Primer_diff_enacbe_analiticno-numericno_RLC-omega-alfa.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Primeri uporabe simbolnega računa za izračun prehodnega pojava RLC vezja \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "<b>Namig:</b> Obstajata dve verziji tega dokumenta. Ena je v obliki html datoteke (končnica html), ki je ni mogoče izvajati, druga pa ima končnico ipny (Jupyter Notebook), ki jo lahko izvajamo z Jupyter aplikacijo. To aplikacijo imate lahko naloženo na vašem računalniku in se izvaja v brskalniku, lahko jo ogledujete s spletno aplikacijo nbViewer, s spletnimi aplikacijami Binder ali Google Colab pa jo lahko tudi zaganjate in spreminjate. Več o tem si preberite v \n",
    "<a href=\"http://lbm.fe.uni-lj.si/index.php?option=com_content&view=article&id=59&Itemid=135&lang=si\">tem članku</a>.\n",
    "<br>    \n",
    "Za izvajanje tega zvezka ne potrebujete posebnega znanja programiranja v Pythonu, lahko pa poljubno spreminjate kodo in se sproti učite tudi uporabe programskega jezika. Več podobnih primerov je na Githubu na https://github.com/osnove/Dodatno/\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rešujemo enačbo\n",
    "\n",
    "$0=\\frac{\\text{d}i}{\\text{d}t}R+\\frac{1}{C}i+L\\frac{{{\\text{d}}^{\\text{2}}}i}{\\text{d}{{t}^{2}}}$.\n",
    "\n",
    "Ta problem je bolj obširno rešen in razložen na strani https://github.com/osnove/Dodatno/blob/master/Primer_diff_enacbe_analiticno-numericno_RLC.ipynb\n",
    "V tem zvezku si bomo ogledali nekoliko drugačen zapis, ki se ga pogosto uporabi pri analizi RLC vezja.\n",
    "\n",
    "\n",
    "Običajno se za analizo odziva obravnavane diferencialne enačbe le to nekoliko predela. In sicer tako, da se jo deli z $L$ in iz $\\frac{{{\\text{d}}^{\\text{2}}}i}{\\text{d}{{t}^{2}}}+\\frac{\\text{d}i}{\\text{d}t}R/L+\\frac{1}{C L}i=0$ zapiše obliko\n",
    "$\\frac{{{\\text{d}}^{\\text{2}}}i}{\\text{d}{{t}^{2}}}+2 \\alpha \\frac{\\text{d}i}{\\text{d}t}+\\omega _0^2 i=0$, kjer je očitno $\\alpha = \\frac{R}{2L}$ in $\\omega _0 = \\frac{1}{\\sqrt {C L}}$. $\\omega_0$ imenujemo krožna resonančna frekvenca, $\\alpha$ pa je faktor dušenja oz. atenuacije. \n",
    "\n",
    "Za več, si poglej na primer https://en.wikipedia.org/wiki/RLC_circuit "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Priprava potrebnih knjižnic:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from sympy import *\n",
    "from IPython.display import *\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline \n",
    "init_printing(use_latex=True)\n",
    "#import warnings\n",
    "#warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Priprava simbolnih spremenljivk, zapis differencialne enačbe in reševanje z uporabo funkcije $dsolve$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\omega_0^{2} i{\\left (t \\right )} + 2 a \\frac{d}{d t} i{\\left (t \\right )} + \\frac{d^{2}}{d t^{2}}  i{\\left (t \\right )} = 0$$"
      ],
      "text/plain": [
       "                                 2          \n",
       "       2            d           d           \n",
       "\\omega₀ ⋅i(t) + 2⋅a⋅──(i(t)) + ───(i(t)) = 0\n",
       "                    dt           2          \n",
       "                               dt           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAa4AAAAhBAMAAABzSlIQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdpndiRBE7yJUzatm\nuzJRAygPAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGrElEQVRYCd1YXWwUVRQ+O7sznf3fYhWRaBYV\no6bgJmhEo3UJYFRURsBojNoVH3hAQ9VoCMY4ifVBhLRIY4gPdk0kxWjCavzBVM2g8cEfTImKGlhY\nqYn6gC0qBCJSz7l37p2f3ZldNvWF08y95zvnO3fumXvundkCtCPvtRPEYpJGq6GtM+tGbHd6SbNu\nqFYN+vZWmbtaJdbzNrb88Lyx73uhG2WmwgUedbND9GQp2DkRfo9JbSw4NsSjVYKd6Vqwj3mSg00I\ntnttCO1giI+5PmxGaOiPmg3NzHgk2MU9mUIzBvcvDKZlmi7HeHBwiGdPiM9TovGuBvCnkGjHlQqZ\nu7dEtywrOWEADKZbLAp3IEDIzPSym9oBvfXwFbcpUA+b2UZPVJ866cYMxstuUyt6wgQ4FkxUSm6f\nDr/Ww17DbWuo7wDIhrAu8wSZmT43ZlANmaKb7Oi4APGCA/3afT7DPsJ5aSQYHZQwQKEF2Or3KdKg\nlqXKlHQJO8fN4HVeSnOEVRSrBNOu8LqUGuExYWQwWRQwqKcqWuZ3pqVB8T2Y+eRx3Ay+K9mtKf3X\n3wmuiandXaY7UC0I9DtAqgJLY4QpL8ZkMEYwTPTFr9fANbEVu29BujPxlSI4MwjwFcSMonATk8M7\nBKfV/g28Q06SV9YmDAlQiVkC5QEyhcTUX4QpEWJyGCmTLUwSON+9kqD2ZGi3OHndLlxUe+Owc6pg\nuxmTwwtsjk49a2xDUIe3iFrSebHrYHgZrVHh0S3VkvuK8nKYKZpGqGQNgBskQ8ln+hCwvOIl1GT8\nJrhf7ity20zU4DNqUGZRo+WoDRUNb0F5ZaooP8JxmC3UyAYM7BSeuJnKYWYoSrX6TPUAYxIMP3Y4\nYw12lNcRuomZLUUqMK/6Q7VqwcwyQARnyT3bYDdmRsLdjMkwO3Zm4L2KDLLsuEO2d+0ftfolilTu\nhYRMXz0BB+62fc8vr7keZaSm1CJm/5BJ3jHcXg4zVSabR9QLRz/Si9I0WzXhNYl6zfTnBl8vfdVp\nV03ARTAfztdvpd3HltNmEvwSr6sB6HheaUIC430SecyAIVdR5IqQLkrO+sxxS4DeCqRyAmifXLKv\nC8r4hzKGl8PEp++XmedCfLElrXPwtPlOog5z4hEErA7VfwDXSMjawzc+VExAgjC5bSbB2dQAdOM1\njCdYnoBb1G8NLKUT0qR117CMJVz9RZclALKiSLZlJP9BRe2DJwlSXg5ThmsW2kkeuAabq0xSmaz+\nhvajEK17C50ULC9iEZlL4vHk03APdNBdyW0zyWmH96F6JV5sGmQXkiALbSqXNCgk8up/wpCLRZYC\n9JCF8nJElnE8ZxtHDFRoU7mECskjPK+BvL1lpO85iNYQcLe08jLWymjAcoS3pMNWlg+iksl7zPqk\nB0pwyNgudabgeq0nJc+QaLKmrYm8YifJcNi22t2AF2LRMEP6mH+v4HqZ6OJuGXQC1Bk5UIqgffzv\n2/VveZ29f9Qa46uLZtCDBXiKtXXNull5n83eX17ruIAir4ECWR7k5lWdc5nSUeLY12pHX/JZ7P3l\nteKPoRhWWcLCI5FWga2+soBJBXHsqIv/cC1TZBAP0EaS2GD6zPZ56LVeKqDIa3lOWLDke+AXhiJ5\nx+jWRna4Eer2eei1KkV4QSmzT1Hs6r82o3iukqiYsH4KIiWG2EZlmqeJ42HVgsg9LPK6qcai1CWj\nBgxYcB4fQ9K8Qw7T424qOEMjmwNa9GgR2RO+iI5jZFDh53NwZx7tupm7Y0Xe+9uFfkMjTK9VFCyK\na59YsIBm+axBBkMBpQCHOpeUCAEs4Z2/jZp+SyOM3wcwbLC8ekuor/GRopNkeJUtZJbmwKWlBATZ\n18uvVpDrRYx4LVvST4Oz5K0l4Bvchuzz+00A+tBch8vCf/W49hd/cxV5XtjZckQobfS7ZIzIC99J\nuJMhZWBef0tv839iSGqdkjSxyE7G2Bf6bbANYI+PotO5EcM62coLFfMnCfuHFGcEtkpNukReAzk0\nLcUrVqCXTdwmrJXEM1boJ5g2uQliFYBRFYef4x9i2II43RLzwtfxCtP2i96GZ9C5fjOIvOKnAIYM\nHGPcwANZvdwezcU8g/EZ1aR2rwVaAWDz16jXbRx13ij7bwLmBV2dmPk0isgLVux/p4Tj6t/jx/oi\n2vLTJj18JB3zayyU13SLzMse+EX4bbpv8SkfULGCBv4/8tK8d9MP/jE36Pbt2tMmi9wcFN+/s27n\nBVHbtndMTU22HRwQqI4xBxb4WSabKJ+UefZk9R8MJK0zOy39oQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$i{\\left (t \\right )} = C_{1} e^{t \\left(- a - \\sqrt{- \\omega_0^{2} + a^{2}}\\right)} + C_{2} e^{t \\left(- a + \\sqrt{- \\omega_0^{2} + a^{2}}\\right)}$$"
      ],
      "text/plain": [
       "             ⎛        _________________⎞         ⎛        _________________⎞\n",
       "             ⎜       ╱          2    2 ⎟         ⎜       ╱          2    2 ⎟\n",
       "           t⋅⎝-a - ╲╱  - \\omega₀  + a  ⎠       t⋅⎝-a + ╲╱  - \\omega₀  + a  ⎠\n",
       "i(t) = C₁⋅ℯ                              + C₂⋅ℯ                             "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var('R C L t  Ug i0 C1 C2') # priprava spremenljivk\n",
    "om=symbols('\\omega_0')\n",
    "alfa=symbols('a')\n",
    "i = Function(\"i\")(t) # funkcija toka\n",
    "de = Eq(i.diff(t,2)+2*alfa*i.diff(t)+(om*om)*i) # zapis diferencialne enačbe\n",
    "display(de) # izpis enačbe\n",
    "des = dsolve(de,i) # rešitev enačbe\n",
    "display(des) # izpis rešitve\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rešitev vsebuje konstanti $C_1$ in $C_2$, ki ju moramo izraziti z začetnimi pogoji. \n",
    "Konstanto $C_1$ bomo zamenjali z začetnim pogojem za tok, t.j. $i(t=0)=0$.\n",
    "\n",
    "To naredimo tako, da izrazimo enačbo za čas $t=0$ in jo rešimo za $C_1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAARBAMAAAD+jZX9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$i_{0} = C_{1} + C_{2}$$"
      ],
      "text/plain": [
       "i₀ = C₁ + C₂"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAVBAMAAAA0iOSIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left [ - C_{2} + i_{0}\\right ]$$"
      ],
      "text/plain": [
       "[-C₂ + i₀]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t0=des.args[1].subs({'t':0})\n",
    "eq_init = Eq(i0, t0)  # izrazimo enačbo pri t=0\n",
    "display(eq_init)  # prikažemo i0\n",
    "\n",
    "init1_solve=solve(eq_init, C1) # rešimo enačbo za C1\n",
    "\n",
    "display(init1_solve)  # prikažemo rešitev"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfwAAAAhBAMAAADOqwRlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$i{\\left (t \\right )} = C_{2} e^{t \\left(- a + \\sqrt{- \\omega_0^{2} + a^{2}}\\right)} + \\left(- C_{2} + i_{0}\\right) e^{t \\left(- a - \\sqrt{- \\omega_0^{2} + a^{2}}\\right)}$$"
      ],
      "text/plain": [
       "             ⎛        _________________⎞                 ⎛        ____________\n",
       "             ⎜       ╱          2    2 ⎟                 ⎜       ╱          2 \n",
       "           t⋅⎝-a + ╲╱  - \\omega₀  + a  ⎠               t⋅⎝-a - ╲╱  - \\omega₀  \n",
       "i(t) = C₂⋅ℯ                              + (-C₂ + i₀)⋅ℯ                       \n",
       "\n",
       "_____⎞\n",
       "   2 ⎟\n",
       "+ a  ⎠\n",
       "      "
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C1_solve = des.subs(C1, init1_solve[0]) # v rešitev vstavimo C1 in dobimo\n",
    "C1_solve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlwAAAAcBAMAAABG9XzGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nFfgP+OM+SrALzyEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$i{\\left (t \\right )} = C_{2} e^{- a t} e^{t \\sqrt{- \\omega_0^{2} + a^{2}}} - C_{2} e^{- a t} e^{- t \\sqrt{- \\omega_0^{2} + a^{2}}} + i_{0} e^{- a t} e^{- t \\sqrt{- \\omega_0^{2} + a^{2}}}$$"
      ],
      "text/plain": [
       "                      _________________                   _________________   \n",
       "                     ╱          2    2                   ╱          2    2    \n",
       "           -a⋅t  t⋅╲╱  - \\omega₀  + a         -a⋅t  -t⋅╲╱  - \\omega₀  + a     \n",
       "i(t) = C₂⋅ℯ    ⋅ℯ                       - C₂⋅ℯ    ⋅ℯ                        + \n",
       "\n",
       "                _________________\n",
       "               ╱          2    2 \n",
       "    -a⋅t  -t⋅╲╱  - \\omega₀  + a  \n",
       "i₀⋅ℯ    ⋅ℯ                       "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(C1_solve)  # drugačen izpis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vstavimo začetni pogoj $i(t=0)=i_0=0$ in za krajši zapis nadomestimo $R_1=\\sqrt {R^2-\\frac{4 l}{C}}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAb8AAAAhBAMAAAC7JrjGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdpndiRBE7yJUzatm\nuzJRAygPAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAG3ElEQVRYCc1ZXYwTVRQ+nXa602472yWriERT\nRY2aBZooCf4tJYDRrMrwYzRE3YIPPizG1UQlGMMk4IMI2UU3xvjg1kQDxgcbI4pZYwqGRDGaEhU1\nsNCwJuoD0FUhEAnrOfevd2Y7/Vn2wbPpvec75zt37rl/c9sFuBxpd6YdvbfpyOaZ/iYvo3uiqX3+\nJpvH25sdm3a3+UZ9TOt9n6FV2J4LjjgzWVcq0VJwrMfzqQd5gF33EZOT8LSH3jrorxNyvI6PuT5v\nROD+aCGYlywH+5infagBoYF7cbDfbjhB48HBuifs6sirn/LCqcjOTLW1YEnUScK7enf15vR2GUw2\nN7rf6YE+3bN6Y1014K++iNZgvS5u9zQ1YFZ0zGAsr5sC9TpdtDwttEGf3giHb+um1vQ9AB1OcMhN\nHpdrD+iYQfOsbqqpx12AOiwjp0dZ8PtU2Fenizp7qk5T8sZUs7SYeanxOpnD2lA2Bu9QMEjBKYll\n/M60MqxTGleOUJXmOpYEw0MKtqjQAusNjjF8DS8galLxGfxEwSAFF1ik4HeWlOEWpTHFKFOl3Ay2\nZ8k2DbGWvVcGrYer99/naWWNRDZm+i1EnCwaWILE5PBByQmqt935EGg9NLu7XKSqDMyMDPwTIFGA\nFRHC5GZMBiOKLbnN1vEhgMOKbPbY3r3ygHTRshyHjyapM5QgY3J4teQE1h9gTEp515TPOAhK0hAp\nSi0NYGfik38TJjcxOQzlyYZisdJlZVMFHTB3K6aRrp4isRyaM9K1Ax5Te48S1JhfSU5gjYMWLirv\n9fwUYQm+idaw9FhFs6j2HrkFE7WE7MgcRl4vQxrXTyGFEjw1huJ25EIFocLsPEAoJT1vwX5MkWT+\n2M9jY0XGZLjeGcUJ0QGeoE3P+AXOwVwwxsZeHjsGoc3I6MR5Y56Ym0hhiijCTUwu7IyahWdVluHQ\nkLDXqMxrRr+wOI28c00X3lW0Pjd50OHIWntJG1y4FhbAVdb9bIfSDGrMQypcKauOjha3KRQqPApx\nHCou5nk49jCqJfy8srKsrZJQ2SiH3G3DLnrILZmIEnksbgegA9fEi1ed2/fsKyC2rIg8Ltfhjv5R\nAmhzz2yUwPwXcNak9J+8Z302DnHClKDGlKMsqTjxzzowrC38VBaSWeXeZJ8rIqAEoa8AiRQpJNEv\n5x3pgjz+oZBbMFHDhcWkG8tYBeAAhzXKxxehcaGrPBu+p6UuJdq9Sx0rxCIyl/hz7S/BI9DmIKQE\nNaYKl2Nj/oAs47yIJG4ZjLSCG77pKiIokQFZYWqSy+70ZwVzAJ4nRG7BRE2GD5Cex/GtBqFFl93k\noY2nSY01Rt7BtFjxiroVwmUElKAmaoXvEsY49TBKfakKW2NViFqJkPUXDHusYGWghyzMrVxihUfz\naAlnAeJFVGpJ5AJZT3pdg14oUfJsnEZDE5xBF6GhmVBVUyUTXDmEVjvtYVkVD0TA/Scc33bCGdxE\nVO4mjaTDBXNWCowsQP+JoymYem9AEslghsonqABY23kjq9tyrPIX0YnXfSaxB73W6lcZkaDF3mNm\nmdHMpbP4KL3gjZLomTlpqYpa7EGvdRzvQrgm2MSNoIsviHm3kdylU1emqijaA78xFEpXjbq2e4+O\nUBenqNdqZCUWCUYmpAHrJ8s29+MZX0vim12fWZyiXusNAK/i3mNX0lvRNfVqK/hLykwxl486MFiE\nK7mZ7WrB0KqRggYC1ep+FwmG8f1CsrZ3L1gXIZRjqEpjUBYxPKubEOyh05ECttbwbQF2xRe0jk1m\nD2xh68UxwMjAic7lOU5b7mMLGHZr273WTg5xubzIl0vbWbKYuIvmQXKi617uj2R57S8X+w21MF04\nYMRhCZrnUPcdYdWYJaTGyh056xJUx665TKqt6Fr1qgxyBtnovoM3lsFyR0Fxm8pEsb0Ku/N/iIfn\nEOZWAUcs0Rp7cKGLkRFIOJjgP6oRu6TUlhXt10CRIH8DZuEgJZhVDZ5SWuvKPgwxL0TYlT2Sj7mQ\nCOrxYAqpK/ATydDNJyae1S/qaVTsKyGPEwlaEwgjDhbL2DtriHvr/azGGYGlUUZXtLKDvR3aC6vw\nTUV51JLYRYBheva4A4fAvFlwrFrc5mxulSYShJEixGgQ7UV0nK+WDFlXA5rVePcOFyGawaP8WBFf\n9oGNrT76cQ7btX7C+93STtRmUGSC5vxR9lPOThzJrs6gsZ7Wg9klByO3Nop+Df5oRGndLxMUkVbB\ndltvpH7EAeHurk8D6/hpfolpwGvNvdFL33n6JG2GGZWky5qzSg1abZucrDSgXL57C/5TYabF5Jkl\nZ3zkZrqj025vB4v8etrx/9/A/wBprsHwCRNqowAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$i{\\left (t \\right )} = - C_{2} e^{t \\left(- a - \\sqrt{- \\omega_0^{2} + a^{2}}\\right)} + C_{2} e^{t \\left(- a + \\sqrt{- \\omega_0^{2} + a^{2}}\\right)}$$"
      ],
      "text/plain": [
       "               ⎛        _________________⎞         ⎛        _________________⎞\n",
       "               ⎜       ╱          2    2 ⎟         ⎜       ╱          2    2 ⎟\n",
       "             t⋅⎝-a - ╲╱  - \\omega₀  + a  ⎠       t⋅⎝-a + ╲╱  - \\omega₀  + a  ⎠\n",
       "i(t) = - C₂⋅ℯ                              + C₂⋅ℯ                             "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "zacetni_pogoji = {'i0': 0}\n",
    "i2=C1_solve.subs(zacetni_pogoji)\n",
    "R_1=symbols('R_1')\n",
    "i2b=i2.subs(sqrt(R**2-4*L/C),R_1)\n",
    "display(i2b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Poiščemo še rešitev za $C_2$ iz odvoda toka. Ker je odvod toka krat induktivnost enaka napetosti na tuljavi $u_L=L \\frac{di}{dt}$, je torej potrebno pogledati, kolikšna je ta napetost ob preklopu. Ker se napetost na kondenzatorju ob preklopu ne more spremeniti, bo ta ob preklopu enaka nič, prav tako bo ob preklopu enaka nič tudi napetost na uporu, saj je tok ob preklopu enak nič. Torej bo vsa napetost ob preklopu na tuljavi in bo kar enaka napetosti vira $u_L(t=0)=U_g$. Iz tega sledi, da bo $u_L(t=0)=L \\frac{di(t=0)}{dt}=U_g$ oziroma $\\frac{di(t=0)}{dt}=U_g/L$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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LMiRlpq+P8UrMLttUz7GBMfUxPgWO1pNLfbR45Z0uU/xDicHlfzsZuoF4oNJyAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$C_{2} \\left(- \\left(a - \\sqrt{- \\omega_0^{2} + a^{2}}\\right) e^{t \\left(a + \\sqrt{- \\omega_0^{2} + a^{2}}\\right)} + \\left(a + \\sqrt{- \\omega_0^{2} + a^{2}}\\right) e^{t \\left(a - \\sqrt{- \\omega_0^{2} + a^{2}}\\right)}\\right) e^{- 2 a t}$$"
      ],
      "text/plain": [
       "   ⎛                                ⎛       _________________⎞                \n",
       "   ⎜  ⎛       _________________⎞    ⎜      ╱          2    2 ⎟   ⎛       _____\n",
       "   ⎜  ⎜      ╱          2    2 ⎟  t⋅⎝a + ╲╱  - \\omega₀  + a  ⎠   ⎜      ╱     \n",
       "C₂⋅⎝- ⎝a - ╲╱  - \\omega₀  + a  ⎠⋅ℯ                             + ⎝a + ╲╱  - \\o\n",
       "\n",
       "                 ⎛       _________________⎞⎞        \n",
       "____________⎞    ⎜      ╱          2    2 ⎟⎟        \n",
       "     2    2 ⎟  t⋅⎝a - ╲╱  - \\omega₀  + a  ⎠⎟  -2⋅a⋅t\n",
       "mega₀  + a  ⎠⋅ℯ                            ⎠⋅ℯ      "
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C2=symbols('C2')\n",
    "C2_solve=i2b.rhs.diff(t) # izračunamo odvod\n",
    "simplify(C2_solve) # izpišemo odvod"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zgornja enačba izraža odvod toka po času. Sedaj jo izrazimo ob času $t=0$ in $C_2$ izrazimo z odvodom pri t=0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAh8AAAAyBAMAAAAO3qQjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwaxR/BT6P0l3p/pM+gqCAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$di0dt = - C_{2} \\left(- a - \\sqrt{- \\omega_0^{2} + a^{2}}\\right) + C_{2} \\left(- a + \\sqrt{- \\omega_0^{2} + a^{2}}\\right)$$"
      ],
      "text/plain": [
       "             ⎛        _________________⎞      ⎛        _________________⎞\n",
       "             ⎜       ╱          2    2 ⎟      ⎜       ╱          2    2 ⎟\n",
       "di0dt = - C₂⋅⎝-a - ╲╱  - \\omega₀  + a  ⎠ + C₂⋅⎝-a + ╲╱  - \\omega₀  + a  ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIcAAAA/BAMAAAA4UMYhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARN0yEM3vmVSJIna7\nZqs/Q23HAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADp0lEQVRYCe2YS2gTURSG/5hpHtNJ2m50JQlR\nqC7UaBdFERxFBRFMVyq60JVvMYjFF2iWogsHRbAImo0IWmxAQUWEgLpuEUEU1ICii0qpGLBadTxn\n5s6jmTvBWAUXHkju+f975us9Z0IzBBnTzGIasc00S8h0deWnwcDirgxBpkOwro39hwRm+NdnEt1Y\nRHsRyUvH+G+zCommJ+koYYaBzVhV5qtJ2UkA1RSS0ak+lUWyyteROstrMJpCznG9VkOizgmpfl6D\n0RSyies7iohMcEJqPq/BCIeoR29NYP1xFAjyHawiH76NBgnkhENmlrU6jJkYMhCdhKUS41JGOETZ\nj3ifWuq3IF8sRUaLkHidbq+CSRQMRCYtxfdbGqHtxKoodPJ9ocEmJmxFhjRCIXSA10BbDVoVqbqt\nMlBahJTwOFEulIzUdppFwVIXcLo1SFsl1Z1ERh3BPKzN22pMqbQGUUbP37wF7QGw7v4b2GrwpZQR\nfovl5XI3dLDycrn7HxKcy783E/O3g7r799oJjrwF52+2M/zrcw4dbLTW0My6y28bHL+Ut6Pl/TWU\nF3GwwfFLOWS5v4Tzjxhq5JKrDIzpvCuHvOEtfzyVQuKIZ7nKhVzrf+hepVbd1E2sL3dX2UlHp/qJ\nMwei1nDF/VZJu5l7VXSum3pJSp8KSeuIbXe2rzqJtw7qIp/teZQlsyydk8TqSH9lg+OIvfje1Vq0\nbMszPhdYorN0IOks0lZ/5Cl9vDMlBnt6dduYAlEfWaYDITFj3C5DsiQSWtL0UufggGkKz4Fcu3QD\nuIllbPsgKyqibpVYeTlFL3XSZwhIZA+GdfVhzyHe8kG6Sa5mj7713ChydtGVgICsHaHPcMw0rdN7\nkDjVR3dyeZZea3IccyKd0IAX7HIczOV25HILKTup+x7gPMgz2jo1TG/tTlucA72AgYF7JKwQJ9kH\n5aNjee0kDFynZ9URGrC7CcSBu1CNZCVSFa4NUb4gZXViue5JlgK3CZAFnotyXrR89DPW6LE8nA+R\ngExA63MfVhyI2n34BAFSdEb/Bzy5+93Z9/sxVMYugRbtbMWToiEst502+lfWR+Y3UF9eKMNb2ndU\nsELHAmEKyIZ7y0bzTp1zEkefLLfpTu6tBNkrlIB4e5Q1QoaKs6bs24LasZ7MSc2WbDdCtPEtsipv\nsJLdwEkiP2qSMrrFPLCwaDwJFpVlpQN3dJktvADkVZPisK0AJKywmc+QP/GDw8pp/+DQpf8EYmRW\n+BIMdqAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\frac{di0dt}{2 \\sqrt{- \\omega_0^{2} + a^{2}}}\\right ]$$"
      ],
      "text/plain": [
       "⎡        di0dt         ⎤\n",
       "⎢──────────────────────⎥\n",
       "⎢     _________________⎥\n",
       "⎢    ╱          2    2 ⎥\n",
       "⎣2⋅╲╱  - \\omega₀  + a  ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "di0dt=symbols('di0dt')\n",
    "t0=C2_solve.subs({'t':0})\n",
    "eq_init2 = Eq(di0dt, t0)  # izrazimo enačbo pri t=0\n",
    "display(eq_init2)  # prikažemo di/dt(t=0)\n",
    "\n",
    "init2_solve=solve(eq_init2, C2) # rešimo enačbo za C2\n",
    "\n",
    "display(simplify(init2_solve))  # prikažemo rešitev C2="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Končno obliko zapisa enačbe za $i(t)$ dobimo z nadomestitvijo konstante $C_2$ z odvodom toka  $\\frac{di(t=0)}{dt}$, tega pa z  $U_g/L$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAW8AAAA3BAMAAAAxuNYJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$i{\\left (t \\right )} = \\frac{Ug e^{- a t}}{L \\sqrt{- \\omega_0^{2} + a^{2}}} \\sinh{\\left (t \\sqrt{- \\omega_0^{2} + a^{2}} \\right )}$$"
      ],
      "text/plain": [
       "                    ⎛     _________________⎞\n",
       "           -a⋅t     ⎜    ╱          2    2 ⎟\n",
       "       Ug⋅ℯ    ⋅sinh⎝t⋅╲╱  - \\omega₀  + a  ⎠\n",
       "i(t) = ─────────────────────────────────────\n",
       "                    _________________       \n",
       "                   ╱          2    2        \n",
       "               L⋅╲╱  - \\omega₀  + a         "
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C3_solve = i2b.subs(C2, init2_solve[0]) # v rešitev vstavimo C2 in dobimo\n",
    "C3_solve\n",
    "i3=C3_solve.subs(di0dt,Ug/L) # Zamenjamo di/dt(0) z Ug/L\n",
    "simplify(i3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Priprava za izračune in izris\n",
    "\n",
    "Pripravimo podatke za izračune in izris. Za to uporabimo funkcijo $lambdify$, ki pretvori zapis enačbe iz simbolnega zapisa v funkcijo za numerične izračune. <a name='pripravi za izračune' />"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R/(2*L)\n",
      "0.999500998253119\n",
      "{C: 0.1, L: 10.02, R: 20.01, Ug: 10, 'a': 0.998502994011976, \\omega_0: 0.999001497504367}\n",
      "Eq(i(t), 15.8133650950391*I*exp(t*(-0.998502994011976 - 0.031555712083352*I)) - 15.8133650950391*I*exp(t*(-0.998502994011976 + 0.031555712083352*I)))\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "'kritično dušenje'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tok pri 0s: 0.000997008+0j A\n"
     ]
    }
   ],
   "source": [
    "podatki = {C:0.1, L: 10.02, R: 20.01,  Ug: 10}  # podam podatke v obliki \"slovarja\" (Dictionary), https://www.w3schools.com/python/python_dictionaries.asp\n",
    "#podatki = {C:1e-5, L: 5e-3, R: 5, Ug: 2}  # podam podatke v obliki \"slovarja\" (Dictionary), https://www.w3schools.com/python/python_dictionaries.asp\n",
    "alpha=R/(2*L)\n",
    "print(alpha)\n",
    "omega=1/sqrt(L*C)\n",
    "alpha_subs=alpha.subs(podatki)  # Izračunam alfa\n",
    "zeta=alpha/omega\n",
    "zeta_subs=zeta.subs(podatki)\n",
    "podatki[\"a\"]= alpha_subs  # Dodam izračunano vrednost R1 v slovar\n",
    "podatki[om]= omega.subs(podatki)  # Dodam izračunano vrednost R1 v slovar\n",
    "print(zeta_subs)\n",
    "\n",
    "print((podatki))\n",
    "\n",
    "ifin=i3.subs(podatki)\n",
    "print(ifin)\n",
    "s='nadkritično dušenje'\n",
    "if zeta_subs<1: \n",
    "    s='podkritično dušenje'\n",
    "if zeta_subs > 0.9 and zeta_subs<1.1: s='kritično dušenje'\n",
    "display(s) # Izpiše slovar\n",
    "\n",
    "tok = lambdify(t, i3.args[1].subs(podatki), 'numpy') # tok kot funkcija časa\n",
    "\n",
    "print('Tok pri 0s: {:g} A'.format(tok(1e-3))) # preverim tok za eno vrednost t\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sledi izris rezultatov. Za primerjavo rišemo primer kritičnega dušenja"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\ProgramData\\Anaconda3\\lib\\site-packages\\numpy\\core\\numeric.py:531: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  return array(a, dtype, copy=False, order=order)\n"
     ]
    },
    {
     "data": {
      "image/png": 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iud1uoksLIU5beTdGr/t/T2RFKd8+8pTVUZRSjVTXU1JHboq/AXwvIveLyP3A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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x237f71bf668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cas = np.linspace(0, 10, 1000) # spremeni po potrebi za drugačen izris\n",
    "\n",
    "def slika():\n",
    "    plt.plot(cas, (tok(cas)), 'C0',label=s)\n",
    "    #ianalit=5*np.exp(-cas)*np.sin(2*cas)\n",
    "    plt.plot(cas, [f(t,20,10,10) for t in cas], 'C3',label='kritično dušenje') # za primerjavo kritičnega odziva\n",
    "    \n",
    "    plt.xlabel('Čas [s]')\n",
    "    plt.ylabel('Tok [A] ')\n",
    "    plt.legend()\n",
    "    plt.show()\n",
    "slika()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Opravi naslednje analize:\n",
    "1. Spreminjaj upornost upora in opazuj spremembe.\n",
    "2. Spreminjaj kapacitivnost kondenzatorja in opazuj spremembe.\n",
    "3. Spreminjaj induktivnost tuljave in opazuj spremembe.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kratka razlaga\n",
    "Opazovali smo odziv zaporedne vezave RLC vezja na priklop na enosmerni napetosti vir. \n",
    "Pred priklopom na vezje v vezju ni bilo toka, po izteku prehodnega pojava pa tudi ne, saj kondenzator \"blokira\" prehod toka. \n",
    "Na njem je po koncu prehodnega pojava celotna napetost vira. Odziv vezja je lahko precej različen, odvisen je od vrednosti elementov R, L in C. Te smo v konkretnem pprimeru \"skrili\" znotraj parametrov $\\alpha = \\frac{R}{2L}$ in $\\omega _0 = \\frac{1}{\\sqrt {C L}}$.\n",
    "\n",
    "Poleg tega tvorimo še en parameter (zeta), ki je $\\zeta = \\frac{\\alpha}{\\omega_0}$. Z vstavitvijo R, L in C pa je $ \\zeta = \\frac{R}{2} \\sqrt\\frac{C}{L}$. Ta faktor imenujemo faktor dušenja (damping factor), od njegove vrednosti je odvisna oblika odziva.\n",
    "\n",
    "Za $ \\zeta < 1$ je odziv vezja podkrtično dušenje, za $ \\zeta > 1$ je dušenje nadkritično, za $ \\zeta =1$ pa kritično.\n",
    "Spreminjate lahko vrednosti parametrov in opazujete odziv vezja.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dodatni izračuni - napetosti na elementih vezja\n",
    "Ko imamo rešitev za tok, lahko enostavno dobimo še zapise za napetosti na elementih vezja. \n",
    "\n",
    "Napetost na uporu je kar $u_R(t) = R i(t)$, napetost na tuljavi $u_L(t) = L \\frac{di}{dt}$ in na kondenzatorju $u_C(t) = 1 / C \\int {i dt}$. \n",
    "\n",
    "Ob integraciji je potrebno upoštevati še dodatno konstanto, ki jo je potrebno določiti iz začetnih pogojev. V konkretnem primeru bi morali določiti konstanto iz pogoja, da mora biti $u_C(t=0) = 0$. Temu dodatnemu delu se lahko izognemo tako, da namesto z integracijo toka  napetost na kondenzatorju določimo iz zveze $u_C(t) = Ug - u_L(t) - u_R(t)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcoAAABLCAMAAADH2TcnAAAANlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABHL6OuAAAAEXRSTlMAdrurVDIi\nzd1EmYkQ72ZQQBUwexcAAAAJcEhZcwAADsQAAA7EAZUrDhsAAArYSURBVHgB7V3nuqysDhYLFtDv\neP83eyiGJtWyhtnP+GMNUgJvIiGEspom/YxzOk9Ojo7m5Ho4z5uVvkk7yAazUjyVsXTpgmSLEjpc\nlP2hzHgsQ1tQbQWASJFs+vUZXnTX+vaw33qaZm4LpFOStQpA65bfZFqSOUJ2Lvp+NKF20OFroa6/\nVi5Rqg5A856v67oxASkzecqv0qQ43O9Tw2QSfCxcCaAum0HDfk0vuhzbsmu0S0a0WI9IJNUg077R\nLWsBhPdcdONDnfIiO3Gk+rVp6G5IzA4agu4jROxCBW/VAGozlU5/u1OOUrHu12ynJaITGOGBiTPw\nGIKmFyv3U64N0LAjf0Od2DFT5E4x/Xr0m/4aHZroT62S9KKrlCFT0FOuDnKJeN7zAZ2axKi9AagN\nf89G+3MlbhRxgodyK5v/KBpLXAaLtm59Q7ESdEcUxduBfEC+Jr0BqM8aLbubqgmTaSSc335m0mVD\nJGLZ0nNfHqRwNz61IbRRMlJ80zS1oIlKvSvJEkCeSt8BtCd0FwdNd09zyrgxSta3Pn1Op6EZYh8L\nOc9/D63FJxgj8x0oWUNDNU1D0CgDay6sfEDQJIPyO4C6XWsnozIruGV1XauI+3JIajoLpWkWJhef\n4dLDCKgEpahiRodyWu5ABHxTNE1Bb2dCimJpIB8QNKlpXgY070o5BdG0sS4TLGUmzMeQDKIcWniY\nXuXWMeKa0opt0Hp0IiS7spnas2IbixbiPCpaOMmV/2EyVjTNVlw0ukwSEM4DZDXpfUB70vChWn8B\nEuMXd9M+ce417bqPPgXKkpikKB8MfemU+5zajTrTFEzJoTA8fQkxVcLHQIQbTHpjnD26gJ/mlqVg\nnwYEvfJ9QG1Sw26Jjktg0omCHXxB0jJpfTlWzIbKmX8M9kN34a/lvc995m5YpnbmFXLxaBEB37w0\nRdd3SXnenwUETeI2x7uAUEJQzOwEUXlQ86gRfC2L07F0/nnZxJDMh7DTwxKZT+YsymYRPmLvWty4\nLk23Ms5QrlT0og3wzUvTW/upOU8Dgiaxil4GNCRt2BVE5UHNo5R9K765QCYZnafigATemeR7X0eG\nHE2DufplJvDxGHyDKP3rU+86VYeeBWQ06W1Ae0JSODpUsoVA8BjRONM5q3CiLs1OEeL2VmLJWPZK\nPg6Lx9frIa3JXAh6GJDZpJcBjQn92auPVPHECoB10mwweWj6BSE+X2goQchSu6vKY9EIvDCm9iYn\nfNnssdKXA+JyV7nOgBSeygEt0KsAsvO7JAbTCUxgxXTC2ItXplhmrvkGs7Milcmpxf86rofX2p/M\nY20LNpyPjVRmQyIZT4AUntoBbdKuCmIb4w6CwyxjxWGo7MU0lH0geOc9E1kdUVsowQqNhD7RNiNr\nOphZ9QmQwlM9oCExs1zjs5VeyIsxEsOkgfdHNr3ATbsjRIglyYZkdo1DMhfX6H1yPTwNviQr7gRI\n4akeEE3YPXvc16NGFpioD6IXI6Z2vV3qQeFYEki9eFzY/iIuII2nfkB71LLDCUkTMEpBv26CXMte\nvYPsA7t0/CJIxGZvcXQBaTz1A5qiJmyfmIvIkYSNlDAh2LhsKVe7k1SmzmLj1S1qCVElkvVSVyJj\n4wLSeOoHNMJo5wWJUi6EkRuluFOzdOH/FCucwiFAF5AxkHffIf7V34JKHUAaT1M9oM6rCIGxJDGt\nZEu/bcd8rPqZO0QW4QNHbFpppug8VYccQBpPUzugBcwT1O5727FVIlMldpBaxP2pbP5YRPsTmb8E\nj9HvZA8cTOm13BFa8Az8XMoQn8AUkPt41q/CQ9RoCNuxWsOmHQtF2bGZOM11XH9cUMwj5VmSMZv1\nVXiQslHR0ZlGS5Rlk3q2vERi21ZNNtUQ7hOD+Vfh2ZQoj5XHwbR0pqhRVIMw7rUhJcp71P+49KY8\nd1KEeILZPm/IlPC2/3FjH6/unxJlD/6ceW/ZmlRrz6bX6Kzzcc7+OcF/U5SED5G0a619HftPlH/+\neV2ucIZeOcqVR3tw3M01LjpO+lH+ncs1f7bgIqCsq/gxx5TPNutO7exsuOyHcjqp1+sE0X3/L0b8\n1rHyzxS24bgK9jNtulWrBvS/Q5SzVKW9PZG0eqUu9M+EXFF+NTDWK0X7F9k5CTdZtfJcTQX71Tj9\njf+nRCnHSjqsUpQbE+Wsl/6nn9nj/wZqjJWiJPwRTqylRYYz6w9Eybz4LzyZrH6jV34MkLJgveDj\nq5neIoWRw0etxxdE+TlAvfL2+GTA9lr5oh+MS17JQwnpzHvAHqw77U6/UNnnAGnHna/ZrT3N9GW5\nF5fe7MNX2eRadk5Nrwo+pwEfBKRXRnwN7ey5iS/LvTi9lSREh+89wtkroGWCD9V5I/6DgPR6pa/9\nao+BLzEWh5dlBFWDun0fxSHMUwlsHI85JcqIjXkw8kVZJvhAlb7oLwC0mGtaJwwomnrKriIwN2bU\nScVGLaSpDEcg7yashngOzLqk5HuZ4P00fLHfACiuQnu1x8CHLxwn7ZT1OFAyhEbc2D1ZBvHBPWUZ\nX/zPF7xRRyL4DYDiNmrqHEKIAdLh0B37EVBor23iBpuDOrZXa1hsdA5xEnyokSXx3wAofr4ndQ4h\nxI1OaES4gESdjHayp66Vktm5bhvsnawxUZ4F79R66fUbACXO98TPISS4Mh3HEEJampj7NA1a4DXA\n/FgfbodhsI5penolXgiSJ5A8gjco3wxWDYh1O2ut2cUaP4fg5rbfYQtf6EqZ0JmcHiSMuYNiFW49\nm7LbK2d2kQQ7acVtXY/g7bI33uoGlBoMuxv+dNgJrC1ZzkZ9r4vp7DUZrLfehrqtI0p5ibTwI/gE\nb9K+Fa4b0BZSfgdmdN1HoK7ksIbKgd3ucTwww3B2J8iFGTHh3OwhkpXzLv7LBbrjvBFQf/63ckAk\nNE84OJG+QiTEsk2JDL4WLpi+b+EcX+hAifD6DqI034RLyPlGIKdXiuOsNGQmhxpYGl87oDa1tgxb\nf0qBi2taMO9V6s4Nac0c2xXYxswASSFExMdLzERJmGRP/4/AFiUVn6M4Phcg+UR09YCmuNXDd8Ke\nlFwOY+R1EoibVHCQGIbGScpwC+175/6+get1yq96WfmWBrcNrii53Nux4ZffvfVUD4hGl7g4W8gl\nuwdPC3vkZOyw4Al8NJvUhK4HR8mArKRb2ZUSC/+PGFhIcQWb9shki1JcnrcxmyT0cSjS1wP1A0pf\nwjGbB7uyWcFv72TPyDbWsmDXib9QWvh1w/dk0XZnl7ygdeWDprwqFu6sBAqOKCk70zngjl9j+NZT\nP6AuNVQyDQc38zzGJXHZW+KeLKhMilJIFaLYryNKI+UjwSoApW6wY5xZstcKc9nIL3ublX0bLyUV\nrPvBxd3pcYovpNYAyDoWG8AY8tUEsudEs/lI8p4soOM1eyCxlt8KACH3c/fx5nkNy8ZfcLP6KrTi\nxGREX/lqpVXzUgGg45xInCXBJap4sVhqyQyHuQgcZ3qM8ofSPg4ouCRsM+SSDWuTcN7Ev5Vw4r76\n9eOAMv+DiJoRfjW3/+nGO6e2glilYyyY/Ev4PAeQ6w8LNQmBoyaU4Rf/YQ7o5aZUQ9YXXWKpun/p\naQ4U9LU+t/+ma/3leJ4D4nrIXLL5/4E2l+Iv33McKJOO69B+rh0/Snc5MGdeJn7UM5Rlv9u6X/kC\nDkz6aHNWqfnFZd2sBvwyBThATUn+H22WizXUy+BwAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$R \\left(- \\frac{Ug e^{t \\left(- a - \\sqrt{- \\omega_0^{2} + a^{2}}\\right)}}{2 L \\sqrt{- \\omega_0^{2} + a^{2}}} + \\frac{Ug e^{t \\left(- a + \\sqrt{- \\omega_0^{2} + a^{2}}\\right)}}{2 L \\sqrt{- \\omega_0^{2} + a^{2}}}\\right)$$"
      ],
      "text/plain": [
       "  ⎛        ⎛        _________________⎞         ⎛        _________________⎞⎞\n",
       "  ⎜        ⎜       ╱          2    2 ⎟         ⎜       ╱          2    2 ⎟⎟\n",
       "  ⎜      t⋅⎝-a - ╲╱  - \\omega₀  + a  ⎠       t⋅⎝-a + ╲╱  - \\omega₀  + a  ⎠⎟\n",
       "  ⎜  Ug⋅ℯ                                Ug⋅ℯ                             ⎟\n",
       "R⋅⎜- ───────────────────────────────── + ─────────────────────────────────⎟\n",
       "  ⎜              _________________                   _________________    ⎟\n",
       "  ⎜             ╱          2    2                   ╱          2    2     ⎟\n",
       "  ⎝       2⋅L⋅╲╱  - \\omega₀  + a              2⋅L⋅╲╱  - \\omega₀  + a      ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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ABBbj+18VJ3R1o8JcXdGRiU9H1oyiKNqdmBXK+usgjhgefMlTFD83xsEXHju2rTvl5Nah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\nQo+LngAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{Ug e^{- 2 a t}}{2 \\sqrt{- \\omega_0^{2} + a^{2}}} \\left(- \\left(a - \\sqrt{- \\omega_0^{2} + a^{2}}\\right) e^{t \\left(a + \\sqrt{- \\omega_0^{2} + a^{2}}\\right)} + \\left(a + \\sqrt{- \\omega_0^{2} + a^{2}}\\right) e^{t \\left(a - \\sqrt{- \\omega_0^{2} + a^{2}}\\right)}\\right)$$"
      ],
      "text/plain": [
       "   ⎛                                ⎛       _________________⎞                \n",
       "   ⎜  ⎛       _________________⎞    ⎜      ╱          2    2 ⎟   ⎛       _____\n",
       "   ⎜  ⎜      ╱          2    2 ⎟  t⋅⎝a + ╲╱  - \\omega₀  + a  ⎠   ⎜      ╱     \n",
       "Ug⋅⎝- ⎝a - ╲╱  - \\omega₀  + a  ⎠⋅ℯ                             + ⎝a + ╲╱  - \\o\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                           _________________  \n",
       "                                                          ╱          2    2   \n",
       "                                                      2⋅╲╱  - \\omega₀  + a    \n",
       "\n",
       "                 ⎛       _________________⎞⎞        \n",
       "____________⎞    ⎜      ╱          2    2 ⎟⎟        \n",
       "     2    2 ⎟  t⋅⎝a - ╲╱  - \\omega₀  + a  ⎠⎟  -2⋅a⋅t\n",
       "mega₀  + a  ⎠⋅ℯ                            ⎠⋅ℯ      \n",
       "────────────────────────────────────────────────────\n",
       "                                                    \n",
       "                                                    \n",
       "                                                    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u_R=R*i3.rhs\n",
    "display(u_R)\n",
    "uR = lambdify(t, u_R.subs(podatki), 'numpy') # napetost na uporu\n",
    "u_L=L*i3.rhs.diff(t)\n",
    "display(simplify(u_L))\n",
    "uL = lambdify(t, u_L.subs(podatki), 'numpy') # napetost na tuljavi\n",
    "#print(uL(0))\n",
    "# u_C=i3.rhs.integrate(t)/C\n",
    "#display(u_C)\n",
    "#uC = lambdify(t, u_C.subs(podatki), 'numpy') # tok kot funkcija časa\n",
    "u_C=Ug-u_L-u_R\n",
    "#display(simplify(u_C))\n",
    "uC = lambdify(t, u_C.subs(podatki), 'numpy') # napetost na kondenzatorju\n",
    "\n",
    "uRLC=lambdify(t, u_C.subs(podatki)+ u_R.subs(podatki)+ u_L.subs(podatki), 'numpy') # vsota vseh napetosti"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\ProgramData\\Anaconda3\\lib\\site-packages\\numpy\\core\\numeric.py:531: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  return array(a, dtype, copy=False, order=order)\n"
     ]
    },
    {
     "data": {
      "image/png": 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sZMSF+2xaq9L6Ut7a8hYn9T+JzOhMn5TZKXFZMPtxKFwFn/3d7GiEEOKQOtMB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\n1dYSvtpS3NJ0mJ0YwSnjMpk+OIUjByQ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      "text/plain": [
       "<matplotlib.figure.Figure at 0x237f83052e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cas = np.linspace(0, 10, 1000)\n",
    "\n",
    "def slika():\n",
    "    #plt.plot(cas, tok(cas), 'C0', label='Tok [A]')\n",
    "    plt.plot(cas, (uR(cas)), 'C1', label='$u_R $')\n",
    "    plt.plot(cas, (uL(cas)), 'C0', label='$u_L $')\n",
    "    plt.plot(cas, (uC(cas)), 'C2', label='$u_C $')\n",
    "    #plt.plot(cas, (uRLC(cas)), '--', label='$u_R + u_L + u_C $') # iz neznanih razlogov včasih nagaja\n",
    "    plt.xlabel('Čas [s]')\n",
    "    plt.ylabel('Napetost [V] ')\n",
    "    plt.legend(loc=(1.01, 0));\n",
    "    plt.show()\n",
    "slika()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.1"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": true,
   "title_cell": "Kazalo",
   "title_sidebar": "Kazalo",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}