Aula 11 - Funções de Singularidade.ipynb

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    "# Funções de Singularidade\n",
    "Jupyter Notebook desenvolvido por [Gustavo S.S.](https://github.com/GSimas)\n",
    "\n",
    "As funções de singularidade (também conhecidas como funções de comutação)\n",
    "são muito úteis na análise de circuitos, pois servem como boas aproximações\n",
    "aos sinais de comutação que surgem em circuitos com operações de\n",
    "comutação, e também por serem úteis na descrição compacta e elegante de\n",
    "alguns fenômenos em circuitos, especialmente a resposta a um degrau de circuitos\n",
    "RC ou RL a serem discutidos nas seções seguintes. Por definição,\n",
    "\n",
    "**Funções de singularidade são funções que são descontínuas ou então que\n",
    "apresentam derivadas descontínuas.**\n",
    "\n",
    "As três funções de singularidade mais usadas na análise de circuitos são:\n",
    "***degrau unitário, impulso unitário e rampa unitária.***\n",
    "\n",
    "## Degrau Unitário\n",
    "\n",
    "**A função degrau unitário u(t) é 0 para valores negativos de t e 1 para\n",
    "valores positivos de t.**\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large u(t) =\n",
    "\\begin{cases} 0, &  t < 0\n",
    "\\\\ 1, &  t > 0\n",
    "\\end{cases}}\n",
    "\\end{align}\n",
    "\n",
    "![](https://i.imgur.com/LZFlqkC.png)\n",
    "\n",
    "Usamos a função degrau para representar uma mudança abrupta na tensão\n",
    "ou corrente, como as mudanças que ocorrem em circuitos de sistemas de controle\n",
    "e computadores digitais. Por exemplo, a tensão:\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large v(t) =\n",
    "\\begin{cases} 0, & t < t_0\n",
    "\\\\ 1, & t > t_0\n",
    "\\end{cases}}\n",
    "\\end{align}\n",
    "\n",
    "pode ser expressa em termos da função degrau unitário como\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large v(t) = V_0u(t-t_0)}\n",
    "\\end{align}\n",
    "\n",
    "![](https://i.imgur.com/oELQ07i.png)\n",
    "\n",
    "Se fizermos t0 = 0, então v(t) é simplesmente a tensão degrau V0u(t). Uma fonte\n",
    "de tensão V0u(t) é mostrada na Figura 7.25a, e seu circuito equivalente é ilustrado\n",
    "na Figura 7.25b.\n",
    "\n",
    "![](https://i.imgur.com/7psYeUu.png)\n",
    "\n",
    "\n",
    "## Impulso Unitário\n",
    "\n",
    "A derivada da função degrau unitário u(t) é a função impulso unitário d(t),\n",
    "que é escrita como segue:\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large \\delta(t) = \\frac{d}{dt}u(t) = \n",
    "\\begin{cases}\n",
    "0, & t < 0\n",
    "\\\\ Indefinido, & t = 0\n",
    "\\\\0, & t > 0\n",
    "\\end{cases}}\n",
    "\\end{align}\n",
    "\n",
    "**A função impulso unitário d(t) é zero em qualquer ponto, exceto em\n",
    "t = 0, onde ela é indefinida.**\n",
    "\n",
    "![](https://i.imgur.com/PPGpq05.png)\n",
    "\n",
    "O impulso unitário pode ser considerado um choque elétrico aplicado ou\n",
    "resultante e ser visualizado como um pulso de área unitária de curtíssima duração,\n",
    "sendo expresso matematicamente como:\n",
    "\n",
    "\\begin{align}\n",
    "\\int_{0^-}^{0^+} \\delta(t)dt = 1\n",
    "\\end{align}\n",
    "\n",
    "A área unitária é conhecida como a intensidade\n",
    "da função impulso, e quando essa função tiver uma intensidade diferente da\n",
    "unidade, a área do impulso será igual à sua intensidade. Por exemplo, a função\n",
    "impulso 10d(t) possui uma área igual a 10.\n",
    "\n",
    "Assim:\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large \\int_{a}^{b} f(t)δ(t - t_0)dt = f(t_0)}\n",
    "\\end{align}\n",
    "\n",
    "## Rampa Unitária\n",
    "\n",
    "**Integrando a função degrau unitário u(t), obtemos a função rampa unitária\n",
    "r(t); escrevemos:**\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large r(t) = \\int_{-∞}^{t} u(x)dx = tu(t)}\n",
    "\\end{align}\n",
    "\n",
    "ou:\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large r(t) =\n",
    "\\begin{cases}\n",
    "0, & t \\le 0\n",
    "\\\\t, & t \\ge 0\n",
    "\\end{cases}}\n",
    "\\end{align}\n",
    "\n",
    "**A função rampa unitária é zero para valores negativos de t e apresenta\n",
    "uma inclinação unitária para valores positivos de t.**\n",
    "\n",
    "Deve-se ter em mente que as três funções de singularidade (impulso, degrau\n",
    "e rampa) estão relacionadas por diferenciação como segue:\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large \\delta(t) = \\frac{du(t)}{dt}}\n",
    "\\\\{\\Large u(t) = \\frac{dr(t)}{dt}}\n",
    "\\end{align}\n",
    "\n",
    "## Função Porta\n",
    "\n",
    "As funções de porta são usadas\n",
    "juntamente com chaves para\n",
    "permitir ou bloquear a passagem\n",
    "de outro sinal."
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    "**Exemplo 7.6**\n",
    "\n",
    "Expresse o pulso de tensão da Figura 7.31 em termos do degrau unitário. Calcule sua\n",
    "derivada e esboce-a.\n",
    "\n",
    "![](https://i.imgur.com/QN8G9xQ.png)"
   ]
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     "text": [
      "Exemplo 7.6\n",
      "[0, 0, 10*Heaviside(0), 10, 10, 10, 10, 10, 10, 10]\n",
      "[0, 0, 0, 0, 0, -10*Heaviside(0), -10, -10, -10, -10]\n",
      "[-10, 2, 2, 5, 5, 10]\n",
      "[0, 0, 1, 1, 0, 0]\n"
     ]
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x23f7d0e9a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAABiCAYAAACrrJNiAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n4W5mVkIOdzOzEnK4m5mVkMPdzKyE/gcTnxamSvyE0AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x23f7cdfa0b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x23f7d02edd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Exemplo 7.6\")\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from sympy import *\n",
    "\n",
    "t = symbols('t')\n",
    "u1 = 10*Heaviside(t-2) #funcao degrau unitario, tambem chamada de funcao Heaviside\n",
    "u2 = -10*Heaviside(t-5)\n",
    "l = list()\n",
    "m = list()\n",
    "n = list()\n",
    "\n",
    "for i in range(10):\n",
    "    l.append(u1.subs(t,i))\n",
    "    m.append(u2.subs(t,i))\n",
    "\n",
    "print(l)\n",
    "print(m)\n",
    "\n",
    "\n",
    "x1 = [-10,2,2,10]\n",
    "y1 = [0,0,1,1]\n",
    "\n",
    "x2 = [-10,5,5,10]\n",
    "y2 = [0,0,-1,-1]\n",
    "\n",
    "x3 = [-10,2,2,5,5,10]\n",
    "y3 = [0,0,1,1,0,0]\n",
    "\n",
    "x4 = [-10,2,2,5,5,10]\n",
    "y4 = [0,0,10,0,-10,0]\n",
    "\n",
    "print(x3)\n",
    "print(y3)\n",
    "\n",
    "plt.figure(1)\n",
    "plt.subplot(311)\n",
    "plt.step(x1,y1)\n",
    "\n",
    "plt.figure(2)\n",
    "plt.subplot(312)\n",
    "plt.step(x2,y2)\n",
    "\n",
    "plt.figure(3)\n",
    "plt.subplot(313)\n",
    "plt.step(x3,y3)\n",
    "\n",
    "plt.figure(4)\n",
    "plt.subplot(313)\n",
    "plt.step(x4,y4)\n",
    "\n",
    "plt.show()\n",
    "\n",
    "#Resposta\n",
    "#10[u(t-2) u(t-5)]\n",
    "#10[delta(t-2) delta(t-5)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Problema Prático 7.6**\n",
    "\n",
    "Expresse o pulso de corrente da Figura 7.33 em termos de degrau unitário. Determine\n",
    "sua integral e esboce-a.\n",
    "\n",
    "![](https://i.imgur.com/1pOaidT.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problema Prático 7.6\n",
      "[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10]\n",
      "[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -20, -20, -20, -20, -20, -20, -20, -20]\n",
      "[0 0 0 0 0 0 0 0 0 0 10 10 -10 -10 -10 -10 -10 -10 -10 -10]\n",
      "[-10.          -8.94736842  -7.89473684  -6.84210526  -5.78947368\n",
      "  -4.73684211  -3.68421053  -2.63157895  -1.57894737  -0.52631579\n",
      "   0.52631579   1.57894737   2.63157895   3.68421053   4.73684211\n",
      "   5.78947368   6.84210526   7.89473684   8.94736842  10.        ]\n"
     ]
    },
    {
     "data": {
      "image/png": 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JahgMkqSGwSBJahySVyUl2Q18c4EvPx54dgnLWSrWdWCs68BY14F5odb1yqpasb9Oh2Qw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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x23f7ce18ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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v0OK3ZseBYzyweA0jerRjxpX9bcdRyq/cN3YQgxNjuGthPvsqdKTDqbT4Laip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\njBa/UkoFGC1+pZQKMFr8SikVYP4fq4o0B7aqxI0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x23f7cf85940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Problema Prático 7.6\")\n",
    "\n",
    "u1 = 10*Heaviside(t)\n",
    "u2 = -20*Heaviside(t-2)\n",
    "\n",
    "l,m,n = list(), list(), list()\n",
    "\n",
    "for i in np.linspace(-10,10,20):\n",
    "    l.append(u1.subs(t,i))\n",
    "    m.append(u2.subs(t,i))\n",
    "\n",
    "print(l)\n",
    "print(m)\n",
    "n = np.add(l,m)\n",
    "n = n.reshape((1,20))\n",
    "n = n[0]\n",
    "print(n)\n",
    "\n",
    "o = np.linspace(-10,10,20)\n",
    "print(o)\n",
    "\n",
    "plt.step(o,n)\n",
    "plt.show()\n",
    "\n",
    "x = np.linspace(0,4,20)\n",
    "\n",
    "o = list()\n",
    "\n",
    "for i in range(len(x)):\n",
    "    if(x[i] < 2):\n",
    "        o.append(10*x[i])\n",
    "    elif(x[i] == 2):\n",
    "        o.append(20)\n",
    "    else:\n",
    "        o.append(20 - 10*(x[i] - 2))\n",
    "\n",
    "plt.plot(x,o)\n",
    "plt.show()\n",
    "\n",
    "#Resposta\n",
    "#10[u(t) – 2u(t – 2) + u(t – 4)] A, \n",
    "#10[r(t) – 2r(t – 2) + r(t – 4)] A·s"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exemplo 7.7**\n",
    "\n",
    "Expresse a função dente de serra mostrada na Figura 7.35 em termos de funções de\n",
    "singularidade.\n",
    "\n",
    "![](https://i.imgur.com/Is8nxZR.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Exemplo 7.7\n",
      "[ 0.          0.55555556  1.11111111  1.66666667  2.22222222  2.77777778\n",
      "  3.33333333  3.88888889  4.44444444  5.        ]\n",
      "[  0.           2.77777778   5.55555556   8.33333333  11.11111111\n",
      "  13.88888889  16.66666667  19.44444444  22.22222222  25.        ]\n",
      "[1, 1, 1, 1, 0, 0, 0, 0, 0, 0]\n"
     ]
    },
    {
     "data": {
      "image/png": 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Rv88iIgCyVbVPRNwA9gLYqKr7DEeznIhsBrAEQI6q3mk6j9VEpB7AElW1fHbd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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x23f7a8e37b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Exemplo 7.7\")\n",
    "\n",
    "#Resposta\n",
    "#5*r(t)*u(-t+2)\n",
    "\n",
    "x = np.linspace(0,5,10)\n",
    "y = 5*x\n",
    "u = Heaviside(-t+2)\n",
    "r = list()\n",
    "\n",
    "for i in x:\n",
    "    r.append(u.subs(t,i))\n",
    "    \n",
    "print(x)\n",
    "print(y)\n",
    "print(r)\n",
    "\n",
    "r = r*y\n",
    "\n",
    "plt.plot(x,r)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Problema Prático 7.7**\n",
    "\n",
    "Consulte a Figura 7.39. Expresse i(t) em termos de funções de singularidade.\n",
    "\n",
    "![](https://i.imgur.com/HlFnxRn.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problema Prático 7.7\n",
      "[-0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0.  2. -8. -2. -2. -2. -2.\n",
      " -2. -2. -2.]\n"
     ]
    },
    {
     "data": {
      "image/png": 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T6YdtTFeYw55VBDuQMKWxvGodFAIrV2eUHxnSurNG+9QyxAXBDiRMYy77coXAypWaJoqr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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x23f7aae3c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Problema Prático 7.7\")\n",
    "\n",
    "def ret(t,start=-10,end=10,points=21):\n",
    "    x = np.linspace(start,end,points)\n",
    "    l = x > t\n",
    "    l = l*x\n",
    "    return l\n",
    "\n",
    "def heaviside(t, start=-10,end=10,points=21):\n",
    "    x = np.linspace(start,end,points)\n",
    "    l = x > t\n",
    "    l = 1*l\n",
    "    return l\n",
    "\n",
    "y = -(2*heaviside(0) - 2*ret(0) + 4*ret(2) - 2*ret(3))\n",
    "print(y)\n",
    "\n",
    "x = np.linspace(-10,10,21)\n",
    "plt.plot(x,y)\n",
    "plt.show()\n",
    "\n",
    "#Resposta\n",
    "#2u(t) - 2r(t) + 4r(t-2) - 2r(t-3) A."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exemplo 7.8**\n",
    "\n",
    "Dado o sinal:\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large g(t) =\n",
    "\\begin{cases}\n",
    "3, & t < 0\n",
    "\\\\-2, & 0 < t < 1\n",
    "\\\\2t - 4, & t >1\n",
    "\\end{cases}}\n",
    "\\end{align}\n",
    "\n",
    "expresse g(t) em termos de funções degrau e de rampa."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Exemplo 7.8\n"
     ]
    }
   ],
   "source": [
    "print(\"Exemplo 7.8\")\n",
    "\n",
    "#3 - 3*u(t)\n",
    "#-2*u(t) + 2*u(t - 1)\n",
    "#2*r(t - 1) - 4*u(t - 1)\n",
    "\n",
    "#3 - 5u(t) - 2u(t - 1) + 2r(t - 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Problema Prático 7.8**\n",
    "\n",
    "Se:\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large h(t) =\n",
    "\\begin{cases}\n",
    "0, & t < 0\n",
    "\\\\-4, & 0 < t < 2\n",
    "\\\\3t - 8, & 2 < t < 6\n",
    "\\\\0, & t > 6\n",
    "\\end{cases}}\n",
    "\\end{align}\n",
    "\n",
    "expresse h(t) em termos de funções de singularidade."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "print(\"Problema Prático 7.8\")\n",
    "\n",
    "#-4u(t) + 4u(t-2)\n",
    "#3r(t-2) - 8*u(t-2) + 8*u(t-6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exemplo 7.9**\n",
    "\n",
    "Calcule as seguintes integrais envolvendo a função impulso.\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large \\int_{0}^{10} (t^2 + 4t - 2)\\delta(t-2)dt}\n",
    "\\\\ {\\Large \\int_{-\\infty}^{\\infty} [\\delta(t - 1)e^{-t} cos(t) + \\delta(t + 1)e^{-t}sen(t)]dt}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Exemplo 7.9\n",
      "Primeira integral: 10\n",
      "Segunda integral: -E*sin(1) + exp(-1)*cos(1)\n"
     ]
    }
   ],
   "source": [
    "print(\"Exemplo 7.9\")\n",
    "\n",
    "u = (t**2 + 4*t - 2)*DiracDelta(t - 2)\n",
    "y = integrate(u,(t,0,10))\n",
    "print(\"Primeira integral:\", y)\n",
    "\n",
    "u = DiracDelta(t-1)*exp(-t)*cos(t) + DiracDelta(t+1)*exp(-t)*sin(t)\n",
    "y = integrate(u,(t,-oo,oo))\n",
    "print(\"Segunda integral:\", y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Problema Prático 7.9**\n",
    "\n",
    "Calcule as seguintes integrais:\n",
    "\n",
    "\\begin{align}\n",
    "{\\Large \\int_{-\\infty}^{\\infty} (t^3 + 5t^2 + 10) \\delta(t + 3)dt}\n",
    "\\\\ {\\Large \\int_{0}^{10} \\delta(t - \\pi)cos(3t)dt}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problema Prático 7.9\n",
      "Primeira Integral: 28\n",
      "Segunda Integral: -1\n"
     ]
    }
   ],
   "source": [
    "print(\"Problema Prático 7.9\")\n",
    "\n",
    "u = (t**3 + 5*t**2 + 10)*DiracDelta(t+3)\n",
    "y = integrate(u,(t,-oo,oo))\n",
    "print(\"Primeira Integral:\",y)\n",
    "\n",
    "u = DiracDelta(t-pi)*cos(3*t)\n",
    "y = integrate(u,(t,0,10))\n",
    "print(\"Segunda Integral:\",y)"
   ]
  }
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