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// Copyright Paul A. Bristow 2016
// Copyright John Z. Maddock 2016
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt or
// copy at http ://www.boost.org/LICENSE_1_0.txt).
/*! brief Example of using Lambert W function to compute current through a diode connected transistor with preset series resistance.
\details T. C. Banwell and A. Jayakumar,
Exact analytical solution of current flow through diode with series resistance,
Electron Letters, 36(4):291-2 (2000)
DOI: doi.org/10.1049/el:20000301
The current through a diode connected NPN bipolar junction transistor (BJT)
type 2N2222 (See https://en.wikipedia.org/wiki/2N2222 and
https://www.fairchildsemi.com/datasheets/PN/PN2222.pdf Datasheet)
was measured, for a voltage between 0.3 to 1 volt, see Fig 2 for a log plot,
showing a knee visible at about 0.6 V.
The transistor parameter isat was estimated to be 25 fA and the ideality factor = 1.0.
The intrinsic emitter resistance re was estimated from the rsat = 0 data to be 0.3 ohm.
The solid curves in Figure 2 are calculated using equation 5 with rsat included with re.
http://www3.imperial.ac.uk/pls/portallive/docs/1/7292572.PDF
*/
#include <boost/math/special_functions/lambert_w.hpp>
using boost::math::lambert_w0;
#include <iostream>
// using std::cout;
// using std::endl;
#include <exception>
#include <stdexcept>
#include <string>
#include <array>
#include <vector>
/*!
Compute thermal voltage as a function of temperature,
about 25 mV at room temperature.
https://en.wikipedia.org/wiki/Boltzmann_constant#Role_in_semiconductor_physics:_the_thermal_voltage
\param temperature Temperature (degrees centigrade).
*/
const double v_thermal(double temperature)
{
constexpr const double boltzmann_k = 1.38e-23; // joules/kelvin.
const double charge_q = 1.6021766208e-19; // Charge of an electron (columb).
double temp =+ 273; // Degrees C to K.
return boltzmann_k * temp / charge_q;
} // v_thermal
/*!
Banwell & Jayakumar, equation 2
*/
double i(double isat, double vd, double vt, double nu)
{
double i = isat * (exp(vd / (nu * vt)) - 1);
return i;
} //
/*!
Banwell & Jayakumar, Equation 4.
i current flow = isat
v voltage source.
isat reverse saturation current in equation 4.
(might implement equation 4 instead of simpler equation 5?).
vd voltage drop = v - i* rs (equation 1).
vt thermal voltage, 0.0257025 = 25 mV.
nu junction ideality factor (default = unity), also known as the emission coefficient.
re intrinsic emitter resistance, estimated to be 0.3 ohm from low current.
rsat reverse saturation current
\param v Voltage V to compute current I(V).
\param vt Thermal voltage, for example 0.0257025 = 25 mV, computed from boltzmann_k * temp / charge_q;
\param rsat Resistance in series with the diode.
\param re Intrinsic emitter resistance (estimated to be 0.3 ohm from the Rs = 0 data)
\param isat Reverse saturation current (See equation 2).
\param nu Ideality factor (default = unity).
\returns I amp as function of V volt.
*/
double iv(double v, double vt, double rsat, double re, double isat, double nu = 1.)
{
// V thermal 0.0257025 = 25 mV
// was double i = (nu * vt/r) * lambert_w((i0 * r) / (nu * vt)); equ 5.
rsat = rsat + re;
double i = nu * vt / rsat;
std::cout << "nu * vt / rsat = " << i << std::endl; // 0.000103223
double x = isat * rsat / (nu * vt);
std::cout << "isat * rsat / (nu * vt) = " << x << std::endl;
double eterm = (v + isat * rsat) / (nu * vt);
std::cout << "(v + isat * rsat) / (nu * vt) = " << eterm << std::endl;
double e = exp(eterm);
std::cout << "exp(eterm) = " << e << std::endl;
double w0 = lambert_w0(x * e);
std::cout << "w0 = " << w0 << std::endl;
return i * w0 - isat;
} // double iv
std::array<double, 5> rss = {0., 2.18, 10., 51., 249}; // series resistance (ohm).
std::array<double, 8> vds = { 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 }; // Diode voltage.
int main()
{
try
{
std::cout << "Lambert W diode current example." << std::endl;
//[lambert_w_diode_example_1
double x = 0.01;
//std::cout << "Lambert W (" << x << ") = " << lambert_w(x) << std::endl; // 0.00990147
double nu = 1.0; // Assumed ideal.
double vt = v_thermal(25); // v thermal, Shockley equation, expect about 25 mV at room temperature.
double boltzmann_k = 1.38e-23; // joules/kelvin
double temp = 273 + 25;
double charge_q = 1.6e-19; // column
vt = boltzmann_k * temp / charge_q;
std::cout << "V thermal "
<< vt << std::endl; // V thermal 0.0257025 = 25 mV
double rsat = 0.;
double isat = 25.e-15; // 25 fA;
std::cout << "Isat = " << isat << std::endl;
double re = 0.3; // Estimated from slope of straight section of graph (equation 6).
double v = 0.9;
double icalc = iv(v, vt, 249., re, isat);
std::cout << "voltage = " << v << ", current = " << icalc << ", " << log(icalc) << std::endl; // voltage = 0.9, current = 0.00108485, -6.82631
//] [/lambert_w_diode_example_1]
}
catch (std::exception& ex)
{
std::cout << ex.what() << std::endl;
}
} // int main()
/*
Output:
//[lambert_w_output_1
Lambert W diode current example.
V thermal 0.0257025
Isat = 2.5e-14
nu * vt / rsat = 0.000103099
isat * rsat / (nu * vt) = 2.42486e-10
(v + isat * rsat) / (nu * vt) = 35.016
exp(eterm) = 1.61167e+15
w0 = 10.5225
voltage = 0.9, current = 0.00108485, -6.82631
//] [/lambert_w_output_1]
*/
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