ECE_3110_Electromagnetic_Fields_I_Wickert/ECE_3110_Chapter_2.ipynb

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    "\\tableofcontents\n",
    "% These TeX commands run at the start to remove section numbering\n",
    "\\renewcommand{\\thesection}{\\hspace*{-1.0em}}\n",
    "\\renewcommand{\\thesubsection}{\\hspace*{-1.0em}}\n",
    "\\renewcommand{\\thesubsubsection}{\\hspace*{-1.0em}}"
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     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline\n",
    "#%matplotlib qt\n",
    "from __future__ import division # use so 1/2 = 0.5, etc.\n",
    "import ssd\n",
    "import scipy.signal as signal\n",
    "from IPython.display import Audio, display\n",
    "from IPython.display import Image, SVG"
   ]
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    "pylab.rcParams['savefig.dpi'] = 100 # default 72\n",
    "#pylab.rcParams['figure.figsize'] = (6.0, 4.0) # default (6,4)\n",
    "#%config InlineBackend.figure_formats=['png'] # default for inline viewing\n",
    "%config InlineBackend.figure_formats=['svg'] # SVG inline viewing\n",
    "#%config InlineBackend.figure_formats=['pdf'] # render pdf figs for LaTeX"
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    "%Graphics embed example\n",
    "%Image('fname.png',width='90%')\n",
    "%Commandline for conversion to article:\n",
    "%>jupyter nbconvert --to latex --template article  ECE_3110_Chapter_2.ipynb\n",
    "% Commandline conversion to pdf in one step:\n",
    "%>jupyter nbconvert --to latex --template article  ECE_3110_Chapter_2.ipynb --post pdf \n",
    "\\newpage"
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   "cell_type": "markdown",
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   "source": [
    "**Helper Function(s)**"
   ]
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    "def cpx_fmt(z,mode1='rect',mode2='f',mode3='deg',d1=2,d2=4):\n",
    "    \"\"\"\n",
    "    Helper function for printing formatted complex numbers as %d1.d2f and\n",
    "    %d1.d2e. The function returns a string that can be displayed using print()\n",
    "    \n",
    "    s_out = cpx_fmt(z,mode1='rect',mode2='f',d1=2,d2=4)\n",
    "    \n",
    "    mode1 = 'rect' or 'polar' form\n",
    "    mode2 = 'f' for floating or 'e' for exponential\n",
    "    mode3 = 'deg' or 'rad' for use in the polar display\n",
    "    \n",
    "    Mark Wickert January 2016\n",
    "    \"\"\"\n",
    "    if mode1 == 'rect':\n",
    "        if mode2 == 'f':\n",
    "            mt = 'f'\n",
    "        else:\n",
    "            mt = 'e'\n",
    "        if z.imag >= 0:\n",
    "            s_o = '%'+str(d1)+'.'+str(d2)+mt+ ' + j%'+str(d1)+'.'+str(d2)+mt \n",
    "            s_out = s_o % (z.real,z.imag)\n",
    "        else:\n",
    "            s_o = '%'+str(d1)+'.'+str(d2)+mt+ ' - j%'+str(d1)+'.'+str(d2)+mt\n",
    "            s_out = s_o % (z.real,abs(z.imag))\n",
    "    else: # polar format otherwise\n",
    "        if mode2 == 'f':\n",
    "            mt = 'f'      \n",
    "        else:\n",
    "            mt = 'e'\n",
    "        s_o = '%'+str(d1)+'.'+str(d2)+mt+ ' /_%'+str(d1)+'.'+str(d2)+mt\n",
    "        if mode3 == 'deg':\n",
    "            s_o += ' (deg)'\n",
    "            s_out = s_o % (abs(z),angle(z)*180/pi)\n",
    "        else:\n",
    "            s_o += ' (rad)'\n",
    "            s_out = s_o % (abs(z),angle(z))            \n",
    "    return s_out"
   ]
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    "\\newpage"
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    "# Square Wave Dispersion Demo\n",
    "In the interactive demo below we plot the first few harmonics of a square wave with time delay adjusters on the 3rd, 5th, and 7th harmonics. The time delay is introduced so we can see the how time dispersion in a transmission line distorts the square wave. Additional attenuation of the various harmonics will likely also be present, but is not modeled in this demo.\n",
    "\n",
    "The math model implemented is:\n",
    "\\begin{align}\n",
    "    x(t) &= A \\sum_{n=1}^{N_\\text{max}} \\frac{1}{2n-1} \\sin\\big(2\\pi\\cdot (2n-1) f_0\\cdot(t - dtn)\\big)\n",
    "\\end{align}\n",
    "where $A = 1$ and $f_0 = 1$ Hz, and here $N_\\text{max} = 6$ which takes us up to the 11th harmonic (i.e., $2\\cdot 6-1 =11$). The $2n-1$ is used to index over the odd integers starting at one when $n=1$."
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       "<function __main__.sq_wave_plot>"
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   "source": [
    "%matplotlib inline\n",
    "# Make an interactive plot of +x and -x waves at time snapshots\n",
    "from IPython.html.widgets import interact\n",
    "\n",
    "def sq_wave_plot(dt3=0,dt5=0,dt7=0,dt11=0):\n",
    "    \"\"\"\n",
    "    Plot the first three harmonics (1,3,5) of a squarewave with different\n",
    "    time delays\n",
    "    \"\"\"\n",
    "    # Assume x is in cm\n",
    "    t = arange(0,2,.005)\n",
    "    # Fundamental\n",
    "    x = sin(2*pi*t)\n",
    "    x += 1/3*sin(2*pi*3*(t-dt3)) # Add third\n",
    "    x += 1/5*sin(2*pi*5*(t-dt5)) # Add fifth\n",
    "    x += 1/7*sin(2*pi*7*(t-dt7)) # Add seventh\n",
    "    x += 1/9*sin(2*pi*9*(t-0)) # Add ninth\n",
    "    x += 1/11*sin(2*pi*11*(t-dt11)) # Add eleventh\n",
    "    figure(figsize=(8,5))\n",
    "    plot(t,x)\n",
    "    ylim([-1.5,1.5])\n",
    "    xlabel(r't (s)')\n",
    "    ylabel(r'Amplitude');\n",
    "    title(r'Square Wave with Dispersion: Harmonics 1, 3, 5, 7, 9, & 11')\n",
    "    grid();\n",
    "    \n",
    "interact(sq_wave_plot,dt3 = (0,0.5,.01), dt5 = (0,0.5,0.01), \n",
    "         dt7 = (0,0.5,0.01), dt11 = (0,0.5,0.01))"
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   "source": [
    "## Transmission Line Parameters\n",
    "The equations found in the text and notes which solve for $R'$, $L'$, $G'$, and $C'$ can esily be written up into a Python function.\n",
    "\n",
    "### Coax Example\n",
    "The example below is for coax."
   ]
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   "source": [
    "def coax_parameters(a,b,f,e_r,sigma_c,sigma=1,mu=1,mu_c=1,display=True):\n",
    "    \"\"\"\n",
    "    Find the distributed parameters of the coax line\n",
    "    \n",
    "    R_p, L_p, G_p, C_p = coax_parameters(a,b,f,e_r,sigma_c,\n",
    "                                         sigma=1,mu=1,mu_c=1,\n",
    "                                         display=True)\n",
    "    Note: By default display is True, so to supress raw variables\n",
    "          use a semicolon to terminate the function call.\n",
    "    \n",
    "    Mark Wickert February 2016\n",
    "    \"\"\"\n",
    "    e_0 = 8.85e-12 # F/m\n",
    "    mu_0 = 4*pi*1e-7 # H/m\n",
    "    Rs = sqrt(pi*f*mu_0*mu_c/sigma_c)\n",
    "    R_p = Rs/(2*pi)*(1/a +1/b) # Ohms/m  \n",
    "    L_p = mu_0*mu/(2*pi)*log(b/a) #H/m\n",
    "    G_p = 2*pi*sigma/log(b/a) # S/m\n",
    "    C_p = 2*pi*e_0*e_r/log(b/a)\n",
    "    # Display calculations as opposed to returning the value\n",
    "    if display:\n",
    "        print(\"R' = %2.5f ohms/m\" % (R_p,))\n",
    "        print(\"L' = %2.2f nH/m\" % (L_p*1e9,))\n",
    "        print(\"G' = %2.3f S/m\" % (G_p,))\n",
    "        print(\"C' = %2.2f pF/m\" % (C_p*1e12,))\n",
    "    return R_p, L_p, G_p, C_p"
   ]
  },
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   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R' = 0.01038 ohms/m\n",
      "L' = 138.63 nH/m\n",
      "G' = 9.065 S/m\n",
      "C' = 80.22 pF/m\n"
     ]
    }
   ],
   "source": [
    "coax_parameters(6e-3,12e-3,1e6,1.0,5.8e7);"
   ]
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   "source": [
    "## Microstrip Analysis and Design Equations\n",
    "Here we code take the text analysis and design equations for zero strip thickness microstrip. These equation assume the line is lossless, that is $\\sigma = 0$ (dielectric is lossless) so $G' = 0$ and the metal conductor is perfect, i.e. $R_s \\approx 0$ so $R' \\approx 0$. \n",
    "\n",
    "Three functions are involved (see Text pages 62--64):\n",
    "\n",
    "1. `Z0, e_eff = mstrip_anal(w/h,er)` which finds the line characteristic impedance $Z_0$ given a particular width to height ratio $s = w/h$ and also returns $\\epsilon_r$ (substrate relative permittivity).\n",
    "2. `s, e_eff = mstrip_dsgn(Z0, er)` which finds the width to height ratio $s = w/h$ given the desired characteristic impedance $Z_0$ and also returns $\\epsilon_r$ (substrate relative permittivity).\n",
    "3. `e_eff = eps_eff(w/h,er)` which finds the effective permittivity of the microstrip given the ratio $s = w/h$ (width to substrate height ratio) and $\\epsilon_r$ (substrate relative permittivity). This is a helper function called by the first two functions.\n",
    "\n",
    "Additionally we may want to obtain $R'$, $L'$, $G'$, and $C'$. Since it assumed that $R' = 0$ and $G' = 0$, that leaves only $L'$ and $C'$ and we may also want to have $\\alpha$ and $\\beta$. These parameters can be found from\n",
    "\\begin{align*}\n",
    "    C' &= \\frac{1}{Z_0 v_p}=\\frac{\\sqrt{\\epsilon_\\text{eff}}}{Z_0 c} \\\\\n",
    "    L' &= Z_0^2C' \\\\\n",
    "    \\alpha &= 0\\ (\\text{since $R' = G' = $}) \\\\\n",
    "    \\beta &= \\frac{2\\pi}{\\lambda} = 2\\pi\\frac{f}{v_p}=2\\pi\\, \\frac{f}{c/\\sqrt{\\epsilon_\\text{eff}}} = \\frac{\\omega}{c}\\sqrt{\\epsilon_\\text{eff}}\n",
    "\\end{align*}\n",
    "where $v_p$ is the velocity of propagation in the medium, which here is $c/\\sqrt{\\epsilon_\\text{eff}}$."
   ]
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   "cell_type": "code",
   "execution_count": 56,
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   "source": [
    "def mstrip_anal(s,er):\n",
    "    \"\"\"\n",
    "    Microstrip zero thickness strips design function\n",
    "    \n",
    "    s = w/h\n",
    "    er = material relative permittivity\n",
    "    \n",
    "    Z0 returned\n",
    "    e_eff returned\n",
    "    \n",
    "    Mark Wickert February 2016\n",
    "    \"\"\"\n",
    "    e_eff = eps_eff(s,er)\n",
    "    t = (30.67/s)**0.75\n",
    "    Z0 = 60/sqrt(e_eff) * \\\n",
    "         log((6+(2*pi-6)*exp(-t))/s + sqrt(1+4/s**2))\n",
    "    return Z0, e_eff\n",
    "\n",
    "def mstrip_dsgn(Z0,er):\n",
    "    \"\"\"\n",
    "    Mircostrip zero thickness strips design function\n",
    "    \n",
    "    Z0 = desired characteristic impedance\n",
    "    er = material relative permittivity\n",
    "    \n",
    "    s = w/h is returned\n",
    "    e_eff returned\n",
    "    \n",
    "    Mark Wickert February 2016\n",
    "    \"\"\"\n",
    "    if Z0 <= (44-2*er):\n",
    "        q = 60*pi*pi/(Z0*sqrt(er))\n",
    "        s = 2/pi*((q-1)-log(2*q-1) + \\\n",
    "                 (er-1)/(2*er)*(log(q-1)+0.29-0.52/er))\n",
    "        \n",
    "    else:\n",
    "        p = sqrt((er+1)/2)*Z0/60+(er-1)/(er+1)*(0.23 + 0.12/er)\n",
    "        s = 8*exp(p)/(exp(2*p)-2)\n",
    "    e_eff = eps_eff(s,er)\n",
    "    return s, e_eff\n",
    "\n",
    "def eps_eff(s,er):\n",
    "    \"\"\"\n",
    "    Find microstrip e_eff given s = w/h and er\n",
    "    \n",
    "    Mark Wickert February 2016\n",
    "    \"\"\"\n",
    "    x = 0.56*((er-0.9)/(er+3))**0.05\n",
    "    y = 1 + 0.02*log((s**4 + 3.7e-4*s**2)/(s**4+0.43)) + \\\n",
    "        0.05*log(1+1.7e-4*s**3)\n",
    "    e_eff = (er+1)/2 + (er-1)/2*(1+10/s)**(-x*y)\n",
    "    return e_eff\n",
    "\n",
    "def mstrip_extra(Z0,er,f = 1e9,h = 1):\n",
    "    \"\"\"\n",
    "    Extra parameters: L', C', beta, \\lambda_g\n",
    "    \n",
    "    Z0 = characteristic impedance in Ohms\n",
    "    er = relative permittivity\n",
    "    f = operatng frequency in Hz\n",
    "    h = substrate height in m\n",
    "    \n",
    "    Mark Wickert February 2016\n",
    "    \"\"\"\n",
    "    c = 3e8 # m/s\n",
    "    s, e_eff = mstrip_dsgn(Z0,er)\n",
    "    C_p = sqrt(e_eff)/(Z0*c) # F/m\n",
    "    L_p = Z0**2*C_p  #H/m\n",
    "    R_p = 0\n",
    "    G_p = 0\n",
    "    alpha = 0\n",
    "    beta = 2*pi/c*sqrt(e_eff) # rad/m\n",
    "    lambda_g = c/sqrt(e_eff)/f # m\n",
    "    # Print formatted results\n",
    "    print('Line width w = %1.3e mm, Rel. perm: e_eff = %2.3f' % (s*h*1000,e_eff))\n",
    "    print('Line width w = %1.3e mils (common PCB unit)' % (s*h*1000/0.0254,))\n",
    "    print('-----------------------------------------------------')\n",
    "    print(\"Resistance per unit length: R' = 0 ohms/m\")\n",
    "    print(\"Inductance per unit length: L' = %1.3e nH/m\" % (L_p*1e9,))\n",
    "    print(\"Conductance per unit length: G' = 0 S/n\")\n",
    "    print(\"Capacitance per unit length: C' = %1.3e pF/m\" % (C_p*1e12,))\n",
    "    print('-----------------------------------------------------')\n",
    "    print('Phase constant: beta = %1.3e rad/m' % beta)\n",
    "    print('Wavelength in free space: lambda = %1.3e cm' % (c/f*1e2,))\n",
    "    print('Wavelength in medium (guide): lambda_g = %1.3e cm' % (lambda_g*1e2,))"
   ]
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      "text/plain": [
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   "source": [
    "1/16*1000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Line width w = 2.931e+00 mm, Rel. perm: e_eff = 3.460\n",
      "Line width w = 1.154e+02 mils (common PCB unit)\n",
      "-----------------------------------------------------\n",
      "Resistance per unit length: R' = 0 ohms/m\n",
      "Inductance per unit length: L' = 3.100e+02 nH/m\n",
      "Conductance per unit length: G' = 0 S/n\n",
      "Capacitance per unit length: C' = 1.240e+02 pF/m\n",
      "-----------------------------------------------------\n",
      "Phase constant: beta = 3.896e-08 rad/m\n",
      "Wavelength in free space: lambda = 1.250e+01 cm\n",
      "Wavelength in medium (guide): lambda_g = 6.720e+00 cm\n"
     ]
    }
   ],
   "source": [
    "mstrip_extra(50,4.6,f= 2.4e9,h=1/16*2.54*1/100) #in*cm/in*m/cm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Quarterwave line length at 2.4 GHz  = 1.680 cm\n",
      "or 0.661 in.\n"
     ]
    }
   ],
   "source": [
    "print('Quarterwave line length at 2.4 GHz  = %1.3f cm' % (6.72/4,))\n",
    "print('or %1.3f in.' % (6.72/4*1/2.54,))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Microstrip Analysis Plots\n",
    "Here we consider three material types: Micro-fiber PTFE, FR4, and Ceramic-filled PTFE over a range of $w/h$ values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
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    "Z0 = zeros_like(s)\n",
    "e_eff = zeros_like(s)\n",
    "er = (2.2,4.6,11.2)\n",
    "for m in range(3):\n",
    "    for k,sk in enumerate(s):\n",
    "        Z0[k],e_eff[k] = mstrip_anal(sk,er[m])\n",
    "    plot(s,Z0)\n",
    "ylim([0,200])\n",
    "xlabel(r'$w/h$')\n",
    "ylabel(r'$Z_0$')\n",
    "title(r'Microstrip Analysis: $Z_0$ vs $w/h$')\n",
    "legend((r'MF-PTFE: $\\epsilon_r = 2.2$',r'FR-4: $\\epsilon_r=4.6$',\n",
    "        r'CF_PTFE: $\\epsilon_r=11.2$'),loc='best')\n",
    "grid();"
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  {
   "cell_type": "code",
   "execution_count": 61,
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    "s = arange(0.01,10,.01)\n",
    "Z0 = zeros_like(s)\n",
    "e_eff = zeros_like(s)\n",
    "er = (2.2,4.6,11.2)\n",
    "for m in range(3):\n",
    "    for k,sk in enumerate(s):\n",
    "        Z0[k],e_eff[k] = mstrip_anal(sk,er[m])\n",
    "    plot(s,e_eff)\n",
    "ylim([0,10])\n",
    "xlabel(r'$w/h$')\n",
    "ylabel(r'$\\epsilon_{eff}$')\n",
    "title(r'Microstrip Analysis: $\\epsilon_{eff}$ vs $w/h$')\n",
    "legend((r'$\\epsilon_r = 2.2$',r'$\\epsilon_r=4.6$',\n",
    "        r'$\\epsilon_r=11.2$'),loc='best')\n",
    "grid();"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Microstrip Design Plots\n",
    "Here we consider three material types: Micro-fiber PTFE, FR4, and Ceramic-filled PTFE over a range of $w/h$ values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
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   "source": [
    "Z0 = arange(20,140,.1)\n",
    "s = zeros_like(Z0)\n",
    "e_eff = zeros_like(Z0)\n",
    "er = (2.2,4.6,11.2)\n",
    "for m in range(3):\n",
    "    for k,Z0k in enumerate(Z0):\n",
    "        s[k],e_eff[k] = mstrip_dsgn(Z0k,er[m])\n",
    "    plot(Z0,s)\n",
    "ylim([0,10])\n",
    "xlabel(r'$Z_0$')\n",
    "ylabel(r'$w/h$')\n",
    "title(r'Microstrip Design: $w/h$ vs $Z_0$')\n",
    "legend((r'MF-PTFE: $\\epsilon_r = 2.2$',r'FR-4: $\\epsilon_r=4.6$',\n",
    "        r'CF_PTFE: $\\epsilon_r=11.2$'),loc='best')\n",
    "grid();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
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   "source": [
    "for m in range(3):\n",
    "    for k,Z0k in enumerate(Z0):\n",
    "        s[k],e_eff[k] = mstrip_dsgn(Z0k,er[m])\n",
    "    plot(Z0,e_eff)\n",
    "ylim([1,10])\n",
    "xlabel(r'$w/h$')\n",
    "ylabel(r'$\\epsilon_{eff}$')\n",
    "title(r'Microstrip Design: $\\epsilon_{eff}$ vs $Z_0$')\n",
    "legend((r'MF-PTFE: $\\epsilon_r = 2.2$',r'FR-4: $\\epsilon_r=4.6$',\n",
    "        r'CF_PTFE: $\\epsilon_r=11.2$'),loc='best')\n",
    "grid();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finding $\\epsilon_r$ Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Lambda Freespace = 12.500 cm\n"
     ]
    }
   ],
   "source": [
    "lambda_0 = 3e8/2.4e9\n",
    "print('Lambda Freespace = %1.3f cm' % (lambda_0*100,))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Lambda Guide = 5.828 cm\n"
     ]
    }
   ],
   "source": [
    "lambda_g = 3e8/2.4e9/sqrt(4.6)\n",
    "print('Lambda Guide = %1.3f cm' % (lambda_g*100,))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
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      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(12.5/5.828)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
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      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(3e8/2.4e9/0.05828)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finding $u_p$, $C'$, and $Z_0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "up = 1.225e+08 m/s\n"
     ]
    }
   ],
   "source": [
    "up = 3e8/sqrt(6)\n",
    "print('up = %1.3e m/s' % up)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "C_p = 8.341e-11 F/m\n"
     ]
    }
   ],
   "source": [
    "C_p = 4*pi*1e-7*8.85e-12*6/0.8e-6\n",
    "print('C_p = %1.3e F/m' % C_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Z0 = 97.935 Ohms\n"
     ]
    }
   ],
   "source": [
    "Z0 = sqrt(0.8e-6/C_p)\n",
    "print('Z0 = %1.3f Ohms' % Z0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Find $\\Gamma_L$ for a Series $RC$ Load"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "GammaL 0.7628 /_-60.7444 (deg)\n"
     ]
    }
   ],
   "source": [
    "f = 100e6\n",
    "Z0 = 100\n",
    "RL = 50\n",
    "CL = 10e-12\n",
    "Z_CL = 1/(1j*2*pi*f*CL)\n",
    "ZL = RL + Z_CL\n",
    "GammaL = (ZL - Z0)/(ZL + Z0)\n",
    "print('GammaL ' + cpx_fmt(GammaL,'polar'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Standing Waves\n",
    "Use Python to investigate voltage and current along a tline. Given that the reflection coefficient in polar form is $\\Gamma = |\\Gamma|e^{j\\theta_r}$ and $d = -z$ is the positive distance back from the line load, we can write\n",
    "\\begin{align*}\n",
    "        |\\tilde{V}(d)| &= |V_0^+|\\left[1+|\\Gamma|^2 + 2|\\Gamma|\\cos\\big(2\\beta d - \\theta_r\\big)\\right] \\\\\n",
    "                |\\tilde{I}(d)| &= \\frac{|V_0^+|}{Z_0}\\left[1+|\\Gamma|^2 - 2|\\Gamma|\\cos\\big(2\\beta d - \\theta_r\\big)\\right] \n",
    "\\end{align*}\n",
    "for $0 \\leq d \\leq l$. The above math follows the text and notes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
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    "d2lam  = arange(0,1,0.005) # d/lambda for plotting\n",
    "V0_p_mag = 1.0 # V\n",
    "Z0 = 50 # Ohms\n",
    "Gamma = 0.8*exp(1j*45*(pi/180))\n",
    "#Gamma = -1.0\n",
    "V_abs = V0_p_mag*(1 + abs(Gamma)**2 + \\\n",
    "                  2*abs(Gamma)*cos(2*(2*pi*d2lam) - angle(Gamma)))**0.5\n",
    "I_abs = V0_p_mag/Z0*(1 + abs(Gamma)**2 - \\\n",
    "                     2*abs(Gamma)*cos(2*(2*pi*d2lam) - angle(Gamma)))**0.5\n",
    "subplot(211)\n",
    "plot(-d2lam,V_abs)\n",
    "ylim([0,2])\n",
    "xlabel(r' Wavelength Normalized Distance $-d/\\lambda$ or $z\\lambda$')\n",
    "ylabel(r'$|\\tilde{V}(d)|$ (V)')\n",
    "title(r'V and I Standing-Wave Patterns: $\\Gamma$ = ' + \\\n",
    "      cpx_fmt(Gamma,'polar',d2=2))\n",
    "grid()\n",
    "subplot(212)\n",
    "plot(-d2lam,I_abs*1e3)\n",
    "ylim([0,2/Z0*1e3])\n",
    "xlabel(r' Wavelength Normalized Distance $-d/\\lambda$ or $z/\\lambda$')\n",
    "ylabel(r'$|\\tilde{I}(d)|$ (mA)')\n",
    "grid()\n",
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   "source": [
    "# Finding $L$ That Makes $S = 5$\n",
    "Consider a load $Z_L$ composed of $R_L = 50\\ \\omega$ in parallel with an inductor, $L$. Assume the line impedance is $Z_0 = 100\\ \\Omega$ and $f = 100$ MHz.\n",
    "\n",
    "\n",
    "* The load impedance is\n",
    "\\begin{align}\n",
    "    Z_L = \\frac{R_L\\cdot j\\omega L}{R_L + j\\omega L}\n",
    "\\end{align}\n",
    "* $S = \\text{VSWR}$ is related to $|\\Gamma|$ via\n",
    "\\begin{align}\n",
    "    S = \\frac{1 + |\\Gamma|}{1 - |\\Gamma|}\\ \\Rightarrow\\ |\\Gamma| = \\frac{S-1}{S+1} = \\frac{4}{6} = \\frac{2}{3}\n",
    "\\end{align} "
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  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exact Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L = 68.489 (nH)\n"
     ]
    }
   ],
   "source": [
    "Gam_mag = 2/3\n",
    "Z0 = 100\n",
    "RL = 50\n",
    "w = 2*pi*100e6\n",
    "L = sqrt((Z0*RL)**2*(1-(Gam_mag)**2)/(w**2*((Z0+RL)**2*Gam_mag**2-(Z0-RL)**2)))\n",
    "print('L = %1.3f (nH)' % (L*1e9,))"
   ]
  },
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   "metadata": {},
   "source": [
    "### Solve Numerically\n",
    "Here we simply plot $|\\Gamma|$ versus $L$ and then use `fsolve` from `scipy.optimize` to solve for the value of $L$ where $|\\Gamma|$ crosses through 2/3."
   ]
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     "metadata": {},
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   ],
   "source": [
    "# Plot Abs[Gamma] versus C\n",
    "L = arange(1,200)*1e-9\n",
    "Gamma_mag = abs((-5000 - 1j*2*pi*100e6*L*50)/(5000 + 1j*2*pi*100e6*L*150))\n",
    "plot(L*1e9,Gamma_mag)\n",
    "title(r'Searching for $L$ Numerically by Plotting $|\\Gamma|$ vs $L$')\n",
    "ylabel(r'$|\\Gamma|$')\n",
    "xlabel(r'$L$ in nH')\n",
    "grid();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.optimize import fsolve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Gamma_abs_objective(L):\n",
    "    \"\"\"\n",
    "    |Gamma| - 2/3 as a function of L\n",
    "    \n",
    "    Mark Wickert February 2016\n",
    "    \"\"\"\n",
    "    L = L*1e-9\n",
    "    RL = 50\n",
    "    Z0 = 100\n",
    "    f = 100e6\n",
    "    w = 2*pi*f\n",
    "    Gamma_abs = abs(((RL*1j*w*L)/(RL+1j*w*L)-Z0)/((RL*1j*w*L)/(RL+1j*w*L)+Z0))\n",
    "    return Gamma_abs - 2/3 # Looking for zero crossing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L_req = 68.5 nH\n"
     ]
    }
   ],
   "source": [
    "L_req = fsolve(Gamma_abs_objective,100,xtol = 1e-5)\n",
    "print('L_req = %1.1f nH' % (L_req,))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
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