# Source code for skrf.tlineFunctions

'''
.. module:: skrf.tlineFunctions
===============================================
tlineFunctions (:mod:skrf.tlineFunctions)
===============================================

This module provides functions related to transmission line theory.

Impedance and Reflection Coefficient
--------------------------------------
These functions relate basic transmission line quantities such as
characteristic impedance, input impedance, reflection coefficient, etc.
Each function has two names. One is a long-winded but readable name and
the other is a short-hand variable-like names. Below is a table relating
these two names with each other as well as common mathematical symbols.

====================  ======================  ================================
Symbol                Variable Name           Long Name
====================  ======================  ================================
:math:Z_l           z_l                     load_impedance
:math:Z_{in}        z_in                    input_impedance
:math:\Gamma_0      Gamma_0                 reflection_coefficient
:math:\Gamma_{in}   Gamma_in                reflection_coefficient_at_theta
:math:\\theta        theta                   electrical_length
====================  ======================  ================================

There may be a bit of confusion about the difference between the load
impedance the input impedance. This is because the load impedance **is**
the input impedance at the load. An illustration may provide some
useful reference.

Below is a (bad) illustration of a section of uniform transmission line
of characteristic impedance :math:Z_0, and electrical length
:math:\\theta. The line is terminated on the right with some
load impedance, :math:Z_l. The input impedance :math:Z_{in} and
input reflection coefficient :math:\\Gamma_{in} are
looking in towards the load from the distance :math:\\theta from the

.. math::
Z_0, \\theta

\\text{o===============o=}[Z_l]

So, to clarify the confusion,

.. math::
\\Gamma_{in}=\\Gamma_l \\text{ at }  \\theta=0

Short names
+++++++++++++
.. autosummary::
:toctree: generated/

theta
zl_2_Gamma0
Gamma0_2_zl
zl_2_zin
zl_2_Gamma_in
Gamma0_2_Gamma_in
Gamma0_2_zin

Long-names
++++++++++++++
.. autosummary::
:toctree: generated/

distance_2_electrical_length
electrical_length_2_distance

reflection_coefficient_at_theta
reflection_coefficient_2_input_impedance
reflection_coefficient_2_input_impedance_at_theta

input_impedance_at_theta

Distributed Circuit and Wave Quantities
----------------------------------------
.. autosummary::
:toctree: generated/

distributed_circuit_2_propagation_impedance
propagation_impedance_2_distributed_circuit

Transmission Line Physics
---------------------------------
.. autosummary::
:toctree: generated/

skin_depth
surface_resistivity
'''

import numpy as npy
from numpy import pi, sqrt, exp, array,tan,sin,cos,inf, log, real,imag,\
interp, linspace, shape,zeros, reshape

from scipy.constants import mu_0
from . import mathFunctions as mf

INF = 1e99
ONE = 1.0 + 1/1e14

[docs]def skin_depth(f,rho, mu_r): ''' the skin depth for a material. see www.microwaves101.com for more info. Parameters ---------- f : number or array-like frequency, in Hz rho : number of array-like bulk resistivity of material, in ohm*m mu_r : number or array-like relative permeability of material Returns ---------- skin depth : number or array-like the skin depth, in m References -------------- .. [1] http://en.wikipedia.org/wiki/Skin_effect ''' return sqrt(rho/(pi*f*mu_r*mu_0))
[docs]def surface_resistivity(f,rho,mu_r): ''' surface resistivity. see www.microwaves101.com for more info. Parameters ---------- f : number or array-like frequency, in Hz rho : number or array-like bulk resistivity of material, in ohm*m mu_r : number or array-like relative permeability of material Returns ---------- surface resistivity: ohms/square ''' return rho/skin_depth(rho=rho,f = f, mu_r=mu_r)
[docs]def distributed_circuit_2_propagation_impedance( distributed_admittance,\ distributed_impedance): ''' Converts distrubuted circuit values to wave quantities. This converts complex distributed impedance and admittance to propagation constant and characteristic impedance. The relation is .. math:: Z_0 = \\sqrt{ \\frac{Z^{'}}{Y^{'}}} \\quad\\quad \\gamma = \\sqrt{ Z^{'} Y^{'}} Parameters ------------ distributed_admittance : number, array-like distributed admittance distributed_impedance : number, array-like distributed impedance Returns ---------- propagation_constant : number, array-like distributed impedance characteristic_impedance : number, array-like distributed impedance See Also ---------- propagation_impedance_2_distributed_circuit : opposite conversion ''' propagation_constant = \ sqrt(distributed_impedance*distributed_admittance) characteristic_impedance = \ sqrt(distributed_impedance/distributed_admittance) return (propagation_constant, characteristic_impedance)
[docs]def propagation_impedance_2_distributed_circuit(propagation_constant, \ characteristic_impedance): ''' Converts wave quantities to distributed circuit values. Converts complex propagation constant and characteristic impedance to distributed impedance and admittance. The relation is, .. math:: Z^{'} = \\gamma Z_0 \\quad\\quad Y^{'} = \\frac{\\gamma}{Z_0} Parameters ------------ propagation_constant : number, array-like distributed impedance characteristic_impedance : number, array-like distributed impedance Returns ---------- distributed_admittance : number, array-like distributed admittance distributed_impedance : number, array-like distributed impedance See Also ---------- distributed_circuit_2_propagation_impedance : opposite conversion ''' distributed_admittance = propagation_constant/characteristic_impedance distributed_impedance = propagation_constant*characteristic_impedance return (distributed_admittance,distributed_impedance)
def electrical_length(gamma, f , d, deg=False): ''' Calculates the electrical length of a section of transmission line. .. math:: \\theta = \\gamma(f) \\cdot d Parameters ---------- gamma : function propagation constant function, which takes frequency in hz as a sole argument. see Notes. l : number or array-like length of line, in meters f : number or array-like frequency at which to calculate deg : Boolean return in degrees or not. Returns ---------- theta : number or array-like electrical length in radians or degrees, depending on value of deg. See Also ----------- electrical_length_2_distance : opposite conversion Notes ------ the convention has been chosen that forward propagation is represented by the positive imaginary part of the value returned by the gamma function ''' # typecast to a 1D array f = array(f, dtype=float).reshape(-1) d = array(d, dtype=float).reshape(-1) if deg == False: return gamma(f)*d elif deg == True: return mf.radian_2_degree(gamma(f)*d )
[docs]def electrical_length_2_distance(theta, gamma, f0,deg=True): ''' Convert electrical length to a physical distance. .. math:: d = \\frac{\\theta}{\\gamma(f_0)} Parameters ---------- theta : number or array-like electical length. units depend on deg option gamma : function propagation constant function, which takes frequency in hz as a sole argument. see Notes f0 : number or array-like frequency at which to calculate deg : Boolean return in degrees or not. Returns ---------- d: physical distance Notes ------ the convention has been chosen that forward propagation is represented by the positive imaginary part of the value returned by the gamma function See Also --------- distance_2_electrical_length: opposite conversion ''' if deg == True: theta = mf.degree_2_radian(theta) return theta/imag(gamma(f0))
[docs]def load_impedance_2_reflection_coefficient(z0, zl): ''' Returns the reflection coefficient for a given load impedance, and characteristic impedance. For a transmission line of characteristic impedance :math:Z_0 terminated with load impedance :math:Z_l, the complex reflection coefficient is given by, .. math:: \\Gamma = \\frac {Z_l - Z_0}{Z_l + Z_0} Parameters ---------- z0 : number or array-like characteristic impedance zl : number or array-like load impedance (aka input impedance) Returns -------- gamma : number or array-like reflection coefficient See Also ---------- Gamma0_2_zl : reflection coefficient to load impedance Notes ------ inputs are typecasted to 1D complex array ''' # typecast to a complex 1D array. this makes everything easier z0 = array(z0, dtype=complex).reshape(-1) zl = array(zl, dtype=complex).reshape(-1) # handle singularity by numerically representing inf as big number zl[(zl==npy.inf)] = INF return ((zl -z0 )/(zl+z0))
[docs]def reflection_coefficient_2_input_impedance(z0,Gamma): ''' calculates the input impedance given a reflection coefficient and characteristic impedance .. math:: Z_0 (\\frac {1 + \\Gamma}{1-\\Gamma}) Parameters ---------- Gamma : number or array-like complex reflection coefficient z0 : number or array-like characteristic impedance Returns -------- zin : number or array-like input impedance ''' # typecast to a complex 1D array. this makes everything easier Gamma = array(Gamma, dtype=complex).reshape(-1) z0 = array(z0, dtype=complex).reshape(-1) #handle singularity by numerically representing inf as close to 1 Gamma[(Gamma == 1)] = ONE return z0*((1.0+Gamma )/(1.0-Gamma))
[docs]def reflection_coefficient_at_theta(Gamma0,theta): ''' reflection coefficient at a given electrical length. .. math:: \\Gamma_{in} = \\Gamma_0 e^{-2j\\theta} Parameters ---------- Gamma0 : number or array-like reflection coefficient at theta=0 theta : number or array-like electrical length, (may be complex) Returns ---------- Gamma_in : number or array-like input reflection coefficient ''' Gamma0 = array(Gamma0, dtype=complex).reshape(-1) theta = array(theta, dtype=complex).reshape(-1) return Gamma0 * exp(-2j* theta)
[docs]def input_impedance_at_theta(z0,zl, theta): ''' input impedance of load impedance zl at a given electrical length, given characteristic impedance z0. Parameters ---------- z0 : characteristic impedance. zl : load impedance theta : electrical length of the line, (may be complex) Returns --------- ''' Gamma0 = load_impedance_2_reflection_coefficient(z0=z0,zl=zl) Gamma_in = reflection_coefficient_at_theta(Gamma0=Gamma0, theta=theta) return reflection_coefficient_2_input_impedance(z0=z0, Gamma=Gamma_in)
[docs]def load_impedance_2_reflection_coefficient_at_theta(z0, zl, theta): Gamma0 = load_impedance_2_reflection_coefficient(z0=z0,zl=zl) Gamma_in = reflection_coefficient_at_theta(Gamma0=Gamma0, theta=theta) return Gamma_in
[docs]def reflection_coefficient_2_input_impedance_at_theta(z0, Gamma0, theta): ''' calculates the input impedance at electrical length theta, given a reflection coefficient and characteristic impedance of the medium Parameters ---------- z0 - characteristic impedance. Gamma: reflection coefficient theta: electrical length of the line, (may be complex) returns zin: input impedance at theta ''' Gamma_in = reflection_coefficient_at_theta(Gamma0=Gamma0, theta=theta) zin = reflection_coefficient_2_input_impedance(z0=z0,Gamma=Gamma_in) return zin
# short hand convenience. # admittedly these follow no logical naming scheme, but they closely # correspond to common symbolic conventions, and are convenient theta = electrical_length distance_2_electrical_length = electrical_length zl_2_Gamma0 = load_impedance_2_reflection_coefficient Gamma0_2_zl = reflection_coefficient_2_input_impedance zl_2_zin = input_impedance_at_theta zl_2_Gamma_in = load_impedance_2_reflection_coefficient_at_theta Gamma0_2_Gamma_in = reflection_coefficient_at_theta Gamma0_2_zin = reflection_coefficient_2_input_impedance_at_theta