r"""Transmission lines: the telegrapher's equations, reflections and the Smith chart.
Heaviside described a line by its series resistance and inductance and its
shunt conductance and capacitance per metre. A wave travels along it with
the propagation constant :math:`\gamma = \sqrt{(R + j\omega L)(G + j\omega C)}`
and sees the characteristic impedance :math:`Z_0`. A load that differs from
:math:`Z_0` reflects part of the wave, which Smith's chart lets an engineer
follow graphically.
"""
from __future__ import annotations
import numpy as np
from engineeringkit._validation import finite as _finite
from engineeringkit._validation import nonnegative as _nonnegative
from engineeringkit._validation import positive as _positive
__all__ = [
"transmission_line_z0",
"transmission_line_velocity",
"propagation_constant",
"distortionless_condition",
"reflection_coefficient",
"standing_wave_ratio",
"input_impedance",
]
[docs]
def transmission_line_z0(L_per_m, C_per_m):
r"""Characteristic impedance :math:`Z_0 = \sqrt{L/C}` of a lossless line.
Parameters
----------
L_per_m : float or array_like
Series inductance per unit length [H/m], positive.
C_per_m : float or array_like
Shunt capacitance per unit length [F/m], positive.
Returns
-------
float or ndarray
Characteristic impedance [ohm].
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/electrical_engineering`.
Examples
--------
>>> round(float(transmission_line_z0(250e-9, 100e-12)), 1)
50.0
"""
_finite(L_per_m=L_per_m, C_per_m=C_per_m)
_positive(L_per_m=L_per_m, C_per_m=C_per_m)
return np.sqrt(L_per_m / C_per_m)
[docs]
def transmission_line_velocity(L_per_m, C_per_m):
r"""Wave velocity :math:`v = 1/\sqrt{LC}` on a lossless line.
Parameters
----------
L_per_m : float or array_like
Series inductance per unit length [H/m], positive.
C_per_m : float or array_like
Shunt capacitance per unit length [F/m], positive.
Returns
-------
float or ndarray
Phase velocity [m/s].
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/electrical_engineering`.
Examples
--------
>>> round(float(transmission_line_velocity(250e-9, 100e-12)) / 1e8, 3)
2.0
"""
_finite(L_per_m=L_per_m, C_per_m=C_per_m)
_positive(L_per_m=L_per_m, C_per_m=C_per_m)
return 1 / np.sqrt(L_per_m * C_per_m)
[docs]
def propagation_constant(R_per_m, L_per_m, G_per_m, C_per_m, omega):
r"""Propagation constant :math:`\gamma = \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)}`.
A sinusoidal wave on the line varies as :math:`e^{-\gamma x}`: its
amplitude decays by :math:`\alpha` nepers per metre and its phase
advances by :math:`\beta` radians per metre, so it travels at
:math:`\omega/\beta`.
Parameters
----------
R_per_m : float
Series resistance per unit length [ohm/m], nonnegative.
L_per_m : float
Series inductance per unit length [H/m], positive.
G_per_m : float
Shunt conductance per unit length [S/m], nonnegative.
C_per_m : float
Shunt capacitance per unit length [F/m], positive.
omega : float or array_like
Angular frequency [rad/s], positive.
Returns
-------
complex or ndarray of complex
:math:`\gamma` [1/m], with :math:`\alpha, \beta \ge 0`.
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/electrical_engineering`.
References
----------
O. Heaviside, "On the extra current," *Phil. Mag.* 2 (1876) 135-145.
Examples
--------
>>> complex(np.round(propagation_constant(0, 1e-6, 0, 1e-12, 1e6), 12))
0.001j
"""
_nonnegative(R_per_m=R_per_m, G_per_m=G_per_m)
_positive(L_per_m=L_per_m, C_per_m=C_per_m, omega=omega)
w = np.asarray(omega, dtype=float)
return np.sqrt((R_per_m + 1j * w * L_per_m) * (G_per_m + 1j * w * C_per_m))
[docs]
def distortionless_condition(R_per_m, L_per_m, G_per_m, C_per_m, rtol=1e-9):
r"""Whether a line meets Heaviside's distortionless condition :math:`R/L = G/C`.
On such a line the attenuation :math:`\alpha = \sqrt{RG}` and the
velocity :math:`1/\sqrt{LC}` are the same at every frequency, so a pulse
shrinks without changing shape.
Parameters
----------
R_per_m : float
Series resistance per unit length [ohm/m], nonnegative.
L_per_m : float
Series inductance per unit length [H/m], positive.
G_per_m : float
Shunt conductance per unit length [S/m], nonnegative.
C_per_m : float
Shunt capacitance per unit length [F/m], positive.
rtol : float, optional
Relative tolerance on :math:`RC = GL` [-], nonnegative.
Returns
-------
bool
``True`` if the line is distortionless.
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/electrical_engineering`.
References
----------
O. Heaviside, "Electromagnetic induction and its propagation," *The
Electrician* (1887); reprinted in *Electrical Papers*, vol. 2 (London:
Macmillan, 1892).
Examples
--------
>>> distortionless_condition(0.01, 1e-6, 1e-8, 1e-12)
True
"""
_nonnegative(R_per_m=R_per_m, G_per_m=G_per_m, rtol=rtol)
_positive(L_per_m=L_per_m, C_per_m=C_per_m)
distortionless = np.isclose(np.multiply(R_per_m, C_per_m), np.multiply(G_per_m, L_per_m), rtol=rtol, atol=0)
return bool(distortionless) if distortionless.ndim == 0 else distortionless
[docs]
def reflection_coefficient(z_load, z0):
r"""Voltage reflection coefficient :math:`\Gamma = (Z_L - Z_0)/(Z_L + Z_0)`.
Parameters
----------
z_load : complex or array_like
Load impedance [ohm], finite with nonnegative real part (a passive
load); an open circuit is the limit :math:`\Gamma = 1`.
z0 : float
Characteristic impedance [ohm], positive.
Returns
-------
complex or ndarray of complex
:math:`\Gamma` [-], inside or on the unit circle.
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/electrical_engineering`.
Examples
--------
>>> complex(reflection_coefficient(150.0, 50.0))
(0.5+0j)
"""
_finite(z_load=z_load)
_positive(z0=z0)
z = np.asarray(z_load, dtype=complex)
if np.any(z.real < 0):
raise ValueError("z_load must have a nonnegative real part")
return (z - z0) / (z + z0)
[docs]
def standing_wave_ratio(gamma):
r"""Voltage standing-wave ratio :math:`(1 + |\Gamma|)/(1 - |\Gamma|)`.
Parameters
----------
gamma : complex or array_like
Reflection coefficient [-], with :math:`|\Gamma| < 1`.
Returns
-------
float or ndarray
VSWR [-], one for a matched line.
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/electrical_engineering`.
Examples
--------
>>> float(standing_wave_ratio(0.5))
3.0
"""
_finite(gamma=gamma)
m = np.abs(np.asarray(gamma, dtype=complex))
if np.any(m >= 1):
raise ValueError("|gamma| must be less than one")
return (1 + m) / (1 - m)
