{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "infrared-logan",
   "metadata": {},
   "outputs": [],
   "source": [
    "# In this tutorial we will numerically evaluate inverse Laplace transforms\n",
    "# using 'invertlaplace()' in the mpmath module.\n",
    "\n",
    "# Inverse Laplace transforms are very useful  in electronics and are often\n",
    "# used to calculate the transient response of circuits.  Typically, one\n",
    "# calculates the response of the circuit in the \"s = jw\" or complex-frequency\n",
    "# domain using methods that a familiar from steady-state circuit analysis.\n",
    "# Then one takes the inverse Laplace transform to that result to deduce\n",
    "# the circuit's transient response.  In this tutorial, we demonstrate the\n",
    "# methode using two examples for which both the transient and s-domain \n",
    "# responses are already known."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "breeding-tuner",
   "metadata": {},
   "outputs": [],
   "source": [
    "# First import the modules that will be required.\n",
    "import mpmath as mp\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "essential-ordering",
   "metadata": {},
   "outputs": [],
   "source": [
    "# This line sets the number of decimal points (dps) to be used in calculations.\n",
    "# We will required mp.dps to be high in order to calculate precise transient \n",
    "# responses.\n",
    "mp.dps = 350"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "spread-valuation",
   "metadata": {},
   "outputs": [],
   "source": [
    "# The main function that we will use is 'mp.invertlaplace()'.  To execute,\n",
    "# it requires the s-domain function fs that is to be transformed into the\n",
    "# time domain and a single time.  In order to extract the full transient\n",
    "# response, we define a function that passes times from a list one by one.\n",
    "# The degree option determines the number of terms to be used in the approximations.\n",
    "# The higher, the better the result and the longer the calculation takes.\n",
    "# The 'if' statement is just used to print the progress of the calculation.\n",
    "# By default, pnt = True and the progress will be printed in increments of\n",
    "# 10%.\n",
    "def trans_response(tt, fs, deg, pnt = True):\n",
    "    trans = []\n",
    "    cnt = 0\n",
    "    for t in tt:    \n",
    "        if len(trans) % int(pts/10) == 0 and pnt == True:\n",
    "            print(cnt*10,'%', sep = '')\n",
    "            cnt += 1\n",
    "        trans += [mp.invertlaplace(fs, t, method = 'dehoog', degree = deg)]\n",
    "    return trans"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "center-incident",
   "metadata": {},
   "outputs": [],
   "source": [
    "# In the first example, the we will take the inverse Laplace transform of\n",
    "# e^(-c*s)/s which should result in a step at t = c.\n",
    "Vg = 2\n",
    "c = 0.5e-6\n",
    "fs = lambda s: Vg*mp.exp(-c*s)/s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "electronic-rates",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2021-03-11 10:31:16.021710\n",
      "0%\n",
      "10%\n",
      "20%\n",
      "30%\n",
      "40%\n",
      "50%\n",
      "60%\n",
      "70%\n",
      "80%\n",
      "90%\n",
      "2021-03-11 10:31:18.694646\n"
     ]
    },
    {
     "data": {
      "image/png": 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BgY3A8yPHJ4D39hizsee5ACTZw/BfA2zatKlHWa915513csstt/zoeLWe7927d83eu4X6ZqnWvXv3Is2aPkGfMW2LdxRfakyfc4eNVQeAAzDcHLxHXa+xe/fuH/u5ms/X8r1bqG/WapVmSaqWz9Qk7wN+p6o+0h1/GqCqfn9kzB8AX6uqu7vj54DtwOZJ544zGAxqfn7+9c1Ikt6AkhypqsG4vj5r9IeBLUkuS7Ie2AUcWjTmEPDJ7tM3VwEvVdV3ep4rSVpBE5duqupMkpuBh4B1wMGqOprkU13/fuB+4DrgGPBDYPdy567ITCRJY01culkLLt1I0tk516UbSdIMM+glqXEGvSQ1zqCXpMadl/8Zm+Q08Bev8/QNwBvtu+rOuX1vtPmCcz5b76qquXEd52XQn4sk80v9z3OrnHP73mjzBec8TS7dSFLjDHpJalyLQX9grQtYA865fW+0+YJznprm1uglST+uxSt6SdIIg16SGjeTQX8um5XPqh5zvqGb65NJHk1y+VrUOU19N5ZP8g+SvJLk46tZ30roM+ck25M8nuRokj9a7Rqnrcfv7bcm+W9JnujmPNM7wCQ5mORUkqeX6J9+flXVTD0Y3u74/wI/DawHngC2LhpzHfAAwx2urgL+dK3rXoU5/0PgJ7vn174R5jwy7n8yvFX2x9e67lX4dX4b8AywqTt+x1rXvQpz/m3g33bP54AXgfVrXfs5zPkDwJXA00v0Tz2/ZvGK/lw2K59VE+dcVY9W1Xe7w8eAS1a5xmnru7H8bwBfAk6tZnErpM+c/ylwb1X9JUBVzfq8+8y5gIuSBHgLw6A/s7plTk9VPcJwDkuZen7NYtAvtRH52Y6ZJWc7nxsZXhHMsolzTrIR+BiwfxXrWkl9fp1/FvjJJF9LciTJJ1etupXRZ87/EXg3cBJ4CvjNqvrb1SlvTUw9v/psDn6+OZfNymdV7/kkuYZh0L9/RStaeX3m/O+B36qqV4YXezOvz5wvAH4J+EfA3wX+JMljVfXNlS5uhfSZ80eAx4FfAX4G+GqSr1fV91a4trUy9fyaxaA/AVw6cnwJw7/pz3bMLOk1nyTvAT4PXFtVL6xSbSulz5wHwD1dyG8Arktypqq+vCoVTl/f39sLVfUD4AdJHgEuB2Y16PvMeTfwb2q4gH0sybeAnwf+z+qUuOqmnl+zuHRzLpuVz6qJc06yCbgX+MQMX92NmjjnqrqsqjZX1WbgvwL/fIZDHvr93v4K8MtJLkjyJuC9wLOrXOc09ZnzXzL8FwxJLgZ+Dji+qlWurqnn18xd0dc5bFY+q3rO+Tbg7cBnuyvcMzXDd/7rOeem9JlzVT2b5EHgSeBvgc9X1diP6c2Cnr/OvwfcleQphssav1VVM3v74iR3A9uBDUlOALcDF8LK5Ze3QJCkxs3i0o0k6SwY9JLUOINekhpn0EtS4wx6SVpBk25i9jpeb1OSh5M8m+SZJJsnnWPQS9LKugvYMcXX+wKwr6rezfBeQRPvd2TQS9IKGncTsyQ/k+TB7n5FX0/y831eK8lW4IKq+mr32t+vqh9OOs+gl6TVdwD4jar6JeBfA5/ted7PAn+V5N4kf5ZkX5J1k06auW/GStIsS/IWhvtH/JeRm/H9na7vHwO/O+a0/1dVH2GY2b8M/CLDW0P8Z+CfAX+43Hsa9JK0un4C+KuqumJxR1Xdy/CeVUs5AfxZVR0HSPJlhpuTLBv0Lt1I0irqbq/8rST/BH60dWDfrT8PM9yPYK47/hWGO44ty6CXpBXU3cTsT4CfS3IiyY3ADcCNSZ4AjjJ+97TXqKpXGK7p/4+Rm7z9p4k1eFMzSWqbV/SS1DiDXpIaZ9BLUuMMeklqnEEvSY0z6CWpcQa9JDXu/wMn/gbprCcM+wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# We use a 'with' statement to temporarily reduce the dps for the first \n",
    "# example within the indented block of code.\n",
    "with mp.workdps(20):\n",
    "    # Here, we call the fucntion that we defined above and then plot the result.\n",
    "    from datetime import datetime\n",
    "    print(datetime.now())\n",
    "    pts = 100\n",
    "    tt = np.linspace(1e-10, 1e-6, pts)\n",
    "    plt.plot(tt, trans_response(tt, fs, 20), 'k.', markersize = 2)\n",
    "    print(datetime.now())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "injured-caribbean",
   "metadata": {},
   "outputs": [],
   "source": [
    "# This next example is the transient response of a coaxial cable to a \n",
    "# voltage step.  The centre conductors of the coax at one end is connected\n",
    "# to a resistor of resistance Rg and the voltage step of size Vg is applied to\n",
    "# the opposite end of the Rg.  The free end of the coaxial cable is left open.\n",
    "# The coaxial cable has a characteristic impedance of Z0 = 50 ohms and a \n",
    "# length of 8 meters.\n",
    "Vg = 1 # volt\n",
    "ell = 8 # meters\n",
    "v0 = 3e8/np.sqrt(2.1) # m/s\n",
    "Rg = 1e3 # ohms\n",
    "Z0 = 50 # ohms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "silent-quebec",
   "metadata": {},
   "outputs": [],
   "source": [
    "# This is the s-domain response of the coax. \n",
    "fs = lambda s: (Vg/s)*mp.coth(s*ell/v0)/(Rg/Z0 + mp.coth(s*ell/v0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "pediatric-advocate",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2021-03-11 10:33:31.172249\n",
      "0%\n",
      "10%\n",
      "20%\n",
      "30%\n",
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      "90%\n",
      "2021-03-11 10:45:05.458700\n"
     ]
    },
    {
     "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": [
    "# Here, we call our function with high 'dps' and 'degree' values and then\n",
    "# plot the calculated transient response.  We expect to get a 'staircase'\n",
    "# response with the size of the steps decreasing as time evolves.  This block\n",
    "# of code took about 12 minutes to execute on my laptop.\n",
    "plt.figure()\n",
    "from datetime import datetime\n",
    "print(datetime.now())\n",
    "pts = 500\n",
    "tt = np.linspace(1e-10, 1e-6, pts)\n",
    "plt.plot(tt, trans_response(tt, fs, 150), 'k.', markersize = 2)\n",
    "print(datetime.now())"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}