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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 Graph showing use of 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 I sat 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 <boost/math/special_functions.hpp>
- using boost::math::isfinite;
- #include <boost/svg_plot/svg_2d_plot.hpp>
- using namespace boost::svg;
- #include <iostream>
- // using std::cout;
- // using std::endl;
- #include <exception>
- #include <stdexcept>
- #include <string>
- #include <array>
- #include <vector>
- #include <utility>
- using std::pair;
- #include <map>
- using std::map;
- #include <set>
- using std::multiset;
- #include <limits>
- using std::numeric_limits;
- #include <cmath> //
- /*!
- 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 Celsius).
- */
- const double v_thermal(double temperature)
- {
- BOOST_CONSTEXPR const double boltzmann_k = 1.38e-23; // joules/kelvin.
- BOOST_CONSTEXPR 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, page 291.
- */
- double i(double isat, double vd, double vt, double nu)
- {
- double i = isat * (exp(vd / (nu * vt)) - 1);
- return i;
- } //
- /*!
- Banwell & Jayakumar, Equation 4, page 291.
- 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 Instrinsic 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.
- */
- //[lambert_w_diode_graph_2
- 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
- //] [\lambert_w_diode_graph_2]
- std::array<double, 5> rss = { 0., 2.18, 10., 51., 249 }; // series resistance (ohm).
- std::array<double, 7> vds = { 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 }; // Diode voltage.
- std::array<double, 7> lni = { -19.65, -15.75, -11.86, -7.97, -4.08, -0.0195, 3.6 }; // ln(current).
- int main()
- {
- try
- {
- std::cout << "Lambert W diode current example." << std::endl;
- //[lambert_w_diode_graph_1
- 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_graph_1]
- // Plot a few measured data points.
- std::map<const double, double> zero_data; // Extrapolated from slope of measurements with no external resistor.
- zero_data[0.3] = -19.65;
- zero_data[0.4] = -15.75;
- zero_data[0.5] = -11.86;
- zero_data[0.6] = -7.97;
- zero_data[0.7] = -4.08;
- zero_data[0.8] = -0.0195;
- zero_data[0.9] = 3.9;
- std::map<const double, double> measured_zero_data; // No external series resistor.
- measured_zero_data[0.3] = -19.65;
- measured_zero_data[0.4] = -15.75;
- measured_zero_data[0.5] = -11.86;
- measured_zero_data[0.6] = -7.97;
- measured_zero_data[0.7] = -4.2;
- measured_zero_data[0.72] = -3.5;
- measured_zero_data[0.74] = -2.8;
- measured_zero_data[0.76] = -2.3;
- measured_zero_data[0.78] = -2.0;
- // Measured from Fig 2 as raw data not available.
- double step = 0.1;
- for (int i = 0; i < vds.size(); i++)
- {
- zero_data[vds[i]] = lni[i];
- std::cout << lni[i] << " " << vds[i] << std::endl;
- }
- step = 0.01;
- std::map<const double, double> data_2;
- for (double v = 0.3; v < 1.; v += step)
- {
- double current = iv(v, vt, 2., re, isat);
- data_2[v] = log(current);
- // std::cout << "v " << v << ", current = " << current << " log current = " << log(current) << std::endl;
- }
- std::map<const double, double> data_10;
- for (double v = 0.3; v < 1.; v += step)
- {
- double current = iv(v, vt, 10., re, isat);
- data_10[v] = log(current);
- // std::cout << "v " << v << ", current = " << current << " log current = " << log(current) << std::endl;
- }
- std::map<const double, double> data_51;
- for (double v = 0.3; v < 1.; v += step)
- {
- double current = iv(v, vt, 51., re, isat);
- data_51[v] = log(current);
- // std::cout << "v " << v << ", current = " << current << " log current = " << log(current) << std::endl;
- }
- std::map<const double, double> data_249;
- for (double v = 0.3; v < 1.; v += step)
- {
- double current = iv(v, vt, 249., re, isat);
- data_249[v] = log(current);
- // std::cout << "v " << v << ", current = " << current << " log current = " << log(current) << std::endl;
- }
- svg_2d_plot data_plot;
- data_plot.title("Diode current versus voltage")
- .x_size(400)
- .y_size(300)
- .legend_on(true)
- .legend_lines(true)
- .x_label("voltage (V)")
- .y_label("log(current) (A)")
- //.x_label_on(true)
- //.y_label_on(true)
- //.xy_values_on(false)
- .x_range(0.25, 1.)
- .y_range(-20., +4.)
- .x_major_interval(0.1)
- .y_major_interval(4)
- .x_major_grid_on(true)
- .y_major_grid_on(true)
- //.x_values_on(true)
- //.y_values_on(true)
- .y_values_rotation(horizontal)
- //.plot_window_on(true)
- .x_values_precision(3)
- .y_values_precision(3)
- .coord_precision(4) // Needed to avoid stepping on curves.
- .copyright_holder("Paul A. Bristow")
- .copyright_date("2016")
- //.background_border_color(black);
- ;
- // ₀ = subscript zero.
- data_plot.plot(zero_data, "I₀(V)").fill_color(lightgray).shape(none).size(3).line_on(true).line_width(0.5);
- data_plot.plot(measured_zero_data, "Rs=0 Ω").fill_color(lightgray).shape(square).size(3).line_on(true).line_width(0.5);
- data_plot.plot(data_2, "Rs=2 Ω").line_color(blue).shape(none).line_on(true).bezier_on(false).line_width(1);
- data_plot.plot(data_10, "Rs=10 Ω").line_color(purple).shape(none).line_on(true).bezier_on(false).line_width(1);
- data_plot.plot(data_51, "Rs=51 Ω").line_color(green).shape(none).line_on(true).line_width(1);
- data_plot.plot(data_249, "Rs=249 Ω").line_color(red).shape(none).line_on(true).line_width(1);
- data_plot.write("./diode_iv_plot");
- // bezier_on(true);
- }
- catch (std::exception& ex)
- {
- std::cout << ex.what() << std::endl;
- }
- } // int main()
- /*
- //[lambert_w_output_1
- Output:
- Lambert W diode current example.
- V thermal 0.0257025
- Isat = 2.5e-14
- voltage = 0.9, current = 0.00108485, -6.82631
- -19.65 0.3
- -15.75 0.4
- -11.86 0.5
- -7.97 0.6
- -4.08 0.7
- -0.0195 0.8
- 3.6 0.9
- //] [/lambert_w_output_1]
- */
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