{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ffc08a3f",
   "metadata": {},
   "source": [
    "# Complex Numbers\n",
    "## In this example we manipulate and plot complex numbers.  The data correspond to the frequency response of an op-amp based low-pass filter\n",
    "*August 9, 2021*\n",
    "\n",
    "- First, import the usual modules for number crunching (numPy) and plotting (matplotlib)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "f0ab89a0",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee25c992",
   "metadata": {},
   "source": [
    "- We will attempt to plot both the magnitude and phase of the filter's transfer function.\n",
    "\n",
    "Here is the circuit that has been analyzed.  It is part of a larger circuit designed by Carl Michal for an Earth's-Field NMR apparatus: https://iopscience.iop.org/article/10.1088/0957-0233/21/10/105902"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "4235b2dc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.image as mpimg\n",
    "img = mpimg.imread('low-pass.png')\n",
    "plt.figure(figsize = (10, 7))\n",
    "imgplot = plt.imshow(img)\n",
    "plt.axis('off');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f36d2f63",
   "metadata": {},
   "source": [
    "- The magnitude and phase of the transfer function $v_{out}/v_{in}$ will be plotted as a function of frequency.  The frequency range will be 10 Hz to 100 kHz. Set up a list of frequencies evenly spaced on a logarithmic scale."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "13e82e99",
   "metadata": {},
   "outputs": [],
   "source": [
    "ff = np.logspace(1, 5, 1000) # Hz\n",
    "w = 2*np.pi*ff # rad/s"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3d6692da",
   "metadata": {},
   "source": [
    "- Enter in component values and calculate the impedance of the capacitor.  Notice the use of 1j to represent $\\sqrt{-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "02d0eb78",
   "metadata": {},
   "outputs": [],
   "source": [
    "R2 = 47e3 # ohms\n",
    "R3 = 12e3 # ohms\n",
    "C2 = 1e-9 # F\n",
    "Zc2 = 1/(1j*w*C2) # ohms"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe1610c8",
   "metadata": {},
   "source": [
    "- Here is the transfer function.  The voltage ratio $v_{out}/v_{in}$ is complex."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "199d608a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.25529677-6.46761144e-03j,  1.25529636-6.52751459e-03j,\n",
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       "        1.25529506-6.71057345e-03j,  1.25529461-6.77272683e-03j,\n",
       "        1.25529416-6.83545584e-03j,  1.25529369-6.89876584e-03j,\n",
       "        1.25529322-6.96266218e-03j,  1.25529273-7.02715031e-03j,\n",
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       "        1.25529018-7.35866127e-03j,  1.25528964-7.42681697e-03j,\n",
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       "        1.24774058-1.18759974e-01j,  1.24759978-1.19853044e-01j,\n",
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       "       -0.11754578-8.14780057e-02j, -0.11598083-7.94542236e-02j,\n",
       "       -0.11442521-7.74769792e-02j, -0.11287935-7.55453726e-02j,\n",
       "       -0.11134364-7.36585138e-02j, -0.10981847-7.18155220e-02j,\n",
       "       -0.1083042 -7.00155267e-02j, -0.10680116-6.82576674e-02j,\n",
       "       -0.10530967-6.65410943e-02j, -0.10383001-6.48649681e-02j,\n",
       "       -0.10236248-6.32284604e-02j, -0.10090731-6.16307542e-02j,\n",
       "       -0.09946475-6.00710434e-02j, -0.09803501-5.85485333e-02j,\n",
       "       -0.0966183 -5.70624410e-02j, -0.0952148 -5.56119948e-02j,\n",
       "       -0.09382468-5.41964348e-02j, -0.0924481 -5.28150130e-02j,\n",
       "       -0.09108519-5.14669928e-02j, -0.08973608-5.01516497e-02j,\n",
       "       -0.08840088-4.88682710e-02j, -0.08707968-4.76161556e-02j,\n",
       "       -0.08577258-4.63946145e-02j, -0.08447965-4.52029702e-02j,\n",
       "       -0.08320094-4.40405572e-02j, -0.08193652-4.29067216e-02j,\n",
       "       -0.08068641-4.18008211e-02j, -0.07945066-4.07222252e-02j,\n",
       "       -0.07822928-3.96703147e-02j, -0.07702228-3.86444820e-02j,\n",
       "       -0.07582966-3.76441309e-02j, -0.07465142-3.66686763e-02j,\n",
       "       -0.07348755-3.57175446e-02j, -0.07233802-3.47901729e-02j,\n",
       "       -0.0712028 -3.38860096e-02j, -0.07008186-3.30045136e-02j,\n",
       "       -0.06897516-3.21451550e-02j, -0.06788264-3.13074142e-02j,\n",
       "       -0.06680426-3.04907820e-02j, -0.06573994-2.96947599e-02j,\n",
       "       -0.06468963-2.89188594e-02j, -0.06365326-2.81626021e-02j,\n",
       "       -0.06263074-2.74255197e-02j, -0.061622  -2.67071536e-02j,\n",
       "       -0.06062696-2.60070548e-02j, -0.05964553-2.53247841e-02j,\n",
       "       -0.05867761-2.46599115e-02j, -0.05772311-2.40120163e-02j,\n",
       "       -0.05678193-2.33806868e-02j, -0.05585396-2.27655204e-02j,\n",
       "       -0.05493912-2.21661233e-02j, -0.05403727-2.15821104e-02j,\n",
       "       -0.05314833-2.10131049e-02j, -0.05227217-2.04587386e-02j,\n",
       "       -0.05140868-1.99186516e-02j, -0.05055775-1.93924917e-02j,\n",
       "       -0.04971925-1.88799151e-02j, -0.04889307-1.83805854e-02j,\n",
       "       -0.04807909-1.78941742e-02j, -0.04727718-1.74203603e-02j,\n",
       "       -0.04648722-1.69588301e-02j, -0.04570909-1.65092772e-02j,\n",
       "       -0.04494266-1.60714020e-02j, -0.0441878 -1.56449124e-02j,\n",
       "       -0.0434444 -1.52295225e-02j, -0.04271231-1.48249536e-02j,\n",
       "       -0.04199142-1.44309332e-02j, -0.04128159-1.40471954e-02j,\n",
       "       -0.0405827 -1.36734805e-02j, -0.03989461-1.33095350e-02j,\n",
       "       -0.03921721-1.29551114e-02j, -0.03855035-1.26099682e-02j,\n",
       "       -0.03789392-1.22738695e-02j, -0.03724778-1.19465852e-02j,\n",
       "       -0.0366118 -1.16278907e-02j, -0.03598587-1.13175669e-02j,\n",
       "       -0.03536984-1.10153997e-02j, -0.03476359-1.07211807e-02j,\n",
       "       -0.03416699-1.04347061e-02j, -0.03357993-1.01557773e-02j,\n",
       "       -0.03300226-9.88420045e-03j, -0.03243388-9.61978661e-03j,\n",
       "       -0.03187465-9.36235134e-03j, -0.03132444-9.11171477e-03j,\n",
       "       -0.03078315-8.86770149e-03j, -0.03025063-8.63014040e-03j,\n",
       "       -0.02972678-8.39886465e-03j, -0.02921148-8.17371154e-03j,\n",
       "       -0.02870459-7.95452242e-03j, -0.02820601-7.74114256e-03j,\n",
       "       -0.02771561-7.53342113e-03j, -0.02723329-7.33121106e-03j,\n",
       "       -0.02675891-7.13436897e-03j, -0.02629238-6.94275506e-03j,\n",
       "       -0.02583357-6.75623308e-03j, -0.02538237-6.57467018e-03j,\n",
       "       -0.02493867-6.39793689e-03j, -0.02450236-6.22590698e-03j,\n",
       "       -0.02407334-6.05845745e-03j, -0.02365148-5.89546839e-03j,\n",
       "       -0.02323669-5.73682296e-03j, -0.02282885-5.58240726e-03j,\n",
       "       -0.02242787-5.43211032e-03j, -0.02203364-5.28582399e-03j,\n",
       "       -0.02164605-5.14344287e-03j, -0.02126501-5.00486427e-03j,\n",
       "       -0.02089041-4.86998813e-03j, -0.02052216-4.73871693e-03j,\n",
       "       -0.02016016-4.61095569e-03j, -0.0198043 -4.48661185e-03j,\n",
       "       -0.0194545 -4.36559523e-03j, -0.01911066-4.24781799e-03j,\n",
       "       -0.01877268-4.13319453e-03j, -0.01844048-4.02164147e-03j,\n",
       "       -0.01811396-3.91307761e-03j, -0.01779304-3.80742381e-03j,\n",
       "       -0.01747761-3.70460302e-03j, -0.01716761-3.60454015e-03j,\n",
       "       -0.01686293-3.50716208e-03j, -0.01656349-3.41239760e-03j,\n",
       "       -0.01626922-3.32017733e-03j, -0.01598002-3.23043370e-03j,\n",
       "       -0.01569582-3.14310092e-03j, -0.01541653-3.05811489e-03j,\n",
       "       -0.01514208-2.97541320e-03j, -0.01487238-2.89493506e-03j,\n",
       "       -0.01460735-2.81662129e-03j, -0.01434693-2.74041422e-03j,\n",
       "       -0.01409104-2.66625773e-03j, -0.01383959-2.59409716e-03j,\n",
       "       -0.01359253-2.52387926e-03j, -0.01334977-2.45555221e-03j,\n",
       "       -0.01311124-2.38906554e-03j, -0.01287688-2.32437010e-03j,\n",
       "       -0.01264661-2.26141804e-03j, -0.01242037-2.20016276e-03j,\n",
       "       -0.0121981 -2.14055891e-03j, -0.01197971-2.08256231e-03j,\n",
       "       -0.01176515-2.02612996e-03j, -0.01155436-1.97122000e-03j,\n",
       "       -0.01134727-1.91779165e-03j, -0.01114382-1.86580525e-03j,\n",
       "       -0.01094395-1.81522214e-03j, -0.01074759-1.76600473e-03j,\n",
       "       -0.01055469-1.71811638e-03j, -0.0103652 -1.67152147e-03j,\n",
       "       -0.01017904-1.62618528e-03j, -0.00999617-1.58207403e-03j,\n",
       "       -0.00981653-1.53915485e-03j, -0.00964006-1.49739571e-03j,\n",
       "       -0.00946671-1.45676546e-03j, -0.00929643-1.41723377e-03j,\n",
       "       -0.00912916-1.37877110e-03j, -0.00896486-1.34134872e-03j,\n",
       "       -0.00880347-1.30493866e-03j, -0.00864494-1.26951368e-03j,\n",
       "       -0.00848923-1.23504728e-03j, -0.00833628-1.20151365e-03j,\n",
       "       -0.00818604-1.16888770e-03j, -0.00803848-1.13714497e-03j,\n",
       "       -0.00789354-1.10626169e-03j, -0.00775117-1.07621469e-03j,\n",
       "       -0.00761135-1.04698145e-03j, -0.00747401-1.01854002e-03j,\n",
       "       -0.00733912-9.90869070e-04j, -0.00720663-9.63947818e-04j,\n",
       "       -0.00707651-9.37756045e-04j, -0.00694871-9.12274074e-04j,\n",
       "       -0.00682319-8.87482752e-04j, -0.00669991-8.63363442e-04j,\n",
       "       -0.00657883-8.39898004e-04j, -0.00645992-8.17068784e-04j,\n",
       "       -0.00634314-7.94858604e-04j, -0.00622844-7.73250745e-04j,\n",
       "       -0.00611579-7.52228934e-04j, -0.00600517-7.31777340e-04j,\n",
       "       -0.00589652-7.11880552e-04j, -0.00578982-6.92523577e-04j,\n",
       "       -0.00568503-6.73691820e-04j, -0.00558212-6.55371083e-04j,\n",
       "       -0.00548106-6.37547547e-04j, -0.00538181-6.20207763e-04j,\n",
       "       -0.00528434-6.03338648e-04j, -0.00518862-5.86927467e-04j,\n",
       "       -0.00509462-5.70961830e-04j, -0.0050023 -5.55429678e-04j,\n",
       "       -0.00491165-5.40319279e-04j, -0.00482263-5.25619215e-04j,\n",
       "       -0.0047352 -5.11318378e-04j, -0.00464935-4.97405957e-04j,\n",
       "       -0.00456505-4.83871432e-04j, -0.00448226-4.70704569e-04j,\n",
       "       -0.00440096-4.57895408e-04j, -0.00432113-4.45434257e-04j,\n",
       "       -0.00424273-4.33311688e-04j, -0.00416574-4.21518524e-04j,\n",
       "       -0.00409015-4.10045837e-04j, -0.00401591-3.98884942e-04j,\n",
       "       -0.00394302-3.88027385e-04j, -0.00387144-3.77464943e-04j,\n",
       "       -0.00380115-3.67189613e-04j, -0.00373212-3.57193609e-04j,\n",
       "       -0.00366435-3.47469355e-04j, -0.0035978 -3.38009481e-04j,\n",
       "       -0.00353244-3.28806814e-04j, -0.00346827-3.19854377e-04j,\n",
       "       -0.00340526-3.11145380e-04j, -0.00334339-3.02673217e-04j,\n",
       "       -0.00328263-2.94431460e-04j, -0.00322298-2.86413858e-04j,\n",
       "       -0.0031644 -2.78614325e-04j, -0.00310688-2.71026941e-04j,\n",
       "       -0.0030504 -2.63645948e-04j, -0.00299494-2.56465741e-04j,\n",
       "       -0.00294049-2.49480869e-04j, -0.00288702-2.42686026e-04j,\n",
       "       -0.00283452-2.36076054e-04j, -0.00278297-2.29645929e-04j,\n",
       "       -0.00273235-2.23390768e-04j, -0.00268265-2.17305818e-04j,\n",
       "       -0.00263385-2.11386454e-04j, -0.00258593-2.05628178e-04j,\n",
       "       -0.00253888-2.00026613e-04j, -0.00249269-1.94577501e-04j,\n",
       "       -0.00244733-1.89276698e-04j, -0.00240279-1.84120175e-04j,\n",
       "       -0.00235906-1.79104011e-04j, -0.00231612-1.74224389e-04j,\n",
       "       -0.00227396-1.69477599e-04j, -0.00223257-1.64860030e-04j,\n",
       "       -0.00219193-1.60368169e-04j, -0.00215202-1.55998599e-04j,\n",
       "       -0.00211284-1.51747993e-04j, -0.00207437-1.47613118e-04j,\n",
       "       -0.0020366 -1.43590826e-04j, -0.00199951-1.39678057e-04j,\n",
       "       -0.00196309-1.35871831e-04j, -0.00192734-1.32169251e-04j,\n",
       "       -0.00189224-1.28567497e-04j, -0.00185777-1.25063828e-04j,\n",
       "       -0.00182393-1.21655576e-04j, -0.0017907 -1.18340144e-04j,\n",
       "       -0.00175808-1.15115007e-04j, -0.00172605-1.11977710e-04j,\n",
       "       -0.0016946 -1.08925862e-04j, -0.00166373-1.05957138e-04j,\n",
       "       -0.00163341-1.03069276e-04j, -0.00160365-1.00260077e-04j,\n",
       "       -0.00157442-9.75273989e-05j, -0.00154573-9.48691603e-05j,\n",
       "       -0.00151756-9.22833349e-05j, -0.0014899 -8.97679520e-05j,\n",
       "       -0.00146275-8.73210944e-05j, -0.00143609-8.49408969e-05j])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vRatio = Zc2**2*(1 + R3/R2)/((R2 + Zc2)**2 - Zc2*R3)\n",
    "vRatio"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b4aff38",
   "metadata": {},
   "source": [
    "- We can use abs(...) to find the magnitude of the complex numbers.  If $z = a+j\\,b$, abs(z) is equivalent to $\\sqrt{a^2 + b^2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "c8581f79",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.25531343, 1.25531333, 1.25531322, 1.25531311, 1.255313  ,\n",
       "       1.25531288, 1.25531277, 1.25531265, 1.25531253, 1.2553124 ,\n",
       "       1.25531228, 1.25531215, 1.25531202, 1.25531189, 1.25531175,\n",
       "       1.25531161, 1.25531147, 1.25531133, 1.25531119, 1.25531104,\n",
       "       1.25531089, 1.25531073, 1.25531058, 1.25531042, 1.25531025,\n",
       "       1.25531009, 1.25530992, 1.25530975, 1.25530957, 1.2553094 ,\n",
       "       1.25530921, 1.25530903, 1.25530884, 1.25530865, 1.25530845,\n",
       "       1.25530825, 1.25530805, 1.25530784, 1.25530763, 1.25530742,\n",
       "       1.2553072 , 1.25530698, 1.25530675, 1.25530652, 1.25530629,\n",
       "       1.25530605, 1.2553058 , 1.25530556, 1.2553053 , 1.25530505,\n",
       "       1.25530478, 1.25530452, 1.25530424, 1.25530397, 1.25530368,\n",
       "       1.2553034 , 1.2553031 , 1.2553028 , 1.2553025 , 1.25530219,\n",
       "       1.25530187, 1.25530155, 1.25530122, 1.25530089, 1.25530055,\n",
       "       1.25530021, 1.25529985, 1.25529949, 1.25529913, 1.25529876,\n",
       "       1.25529838, 1.25529799, 1.2552976 , 1.25529719, 1.25529679,\n",
       "       1.25529637, 1.25529595, 1.25529551, 1.25529507, 1.25529463,\n",
       "       1.25529417, 1.2552937 , 1.25529323, 1.25529275, 1.25529226,\n",
       "       1.25529176, 1.25529125, 1.25529073, 1.2552902 , 1.25528966,\n",
       "       1.25528911, 1.25528855, 1.25528798, 1.2552874 , 1.25528681,\n",
       "       1.25528621, 1.2552856 , 1.25528497, 1.25528434, 1.25528369,\n",
       "       1.25528303, 1.25528236, 1.25528167, 1.25528097, 1.25528026,\n",
       "       1.25527954, 1.2552788 , 1.25527805, 1.25527729, 1.25527651,\n",
       "       1.25527572, 1.25527491, 1.25527408, 1.25527324, 1.25527239,\n",
       "       1.25527152, 1.25527063, 1.25526973, 1.25526881, 1.25526787,\n",
       "       1.25526692, 1.25526595, 1.25526496, 1.25526395, 1.25526292,\n",
       "       1.25526188, 1.25526081, 1.25525972, 1.25525862, 1.25525749,\n",
       "       1.25525634, 1.25525518, 1.25525398, 1.25525277, 1.25525154,\n",
       "       1.25525028, 1.255249  , 1.25524769, 1.25524636, 1.25524501,\n",
       "       1.25524363, 1.25524222, 1.25524079, 1.25523933, 1.25523785,\n",
       "       1.25523633, 1.25523479, 1.25523322, 1.25523162, 1.25522999,\n",
       "       1.25522833, 1.25522664, 1.25522492, 1.25522317, 1.25522138,\n",
       "       1.25521956, 1.25521771, 1.25521582, 1.2552139 , 1.25521194,\n",
       "       1.25520995, 1.25520791, 1.25520584, 1.25520373, 1.25520159,\n",
       "       1.2551994 , 1.25519717, 1.2551949 , 1.25519259, 1.25519023,\n",
       "       1.25518783, 1.25518539, 1.2551829 , 1.25518036, 1.25517778,\n",
       "       1.25517515, 1.25517247, 1.25516974, 1.25516696, 1.25516413,\n",
       "       1.25516124, 1.2551583 , 1.25515531, 1.25515226, 1.25514915,\n",
       "       1.25514599, 1.25514277, 1.25513948, 1.25513614, 1.25513274,\n",
       "       1.25512927, 1.25512573, 1.25512213, 1.25511847, 1.25511473,\n",
       "       1.25511093, 1.25510705, 1.2551031 , 1.25509908, 1.25509499,\n",
       "       1.25509081, 1.25508656, 1.25508224, 1.25507783, 1.25507334,\n",
       "       1.25506876, 1.2550641 , 1.25505935, 1.25505452, 1.25504959,\n",
       "       1.25504458, 1.25503947, 1.25503426, 1.25502896, 1.25502356,\n",
       "       1.25501806, 1.25501245, 1.25500674, 1.25500093, 1.25499501,\n",
       "       1.25498897, 1.25498283, 1.25497657, 1.25497019, 1.2549637 ,\n",
       "       1.25495708, 1.25495034, 1.25494348, 1.25493649, 1.25492937,\n",
       "       1.25492211, 1.25491472, 1.25490719, 1.25489953, 1.25489172,\n",
       "       1.25488376, 1.25487566, 1.2548674 , 1.254859  , 1.25485043,\n",
       "       1.25484171, 1.25483282, 1.25482377, 1.25481455, 1.25480516,\n",
       "       1.25479559, 1.25478585, 1.25477592, 1.25476581, 1.25475551,\n",
       "       1.25474502, 1.25473433, 1.25472345, 1.25471236, 1.25470106,\n",
       "       1.25468956, 1.25467784, 1.2546659 , 1.25465374, 1.25464136,\n",
       "       1.25462874, 1.25461589, 1.2546028 , 1.25458947, 1.25457589,\n",
       "       1.25456205, 1.25454796, 1.2545336 , 1.25451898, 1.25450409,\n",
       "       1.25448891, 1.25447346, 1.25445772, 1.25444168, 1.25442535,\n",
       "       1.25440871, 1.25439176, 1.2543745 , 1.25435692, 1.254339  ,\n",
       "       1.25432076, 1.25430217, 1.25428324, 1.25426396, 1.25424432,\n",
       "       1.25422431, 1.25420393, 1.25418317, 1.25416202, 1.25414048,\n",
       "       1.25411853, 1.25409618, 1.25407342, 1.25405023, 1.2540266 ,\n",
       "       1.25400254, 1.25397803, 1.25395306, 1.25392763, 1.25390172,\n",
       "       1.25387533, 1.25384845, 1.25382107, 1.25379318, 1.25376477,\n",
       "       1.25373583, 1.25370636, 1.25367633, 1.25364574, 1.25361459,\n",
       "       1.25358285, 1.25355052, 1.25351759, 1.25348405, 1.25344988,\n",
       "       1.25341508, 1.25337962, 1.25334351, 1.25330672, 1.25326925,\n",
       "       1.25323108, 1.2531922 , 1.2531526 , 1.25311225, 1.25307116,\n",
       "       1.2530293 , 1.25298666, 1.25294323, 1.25289898, 1.25285391,\n",
       "       1.252808  , 1.25276124, 1.25271361, 1.25266508, 1.25261566,\n",
       "       1.25256531, 1.25251402, 1.25246178, 1.25240856, 1.25235436,\n",
       "       1.25229914, 1.25224289, 1.2521856 , 1.25212723, 1.25206778,\n",
       "       1.25200722, 1.25194553, 1.2518827 , 1.25181869, 1.25175348,\n",
       "       1.25168707, 1.25161941, 1.25155049, 1.25148029, 1.25140877,\n",
       "       1.25133593, 1.25126172, 1.25118614, 1.25110914, 1.25103071,\n",
       "       1.25095081, 1.25086942, 1.25078652, 1.25070207, 1.25061605,\n",
       "       1.25052842, 1.25043915, 1.25034822, 1.2502556 , 1.25016124,\n",
       "       1.25006513, 1.24996722, 1.24986749, 1.24976589, 1.2496624 ,\n",
       "       1.24955698, 1.24944959, 1.2493402 , 1.24922877, 1.24911525,\n",
       "       1.24899962, 1.24888183, 1.24876184, 1.24863961, 1.2485151 ,\n",
       "       1.24838826, 1.24825906, 1.24812744, 1.24799337, 1.24785679,\n",
       "       1.24771767, 1.24757594, 1.24743157, 1.2472845 , 1.24713469,\n",
       "       1.24698207, 1.24682661, 1.24666824, 1.24650692, 1.24634258,\n",
       "       1.24617517, 1.24600463, 1.24583091, 1.24565394, 1.24547366,\n",
       "       1.24529001, 1.24510293, 1.24491236, 1.24471822, 1.24452045,\n",
       "       1.24431899, 1.24411376, 1.24390469, 1.24369172, 1.24347476,\n",
       "       1.24325375, 1.2430286 , 1.24279924, 1.24256559, 1.24232757,\n",
       "       1.24208511, 1.2418381 , 1.24158647, 1.24133014, 1.24106901,\n",
       "       1.240803  , 1.240532  , 1.24025594, 1.23997471, 1.23968822,\n",
       "       1.23939637, 1.23909905, 1.23879617, 1.23848762, 1.23817329,\n",
       "       1.23785308, 1.23752687, 1.23719456, 1.23685602, 1.23651115,\n",
       "       1.23615982, 1.23580191, 1.23543729, 1.23506585, 1.23468745,\n",
       "       1.23430197, 1.23390926, 1.2335092 , 1.23310165, 1.23268646,\n",
       "       1.2322635 , 1.23183261, 1.23139365, 1.23094647, 1.23049091,\n",
       "       1.23002682, 1.22955403, 1.22907238, 1.22858172, 1.22808186,\n",
       "       1.22757264, 1.22705387, 1.2265254 , 1.22598702, 1.22543856,\n",
       "       1.22487982, 1.22431062, 1.22373076, 1.22314004, 1.22253826,\n",
       "       1.22192521, 1.22130068, 1.22066446, 1.22001633, 1.21935607,\n",
       "       1.21868345, 1.21799825, 1.21730022, 1.21658913, 1.21586475,\n",
       "       1.21512681, 1.21437508, 1.2136093 , 1.2128292 , 1.21203452,\n",
       "       1.211225  , 1.21040035, 1.20956031, 1.20870458, 1.20783288,\n",
       "       1.20694493, 1.2060404 , 1.20511902, 1.20418046, 1.20322442,\n",
       "       1.20225057, 1.20125859, 1.20024816, 1.19921893, 1.19817057,\n",
       "       1.19710273, 1.19601507, 1.19490721, 1.19377881, 1.1926295 ,\n",
       "       1.19145889, 1.19026662, 1.18905229, 1.18781551, 1.18655589,\n",
       "       1.18527303, 1.18396651, 1.18263592, 1.18128085, 1.17990085,\n",
       "       1.17849551, 1.17706438, 1.17560702, 1.17412298, 1.17261179,\n",
       "       1.17107301, 1.16950616, 1.16791077, 1.16628636, 1.16463245,\n",
       "       1.16294853, 1.16123413, 1.15948873, 1.15771183, 1.15590291,\n",
       "       1.15406147, 1.15218697, 1.1502789 , 1.14833671, 1.14635988,\n",
       "       1.14434786, 1.1423001 , 1.14021607, 1.1380952 , 1.13593693,\n",
       "       1.13374073, 1.131506  , 1.12923221, 1.12691877, 1.12456512,\n",
       "       1.12217068, 1.1197349 , 1.11725718, 1.11473696, 1.11217366,\n",
       "       1.10956672, 1.10691556, 1.1042196 , 1.10147828, 1.09869103,\n",
       "       1.09585728, 1.09297648, 1.09004807, 1.08707149, 1.0840462 ,\n",
       "       1.08097166, 1.07784734, 1.0746727 , 1.07144722, 1.06817041,\n",
       "       1.06484175, 1.06146076, 1.05802696, 1.05453989, 1.05099908,\n",
       "       1.04740411, 1.04375454, 1.04004997, 1.03629001, 1.03247428,\n",
       "       1.02860243, 1.02467412, 1.02068904, 1.01664688, 1.01254739,\n",
       "       1.00839032, 1.00417544, 0.99990255, 0.99557149, 0.99118211,\n",
       "       0.98673431, 0.98222799, 0.97766311, 0.97303964, 0.9683576 ,\n",
       "       0.96361703, 0.95881802, 0.95396068, 0.94904516, 0.94407165,\n",
       "       0.93904039, 0.93395163, 0.92880569, 0.92360292, 0.91834369,\n",
       "       0.91302845, 0.90765766, 0.90223184, 0.89675154, 0.89121736,\n",
       "       0.88562996, 0.87999001, 0.87429826, 0.86855546, 0.86276245,\n",
       "       0.85692009, 0.85102928, 0.84509098, 0.83910618, 0.83307591,\n",
       "       0.82700126, 0.82088334, 0.81472331, 0.80852237, 0.80228177,\n",
       "       0.79600278, 0.78968672, 0.78333493, 0.77694881, 0.77052977,\n",
       "       0.76407926, 0.75759877, 0.75108981, 0.74455392, 0.73799267,\n",
       "       0.73140764, 0.72480045, 0.71817274, 0.71152615, 0.70486236,\n",
       "       0.69818305, 0.69148992, 0.68478469, 0.67806908, 0.6713448 ,\n",
       "       0.66461361, 0.65787722, 0.65113739, 0.64439585, 0.63765433,\n",
       "       0.63091457, 0.62417829, 0.61744721, 0.61072302, 0.60400743,\n",
       "       0.59730211, 0.59060873, 0.58392893, 0.57726434, 0.57061657,\n",
       "       0.56398719, 0.55737777, 0.55078983, 0.54422489, 0.53768443,\n",
       "       0.53116989, 0.52468269, 0.51822422, 0.51179583, 0.50539885,\n",
       "       0.49903456, 0.49270421, 0.48640902, 0.48015015, 0.47392877,\n",
       "       0.46774596, 0.4616028 , 0.45550031, 0.44943947, 0.44342125,\n",
       "       0.43744654, 0.43151622, 0.42563113, 0.41979204, 0.41399973,\n",
       "       0.40825489, 0.40255822, 0.39691034, 0.39131185, 0.38576333,\n",
       "       0.38026529, 0.37481822, 0.36942257, 0.36407877, 0.35878719,\n",
       "       0.35354817, 0.34836204, 0.34322907, 0.33814951, 0.33312357,\n",
       "       0.32815143, 0.32323325, 0.31836916, 0.31355925, 0.30880358,\n",
       "       0.30410219, 0.2994551 , 0.29486229, 0.29032373, 0.28583935,\n",
       "       0.28140907, 0.27703278, 0.27271035, 0.26844162, 0.26422643,\n",
       "       0.26006458, 0.25595588, 0.25190008, 0.24789694, 0.24394621,\n",
       "       0.24004761, 0.23620084, 0.2324056 , 0.22866158, 0.22496843,\n",
       "       0.22132582, 0.21773339, 0.21419077, 0.21069759, 0.20725345,\n",
       "       0.20385797, 0.20051074, 0.19721135, 0.19395938, 0.1907544 ,\n",
       "       0.18759598, 0.18448368, 0.18141706, 0.17839567, 0.17541906,\n",
       "       0.17248677, 0.16959834, 0.1667533 , 0.1639512 , 0.16119156,\n",
       "       0.15847391, 0.15579778, 0.1531627 , 0.15056819, 0.14801377,\n",
       "       0.14549898, 0.14302334, 0.14058637, 0.1381876 , 0.13582654,\n",
       "       0.13350274, 0.13121572, 0.12896501, 0.12675013, 0.12457063,\n",
       "       0.12242604, 0.1203159 , 0.11823973, 0.1161971 , 0.11418754,\n",
       "       0.1122106 , 0.11026582, 0.10835278, 0.10647101, 0.10462009,\n",
       "       0.10279957, 0.10100902, 0.09924802, 0.09751613, 0.09581294,\n",
       "       0.09413802, 0.09249097, 0.09087137, 0.08927882, 0.0877129 ,\n",
       "       0.08617324, 0.08465942, 0.08317107, 0.08170779, 0.0802692 ,\n",
       "       0.07885493, 0.07746461, 0.07609785, 0.07475431, 0.07343362,\n",
       "       0.07213542, 0.07085936, 0.0696051 , 0.06837228, 0.06716058,\n",
       "       0.06596966, 0.06479918, 0.06364882, 0.06251827, 0.06140719,\n",
       "       0.06031529, 0.05924224, 0.05818775, 0.05715151, 0.05613323,\n",
       "       0.05513261, 0.05414937, 0.05318322, 0.05223388, 0.05130107,\n",
       "       0.05038453, 0.04948397, 0.04859915, 0.04772979, 0.04687564,\n",
       "       0.04603644, 0.04521195, 0.04440192, 0.04360612, 0.04282429,\n",
       "       0.0420562 , 0.04130163, 0.04056034, 0.03983211, 0.03911672,\n",
       "       0.03841396, 0.0377236 , 0.03704544, 0.03637926, 0.03572487,\n",
       "       0.03508206, 0.03445064, 0.03383041, 0.03322118, 0.03262275,\n",
       "       0.03203495, 0.03145759, 0.03089049, 0.03033348, 0.02978637,\n",
       "       0.029249  , 0.02872121, 0.02820281, 0.02769366, 0.02719358,\n",
       "       0.02670243, 0.02622005, 0.02574628, 0.02528098, 0.02482399,\n",
       "       0.02437517, 0.02393438, 0.02350148, 0.02307633, 0.0226588 ,\n",
       "       0.02224874, 0.02184604, 0.02145055, 0.02106216, 0.02068074,\n",
       "       0.02030616, 0.01993831, 0.01957706, 0.0192223 , 0.01887392,\n",
       "       0.01853181, 0.01819584, 0.01786592, 0.01754193, 0.01722378,\n",
       "       0.01691135, 0.01660455, 0.01630328, 0.01600743, 0.01571692,\n",
       "       0.01543164, 0.01515151, 0.01487643, 0.01460631, 0.01434107,\n",
       "       0.01408061, 0.01382486, 0.01357373, 0.01332713, 0.01308498,\n",
       "       0.01284721, 0.01261374, 0.01238449, 0.01215938, 0.01193834,\n",
       "       0.01172131, 0.01150819, 0.01129894, 0.01109347, 0.01089172,\n",
       "       0.01069362, 0.01049911, 0.01030812, 0.01012059, 0.00993646,\n",
       "       0.00975566, 0.00957814, 0.00940384, 0.0092327 , 0.00906465,\n",
       "       0.00889966, 0.00873766, 0.0085786 , 0.00842242, 0.00826907,\n",
       "       0.00811851, 0.00797068, 0.00782553, 0.00768302, 0.00754309,\n",
       "       0.00740571, 0.00727082, 0.00713837, 0.00700834, 0.00688066,\n",
       "       0.00675531, 0.00663223, 0.00651139, 0.00639274, 0.00627625,\n",
       "       0.00616188, 0.00604959, 0.00593934, 0.00583109, 0.00572481,\n",
       "       0.00562046, 0.00551801, 0.00541743, 0.00531867, 0.00522171,\n",
       "       0.00512651, 0.00503305, 0.00494128, 0.00485119, 0.00476273,\n",
       "       0.00467589, 0.00459062, 0.00450691, 0.00442472, 0.00434402,\n",
       "       0.0042648 , 0.00418702, 0.00411065, 0.00403567, 0.00396206,\n",
       "       0.00388979, 0.00381884, 0.00374918, 0.00368079, 0.00361364,\n",
       "       0.00354771, 0.00348299, 0.00341945, 0.00335706, 0.00329581,\n",
       "       0.00323568, 0.00317664, 0.00311868, 0.00306177, 0.0030059 ,\n",
       "       0.00295105, 0.0028972 , 0.00284433, 0.00279243, 0.00274147,\n",
       "       0.00269144, 0.00264232, 0.00259409, 0.00254675, 0.00250027,\n",
       "       0.00245463, 0.00240983, 0.00236585, 0.00232267, 0.00228027,\n",
       "       0.00223865, 0.00219779, 0.00215767, 0.00211828, 0.00207961,\n",
       "       0.00204165, 0.00200438, 0.00196779, 0.00193187, 0.0018966 ,\n",
       "       0.00186198, 0.00182798, 0.00179461, 0.00176185, 0.00172968,\n",
       "       0.0016981 , 0.0016671 , 0.00163666, 0.00160678, 0.00157744,\n",
       "       0.00154864, 0.00152036, 0.0014926 , 0.00146535, 0.0014386 ])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "abs(vRatio)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a96f30cf",
   "metadata": {},
   "source": [
    "- We can use np.angle(...) to find the phase of the complex numbers.  If $z = a+j\\,b$, np.angle(z) is equivalent to $\\arctan(b/a)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "a4cd3a55",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.00515221, -0.00519993, -0.00524809, -0.0052967 , -0.00534576,\n",
       "       -0.00539528, -0.00544525, -0.00549568, -0.00554659, -0.00559796,\n",
       "       -0.00564981, -0.00570214, -0.00575495, -0.00580826, -0.00586205,\n",
       "       -0.00591635, -0.00597115, -0.00602645, -0.00608227, -0.00613861,\n",
       "       -0.00619546, -0.00625285, -0.00631076, -0.00636921, -0.00642821,\n",
       "       -0.00648774, -0.00654784, -0.00660848, -0.00666969, -0.00673147,\n",
       "       -0.00679382, -0.00685674, -0.00692025, -0.00698435, -0.00704904,\n",
       "       -0.00711433, -0.00718022, -0.00724672, -0.00731385, -0.00738159,\n",
       "       -0.00744996, -0.00751896, -0.0075886 , -0.00765889, -0.00772983,\n",
       "       -0.00780142, -0.00787368, -0.00794661, -0.00802021, -0.0080945 ,\n",
       "       -0.00816947, -0.00824514, -0.0083215 , -0.00839858, -0.00847637,\n",
       "       -0.00855488, -0.00863411, -0.00871408, -0.0087948 , -0.00887626,\n",
       "       -0.00895847, -0.00904144, -0.00912519, -0.00920971, -0.00929501,\n",
       "       -0.0093811 , -0.00946799, -0.00955568, -0.00964419, -0.00973352,\n",
       "       -0.00982367, -0.00991466, -0.01000649, -0.01009917, -0.01019271,\n",
       "       -0.01028712, -0.0103824 , -0.01047856, -0.01057562, -0.01067357,\n",
       "       -0.01077243, -0.01087221, -0.01097291, -0.01107454, -0.01117712,\n",
       "       -0.01128064, -0.01138512, -0.01149058, -0.011597  , -0.01170442,\n",
       "       -0.01181283, -0.01192224, -0.01203266, -0.01214411, -0.01225659,\n",
       "       -0.01237012, -0.01248469, -0.01260033, -0.01271703, -0.01283482,\n",
       "       -0.0129537 , -0.01307368, -0.01319477, -0.01331698, -0.01344032,\n",
       "       -0.01356481, -0.01369045, -0.01381725, -0.01394523, -0.0140744 ,\n",
       "       -0.01420475, -0.01433632, -0.01446911, -0.01460312, -0.01473838,\n",
       "       -0.01487489, -0.01501266, -0.01515171, -0.01529205, -0.01543369,\n",
       "       -0.01557664, -0.01572091, -0.01586652, -0.01601348, -0.0161618 ,\n",
       "       -0.01631149, -0.01646257, -0.01661505, -0.01676894, -0.01692426,\n",
       "       -0.01708101, -0.01723922, -0.01739889, -0.01756004, -0.01772269,\n",
       "       -0.01788684, -0.01805251, -0.01821971, -0.01838847, -0.01855879,\n",
       "       -0.01873068, -0.01890417, -0.01907926, -0.01925598, -0.01943433,\n",
       "       -0.01961433, -0.019796  , -0.01997936, -0.02016441, -0.02035117,\n",
       "       -0.02053967, -0.02072991, -0.02092191, -0.0211157 , -0.02131127,\n",
       "       -0.02150866, -0.02170788, -0.02190894, -0.02211186, -0.02231667,\n",
       "       -0.02252337, -0.02273198, -0.02294253, -0.02315502, -0.02336949,\n",
       "       -0.02358594, -0.0238044 , -0.02402488, -0.0242474 , -0.02447198,\n",
       "       -0.02469864, -0.02492741, -0.02515829, -0.02539131, -0.02562649,\n",
       "       -0.02586384, -0.0261034 , -0.02634517, -0.02658918, -0.02683545,\n",
       "       -0.02708401, -0.02733486, -0.02758804, -0.02784356, -0.02810145,\n",
       "       -0.02836173, -0.02862442, -0.02888955, -0.02915712, -0.02942718,\n",
       "       -0.02969974, -0.02997482, -0.03025245, -0.03053265, -0.03081545,\n",
       "       -0.03110087, -0.03138892, -0.03167965, -0.03197307, -0.03226921,\n",
       "       -0.03256809, -0.03286974, -0.03317418, -0.03348144, -0.03379155,\n",
       "       -0.03410453, -0.03442041, -0.03473922, -0.03506097, -0.03538571,\n",
       "       -0.03571346, -0.03604424, -0.03637808, -0.03671502, -0.03705507,\n",
       "       -0.03739828, -0.03774467, -0.03809426, -0.03844709, -0.03880319,\n",
       "       -0.03916259, -0.03952531, -0.0398914 , -0.04026087, -0.04063377,\n",
       "       -0.04101012, -0.04138996, -0.04177332, -0.04216022, -0.04255071,\n",
       "       -0.04294482, -0.04334257, -0.04374401, -0.04414917, -0.04455808,\n",
       "       -0.04497078, -0.0453873 , -0.04580767, -0.04623195, -0.04666015,\n",
       "       -0.04709231, -0.04752848, -0.04796869, -0.04841297, -0.04886137,\n",
       "       -0.04931392, -0.04977067, -0.05023164, -0.05069689, -0.05116644,\n",
       "       -0.05164034, -0.05211863, -0.05260135, -0.05308854, -0.05358024,\n",
       "       -0.0540765 , -0.05457735, -0.05508284, -0.05559301, -0.05610791,\n",
       "       -0.05662758, -0.05715205, -0.05768139, -0.05821563, -0.05875481,\n",
       "       -0.05929899, -0.05984821, -0.06040252, -0.06096196, -0.06152658,\n",
       "       -0.06209643, -0.06267155, -0.06325201, -0.06383784, -0.06442909,\n",
       "       -0.06502582, -0.06562807, -0.06623591, -0.06684937, -0.06746851,\n",
       "       -0.06809339, -0.06872406, -0.06936056, -0.07000296, -0.07065131,\n",
       "       -0.07130566, -0.07196607, -0.0726326 , -0.0733053 , -0.07398423,\n",
       "       -0.07466945, -0.07536101, -0.07605898, -0.07676341, -0.07747437,\n",
       "       -0.07819191, -0.07891609, -0.07964698, -0.08038464, -0.08112912,\n",
       "       -0.08188051, -0.08263885, -0.08340421, -0.08417666, -0.08495626,\n",
       "       -0.08574308, -0.08653719, -0.08733865, -0.08814754, -0.08896391,\n",
       "       -0.08978784, -0.09061941, -0.09145867, -0.0923057 , -0.09316058,\n",
       "       -0.09402337, -0.09489415, -0.095773  , -0.09665998, -0.09755517,\n",
       "       -0.09845866, -0.09937051, -0.1002908 , -0.10121961, -0.10215703,\n",
       "       -0.10310312, -0.10405797, -0.10502167, -0.10599428, -0.1069759 ,\n",
       "       -0.10796661, -0.1089665 , -0.10997564, -0.11099412, -0.11202203,\n",
       "       -0.11305946, -0.11410649, -0.11516322, -0.11622973, -0.11730611,\n",
       "       -0.11839246, -0.11948887, -0.12059542, -0.12171222, -0.12283936,\n",
       "       -0.12397693, -0.12512504, -0.12628377, -0.12745323, -0.12863351,\n",
       "       -0.12982472, -0.13102695, -0.13224032, -0.13346491, -0.13470084,\n",
       "       -0.13594821, -0.13720713, -0.13847769, -0.13976002, -0.14105422,\n",
       "       -0.14236039, -0.14367865, -0.14500912, -0.14635189, -0.1477071 ,\n",
       "       -0.14907484, -0.15045525, -0.15184843, -0.1532545 , -0.15467358,\n",
       "       -0.15610579, -0.15755126, -0.1590101 , -0.16048244, -0.1619684 ,\n",
       "       -0.16346811, -0.1649817 , -0.16650929, -0.16805102, -0.16960701,\n",
       "       -0.17117739, -0.1727623 , -0.17436187, -0.17597624, -0.17760554,\n",
       "       -0.17924992, -0.1809095 , -0.18258444, -0.18427486, -0.18598092,\n",
       "       -0.18770276, -0.18944052, -0.19119435, -0.1929644 , -0.19475082,\n",
       "       -0.19655376, -0.19837336, -0.20020979, -0.2020632 , -0.20393374,\n",
       "       -0.20582158, -0.20772687, -0.20964976, -0.21159044, -0.21354904,\n",
       "       -0.21552576, -0.21752073, -0.21953415, -0.22156617, -0.22361696,\n",
       "       -0.22568671, -0.22777557, -0.22988373, -0.23201137, -0.23415866,\n",
       "       -0.23632578, -0.23851292, -0.24072026, -0.24294797, -0.24519626,\n",
       "       -0.2474653 , -0.2497553 , -0.25206643, -0.25439889, -0.25675288,\n",
       "       -0.25912859, -0.26152623, -0.26394598, -0.26638806, -0.26885267,\n",
       "       -0.27134001, -0.27385028, -0.27638371, -0.27894049, -0.28152085,\n",
       "       -0.28412499, -0.28675313, -0.28940549, -0.29208229, -0.29478374,\n",
       "       -0.29751008, -0.30026153, -0.30303832, -0.30584067, -0.30866881,\n",
       "       -0.31152299, -0.31440343, -0.31731036, -0.32024404, -0.32320469,\n",
       "       -0.32619257, -0.32920791, -0.33225096, -0.33532197, -0.33842119,\n",
       "       -0.34154888, -0.34470528, -0.34789066, -0.35110526, -0.35434936,\n",
       "       -0.35762321, -0.36092708, -0.36426123, -0.36762593, -0.37102146,\n",
       "       -0.37444808, -0.37790606, -0.38139569, -0.38491724, -0.38847099,\n",
       "       -0.39205722, -0.39567622, -0.39932827, -0.40301366, -0.40673268,\n",
       "       -0.41048562, -0.41427278, -0.41809444, -0.42195091, -0.42584248,\n",
       "       -0.42976945, -0.43373214, -0.43773083, -0.44176584, -0.44583748,\n",
       "       -0.44994605, -0.45409186, -0.45827523, -0.46249647, -0.4667559 ,\n",
       "       -0.47105383, -0.47539058, -0.47976647, -0.48418183, -0.48863697,\n",
       "       -0.49313222, -0.4976679 , -0.50224434, -0.50686187, -0.51152081,\n",
       "       -0.5162215 , -0.52096426, -0.52574943, -0.53057733, -0.53544829,\n",
       "       -0.54036266, -0.54532076, -0.55032292, -0.55536949, -0.56046078,\n",
       "       -0.56559714, -0.5707789 , -0.57600639, -0.58127994, -0.58659989,\n",
       "       -0.59196656, -0.59738029, -0.60284141, -0.60835025, -0.61390713,\n",
       "       -0.61951237, -0.62516631, -0.63086927, -0.63662156, -0.6424235 ,\n",
       "       -0.64827541, -0.65417761, -0.66013039, -0.66613408, -0.67218896,\n",
       "       -0.67829536, -0.68445355, -0.69066383, -0.69692649, -0.70324182,\n",
       "       -0.70961009, -0.71603158, -0.72250655, -0.72903526, -0.73561798,\n",
       "       -0.74225495, -0.74894642, -0.75569261, -0.76249377, -0.7693501 ,\n",
       "       -0.77626183, -0.78322915, -0.79025226, -0.79733135, -0.80446659,\n",
       "       -0.81165814, -0.81890617, -0.82621081, -0.8335722 , -0.84099046,\n",
       "       -0.84846569, -0.85599799, -0.86358744, -0.87123412, -0.87893807,\n",
       "       -0.88669933, -0.89451793, -0.90239387, -0.91032715, -0.91831775,\n",
       "       -0.92636561, -0.93447068, -0.94263289, -0.95085213, -0.95912829,\n",
       "       -0.96746123, -0.97585081, -0.98429683, -0.99279911, -1.00135742,\n",
       "       -1.00997153, -1.01864117, -1.02736605, -1.03614586, -1.04498027,\n",
       "       -1.05386892, -1.06281142, -1.07180737, -1.08085633, -1.08995785,\n",
       "       -1.09911143, -1.10831657, -1.11757272, -1.12687932, -1.13623578,\n",
       "       -1.14564147, -1.15509575, -1.16459793, -1.17414733, -1.18374319,\n",
       "       -1.19338477, -1.20307128, -1.21280189, -1.22257578, -1.23239205,\n",
       "       -1.24224982, -1.25214816, -1.26208611, -1.27206269, -1.2820769 ,\n",
       "       -1.29212769, -1.30221402, -1.3123348 , -1.3224889 , -1.33267521,\n",
       "       -1.34289256, -1.35313976, -1.36341563, -1.37371891, -1.38404838,\n",
       "       -1.39440276, -1.40478076, -1.41518108, -1.42560239, -1.43604335,\n",
       "       -1.44650261, -1.45697879, -1.46747051, -1.47797637, -1.48849496,\n",
       "       -1.49902485, -1.50956463, -1.52011284, -1.53066805, -1.54122881,\n",
       "       -1.55179365, -1.56236112, -1.57292976, -1.5834981 , -1.59406469,\n",
       "       -1.60462805, -1.61518674, -1.6257393 , -1.63628428, -1.64682024,\n",
       "       -1.65734575, -1.66785937, -1.67835968, -1.6888453 , -1.69931481,\n",
       "       -1.70976685, -1.72020004, -1.73061304, -1.74100451, -1.75137313,\n",
       "       -1.7617176 , -1.77203665, -1.78232901, -1.79259345, -1.80282875,\n",
       "       -1.8130337 , -1.82320715, -1.83334793, -1.84345493, -1.85352705,\n",
       "       -1.8635632 , -1.87356234, -1.88352344, -1.89344552, -1.90332759,\n",
       "       -1.91316872, -1.92296798, -1.9327245 , -1.94243741, -1.95210587,\n",
       "       -1.96172909, -1.97130628, -1.98083671, -1.99031963, -1.99975438,\n",
       "       -2.00914027, -2.01847667, -2.02776297, -2.03699859, -2.04618298,\n",
       "       -2.0553156 , -2.06439596, -2.07342357, -2.082398  , -2.0913188 ,\n",
       "       -2.1001856 , -2.10899801, -2.11775568, -2.12645829, -2.13510555,\n",
       "       -2.14369716, -2.15223288, -2.16071247, -2.16913573, -2.17750246,\n",
       "       -2.18581249, -2.19406569, -2.20226192, -2.21040107, -2.21848305,\n",
       "       -2.2265078 , -2.23447526, -2.24238539, -2.25023819, -2.25803363,\n",
       "       -2.26577175, -2.27345257, -2.28107614, -2.28864251, -2.29615176,\n",
       "       -2.30360398, -2.31099926, -2.31833773, -2.32561949, -2.3328447 ,\n",
       "       -2.3400135 , -2.34712605, -2.35418252, -2.36118308, -2.36812794,\n",
       "       -2.37501727, -2.38185131, -2.38863025, -2.39535432, -2.40202375,\n",
       "       -2.40863879, -2.41519968, -2.42170666, -2.42816001, -2.43455998,\n",
       "       -2.44090684, -2.44720087, -2.45344236, -2.45963157, -2.46576882,\n",
       "       -2.47185438, -2.47788856, -2.48387166, -2.48980398, -2.49568583,\n",
       "       -2.50151754, -2.50729939, -2.51303173, -2.51871485, -2.5243491 ,\n",
       "       -2.52993478, -2.53547222, -2.54096175, -2.5464037 , -2.5517984 ,\n",
       "       -2.55714617, -2.56244734, -2.56770226, -2.57291124, -2.57807463,\n",
       "       -2.58319276, -2.58826595, -2.59329455, -2.59827888, -2.60321929,\n",
       "       -2.60811609, -2.61296963, -2.61778024, -2.62254824, -2.62727398,\n",
       "       -2.63195777, -2.63659996, -2.64120086, -2.64576081, -2.65028013,\n",
       "       -2.65475915, -2.65919818, -2.66359757, -2.66795762, -2.67227866,\n",
       "       -2.67656101, -2.68080498, -2.68501089, -2.68917906, -2.6933098 ,\n",
       "       -2.69740342, -2.70146024, -2.70548055, -2.70946467, -2.7134129 ,\n",
       "       -2.71732554, -2.72120291, -2.72504529, -2.72885298, -2.73262628,\n",
       "       -2.73636549, -2.7400709 , -2.7437428 , -2.74738147, -2.75098721,\n",
       "       -2.75456029, -2.75810101, -2.76160964, -2.76508646, -2.76853175,\n",
       "       -2.77194578, -2.77532883, -2.77868116, -2.78200306, -2.78529477,\n",
       "       -2.78855658, -2.79178874, -2.79499151, -2.79816516, -2.80130994,\n",
       "       -2.8044261 , -2.80751391, -2.8105736 , -2.81360544, -2.81660966,\n",
       "       -2.81958651, -2.82253625, -2.8254591 , -2.82835531, -2.83122511,\n",
       "       -2.83406875, -2.83688645, -2.83967845, -2.84244498, -2.84518627,\n",
       "       -2.84790254, -2.85059401, -2.85326092, -2.85590347, -2.8585219 ,\n",
       "       -2.86111641, -2.86368722, -2.86623455, -2.8687586 , -2.8712596 ,\n",
       "       -2.87373773, -2.87619322, -2.87862626, -2.88103706, -2.88342582,\n",
       "       -2.88579274, -2.88813802, -2.89046184, -2.89276442, -2.89504593,\n",
       "       -2.89730657, -2.89954653, -2.901766  , -2.90396515, -2.90614419,\n",
       "       -2.90830328, -2.91044262, -2.91256237, -2.91466272, -2.91674384,\n",
       "       -2.91880592, -2.92084911, -2.9228736 , -2.92487955, -2.92686713,\n",
       "       -2.92883652, -2.93078786, -2.93272134, -2.93463711, -2.93653533,\n",
       "       -2.93841617, -2.94027977, -2.94212631, -2.94395593, -2.94576878,\n",
       "       -2.94756503, -2.94934483, -2.95110831, -2.95285564, -2.95458695,\n",
       "       -2.9563024 , -2.95800213, -2.95968629, -2.96135501, -2.96300843,\n",
       "       -2.9646467 , -2.96626996, -2.96787834, -2.96947198, -2.97105101,\n",
       "       -2.97261556, -2.97416578, -2.97570178, -2.9772237 , -2.97873167,\n",
       "       -2.98022582, -2.98170627, -2.98317314, -2.98462657, -2.98606667,\n",
       "       -2.98749357, -2.98890738, -2.99030823, -2.99169624, -2.99307152,\n",
       "       -2.99443419, -2.99578436, -2.99712215, -2.99844768, -2.99976105,\n",
       "       -3.00106237, -3.00235176, -3.00362933, -3.00489518, -3.00614942,\n",
       "       -3.00739216, -3.0086235 , -3.00984355, -3.01105241, -3.01225018,\n",
       "       -3.01343696, -3.01461286, -3.01577798, -3.01693241, -3.01807625,\n",
       "       -3.0192096 , -3.02033255, -3.0214452 , -3.02254765, -3.02363998,\n",
       "       -3.0247223 , -3.02579468, -3.02685723, -3.02791003, -3.02895318,\n",
       "       -3.02998676, -3.03101085, -3.03202555, -3.03303094, -3.03402711,\n",
       "       -3.03501414, -3.03599212, -3.03696112, -3.03792124, -3.03887254,\n",
       "       -3.03981512, -3.04074905, -3.04167442, -3.04259129, -3.04349976,\n",
       "       -3.04439988, -3.04529175, -3.04617544, -3.04705102, -3.04791857,\n",
       "       -3.04877816, -3.04962986, -3.05047375, -3.05130989, -3.05213837,\n",
       "       -3.05295924, -3.05377258, -3.05457846, -3.05537694, -3.0561681 ,\n",
       "       -3.056952  , -3.05772871, -3.05849829, -3.05926081, -3.06001633,\n",
       "       -3.06076492, -3.06150665, -3.06224156, -3.06296974, -3.06369123,\n",
       "       -3.0644061 , -3.06511442, -3.06581623, -3.06651161, -3.0672006 ,\n",
       "       -3.06788327, -3.06855968, -3.06922988, -3.06989394, -3.0705519 ,\n",
       "       -3.07120382, -3.07184976, -3.07248977, -3.07312391, -3.07375223,\n",
       "       -3.07437479, -3.07499163, -3.07560282, -3.07620839, -3.07680841,\n",
       "       -3.07740292, -3.07799198, -3.07857563, -3.07915393, -3.07972692,\n",
       "       -3.08029465, -3.08085718, -3.08141454, -3.08196678, -3.08251396])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.angle(vRatio)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cbe104d1",
   "metadata": {},
   "source": [
    "- Here's a plot of the magnitude of the voltge ratio."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "fb63de6e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.loglog(ff, abs(vRatio), 'k-', linewidth = 2.5)\n",
    "plt.xlabel('Frequency (Hz)')\n",
    "plt.ylabel('Voltage Ratio')\n",
    "plt.axis((10, 1e5, 1e-3, 2));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3099ba3d",
   "metadata": {},
   "source": [
    "- Here's a plot of the phase in radians scaled by $\\pi$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "cdaf767a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.semilogx(ff, np.angle(vRatio)/np.pi, 'r-', linewidth = 1.5)\n",
    "plt.xlabel('Frequency (Hz)')\n",
    "plt.ylabel('Phase divided by $\\pi$')\n",
    "plt.axis((10, 1e5, -1, 0.05));"
   ]
  }
 ],
 "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": "3.8.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}