Example_2_transfer_function_derivation.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "from sympy.utilities.lambdify import lambdify\n",
    "import numpy.linalg as la\n",
    "import numpy as np\n",
    "from sympy import init_printing\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "z, r,tau=sympy.symbols('z r tau')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t=1.-r**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "e=sympy.exp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "dim=2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "T_denom = (1.-r**2* e(z)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAAWBAMAAADultUCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- r^{2} e^{2 z} + 1.0$$"
      ],
      "text/plain": [
       "   2  2⋅z      \n",
       "- r ⋅ℯ    + 1.0"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T_denom"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "## transfer function in the program\n",
    "T_program = -r*sympy.eye(dim) + ((1.-r**2)/T_denom) * sympy.Matrix([[r*e(z)**2,e(z)],[e(z),r*e(z)**2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## coefficients of transfer between the two internal nodes\n",
    "M1 = sympy.Matrix([[0,r],[r,0]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## from inputs to internal nodes\n",
    "M2 = sympy.Matrix([[t,0],[0,t]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## from internal nodes to outputs\n",
    "M3 = sympy.Matrix([[0,t],[t,0]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## from inputs to outputs. Note the phase shift \n",
    "M4 = sympy.Matrix([[-r,0],[0,-r]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M = np.vstack([np.hstack([M1,M2]),np.hstack([M3,M4])])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 r -r**2 + 1.0 0]\n",
      " [r 0 0 -r**2 + 1.0]\n",
      " [0 -r**2 + 1.0 -r 0]\n",
      " [-r**2 + 1.0 0 0 -r]]\n"
     ]
    }
   ],
   "source": [
    "print M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAyBAMAAAD8cdteAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nbJmP4g8oQll1eefr9QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}e^{z} & 0\\\\0 & e^{z}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ z    ⎤\n",
       "⎢ℯ   0 ⎥\n",
       "⎢      ⎥\n",
       "⎢     z⎥\n",
       "⎣0   ℯ ⎦"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E = sympy.Matrix([[e(z),0],[0,e(z)]])\n",
    "E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAAyBAMAAACOvrnoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & - r e^{z}\\\\- r e^{z} & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡           z⎤\n",
       "⎢  1    -r⋅ℯ ⎥\n",
       "⎢            ⎥\n",
       "⎢    z       ⎥\n",
       "⎣-r⋅ℯ     1  ⎦"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.eye(2)-M1*E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "##inverse of sympy.eye(2)-M1*E\n",
    "inv = sympy.Matrix([[1,r*e(z)],[r*e(z),1]])/(1-r**2*e(2*z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWQAAABNCAMAAACmEe5YAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\na7tiKQzWA+J/D0aOXq0S/qsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{r \\left(- r^{2} + 1.0\\right)^{2} e^{2 z}}{- r^{2} e^{2 z} + 1} - r & \\frac{\\left(- r^{2} + 1.0\\right)^{2} e^{z}}{- r^{2} e^{2 z} + 1}\\\\\\frac{\\left(- r^{2} + 1.0\\right)^{2} e^{z}}{- r^{2} e^{2 z} + 1} & \\frac{r \\left(- r^{2} + 1.0\\right)^{2} e^{2 z}}{- r^{2} e^{2 z} + 1} - r\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡              2                           2       ⎤\n",
       "⎢  ⎛   2      ⎞   2⋅z          ⎛   2      ⎞   z    ⎥\n",
       "⎢r⋅⎝- r  + 1.0⎠ ⋅ℯ             ⎝- r  + 1.0⎠ ⋅ℯ     ⎥\n",
       "⎢──────────────────── - r      ────────────────    ⎥\n",
       "⎢      2  2⋅z                      2  2⋅z          ⎥\n",
       "⎢   - r ⋅ℯ    + 1               - r ⋅ℯ    + 1      ⎥\n",
       "⎢                                                  ⎥\n",
       "⎢                2                       2         ⎥\n",
       "⎢    ⎛   2      ⎞   z        ⎛   2      ⎞   2⋅z    ⎥\n",
       "⎢    ⎝- r  + 1.0⎠ ⋅ℯ       r⋅⎝- r  + 1.0⎠ ⋅ℯ       ⎥\n",
       "⎢    ────────────────      ──────────────────── - r⎥\n",
       "⎢        2  2⋅z                  2  2⋅z            ⎥\n",
       "⎣     - r ⋅ℯ    + 1           - r ⋅ℯ    + 1        ⎦"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "##derived transfer function\n",
    "T = M3*E*inv*M2+M4\n",
    "T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{r \\left(- r^{2} + 1.0\\right) e^{2 z}}{- r^{2} e^{2 z} + 1.0} - r & \\frac{\\left(- r^{2} + 1.0\\right) e^{z}}{- r^{2} e^{2 z} + 1.0}\\\\\\frac{\\left(- r^{2} + 1.0\\right) e^{z}}{- r^{2} e^{2 z} + 1.0} & \\frac{r \\left(- r^{2} + 1.0\\right) e^{2 z}}{- r^{2} e^{2 z} + 1.0} - r\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  ⎛   2      ⎞  2⋅z          ⎛   2      ⎞  z    ⎤\n",
       "⎢r⋅⎝- r  + 1.0⎠⋅ℯ             ⎝- r  + 1.0⎠⋅ℯ     ⎥\n",
       "⎢─────────────────── - r      ───────────────    ⎥\n",
       "⎢     2  2⋅z                     2  2⋅z          ⎥\n",
       "⎢  - r ⋅ℯ    + 1.0            - r ⋅ℯ    + 1.0    ⎥\n",
       "⎢                                                ⎥\n",
       "⎢    ⎛   2      ⎞  z        ⎛   2      ⎞  2⋅z    ⎥\n",
       "⎢    ⎝- r  + 1.0⎠⋅ℯ       r⋅⎝- r  + 1.0⎠⋅ℯ       ⎥\n",
       "⎢    ───────────────      ─────────────────── - r⎥\n",
       "⎢       2  2⋅z                 2  2⋅z            ⎥\n",
       "⎣    - r ⋅ℯ    + 1.0        - r ⋅ℯ    + 1.0      ⎦"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## transfer function in code\n",
    "T_program"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
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