Source code for piel.models.physical.electrical.transmission_lines.microstrip
import numpy as np
from piel.types.constants import mu_0
[docs]
def epsilon_e(epsilon_r, width_m, dielectric_thickness_m):
"""
Calculate the effective dielectric constant (ε_e) for a microstrip.
The effective dielectric constant accounts for the field distribution
between the microstrip and the substrate, influencing signal propagation.
Parameters
----------
epsilon_r : float
Relative permittivity (dielectric constant) of the substrate.
width_m : float
Width of the microstrip line (meters).
dielectric_thickness_m : float
Thickness of the substrate (meters).
Returns
-------
epsilon_e : float
Effective dielectric constant.
References
----------
Equation (2):
ε_e = (ε_r + 1)/2 + (ε_r - 1)/2 * 1/sqrt(1 + 12*dielectric_thickness_m/width_m)
"""
epsilon_e = (epsilon_r + 1) / 2 + (epsilon_r - 1) / 2 / np.sqrt(
1 + 12 * dielectric_thickness_m / width_m
)
return epsilon_e
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def Z_0(width_m, dielectric_thickness_m, epsilon_e):
"""
Calculate the characteristic impedance (Z₀) of a microstrip.
The characteristic impedance represents the inherent resistance that
the transmission line presents to the signal propagating through it.
Parameters
----------
width_m : float
Width of the microstrip line (meters).
dielectric_thickness_m : float
Thickness of the substrate (meters).
epsilon_e : float
Effective dielectric constant of the microstrip.
Returns
-------
characteristic_impedance_ohms : float
Characteristic impedance in Ohms.
References
----------
Equation (1):
Z₀ = 120π / [√ε_e * (width_m/dielectric_thickness_m + 1.393 + 0.667 ln(width_m/dielectric_thickness_m + 1.444))]
"""
ratio = width_m / dielectric_thickness_m
denominator = ratio + 1.393 + 0.667 * np.log(ratio + 1.444)
characteristic_impedance_ohms = 120 * np.pi / (np.sqrt(epsilon_e) * denominator)
return characteristic_impedance_ohms
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def alpha_c(surface_resistance_ohms, characteristic_impedance_ohms, width_m):
"""
Calculate the attenuation constant (α_c) in decibels per meter (dB/m).
The attenuation constant measures how much signal is lost per meter due
to resistive (ohmic) losses in the conductor of the microstrip line.
Parameters
----------
surface_resistance_ohms : float
Surface resistance of the conductor (Ohms).
characteristic_impedance_ohms : float
Characteristic impedance of the microstrip (Ohms).
width_m : float
Width of the microstrip line (meters).
Returns
-------
alpha_c : float
Attenuation constant in dB/m.
References
----------
Equation (3):
α_c (dB/m) = 8.68588 * (R_s / (Z₀ * width_m))
"""
return 8.68588 * (
surface_resistance_ohms / (characteristic_impedance_ohms * width_m)
)
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def R_s(frequency_Hz, conductivity_S_per_m, permeability_free_space=mu_0.value):
"""
Calculate the surface resistivity (R_s) of a conductor at a given frequency.
The surface resistivity is a measure of how much a conductor resists current
flow along its surface, and it increases with frequency due to the skin effect.
Parameters
----------
frequency_Hz : float
Frequency at which the resistivity is calculated (Hz).
conductivity_S_per_m : float
Electrical conductivity of the conductor (S/m).
permeability_free_space : float, optional
Permeability of free space (H/m). Default is the value from mu_0.
Returns
-------
surface_resistance_ohms : float
Surface resistivity in Ohms.
Formula
-------
R_s = sqrt(ω * μ₀ / (2 * σ))
Where:
ω = 2π * frequency_Hz (angular frequency in rad/s)
μ₀ = Permeability of free space (H/m)
σ = Conductivity (S/m)
"""
frequency_rad_s = 2 * np.pi * frequency_Hz
return np.sqrt(
frequency_rad_s * permeability_free_space / (2 * conductivity_S_per_m)
)
