Source code for

.. module::
MLine (:mod:``)

Microstripline class

This class was made from the technical documentation [#]_ provided
by the qucs project [#]_ .
The variables  and properties of this class are coincident with
their derivations.

In addition, Djordjevic/Svensson widebande debye dielectric model is considered
to provide more realistic modelling of broadband microstrip as well as causal
time domain response.

* Quasi-static characteristic impedance model:
    In the case another model is used for effective dielectric constant,
    Hammerstad and Jensen method is used for the impedance.
    + Kirschning and Jansen
    + Hammerstad and Jensen
    + None
* Quasi-static effective dielectric constant:
    + Kirschning and Jansen
    + Hammerstad and Jensen
    + Yamashita
    + Kobayashi
    + None
* Strip thickness correction:
    + Hammerstad and Jensen, add a certain amount to W if T > 0.

.. [#]
.. [#]
.. C. Svensson, G.E. Dermer,
    Time domain modeling of lossy interconnects,
    IEEE Trans. on Advanced Packaging, May 2001, N2, Vol. 24, pp.191-196.
.. Djordjevic, R.M. Biljic, V.D. Likar-Smiljanic, T.K. Sarkar,
    Wideband frequency-domain characterization of FR-4 and time-domain causality,
    IEEE Trans. on EMC, vol. 43, N4, 2001, p. 662-667.
import numpy as npy
from scipy.constants import  epsilon_0, mu_0, c
from numpy import real, imag, pi, sqrt,log, exp, log10, zeros, ones, tanh, \
    cosh, arctan, absolute
from .media import Media
from ..tlineFunctions import skin_depth, surface_resistivity

[docs]class MLine(Media): ''' Microstripline initializer Parameters ------------- frequency : :class:`~skrf.frequency.Frequency` object frequency band of the media z0 : number, array-like, or None the port impedance for media. Only needed if its different from the characterisitc impedance of the transmission w : number, or array-like width of conductor, in m. h : number, or array-like height of subtrate between ground plane and conductor, in m. t : number, or array-like, optional conductor thickness, in m. ep_r : number, or array-like relative permittivity of dielectric at frequency f_epr_tand mu_r : number, or array-like relative permeability of dielectric (assumed frequency invariant) diel : number, or array-like dielectric dispersion model: djordjevicsvensson or frequencyinvariant. rho: number, or array-like, optional resistivity of conductor (None) tand: number, or array-like dielectric loss factor at frequency f_epr_tand rough: number, or array-like RMS rhougness of conductor in m. disp: number, or array-like microstripline dispersion model in * kirschningjansen * hammerstadjensen * yamashita * kobayashi * none f_low: number, or array-like lower frequency for wideband Debye Djordjevic/Svensson dielectric model f_high: number, or array-like higher frequency for wideband Debye Djordjevic/Svensson dielectric model f_epr_tand: number, or array-like measurement frequency for ep_r and tand of dielectric '''
[docs] def __init__(self, frequency=None, z0=None, w=3, h=1.6, t=None, ep_r=4.5, mu_r=1, diel='djordjevicsvensson', rho=1.68e-8, tand=0, rough=0.15e-6, disp='kirschningjansen', f_low=1e3, f_high=1e12, f_epr_tand=1e9, *args, **kwargs): Media.__init__(self, frequency=frequency,z0=z0) self.w, self.h, self.t = w, h, t self.ep_r, self.mu_r, self.diel = ep_r, mu_r, diel self.rho, self.tand, self.rough, self.disp = rho, tand, rough, disp self.f_low, self.f_high, self.f_epr_tand = f_low, f_high, f_epr_tand
def __str__(self): f=self.frequency output = \ 'Microstripline Media. %i-%i %s. %i points'%\ (f.f_scaled[0],f.f_scaled[-1],f.unit, f.npoints) + \ '\n W= %.2em, H= %.2em'% \ (self.w,self.h) return output def __repr__(self): return self.__str__() @property def delta_w1(self): ''' intermediary parameter. see qucs docs on microstrip lines. ''' w, h, t = self.w, self.h, self.t if t > 0.: # Qucs formula 11.22 is wrong, normalized w has to be used instead (see Hammerstad and Jensen Article) return t/pi * log(1. + 4*exp(1.)*tanh(sqrt(6.517*w/h))**2/t) * h else: return 0. @property def delta_wr(self): ''' intermediary parameter. see qucs docs on microstrip lines. ''' delta_w1, ep_r = self.delta_w1, real(self.ep_r_f) if self.t > 0.: return delta_w1/2 * (1+1/cosh(sqrt(ep_r-1))) else: return 0. @property def ep_r_f(self): ''' Frequency dependant relative permittivity of dielectric ''' ep_r, tand = self.ep_r, self.tand f_low, f_high, f_epr_tand = self.f_low, self.f_high, self.f_epr_tand f = self.frequency.f if self.diel == 'djordjevicsvensson': # compute the slope for a log frequency scale, tanD dependant. m = (ep_r*tand) * (pi/(2*log(10))) # value for frequency above f_high ep_inf = (ep_r - 1j*ep_r*tand - m*log((f_high + 1j*f_epr_tand)/(f_low + 1j*f_epr_tand))) return ep_inf + m*log((f_high + 1j*f)/(f_low + 1j*f)) elif self.diel == '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): ''' Frequency dependant dielectric loss factor ''' ep_r = self.ep_r_f return -imag(ep_r) / real(ep_r) @property def ep_reff(self): ''' Quasistatic effective relative permittivity of dielectric. Accounting for the filling factor between air and substrate. ''' h, ep_r = self.h, self.ep_r_f w1 = self.w + self.delta_w1 wr = self.w + self.delta_wr return ep_re(wr, h, ep_r) * (ZL1(w1, h)/ZL1(wr, h))**2 @property def G(self): ''' intermediary parameter. see qucs docs on microstrip lines. ''' ep_r, ep_reff, ZL = self.ep_r_f, self.ep_reff, self.Z0 ZF0 = npy.sqrt(mu_0/epsilon_0) return (pi**2)/12 * (ep_r-1)/ep_reff * sqrt(2*pi*ZL/ZF0) @property def ep_reff_f(self): ''' Frequency dependant effective relative permittivity of dielectric, accounting for microstripline dispersion. ''' ep_r, ep_reff = self.ep_r_f, self.ep_reff w, h = self.w + self.delta_wr, self.h f = self.frequency.f if self.disp == 'hammerstadjensen': ZL, G = self.Z0, self.G fp = ZL/(2*mu_0*h) return ep_r - (ep_r - ep_reff) / (1+G*(f/fp)**2) elif self.disp == 'kirschningjansen': fn = self.frequency.f * h * 1e-6 P1 = 0.27488+(0.6315+0.525/(1+0.0157*fn)**20)*w/h -0.065683*exp(-8.7513*w/h) P2 = 0.33622*(1-exp(-0.03442*ep_r)) P3 = 0.0363*exp(-4.6*w/h)*(1-exp(-(fn/38.7)**4.97)) P4 = 1+2.751*(1-exp(-(ep_r/15.916)**8)) Pf = P1*P2*((0.1844+P3*P4)*fn)**1.5763 return ep_r - (ep_r-ep_reff)/(1+Pf) elif self.disp == 'yamashita': k = sqrt(ep_r/ep_reff) F = 4*h*f*sqrt(ep_r-1)/c * (0.5+(1+2*log(1+w/h))**2) return ep_reff * ((1+1/4*k*F**1.5)/(1+1/4*F**1.5))**2 elif self.disp == 'kobayashi': f50 = c/(2*pi*h*(0.75+(0.75-0.332/(ep_r**1.73)*w/h))) \ * arctan(ep_r*sqrt((ep_reff-1)/(ep_r-ep_reff)))/ \ sqrt(ep_r-ep_reff) m0 = 1+1/(1+sqrt(w/h))+0.32*(1/(1+sqrt(w/h)))**3 mc = 1+1.4/(1+w/h)*(0.15-0.235*exp(-0.45*f/f50)) if(w/h >= 0.7): mc = 1 m = m0*mc return ep_r - (ep_r-ep_reff)/(1+f/f50)**m elif self.disp == 'none': return ones(self.frequency.f.shape)*self.ep_reff else: raise ValueError('Unknown microstripline dispersion model') @property def Z0(self): ''' Quasistatic characteristic impedance ''' h, ep_reff = self.h, real(self.ep_reff) wr = self.w + self.delta_wr return ZL1(wr, h)/sqrt(ep_reff) @property def Z0_f(self): ''' Frequency dependant characteristic impedance ''' ZL, ep_reff, ep_reff_f = self.Z0, real(self. ep_reff), real(self.ep_reff_f) wr, h = self.w + self.delta_wr, self.h if self.disp == 'hammerstadjensen': return ZL * sqrt(ep_reff/ ep_reff_f) * (ep_reff_f-1)/(ep_reff-1) elif self.disp == 'kirschningjansen': u = wr/h fn = self.frequency.f * self.h * 1e-6 ep_r = real(self.ep_r_f) R1 = 0.03891*ep_r**1.4 R2 = 0.267*u**7 R3 = 4.766*exp(-3.228*u**0.641) R4 = 0.016+(0.0514*ep_r)**4.524 R5 = (fn/28.843)**12 R6 = 22.20*u**1.92 R7 = 1.206-0.3144*exp(-R1)*(1-exp(-R2)) R8 = 1+1.275*(1-exp(-0.004625*R3*ep_r**1.674)*(fn/18.365)**2.745) R9 = 5.086*R4*R5/(0.3838+0.386*R4)*exp(-R6)/(1+1.2992*R5)*(ep_r-1)**6/(1+10*(ep_r-1)**6) R10 = 0.00044*ep_r**2.136+0.0184 R11 = (fn/19.47)**6/(1+0.0962*(fn/19.47)**6) R12 = 1/(1+0.00245*u**2) R13 = 0.9408*ep_reff_f**R8-0.9603 R14 = (0.9408-R9)*ep_reff**R8-0.9603 R15 = 0.707*R10*(fn/12.3)**1.097 R16 = 1+0.0503*ep_r**2*R11*(1-exp(-(u/15)**6)) R17 = R7 * (1-1.1241*R12/R16*exp(-0.026*fn**1.15656-R15)) return ZL * (R13/R14)**R17 elif self.disp == 'yamashita': return ZL * npy.sqrt(ep_reff/ ep_reff_f) * (ep_reff_f-1)/(ep_reff-1) elif self.disp == 'kobayashi': return ZL * npy.sqrt(ep_reff/ ep_reff_f) * (ep_reff_f-1)/(ep_reff-1) elif self.disp == 'none': return npy.ones(self.frequency.f.shape)*self.Z0 else: raise ValueError('Unknown microstripline dispersion model') @property def alpha_conductor(self): ''' Losses due to conductor resistivity Returns -------- alpha_conductor : array-like lossyness due to conductor losses See Also ---------- surface_resistivity : calculates surface resistivity ''' if self.rho is None or self.t is None: raise(AttributeError('must provide values conductivity to calculate this. see initializer help')) else: f = self.frequency.f ZF0 = npy.sqrt(mu_0/epsilon_0) ZL, rho, mu_r = real(self.Z0_f), self.rho, self.mu_r ep_reff= real(self.ep_reff) w = self.w + self.delta_wr rough = self.rough Kr = 1 + 2/pi*arctan(1.4*(rough/skin_depth(f, rho, mu_r))**2) Ki = exp(-1.2*(ZL/ZF0)**0.7) Rs = surface_resistivity(f=f, rho=rho, mu_r=1) return sqrt(ep_reff) * Rs*Kr*Ki/(ZL*w) @property def alpha_dielectric(self): ''' Losses due to dielectric ''' ep_r, ep_reff, tand = real(self.ep_r_f), real(self.ep_reff), self.tand_f f = self.frequency.f return pi*f/c * ep_r/sqrt(ep_reff) * (ep_reff-1)/(ep_r-1) * tand @property def beta_phase(self): ''' Phase parameter ''' ep_reff, f = real(self.ep_reff_f), self.frequency.f return 2*pi*f*sqrt(ep_reff)/c @property def gamma(self): ''' Propagation constant See Also -------- alpha_conductor : calculates losses to conductor alpha_dielectric: calculates losses to dielectric beta : calculates phase parameter ''' f = self.frequency.f alpha = zeros(len(f)) beta = zeros(len(f)) beta = self.beta_phase alpha = self.alpha_dielectric if self.rho is not None: alpha += self.alpha_conductor return alpha + 1j*beta
def a(u): ''' intermediary parameter. see qucs docs on microstrip lines. ''' return 1. + 1./49.*log((u**4+(u/52)**2)/(u**4+0.432)) + 1./18.7*log(1+(u/18.1)**3) def b(ep_r): ''' intermediary parameter. see qucs docs on microstrip lines. ''' return 0.564 * ((ep_r-0.9)/(ep_r+3.))**0.053 def f(u): ''' intermediary parameter. see qucs docs on microstrip lines. ''' return 6 + (2*pi-6)*exp(-(30.666/u)**0.7528) def ZL1(w, h): ''' intermediary parameter. see qucs docs on microstrip lines. ''' u = w/h ZF0 = sqrt(mu_0/epsilon_0) return ZF0/(2*pi)*log(f(u)/u + sqrt(1.+(2./u)**2)) def ep_re(w, h, ep_r): ''' intermediary parameter. see qucs docs on microstrip lines. ''' u = w/h return (ep_r+1)/2 + (ep_r-1)/2 * (1.+10./u)**(-a(u)*b(ep_r))