SchelkunoffCoaxialFormulation

class pmrf.models.components.lines.coaxial.SchelkunoffCoaxialFormulation(inner_impedance: AbstractSurfaceImpedance = SchelkunoffRodSurfaceImpedance(), shield_impedance: AbstractSurfaceImpedance = SchelkunoffTubeSurfaceImpedance())

Bases: AbstractCoaxialFormulation

Coaxial line formulation using Schelkunoff’s exact cylindrical solution.

Mathematical Formulation

The external inductance and shunt admittance are

\[L' = \frac{\mu_0 \mu_r}{2\pi} \ln\left(\frac{b}{a}\right) \qquad Y = \frac{j\omega 2\pi\varepsilon}{\ln(b/a)},\]
The inner conductor uses SchelkunoffRodSurfaceImpedance – Schelkunoff’s eq. (65) rather than an equivalent circuit:
\[Z_{inner} = \frac{\zeta_c}{2\pi a}\, \frac{I_0(\gamma a)}{I_1(\gamma a)},\qquad \gamma = \sqrt{j\omega\mu\sigma}.\]

The shield uses Schelkunoff’s tube solution, referred to its inner surface. An unspecified shield_thickness gives \(Z_{outer}=\zeta_c K_0(\gamma b)/(2\pi bK_1(\gamma b))\). Both formulations are configurable fields.

Validity

Exact for homogeneous cylindrical conductors carrying axially symmetric current. The TEM line model applies below the TE11 cutoff \(f_c \approx c / \left[\pi (a + b) \sqrt{\varepsilon_r \mu_r}\right]\).

References

Schelkunoff, S. A. (1934). The Electromagnetic Theory of Coaxial Transmission Lines and Cylindrical Shields. Bell System Technical Journal, 13(4), 532-579. Eq. (65), (74).

immittance(freq: Frequency, *, d_in, d_out, dielectric: DielectricProperties, conductor: ConductorProperties, outer_conductor=None, shield_thickness=None) → ImmittanceResult

Calculate the per-unit-length immittance.

Parameters:

  • freq (Frequency) – The frequency axis.

  • d_in (ArrayLike) – Inner conductor diameter in meters.

  • d_out (ArrayLike) – Outer conductor inner diameter in meters.

  • dielectric (DielectricProperties) – Evaluated relative permittivity and permeability.

  • conductor (ConductorProperties) – Evaluated inner-conductor properties.

  • outer_conductor (ConductorProperties | None, default=None) – Evaluated shield properties, or None to reuse conductor.

  • shield_thickness (ArrayLike | None, default=None) – Shield wall thickness, or None for an infinitely thick shield.

Returns:

The series impedance and shunt admittance.

Return type:

ImmittanceResult

inner_impedance: AbstractSurfaceImpedance = SchelkunoffRodSurfaceImpedance()

Surface-impedance formulation for the inner conductor

shield_impedance: AbstractSurfaceImpedance = SchelkunoffTubeSurfaceImpedance()

Surface-impedance formulation for the shield, called with the shield’s inner radius and its wall thickness, infinite when none is given