{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "a38fac8e",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from math import exp\n",
    "from scipy.integrate import solve_ivp\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a5b7ec98",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import symbols, exp as sexp, diff, Derivative, Eq"
   ]
  },
  {
   "attachments": {
    "bc547-pulse.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "9a5dc2ed",
   "metadata": {},
   "source": [
    "![bc547-pulse.png](attachment:bc547-pulse.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae0f8575",
   "metadata": {},
   "source": [
    "The object was to measure the voltage at the base of the transmitter (and later on at the base of R2) to determine the characteristics of the transistor.  Specifically to estimate the thermal coefficient $V_T$ and the scale current $I_{ES}$. The capacitor was to produce a comparatively slowly changing transistor base voltage with hopefully enough time for me to make measurements using an Arduino.  (This has since become slightly more complicated because the Arduino analog inputs have a time lag of 100 $\\mu S$ which may add some measurement bias)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "17ce4b69",
   "metadata": {},
   "outputs": [],
   "source": [
    "r1, r2, c, issym, vt, vin, vout, vbe, vc, t = symbols(\"R1 R2 C I_{ES} V_T V_{in} V_{out} V_{BE} V_C t\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "771d4afc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - C \\frac{d}{d t} V_{C} - I_{ES} \\left(e^{\\frac{V_{C}}{V_{T}}} - 1\\right) + \\frac{- V_{C} + V_{in}}{R_{1}} = 0$"
      ],
      "text/plain": [
       "Eq(-C*Derivative(V_C, t) - I_{ES}*(exp(V_C/V_T) - 1) + (-V_C + V_{in})/R1, 0)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " Eq((vin - vc)/r1 - c * Derivative(vc,t) - issym*(sexp(vc/vt)-1),0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "09b9a562",
   "metadata": {},
   "source": [
    "The above is a differential equation for the R1, C and base-emitter junction.  $V_C$ is the voltage across the capacitor.  Vin is a choice of 0 or 5 Volts as it is connected to one of the outputs of the Arduino digital pins.  By turning it on or off I was hoping to sweep through a range of voltages controlled by the time constand of R1 $\\times$ C.  In my case R1 = 6K8 and C = $100 \\mu F$.  I think R2 is about 470 $\\Omega$.\n",
    "\n",
    "It's basically a non linear differential equation which is impossible to solve using analytical methods.  Hence the attempt to solve it below using numerical methods."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29143ef7",
   "metadata": {},
   "source": [
    "Now I try to build my theoretical model using the equation above and some scientific libraries to integrate the equation to find what the solution would look like over time.\n",
    "I am basically plotting the voltage across the capacitor as function of time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "4e3423e0",
   "metadata": {},
   "outputs": [],
   "source": [
    "def model(t, y, vin):\n",
    "    vcc=vin\n",
    "    vt =  0.0321850033399526\n",
    "    isval = 1.602564611659814e-12\n",
    "    r=6800\n",
    "    c=100e-6\n",
    "    \n",
    "    vc = y\n",
    "    try:\n",
    "        vcp =  (vcc-vc)/(r*c) - isval * (exp(vc/vt)-1) /(c)\n",
    "    except OverflowError:\n",
    "        print (vc)\n",
    "        return np.inf\n",
    "    #print (vcp)\n",
    "    return vcp\n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "6b52e0bf",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[56.81583729]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/lib/python3.9/site-packages/scipy/integrate/_ivp/rk.py:109: RuntimeWarning: invalid value encountered in true_divide\n",
      "  return norm(self._estimate_error(K, h) / scale)\n"
     ]
    }
   ],
   "source": [
    "teval = np.linspace(0, .4,100)\n",
    "res = solve_ivp(model, (0,.4), (0,),args=(5,), t_eval = teval, first_step=0.001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "c446244e",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = res.t\n",
    "y = res.y[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "04001485",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Charging of $V_C$ over time(t)')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.plot(x,y)\n",
    "ax.grid()\n",
    "ax.set_xlabel(\"x (secs)\")\n",
    "ax.set_ylabel(\"Vc (cap)\")\n",
    "ax.set_title(\"Charging of $V_C$ over time(t)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "5b459d43",
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "teval = np.linspace(0, 1,100)\n",
    "res = solve_ivp(model, (0,1), (.7,),args=(0,), t_eval = teval, first_step=0.001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "a829b00b",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = res.t\n",
    "y = res.y[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c46ad600",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Discharging of C over time(t)')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.plot(x,y)\n",
    "ax.grid()\n",
    "ax.set_xlabel(\"x (secs)\")\n",
    "ax.set_ylabel(\"Vc (cap)\")\n",
    "ax.set_title(\"Discharging of C over time(t)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7fae56b9",
   "metadata": {},
   "source": [
    "The results are unsurprising as the capacitor charges up to ( or discharges from ) 0.7 volts approximately.  Anything higher and the transistor conducts so the results are at least realistic."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "551150e0",
   "metadata": {},
   "source": [
    "Now I look at the results obtained from direct measurement via the Arduino analog input pin."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "ea25cc0b",
   "metadata": {},
   "outputs": [],
   "source": [
    "measurements = np.array([[ 0, 11, 0.05], \n",
    "[ 4, 17, 0.08], \n",
    "[ 8, 23, 0.11], \n",
    "[ 12, 29, 0.14], \n",
    "[ 16, 34, 0.17], \n",
    "[ 20, 40, 0.20], \n",
    "[ 24, 45, 0.22], \n",
    "[ 28, 50, 0.24], \n",
    "[ 32, 57, 0.28], \n",
    "[ 36, 62, 0.30], \n",
    "[ 40, 68, 0.33], \n",
    "[ 44, 73, 0.36], \n",
    "[ 48, 78, 0.38], \n",
    "[ 52, 84, 0.41], \n",
    "[ 56, 90, 0.44], \n",
    "[ 60, 95, 0.46], \n",
    "[ 65, 100, 0.49], \n",
    "[ 69, 105, 0.51], \n",
    "[ 73, 110, 0.54], \n",
    "[ 77, 116, 0.57], \n",
    "[ 81, 121, 0.59], \n",
    "[ 85, 127, 0.62], \n",
    "[ 89, 135, 0.66], \n",
    "[ 93, 144, 0.70], \n",
    "[ 97, 159, 0.78], \n",
    "[ 101, 183, 0.89], \n",
    "[ 105, 193, 0.94], \n",
    "[ 109, 195, 0.95], \n",
    "[ 113, 196, 0.96], \n",
    "[ 117, 198, 0.97], \n",
    "[ 121, 198, 0.97], \n",
    "[ 125, 197, 0.96], \n",
    "[ 129, 198, 0.97], \n",
    "[ 133, 198, 0.97], \n",
    "[ 137, 198, 0.97], \n",
    "[ 141, 198, 0.97], \n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "c3d98428",
   "metadata": {},
   "outputs": [],
   "source": [
    "dfmeasure =  pd.DataFrame(measurements)\n",
    "dfmeasure.columns = ['time', 'ADC', 'volts']\n",
    "# ADC is the measure obtained from Arduino analogRead which returns a value from 0 to 1023.\n",
    "# I divide that by 1024 and multiply by 5 to get a voltage."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "79933ebd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.set_title('Arduino measure charging of cap across transistor base')\n",
    "\n",
    "\n",
    "ax1 = dfmeasure.plot('time', 'volts', kind='scatter', ax=ax)\n",
    "ax1.set_xlabel('Time (mSec)')\n",
    "ax1.set_ylabel('voltage $V_C$')\n",
    "ax1.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9edca529",
   "metadata": {},
   "source": [
    "Well it is linear for part of the range but then gets weird soon after around the 100 mSec mark.  What the prediction in cell #8 got right was when the voltage peaked but it didn't predict the \"dog's leg\"."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e6af97a",
   "metadata": {},
   "source": [
    "Now we will look and see what happens from multiple runs with data plotted on same graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "1cb234ea",
   "metadata": {},
   "outputs": [],
   "source": [
    "measurements = [\n",
    "  [\n",
    "    [ 0, 16, 0.08], [ 4, 21, 0.10], [ 8, 27, 0.13], [ 12, 34, 0.17], \n",
    "    [ 16, 38, 0.19], [ 20, 45, 0.22], [ 24, 50, 0.24], [ 28, 56, 0.27], \n",
    "    [ 32, 61, 0.30], [ 36, 67, 0.33], [ 40, 73, 0.36], [ 44, 77, 0.38], \n",
    "    [ 48, 83, 0.41], [ 52, 89, 0.43], [ 56, 94, 0.46], [ 60, 99, 0.48], \n",
    "    [ 65, 105, 0.51], [ 69, 110, 0.54], [ 73, 115, 0.56], [ 77, 121, 0.59], \n",
    "    [ 81, 126, 0.62], [ 85, 133, 0.65], [ 89, 140, 0.68], [ 93, 152, 0.74], \n",
    "    [ 97, 172, 0.84], [ 101, 208, 1.02], [ 105, 240, 1.17], [ 109, 243, 1.19], \n",
    "    [ 113, 245, 1.20], [ 117, 246, 1.20], [ 121, 248, 1.21], [ 125, 247, 1.21], \n",
    "    [ 129, 248, 1.21], [ 133, 247, 1.21], [ 137, 247, 1.21], [ 141, 248, 1.21], \n",
    "    [ 145, 248, 1.21], [ 150, 248, 1.21], [ 154, 248, 1.21], \n",
    "  ],\n",
    "  [\n",
    "    [ 0, 15, 0.07], [ 4, 22, 0.11], [ 8, 27, 0.13], [ 12, 33, 0.16], \n",
    "    [ 16, 39, 0.19], [ 20, 44, 0.21], [ 24, 49, 0.24], [ 28, 55, 0.27], \n",
    "    [ 32, 61, 0.30], [ 36, 66, 0.32], [ 40, 72, 0.35], [ 44, 77, 0.38], \n",
    "    [ 48, 83, 0.41], [ 52, 89, 0.43], [ 56, 94, 0.46], [ 60, 99, 0.48], \n",
    "    [ 65, 104, 0.51], [ 69, 109, 0.53], [ 73, 115, 0.56], [ 77, 120, 0.59], \n",
    "    [ 81, 126, 0.62], [ 85, 132, 0.64], [ 89, 140, 0.68], [ 93, 151, 0.74], \n",
    "    [ 97, 172, 0.84], [ 101, 207, 1.01], [ 105, 239, 1.17], [ 109, 243, 1.19], \n",
    "    [ 113, 246, 1.20], [ 117, 246, 1.20], [ 121, 246, 1.20], [ 125, 247, 1.21], \n",
    "    [ 129, 247, 1.21], [ 133, 247, 1.21], [ 137, 247, 1.21], \n",
    "  ],\n",
    "  [\n",
    "    [ 0, 14, 0.07], [ 4, 21, 0.10], [ 8, 27, 0.13], [ 12, 33, 0.16], \n",
    "    [ 16, 39, 0.19], [ 20, 45, 0.22], [ 24, 49, 0.24], [ 28, 56, 0.27], \n",
    "    [ 32, 61, 0.30], [ 36, 67, 0.33], [ 40, 72, 0.35], [ 44, 77, 0.38], \n",
    "    [ 48, 83, 0.41], [ 52, 88, 0.43], [ 56, 94, 0.46], [ 60, 100, 0.49], \n",
    "    [ 65, 104, 0.51], [ 69, 110, 0.54], [ 73, 115, 0.56], [ 77, 120, 0.59], \n",
    "    [ 81, 126, 0.62], [ 85, 132, 0.64], [ 89, 140, 0.68], [ 93, 151, 0.74], \n",
    "    [ 97, 172, 0.84], [ 101, 208, 1.02], [ 105, 241, 1.18], [ 109, 245, 1.20], \n",
    "    [ 113, 248, 1.21], [ 117, 249, 1.22], [ 121, 249, 1.22], [ 125, 249, 1.22], \n",
    "    [ 129, 249, 1.22], \n",
    "  ]\n",
    "]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "546201f8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "markers = ['x', 'o', '.']\n",
    "mcolours = ['red', 'green', 'blue']\n",
    "ax.set_title('Arduino measure charging of cap across transistor base')\n",
    "for idx in range(0,3):\n",
    "    pdmeasure = pd.DataFrame(measurements[idx])\n",
    "    pdmeasure.columns = ['time', 'ADC', 'volts']\n",
    "    ax1 = pdmeasure.plot('time', 'volts', kind='scatter', ax=ax, \n",
    "                         marker=markers[idx], color=mcolours[idx])\n",
    "ax1.set_xlabel('Time (mSec)')\n",
    "ax1.set_ylabel('voltage $V_C$')\n",
    "ax1.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e33d6be3",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}