HammerstadJensenMicrostripFormulation

class pmrf.models.components.lines.microstrip.HammerstadJensenMicrostripFormulation

Bases: AbstractMicrostripFormulation

Hammerstad–Jensen quasi-static microstrip formulation.

Mathematical Formulation

For normalized width \(u=W/H\), the impedance of the homogeneous geometry is

\[Z_L(u) = \frac{Z_0}{2\pi}\ln\left[\frac{F(u)}{u} + \sqrt{1 + \left(\frac{2}{u}\right)^2}\right]\]

\[F(u) = 6 + (2\pi - 6)\exp\left[-(30.666/u)^{0.7528}\right].\]

With normalized thickness \(t_n=T/H\), finite conductor thickness changes the normalized widths in air and in the dielectric:

\[\Delta u_1 = \frac{t_n}{\pi}\ln\left[1 + \frac{4e}{t_n} \tanh^2\left(\sqrt{6.517u}\right)\right]\]

\[\Delta u_r = \frac{\Delta u_1}{2}\left[1 + \operatorname{sech}\left(\sqrt{\varepsilon_r-1}\right)\right], \qquad u_1=u+\Delta u_1,\quad u_r=u+\Delta u_r.\]

The empirical exponents are

\[a = 1 + \frac{1}{49}\ln\left[\frac{u_r^4+(u_r/52)^2} {u_r^4+0.432}\right] + \frac{1}{18.7}\ln\left[1+(u_r/18.1)^3\right]\]

\[b = 0.564\left(\frac{\varepsilon_r-0.9}{\varepsilon_r+3}\right)^{0.053}.\]

Defining

\[e = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2}(1 + 10/u_r)^{-ab},\]

the returned quantities are

\[Z_c=\frac{Z_L(u_r)}{\sqrt{e}},\qquad \varepsilon_e=e\left[\frac{Z_L(u_1)}{Z_L(u_r)}\right]^2, \qquad W_{eff}=u_rH,\]

\(W_{eff}\) is passed to the dispersion formulation. Conductor loss is calculated separately by the current distribution.

The fractional power in \(b\) is evaluated through the principal logarithm. The dielectric constraint \(\Re(\varepsilon_r)>1\) keeps its argument away from the negative-real branch cut for passive materials.

Validity

The quoted error in \(\varepsilon_e\) is below 0.2% for \(\varepsilon_r<128\) and \(0.01\leq u\leq100\). The quoted error in \(Z_L\) is below 0.01% for \(u\leq1\) and 0.03% for \(u\leq1000\). The formulation is quasi-static and requires a non-magnetic substrate. Inputs outside the fitted ranges are extrapolated.

References

Hammerstad, E., & Jensen, O. (1980). Accurate Models for Microstrip Computer-Aided Design. IEEE MTT-S International Microwave Symposium Digest, 407-409.

quasi_static(*, w, h, t, ep_r) → PlanarQuasiStaticResult

Calculate the quasi-static solution.

Parameters:

• w (ArrayLike) – Width of the microstrip trace in meters.

• h (ArrayLike) – Height of the dielectric substrate in meters.

• t (ArrayLike | None) – Trace thickness in meters, or None when unspecified.

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

Returns:

The effective permittivity, impedance and effective width.

Return type:

PlanarQuasiStaticResult

thickness_aware: ClassVar[bool] = True

Finite thickness enters through the normalized-width corrections.