EcuacionCable.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ecuación de cable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pylab as plt\n",
    "import math\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![title](cable.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Resolviendo el circuito, se obtiene la siguiente ecuación\n",
    "\n",
    "\\begin{equation}\n",
    "\\tau \\frac{\\partial V}{\\partial t} = \\lambda^2 \\frac{\\partial^2 V}{\\partial x^2}-V\n",
    "\\end{equation}\n",
    "\n",
    "Donde \n",
    "\n",
    "\\begin{equation}\n",
    "\\lambda=\\sqrt{\\frac{a R_m}{2 R_l}}\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation}\n",
    "\\tau=c_m R_m\n",
    "\\end{equation}\n",
    "\n",
    "Tomando el caso cuando el voltaje es estable temporalmente ($t\\to \\infty$) se simplifica a:\n",
    "\n",
    "\\begin{equation}\n",
    " \\lambda^2 \\frac{\\partial^2 V}{\\partial x^2}=V\n",
    "\\end{equation}\n",
    "\n",
    "Con condición inicial\n",
    "\n",
    "\\begin{equation}\n",
    "I_0 = \\frac{-\\pi a^2}{r_l}\\frac{\\partial V_0}{\\partial x}\n",
    "\\end{equation}\n",
    "\n",
    "Obtendiendo la solución:\n",
    "\n",
    "\\begin{equation}\n",
    "V(x)=\\frac{\\lambda r_l I_0}{\\pi a^2}e^{\\frac{-x}{\\lambda}}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Generamos un vector x\n",
    "long=np.linspace(0,10,1000)\n",
    "\n",
    "#Definimos la función\n",
    "def voltage(x,a=1,b=1):\n",
    "    return a*np.exp(-x/b)\n",
    "\n",
    "#Evaluamos la función\n",
    "v=voltage(long)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Voltaje a lo largo del cable')"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hIDLImdmz7l5/xPkUCtKbxp37+cGjq7k7vBT3u08dzd+9ZTzTaspirkxEjoZC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\n2RQtYgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f392edef518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(long,v)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('Voltaje')\n",
    "plt.legend()\n",
    "plt.title('Voltaje a lo largo del cable')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Método de Euler \n",
    "Dada una ecuación diferencial \n",
    "\n",
    "\\begin{equation}\n",
    "\\frac{dx}{dt}=f(x,t; parámetros)\n",
    "\\end{equation}\n",
    "\n",
    "Con valor inicial \n",
    "\n",
    "\\begin{equation}\n",
    "x(t_0)=x_0\n",
    "\\end{equation}\n",
    "\n",
    "Se puede aproximar la derivada como:\n",
    "\n",
    "\\begin{equation}\n",
    "\\frac{dx}{dt}\\simeq \\frac{x(t+h)-x(t)}{h}\n",
    "\\end{equation}\n",
    "\n",
    "Donde h es el tamaño del paso y por lo tanto despejando $x(t+h)$ tenemos:\n",
    "\n",
    "\\begin{equation}\n",
    "x(t+h) \\simeq x(t+h)\\frac{dx}{dt}=x(t)+hf(x,t)\n",
    "\\end{equation}\n",
    "\n",
    "Podemos aproximar la solución de la forma:\n",
    "\n",
    "\\begin{equation}\n",
    "x_{n+1}= x_n +hf(x_n,t_n)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Aproximación\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Reescribiendo la ecuación:\n",
    "\n",
    "\\begin{equation}\n",
    "\\lambda^2 \\frac{\\partial^2 V}{\\partial x^2}=V\n",
    "\\end{equation}\n",
    "\n",
    "Como un sistema de ecuaciones de primer orden:\n",
    "    \n",
    "definimos:\n",
    "$$z= V(x)$$ \n",
    "$$w=\\frac{dV(x)}{dx}$$ \n",
    "\n",
    "$$\\to$$\n",
    "\n",
    "$$\\frac{dz(x)}{dx}=\\frac{dV(x)}{dx}= w(x)$$\n",
    "$$\\frac{dw(x)}{dx}=\\frac{d^2 V(x)}{dx^2}=\\frac{V}{\\lambda^2}=\\frac{z(x)}{\\lambda^2}$$\n",
    "\n",
    "$$\\to$$\n",
    "\n",
    "$$\\frac{dz(x)}{dx}=w(x)$$\n",
    "$$\\frac{dw(x)}{dx}=\\frac{z(x)}{\\lambda^2}$$\n",
    "\n",
    "Con la condición inicial de:\n",
    "\n",
    "\\begin{equation}\n",
    "I_0 = \\frac{-\\pi a^2}{r_l}\\frac{\\partial V_0}{\\partial x}\n",
    "\\end{equation}\n",
    "\n",
    "$$\\to$$\n",
    "\n",
    "\\begin{equation}\n",
    "w(0)=\\frac{\\partial V_0}{\\partial x} = \\frac{-r_l I_0}{\\pi a^2}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def dz(z,w,l):\n",
    "    return w\n",
    "def dw(z,w,l):\n",
    "    return z/(l**2)\n",
    "\n",
    "dt1=0.01\n",
    "\n",
    "# Integramos usando el método de Euler  \n",
    "\n",
    "def euler(dz,dw,dt):\n",
    "    x=np.linspace(0,10,1000)\n",
    "    z=np.zeros(len(x))\n",
    "    w=np.zeros(len(x))\n",
    "    z[0]=1\n",
    "    w[0]=-1\n",
    "    l=1\n",
    "    for i in range(0,len(x)-1):\n",
    "        z[i+1] = z[i] + dz(z[i],w[i],l)*dt\n",
    "        w[i+1] = w[i] + dw(z[i],w[i],l)*dt\n",
    "    return x,z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'')"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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W31m7j0REpJ9KQURE+kWtFB4KO0AI9J2jQd85GgL/zpE6piAiIqcWtS0FERE5\nhciUgpktN7NNZtZoZneHnSdoZlZjZr80s41mtt7M7gw701gws7iZvW5mPwk7y1gws1Ize8rM3kr9\nt7407ExBM7M/S/2dXmdmj5lZIuxMo83MHjazvWa2Lm3ZJDP7mZltSf0sC+KzI1EKZhYHHgCuA+YD\nN5vZ/HBTBa4b+Ly7nwcsBT4dge8McCewMewQY+gbwE/dfR5wIRP8u5tZFfBZoN7dF5Icln8iDrn/\nKLB8wLK7gefcvQ54LjU/6iJRCsASoNHdt7p7J/A4sCLkTIFy913u/lpq+gjJXxZV4aYKlplVAx8A\nvhN2lrFgZsXAlSTvS4K7d7r7wXBTjYksIC91t8Z8Tr6jY8Zz919x8l0oVwD/kJr+B+D3g/jsqJRC\nFdCUNt/MBP8Fmc7MaoHFwCvhJgncfcB/BzLzbvdnbjbQAjyS2mX2HTMrCDtUkNx9B/BV4F1gF3DI\n3f8t3FRjZoq774LkP/qAyiA+JCqlYIMsi8RpV2ZWCPwz8Dl3Pxx2nqCY2QeBve6+JuwsYygLeA/w\nbXdfDBwloF0K40VqP/oKYBYwHSgwsw+Fm2piiUopNAM1afPVTMBNzoHMLJtkIfyju/8w7DwBuxy4\nwcy2kdw9uMzMvh9upMA1A83u3rcF+BTJkpjIrgXecfcWd+8CfghcFnKmsbLHzKYBpH7uDeJDolIK\nq4E6M5tlZjkkD0ytDDlToMzMSO5r3ujuXws7T9Dc/S/cvdrda0n+9/2Fu0/of0G6+26gyczOTS26\nBtgQYqSx8C6w1MzyU3/Hr2GCH1xPsxL4SGr6I8DTQXxIoPdoHi/cvdvM7gCeJXm2wsPuvj7kWEG7\nHPgw8KaZ/Ta17Aup+2bLxPEZ4B9T/9jZCtwacp5AufsrZvYU8BrJM+xeZwJe2WxmjwFXAeVm1gx8\nEfgS8ISZfZxkOQZyf3td0SwiIv2isvtIRESGQaUgIiL9VAoiItJPpSAiIv1UCiIi0k+lICIi/VQK\nIiLST6UgcpbM7GIze8PMEmZWkBrrf2HYuURGQheviYwCM/sbIAHkkRyP6G9DjiQyIioFkVGQGmZi\nNdABXObuPSFHEhkR7T4SGR2TgEKgiOQWg0hG0paCyCgws5Ukh+yeBUxz9ztCjiQyIpEYJVUkSGb2\np0C3u/8gdT/w35jZMnf/RdjZRM6UthRERKSfjimIiEg/lYKIiPRTKYiISD+VgoiI9FMpiIhIP5WC\niIj0UymIiEg/lYKIiPT7/9C4Q4In5DgeAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f392edefef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x,z=euler(dz,dw,dt1)\n",
    "plt.plot(x, z)\n",
    "#plt.plot(x,w, label=\"w\")\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('func')\n",
    "plt.legend()\n",
    "plt.title('')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Comparando la solución original con las soluciones dadas por diferentes \"dt\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "dt0=0.1\n",
    "dt1=0.06\n",
    "dt2=0.03\n",
    "dt3=0.015\n",
    "\n",
    "x0,z0=euler(dz,dw,dt0)\n",
    "x1,z1=euler(dz,dw,dt1)\n",
    "x2,z2=euler(dz,dw,dt2)\n",
    "x3,z3=euler(dz,dw,dt3)\n",
    "\n",
    "\n",
    "long=np.linspace(0,10,1000)\n",
    "def voltage(x,a=1,b=1):\n",
    "    return a*np.exp(-x/b)\n",
    "v=voltage(long)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Voltaje a lo largo del cable')"
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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im0Vkk4j8NZrlwe5q62huqykcOXZk0ZAOw1IB5oxOZeXuCkIhbf9USp3eohYU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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f392e293908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(long,v,label='Sol exacta')\n",
    "plt.plot(x0,z0,label='dt=0.1')\n",
    "plt.plot(x1,z1,label='dt=0.06')\n",
    "plt.plot(x2,z2,label='dt=0.03')\n",
    "plt.plot(x3,z3,label='dt=0.015')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('Voltaje')\n",
    "plt.legend()\n",
    "plt.title('Voltaje a lo largo del cable')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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