = \epsilon'_r + j \cdot \epsilon'_r \cdot \tan\delta
if `model='djordjevicsvensson'`, the complex relative permittivity is:
.. math::
\epsilon_r(f) = \epsilon_\inf + m \cdot \ln{\frac{f_{high} + j f_\epsilon}{f_{low} + j f_\epsilon}}
where :math:`\epsilon_\inf` is the permittivity value when frequency approaches infinity.
In this case, the value of :math:`\epsilon_r` and :math:`\tan\delta` (tanD)
are given for frequency :math:`f_\epsilon`:
.. math::
\epsilon_r(f_\epsilon) = \epsilon'_r+ j \cdot \epsilon'_r \cdot \tan\delta
tanD : number, array-like, default 0.0
Dielectric relative permittivity loss tangent :math:`\tan\delta`. See `ep_r`.
z0 : number, array-like, default 50.0
Quasi-static nominal impedance of the medium.
Because of the dispersion introduced by the conductor and dielectric
losses, the characteristic impedance can be frequency-dependent with an
imaginary part. The characteristic impedance is computed with an RLGC
model.
If the impedance parameter is array-like, the characteristic impedance
is assigned to this value without modification.
.. math::
R = 2 Z_n \alpha_{conductor}
.. math::
L = \frac{Z_n \sqrt{\epsilon_r}}{C_0}
.. math::
G = \frac{2}{Z_n} \alpha_{dielectric}
.. math::
C = \frac{\sqrt{\epsilon_r}}{C_0 Z_n}
.. math::
Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}
where :math:`Z_n` is the nominal impedance and :math:`Z_0` is the
characteristic impedance.
Z0 : number, array-like, or None
deprecated parameter, only emit a deprecation warning.
f_low : number, default 1e3, optional
Low frequency in Hz for for Djordjevic/Svennson dispersion model.
See `ep_r`.
f_high : number, default 1e12, optional
High frequency in Hz for for Djordjevic/Svennson dispersion model.
See `ep_r`.
f_ep : number, default 1e9, , optional
Specification frequency in Hz for for Djirdjevic/Svennson dispersion model.
`ep_r` and `tanD` parameters are specified for this frequency. See `ep_r`.
model : string, 'frequencyinvariant' or 'djordjevicsvensson', optional
Use Djirdjevic/Svennson wideband Debye dispersion model or not.
Default is frequency invariant behaviour.
\*args, \*\*kwargs : arguments and keyword arguments
Examples
--------
>>> from skrf.media.definedAEpTandZ0 import DefinedAEpTandZ0
>>> from skrf.frequency import Frequency
>>> f = Frequency(75,110,101,'ghz')
>>> DefinedAEpTandZ0(frequency=f, A=1, f_A=1e9, ep_r=3, z0=50)
>>> DefinedAEpTandZ0(frequency=f, A=1, f_A=1e9, ep_r=3, tand=0.02, z0=50)
>>> DefinedAEpTandZ0(frequency=f, A=1, f_A=1e9, ep_r=3, tand=0.02, z0=50,
f_low=1e3, f_high=1e12, f_Ep=1e9,
model='djordjevicsvensson')
References
----------
.. [#Svensson] C. Svensson and G. E. Dermer, "Time Domain Modeling of Lossy Interconnects,"
IEEE Trans. Advanced Packaging, Vol. 24, No. 2, May 2001.
https://doi.org/10.1109/6040.928754
.. [#Djordjevic] A. R. Djordjevic, R. M. Biljic, V. D. Likar-Smiljanic, and T. K. Sarkar,
"Wideband Frequency-Domain Characterization of FR-4 and Time-Domain Causality,"
IEEE Trans. Electromagnetic Compatibility, Vol. 43, No. 4, November 2001.
https://doi.org/10.1109/15.974647
"""
[docs]
def __init__(self, frequency: Frequency| None = None,
A: float = 0.0, f_A: float = 1.0,
ep_r: NumberLike = 1.0, tanD: NumberLike = 0.0,
z0_port: NumberLike | None = None,
z0: float = 50.0,
Z0: NumberLike | None = None,
f_low: float = 1.0e3, f_high: float = 1.0e12, f_ep: float = 1.0e9,
model: str = 'frequencyinvariant', *args, **kwargs):
Media.__init__(self, frequency=frequency, z0_port=z0_port)
self.A, self.f_A = A, f_A
self.ep_r, self.tanD = ep_r, tanD
self.f_low, self.f_high, self.f_ep = f_low, f_high, f_ep
self.model = model
if isinstance(z0, (Sequence, ndarray)):
# keep raw impedance for characteristic impedance
self.z0_characteristic = z0
else:
# z0 is the nominal impedance. Compute characteristic impedance with
# an RLGC model based on alpha conductor and alpha dielectric.
self.Zn = z0
self.R = 2 * self.Zn * self.alpha_conductor
self.L = self.Zn * sqrt(self.ep_r) / c
self.G = 2 / self.Zn * self.alpha_dielectric
self.C = sqrt(ep_r) / c / self.Zn
self.z0_characteristic = sqrt(
(self.R + 1j * self.frequency.w * self.L) /
(self.G + 1j * self.frequency.w * self.C))
if Z0 is not None:
# warns of deprecation
warnings.warn(
'Use of `Z0` in DefinedAEpTandZ0 initialization is deprecated.\n'
'`Z0` has no effect. Use `z0` instead\n'
'`Z0` will be removed in version 1.0',
DeprecationWarning, stacklevel = 2)
def __str__(self):
f = self.frequency
output = 'DefinedAEpTandZ0 medium. %i-%i %s. %i points'%\
(f.f_scaled[0], f.f_scaled[-1], f.unit, f.npoints)
return output
def __repr__(self):
return self.__str__()
@property
def ep_r_f(self) -> NumberLike:
"""
Frequency dependent complex relative permittivity of dielectric.
"""
ep_r, tand = self.ep_r, self.tanD
f_low, f_high, f_ep = self.f_low, self.f_high, self.f_ep
f = self.frequency.f
if self.model == 'djordjevicsvensson':
# compute the slope for a log frequency scale, tanD dependent.
k = log((f_high + 1j * f_ep) / (f_low + 1j * f_ep))
fd = log((f_high + 1j * f) / (f_low + 1j * f))
ep_d = -tand * ep_r / imag(k)
# value for frequency above f_high
ep_inf = ep_r * (1. + tand * real(k) / imag(k))
# compute complex permitivity
return ep_inf + ep_d * fd
elif self.model == 'frequencyinvariant':
return ones(self.frequency.f.shape) * (ep_r - 1j*ep_r*tand)
else:
raise ValueError('Unknown dielectric dispersion model')
@property
def tand_f(self) -> NumberLike:
"""
Frequency dependent dielectric loss factor.
"""
ep_r = self.ep_r_f
return -imag(ep_r) / real(ep_r)
@property
def alpha_conductor(self) -> NumberLike:
"""
Losses due to conductor resistivity.
"""
A, f_A, f = self.A, self.f_A, self.frequency.f
return A * log(10)/20 * sqrt(f/f_A)
@property
def alpha_dielectric(self) -> NumberLike:
"""
Losses due to dielectric
"""
ep_r, tand = real(self.ep_r_f), self.tand_f
f = self.frequency.f
return pi*sqrt(ep_r)*f/c * tand
@property
def beta_phase(self) -> NumberLike:
"""
Phase parameter
"""
ep_r, f = real(self.ep_r_f), self.frequency.f
return 2*pi*f*sqrt(ep_r)/c
@property
def gamma(self) -> NumberLike:
"""
Propagation constant.
See Also
--------
alpha_conductor : calculates conductor losses
alpha_dielectric: calculates dielectric losses
beta_phase : calculates phase parameter
"""
beta = self.beta_phase
alpha = self.alpha_conductor + self.alpha_dielectric
return alpha + 1j*beta
@property
def z0_characteristic(self) -> NumberLike:
"""
Characteristic Impedance, :math:`z_0`
Returns
-------
z0_characteristic : np.ndarray
Characteristic Impedance in units of ohms
"""
return self._z0
@z0_characteristic.setter
def z0_characteristic(self, val):
self._z0 = val