20230327 Sallen and Key.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "13c3d6b2",
   "metadata": {},
   "source": [
    "This notebook derives the transfer function of a Sallen and Key low-pass filter, and then for the Butterworth conditions, displayes a Bode plot ans step reponse.  You will want to sue the lateset veriodn of SymPy so tha thte inverse Laplace transforms do not take too long. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 245,
   "id": "f094c7cd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: schemdraw in c:\\users\\000000911\\anaconda3\\lib\\site-packages (0.15)\n"
     ]
    }
   ],
   "source": [
    "# Install a pip package in the current Jupyter kernel\n",
    "import sys\n",
    "!{sys.executable} -m pip install schemdraw --user"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "id": "7d924150",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import schemdraw\n",
    "import schemdraw.elements as elm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 336,
   "id": "d9847954",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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       " <defs>\n",
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       "<schemdraw.backends.mpl.Figure object at 0x0000020A72929070>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "elm.style(elm.STYLE_IEC)\n",
    "with schemdraw.Drawing(file='sallenkey.jpg',inches_per_unit=.5) as d:\n",
    "    V1 = d.add(elm.SourceV().up().label('$V_{in}$'))\n",
    "    R1 = d.add(elm.Resistor(lftlabel='1').right().label('$Z_1$'))\n",
    "    d.add(elm.DOT,toplabel='2')\n",
    "    C1 = d.add(elm.Resistor().down().label('$Z_2$'))     \n",
    "    d.add(elm.GND(xy=V1.start))\n",
    "    R2 = d.add(elm.Resistor(xy=R1.end).right().label('$Z_3$'))\n",
    "    C2 = d.add(elm.Resistor().down().label('$Z_4$'))\n",
    "    d.add(elm.GND)\n",
    "    L2 = d.add(elm.Line(xy=C2.start,l=d.unit/2).right())\n",
    "    d.add(elm.DOT,xy=C2.start,toplabel='3')\n",
    "    op1 = d.add(elm.OPAMP,xy=L2.end,fill='red',d='right',anchor='in2')\n",
    "    L3 = d.add(elm.Line(xy=op1.in1,l=d.unit/2).left())\n",
    "    L4 = d.add(elm.Line(l=d.unit/2).up())\n",
    "    L5= d.add(elm.Line(l=1.5*d.unit ).right())\n",
    "    L6 = d.add(elm.Line(toy=op1.out,l=d.unit/2).down())\n",
    "    d.add(elm.DOT,rgtlabel='$V_{out}$')\n",
    "    L7 = d.add(elm.Line(tox=op1.out).left())\n",
    "    L8 = d.add(elm.Line(xy=L6.end,l=1.5*d.unit).down())\n",
    "    L9 = d.add(elm.Line(tox=C1.end).left())\n",
    "    L10 = d.add(elm.Line(toy=C1.end).up())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c03c2bc7",
   "metadata": {},
   "source": [
    "Use Ohm's law to put the aboce equaiton to terms of voltages and impedences\n",
    "$$\n",
    "\\frac{V_1-V_2}{Z_1}=\\frac{V_2-V_{out}}{Z_2}+\\frac{V_2-V_3}{Z_3}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "\\frac{V_2-V_3}{Z_3}=\\frac{V_3}{Z_4}\n",
    "$$\n",
    "Assume an ideal opanmp and $$V_{out}=V_3$$\n",
    "Rearrange to get into Matrix form:\n",
    "\n",
    " \n",
    "$$\n",
    "\\frac{V_1}{Z_1}=V_2 \\left(\\frac{1}{Z_1}+\\frac{1}{Z_2}+\\frac{1}{Z_3}\\right)-V_3\\left(\\frac{1}{Z_2}+\\frac{1}{Z_3} \\right)\n",
    "$$\n",
    "and \n",
    "$$\n",
    "0=\\frac{-V_2}{Z_3}+\\frac{V_3}{Z_4}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "id": "0075367c",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import symbols\n",
    "from sympy.matrices import Matrix\n",
    "from sympy import *\n",
    "import sympy as sym\n",
    "Z1, Z2, Z3, Z4 = symbols(\"Z1, Z2, Z3, Z4\")\n",
    "I1, I2, I3, I4, V1, V2, V3,Vin= symbols(\"I1, I2, I3, I4, V1, V2, V3,Vin\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "id": "bd6a51d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{V_{1} - V_{2}}{Z_{1}}$"
      ],
      "text/plain": [
       "V₁ - V₂\n",
       "───────\n",
       "   Z₁  "
      ]
     },
     "execution_count": 249,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1=(V1-V2)/Z1\n",
    "I1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 250,
   "id": "bd942056",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAuCAYAAAB+vtSwAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEUUlEQVRoBe2a7U0cMRCGD5QCEugAOgClAqCDoFQQ6CCIX/APkQ4gHQQ6COkgoYOQCpLQAXkfY1v23tyxLLv2StmR5uwdf41fz/jzVh4eHmanp6cfZ7PZe/GWGLoRX0h+7b78j75JvxJviO/EN5IdKqxGQ+i+AiiQKn+t4K/4VvFtZBYpbVfyQ4X7VnoNWd+6r4ZOqOJ7xWGsYBmNChAU7Vv3CIpHAZfAYkxS47jZmZlYX9ib7hYoID9nLV62rvC2fv9NDQClF91NUFT3HCiSnQuQIxoeKTlQpNuLdW+C8tN3OKtYYBxIfjFSMIJaven+KtTow4D2ZpALEADaVHiZyJh3jv13APCD8jBR16K2uqMvK+gfMfG34jPpHqeFJijflQEK+xXiuE1z+UUW9yeKY0U/xBFMClqkvAD6TbxwQjfK7atcVNpIR9Rad+W9U31ukBWyeKDPG7GjDBRluBeT4EZf8UVuc6C0KzGbPOhcjGxLvFR5pWNNC/dBVNaFqFdM0ad0b66e6yqD1UTKQPFSzHBDDTCSmdvEUtq8KR5GJhFXjz6pu/rVHLR30ppBjbQQFOXAEvZiziQieZxfvBiQMMlmg0mpIlEHilpaqHvQQrqGow3Hmaw/qyFTEoYJK0MvSc+iqpD5B7R7d4msoXYfrXWX3p9U5Y54zwMUW7BAYcK8VMYwX8TMzYjy4L+At604c0Vtaq07inqd2XuxcMTFJR4In9sbVQIgRwrdKuS/ZwrDaD23ysHzSzfmyV/iHcWdqyukH+xxONM5N7IsRenLyVfEMow/suKAMohns/jyWsqnSk+sGR3TgdvVN/IvYkfWRBvSloWYKagTRlKjzmqiYJwR9lzH0vW3V4/NW+b+nd1nnP3tR6tO7tNP0+OtZQLFGJsJlAkUAwFDtHJycvJ4c20k/q+iafUxRn6aUyZQDAQM0WQpEygGAoao69nHqKqdSGcOnmY5N3EI42CWHs429M3hkgurJ+97lW8QKgqKOsqJFEDiMT30SmnhgAlYVS+sioKiznJC5WY++zeDB4Yb9TVxdmL1aUWD0qCsWYBI9lW9xm0AJHWnomCExkqvPmeh4RAKBC6rcCvuSmtffDu1qu5oBQL3u9yqz80xTrtKP6UtJXZTgPDQBiDc82ZPDDFTpUgVSxEIPInwNzFeDUZ3hVkcFIHA/MHEeq149kat7/jsqjhLdJVH/KKrD532gPAHwgwQySGsJlhO50d8V9MLforNKQKB3Sp7Ef5oOPccKxkWlL4OHHhZ6B6TMm/cADsoFbEUdQRXwGV4c+Gp0iKW5nQni8VUecQvAoo6t3C36kf+s/Jw3mGL70jx5ooESEUe8QcHRZ1jlcHk2anyoqjA0Zp+cSmsCApzyeNX8qsylC/2iF989Un62ioqQAAO1+LMFC2pVeGOmQa3lI56uWIeEDZ3bmL23zOFg56PRmspHgAsJP2bKi4GSINazJgtpdoj/j9qv6z/gmPsygAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\frac{V_{2} - V_{3}}{Z_{2}}$"
      ],
      "text/plain": [
       "V₂ - V₃\n",
       "───────\n",
       "   Z₂  "
      ]
     },
     "execution_count": 250,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2=(V2-V3)/Z2\n",
    "I2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 251,
   "id": "fcc97c25",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{V_{2} - V_{3}}{Z_{3}}$"
      ],
      "text/plain": [
       "V₂ - V₃\n",
       "───────\n",
       "   Z₃  "
      ]
     },
     "execution_count": 251,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I3=(V2-V3)/Z3\n",
    "I3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 252,
   "id": "c6cda108",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{V_{3}}{Z_{4}}$"
      ],
      "text/plain": [
       "V₃\n",
       "──\n",
       "Z₄"
      ]
     },
     "execution_count": 252,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I4=V3/Z4\n",
    "I4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 253,
   "id": "54b48d22",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{Z_{3}} + \\frac{1}{Z_{2}} + \\frac{1}{Z_{1}}$"
      ],
      "text/plain": [
       "1    1    1 \n",
       "── + ── + ──\n",
       "Z₃   Z₂   Z₁"
      ]
     },
     "execution_count": 253,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a11=(-I1+I2+I3).expand().coeff(V2, 1)\n",
    "a11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 254,
   "id": "9c58985e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{1}{Z_{3}} - \\frac{1}{Z_{2}}$"
      ],
      "text/plain": [
       "  1    1 \n",
       "- ── - ──\n",
       "  Z₃   Z₂"
      ]
     },
     "execution_count": 254,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a12=(-I1+I2+I3).expand().coeff(V3, 1)\n",
    "a12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 255,
   "id": "7fe28ce0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{1}{Z_{3}}$"
      ],
      "text/plain": [
       "-1 \n",
       "───\n",
       " Z₃"
      ]
     },
     "execution_count": 255,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a21=(-I3+I4).expand().coeff(V2, 1)\n",
    "a21"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 256,
   "id": "e715b52b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{Z_{4}} + \\frac{1}{Z_{3}}$"
      ],
      "text/plain": [
       "1    1 \n",
       "── + ──\n",
       "Z₄   Z₃"
      ]
     },
     "execution_count": 256,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a22=(-I3+I4).expand().coeff(V3, 1)\n",
    "a22"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 257,
   "id": "0b45c335",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{Z_{3}} + \\frac{1}{Z_{2}} + \\frac{1}{Z_{1}} & - \\frac{1}{Z_{3}} - \\frac{1}{Z_{2}}\\\\- \\frac{1}{Z_{3}} & \\frac{1}{Z_{4}} + \\frac{1}{Z_{3}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1    1    1     1    1 ⎤\n",
       "⎢── + ── + ──  - ── - ──⎥\n",
       "⎢Z₃   Z₂   Z₁    Z₃   Z₂⎥\n",
       "⎢                       ⎥\n",
       "⎢    -1         1    1  ⎥\n",
       "⎢    ───        ── + ── ⎥\n",
       "⎣     Z₃        Z₄   Z₃ ⎦"
      ]
     },
     "execution_count": 257,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_SK = Matrix([[a11,a12], [a21, a22]])\n",
    "A_SK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 258,
   "id": "fa4b5bce",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{Vin}{Z_{1}}$"
      ],
      "text/plain": [
       "Vin\n",
       "───\n",
       " Z₁"
      ]
     },
     "execution_count": 258,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b0=(I1+I2+I3).expand().coeff(V1, 1)*Vin\n",
    "b0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 259,
   "id": "e37719d8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAz0lEQVQoFXWS4Q2CMBCFhQkIbqAjqBvgBsoGOkf/GUbQFXQENjAyAhtA2AC/V3tGG2hyeXdfH71LSzKO48KWc64KeYeuiQrWiiVmBLyoL+hDG2iGiO3J2zTAM5qZKbAB1UdX1d6IHolGIFpP6kKnm7EA9JFJpZ8PLdIwy4TnD+U6MQ9IM82tb+s5g/GlTpyazQzWrdOM1lL3Fi9jn3tktyZWsYvaTqzteu7A7YRxA2vU1RtJboAePZiZXG1L4iT2+9ba0E8xEPopdoTe3r/YGx/SQ0OZAIYmAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 259,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b0=(I1+I2+I3).expand().coeff(V1, 1)*Vin\n",
    "b1=(-I3+I4).expand().coeff(V1, 1)\n",
    "b1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 260,
   "id": "52d55ee2",
   "metadata": {},
   "outputs": [],
   "source": [
    " B=Matrix([b0,b1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 261,
   "id": "e793c36c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{Vin Z_{2} Z_{3} + Vin Z_{2} Z_{4}}{Z_{1} Z_{2} + Z_{1} Z_{3} + Z_{2} Z_{3} + Z_{2} Z_{4}}\\\\\\frac{Vin Z_{2} Z_{4}}{Z_{1} Z_{2} + Z_{1} Z_{3} + Z_{2} Z_{3} + Z_{2} Z_{4}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    Vin⋅Z₂⋅Z₃ + Vin⋅Z₂⋅Z₄    ⎤\n",
       "⎢─────────────────────────────⎥\n",
       "⎢Z₁⋅Z₂ + Z₁⋅Z₃ + Z₂⋅Z₃ + Z₂⋅Z₄⎥\n",
       "⎢                             ⎥\n",
       "⎢          Vin⋅Z₂⋅Z₄          ⎥\n",
       "⎢─────────────────────────────⎥\n",
       "⎣Z₁⋅Z₂ + Z₁⋅Z₃ + Z₂⋅Z₃ + Z₂⋅Z₄⎦"
      ]
     },
     "execution_count": 261,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sol_SK=A_SK.solve(B)\n",
    "Sol_SK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5332b774",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 262,
   "id": "8835d50c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\frac{R_{2} Vin}{C_{1} s} + \\frac{Vin}{C_{1} C_{2} s^{2}}}{R_{1} R_{2} + \\frac{R_{1}}{C_{1} s} + \\frac{R_{2}}{C_{1} s} + \\frac{1}{C_{1} C_{2} s^{2}}}\\\\\\frac{Vin}{C_{1} C_{2} s^{2} \\left(R_{1} R_{2} + \\frac{R_{1}}{C_{1} s} + \\frac{R_{2}}{C_{1} s} + \\frac{1}{C_{1} C_{2} s^{2}}\\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡           R₂⋅Vin     Vin                ⎤\n",
       "⎢           ────── + ────────             ⎥\n",
       "⎢            C₁⋅s           2             ⎥\n",
       "⎢                    C₁⋅C₂⋅s              ⎥\n",
       "⎢     ──────────────────────────────      ⎥\n",
       "⎢              R₁     R₂       1          ⎥\n",
       "⎢     R₁⋅R₂ + ──── + ──── + ────────      ⎥\n",
       "⎢             C₁⋅s   C₁⋅s          2      ⎥\n",
       "⎢                           C₁⋅C₂⋅s       ⎥\n",
       "⎢                                         ⎥\n",
       "⎢                   Vin                   ⎥\n",
       "⎢─────────────────────────────────────────⎥\n",
       "⎢       2 ⎛         R₁     R₂       1    ⎞⎥\n",
       "⎢C₁⋅C₂⋅s ⋅⎜R₁⋅R₂ + ──── + ──── + ────────⎟⎥\n",
       "⎢         ⎜        C₁⋅s   C₁⋅s          2⎟⎥\n",
       "⎣         ⎝                      C₁⋅C₂⋅s ⎠⎦"
      ]
     },
     "execution_count": 262,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R1, R2, C1, C2, = symbols(\"R1, R2, C1, C2 \",real=True, positive=True)\n",
    "s= symbols(\"s\", positive=True)\n",
    "Sol_SK=Sol_SK.subs(Z1,R1)\n",
    "Sol_SK=Sol_SK.subs(Z2,1/s/C1)\n",
    "Sol_SK=Sol_SK.subs(Z3,R2)\n",
    "Sol_SK=Sol_SK.subs(Z4,1/s/C2)\n",
    "Sol_SK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 263,
   "id": "a8d0e192",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXQAAAAyCAYAAABMFSCWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAPCklEQVR4Ae2d4bXctBLHN/fcApJLBSQdBFJBLh2QpAJCB3DyLd/ugQ4CFbwHHQAdQDpIOkheOsj7/7QeY+/aXkm2vF575hyvZFsaSX9Jo9FI1t77/PnzzmkaBF6/fv2HOP0o9+00HJ2LI+AIbB0ByZP7wuA3Xc/k/zSEx9XQS38Xj4CABvDfXJjHY+YhHQFH4DQClRD/USH/kR/h3kvXvW/8RTQClTB/L/eX6EgeMBkB4fuDIn2j67aK/KdcNJZmI3+jcL9X71fnOAZlq1T4PlYKKGdfyT+oDZfNSZu78vJWVxDqevOo/fbfu3tucvkXjByfQH6peN/L/SonvsdJR0BYYyf8Uy7CvSbdP9TNO12/y/+sfrFCj2MwXaUKSxSCX3V91PW1LoT6Az1fjEBXfgIpTz/J81BuZ/t2k0sFVI4jUBEgANwJbg5PjzOMgDCns0GsV7RI797rAVr7t1XdtN6v5cYxmLYmhecnXdinvxfn/0zLfVpuyiNa+mO5KJJH5AL9CJKkBwiVO4GLIHGaB4GmuaUrxZuuhyt75hisrEITi4NQ/0lyp2lqDCxcoCciacGrEfJG7s/2zN1ZEHhBKsL9aCeRnqG9c2GOWfMg6xjQCDZKatusEdG+sQ60yAV6C46kG8D0RdAkyCYJHAT2ISc1csxfLGZhP2/Z1g/DruDeMVhBJY4sAlr6y6rd16x8l0sNRbxHILLbgunOXXwsDzkWAeFupoZdVQew/EIXwpwFLeygR5q7nq+GHIPVVOWogqgdMAulrbfW8Fyg58HK4gma4OJWwfOKczGxTPNmV1EwqchlYGWHAkK908xShXmu92vYjZSMQVX+Vyo/BE7Qd3ru7XePxaX+olDy7ct9q0sX6IlVKeCY7tIpfGdLInYTBEdDZ0dCLbirhoxmzlZGhBZT0Zqq+jLNHuF/6ZSMgQrMAhpKSCD538jzj67e/cz7kP67cATY0QWx4yWs5V2FW/9JQQChgVBZ9dQ+BZAZw/bZjk1Qm/ZZZ4l60kVjrweB+uVlepIxUDGxtdqgRqmZprOXGV5OF4qA6o8ZFkK9HqyvL7QsydlW4en0U0w76Rg2MibnwyPkIdAQSEf7z8XRhBV29NXSCAzo8H+vFphtF4yNAHwdzQD9fksaOtNODs7iwlxC52faGU2Kh0bDwNAlVKL5eMAsBMzE9d+O2E+qZ3wlGqiqK7tdi5uFgbD4RRfanBECnqMqfJZpiFyua/0hKDVbEuhTTDtNE3QNfcYOIMGDKYVFzT6qBTkBFJ56OjK/9EW+hOdTYSA+KCXf6vKjKrornl1T0M3eWfav6pOBmisslm9JoE8x7QS01qLcsqv78nOnBou9l7pjzz+28Fd61vrsWfe843ohP1tKmX6u5oAulWUSDMSHQQ5eizp4Svk5OwkbdotgvrC2Fe71zO7PnseBDGBOC+sh112BVAjMChQkCLBGmD/0jukbDYMtYK0dBY1wg97S/LsSJ98Hz3OmnRzck2yLVNoIGbBsaviMquBsNOqUwDnSsIzO6apcUW1M4ajPVdIUGIgHfRZzY9Dkqvud3LUsFo+qe+Fg5qxRfM4UGdNZkC1HAl0FQ/iweHinq3Wgut5x6BFbnojMSJ9MpfnHZEh5SJ52Kg7Clyu5AygumuXPcgdPCdT77FMC50gjBlsPszwE1DYQ5vRbBHrQ5ORnAIwaLBXOadkIBJOj6vb2yvKpGzans9hHRT+VHwGEFlmT7m0aSwNJsiMrblH+dSZPeJQP8p4z7SQe1LLX7h+d/lW61pGOFlT1jkECPEedEjhHGkMlVfq3uhY1RVV+ML9Q3ygpwa97lJbJSXwXV/6qkCz+o4Th2vVS+W317ypstrPg8meX6UIimtXgcVND/0uZR2h9eaKisTM9V5hUTbU0/5PYK8+UL3faSVwotdz7WG1ziz1rujfNm0w/nRbqG2ynSGOfQvevzWK6357hadVO59JEF1d+IBcGD2aCfpHln6ns50zGZNKjK3KhCkeDQYOM+RyY7X62VUbe01Sa/+kchDIikJl2YqvmPGHKS0enPDFkwtDAi4nTDPOCG6V7tFWsygv5GXtK4BxpNMvkfkfAETgzApIfNtO6udYNgo4pKPtSzaQylEUEGoIxikrzb2ZCaTEwfaie8Vkz2rgVlqkmGgRuTXofu5hGXMj47e/if4PAPgxe4WOnBI5dmJkjjcMi+L0j4AgsA4H7mFxMoEUJ6UpAHmmZA+UpzT8krXxhm0aAh7zJxfyAgA/p637stNP2p8Zq9CFf/FR5Cffym/120lMC50ijLpB7HAFHYGkIoGjeIND5yACK0c73IRu/EiRors919Z1kl82/4s1iFmQ27COzkMLx7lZuc6DBnILmOxWZhp7DL2wVU8SSpwQmpxGLb06BPY4j4AjMigCKZtDQb0hWnTvKNowQ0BXMDnKZ4ttCXJ/Ay+Yv3kmnxCk/aOkI8aM/ENazsWTlCGVPZAZGrQ+SKgynPCUwOQ3lKQlfK7PyzmzO6t0e4xpGNitrvuOQrEGTkt6zrfPspHzcG8qEl79M/YP5UtrAUP3P+e5UWzzIS9DQg2Q/eNF5K+Zo2wj+oAnrHpeOalp4V7xs/mLG1iq+2LJdG5hQeMaiZsgDCcqP/Z+93pgzgqDhXhda+hKIge9oBqT82SBos486r3oXg20dXp7kNBQnCt9mIviVty6BzXPaAVsDqYtkUrxBQZrMsFAE5dPL34Ht2PqH5aW0gY7iL+LRlXIRhKWAPBIqHTl8onC1IO143/VoDH86ju2x7OJdP1O+sJ8jEDA9kOYPujeBWYcb4WFg2qXyVHjTZI/2n4udvQu8c/M2Io1ofHPz5vEcAUdgNgQ+ItDReqEhLRtBRri7EDLtJ5u/0jx5Shz50lV/7CN/CXMLJTZTSzArJEBgZoaurZ5PKj7N/KNpp1JWGjH4pmbEwzsCjsBZEEAufbpWp8ZcgUDAtIGdt3Xmie7RItF673gvN4kUZxL+4oOgY9A5PCWO/LV26Cgs4Q4HAz0aRbYdMlrrVz6Y9bBg3Ee1ICeAwlMW+EfPgqZKQ3z68CVrTo6AIzAxAupz9PWTmz4Skn1/TWAx5gwRtvWh7WIaMMGNKQBBP8oWPZa/4iMY0fS7Ton7Ts9vFYZPzk3YMjB12jkVJpcMk6j4St9mJjZAckrgO112v8OviwHqhVz4Hg2oPOwjxZkkDfEZwrcveX/uCDgC4xDI2pTQkySy72MQ6ARQp0ZgTS0EYR0ol38lbLCPh2151f1ObtiVIxdtNlqjrbKT4zC4QUxtTpLyFTUIKlw25lOkIR4I8158TxZ0hgDKI4212OmfMxRhVBJbLz/glcagNP+eBvBS6Z7c9NETt35c5Z37T7VAr98uyKOMImwwpyBwMAlACMAoYRlCT/dj2zrJ0yqoAL4oBVyTkfLIziWmpXe6Jj/9c7KM7hl5+Seuf2At3QZK8x9oY8iyqE0fAzx4ZTLpw73Pn/O3/goIGJGpW10IXLarfdDzrG1ritsi8fmfHqCdtUjPz7K9TekCFmYSylyUlEZRbMm80lgUvk1AlTfqnW8KwAFB3jkL03MGfLT3R/LboKvby6atl5/aK41Baf6pLVD5wYTKiascWxJNCs+aIX3l2SgNXYzoQMW0ZfEf+7l+NCiRASnv15FhRwUrjS2ZWyC+TczOfjpnMzNn8G+9/EBeGoPS/KObjfrimE0JpqG/HSXQo3O7noBoicxGnAoioMaNpkIDb5lYepJkbaNrS2hP8OU/3nr5qaHSGJTmn9LKlBcEMm2+a9NHDKuw/Vl89rtcYmJ4mIAAO4CYEt3XNamt2PHdI1A1buzmRU7/bOKstOhEth318HTOZtDZ/FsvP0CXxqA0/2ZjUVqDbazKy9hNCSiZ4QNO19Cb6J/22wIGAB59yn86uoeIQMDWJ7CNnyR1CAbWTvv6UGTFY3CmI4W4cqlTOp+lPxS95DtLf6vlB9vSGJTmH9rHqTam92jm1HP2po+KB+tN4Ut0F+gB+rgfgce5NQiQF7pcoMfBlhqKBR4oC1/VD42bj7n6Tv/cKQwd6VZucyBgLWjK0znFLou2Xn5AK41BNv+qfb2qapZ2BOWeADv2PxpIG0UECv3FBfoejJRf7LVDX3+m8PKwxwjc8EgdJ2rHCh1MF4MscbC7WwNHsA+SwqPVlDqdczDtgZdbLz/QlMYgm7/ylvQx0FAb07spNn3wfQ7mydBfrkDPKQkBBABCxARHUmQPfBIBFjmjSHWApmVa0k73zKDYMjs4GCgM7wlHHTLl5QtezC1LoK2XnzoojUE2f+XtpdpKs+/TbjhhFGWiphnbGH2gNs+5QK+rIM6jimLxAYFgB2LFRfRQsQiMOZ0zNo2d6hG7ZcnTOaPzchBw6+UHjtIYjOGP/f3vgzrrvC3dxsSf7y+g+jgRF+h7QFJ/GZUZqU9O61MZe/iwMAkMZufshETYUwd3nS8HHhJP1zsLIn+p0zktiVTXZgpbLT94FcVgDH+1Fz4sDCa+qmIR8Jg83lb3O/nnamOs+3AOV50fF+hWCwmuAGREBERbHEmI7UGHEBC2NvuhU5gGUkfRMxYzgzCXWzfkOsBpj5lZ6pDig/A87Kj1+zk9ysumyw/WpTGYir/4YGah7cx+AqzSph0/1NX6sHPUp/9itlkSoAgbbFcP5M8RLJvFLqbgwvS+wiG4abSGL7bP5mFGuj0mxaWTMSAcfUKtZ3RCOgM8SQNKOuVyH6Xs79bLD7qlMLCaG8mfdkn/P/r4TXyLtzGlwYI+7bZl+nWBbrWb4QpMth0xZW+NkhmsPMqECKg+egX6hMksltXWy0/FlMRAvINmLDfsZ6/ud3KZXRUnpYNCgkA/Uiaviqe+7gQ4ix1bOhXs5Ag4AitHoOrraOZv5Oe/jdHGUeiid85MAFHnzAC+rqGPRFcVymfq38gN57WPZOfRRyCgOmBgRWtCg6GjsTVxstM/xWvRtPXyUzmlMRD/s55QqvSDGVJuy9RiDdMFuiExwhW47E1npdtNLyNw9KiOgCPQj4DkC4oKa0OHi7B1JDe51FDkewQwo+WtXGy3To6AI+AITIqAZAszTrTzp0OMXaAPoZPwrho1OT8E4J0cAUfAEZgEAckUdmP9quup/Lbjq5P3/wHkXP1NOpqKSAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle C_{1} C_{2} s^{2} \\left(R_{1} R_{2} + \\frac{R_{1}}{C_{1} s} + \\frac{R_{2}}{C_{1} s} + \\frac{1}{C_{1} C_{2} s^{2}}\\right)$"
      ],
      "text/plain": [
       "       2 ⎛         R₁     R₂       1    ⎞\n",
       "C₁⋅C₂⋅s ⋅⎜R₁⋅R₂ + ──── + ──── + ────────⎟\n",
       "         ⎜        C₁⋅s   C₁⋅s          2⎟\n",
       "         ⎝                      C₁⋅C₂⋅s ⎠"
      ]
     },
     "execution_count": 263,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num,de=fraction(Sol_SK[1])\n",
    "de"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 264,
   "id": "665bea6c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAT8AAAAWCAYAAAC2cpvhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGv0lEQVR4Ae2c7XHcNhCGTxoVYCsVROnAHxXY7kB2KojcgTz+Jf3z2B3IqcCxO5DTgeMO5A6cqAPlfSgsBkfiKIIkeDwKOwMRBLCL3RfAcgHytHdzc7MqtDsInJ+fP5C2b53GR+76h8qvd8eKomlBYPsIHGxfhaJBIgLv5eheG4/yF8r/o/SblZVrQaAgcDcC+3c3KS1mhsCJHN7zQKf3yh+p7FFQVrIFgYLAHQgU53cHQDOsJur7NkO9ikoFgZ1CYK+c+e3UeDWUVcRH5Hesa9n2NtApBQWBzQhEz/y0kDhUP1F6oRQepF+q7qMSB+2vdX2jazLlki+5p1IGnW1b+FV59Mceowu1+2I3qdcp+uiqk3Rhq3us9LgrT+520inL3JkT7rkxDOUXPEM0+ufdWvksCY+Vr3xaI/Jzk4y3ie+UcHRVQ7pVnoVmzoWD94+Up5B4cFDZ5KOL+uAV9ldd0dWT7nHaV0pflH/pK3pkpuijTS1nCy87Xirvx6iNJ3ed9FjE2ObGqav8gmdXpOLthB8P4j+V/lV6okSw8NDWiz/zo6HSpSo5U3qm/AdrpPuKdG8RE06EqKoziTerfFNE/djBP7askep+qAC92SZiQy+aoo82xZzub3R9oXTNvStrY4vWie+5ElF+bxL/Ysa2NwiOseA5FMEmf19MxcfaIDjAp32qSw63vX+rEofwK0z1hsE9oeMrtcGRpFBu+aZLuOW1svB6GN70zE/RR1Q14c4YEfHh/MzRM7i9jiDEx9ORNISWNLZDcIC34DkUwSb/GJg2pFbOT4uIQ3MWUpctFCHkXw1JLQW55de6/p179fm9Vk4ZNpLYEqc671DcFH2E/YV5vuljMnD1JHtwgJOT+s06d2oGbRP3mip5bgueeXCNST0Q2EQSnNX8UN62tbG2VobTIPLoRLnlR5SonFu93OlB1Dr4vE8ysvYhXXEoP50NvMUlyrvmXteHrnzrF+mSde5EDNwa7hFdRi8qeI4OaatAIj+LGDo5NA0Qi7ARVbX0klu+71q62XZ0pTwOHfpFiUVKxEpkm6K7WNYpdx+Sz1klzq7S0/WHMzQc1xXa7p3plGvueOvuCe4FTz/i+TM4P97gQl2ivtuWwV9NSrZgr5T49CX2yUVv+U72W9cdDgxq+x2rvd1Fl2pb62Twxgf+6FbXtWmzgX6NkvvoaofaoSMvIEIHzVkeEescaRFjOyPcZ43n0uYxzu+QVSXDoo6hvuIAQMm2YGxDLNrCCcaot3wJS/0dK7rwhsfb4nQl4uPzFxwpzsSTyrvY4Nsrk9yHeJLskE5Efzi8xuc6KutFkkl0ZmMVyrDxsagjrPsuvrZPgozX4x0y1/OS5eeO6pIwUftk3NVf0tiqfWfc1fbe4Zk6Zil4MlcyYYroKB2olO3gJse1xiTleDIx0W1LxpUFYk+stfbuprd88Z9I9mcl+6yG7R9lj5QqHWodMtkbEazamn0WPXo2J+cuG3x7ZZL7EE8nO6QL564f1J4te+WouFdac9iqSybJiDm3lcoZOz6Vod9UWsTYyvZk3MVz7/DU5Mg6jzNhunFO76umcizquOEYIlxP1S7mdCJNfdEQ+Uywb15SS0Z6WVTD07tOVsdi7U0D+kixg/O+PSnJ9hrsTnVvzru37pkYlzS2c8B97nguah7j/IimoLbobaUFSLt3Vcu0P73lq8+1X5ioW8DnKR1zwLY9i32G89SpzK87KpIMIrhU6tVHFzvUhm1gqN9oW95UIxPa7/zYzgz3WeO5tHm8L4PYxrKoWXyNL/1VxgF85fh0vU5YGFVT8YwiX3JwVjjoxksV1RG18sJiE3mnQgO1JxLsEul6eWP1ITmb7EAnzpE8qS321h8Avn7bGem3hLGdDe67hOcOzmO++oAOby+rlf9tr4xha4WTwymYk2ObGJ656bZJ4mWR4jw3/meRgfLRCcfQ+Ahbcu1pGSp2pfK13x3rHv4nSp+UeClSr99ow4h9tNmBU2Qhgj1jATX0vC0e769sw+6+Z36VIpIxZO60YTLF2I6K+z3Bs23MRsWTCTYEU/HalxKsLeYpu0Ye2pcr/qXV0HR2dnasdDVUToxfco+ULqzO3R/Z/VhXyc1mAzpOZUcqHs7u01S+MdpPhUnusQ2xWDqeU43ZFJgeyAPOluS17QnDYTRPFIhzv8FvPytJE/2ZuR1EmqRJaeaYDMFisXhuccyyYOq3vX1G24GBMyKkxDnxucRPlff5bEKs6yQ5/6nkwXppFQbzNnQUym0DSk5hxyhgTChkCkymGNsJIWvtaiI8s6/HViNHrvwfwC7pN7ZOC+kAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle C_{1} C_{2} R_{1} R_{2} s^{2} + C_{2} R_{1} s + C_{2} R_{2} s + 1$"
      ],
      "text/plain": [
       "             2                        \n",
       "C₁⋅C₂⋅R₁⋅R₂⋅s  + C₂⋅R₁⋅s + C₂⋅R₂⋅s + 1"
      ]
     },
     "execution_count": 264,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "de=de.expand()\n",
    "de"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 265,
   "id": "3f68eb76",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle C_{1} C_{2} R_{1} R_{2}$"
      ],
      "text/plain": [
       "C₁⋅C₂⋅R₁⋅R₂"
      ]
     },
     "execution_count": 265,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s2=de.coeff(s,2)\n",
    "s2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 266,
   "id": "83bb4f8e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle C_{2} R_{1} + C_{2} R_{2}$"
      ],
      "text/plain": [
       "C₂⋅R₁ + C₂⋅R₂"
      ]
     },
     "execution_count": 266,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s1=de.coeff(s,1)\n",
    "s1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 267,
   "id": "79da082d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle C_{1} C_{2} R_{1} R_{2} \\left(s^{2} + \\frac{s \\left(C_{2} R_{1} + C_{2} R_{2}\\right)}{C_{1} C_{2} R_{1} R_{2}} + \\frac{1}{C_{1} C_{2} R_{1} R_{2}}\\right)$"
      ],
      "text/plain": [
       "            ⎛ 2   s⋅(C₂⋅R₁ + C₂⋅R₂)        1     ⎞\n",
       "C₁⋅C₂⋅R₁⋅R₂⋅⎜s  + ───────────────── + ───────────⎟\n",
       "            ⎝        C₁⋅C₂⋅R₁⋅R₂      C₁⋅C₂⋅R₁⋅R₂⎠"
      ]
     },
     "execution_count": 267,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s0=de.coeff(s,0)\n",
    "s0\n",
    "sym.init_printing(order='lex')\n",
    "D=s2*(s**2+s*(s1)/s2+s0/s2)\n",
    "D\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5849f944",
   "metadata": {},
   "source": [
    "$$ \\left( \\frac{R_1C_2+R_2C_2}{R_1R_2C_1C_2}\\right)^2<\\frac{4}{R_1R_2C_1C_2}  $$\n",
    "\n",
    "$$ \\left( \\frac{R_1C_2+R_2C_2}{1}\\right)^2<\\frac{4R_1R_2C_1C_2}{1}  $$\n",
    "Notice that the varible C1 does not appear in the left hand side of the eppression.  This means that the damping can be set to any value, as ooposed to the RCRC circuit.\n",
    "There are 4 variables.  Simplify the analisys by: m has to be a real, positve number.\n",
    "$$R_2=Rm$$\n",
    "$$C_2+C/m$$\n",
    "$$R_1=R$$\n",
    "$$C_1=C$$\n",
    "$$ \\left(RC/m+RC \\right)^2<4R^2C^2  $$\n",
    "\n",
    "If we let $$0<m<\\infty$$ then we can control the types of poles the sytem prodcues."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8453acd7",
   "metadata": {},
   "source": [
    "It is easier to use this notation to design low pass filters.\n",
    "$$\n",
    "H(s)=\\frac{\\omega_0^2}{s^2+2 \\zeta \\omega_0 s+ \\omega_0^2}\n",
    "$$\n",
    "\n",
    "$$\n",
    "2 \\zeta \\omega_0 =\\frac{1}{C_1} \\times \\frac{R_1+R_2}{R_1R_2}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\omega_0=\\sqrt{R_1R_2C_1C_2}\n",
    "$$\n",
    "There are four variables and two unknowns. Use the scale facotor method to link the resistances and capacitances\n",
    "$$\n",
    "R_1=mR,\n",
    "R_2=R/m, C_1=nC, C_2=C/n\n",
    "$$\n",
    "The variables n and m are poitive real numbers.\n",
    "$$\n",
    "\\omega_0=2 \\pi f_0 =\\frac{1}{RC}\n",
    "$$\n",
    "\n",
    "$$\n",
    "2 \\zeta \\omega_0=\\frac{1}{C_1} \\times \\frac{R_1+R_2}{R_1R_2}\n",
    "$$\n",
    "\n",
    "$$\n",
    "2 \\zeta \\omega_0=\\frac{1}{nC} \\times \\frac{Rm+R/m}{R^2}=\\frac{m+1/m}{nCR}\n",
    "$$\n",
    "Substitute \n",
    "$$\n",
    "\\zeta=\\frac{m+1/m}{2n}=\\frac{m^2+1}{2mn}\n",
    "$$\n",
    "A butterworth filter has comple poles and -3dB at the curoff frequency (as opposed toi -6dB or critcally damped.)\n",
    "Assume an m and calculate n:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 284,
   "id": "3787c00c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 4.23606797750222$"
      ],
      "text/plain": [
       "4.236067977502216"
      ]
     },
     "execution_count": 284,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "m=1\n",
    "zeta=1/np.sqrt(2)\n",
    "# set n to 10**.5 so tha the capacitors come our one order of magnitude different\n",
    "# Also set base capacitor to .31622776601e-6 so the capacitros are in powers of 10.\n",
    "n=3.16227766017\n",
    "# Selcting n first to get the cpacitors to scale properly, you have to use the quadratic equaiton to solve for m,\n",
    "R=np.roots([1,-2*zeta*n,1])\n",
    "m=R[0]\n",
    "m\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 312,
   "id": "b49c1981",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAPCAYAAABjhcQWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFdUlEQVRoBeWa33EUORDGB4oADJcBZAB2BpCBfRcBRwZc3ZvfKMgAiADsDCADDmcAGeDbDMz3k9WiZ0Yatbx+o6tk9bQ+tfqfNNpd37m6upp+Bzo9PX0oP5+qfRS/a/kcxbXm/w7y24zRHSvArPRFDuCBehL2WvLPtaBK/ljyM7Un4rcS+nox/4PwFyYTzzr/5OdD9Zc8e0wem0awNsd6zT0Wj70t2glzP4rzSjRn08dRrNP3Q3MfqZGH714PvGSh2GXcSG43/ZG+UCy9vZpT1XkvO0LBkXQzcsqLfFJ/onbucO/FUyQUCwGokuYwRsLRm4pYPc80gjplzFv1z3iGxGPoV2RqpfjFoy+ERU+FjiRD3yqRknEy2iaI4qZs06aP0psoihXuqya8Uu9jbvEotmd93XgIF8otRmadEX9uLUapALX23zQZQMElx/Vsyf9XvAWDk+5EDWNfquMUbBGOcNqZHnAEowRRPMVWil48eilY7GH+fWSZRrA2Z9ZLbyl0G5AsbSL15vcUxUlHxEdbqovVuvh9sLCFkxnb3qp5+6PxCOU2G9m1MeMm2eRtSWLJVrHUwKbOu1khr0SKi5ZIygpvsmivuRzRFOc7PwejaU7GyfNNMgrTE0VLIpJDeWAE63UZ/8WYRc/rzW+CEE5zoj5OA1g2d7meODux6an0+DhF4xHK7YCNmHVrMUonoBYn4f60mbJBLMbOGyUSys7tFTHrPt7A+YCPYFf2ao1ywtmgZJwir+yZPooTNOojaqNYimq2aZkssrcG4+ZHKB7yJ5rbqI23GqNUgMlF90dG42g64sXXAuLQVZb74XfN5RT8S80u02eSEZBE4tPr3J5dn17tGi+nwQjW6Wmy0sfpSvHb3a+K3cCFfMxKu1it4zdb1RYJH9iA8OHY2Rx6zWvltmuj1+N56WzFsqtzVoBSROIx8EiN5P+ndhOyYB5KZ0mw+P/VnqvZLl7p1hg24FCZtwJlwQi2ooMNRutRCzfiYwRrxbXbMMj0VCFb8chjW7k13cM5kzE3jtFd74mMvFB7o8bu+qDGp69jj+nxwpsjnC7L0/Oj5r93mJq6MwnPhXlTG1zIRrBlqnRT4NypymlcBh3Twkke9nEE65ZusX+0BrK8GQ/Z0cztPjZqbjWWUZ2zAvTOSQGn1E6N16YF3EN6vN1bPI6vGNDF0bwircN9k1d36/VS5oxgy6RfDPedmn2/ENdcD1fT0fKxh71cLu6e7XTkKlOlkXgI28ptz8ba2nvFKBWgDOK04tW3JHsFc3SHSHooWsj666f5X3bNjDSPrwseqPefkmcYexjB2pxFz6m+lXCDV3EjPkaxDlfb7CarFcikuc3YaaybW7f2UM5ykPaKkZ2A7Fpet+aoJeCmPa+2LV2zQGpdnHikvpx84h/SlgaMYJdzedZ87ELvVrAjuBEfo1hwK58lsxOQ8RkF4hHNbdTGsn4gll2dVoAkg3vXMimHebWV48WKOsOrtBbIJ5Lz9UzRJ56T90j98kPHameNYOtmJan51DsBe7iwj1o1iuUOZ+t6F4gbd7hZfoLxiOY2aqO3y2xtxbKr0wqQ5H/ymuXcsZ4P1F4sHc84uxDb7izTheeO8Vl9+ZQpHl1/qj03oGQUKUHnS2d+VipNstm6Uaxw6LpSY+fXCDsgErNFmzjpD/nIAlGscHxou1RP7BOJX8WNAcmjsQvlVvrC/lxblv7uHSP/zwjc88orUDwO8itBOa1YUs8UDAQeAy7UeKXyM97sU6+eKUAzkkLlN07wicR/E8M6NWLHs/MT3QDLBuCCPCPJWI/i7H0dFMVt+ugX19pdrDDEC9xO7YfakdosbnqehBuJXSi3WW/XRnCQbNg7Rj8BJXvzJJsDA7UAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 31622.7766023676$"
      ],
      "text/plain": [
       "31622.776602367583"
      ]
     },
     "execution_count": 312,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#set wo\n",
    "wo=1\n",
    "# This value of C will scale to powers of 10 if n=3.16227766017\n",
    "#You will have to change the exponent of this vlaue to get reasonable R vlaues.\n",
    "C=31.622776601e-6\n",
    "#Solve for R\n",
    "R=1/C/wo\n",
    "R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 318,
   "id": "e26406ed",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPCAYAAAD6fR2jAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAF/klEQVRoBe2a23EVORCGjykHYEwEmAwWk4HJYFkiADJga5/w2xZkAETAJQMgAi4ZQAZryID9Pp2RVqOjOdMCv1C1qtKR1Pr7Mt2ty4x98P37982vUM7Pz3+jfqptZXzE+Jj2S03/v/9reOCwNtMAM35FvUn/Wz1nH9oJzYOJbuAdP4b+dqKlJooTDPZpxXtM/x60Hd3Q30FXZ05A+5ab22b+C/bxnLJ5AS3zNlP/DcH4TH9OlFPaC8ct74Rb9cUkJzXwXIp/FTbpX7Wz1j/xacMD+LPtBQItGgv1j2C7sTh49OiRQXxO1ck6W+OuInyWAIzFmWjFaPq/QzNZ79B/TbuhHcF9hOUpPE8mXnW/o5r8s92M8Wfox1TlO6e+v6G3dpo82mTCpEVB69id8wbtYmFeXu25nUH0ddxD6m36WV7oGZUBj9hL8+8kM2Sn2LZgj378RHsnz002hmIxiN0biysI+6YhVJPqRTao096Hdh+cCZdL3vH+ygTaKM6AeGSmxJOfvjvTB2q9qpyy6DAXxQH1BtXkmiXeFpYSz10u2ybZBJgl84RtGxOtLC4n1UOjHhM4l+gzyn/Z/tWGqJ3Z3tRii4uoV0ZiMYLVZ4uxuNKzZIFmYhiEEnAd28FGcSZxLyHkP0P2UUf2XhI8ynT3fFYDobtrld2snmv6Z4w/d3SbyEfQXcmW6DNu0bHfEZlRO4tmbNcvs/iVyc1mJBYhLPpWYzG781XG7HQRZgCu1hOTAkllp4rgwOTEuqjlTf1/pvaUtt69OtAdkruWO01vUeyAOwT1eTwv8Se7I8/Ykb2XNCgzZGej8C46PC3anT0cC3jDWHSvxiKcfM2DbDDE1Ze2f/qznabG9nDQTBBh3uHacm0i5F2mzMPjcacDxDjvnc8dIxcT9gs0V/ldqonsPe8VtNVEBlPuQfDURXkb5mtdZR56yBeFIdDZJ5O5ITvBe9yWDaJWz1w4FiNYdKzGYjj5MMBA6OxbVIPhHW2nBHC+MCinLSnQEPMqy/OOX+oACbQmn0dkeRFgnHlOoec3QbFfqb5Fq3OowKM96irysoBpbtUXGR9pf1Tmkp3Qtd0E611xskkjsYhiV2NxJWuPtjyEF/8nVFefLygf6Xu+z0oAd08GcCUB6RvolFy0M2cxZ5LluQ19593N0opmnB/WY7PdiV+Ce15hGIaLl+bX8JYXo8wJLeSLjI+0PyFzyU4/q7T+aE0ZicUqFn2hWAwnX201SlwFJoTHWlZYQ1K/h4Mm33Wqb9oPqR7hrtL3VMss+baknV8xJ/DKl0uPz88I2udREC7INbE9xpeOuSILTMgXhSHQicpcshO615TucVurBxeOxQgWHXtjcVgbsa+P0nQc0rb3ng/wuXtZ3SFCOHWB9aHbC7BJaCmGg3vD2M8y3Q/KzB0xZ5LQTYvBtlfqJO3NFxqyDJw6d96SoYWfsQhc6fyozCU7ofusyS8rqtM0+FAsBK9hnacKVeZSOTlcmunQ3T1UvPMBusFGcQ1bGRrYtz5AoWx3rItqnLvHdsDmBeExvC/BSkJnAb0WeV4j/JZYdjz6SS6tMn72GXtqh2XusxMF2nsLjMdxXfSvp4V0F+zOXbYC92JRTc+6LXY1FiPJZzK0SaH2fJTlt8kQbnKcHyyv00+JRnvE+Iza7nDPFpwkNuulm46Y1tnSledqrLHSdwoYnWjQ2qCYkPnuFHrGHeH7CUMy1+xk3kWy87zQv0qnrReWzxaKBXxRrMf93li0yecnDMsxNSVEGm1/2mBsJkNMGC+1GR/FuTLb3UxjlZV3sq1mkgqatRzR9PPX+tqJHvs61j8DJjtote8ParooK3CiGQRfGEqi09cmbVBGe1c6g5ZfOqLPiKhZuQz/bgbsnCmfBvrDWpeRWISw2LgaiwP/qwVgzlB3Eg0z+K6cN8zl1S7O+RJs+hpioGcrbACX73fZGYvf45Cprhx0F4eJ2/0TG1jlZpli2++BGzD+jdMkqxNamnp6pU3UkC8UhI7L9m/Yzvwg2OBi8tm02+IL0nvoaUHRjsRiFNuNxb8YjAz2dnY1VAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle 133956.231324996$"
      ],
      "text/plain": [
       "133956.23132499566"
      ]
     },
     "execution_count": 318,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1=m*R\n",
    "r1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 319,
   "id": "90754aff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAPCAYAAAAVv3adAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGFUlEQVRoBeWa23UWNxCANz4pwDgVYDpwoAPSQQgVYDqAkyf7jQMdABVw6QBSAZcOIBUE3IHzffKO2JVW+8v2b78w5+iXdjSau0batYejo6N92iFt9/T0dLjOhsyDUp560PZL/M/w/DP549dhGA5oz23Hx8d0FZyAv1FhQYB37UP6h435pwX+FbSfJ7h/eN7lOXCOhd/Puv7fUZc3rmV8Uq4Etw/u8Yi/Tf/NZ/Ahe5yadyPfRRuZ028BewwegKtkB8HIq6kjdBf2x5qeId9+G3Tw0JcRc2Pm81Pw7+krGGXO7Dbx7tBc8LVaMQx3wUWwFqYHmVWBGxVzzsAmZeh9tt2iBRh8wQRW/lvaE2ibwWM+A3Qa/ZImH5NJB1QAnXg31h8xydhN8UkcbdFhI21lI/TK/UST5zPp6LXhX3oTP/uScZeO8gAu449KzzOW1e+l6EZ7jGsk3sD4T6S8o79HM4biVu028STKAfFZAJeCSJ8YnWF//IJ/9OOpGmmc1W0aUBXJARlXfIbmXrW6E8HaE0jT+lEfg78EJll2lATQ67xDhuraqugtG032PdanpGMsP235yNAqmP0JrldH2VzIH8ho6SnPDFui02eH8DLRIjcizn8zF4m3avcOhB+yZvOBpXMWrJgGb4BlbJsBc2a/8y+mE+CtLDkg07lrGFu5vyB/t5Clw3bBV5USXNNG1mhjuYlkbfW/uyDHuSuBDXpmmVuk08ZZ7OFd5UEW3BjssCiyNpOAs0I8yYh6cB+aWWJNSExW74XnVmbCY9tDE+zrik5lQip/0UZ4BG0ci1Nd/xsfPPavCxb1XBC+FTrsf0+7YR8yGLsRhemd9wzT+E1H7XQOJu7+A/rFux14y/qaAJ1ukK0Y92kGw3vdG3BZWZ4TgLN0G8zfaMr2jueu2hrAr3Wcq+NQyuO5aSNzbiqX7flTgDYIVQU9Q2/+hXe3P9b0nEraNl3B29MkXWWQ0ypG0yVpXCXeyERGFcBYh+r4pWMm6KMi3IYuJy/j7zTf+qYVVtrX4E5cTC9/j8RNF37JLwXIMOmUl3WU4ajDJhu1QYeXkBIZZPignN/03O2PTj177emmCwOQr636wJdTC8VHWjfsTClHY7yjVJVppPOzQjOrmQuHWzFLutfweDmhGRibYCnp5M/YhFb2WkWVdBvgS8VbZOYXhJHpqo0jzQN71ubkY2wgwpa1jTmyqDt4nMcfPXoqZNt0SXF09UXoGc3T5BXNLwRx5CaatZ9Z4kHo/WzRaTD1COhNiCUefn4wMTfdf1y7jzyr0ZUAvLXD68DsCO61EToT7CbNzwePaJ4Q6hsvakv2M30hqPxxDj27YtbLr6U96z0B9InXKWO8Ecqj1oytnAYznbpLX81NJTAf9x+VaEFKKGjfQeAnidbH4i4DWkJaeOQZDOXO3rB57rIx+EKvjbO3fnBxRVn1U/CY9qzt8kevntumC13hm64T9OU93KPWE8BmIq5CTjwYGWidXzKUgfg70Hg8TUElrE7irSDelzwqU3LRL0EExcr3bYFgTxy8lvRYIO9HwdONdYs+VzrGoet5bGwJ1R++9a1tvNbaXn906YkQE3ljzHrpsCnuwp5cA8++2V7ETpcPOfEYa7hQJQMCTKbq3gf+u3j6HEiePcbKBAWV/gxmRQw+LxiHMc4HuGOCJnCX7pFlUhiIUqbJqC5uiEou+MpGcK7xI/JNxsn59G5cdW9VcKZWocsfyFHHLj2vgE4DtHdpc0X+VLq5qISdCULHCcmRZ8ONv66JdYkYx1hmVSyOnYGxNH/R0qU8EZKg4Gd3Rp7jC/w0kdN65k5pabeN68suPmWkijmdZJ1Vws3gdSHJjR6cl+81mysbWSO/coPKX15rlbqpI2u7/QHtEizpeRV0blyraQZsdiMqv+XLyu5f/K8PgcU608CWnzycngG0Joz07nDBZPsAPr8hMjbxVEYwGZ6AmwWFZ3lEBZLGYD4Gf0I/A3BfQJjQ5b0qqqu6KE8ZVi//pJPerMe1yloC386qKgVu1UbmY2OFjYvfKRUI7UYdR7puf4Qhm/S8Cjpk6utpcVDv6p8E1uz+H09p8ZSqpxN3AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 7465.12491544431$"
      ],
      "text/plain": [
       "7465.12491544431"
      ]
     },
     "execution_count": 319,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r2=R/m\n",
    "r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 320,
   "id": "db184dc6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 9.99999999978889 \\cdot 10^{-5}$"
      ],
      "text/plain": [
       "9.999999999788891e-05"
      ]
     },
     "execution_count": 320,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c1=n*C\n",
    "c1\n",
    "c1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 321,
   "id": "788a0df4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 9.99999999977864 \\cdot 10^{-6}$"
      ],
      "text/plain": [
       "9.99999999977864e-06"
      ]
     },
     "execution_count": 321,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c2=C/n\n",
    "c2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 322,
   "id": "2130c5a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUIAAAAuCAYAAABd7BAJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAIUUlEQVR4Ae2d63HcNhSFVxoVYCsdrDuQ5A7kDqSkgtgdyONf0j+P04GsCvzowE4HiTuQO7CzHWzOgQCa5PIFkniIezCDBQkCuBffBS8Bkrt7sN1uVwoikIrAzc3NCWR/QjzF9iaVHpK73wSO9rv76n0KAnB4TyD3DvEn4hniGlFBBJIRkCNMhn5/BduZ3yUJYPsKCWeFCiKQjMBhMskSLAIiIAKZEJAjzMQQUkMERCAdATnCdOwlWQREIBMCcoSZGEJqiIAIpCMgR5iOvSSLgAhkQkCOMBNDSA0REIF0BOQI07GXZBEQgUwIyBFmYgipIQIikI6AHGE69pIsAiKQCQE5wkwMscdq/Gb7frzHDNT1xAQO9KMLiS2wp+Lx1Tr+0ALDOSK/e/wN8TviFxx7j1RBBKIRkCOMhlqCREAEciWgpXGulpFeIiAC0QjIEUZDLUEiIAK5EpAjzNUy0ksERCAaATnCaKglSAREIFcCcoS5WkZ6iYAIRCMgRxgNtQSJgAjkSkCOMFfLSC8REIFoBOQIo6GWIBEQgVwJHFxfX+v/PHO1jvQSARGIQkDfLImCWUJEQARyJqClcc7WkW4iIAJRCMgRRsEsISIgAjkT0B+852ydDt3wCy38xZY3tsjapn8if9NRTYdEQAQaCMgRNkB5JFnv4PReOV2xfYvtfxGfuTylIiACwwi0PiyxM46XaOYFYnmWYX4vDsc5C3mF9PUwUbulQslAu1eQRr35W3cMXxHZB86iXLhFuc9uxzeNIaNLJ8jn0/4XSNm3FVLa4x7xFNv8bb9kAfLJefaxg3aD2zUZtB7BYtoDaOBhcDxBUf4WJs+Twq81OkI74Ljseov4vlwB2xfIc06Gs5JRP6IZSQadxVfIor5FwL5zGp+xfVkcGLGB+sFlNKkFuXQ0H5EaYyJ1fUrqCKEHnVXosZOEeZMdYuSJ6TTK4McL8x3iT8QzRDrDp8hvdoS2Ar0lT6pL7DfOLJDPZRhPxGfY5q8KDw4xZFAZyGFnuVR8je2/mFcOyPuCfc4Yvfvg2okhw8nqS6HLO5S5QDpqaYx6ZLFGOvbCxsEWdOyQAfQLblfKmRqm8qR8tCGm1hBz8LRMeaHmuVJxhIdWjkv+xgY9Zt+sggN+A+W8nKAVEkMGRZWXxVZ0JTmu7I3biSGjVzPYgc6BM/XT3sLtBXjSMY4NS7LrWAblelN5si0x/UV0Dp6/WqttHbl9nEz0kjyhOBPcuPyWlFPMjy3HWrNjyCgJ/4PbkLkzq0Ue+8nIZfMYZ86mGWLIeJDU8gn9OXun7Sr3PFqKB8mGDsHHTknx5MxLugTbFNNgaBsbNo7QnkycMn7H9pAHCHQeXB4PDjFk1JQxjq6Wt7J6cEY7+f4g2ggqA7rSwfywfeCSl8v84iJl+8I8cw/U7q+QTnHuVtywxMoMOnZqmgRlTlnoUyf3mj6z7y6NaWqeQwzkZoTuNYxBzg0d48m4M9PqERhDhlEB+rkl6wrbPEkZ+LeRnD1xNtt6/xPHBoXQMtA+72HSyRnOVh5PUMMR++wL7cUydA4MPDb6Kb5pwf9jaXbt5O6PZ1SNxTDF2MyBZ68RnCPk/SWGIbPBh5KlT3SW6/ffEfk6Tdt9qtEybPtvrEg6AIaul4fdU2LqY2ZHto071GP9xhmTLdPXD8pm8JYxtB8oRx3PkZYvNnRwnMm6wAdB5M60CKjjTqIiL/DGYuw6kHtgnKb5rJmCE8dd7/mYEc9emzlHeMySULzRQdRbIQhEzgpZh7MRNwMjoLYwWgYa9H15mPpUHuZYfTkT5KsXNGJl5uTRD1Q1wVsGann1Azrxakrn1/QK0FOjhecH2uQs0tmrXNvZp8mRfkO9tteMXD3vsQPhXjys3qHtuuriXgbGbZSdmyebzZop9POyWwY8ybQzHNmjXC52ObGiEXSKVysOerdkY8oTxV3FirK1jdEy0M5LtP8J0bw8jH0uEZl3gmj0qMk6wf7O7BZlXR/drLKoZtsZ0g9Xx1sGKg7qB3ThvVq+8sNlvXFa3EesOG+niE+KNpoc3Qr5tB9fn9l51ain/cXYFX335o46c/Mk7qyZQr9g4zgQz54hvFod2hLlbyf0VXoOZZucT1+9KTI42P7pE8Dj0M3Ndjibqgd3jANtdJggw6cfvPd3ACW5BCe7K+w7Rz5a9wAVF2NXsgHjHLjnznRx49g5Qs6wGDpndRgkLPfWlPT/GC0DcivfboFoGoJX7yaH7JZwTa/3PLdq3zv10QZndr5hlIwh/UAZLjvK+u0si32VDVx+EXbNjHvWTJc4jo0jRMe41OXJzZOQ3xipBOSdIxoniHRTOThwB/VmkYF26LjosHceyuDYGvl82NEWCgfDAijPGSLrDA5zyUA7bf2gTrzvVASUZX/rF4PieMoN6LYIu4JhNtwfE9NHOI759gjD8UPy8Fn5rjE6xaUXHR6dg3N4XEaW789hdzegLk9WOtLOr3hNlEG96CR2XvpGu+4qWlbuHvmVr4xhn/XPED8g8sZ7/XhrP2aU0dUPOkielORPezDs6PmQPd8n+sZ+j7lHaJRA/Sljp4tHcLuyA9B/Vu5TeVqdcmfaZbeseMIe7o0LnlvkytUkL+LmR2RW2+12loj/PrlAvJ+rvXo7aHuNeOvy7f7a7c+Vot1F9MOXh+33lW+9qeWtHR+9XescUvGkHjGYxpBRZhqa5xE8YvYB3txdeVK/PDyJVeb92KBzjNFC5jymcojOkwrHYBpDRgP8oDwrS+MG4b1ZFgofXnDKyekwX7/4gXzf1zBQrTmgrf9w5En9KPL5VHWWsJR+zAIjUiNLsWskXIPERGIa/Hwc1NkZC/0P4Wwnmj5ynfcAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{C_{1} C_{2} R_{1} R_{2} s^{2} + C_{2} R_{1} s + C_{2} R_{2} s + 1}$"
      ],
      "text/plain": [
       "                  1                   \n",
       "──────────────────────────────────────\n",
       "             2                        \n",
       "C₁⋅C₂⋅R₁⋅R₂⋅s  + C₂⋅R₁⋅s + C₂⋅R₂⋅s + 1"
      ]
     },
     "execution_count": 322,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H_LP=Sol_SK[1].expand()/Vin\n",
    "H_LP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 323,
   "id": "93c0bcbe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 10000.0$"
      ],
      "text/plain": [
       "10000.000000000004"
      ]
     },
     "execution_count": 323,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "import sympy as sympy\n",
    "num, de=fraction(H_LP)\n",
    "de=de.subs(R1,r1)\n",
    "de=de.subs(C1,c1)\n",
    "de=de.subs(R2,r2)\n",
    "de=de.subs(C2,c2)\n",
    "\n",
    "num=num.subs(R1,r1)\n",
    "num=num.subs(C1,c1)\n",
    "num=num.subs(R2,r2)\n",
    "num=num.subs(C2,c2)\n",
    "\n",
    "poles=roots(de,s)\n",
    "zeros=roots(num,s)\n",
    "de_o = degree(de)\n",
    "num_o = degree(num) \n",
    "Gain=float(num.coeff(s,num_o)/de.coeff(s,de_o))\n",
    "\n",
    "de*Gain\n",
    "(1/10e3/1e-6)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 324,
   "id": "52d2ff26",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARYAAAAtCAYAAABrh0SVAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAKhklEQVR4Ae2d7XXVOBCGTU4KCFAB0AEkFWzoAJYKgA7g7C/4x2E7ACrgowPogI8OYCsgpAP2fRyNkGzZlu0bckNG5yiS5dFo5tXMSPK9ub708+fPxpMjMIXA06dPb4rmrfIt1Y+n6P3+xUZg92Kr79qPIaAAsqf7r5SPlPeVryt7cgQmEfDAMgnRxSUIO5O7IKD6IxXsWjw5ApMI7ExSOIEj4Ag4AjMR8MAyEzAndwQcgWkEPLBMY+QUjoAjMBMBDywzAXNyR8ARmEbAA8s0Rk7hCDgCMxHwwDITMCd3BByBaQQ8sExj5BSOgCMwEwEPLDMBc3JHwBGYRsADyzRGTuEIOAIzEfDAMhOwC0x+Neh+5QJj4KpXInDJ/wmxEqkLSqav8vOPh6RDZf536IvyN+X3uvdSpSdHoIeAB5YeJN7gCDgCaxHwo9BaBL2/I+AI9BDwwNKDxBscAUdgLQIeWNYi6P0dAUegh4AHlh4k3uAIOAJrEfDAshZB7+8IOAI9BDyw9CDxBkfAEViLgAeWtQh6f0fAEegh4IGlB4k3OAKOwFoELj158sTf/7EWRe/vCDgCGQL+zdsMDr9wBByBTSDgR6FNoOg8HAFHIEPAA0sGh184Ao7AJhDYyheW6b9m+S/af4KC10N5X+3+as9NzLrzcAROGYGtDCzS+bmCyEPTXfUXqn9WvmFtXm4/Apq3m8r8zEJMumbRuKKSn1440yQZWLT4OYg3qvuitcHZ6D28FcCLX/6tvs+DbN9VEgQIELMNSH34pOq2yg/wU4kBfFXmheSZoXL/dyWNvRibVMbA56HKGDwL96tfwD7GT/fA7nHgz/uXeQ/zY7UXcQy8BscO/ExuggT8med2rlSPSW0/dAGNjUWdVHyxvOjNfk6omua12qxv26brKn0C3aicorkjpvZ7MzZmWh6L5rI11PBMaE2XUV8QTxZNS/yI1rnamUv+ok+0OxbdZMJXvfxbPNhRPFP5DpQCz88qCRBzgwsG8Qk+Z52CHquwKeiAMXcdZs0c9PgxpmTHCV+ovM01SXUM3ubFAnfV2OoLHUHJHBZ+OCc/+nRXuZ17XVs6ChWMDxvgPjZyHNrbQtfIiQ7wNpm4JsddaqCr1adGzgPxZ7ySfbKTsYDcaOxq3UU76QuBH3To86/KRiU4/aeSwFuSCbIzT5Jt0l4ssDDRi1/+rYEeqP+eymhYqhPtuSYiR8NWfTKpX/eXyTDkb2rPnHGSUUKgvhjK9QLvhKpfFf0qbLocxY+Xq/fS0nGG+IUBCCIxCNAmehyO+cJp29VY17U60u+B6AkkNte2U+GZmLWp2qYvomvtyhoGSmRhd2K8IMN4u85VpY/6VcupMXu2qTYCXaMy1aeKp/pAV+MLLFYcCdugEsYDr0+qz/YZ+s9JGmeRPzCG+k7ay84cYUZoMZ6S039U+6EEwUgWJfUlirMq3lrE4FcnZFgsxy82y2tBFyaFvDpV8MN4vhbwx4Ex/taBZgjCHGfyi8cqXdSfuWWOs8VE7ex0u05fq0+tnNhnKXG0ywKyiGp51voCencDJ7IwziqfgUlF2hMN+VTS7oa4MuGZYQS+Bhz30+hfNawmF8NnlSqeyauYbBfRPemUbdFXijfFjwDCA9Qh559lWOIDv/jMAdnVhoOQ0mcFJy11f3FgdrdDMqZcqvSplVN0PZtUG/b2LB2Uei1PkU76QuAF2yP+dBLPZEj7yuh7LtPqwCKQaozzShcd9WMCDUTO0ThcNC7VCSq0tatWuG5UWrDqstzqa8nNEWip8/V0q+EnmqFjCDuERvdZHRcn9ceJ2uOJ6qWFhTHao4Ho+JV/5pRnLOm4OBDHXGS6p4xNYA9v1ZY5lq4X6aN+k3JqvEZ0yEcgjs9WaC+lEk+1VfmC6AiksO35hdrsbQjI0kvqN+o3vQ5n1LA6sEhuAycGhYIuGeAC571oCBqtgamME09fXQMqTghN6wSqs7JNTrhoti4FfTCmjQTFNfwCnuC7GMvAgzk7UGYOeS5QSsx7/Cg3yM3RLH7ip/tmG/tqjzKp/kOZT0h6u4p0IN3HPor6hHs1chpLnJY8mCZ4zvEF9EK2bjJ7N1zifY096jeRcAsqO79JBovCjcDBCA5VpqsWBsUTckvUAZ3S8gP1OTaCc1by0XJxRV+oxxp+PCh9J3niQ8O5MjB39FdmB/FamU+Z7EgU2amNABLnTHUCK7uQduema3MedgldfN6I7lVCE/l2KoP6qG+VnPATrdlltkvqjAVdNc9u33BtvnCfa/GLwUV1gorhlS1CumfyjflNGOLsi90NiHA0wsMi+PcujYAi+mIUH1TPHtLpOjvHd/uOXasvRhsnK6FtZdH97kM5SDCWoW12wmJ+VXw5CmzyCLSYX8CGY8fGdBUvghTOwNHlcqiPAYXDsLDgKGY7mROFziwo6Dr4rEE8wLVKH9FOyYldlORQczl1eYrK9Cl1yHxBfdnBXhMhD4oJKAQcHiaTCdJFWUQ76Dfq00uiPxN/WB1YJLidF231SZWztgiS6DEEVkueObQBgGvluA1OGcyti08pcDRqZ7L4uHnxSr1AFpxnT2NG/efySOnFZzE/9cVJ+XgzC+Ip/6m6+uIAjcp01aSJoxBzSW6PLqLBARjvlspSanHRfe7ZKl2iQ+deUr9BfXSvWs6E8aAzQ1PDUzQEL8jN7qlbsrZoC6JF78xe1WZHsUgHA7Uv8hv1y/ibMGo/VX/YtYFWlmwfSwZgUTrbXkopggjPTzBE6o9U731xSu3nPYHJgXRjZ5YmDJ8gRzsGUxtUF/ETf4zohsq4U1G9nS+VmQGnQhbq7CIa9anZmeyLtLSCtzYhHhachmzHhu/Jp75T+syRE31wevAwmWzstKzlOaRP0RfSAUId22AXT9DJktrOjd9sKrDgIBZpUzBYrThmtCCphOaOyvbblCqZBED8I39syvRLAaGudr7qjt7R0bs0pesl/NQHQyW4dYMXztl9rlEaNm1jHktGTxAhpQvIy8KY0LCYpHRs1buBFzpsh91wStvoukafOXIylslfCoTcJ9XyrPUF8H+lfE06mX/s6Rp8sl2e7p87vykFlqtSjESEbRVur/RHCqI4TkGwiMqr3hqRSoKGbYWh/Vv5L2VLgJY9b6CP2uifjWUdtqycjc2A/GBDHkqD4wx0KPITpqzCGDrBIMNdbYdqKx0Lx8buBqdGPJg/xueB8rFKS3xVnRy34qrbt45jQFUbxwfk41lDy18l/LCd9gGnMVR7rT5z5IQ945FS+U9afv2t4ikZa30BXbqBjLkCx+7OaZv9pmgv8Z8QpYytGigB0CjHNjR7+bfo+GdADCEajK4bXdOHyMrkfFc+UOZ4E0FSndUG/tBAT2JVmrtynvSc8Vdj4ACLnrGo7ypsTEzxwbkxKDAgEYQ/qr118Npx2p76U8GPuWK8UuouDrU6InsMDKrDn6CQ7SwYUG3cM4dkocKROAIz/1lSG7ZjNgFtZjsQi2aOPnPl5Kgz+vG2xq/iKTr0GPWFoA80JNO7990dborfxv1GPBf7Q5Bp1F5iYIH4T05rgfyTsXHdLh4Cp+0POxcIUlZJsidHwBE48YVT84f/AZMyNCze2kkKAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{1.0 s^{2} + 1.41421356237309 s + 1}$"
      ],
      "text/plain": [
       "               1               \n",
       "───────────────────────────────\n",
       "     2                         \n",
       "1.0⋅s  + 1.41421356237309⋅s + 1"
      ]
     },
     "execution_count": 324,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G=sympy.Rational(num.coeff(s,num_o)/de.coeff(s,de_o)).limit_denominator(1000) \n",
    "dem=de*G\n",
    "numm=num*G\n",
    "for i in de.atoms(sympy.Float):\n",
    "        r = sympy.Rational(i).limit_denominator(1000)\n",
    "        de = de.subs(i, r)\n",
    "for i in num.atoms(sympy.Float):\n",
    "        r = sympy.Rational(i).limit_denominator(1000)\n",
    "        num = num.subs(i, r) \n",
    "#N=sympy.Rational(num*G).limit_denominator(1000) \n",
    "#num.coeff(s,num_o)\n",
    "#num\n",
    "numm/dem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 333,
   "id": "a2a9c37a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$\\displaystyle \\frac{\\left(51 e^{\\frac{7 t}{10}} - 7 \\sqrt{51} \\sin{\\left(\\frac{\\sqrt{51} t}{10} \\right)} - 51 \\cos{\\left(\\frac{\\sqrt{51} t}{10} \\right)}\\right) e^{- \\frac{7 t}{10}} \\theta\\left(t\\right)}{51}$"
      ],
      "text/plain": [
       "⎛    7⋅t                                   ⎞  -7⋅t      \n",
       "⎜    ───                                   ⎟  ─────     \n",
       "⎜     10            ⎛√51⋅t⎞         ⎛√51⋅t⎞⎟    10      \n",
       "⎜51⋅ℯ    - 7⋅√51⋅sin⎜─────⎟ - 51⋅cos⎜─────⎟⎟⋅ℯ     ⋅θ(t)\n",
       "⎝                   ⎝  10 ⎠         ⎝  10 ⎠⎠            \n",
       "────────────────────────────────────────────────────────\n",
       "                           51                           "
      ]
     },
     "execution_count": 333,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t=symbols(\"t\",real=true)\n",
    "s=symbols(\"s\",positive=true)\n",
    "G=1/s\n",
    "H=numm/dem\n",
    "# We have to turn the floats into rationals or the inverse laplace might not work.\n",
    "#limit_denominator(1000) too slow\n",
    "#limit_denominator(1) fast but not accurate \n",
    "#limit_denominator(10)just right\n",
    "for i in H.atoms(sympy.Float):\n",
    "        r = sympy.Rational(i).limit_denominator(10)\n",
    "        H = H.subs(i, r)\n",
    "F=apart(H*G,s)\n",
    "for i in F.atoms(sympy.Float):\n",
    "        r = sympy.Rational(i).limit_denominator(10)\n",
    "        F = F.subs(i, r)\n",
    "z=inverse_laplace_transform(F,s,t).simplify()\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 334,
   "id": "65eb7c40",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ 1.0\\right]$"
      ],
      "text/plain": [
       "[1.0]"
      ]
     },
     "execution_count": 334,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Go though list of poles and zeros and make a list for the Bod plot function.\n",
    "D=[]\n",
    "for i in range(de_o+1):\n",
    "    D.append(float(dem.coeff(s,de_o-i)))\n",
    "    if (de_o-i)==0:\n",
    "        wo=np.sqrt(float(dem.coeff(s,de_o-i)))\n",
    "N=[]    \n",
    "for i in range(num_o+1):\n",
    "    N.append(float(dem.coeff(s,num_o-i)))\n",
    "N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 335,
   "id": "01264371",
   "metadata": {},
   "outputs": [
    {
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   "source": [
    "from scipy import signal\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "# N and D are a list of zeros and poles\n",
    "#N=[10000]\n",
    "#D=[1.0, 141.421356237309, 10000]\n",
    "# autmoatically find the plotting range.\n",
    "wo_min=(np.log10(wo))\n",
    "wo_max=(np.log10(wo))\n",
    "w=np.logspace(wo_min-2,wo_max+2,50)\n",
    "#Create system that in zero, pole and gain format.\n",
    "system = signal.TransferFunction(N, D)\n",
    "#Create Bode plot object\n",
    "w, mag, phase = signal.bode(system,w)\n",
    "#Create Bode Plot\n",
    "axis_color = 'lightgoldenrodyellow'\n",
    " \n",
    "fig, (ax, ax1) = plt.subplots(2,1,figsize=(8,8)) \n",
    "fig.subplots_adjust(left=0.25, bottom=0.25)\n",
    "[line] = ax.semilogx(w,mag, linewidth=2, color='blue')\n",
    "\n",
    "ax.set_xlabel('$\\omega$(r/s)',fontsize=12)\n",
    "ax.set_ylabel('Magnitude (dB)', color='b', fontsize=12) \n",
    " \n",
    "[line1] = ax1.semilogx(w,phase, linewidth=2, color='blue')\n",
    "ax1.set_ylabel('Phase($^o$)', color='b', fontsize=12 ) \n",
    "ax1.set_xlabel('$\\omega$(r/s)',fontsize=12)\n",
    "ax1.yaxis.set_ticks(np.arange(-180,225, 45.0))\n",
    "ax.yaxis.set_ticks(np.arange(-80,90, 20.0))\n",
    "ax.grid()\n",
    "ax1.grid()\n",
    "plt.tight_layout(pad=2, w_pad=2, h_pad=1.0)\n",
    "plt.show()"
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  },
  {
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   "id": "bfb1d96f",
   "metadata": {},
   "source": [
    "How to design a 2'nd order filter can be found here here:\n",
    "\n",
    "[Sallen and Key Filter](https://en.wikipedia.org/wiki/Sallen%E2%80%93Key_topology)\n",
    "\n",
    "Also Analog Devices has a design wizard that is very useful:\n",
    "[ADI Filter wizard](https://tools.analog.com/en/filterwizard/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 332,
   "id": "e552d80e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(t)=(51*exp(7*t/10) - 7*sqrt(51)*sin(sqrt(51)*t/10) - 51*cos(sqrt(51)*t/10))*exp(-7*t/10)*Heaviside(t)/51\n"
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      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import matplotlib.pyplot as plt\n",
    "f=z\n",
    "print('f(t)={}'.format(f)) \n",
    "N=1000\n",
    "wo=np.sqrt(r1*r2*c1*c2)\n",
    "t_start=-.2*wo\n",
    "t_end=10*wo\n",
    "time=np.linspace( t_start,t_end,N)\n",
    "flam = lambdify(t, f, \"numpy\")\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(1,1,1)\n",
    "fig.subplots_adjust(left=0.25, bottom=0.25)\n",
    "legend='$g(t)*h(t) $'\n",
    "[line] = ax.plot(time, flam(time), linewidth=2, color='blue',label=legend)\n",
    "ax.set_xlabel('Time (s)',fontsize=12)\n",
    "ax.set_ylabel('Arbritrary units',fontsize=12) \n",
    "ax.legend()\n",
    "ax.set_title('Step input convolved with a second-order low-pass system ')\n",
    "ax.grid()\n",
    "plt.show() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 341,
   "id": "8893438b",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
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   ],
   "source": [
    "elm.style(elm.STYLE_IEEE)\n",
    "with schemdraw.Drawing(file='sallenkey.jpg',inches_per_unit=.5) as d:\n",
    "    V1 = d.add(elm.SourceV().up().label('$V_{in}$'))\n",
    "    R1 = d.add(elm.Resistor(lftlabel='1').right().label('$R_1=134k$'))\n",
    "    d.add(elm.DOT,toplabel='2')\n",
    "    C1 = d.add(elm.Capacitor().down().label('$C_1=100 \\mu $'))     \n",
    "    d.add(elm.GND(xy=V1.start))\n",
    "    R2 = d.add(elm.Resistor(xy=R1.end).right().label('$R_2=7.5k$'))\n",
    "    C2 = d.add(elm.Capacitor().down().label('$C_2=10 \\mu$'))\n",
    "    d.add(elm.GND)\n",
    "    L2 = d.add(elm.Line(xy=C2.start,l=d.unit/2).right())\n",
    "    d.add(elm.DOT,xy=C2.start,toplabel='3')\n",
    "    op1 = d.add(elm.OPAMP,xy=L2.end,fill='red',d='right',anchor='in2')\n",
    "    L3 = d.add(elm.Line(xy=op1.in1,l=d.unit/2).left())\n",
    "    L4 = d.add(elm.Line(l=d.unit/2).up())\n",
    "    L5= d.add(elm.Line(l=1.5*d.unit ).right())\n",
    "    L6 = d.add(elm.Line(toy=op1.out,l=d.unit/2).down())\n",
    "    d.add(elm.DOT,rgtlabel='$V_{out}$')\n",
    "    L7 = d.add(elm.Line(tox=op1.out).left())\n",
    "    L8 = d.add(elm.Line(xy=L6.end,l=1.5*d.unit).down())\n",
    "    L9 = d.add(elm.Line(tox=C1.end).left())\n",
    "    L10 = d.add(elm.Line(toy=C1.end).up())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8a6cbbb",
   "metadata": {},
   "source": [
    "An LTspice file can be found here:\n",
    "[Sallen and Key Low Pass Filter](https://github.com/davidparent/Circuits-and-Systems/blob/main/20230328%20Sallen%20Key%20Butterworth%20radian%20step.asc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8e4f4841",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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