Jupyter notebooks/12.1 - Exercícios com função de transferência e integral de convolução.ipynb

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   "source": [
    "<a href=\"https://colab.research.google.com/github/edsonportosilva/ElectricCircuits/blob/master/Jupyter%20notebooks/Circuitos%20El%C3%A9tricos%20I%20-%20Semana%2012.1%20-%20Exerc%C3%ADcios%20com%20fun%C3%A7%C3%A3o%20de%20transfer%C3%AAncia%20e%20integral%20de%20convolu%C3%A7%C3%A3o.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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   "source": [
    "from IPython.core.display import HTML\n",
    "HTML(\"\"\"\n",
    "<style>\n",
    ".output_png {\n",
    "    display: table-cell;\n",
    "    text-align: center;\n",
    "    vertical-align: middle;\n",
    "}\n",
    "</style>\n",
    "\"\"\")\n",
    "\n",
    "if 'google.colab' in str(get_ipython()):    \n",
    "    ! git clone -b master https://github.com/edsonportosilva/ElectricCircuits\n",
    "    from os import chdir as cd\n",
    "    cd('/content/ElectricCircuits/')\n",
    "    ! pip install -e ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "h6sSBJ9bUuWs"
   },
   "source": [
    "# *Circuitos Elétricos I - Semana 12.1*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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    "id": "AEIxuYElUuWt"
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   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import sympy as sp\n",
    "from circuit.utils import round_expr, symdisp, symplot\n",
    "\n",
    "from sympy.polys.partfrac import apart\n",
    "\n",
    "# temp workaround\n",
    "import warnings\n",
    "from matplotlib import MatplotlibDeprecationWarning\n",
    "warnings.filterwarnings('ignore', category=MatplotlibDeprecationWarning)\n",
    "\n",
    "plt.rcParams['figure.figsize'] = 6, 4\n",
    "plt.rcParams['legend.fontsize'] = 13\n",
    "plt.rcParams['lines.linewidth'] = 2\n",
    "plt.rcParams['axes.grid'] = False"
   ]
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   "source": [
    "# transformada de Laplace\n",
    "def L(f,t,s):\n",
    "    return sp.laplace_transform(f, t, s, noconds=True)\n",
    "\n",
    "# transformada inversa de Laplace\n",
    "def invL(F,s,t):\n",
    "    return sp.re(sp.inverse_laplace_transform(F, s, t, noconds=True))\n",
    "\n",
    "# funções para auxílio na expansão em frações parciais\n",
    "def adjustCoeff(expr):    \n",
    "    coeff = expr.as_numer_denom()\n",
    "    c0 = sp.poly(coeff[1].cancel()).coeffs()[0]\n",
    "    \n",
    "    return (coeff[0].cancel()/c0)/(coeff[1].cancel()/c0)\n",
    "\n",
    "def partFrac(expr, Ndigits):\n",
    "    expr = expr.cancel()\n",
    "    expr = apart(adjustCoeff(expr), s, full=True).doit()\n",
    "    \n",
    "    return sp.N(expr, Ndigits)\n",
    "\n",
    "sp.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "0UcuJTrZUuWu"
   },
   "source": [
    "#### Definindo algumas variáveis simbólicas de interesse"
   ]
  },
  {
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   "outputs": [],
   "source": [
    "s     = sp.symbols('s')\n",
    "a     = sp.symbols('a', real=True, positive=True)\n",
    "omega, t = sp.symbols('omega, t', real=True)\n",
    "infty = sp.oo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "DOu_DBGlUuWv"
   },
   "source": [
    "### Problema 1\n",
    "\n",
    "O circuito da figura a seguir está em regime estacionário no momento em que chave é aberta. Sabe-se que $v(t)=\\mathrm{12\\;V}$.\n",
    "\n",
    "<img src=\"https://github.com/edsonportosilva/ElectricCircuits/blob/master/Jupyter%20notebooks/figures/J15C1.png?raw=1\" width=\"600\">\n",
    "\n",
    "Simulação do circuito: https://tinyurl.com/y84pozs3\n",
    "\n",
    "a. Determine $I_0(s)$ e $I_1(s)$.\\\n",
    "b. Verifique a consistência das respostas do item a. com os teoremas do valor inicial e do valor final.\\\n",
    "c. Determine a função de transferência $H_0(s)$ entre $V(s)$ e $I_0(s)$.\\\n",
    "d. Determine a função de transferência $H_1(s)$ entre $V(s)$ e $I_1(s)$.\\\n",
    "e. Determine a função de transferência $H_c(s)$ entre $V(s)$ e $V_c(s)$.\\\n",
    "f. Determine a $v_c(t)$ para o caso em que a tensão aplicada $v(t)$ em $\\mathrm{V}$ corresponde ao gráfico a seguir:\n",
    "\n",
    "<img src=\"https://github.com/edsonportosilva/ElectricCircuits/blob/master/Jupyter%20notebooks/figures/J15Fig1.png?raw=1\" width=\"400\">"
   ]
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   "source": [
    "a. Determinando $I_0(s)$ e $I_1(s)$:"
   ]
  },
  {
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     "text": [
      "Correntes no domínio de Laplace: \n",
      "\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle I_0(s) =- \\frac{150.0}{s^{3} + 35.0 s^{2} + 375.0 s + 3125.0}\\;As$"
      ],
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       "<IPython.core.display.Math object>"
      ]
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle I_1(s) =\\frac{6.0 s^{2} + 210.0 s + 750.0}{5.0 s^{3} + 175.0 s^{2} + 1875.0 s + 15625.0}\\;As$"
      ],
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   ],
   "source": [
    "I0, I1, s = sp.symbols('I_0, I_1, s')\n",
    "\n",
    "# define os sistema de equações\n",
    "eq1 = sp.Eq((-2*s**2)*I0 + (2*s**2 + 10*s + 250)*I1, 2.4*s + 12)             \n",
    "eq2 = sp.Eq((4*s+50)*I0 -2*s*I1, -2.4)  \n",
    "\n",
    "# resolve o sistema\n",
    "soluc = sp.solve([eq1, eq2],[I0, I1], dict=True)\n",
    "soluc\n",
    "\n",
    "I0 = [sol[I0] for sol in soluc]\n",
    "I1 = [sol[I1] for sol in soluc]\n",
    "\n",
    "I0 = I0[0]\n",
    "I1 = I1[0]\n",
    "\n",
    "print('Correntes no domínio de Laplace: \\n')\n",
    "symdisp('I_0(s) =', I0, 'As')\n",
    "symdisp('I_1(s) =', I1, 'As')"
   ]
  },
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   "metadata": {
    "id": "DX6d4RX1UuWw"
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   "source": [
    "b. Checando a consistência das soluções\n",
    "\n",
    "**Teorema do valor inicial (TVI)**\n",
    "\n",
    "$$\n",
    "f(0^+) = \\lim_{t \\to 0^+}f(t) = \\lim_{s \\to \\infty}sF(s)\n",
    "$$\n"
   ]
  },
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i_0(0^+) = 0\\; A$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i_1(0^+) = \\frac{6}{5}\\; A$"
      ],
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       "<IPython.core.display.Math object>"
      ]
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   ],
   "source": [
    "i0_0_tvi = sp.limit(s*I0, s, infty)\n",
    "i1_0_tvi = sp.limit(s*I1, s, infty)\n",
    "\n",
    "\n",
    "symdisp('i_0(0^+) = ', i0_0_tvi, ' A' )\n",
    "symdisp('i_1(0^+) = ', i1_0_tvi, ' A' )"
   ]
  },
  {
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   "source": [
    "**Teorema do valor final (TVF)**\n",
    "\n",
    "$$\n",
    "f(\\infty) = \\lim_{t \\to \\infty}f(t) = \\lim_{s \\to 0}sF(s)\n",
    "$$\n",
    "\n"
   ]
  },
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i_0(\\infty) = 0\\; A$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle i_1(\\infty) = 0\\; A$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
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   ],
   "source": [
    "i0_inf_tvf = sp.limit(s*I0, s, 0)\n",
    "i1_inf_tvf = sp.limit(s*I1, s, 0)\n",
    "\n",
    "\n",
    "symdisp('i_0(\\infty) = ', i0_inf_tvf, ' A' )\n",
    "symdisp('i_1(\\infty) = ', i1_inf_tvf, ' A' )"
   ]
  },
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle V_c(s) =\\frac{300.0 s^{2} + 10500.0 s + 37500.0}{s \\left(1.0 s^{3} + 35.0 s^{2} + 375.0 s + 3125.0\\right)}\\;Vs$"
      ],
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       "<IPython.core.display.Math object>"
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   ],
   "source": [
    "C = 4e-3  # F\n",
    "\n",
    "# Calculando Vc\n",
    "Vc = (1/(s*C))*I1\n",
    "\n",
    "Vc = Vc.simplify()\n",
    "\n",
    "symdisp('V_c(s) =', adjustCoeff(Vc).simplify(), 'Vs')"
   ]
  },
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   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAOCAYAAADNGCeJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJElEQVQ4EaWT7Q2CMBCG0TgA0Q10Az82wA10BR3Bv/yVDdAJjG4gK8gGOoJxA3zepm0Q1IBeUt7rfbzt3dFOURSBkziOx+hH1gT94ewOsQ3RN3Y/Be/aY89l66GE4J4lhwKUUBNLlIJz50Tfol9kY2VdPg/WkrXGcXCBb1CJivFCjm6pClRN0NWnoUTEXSFQJWXJ2ITYh23IlHQjqdZLyxz2ykd80yFZfvBraAH+vM3NalwQiEgDMxP+iwwSNf4EaQK2GoDivUCQslEPffk/3QyCFUR90P9zOqXxABQsgWABjEB/I3T1rV2ZJKnhM9A9KXFIdMC9erOBcVEC+PI/2dPV8Axd/SpLhC0xZCjmOeCNbMQR2w39DO6s7QyqHPWrKuahPwGbS2D0goaxHgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle 12$"
      ],
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       "12"
      ]
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   "source": [
    " sp.limit(s*Vc, s, 0)"
   ]
  },
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-25. +0.j,  -5.+10.j,  -5.-10.j,   0. +0.j])"
      ]
     },
     "execution_count": 10,
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   "source": [
    "np.roots([1, 35, 375, 3125, 0])"
   ]
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     "data": {
      "image/png": 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2buRXEayBFSx+LW5t/KxFLl18mMy60LF7W0XA+sUZSf7/Ita4pRVGlaYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{-7.5 + 15.0 i}{s + 5.0 + 10.0 i} + \\frac{-7.5 - 15.0 i}{s + 5.0 - 10.0 i} + \\frac{0.12}{0.04 s + 1.0} + \\frac{12.0}{s}$"
      ],
      "text/plain": [
       " -7.5 + 15.0⋅ⅈ      -7.5 - 15.0⋅ⅈ         0.12       12.0\n",
       "──────────────── + ──────────────── + ──────────── + ────\n",
       "s + 5.0 + 10.0⋅ⅈ   s + 5.0 - 10.0⋅ⅈ   0.04⋅s + 1.0    s  "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "partFrac(Vc, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T12:56:55.910320Z",
     "start_time": "2021-10-05T12:56:53.544784Z"
    },
    "id": "Om6pHBZpUuW0"
   },
   "outputs": [],
   "source": [
    "#vc = invL(partFrac(Vc,4), s, t)\n",
    "\n",
    "#symdisp('v_c(t) = ', vc, ' V')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "ICrkbuDGUuW0",
    "outputId": "b6d2d1a5-5e04-4d71-dd1c-8587d73768ee"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle v_c(t) = \\left(\\left(30 \\sin{\\left(10 t \\right)} - 15 \\cos{\\left(10 t \\right)}\\right) e^{- 5 t} + 12 + 3 e^{- 25 t}\\right) \\theta\\left(t\\right)\\;V$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vc = (12 + sp.exp(-5*t)*( -15*sp.cos(10*t) + 30*sp.sin(10*t) ) + 3*sp.exp(-25*t) )*sp.Heaviside(t)\n",
    "\n",
    "symdisp('v_c(t) = ', vc, 'V')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T12:56:59.179862Z",
     "start_time": "2021-10-05T12:56:55.912317Z"
    },
    "id": "pTNu5WsWUuW0",
    "outputId": "47d0fee5-51e7-47f1-f2a9-4523405f2992"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# plota funções no domínio do tempo\n",
    "intervalo = np.arange(-1, 4, 0.01)\n",
    "symplot(t, vc, intervalo, 'vc(t)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "966-3kcCUuW1"
   },
   "source": [
    "c-e) Determinando $H_0(s)$, $H_0(s)$ e $H_c(s)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T12:56:59.211841Z",
     "start_time": "2021-10-05T12:56:59.181862Z"
    },
    "id": "JRE0vhWTUuW1",
    "outputId": "acbeb82a-8483-49da-c934-c579f690456d"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle H_0(s) =- \\frac{12.5 s}{s^{3} + 35.0 s^{2} + 375.0 s + 3125.0}\\; $"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "V = 12/s\n",
    "\n",
    "H0 = I0/V\n",
    "\n",
    "symdisp('H_0(s) =', H0,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T12:56:59.273804Z",
     "start_time": "2021-10-05T12:56:59.214840Z"
    },
    "id": "XZKe1Oh2UuW1",
    "outputId": "b9a8e09f-1efe-4f41-90f0-9caefdb802cd"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle H_1(s) =\\frac{0.1 s^{3} + 3.5 s^{2} + 12.5 s}{1.0 s^{3} + 35.0 s^{2} + 375.0 s + 3125.0}\\; $"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H1 = I1/V\n",
    "H1 = adjustCoeff(H1)\n",
    "\n",
    "symdisp('H_1(s) =', H1,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T12:56:59.321773Z",
     "start_time": "2021-10-05T12:56:59.275803Z"
    },
    "id": "7n4eb_QwUuW1",
    "outputId": "b03aa34c-7dc3-480f-d101-6d2de3d65866"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle H_c(s) =\\frac{25.0 s^{2} + 875.0 s + 3125.0}{1.0 s^{3} + 35.0 s^{2} + 375.0 s + 3125.0}\\; $"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Hc = Vc/V\n",
    "Hc = adjustCoeff(Hc)\n",
    "\n",
    "symdisp('H_c(s) =', Hc,)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "pCiEjYDQUuW2"
   },
   "source": [
    "f. $v_c(t)=?$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T14:00:07.360045Z",
     "start_time": "2021-10-05T14:00:01.893218Z"
    },
    "id": "Sr9nC17dUuW2",
    "outputId": "eeafbcfa-cc13-4505-85da-3e94303857c7"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle v(t) =2 \\theta\\left(t\\right) + \\theta\\left(t - 2\\right) - 3 \\theta\\left(t - 1\\right)\\; V$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = 2*(sp.Heaviside(t)-sp.Heaviside(t-1)) - 1*(sp.Heaviside(t-1)-sp.Heaviside(t-2))\n",
    "\n",
    "symdisp('v(t) =', v, ' V')\n",
    "\n",
    "# plota funções no domínio do tempo\n",
    "intervalo = np.arange(-1, 3, 0.01)\n",
    "symplot(t, v, intervalo, 'v(t)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T14:33:16.718436Z",
     "start_time": "2021-10-05T14:33:15.212369Z"
    },
    "id": "piKX_yB8UuW2",
    "outputId": "db1c2186-56ba-41c0-c746-8c4f2171ac3d"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle V_c(s) =\\frac{\\left(25.0 s^{2} + 875.0 s + 3125.0\\right) \\left(2 e^{2 s} - 3 e^{s} + 1\\right) e^{- 2 s}}{s \\left(1.0 s^{3} + 35.0 s^{2} + 375.0 s + 3125.0\\right)}\\; Vs$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# determina Vc(s) via função de transferência\n",
    "V = L(v, t, s)\n",
    "Vc = Hc*V\n",
    "\n",
    "symdisp('V_c(s) =', Vc.simplify(), ' Vs')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T14:35:02.416911Z",
     "start_time": "2021-10-05T14:34:59.676607Z"
    },
    "id": "YWU-ZiLgUuW2",
    "outputId": "6bf2df29-6ea2-41e7-abac-fca434bbad78"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle P(s) =\\frac{-0.625 + 1.25 i}{s + 5.0 + 10.0 i} + \\frac{-0.625 - 1.25 i}{s + 5.0 - 10.0 i} + \\frac{0.25}{s + 25.0} + \\frac{1}{s}\\; Vs$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# função auxiliar Va(s)\n",
    "P = (25*s**2 + 875*s + 3125)/(s**4 + 35*s**3 + 375*s**2 + 3125*s)\n",
    "#P = partFrac(P, 10)\n",
    "\n",
    "symdisp('P(s) =', partFrac(P, 10), ' Vs')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T14:35:02.416911Z",
     "start_time": "2021-10-05T14:34:59.676607Z"
    },
    "id": "_BDe5ZSZUuW2",
    "outputId": "67baf491-d24b-4bc3-b51c-e85fd24c69e3"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle p(t) =\\theta\\left(t\\right) + \\frac{5 e^{- 5 t} \\sin{\\left(10 t \\right)} \\theta\\left(t\\right)}{2} - \\frac{5 e^{- 5 t} \\cos{\\left(10 t \\right)} \\theta\\left(t\\right)}{4} + \\frac{e^{- 25 t} \\theta\\left(t\\right)}{4}\\; V$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# encontra va(t)\n",
    "p = invL(P.apart(), s, t)\n",
    "p = p.expand()\n",
    "\n",
    "symdisp('p(t) =', p, ' V')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T14:37:29.299064Z",
     "start_time": "2021-10-05T14:37:29.220114Z"
    },
    "id": "wO_nqlEJUuW2",
    "outputId": "e16f51f0-bd09-4402-d2b2-50d078d947f4"
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle v_c(t) =2 \\theta\\left(t\\right) + \\theta\\left(t - 2\\right) + \\frac{\\theta\\left(t - 2\\right) e^{- 25 \\left(t - 2\\right)}}{4} + \\frac{5 \\theta\\left(t - 2\\right) e^{- 5 \\left(t - 2\\right)} \\sin{\\left(10 \\left(t - 2\\right) \\right)}}{2} - \\frac{5 \\theta\\left(t - 2\\right) \\cos{\\left(10 \\left(t - 2\\right) \\right)} e^{- 5 \\left(t - 2\\right)}}{4} - 3 \\left(\\theta\\left(t - 1\\right) + \\frac{\\theta\\left(t - 1\\right) e^{- 25 \\left(t - 1\\right)}}{4} + \\frac{5 \\theta\\left(t - 1\\right) e^{- 5 \\left(t - 1\\right)} \\sin{\\left(10 \\left(t - 1\\right) \\right)}}{2} - \\frac{5 \\theta\\left(t - 1\\right) \\cos{\\left(10 \\left(t - 1\\right) \\right)} e^{- 5 \\left(t - 1\\right)}}{4}\\right) + 5 e^{- 5 t} \\sin{\\left(10 t \\right)} \\theta\\left(t\\right) - \\frac{5 e^{- 5 t} \\cos{\\left(10 t \\right)} \\theta\\left(t\\right)}{2} + \\frac{e^{- 25 t} \\theta\\left(t\\right)}{2}\\; V$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vc = 2*p - 3*p.subs({t:sp.UnevaluatedExpr(t-1)}) + p.subs({t:sp.UnevaluatedExpr(t-2)})\n",
    "\n",
    "symdisp('v_c(t) =', vc, ' V')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-10-05T14:41:14.890953Z",
     "start_time": "2021-10-05T14:41:01.790522Z"
    },
    "id": "L2RHQL_fUuW3",
    "outputId": "a1de4d18-c877-4f47-bd8a-dc6d95c3e166"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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HpPrfzxCNM/7EqRnJRCQXmA88YIzpdEjD6NGjzdq1ax05bzxVVFRQXl6e7DA6tHZ3NRf897sMzBXe/78XIbO/gax5lcBXn4ITLo/vydf8E8/sazCFQzG3LQWPN+whFfMrKD+7vPOdavcjr9yKrH8DADPsS5gLHrTljziqbvTx1f/6X97M/B512YPo9f3U/4Ey1f9+hmiczhGRJcaYU4/+3JEeuYikAy8Cz4ZL4so5zX5bbvCK4PGIHcEBeHKLwRPnm3nHT4eiUqRyC7J+Loy5JOwhnlCcHdm5DJ6/Bqq2QVYBXPAgctLXkQTcmEz3CjXGloi8/vq4n08pJ8U8/FDsv7IngdXGmN/EHpKKlC9gf5pKC+W5lpud/eJ/co8XTrvFvn/3V7EP91v2LDx5gU3igyfBzR/BxKsTNrrEI0ItdgpTTeTKbZwYRz4FuAb4kogsD76mOdCuCsMfTJ7e0J9ibZwfCDraxG9A3kC7MPOKv3atDV8TvPaf8I9bwN8Ip1wH//4aFAxyNNRwvJ62ibxBpyBQruLEqJX3AR2UmwTNfptsPIIdY9142I6fzipMTAAZOTD1PnhpJrz9EzjhUsjMi/z4w7vgb/9uZ1P0ZsD0X8PJ18Yr2k55RfDjpd5kkC1N0FSrQxGVa+iTnS7mCyZyr3DkGHJPAv9Yx19pl4Kr2QP/+lHkx215H37/RZvE8wfBda8nLYkDLbX7mmCvnKaapMWiVLQ0kbtYc0tpRRJfVgnxeOCSh8GbCUv+BEue7nx/vw/e+w08fSnU7oXSs2DmfBh8SmLi7USaR6gN3vCkURO5cg9N5C7mP6JHbkesxPWpzo4MGA/THrLvX73DPoXZTo05v2o1PHmeLcMYP5z5XbjmZchNQszt8LSpk9NUndxglIqCrtnpYr5Qj1ywvVtIzIiV9pzyTftTwbz7Ye734NOXYMJVdqWifWtg7eucvPVDu2/+YLj04dbl41KEV4QatEeu3EcTuYuFbnZ6PSSvtNLWF++EolKYcyds/ci+2vB5e5E25VY44zuQlZ+cGDvh9Qi1Aa2RK/fRRO5i/kCKlFbaGv9VGHk+rPybTeT1lXZSq7KzWbAnmzO/dHFy4+uER2gtrWiPXLmIJnIXa/tkZ0JmPoxUVj5M+pZ9teFL8Xks0rweanzB0orWyJWL6M1OFws92en1AHUH7Ie9+iQvIJdr+3Sn9siVm2gidzGfv83NzrqD9kNN5F3m9dA6Ja/WyJWLaCJ3MV/bGnlLj7x38gJyOa8INUZ75Mp9NJG7mK/tqJWWHrkm8q6y48i1Rq7cRxO5i4We7MzAZxOPeO0yZqpLvB7tkSddcwPsXgnVu5MdiavoqBUXC/XI800w6WQXJXaelW7Ge8STnZrIE27Fc/D6D6A++NPl6Glw8W8hb0By43IB/VfvYqEaeR7BMoDe6IyJV9qUVrRHnljLZsFLN9okXngcpGXB2jnwx6lQuSXZ0aU8TeQuFhq1kh/QRO6EI0orWiNPnAMb7Zz0ABf8DG7/BL6z3C4wUrUNnv0aNOqfR2c0kbtYS4/cHLYf6I3OmHi0R54cb90HvgY7N8/pt9pVofJL4Bt/h+LjYf9aW3JRHdJE7mKhJztzQzVyTeQxObJHrok8IfZ8BqtfgbRsmPqTI7/Lyoev/dkuOrLsGdhUkZQQ3UATuYuF5lrJ1dKKI44Yfqg98sT4+Pd2O/Ebthd+tOLRcPb37Pt/3g7Nup5qezSRu1ho9sOcUCLP1h55LNI8Qh2ZGAR89XYRDBU/TXXwyd/s+8nf7ni/KXdAv7H2pufCxxMRmetoInex0M3OnECoRq498lh4RQCh2aPllYRY9zo018KgU23PuyPedLjgp/b9e79tffhNtdBE7mKhm505/lBpRXvksQgNwW/y6HwrCbHqRbsd/9Xw+w7/kn01VsG7v4pvXC6kidzFQom8VyB0s1N75LHwBhdgbvZonTzufI2w8R37fswlkR0Tuhm66A9QtT0+cbmUJnIXC5VWevm1tOIEj9hE3qSllfj7/ANbVuk/DgoGR3ZMyQQYewX4m7RXfhRN5C4WutmZFSqtZBclMRr3C/XIW0or+hBK/Kx/y25HnhfdceU/APHY4YgHNzsfl0tpIncxXyBAGj6yAnX2L3dWYbJDcrW0YCJv1Bp5/G15126HnRPdccWjYMIMCPjg3Yecj8ulNJG7mD9gKCI0YVZvnTArRq2lFa2Rx1XdQdi9yj7oM2Ry9Mef/T3wpMGKv8L+9c7H50L6L9/Fmv0BCkWf6nSKV3vkibH1I8DYuVTSs6M/vneZfYDIBKDiQcfDcyNN5C7m8xt668yHjvEcnci1Rh4f2xba7dAzut7GF++yPfpVL9refQ+nidzFmgOGQtFE7hRvsLTSKNojj6sdS+120Cldb6NgMJx6vX2vvXJN5G7mDwToLTpixSktpRXRVYLiJhCAncvt+4Enx9bWmf/HTra15lXYuSzm0NxME7mL+fyGIi2tOCaUyBt0HHn8HFhv53rPHwR5/WNrK69/6xwt8x6IPTYX00TuYs3+AEV6s9MxodJKg2iNPG5CZZWBE51pb8odkJELG96ErQucadOFNJG7mC9gWksr2iOPWehmZ4PWyONnpwP18bZy+sBpt9j3837qTJsupIncxXx+QyE6z4pTvMF/Da09ck3kjmu50Rljfbyt02+FrALY8l6PXXxCE7mL+Y642amllVi1lla0Rh4XvibY/Yl9X3KSc+1mF8IZ37Hv594N/mbn2nYJTeQupjc7nXVMaUV75M7av9ZOeNV7mE2+Tjr9Vigqg32r4aPfOdu2CziSyEXkQhFZKyIbRORuJ9pU4dmbnToXuVNCc63UtfTI9Wano/auttv+Y51vOz0bpgdnRJz/Czi4yflzpLC0WBsQES/wO+A8YDuwSEReMcZ8FmvbKoyAjwJPHQYPohNmxSzUI6/XHnl87A2mhH4nxKf9EVNh/JWw8m/w4g1w/Rt2daFEqD9kx7LvXQ2HPodDW6GhypbnmmptucebYePxZtifSLJ72+c/eoW2fewru7f9rFdvyMyHYMmvMzEncmAysMEYswlARJ4DLgM6TOTemh2sevBsB04dX4XNzaz6KEF/EbrgDzQA0JyWS4ZOmBWzUI28mXTwpEOg2S6AkJaZ5MjcqbqhmR++tIr12+p5Yv0C7jzwIScDD69MY+GG+AwVzAlcxc+88ynesYRXfnsLzxV8K+JjKyttnJFIM02c0PgJJzYsYkLjEgb5tnU15E758FLjyafak0+1p6DjeBw41yCg7e9iO/CFo3cSkZnATIBTSjyMa1zuwKkToDHZAXQimLurs0pYWVGR1FAiUVNTQ0UKx7ltWxMAjY1NNHuySA8088G8N2jOyE9yZO1L9eu5aLePV1YE/wEdPEBxxibwwCu7CtloDsTtvJVyE89n3M+lNbN5t7KIF/xRdBoPdh7XCNnODO87XOF9j97S+hNbo0nnMzOUzwJD+dz0Y5vpRyV51JosasnCh5d0fKTjJ4smCqSWAmookhoKqaFQaiiSanpTTZFUUyQ1FFFNjjRSGKikMFDZaVxOJPL2+v3mmA+MeQJ4AmBE2RCz6txHHTh1fG3ctInhw4YlO4xODe6dw2e7migvL092KGFVVFSkdJzLmtfBxvWkpWeQnlMEVdVMmTQBikqTHVq7Uv167l+yHZavYGwfDz+6eAxDnt9HwJPB/dddgvHE8yfdL7BlXRrDP/4xv8x4km99cSIHh4RfwGLFihWceOKJx3zu8dXR9/M59N/wPPn7lrZ8Xls4ioODz6VyYDk1fSZgvBmUAqUxRn84+PocEH8j6Y2VpDUeIr3xIPzksnaPcSKRbweGtPn1YGBnZwd4M3ox7qz2A0ol+/0VjDurPNlhhOXfV5HsELqF0CP6AQNk5toPtU7eZQ3NfgCKsz2clrcfAE/xKM4YVRL/k4+4HTIq8bz/W8a8ewtMewhO/Van9ebm7V6mjOjb+sHO5bD0aVj5AjQGl1PMyLOLRZ98LTkDJ5IjckTyi49BYfdwIpEvAkaKSBmwA5gBfN2BdpVKqFAi9xvsY9+gY8ljEErkGV7if6OzPefeC95MmP9zeO0/7fJyF/2885+wDu+yU+N+8nzrmHeAwZPh5Gth7Jdb/5NPITEncmOMT0RuA94AvMD/GmM+jTkypRKs/R65DkHsqtZELq1DD/uNSVwAInDOD+xCFHPugnVzYf0bMPxLUHY29BlhVxqq3Qv71nLyyrlQsYGWynB2kV1W7uRroX8C/wPqAid65Bhj5gBznGhLqWQJjVoxxthhX9D6I3WyBQK2V9twyCagvAHJjiis+mT3yENOnGET99s/sb3tDW/Z11HywQ4NHHk+TPgajLwA0rMSHm5XOJLIleoOQuPIAwBZwUTekAKJfO1ceP1uqNzS+lnZ2eT2vgwoT1JQ4TU0BwBI9wjsW2c/LB6VnGDyS+DLj8P5D9j5y3etsGO9Mbbn3Xs4nxzMZMLFM1OydBKOJnKlgrzB+2C2tJIiPfIFj8Pr37fv8wZC4RDY9Qlsns/JWz6A4mb4wo0RPTSSaKEeeZ7UQ/VO29stHJrcoHL6wCnfbPergxUVrkzioIlcqRZH1shToEe+7o3WJD71PjsxlMdrnyKc91M8i/5gvz+4CS76Rcol81CNvL9/l/2g9zAbv3KcPg6oVJA3+HSs39BaWklWj7zuILx8s33/pR/Bmd9tTYLZhTD9V3x6QnAB4o9/D3O/B+aYxzeSqiWR+3bbD/qMSGI03ZsmcqWCQvORm1TokVc8CHUHYOiZdm3KduzrdyZc9WwwmT8BH/x3YmMMI1Qj7+vbYT/QRB43msiVCvJIm9JKMnvkh7bB4j+BeGDaL6GzeXRGnQ9feRIQeOs++PTlBAUZXn2T7ZH3bQ6WVjSRx40mcqWCWmvkbYcfJmEc+UeP2gm7xl4R2ZSvJ1wK5/3Evn/pRti+JL7xRajBZxN5UWOwR953ZBKj6d40kSsVdMTNzpbhh1WJDaK5AVb81b6fcnvkx53xHfvgiq8Bnvs6HO50loyEsD1yQ0FjMBbtkceNJnKlglpKK5C84YdrXrX/eZScBCUTIj9OBKb/BkrPgprdNpk31cUtzEg0+gIUU0VGoB6yCnUVqzjSRK5UUFoqDD9cNstuJ34j+mO96XDl03as9s5l8MptSR3JUt/kZ5i0qY+n2PDI7kQTuVJBnvZKK42HE5cMD22zq8B7M+0Me12R0we+/ryd9GvVi/DerxwNMRoNPj9lnmAi1/p4XGkiVyrI23bUSlqmTagBHzTXJyaAz/4BGDh+mn1svKv6jWkdyTLvp/DZK05FGJUje+TDkxJDT6GJXKmgI252QuKHIK573W6Pvzj2tkZfCFPvte9fvAE2vxt7m1EIBAyNvgBlLYlce+TxpIlcqSBP2+GHkNg6eUMVbP0IxAsjznWmzSl3wKnXg78R/jIDti1ypt0INPrsw0AjPDqGPBE0kSsVdERpBdr0yBMwlnzjPFvGOe702MoqbYnAtF/DhKuguRae/UrCknl9s580fAyWvRhESytxppNmKRUUKq00+mFXVT29vblkAgcO7qMpL7518oJVr9ELOHzcOdRWRXaugw0BdkWy7zm/pqiumqwNczBPX0LltMdoHH5hbAGHsfdwI6NkO+n4acjsR1Z6dlzP19NpIlcqKC04j+2mqgCnPziPx9MbuNAL9zz3IXMD/ridVwiwOPN1egl8+a18Nr45L/KDKyLbN40ZPJBWx1VU0PuVf+dx38X82vc1mh1KATnUM9GzgUmetZwkGxjt2cacTLvye12vgbhjeQb30kSuVNDYgfmcOrSIjbsPkZmZic+XCwEYnN3MAG/8UtGIwBb6+KrZRV9qc4cxIMLx1o2NjWRmZkZ8nt+a/2B/YDA3+v/CTWmvcn76Cv7bex0fy4lRj/HON9VMNJ8yMfApJ5nVjDRb8NpHqVo0kEFV3kh2D/kyvaNqXUVLE7lSQb0y0njh5jOoqKigvLwc5r4JC9/hh+cO4odnOHQDsj0Ln4C5UDJhKguumBrxYS1xRmUqbL0GXrqJYZWbecT3XzBggn28f8RUuzDx0Und74MDG2D3Stj+MWz5APYetSyvJw1KJtoa/5DJ0H8cWUWlZHm8rK6oiDJGFS1N5Ep1JFHDDz//wG6HnhHf84QcdxrcuhAWPAYfPmJXi59zp/0uswAKj4P0bPA3Qe0+qNlrJ/Fqy5sJgydB6Zk27sGnQkZOYuJXx9BErlRHEjH80Bj4/EP7fuiU+J3naGmZcOYd8IWb7INIa1+zY83rK2HPymP3LzwO+o+HgSfZOAed4pqFiXsCTeRKdSQRPfIDG6F2L+T0S84QvfQsOPEq+zIGavfD4e3gawRPOuQWQ6++kNEr8bGpiGkiV6ojiZgBsW1ZJdmTSonYxJ1bnNw4VNT0gSClOpKVgNJKMsoqqtvRRK5URxLRI9+x2G6HTI7fOVS3p4lcqY7E+2Zn/SE7rM+bGdmSbkp1QBO5Uh2J983OncvstmSCXRRCqS7SRK5UR+LdI9+51G4Hnhyf9lWPoYlcqY6kZ9uyh78xPotL7Agm8kGayFVsNJEr1RGR1ill6yudb78lkZ/ifNuqR9FErlRn4pXID++C6p22fNNb5+pWsdFErlRnsgvt1ulE3lIfPwk8+s9QxUb/BinVmXj1yHevstuSE51tV/VImsiV6ky8EnloGth+On5cxU4TuVKdaUnkh5xtd+9qu+1/grPtqh5JE7lSnYlHjby5wc56KB7oO8q5dlWPFVMiF5GHRGSNiHwiIi+JSKFDcSmVGrIK7dbJRL5/LRi/Ha2iixIrB8TaI38TGGeMmQCsA34Qe0hKpZB41Mi1rKIcFlMiN8b8yxjjC/5yATA49pCUSiHxSOR7Qjc6NZErZzhZI78emOtge0olXzx75JrIlUPEGNP5DiJvAQPa+eqHxph/BPf5IXAqcIXpoEERmQnMBCguLj5l9uzZscSdEDU1NeTm5iY7jLA0Tme1jTOrfjenLbyR+qx+LDztD460f9pH15PVeICFkx+jvtdAR+JMZRqnc84555wlxphTj/nCGBPTC/gm8BHQK9JjRo0aZdzgnXfeSXYIEdE4nXVEnHUHjbk335ifDXam8VB79/c3xu+LqSlXXs8U5oY4gcWmnZwa05qdInIh8H3gbGNMXSxtKZWSMgsAsXOS+5tjnzd87xq7LR4NHm/M4SkFsdfIHwXygDdFZLmIPO5ATEqlDo+ndSx5Q1Xs7YWe6NQVgZSDYuqRG2NGOBWIUikru8je7KyvhJy+sbW15zO77Tcm9riUCtInO5UKx8nH9HXEiooDTeRKhePUEERjtLSi4kITuVLhOPWY/uGdts6eXQS5/WMOS6kQTeRKheNUj7ylrDLWLiOnlEM0kSsVjmOJPFRW0fq4cpYmcqXCcSqR64gVFSeayJUKx7EeeSiR641O5SxN5EqF06uP3dbt73obfh/sW2vfa49cOUwTuVLhhB4Cqo0hkR/cBP5GKBgCWfnOxKVUkCZypcJxIpG3lFX0RqdyniZypcLpFUzkdfvtQz1dsVdvdKr40USuVDgZvSA9B/xN0FjdtTb26BOdKn40kSsViZbyyr6uHa9zrKg40kSuVCRCibzuQPTHNtXZm53ihb4jnY1LKTSRKxWZnGK77UqPfN8awNgknpbpaFhKgSZypSLTK4aRK1pWUXGmiVypSMRSI9ehhyrONJErFYnQtLM1e6M/do9OlqXiSxO5UpHICyXy3dEfq6UVFWeayJWKRKhHXr0nuuPqDtrkn54DhUOdj0spNJErFZncAXYbbY+8pT5+PHj0n5uKD/2bpVQk8tr0yKN5TH+P3uhU8aeJXKlIZObZ8oivHhoPR35caFUgTeQqjjSRKxWpvC6MXAnd6NQRKyqONJErFamWG54R1smNOXLBZaXiRBO5UpHKC97wrN4V2f5V22wZpldfyC2OX1yqx9NErlSk8gfZbdX2yPbfvdJuB4yPTzxKBWkiVypSBUPs9vCOyPZvSeTj4hOPUkGayJWKVEFXe+QT4hOPUkGayJWKVMFgu62KskfeX3vkKr40kSsVqfxQIt8Wft+GKjj0OXgzdTEJFXeayJWKVE5fm5gbDkFjTef7hmY87DcGvOlxD031bJrIlYqUSOR1cr3RqRJIE7lS0Sgqs9vKzZ3vpzc6VQJpIlcqGr2H2e2BjZ3vt2uF3eoYcpUAmsiVikYokR/c1PE+TbW2Ri5eKDkxMXGpHk0TuVLR6DPcbjtL5DuWgvFD/7GQkZOYuFSP5kgiF5E7RcSISF8n2lMqZUXSI9++yG6HTI5/PErhQCIXkSHAecDW2MNRKsUVDgXx2LHkzfXt7xNK5IM1kavEcKJH/lvge0AUy6Yo5VJpGdBnJJgA7Ftz7PfGwLaP7fshkxIbm+qxxESzbNXRB4tcCpxrjLldRLYApxpj9new70xgJkBxcfEps2fP7vJ5E6Wmpobc3NxkhxGWxumscHGe8OlD9Nv3PmtG/we7S6Ye8V1W/S5OW3gTTen5fHjGn+3Y8yTFmSo0Tuecc845S4wxpx7zhTGm0xfwFrCqnddlwEKgILjfFqBvuPaMMYwaNcq4wTvvvJPsECKicTorbJzzHzLm3nxj5t597HfLn7PfPXtVXGJrq9tczxThhjiBxaadnJoW7n8AY8zU9j4XkfFAGbBCbK9jMLBURCYbY6JcalwpFwmNDQ899NPW5nft9rjTEheP6vHCJvKOGGNWAv1Cvw5XWlGq2wiNDd+5DPw+8Ab/GRkDG9+270ecm5zYVI+k48iVilbeAOg9HJpqWp/gBNj7mV0GLre/Tl2rEsqxRG6MKdXeuOoxSqfY7efvt3726ct2O/K8uN7kVOpo2iNXqitKz7Lbdf+yW2Ng5d/s+/FXJicm1WN1uUauVI826gJIy7Y98oOb7SRalZshd0BrklcqQTSRK9UVWQVwwqXwyfMw587W+clPvxU83uTGpnocLa0o1VVT7oC0LNjwln3Ks2AITPpWsqNSPZAmcqW6qv8J8JUnoXgMHHc6XPsPne1QJYWWVpSKxZiL7UupJNIeuVJKuZwmcqWUcjlN5Eop5XKayJVSyuU0kSullMtpIldKKZfTRK6UUi6niVwppVwupjU7u3xSkWpgbcJPHL2+gBum5tU4naVxOkvjdM5QY0zx0R8m68nOtaa9BURTjIgs1jido3E6S+N0llvibI+WVpRSyuU0kSullMslK5E/kaTzRkvjdJbG6SyN01luifMYSbnZqZRSyjlaWlFKKZfTRK6UUi6XkEQuIleKyKciEhCRDof3iMiFIrJWRDaIyN2JiO2o8/cWkTdFZH1wW9TBfltEZKWILBeRxQmKrdNrI9Yjwe8/EZGTExFXF+IsF5Gq4LVbLiI/TlKc/ysie0VkVQffp8r1DBdn0q+niAwRkXdEZHXw3/nt7eyT9OsZYZxJv55dYoyJ+wsYA4wGKoBTO9jHC2wEhgEZwArghETE1yaGXwJ3B9/fDfyig/22AH0TGFfYawNMA+YCApwGLEzktYsiznLg1UTH1k6sXwROBlZ18H3Sr2eEcSb9egIlwMnB93nAuhT9+xlJnEm/nl15JaRHboxZbYwJ9yTnZGCDMWaTMaYJeA64LP7RHeEy4Ong+6eByxN8/o5Ecm0uA/5srAVAoYiUpGCcKcEY8y5wsJNdUuF6RhJn0hljdhljlgbfVwOrgUFH7Zb06xlhnK6USjXyQcC2Nr/eTuIvcn9jzC6wf+hAvw72M8C/RGSJiMxMQFyRXJtUuH6RxnC6iKwQkbkiMjYxoUUtFa5npFLmeopIKTARWHjUVyl1PTuJE1LoekbKsUf0ReQtYEA7X/3QGPOPSJpo5zPHx0Z2FmcUzUwxxuwUkX7AmyKyJthzipdIrk1Crl8YkcSwFDtfRI2ITANeBkbGO7AuSIXrGYmUuZ4ikgu8CNxhjDl89NftHJKU6xkmzpS5ntFwLJEbY6bG2MR2YEibXw8GdsbY5jE6i1NE9ohIiTFmV/DHvr0dtLEzuN0rIi9hSwrxTOSRXJuEXL8wwsbQ9h+OMWaOiPw/EelrjEm1yYpS4XqGlSrXU0TSscnxWWPM39vZJSWuZ7g4U+V6RiuVSiuLgJEiUiYiGcAM4JUEx/AK8M3g+28Cx/wkISI5IpIXeg+cD7Q7osBBkVybV4Brg6MDTgOqQmWiBAobp4gMEBEJvp+M/Tt4IMFxRiIVrmdYqXA9g+d/ElhtjPlNB7sl/XpGEmcqXM8uScQdVeDL2P+RG4E9wBvBzwcCc9rsNw17J3kjtiST6LvafYC3gfXBbe+j48SOyFgRfH2aqDjbuzbATcBNwfcC/C74/Uo6GB2UAnHeFrxuK4AFwBlJivOvwC6gOfh381spej3DxZn06wmciS2TfAIsD76mpdr1jDDOpF/Prrz0EX2llHK5VCqtKKWU6gJN5Eop5XKayJVSyuU0kSullMtpIldKKZfTRK5cRUQKReSWZMcB9jFvEakXkeVh9ntWRA6KyFcTFJrqYTSRK7cpBFIikQdtNMac1NkOxpirSfzDbaoH0USu3ObnwPDgXNEPAYjIXSKyKDjP9U+Cn5WKyBoR+aOIrAr2iqeKyAdi55ufHNzvPhF5RkTmBT//dvBzEZGHgseuFJGrwgUWfOr3teCES6siOUYpJzg214pSCXI3MC7UCxaR87GTGk3GPj34ioh8EdgKjACuBGZipw/4OvbpvkuB/0vrNMUTsHNk5wDLROQ14HTgJOBEoC+wSETeNZ0/Vn4hsNMYMz0YW4Ejv2OlwtAeuXK784OvZdiZ646ndba6zcaYlcaYAPax67eNfZR5JVDapo1/GGPqjZ0Y6R3sfwpnAn81xviNMXuA+cCkMLGsBKaKyC9E5CxjTJUzv0WlOqeJXLmdAA8aY04KvkYYY54MftfYZr9Am18HOPKn0aPnqTC0P+1qp4wx64BTsAn9QXHLMmHK9TSRK7epxi7TFfIGcH1wjmlEZFBwnvhoXCYiWSLSB7vU1yLstMRXiYhXRIqxS6593FkjIjIQqDPGzAJ+hV2iTam40xq5chVjzIHgDctVwFxjzF0iMgb4KDj7aA3wDcAfRbMfA68BxwH3G7toyEvYOvkKbA/9e8aY3WHaGQ88JCIB7GyFN0fze1Oqq3T2Q9Wjich9QI0x5lddOLYUu1DvuAj2fSq47wvRnkepcLS0olTX+YGCSB4IAs4GGhIRlOp5tEeulFIupz1ypZRyOU3kSinlcprIlVLK5TSRK6WUy2kiV0opl/v/c4s0yYt46agAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# plota funções no domínio do tempo\n",
    "intervalo = np.arange(-1, 3, 0.01)\n",
    "symplot(t, [v, vc], intervalo, ['v(t)','vc(t)'])"
   ]
  }
 ],
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