rjesavanje_jednadzbi.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Rješavanje jednadžbi za dobivanje prijenosne funkcije\n",
    "\n",
    "U ovom poglavlju biti će opisano rješavanje linearnih jednadžbi supstitucijom, rješavanje sustava linearnih jednadžbi te supstitucije varijabli.\n",
    "\n",
    "Da bi smo dobili prijenosnu funkciju sustava potrebno je rješiti sustav jednadžbi koje definiraju sustav, u elektrotehnici su to npr. jednadžbe dobivene metodom konturnih struji, napona čvorova ili nekom metodom koja opisuje strujno-naponske zavisnosti unutar sustava.\n",
    "\n",
    "Rješavanju jednadžbi potrebno je pristupiti analitički, te standardnim postupcima supstitucije doći do rješenja.\n",
    "\n",
    "Python funkcije koje se koriste za ove operacije dio su ```Sympy``` knjižnice.\n",
    "```python\n",
    "* Eq\n",
    "* symbols\n",
    "* Function\n",
    "* sympy.solvers solve\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Za početak potrebno je obrisati sve varijable iz memorije."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import os\n",
    "clear = lambda: os.system('cls')\n",
    "clear()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Potom importati sve funkcije koje koristimo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.interactive import printing\n",
    "printing.init_printing(use_latex=True)\n",
    "\n",
    "from sympy import Eq, symbols, Function\n",
    "from sympy.solvers import solve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Primjer: Linearni električni sustav (Izvor: N.Stojkovic, N.Mijat - Teorija mreža i linija (Zbirka zadataka), zadatak 3.05.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Za početak je potrebno definirati sva imena matematičkih varijabli koje koristimo kao Sympy simbole."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "Uul, Uiz = symbols('Uul Uiz')\n",
    "Z1, Z2 = symbols('Z1 Z2')\n",
    "I1, I2 = symbols('I1 I2')\n",
    "r = symbols('r')\n",
    "Tf = symbols('Tf')\n",
    "s = symbols('s')\n",
    "L1, L2 = symbols('L1 L2')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.) Postavimo jednadžbe koje opisuju sustav."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$Uul = I_{2} Z_{1} + I_{2} Z_{2} + Uiz$$"
      ],
      "text/plain": [
       "Uul = I₂⋅Z₁ + I₂⋅Z₂ + Uiz"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eq = Eq(Uul, I2*Z1 + Uiz + I2*Z2)\n",
    "display(eq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "I2 =\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{Uul}{r}$$"
      ],
      "text/plain": [
       "Uul\n",
       "───\n",
       " r "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "i2 = (1/r) * Uul\n",
    "print('I2 =')\n",
    "display(i2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Za supstituiranje I2 u gornjoj jednadžbi s izrazom za I2 potrebno je koristiti funkciju ```.subs()```\n",
    "\n",
    "Funkcija ```.subs()``` kao argument ima varijablu koju supstituiramo te izraz koji unosi ta supstitucija.\n",
    "Simbol (Matematicka varijabla) i python varijabla ne smiju imati isto ime, no ukoliko nam je simbol ```I2```, a python varijabla ```i2``` tada možemo supstituirati simbol s izrazom za ```I2``` koji je spremljen kao python varijabla ```i2```.\n",
    "Iz tog razloga je gornja jednadžba zapisana kao ```i2=...``` a ne ```I2=...```.\n",
    "\n",
    "```.subs(VARIJABLA, IZRAZ)```, u ovom slučaju:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$Uul = Uiz + \\frac{Uul Z_{1}}{r} + \\frac{Uul Z_{2}}{r}$$"
      ],
      "text/plain": [
       "            Uul⋅Z₁   Uul⋅Z₂\n",
       "Uul = Uiz + ────── + ──────\n",
       "              r        r   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eq = eq.subs(I2, i2)\n",
    "display(eq)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Isto smo mogli napraviti s:\n",
    "\n",
    "```python\n",
    "eq = eq.subs(I2, (1/r) * Uul)\n",
    "```\n",
    "\n",
    "\n",
    "jer je u memoriji računala (1/r) * Uul spremljeno kao i2."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sada kada smo dobili izraz iz kojeg mozemo izlučiti ```Uul``` te ```Uiz```, samim time dobiti prijenosnu funkciju ```Tf = Uiz/Uul``` potrebno je rješiti jednadžbu za varijablu ```Uul``` uz pomoć funkcije ```solve()```, to će nam dati izraz u obliku ```Uul = ...```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ \\frac{Uiz r}{- Z_{1} - Z_{2} + r}\\right ]$$"
      ],
      "text/plain": [
       "⎡   Uiz⋅r    ⎤\n",
       "⎢────────────⎥\n",
       "⎣-Z₁ - Z₂ + r⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Uul = solve(eq, Uul)\n",
    "display(Uul)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vidimo da je zapis u obliku matrice, s matricom ne možemo raditi daljnje potrebne operacije stoga je potrebno izvući izraz iz matrice i spremiti ga u \"običnu\" varijablu.  \n",
    "S obzirom da smo jednadžbu rješili samo za jednu varijablu, rezultat je izražen kao samo jedno polje te je dovoljno napraviti sljedeće:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{Uiz r}{- Z_{1} - Z_{2} + r}$$"
      ],
      "text/plain": [
       "   Uiz⋅r    \n",
       "────────────\n",
       "-Z₁ - Z₂ + r"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Uul = Uul[0]\n",
    "display(Uul)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sada kada imamo definiran izraz za ```Uul``` možemo jednostavno definirati ```Tf``` kao ```Uiz/Uul``` te dobiti prijenosnu funkciju."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{- Z_{1} - Z_{2} + r}{r}$$"
      ],
      "text/plain": [
       "-Z₁ - Z₂ + r\n",
       "────────────\n",
       "     r      "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Tf = Uiz/Uul\n",
    "display(Tf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```Z1``` i ```Z2``` možemo supstituirati kao ```sL1``` i ```sL2``` putem funkcije ```.subs()``` kao u ranijem koraku."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{- L_{1} s - L_{2} s + r}{r}$$"
      ],
      "text/plain": [
       "-L₁⋅s - L₂⋅s + r\n",
       "────────────────\n",
       "       r        "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Tf = Tf.subs(Z1, s*L1)\n",
    "Tf = Tf.subs(Z2, s*L2)\n",
    "display(Tf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Potom vrijednosti ```L1```, ```L2``` i ```r``` možemo unositi istim postupkom supstitucije, npr. ```L1=2```, ```L2=3```, ```r=1/2```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 10.0 s + 1.0$$"
      ],
      "text/plain": [
       "-10.0⋅s + 1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Tf = Tf.subs(L1, 2)\n",
    "Tf = Tf.subs(L2, 3)\n",
    "Tf = Tf.subs(r, 1/2)\n",
    "display(Tf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Isto smo mogli napraviti na način:\n",
    "```python\n",
    "l1 = 2\n",
    "l2 = 3\n",
    "R = 1/2\n",
    "Tf = Tf.subs(L1, l1)\n",
    "Tf = Tf.subs(L2, l2)\n",
    "Tf = Tf.subs(r, R)\n",
    "display(Tf)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Primjer: Linearni električni sustav (Izvor: N.Stojkovic, N.Mijat - Teorija mreža i linija (Zbirka zadataka), zadatak 3.12.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAASCAYAAABrXO8xAAAABHNCSVQICAgIfAhkiAAAAM5JREFUOI3N0rEyQ0EUxvEfJcMVWhoZ8Qpq75PnoDJ0Kl5BK0U67yAUiUYmpZGYoRDNuePOkXszYxS+5ps95/x3Z3c//lC7uMIz3jHCOVpNUBsTzHGDU/RjfY+dOvA2hrqpfhb1y0XQfjSHWE29DUwxw7o0cBzew2cCX3GHNRxl8DD8oeYaj+GdDBbhLzVgWd/K4DKthM8zWO5YWKzN6lwVHIR3asCD8B9v0Lb8O97Ed2T9KgDlqdXInfiO3EBD5GAP1xjjA0+4wHYT9I/1BRgnMa7SDNsAAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import os\n",
    "clear = lambda: os.system('cls')\n",
    "clear()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.interactive import printing\n",
    "printing.init_printing(use_latex=True)\n",
    "\n",
    "from sympy import Eq, symbols, Function\n",
    "from sympy.solvers import solve\n",
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "U1, U2 = symbols('U1 U2')\n",
    "Uiz, Uul = symbols('Uiz Uul')\n",
    "I1, I2 = symbols('I1 I2')\n",
    "Tf = symbols('Tf')\n",
    "s = symbols('s')\n",
    "R1, R2, L, C = symbols('R1 R2 L C')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1) Definiramo jednadžbe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$U_{1} \\left(C s + \\frac{1}{L s}\\right) - \\frac{Uul}{L s} = 0$$"
      ],
      "text/plain": [
       "   ⎛       1 ⎞   Uul    \n",
       "U₁⋅⎜C⋅s + ───⎟ - ─── = 0\n",
       "   ⎝      L⋅s⎠   L⋅s    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$U_{2} \\left(\\frac{1}{R_{2}} + \\frac{1}{R_{1}}\\right) - \\frac{Uiz}{R_{1}} = 0$$"
      ],
      "text/plain": [
       "   ⎛1    1 ⎞   Uiz    \n",
       "U₂⋅⎜── + ──⎟ - ─── = 0\n",
       "   ⎝R₂   R₁⎠    R₁    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eq1 = Eq(U1 * ((1/(s*L))+s*C) - Uul*(1/(s*L)), 0)\n",
    "eq2 = Eq(U2 * ((1/R1)+(1/R2)) - Uiz*(1/R1), 0)\n",
    "\n",
    "display(eq1)\n",
    "display(eq2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Po definiciji operacijskog pojačala ```Uiz=A(U1-U2)```, gdje A teži ka beskonačno dolazimo do izraza ```U1=U2```  \n",
    "U idućem koraku rješimo gornje jednadžbe za varijable ```U1``` i ```U2```."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{Uul}{C L s^{2} + 1}$$"
      ],
      "text/plain": [
       "   Uul    \n",
       "──────────\n",
       "     2    \n",
       "C⋅L⋅s  + 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{R_{2} Uiz}{R_{1} + R_{2}}$$"
      ],
      "text/plain": [
       " R₂⋅Uiz\n",
       "───────\n",
       "R₁ + R₂"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u1 = solve(eq1, U1)\n",
    "u2 = solve(eq2, U2)\n",
    "u1 = u1[0]\n",
    "u2 = u2[0]\n",
    "display(u1)\n",
    "display(u2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vratimo se na definiciju ```U1 = U2```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$U_{1} = U_{2}$$"
      ],
      "text/plain": [
       "U₁ = U₂"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eq3 = Eq(U1, U2)\n",
    "display(eq3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Te uvrstimo supstituciju s gore dobivenim izrazima za ```U1``` i ```U2```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{Uul}{C L s^{2} + 1} = \\frac{R_{2} Uiz}{R_{1} + R_{2}}$$"
      ],
      "text/plain": [
       "   Uul        R₂⋅Uiz\n",
       "────────── = ───────\n",
       "     2       R₁ + R₂\n",
       "C⋅L⋅s  + 1          "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eq3 = eq3.subs(U1, u1)\n",
    "eq3 = eq3.subs(U2, u2)\n",
    "display(eq3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Potom rješimo jednadžbu za varijablu ```Uiz``` te podijelimo s ```Uul```."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAnCAYAAACL4Y8gAAAABHNCSVQICAgIfAhkiAAABZtJREFUaIHtmllsVkUUx38f4FLTxIhEWyO22LqhVUpLECi1Ron6YNwQMAYX4pYIEo3B5UHUGFF5covxQfOJEKOiD2rUSiGEROPWtFQF6xpSbUQsqCgYaq0P5wydbzp36/1aC95/cvPdM/fMnXPPzJnl/D/IcEDjUeAdSz4K2A5U/TfmDBvWAnfEVe4BbveUrwC+Tdhws9YzWAm84OhsAPr16gW+AhYlbKcYSGPHmcBO4MgoxUptYLbn2TrglZgNGvwEzNf7I4BfPe/eBdwDlAEVwMPAP2p0GuSB+xPop7WjFbg1SukKoA8o9TzrAZbp/RxgH3CI9Xwi0jkm/MtUPkXlK5Fezll1qlSn1iqr1LIFUcZGIE98BxfDjuXA+27hGEeuA74E/vAYMB7pJYApwBYklLDKdgPfqVwL7AG+VrlB6/c77f0OdKhcDjyGjJy26G8qGophx0fANKDELvQ5+FNP5Xr9tR3c7ujUqoH9lk6HGgkyIro97ZUCvyGd0Q1cgiwYnZbeG0gIr/XYVgxE2TER2IgMqs3A5Z53dCMRfVxYQzuAJZ7ylcA3lryFwQvh68DTlvwy8IwlNwPPOnVatKwamKo6T3raPxe4mHAH34tEnrl6kWnMLvOtLXHsKAfO0vtjgC5kTbFxEjK4zggy8HhVON/zrA1YpfclwN/IR9voAm605E7gZktew+BFsge41pIrkDXgdI8NTYQ7eDziIHO9BjzhlJUE1E1iB0hknuCUTUf8d6xdaE8RZsFye2YGEu6vqlwFjEXmaoNGpIM2W++opnAaaQMmW/IkxCmfWWXbtM7Vg78pEjuRKDPXbk/ZXk+9pHbUI77qcsprkGlie5CBOeALbWgOsj1ZpJXetPTKkHn1MpWnI87uY2CEzFDZ7qwaLZug8lynjsEjFHaeQRPJ5uA88XYRSew4GpkeZ3reswp4LqqxSchH7EBW1TZgKXCoo3c3Mjp+1BcvB7Zaz29xZIMPgMV6vwLZzLs4Dwk1NzybGB4Hx7XjMGATsNCjW4L46+wE9g0LLkA+ZuwQ6jYxfLuIKOSAlwjusMXAeyNmTQRuQxaRJGhBomoP8AMyBY0kGpBpsd26aqznNzFwoMqQIUOGDKMEJrPVH6qV4X+NOImYDCkQJxFz0GA0cG6+RMxIIBYvl5Yf83FuICNsDXJI+Av4HniRwuPwJgYydkNFPXJEz0UpJkQjko/pRnwz16MTi5dLw0sFcW7XIOnN54FZyOiajTjY5IdzyFl+aYx2ghCWiPEhT3xK6SLgIWR+D3IwRPByaXkpH+c2E3Guj6UGSRMCnKztNATo5RA+sBOJgJ8pzEuEJWKCkCcZKWoQ5mAvL2cwD6FMTCKmHEmQ9xHvnP04wjzb+Bj58ChcRTDZCuLcrUiGqwLJWplkflQiJgj5IdSBcAdfiLAo+1Of46yHNi81RpX2MZgfC0IlhZzbqQgJGGf0T0Xme5dstQ1/G1iv8jbgQ72fhfw1oAO4VMsWUphAHynYvNyg/5BE8VJR+02Xc1uA9HZ1DMM2AKtDni9BRngLMsdNCNENQhrOzkbYCA7l5aJ4qaj9psu5Xa+NlcUwehfB87RBFRJN7UiUnRbjvTbScHY2whzs5eVAmIx+ZOTaaEV2Ej64+807gc8teZq+c15AfdM5J6reOQF6LsYhO475UYoRyFP8OfgGhOXZD0N61iHbMZfmWYf/6Okj/pqRUWXC9xNk3nwKuA4Jn2rEMesZiIY6/e1FQstcZoTepfUnI7uNB5DQ3hjwkcOBUoT4naJypd67B5pG4F3fC5LwY2H7TZtzA9k+LUNG+5/IVNAKPAgcbrXd77nMInUfQj7uBX4B3mKgc9IgT/wR3BRgY97SKQovF7XfTMO5HehIzcvF3W8OhXM7GJCal4si/jJkyJAhQ4ZRhX8BDVKYlAx61ocAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{Uul \\left(R_{1} + R_{2}\\right)}{R_{2} \\left(C L s^{2} + 1\\right)}$$"
      ],
      "text/plain": [
       " Uul⋅(R₁ + R₂) \n",
       "───────────────\n",
       "   ⎛     2    ⎞\n",
       "R₂⋅⎝C⋅L⋅s  + 1⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tf = solve(eq3, Uiz)\n",
    "tf = tf[0]\n",
    "display(tf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{R_{1} + R_{2}}{R_{2} \\left(C L s^{2} + 1\\right)}$$"
      ],
      "text/plain": [
       "    R₁ + R₂    \n",
       "───────────────\n",
       "   ⎛     2    ⎞\n",
       "R₂⋅⎝C⋅L⋅s  + 1⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tf = tf/Uul\n",
    "display(tf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Uvrstimo zadane vrijednosti parametara te supstituiramo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "r1 = 9\n",
    "r2 = 1\n",
    "l = 2\n",
    "c = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{10}{2 s^{2} + 1}$$"
      ],
      "text/plain": [
       "   10   \n",
       "────────\n",
       "   2    \n",
       "2⋅s  + 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tf = tf.subs(R1, r1)\n",
    "tf = tf.subs(R2, r2)\n",
    "tf = tf.subs(L, l)\n",
    "tf = tf.subs(C, c)\n",
    "display(tf)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}