DjordjevicSarkarDielectric

class pmrf.materials.DjordjevicSarkarDielectric(ep_r: Any = 1.0, tand: Any = 0.0, f_low: Any = 1000.0, f_high: Any = 1000000000000.0, f_ref: Any = 1000000000.0, sigma: Any = 0.0, *, name: str | None = None, metadata: Any = None)

Bases: AbstractDielectric

Causal wideband Debye (Djordjevic-Sarkar / Svensson-Dermer) dielectric.

A continuous distribution of relaxation times between f_low and f_high gives a permittivity that falls slowly with frequency while keeping an almost constant loss tangent, and which satisfies the Kramers-Kronig relations.

Mathematical Formulation

With \(k = \ln\left(\frac{f_{high} + jf_{ref}}{f_{low} + jf_{ref}}\right)\) and \(f_d(f) = \ln\left(\frac{f_{high} + jf}{f_{low} + jf}\right)\):

\[\varepsilon_d = \frac{-\tan\delta \cdot \varepsilon_r'}{\mathrm{Im}(k)} \qquad \varepsilon_\infty = \varepsilon_r'\left(1 + \tan\delta\frac{\mathrm{Re}(k)}{\mathrm{Im}(k)}\right)\]

\[\varepsilon_r(f) = \varepsilon_\infty + \varepsilon_d f_d(f) - j\frac{\sigma}{\omega\varepsilon_0}\]

Here \(\varepsilon_r'\) and \(\tan\delta\) are the values measured at the reference frequency f_ref.

References

Djordjevic, A. R., Biljic, R. M., Likar-Smiljanic, V. D., & Sarkar, T. K. (2001). Wideband frequency-domain characterization of FR-4 and time-domain causality. IEEE Transactions on Electromagnetic Compatibility, 43(4), 662-667.

Svensson, C., & Dermer, G. E. (2001). Time domain modeling of lossy interconnects. IEEE Transactions on Advanced Packaging, 24(2), 191-196.

Parameters:

  • ep_r (Param, default=1.0) – Real relative permittivity at f_ref.

  • tand (Param, default=0.0) – Dielectric loss tangent at f_ref.

  • f_low (Param, default=1e3) – Lower bound of the relaxation-time distribution in Hz.

  • f_high (Param, default=1e12) – Upper bound of the relaxation-time distribution in Hz.

  • f_ref (Param, default=1e9) – Frequency in Hz at which ep_r and tand were measured.

  • sigma (Param, default=0.0) – Static bulk conductivity in S/m.

properties(freq: Frequency) → DielectricProperties

Evaluate the dielectric properties over a frequency axis.

Parameters:

freq (Frequency) – The frequency axis to evaluate over.

Returns:

Complex permittivity, permeability, and static conductivity.

Return type:

jnp.ndarray

ep_r: Param = 1.0

Real relative permittivity at f_ref

f_high: Param = 1000000000000.0

Upper bound of the relaxation-time distribution

f_low: Param = 1000.0

Lower bound of the relaxation-time distribution

f_ref: Param = 1000000000.0

Frequency at which ep_r and tand were measured

sigma: Param = 0.0

Static bulk conductivity in S/m

tand: Param = 0.0

Dielectric loss tangent at f_ref