KirschningJansenMicrostripDispersion

class pmrf.models.components.lines.microstrip.KirschningJansenMicrostripDispersion

Bases: AbstractMicrostripDispersion

Kirschning–Jansen modal dispersion for microstrip.

Mathematical Formulation

The model expresses the dispersed effective permittivity as

\[\varepsilon_e(f) = \varepsilon_r - \frac{\varepsilon_r-\varepsilon_e(0)}{1 + P(f)},\]
where \(P=P_1P_2(0.1844+P_3P_4)^{1.5763}f_n^{1.5763}\) and the \(P_i\) are
\[P_1=0.27488+\left[0.6315+\frac{0.525}{(1+0.0157f_n)^{20}}\right]u -0.065683e^{-8.7513u}\]
\[P_2=0.33622(1-e^{-0.03442\varepsilon_r}),\quad P_3=0.0363e^{-4.6u}[1-e^{-(f_n/38.7)^{4.97}}]\]
\[P_4=1+2.751[1-e^{-(\varepsilon_r/15.916)^8}].\]
The normalized frequency is \(f_n=f[\mathrm{Hz}]H[\mathrm{m}]10^{-6}\) (GHz-mm). The normalized width is the thickness-corrected \(u = W_{eff}/H\).

Characteristic impedance is corrected by

\[Z_c(f)=Z_c(0)\left(\frac{R_{13}}{R_{14}}\right)^{R_{17}},\]
with
\[R_1=\min(0.03891\varepsilon_r^{1.4},20),\quad R_2=\min(0.2671u^7,20),\quad R_3=4.766e^{-3.228u^{0.641}}\]
\[R_4=0.016+(0.0514\varepsilon_r)^{4.524},\quad R_5=(f_n/28.843)^{12},\quad R_6=\min(22.20u^{1.92},20)\]
\[R_7=1.206-0.3144e^{-R_1}(1-e^{-R_2})\]
\[R_8=1+1.275\left[1-e^{-0.004625R_3\varepsilon_r^{1.674} (f_n/18.365)^{2.745}}\right]\]
\[R_9=\frac{5.086R_4R_5e^{-R_6}}{(0.3838+0.386R_4)(1+1.2992R_5)} \frac{(\varepsilon_r-1)^6}{1+10(\varepsilon_r-1)^6}\]
\[R_{10}=0.00044\varepsilon_r^{2.136}+0.0184,\quad R_{11}=\frac{(f_n/19.47)^6}{1+0.0962(f_n/19.47)^6},\quad R_{12}=\frac{1}{1+0.00245u^2}\]
\[R_{13}=0.9408\varepsilon_e(f)^{R_8}-0.9603,\quad R_{14}=(0.9408-R_9)\varepsilon_e(0)^{R_8}-0.9603\]
\[R_{15}=0.707R_{10}(f_n/12.3)^{1.097},\quad R_{16}=1+0.0503\varepsilon_r^2R_{11}[1-e^{-(u/15)^6}]\]
\[R_{17}=R_7\left[1-\frac{1.1241R_{12}}{R_{16}} e^{-0.026f_n^{1.15656}-R_{15}}\right].\]

The papers validate the fit from \(\varepsilon_r=2.2\), while modal dispersion must vanish at the homogeneous \(\varepsilon_r=1\) limit. In the extrapolation interval ParamRF therefore applies the smooth weight

\[x=\operatorname{clip}\left(\frac{\Re(\varepsilon_r)-1}{1.2},0,1\right), \qquad q=x^2(3-2x),\]
\[\varepsilon_e=(1-q)\varepsilon_e(0)+q\varepsilon_{e,KJ},\qquad Z_c=(1-q)Z_c(0)+qZ_{c,KJ}.\]
This smooth extension to the homogeneous limit is specific to ParamRF.

Validity

The published fits cover \(1\leq\varepsilon_r\leq20\), \(0.1\leq W/H\leq100\), and \(0\leq H/\lambda_0\leq0.13\), with numerical fits anchored at \(\varepsilon_r\geq2.2\). ParamRF uses the extension above for \(1<\varepsilon_r<2.2\) and extrapolates outside the remaining ranges.

References

Kirschning, M., & Jansen, R. H. (1982). Accurate Model for Effective Dielectric Constant of Microstrip with Validity up to Millimeter-Wave Frequencies. Electronics Letters, 18(6), 272-273.

Jansen, R. H., & Kirschning, M. (1983). Arguments and an Accurate Model for the Power-Current Formulation of Microstrip Characteristic Impedance. Archiv fuer Elektronik und Uebertragungstechnik, 37, 108-112.

disperse(freq: Frequency, *, ep_eff_0, zc_0, ep_r, w_eff, h) → tuple[Array, Array]

Return frequency-dependent \((\varepsilon_e, Z_c)\).

Parameters:

  • freq (Frequency) – The frequency axis.

  • ep_eff_0 (jnp.ndarray) – Quasi-static complex effective relative permittivity.

  • zc_0 (jnp.ndarray) – Quasi-static characteristic impedance in ohms.

  • ep_r (jnp.ndarray) – Complex relative permittivity of the substrate.

  • w_eff (jnp.ndarray) – Electromagnetic effective conductor width in meters.

  • h (ArrayLike) – Substrate height in meters.

Returns:

The dispersed effective permittivity and characteristic impedance.

Return type:

tuple of jnp.ndarray