{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "9f9c1606",
   "metadata": {},
   "outputs": [],
   "source": [
    "from system_eqns import systemEqns\n",
    "from sympy import Function, Derivative, symbols, Matrix, Eq, simplify, sin, cos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "6d83b5a0",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d}{d t} \\operatorname{Vc_{1}}{\\left(t \\right)} = - \\frac{\\operatorname{Vc_{2}}{\\left(t \\right)}}{C R}$"
      ],
      "text/plain": [
       "Eq(Derivative(Vc1(t), t), -Vc2(t)/(C*R))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d}{d t} \\operatorname{Vc_{2}}{\\left(t \\right)} = \\frac{- V_{in} + \\operatorname{Vc_{1}}{\\left(t \\right)} - \\operatorname{Vc_{2}}{\\left(t \\right)}}{C R}$"
      ],
      "text/plain": [
       "Eq(Derivative(Vc2(t), t), (-V_{in} + Vc1(t) - Vc2(t))/(C*R))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(a_2*omega, -a_1*omega), (a_3/(C*R), a_4/(C*R))]\n",
      "[(a_4*omega, -a_3*omega), (-a_1/(C*R), -a_2/(C*R)), (a_5/(C*R), a_6/(C*R)), (a_3/(C*R), a_4/(C*R))]\n",
      "[a_1, a_2, a_3, a_4, a_5, a_6]\n",
      "a_2*omega -a_1*omega\n",
      "a_3/(C*R) a_4/(C*R)\n",
      "a_4*omega -a_3*omega\n",
      "-a_1/(C*R) -a_2/(C*R)\n",
      "a_5/(C*R) a_6/(C*R)\n",
      "a_3/(C*R) a_4/(C*R)\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\omega & 0 & 0 & \\frac{1}{C R} & 0 & 0\\\\0 & \\omega & \\frac{1}{C R} & 0 & 0 & 0\\\\0 & - \\frac{1}{C R} & - \\omega & \\frac{1}{C R} & 0 & \\frac{1}{C R}\\\\- \\frac{1}{C R} & 0 & \\frac{1}{C R} & \\omega & \\frac{1}{C R} & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  -omega,        0,       0, 1/(C*R),       0,       0],\n",
       "[       0,    omega, 1/(C*R),       0,       0,       0],\n",
       "[       0, -1/(C*R),  -omega, 1/(C*R),       0, 1/(C*R)],\n",
       "[-1/(C*R),        0, 1/(C*R),   omega, 1/(C*R),       0]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[a_1, a_2, a_3, a_4, a_5, a_6]\n"
     ]
    }
   ],
   "source": [
    "A_0, A_1, A_2, B_1, B_2, R1, R2, R3, Vin, C1, C2 = symbols(\"A_0 A_1 A_2 B_1 B_2 R_1 R_2 R_3 V_{in} C_1 C_2\")\n",
    "Vc1 = Function('Vc1')\n",
    "Vc2 = Function('Vc2')\n",
    "Vc3 = Function('Vc3')\n",
    "R1,R2, Vin, Vc1, Vc2, C1,C2\n",
    "Vin, Vc, C, R, i, t, f, A, tau, w = symbols(\"V_{in} V_c C R i t f A tau omega\")\n",
    "\n",
    "m = Matrix([[R1, -R1], [-R1, R1-R2]])\n",
    "v = Matrix([Vin-Vc1(t), Vc2(t)])\n",
    "mtmp = m.inv()*v\n",
    "\n",
    "\n",
    "eqn_1 = Eq(C1 * Vc1(t).diff()/C, mtmp[0]/C)\n",
    "eqn1 = eqn_1.subs([(R1, R), (R2, R), (C1,C)])\n",
    "\n",
    "eqn_2 = Eq(Vc2(t).diff(), simplify(mtmp[1]/C))\n",
    "eqn2 = eqn_2.subs([(R1, R), (R2, R)])\n",
    "\n",
    "display(eqn1)\n",
    "display(eqn2)\n",
    "\n",
    "sysEqns = systemEqns()\n",
    "\n",
    "res1 = sysEqns.extractCoeffsFromDiffEqn(eqn1)\n",
    "print (res1)\n",
    "res2 = sysEqns.extractCoeffsFromDiffEqn(eqn2)\n",
    "print (res2)\n",
    "print (sysEqns.symbols_used)\n",
    "\n",
    "M1, syms_used = sysEqns.orderedMatrix([res1, res2])\n",
    "display(M1)\n",
    "print (syms_used)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "09a4e294",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d}{d t} \\operatorname{Vc_{1}}{\\left(t \\right)} = - \\frac{\\operatorname{Vc_{2}}{\\left(t \\right)}}{C R}$"
      ],
      "text/plain": [
       "Eq(Derivative(Vc1(t), t), -Vc2(t)/(C*R))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqn1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "1a0e52b2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eq(Derivative(Vc1(t), t), -Vc2(t)/(C*R))\n"
     ]
    }
   ],
   "source": [
    "print(eqn1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "96a64ff7",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\omega & 0 & 0 & \\frac{1}{C R}\\\\0 & \\omega & \\frac{1}{C R} & 0\\\\0 & - \\frac{1}{C R} & - \\omega & \\frac{1}{C R}\\\\- \\frac{1}{C R} & 0 & \\frac{1}{C R} & \\omega\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  -omega,        0,       0, 1/(C*R)],\n",
       "[       0,    omega, 1/(C*R),       0],\n",
       "[       0, -1/(C*R),  -omega, 1/(C*R)],\n",
       "[-1/(C*R),        0, 1/(C*R),   omega]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqmat = M1[:4,:4]\n",
    "sqmat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "4c6168d6",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- C R \\omega & 0 & 0 & 1\\\\0 & C R \\omega & 1 & 0\\\\0 & -1 & - C R \\omega & 1\\\\-1 & 0 & 1 & C R \\omega\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-C*R*omega,         0,          0,         1],\n",
       "[         0, C*R*omega,          1,         0],\n",
       "[         0,        -1, -C*R*omega,         1],\n",
       "[        -1,         0,          1, C*R*omega]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqmat *C*R\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "94d52194",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{C^{4} R^{4} \\omega^{3}}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{C R}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{C^{2} R^{2} \\omega}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{C^{3} R^{3} \\omega^{2} - C R}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1}\\\\\\frac{C R}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{C^{4} R^{4} \\omega^{3}}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{C^{3} R^{3} \\omega^{2} - C R}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & - \\frac{C^{2} R^{2} \\omega}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1}\\\\- \\frac{C^{2} R^{2} \\omega}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{- C^{3} R^{3} \\omega^{2} + C R}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{- C^{4} R^{4} \\omega^{3} + C^{2} R^{2} \\omega}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{C^{3} R^{3} \\omega^{2}}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1}\\\\\\frac{- C^{3} R^{3} \\omega^{2} + C R}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{C^{2} R^{2} \\omega}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{C^{3} R^{3} \\omega^{2}}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} & \\frac{C^{4} R^{4} \\omega^{3} - C^{2} R^{2} \\omega}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[        -C**4*R**4*omega**3/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1),                         C*R/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1),                         C**2*R**2*omega/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1),             (C**3*R**3*omega**2 - C*R)/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1)],\n",
       "[                        C*R/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1),          C**4*R**4*omega**3/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1),              (C**3*R**3*omega**2 - C*R)/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1),                       -C**2*R**2*omega/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1)],\n",
       "[           -C**2*R**2*omega/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1), (-C**3*R**3*omega**2 + C*R)/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1), (-C**4*R**4*omega**3 + C**2*R**2*omega)/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1),                     C**3*R**3*omega**2/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1)],\n",
       "[(-C**3*R**3*omega**2 + C*R)/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1),             C**2*R**2*omega/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1),                      C**3*R**3*omega**2/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1), (C**4*R**4*omega**3 - C**2*R**2*omega)/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1)]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqmat**-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "b6558bc4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\- \\frac{A}{C R}\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[       0],\n",
       "[       0],\n",
       "[-A/(C*R)],\n",
       "[       0]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = Matrix([0,0, -A/(C*R), 0]);v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "8fe7a8d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{A C R \\omega}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1}\\\\- \\frac{A \\left(C^{3} R^{3} \\omega^{2} - C R\\right)}{C R \\left(C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1\\right)}\\\\- \\frac{A \\left(- C^{4} R^{4} \\omega^{3} + C^{2} R^{2} \\omega\\right)}{C R \\left(C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1\\right)}\\\\- \\frac{A C^{2} R^{2} \\omega^{2}}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                                    -A*C*R*omega/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1)],\n",
       "[             -A*(C**3*R**3*omega**2 - C*R)/(C*R*(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1))],\n",
       "[-A*(-C**4*R**4*omega**3 + C**2*R**2*omega)/(C*R*(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1))],\n",
       "[                           -A*C**2*R**2*omega**2/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1)]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coeffs = (sqmat**-1)*v; coeffs"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea7339f8",
   "metadata": {},
   "source": [
    "Lets try subsituting in the original values, for a1, a2, a3, a4 and see what happens"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "ce122389",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{A C R \\omega \\cos{\\left(\\omega t \\right)}}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} - \\frac{A \\left(C^{3} R^{3} \\omega^{2} - C R\\right) \\sin{\\left(\\omega t \\right)}}{C R \\left(C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1\\right)}$"
      ],
      "text/plain": [
       "-A*C*R*omega*cos(omega*t)/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1) - A*(C**3*R**3*omega**2 - C*R)*sin(omega*t)/(C*R*(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vc1 = coeffs[1]*sin(sysEqns.w * t) + coeffs[0]*cos(sysEqns.w*t)\n",
    "vc1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "47108734",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{A C^{2} R^{2} \\omega^{2} \\sin{\\left(\\omega t \\right)}}{C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1} - \\frac{A \\left(- C^{4} R^{4} \\omega^{3} + C^{2} R^{2} \\omega\\right) \\cos{\\left(\\omega t \\right)}}{C R \\left(C^{4} R^{4} \\omega^{4} - C^{2} R^{2} \\omega^{2} + 1\\right)}$"
      ],
      "text/plain": [
       "-A*C**2*R**2*omega**2*sin(omega*t)/(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1) - A*(-C**4*R**4*omega**3 + C**2*R**2*omega)*cos(omega*t)/(C*R*(C**4*R**4*omega**4 - C**2*R**2*omega**2 + 1))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vc2 = coeffs[3]*sin(sysEqns.w * t) + coeffs[2]*cos(sysEqns.w*t)\n",
    "vc2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "c9d136fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The following shoulod simplify to zero as a self check our calculations are correct \n",
    "simplify(vc1.diff(t) + vc2/(C*R))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "55f9c8c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle A \\sin{\\left(\\omega t \\right)}$"
      ],
      "text/plain": [
       "A*sin(omega*t)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vc3 = A*sin(sysEqns.w*t); vc3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "91f1dec6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify((vc3-vc1+vc2)/(R*C)+vc2.diff(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "74a52129",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d}{d t} \\operatorname{Vc_{1}}{\\left(t \\right)} = \\frac{A \\sin{\\left(\\omega t \\right)} - \\operatorname{Vc_{1}}{\\left(t \\right)} - \\operatorname{Vc_{3}}{\\left(t \\right)}}{C R}$"
      ],
      "text/plain": [
       "Eq(Derivative(Vc1(t), t), (A*sin(omega*t) - Vc1(t) - Vc3(t))/(C*R))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqn1 = Eq(Derivative(Vc1(t), t), (A*sin(w*t) - Vc1(t) - Vc3(t))/(C*R))\n",
    "eqn1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "a437d3c6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\omega & 0 & 0 & \\frac{1}{C R} & 0 & 0\\\\0 & \\omega & \\frac{1}{C R} & 0 & 0 & 0\\\\0 & - \\frac{1}{C R} & - \\omega & \\frac{1}{C R} & 0 & \\frac{1}{C R}\\\\- \\frac{1}{C R} & 0 & \\frac{1}{C R} & \\omega & \\frac{1}{C R} & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  -omega,        0,       0, 1/(C*R),       0,       0],\n",
       "[       0,    omega, 1/(C*R),       0,       0,       0],\n",
       "[       0, -1/(C*R),  -omega, 1/(C*R),       0, 1/(C*R)],\n",
       "[-1/(C*R),        0, 1/(C*R),   omega, 1/(C*R),       0]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "3233946f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{C R}$"
      ],
      "text/plain": [
       "1/(C*R)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M1[2,3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "09d730de",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\omega & 0 & 0 & \\frac{1}{C R} & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[-omega, 0, 0, 1/(C*R), 0, 0]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M1.row(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "f461c002",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-omega, 0, 0, 1/(C*R), 0, 0]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(M1.row(0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d08bf393",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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