20230323 First Order RC Low Pass Filter.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "f104991b",
   "metadata": {},
   "source": [
    "# First order low-pass RC filter\n",
    "This notebook shows how to use SymPy, Schemdraw, SciPy, and matplotlib to simulate the frequency response and step response of a first-order RC filter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "0738f0ef",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import schemdraw\n",
    "import schemdraw.elements as elm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "00da27d7",
   "metadata": {},
   "outputs": [
    {
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     "metadata": {},
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   ],
   "source": [
    "with schemdraw.Drawing(file='RC_LP.jpg') as d:\n",
    "    V1 = d.add(elm.SourceV(rgtlabel='1').up().label('$V_{in}$'))\n",
    "    R1 = d.add(elm.Resistor().right().label('$R_1$'))\n",
    "    C1 = d.add(elm.Capacitor(rgtlabel='2').down().label('$C_1$'))     \n",
    "    L1=d.add(elm.LINE, d='left', to=V1.start) \n",
    "    d.add(elm.GND)\n",
    "    d.add(elm.DOT,xy=V1.start)\n",
    "     "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "597bc169",
   "metadata": {},
   "source": [
    "We can use voltage division and impedances to write the transfer function.\n",
    "$$ H(s)=\\frac{\\frac{1}{R_1C_1}}{s+\\frac{1}{R_1C_1}}=\\frac{\\omega_0}{s+\\omega_0} $$\n",
    "This can be put into a signal flow graph as shown below:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "056b84cf",
   "metadata": {},
   "outputs": [
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       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-32\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_5\">\n",
       "    <g clip-path=\"url(#pf1e68ba49d)\">\n",
       "     <!-- $V_{in}$ -->\n",
       "     <g transform=\"translate(19.5684 38.529219)scale(0.125 -0.125)\">\n",
       "      <use xlink:href=\"#DejaVuSans-Oblique-56\" transform=\"translate(0 0.09375)\"/>\n",
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       "   </g>\n",
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       "   </g>\n",
       "   <g id=\"text_7\">\n",
       "    <g clip-path=\"url(#pf1e68ba49d)\">\n",
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       "      <use xlink:href=\"#DejaVuSans-Oblique-3c9\" transform=\"translate(61 56.71875)scale(0.7)\"/>\n",
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       "      <use xlink:href=\"#DejaVuSans-2b\" transform=\"translate(50.107422 -30.604531)scale(0.7)\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-Oblique-3c9\" transform=\"translate(122.397461 -30.604531)scale(0.7)\"/>\n",
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       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_8\">\n",
       "    <g clip-path=\"url(#pf1e68ba49d)\">\n",
       "     <!-- 2 -->\n",
       "     <g transform=\"translate(138.036838 29.420391)scale(0.125 -0.125)\">\n",
       "      <use xlink:href=\"#DejaVuSans-32\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pf1e68ba49d\">\n",
       "   <rect x=\"7.2\" y=\"7.2\" width=\"160.8234\" height=\"46.24\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<schemdraw.backends.mpl.Figure object at 0x000002A35E8F9F70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from schemdraw import dsp \n",
    "with schemdraw.Drawing(file='signal.jpg') as d:    \n",
    "    a1=d.add(dsp.Arrow(l=d.unit/2, label='1').label('$V_{in}$', loc='left'))\n",
    "    g1= d.add(dsp.Box(w=1.1, h=1, label='$\\\\frac{\\omega_0}{s+\\omega_0}$', anchor='W'))\n",
    "    a2=d.add(dsp.Arrow('right', l=d.unit/2, xy=g1.E,label='2'))\n",
    "\n",
    "     "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8cee674d",
   "metadata": {},
   "source": [
    "We can now call Node 2 Vout."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "871698bd",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "   </g>\n",
       "   <g id=\"text_4\">\n",
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      "text/plain": [
       "<schemdraw.backends.mpl.Figure object at 0x000002A35E8DB3D0>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "with schemdraw.Drawing(file='signal.jpg') as d:   \n",
    "    a1=d.add(dsp.Arrow(l=d.unit/2).label('$V_{in}$', loc='left'))\n",
    "    g1= d.add(dsp.Box(w=1.1, h=1, label='$\\\\frac{\\omega_0}{s+\\omega_0}$', anchor='W'))\n",
    "    a2=d.add(dsp.Arrow('right', l=d.unit/2, xy=g1.E,label='$V_{out}$'))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6c52763",
   "metadata": {},
   "source": [
    "We can write:\n",
    "$$V_{out}=\\frac{\\omega_0}{s+\\omega_0} \\times V_{in}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "551c1ea7",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import symbols\n",
    "from sympy import *\n",
    "s= symbols(\"s\", positive=True)\n",
    "t= symbols(\"t\", real=True)\n",
    "R1, C1 = symbols(\"R1,C1,\",real=True, positive=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f3fc7d1b",
   "metadata": {},
   "source": [
    "The transfer function in Sympy is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "0fcefa14",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{C_{1} R_{1} \\left(s + \\frac{1}{C_{1} R_{1}}\\right)}$"
      ],
      "text/plain": [
       "1/(C1*R1*(s + 1/(C1*R1)))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H=1/R1/C1/(s+1/R1/C1)\n",
    "H"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73ca4d68",
   "metadata": {},
   "source": [
    "Now that we have the transfer function, we can find a function for the step response."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "526bf952",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Prepare for Bode Plot. (Need poles and zeros.)\n",
    "num,de=fraction(H)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f4dd96bb",
   "metadata": {},
   "outputs": [],
   "source": [
    "#We need numerical poles and zeros toplot Bode plot.\n",
    "# Set R and C vlaues for plotting\n",
    "res=100e3\n",
    "cap=10e-6\n",
    "de=de.subs(R1,res)\n",
    "de=de.subs(C1,cap)\n",
    "num=num.subs(R1,res)\n",
    "num=num.subs(C1,cap)\n",
    "poles=roots(de,s)\n",
    "zeros=roots(num,s)\n",
    "num_coeffs = Poly(num, s).coeffs()\n",
    "de_coeffs = Poly(de, s).coeffs()\n",
    "Gain=float(num_coeffs[0]/de_coeffs[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "315c13e6",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Go though list of poles and zeros and make a list for the Bod plot function.\n",
    "import numpy as np\n",
    "plist=[]\n",
    "for root, multiplicity in poles.items():\n",
    "    plist.append(float(root))\n",
    "plist=np.array(plist) \n",
    "nlist=[]\n",
    "for root, multiplicity in zeros.items():\n",
    "    nlist.append(float(root))\n",
    "nlist=np.array(nlist) \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "fb8d38ad",
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 800x800 with 2 Axes>"
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     "metadata": {},
     "output_type": "display_data"
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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",
    "N=nlist\n",
    "D=plist\n",
    "# autmoatically find the plotting range.\n",
    "wo_min=(np.log10(abs(np.amin(D))))\n",
    "wo_max=(np.log10(abs(np.amax(D))))\n",
    "w=np.logspace(wo_min-2,wo_max+2,50)\n",
    "#Create system that in zero, pole and gain format.\n",
    "system = signal.lti(N, D,[Gain])\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()\n",
    " \n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1d245213",
   "metadata": {},
   "source": [
    "Now that we have the transfer function, we can find a function for the step response."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "d0a355a2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\theta\\left(t\\right) - e^{- \\frac{t}{C_{1} R_{1}}} \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "Heaviside(t) - exp(-t/(C1*R1))*Heaviside(t)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G=1/s\n",
    "F=H*G\n",
    "z=inverse_laplace_transform(F,s,t).evalf().simplify()\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e29a280",
   "metadata": {},
   "source": [
    "Note: In SymPy $$\\Theta(t)=u(t)$$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "b033da2e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "z(t)=g(t)*(t)=Heaviside(t) - exp(-t/(C1*R1))*Heaviside(t)\n",
      "f(t)=Heaviside(t) - exp(-1.0*t)*Heaviside(t)\n"
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      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print('z(t)=g(t)*(t)={}'.format(z))\n",
    "f=z\n",
    "f=f.subs(R1,res)\n",
    "f=f.subs(C1,cap)\n",
    "print('f(t)={}'.format(f)) \n",
    "N=1000\n",
    "Tau=res*cap\n",
    "t_start=-.2*Tau\n",
    "t_end=5*Tau\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 low-pass system ')\n",
    "ax.grid()\n",
    "plt.show() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f18d5f49",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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