Conformal mapping of a CPW coupler.ipynb

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   "cell_type": "code",
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   "source": [
    "from __future__ import print_function\n",
    "from matplotlib import pyplot as plt\n",
    "import numpy as np\n",
    "import scipy.integrate as integrate\n",
    "from scipy.optimize import root\n",
    "from scipy.constants import epsilon_0, mu_0\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Branch points in a coplanar waveguide coupler\n",
    "The coplanar waveguide is a long structure, where the waves propagate in one direction and are confined in the other two direction. In the following document we assume that the structure is infinite and translationally invariant in the propagation direction. This means that all cross-sections look completely the same.\n",
    "\n",
    "We want to find the following characteristics of this structure:\n",
    "\n",
    "$\\phi(x,y)$, the electric potential\n",
    "\n",
    "$\\psi(x,y)$, the scalar magnetic potential\n",
    "\n",
    "$\\vec{E}(x,y) = -\\nabla \\phi(x,y)$, the electric field\n",
    "\n",
    "$\\vec{H}(x,y) = -\\nabla \\psi(x,y)$, the magnetic field\n",
    "\n",
    "$\\hat{C}$, the capacitance matrix of the two waveguides\n",
    "\n",
    "$\\hat{L}$, the inductance matrix of the two waveguides\n",
    "\n",
    "First, we need to define the \"special points\" of the waveguide coupler cross-section. We define 5 geometric distances: the gap between central line and ground plane conductors of the waveguides ($s_1$ and $w_1$ for the first waveguide and $s_2$ and $w_2$ for the second waveguide). The distance between the adjacent gaps of the couplers is called $w_3$.\n",
    "\n",
    "The following function defines the \"branch points\" of the future conformal transform. These points mark ends of metallization. We will denote them $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$ from left to right"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 2 coplanar waveguides on the real axis\n",
    "def z_branch_points_cpw(s1,s2,w1,w2,w3):\n",
    "    z_B = 0\n",
    "    z_C = z_B+s2\n",
    "    z_D = z_C+w2\n",
    "    z_E = z_D+s2\n",
    "    z_F = z_E+w3\n",
    "    z_G = z_F+s1\n",
    "    z_H = z_G+w1\n",
    "    z_I = z_H+s1    \n",
    "    return [np.complex(x) for x in [z_B, z_C, z_D, z_E, z_F, z_G, z_H, z_I]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For conformal mapping, we shall use the convention $z=x+iy$. The coplanar waveguide coupler geometry is thus defined in the complex plane of the $z$ variable. For two relatively strongly coupled 50-Ohm lines on a silicon substrate we have the following geometry:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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     "execution_count": 3,
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Hjh113759Voca0urVq2tcu84wjKjb19xutw4cOLDGXFp2u1179eoVs0n7559/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\nXfXABlEAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x5824cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "s1,s2,w1,w2,w3 = [12,6,10,20,5]\n",
    "z_branch_points = z_branch_points_cpw(s1,s2,w1,w2,w3)\n",
    "\n",
    "x = np.linspace(-(2*s2+2*s1+w1+w2+w3)*0.5,(2*s2+2*s1+w1+w2+w3)*1.5,251)\n",
    "y = np.linspace(0, (np.max(x)-np.min(x))*0.4, 201)\n",
    "z_B,z_C,z_D,z_E,z_F,z_G,z_H,z_I = z_branch_points\n",
    "\n",
    "plt.figure(figsize=(10,5))\n",
    "plt.clf()\n",
    "axes=plt.subplot(2,1,1)\n",
    "vertices = [i for i in z_branch_points]\n",
    "lines = [(z_B-100,z_B,'-'),(z_B,z_C,':'),(z_C,z_D,'-'),(z_D,z_E,':'),(z_E,z_F,'-'),\\\n",
    "         (z_F,z_G,':'),(z_G,z_H,'-'),(z_H,z_I,':'),(z_I,z_I+100,'-')]\n",
    "for z1,z2,style in lines:\n",
    "    plt.plot([z1.real, z2.real], [z1.imag, z2.imag], color='black', linestyle=style, linewidth=(3 if style=='-' else 1) )\n",
    "\n",
    "plt.xlim(np.min(x), np.max(x))\n",
    "plt.ylim(-0.5*np.max(y), np.max(y))\n",
    "\n",
    "plt.scatter([j.real for j in vertices], [j.imag for j in vertices])\n",
    "\n",
    "annotations = ['B', 'C', 'D', 'E', 'F', 'G', 'H', 'I']\n",
    "annotation_positions = [(a*0.10+0.15, 0.05) for a in range(10)]\n",
    "annotation_points = z_branch_points\n",
    "\n",
    "for annotation, annotation_position, annotation_point in zip(annotations, annotation_positions, annotation_points):\n",
    "    axes.annotate(annotation, xy=(annotation_point.real, annotation_point.imag), xytext=annotation_position, \n",
    "              xycoords='data',  textcoords='axes fraction', arrowprops={'facecolor':'black', 'shrink':0.05})\n",
    "axes.annotate('$s_1$', xy=((z_B+z_C).real/2-2,5), fontsize='xx-large')\n",
    "axes.annotate('$w_1$', xy=((z_C+z_D).real/2-2,5), fontsize='xx-large')\n",
    "axes.annotate('$s_1$', xy=((z_D+z_E).real/2-2,5), fontsize='xx-large')\n",
    "axes.annotate('$w_3$', xy=((z_E+z_F).real/2-2,5), fontsize='xx-large')\n",
    "axes.annotate('$s_2$', xy=((z_F+z_G).real/2-2,5), fontsize='xx-large')\n",
    "axes.annotate('$w_2$', xy=((z_G+z_H).real/2-2,5), fontsize='xx-large')\n",
    "axes.annotate('$s_2$', xy=((z_H+z_I).real/2-2,5), fontsize='xx-large')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since this structure is mirror-symmetric wrt the $x$ axis, it is enough to solve the problem in the upper half-plane. In a real structure the upper and lower half-planes usually have different dielectric constants. However one can still show that the analytical expression for the fields, potentials, capacitances and inductances remain the same up to a scaling factor for the coordinates.\n",
    "\n",
    "The central problem we shall be solving is Laplace's equations for the electric and scalar magnetic potentials in the upper half-plane:\n",
    "$$ \\Delta \\phi = 0 $$\n",
    "$$ \\Delta \\psi = 0 $$\n",
    "For the electrostatic potential $\\phi$ the boundary conditions are that the potential on each conductor is constant. The field lines originate from charges, in our case - surface charges on the conductor. Since we want the conductors terminating at the point $B$ and $I$ to be the ground plane and the conductor $EF$ to be an effective ground for the two coplanar modes, we set\n",
    "\\begin{align}\n",
    "&\\phi(x,y)=0  && \\mbox{if} && x<z_B &&\\mbox{and} && y=0, \\\\\n",
    "&\\phi(x,y)=\\phi_1  && \\mbox{if} && z_C<x<z_D && \\mbox{and}&& y=0, \\\\\n",
    "&\\phi(x,y)=0 &&  \\mbox{if} && z_E<x<z_F && \\mbox{and}&& y=0, \\\\\n",
    "&\\phi(x,y)=\\phi_2  && \\mbox{if} && z_F<x<z_G && \\mbox{and}&& y=0, \\\\\n",
    "&\\phi(x,y)=0 &&  \\mbox{if} && x>z_I && \\mbox{and} && y=0, \\\\\n",
    "\\end{align}\n",
    "\n",
    "These boundary conditions are incomplete: they would beonly complete if we define the value of $\\phi$ or it's derivative on the boundary of the upper-half plane. But for our purposes this is fine.\n",
    "\n",
    "The scalar magnetic potential is a less common concept. The idea behind it is that in the upper half-plane we don't have any volume density of currents and thus no curly magnetic fields. In absence of a curly component the magnetic field in a domain can be expressed as the antigradient of some function, which is called the scalar potential (analogous to the electric potential). Leaving out the details we suppose that the source of magnetic field lines is the external magnetic flux density in the gaps and that the boundary conditions for the scalar potential can be formulated as follows:\n",
    "\n",
    "\\begin{align}\n",
    "&\\psi(x,y)=\\psi_1  && \\mbox{if} && z_B<x<z_C && \\mbox{and}&& y=0, \\\\\n",
    "&\\psi(x,y)=0 &&  \\mbox{if} && z_D<x<z_E && \\mbox{and}&& y=0, \\\\\n",
    "&\\psi(x,y)=\\psi_2  && \\mbox{if} && z_F<x<z_G && \\mbox{and}&& y=0, \\\\\n",
    "\\end{align}\n",
    "Is in the former case, these boundary conditions are incomplete, but we'll try to find a conformal transform just using these boundary conditions and see what happens."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conformal transformation of the planar structure to a rectangle\n",
    "The whole idea of conformal mapping resides on two main pillars: the Euler-d'Alambert condition and the Schwarz-Cristoffel formula.\n",
    "## Euler-d'Alambert condition\n",
    "Suppose we have some analytical function of a complex variable, $w(z)$. Analytical means it has a derivative in some domain $\\Omega \\subset \\mathbb{C}$: \n",
    "\n",
    "\\begin{equation} \n",
    "w'(z) = \\lim_{\\Delta z \\to 0} \\frac{w(z+\\Delta z) - w(z)}{\\Delta z} \n",
    "\\end{equation}\n",
    "\n",
    "For the limit to exist, it must be equal independent from the argument of $\\Delta z$, so a function of two real variables $x$ and $y$, $w(x+iy)$ must satisfy the Euler-d'Alambert condition\n",
    "\n",
    "\\begin{equation} \n",
    "\\frac{\\partial w}{\\partial x} = \\lim_{\\Delta x \\to 0} \\frac{w(z+\\Delta x) - w(z)}{\\Delta x} = \\lim_{\\Delta y \\to 0} \\frac{w(z+i\\Delta y) - w(z)}{i\\Delta y} = -i\\frac{\\partial w}{\\partial y}\n",
    "\\end{equation}\n",
    "\n",
    "Appyling this rule twice, we arrive at\n",
    "\n",
    "\\begin{equation}\n",
    "\\frac{\\partial}{\\partial x}\\frac{\\partial w}{\\partial x} = \\frac{\\partial}{\\partial x}(-i)\\frac{\\partial w}{\\partial y} = \n",
    "(-i)^2\\frac{\\partial}{\\partial y}\\frac{\\partial w}{\\partial y}\n",
    "\\end{equation}\n",
    "\n",
    "which translates to the fact than an analytical function of a complex variable is in fact a solution to Laplace's equation (also called a harmonic function) of two real variables:\n",
    "\n",
    "\\begin{equation}\n",
    "\\frac{\\partial^2 w}{\\partial x^2}+\\frac{\\partial^2 w}{\\partial y^2}=0\n",
    "\\end{equation}\n",
    "\n",
    "What we want to to find an analytic function $w(z)$ of which either the real or imaginary part fits to our boundary conditions.\n",
    "## Schwarz–Christoffel mapping\n",
    "How do we find such a analytic function? The Schwarz-Cristoffel formula is a powerful tool to perform a conformal mapping of the real axis onto the edges of a simple polygon.\n",
    "The Schwarz-Cristoffel formula is following:\n",
    "\\begin{equation}\n",
    "w(z) = \\int_{0}^{z}\\prod_{i}(z-z_i)^{\\alpha_i/\\pi-1}\\mathrm{d}z\n",
    "\\end{equation}\n",
    "$z_i$ are the coordinates of the points on the real axis that are transformed into the verteces of the polygon and $\\alpha_i$ are the angles at these verteces. Why does this work? Let's take the derivative of the formula in differential form and look at its argument:\n",
    "\\begin{equation}\n",
    "\\operatorname{Arg}\\mathrm{d} w = \\sum_{i}(\\alpha_i/\\pi-1)\\operatorname{Arg}(z-z_i)+\\operatorname{Arg}\\mathrm{d}z\n",
    "\\end{equation}\n",
    "As we are moving along the positive direction of the real axis, $\\operatorname{Arg}\\mathrm{d}z$ remains zero. As long as all $z_i$ are real, $\\operatorname{Arg}(z-z_i)$ is either $\\pi$ or $0$, depending on whether $z$ has passed $z_i$ of not. While $z_i<z<z_{i+1}$, the value of $\\operatorname{Arg}\\mathrm{d} w$ is constant: intervals between the points $z_i$ on the real axis are transformed into straight lines. When passing one of the points $z_i$, the argument becomes $\\alpha_i-\\pi$ larger, which corresponds to a vertex in $w$-space where the angle is equal to $\\alpha_i$.\n",
    "\n",
    "Our transform for the CPW coupler looks in general like this:\n",
    "\\begin{equation}\n",
    "w(z) =\\int_{0}^{z}\\frac{(z-z_{F'})(z-z_{H'})}{(z-z_B)^{1/2}(z-z_C)^{1/2}(z-z_D)^{1/2}(z-z_E)^{1/2}(z-z_F)^{1/2}(z-z_G)^{1/2}(z-z_H)^{1/2}(z-z_I)^{1/2}}\\mathrm{d}z\n",
    "\\end{equation}\n",
    "\n",
    "Non-integer powers of polynomials are multivalued functions of their complex argument. At the zero of the argument of these functions, they have a so-called \"branch point\". Thus in our formula $z_B$, $z_C$, $z_D$, $z_E$, $z_F$, $z_G$, $z_H$, $z_I$ are branch points of the function under the integral and $z_{F'}$ and $z_{H'}$ are its zeros.\n",
    "We will need $z_{F'}$ and $z_{H'}$ a bit later. Their pupose in thoroughly explained in https://arxiv.org/ftp/arxiv/papers/1511/1511.02639.pdf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculate the confromal transform on all branch points\n",
    "Our Schwarz-Christoffel integral looks awful. It has a degree-2 polynomial in the numerator and the sqaure root of a degree-8 polynomial in the denumerator. In case it can be simplified at least to an at most degree-4 polynomial under the square root, it could be written as an elliptic integral. We won't try to simplify it and shall evaluate w(z) numerically.\n",
    "Numerical integration has a problem here: at the branch points the integrand diverges, and standard numerical integration routines typically fail. To curcumvent this issue, we can apply integration by parts:\n",
    "\n",
    "\\begin{equation}\n",
    "w(z_1)-w(z_0) = \\int_{z_0}^{z_1} u(z)\\frac{\\mathrm{d}v}{\\mathrm{d}z} \\mathrm{d}z= u(z_1)v(z_1) - u(z_0)v(z_0)-\\int_{z_0}^{z_1} \\frac{\\mathrm{d}u}{\\mathrm{d}z}v(z) \\mathrm{d}z\n",
    "\\end{equation}\n",
    "\n",
    "We select two of the eight branch points and call them $j$ and $k$. All the other branch points are indexed with $i$ and the two zeros are indexed with $l$. As the part $u(z)$ and $v(z)$ we select\n",
    "\\begin{equation}\n",
    "u(z) = \\prod_{l}(z-z_l)\\prod_{i \\neq j,k}(z-z_i)^{-1/2}\n",
    "\\end{equation}\n",
    "\\begin{equation}\n",
    "v(z) = \\operatorname{Log}\\left(z-\\frac{z_j+z_k}{2}+(z-z_j)^{1/2}(z-z_k)^{1/2}\\right)\n",
    "\\end{equation}\n",
    "with the derivatives\n",
    "\\begin{equation}\n",
    "u'(z) = \\sum_{l} \\prod_{l' \\neq l}(z-z_l')\\prod_{i \\neq j,k}(z-z_i)^{-1/2} - \\sum_{i \\neq j,k} (z-z_i)^{-1}\\prod_{l}(z-z_l)\\prod_{i' \\neq j,k} (z-z_{i'})^{1-2}\n",
    "\\end{equation}\n",
    "\\begin{equation}\n",
    "v'(z) = (z-z_j)^{-1/2}(z-z_k)^{-1/2}\n",
    "\\end{equation}\n",
    "\n",
    "The following calculates the conformal transform exactly this way. This helps use mitigate two singularities of 8, which is not bad. Actually, we could use a formula that makes only one singularity go away, since it is only useful near the singularities anyway."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def conformal_transform_difference_by_parts(z0,z1,branch_point1,branch_point2,branch_points,zero_points):\n",
    "    # 1) solving integral part (positive)\n",
    "    f1 = lambda z: np.log(z-(branch_points[branch_point1] + branch_points[branch_point2])/2 + \\\n",
    "                       np.sqrt((z-branch_points[branch_point1])*(z-branch_points[branch_point2]))) * \\\n",
    "                       np.prod([z-zero_point for zero_point in zero_points]) / \\\n",
    "                       np.prod([np.sqrt(z-branch_point) for branch_point_id, branch_point \\\n",
    "                       in enumerate(branch_points) if branch_point_id not in [branch_point1, branch_point2]])\n",
    "    f2 = lambda z: np.log(z-(branch_points[branch_point1] + branch_points[branch_point2])/2 + \\\n",
    "                       np.sqrt((z-branch_points[branch_point1])*(z-branch_points[branch_point2]))) * \\\n",
    "                       np.sum([np.prod([(z-zero_point) for zero_point_id, zero_point in enumerate(zero_points) \\\n",
    "                       if zero_point_id != diff_zero_point_id])/ \\\n",
    "                       np.prod([np.sqrt(z-branch_point) for branch_point_id, branch_point \\\n",
    "                       in enumerate(branch_points) if branch_point_id not in [branch_point1, branch_point2]]) \\\n",
    "                       for diff_zero_point_id, diff_zero_point in enumerate(zero_points)]+ \\\n",
    "                       [np.prod([z-zero_point for zero_point in zero_points]) / \\\n",
    "                       np.prod([np.sqrt(z-branch_point) for branch_point_id, branch_point \\\n",
    "                       in enumerate(branch_points) if branch_point_id not in [branch_point1, branch_point2]]) / \\\n",
    "                       (z-diff_branch_point)*(-0.5) for diff_branch_point_id, diff_branch_point in enumerate(branch_points) \\\n",
    "                       if diff_branch_point_id not in [branch_point1, branch_point2] ])\n",
    "    \n",
    "    integral_part = f1(z1)-f1(z0)\n",
    "    numerical_part = np.complex(*tuple(integrate.quad(lambda t: f2(t*(z1-z0)+z0).real, 0, 1)))+1j*np.complex(*tuple(integrate.quad(lambda t: f2(t*(z1-z0)+z0).imag, 0, 1)))\n",
    "    \n",
    "    return integral_part - (z1-z0)*numerical_part   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the following code we will need the values of the conformal transform at the zeros and branch points of the integrand. An easy way of doing that is calculating the conformal transform difference between neighboring branch points with the formula above and then putting the differences together."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def w_special_points(z_branch_points, z_zero_points):\n",
    "    w_branch_points = []\n",
    "    for branch_point_id, z_branch_point in enumerate(z_branch_points):\n",
    "        if branch_point_id==0:\n",
    "            w_branch_points.append(0)\n",
    "        else:            \n",
    "            w_branch_points.append(w_branch_points[branch_point_id-1] + \\\n",
    "            conformal_transform_difference_by_parts(\\\n",
    "            z_branch_points[branch_point_id-1],z_branch_points[branch_point_id],\\\n",
    "            branch_point_id-1,branch_point_id,z_branch_points,z_zero_points))\n",
    "    \n",
    "    w_zero_points = []        \n",
    "    for zero_point_id, z_zero_point in enumerate(z_zero_points):\n",
    "        nearest_branch_points = np.argsort(np.abs(np.asarray(z_branch_points)-z_zero_point))[:2]\n",
    "        w_zero_points.append(w_branch_points[nearest_branch_points[0]] + \\\n",
    "            conformal_transform_difference_by_parts(\\\n",
    "                z_branch_points[nearest_branch_points[0]],\\\n",
    "                z_zero_point,\\\n",
    "                nearest_branch_points[0],\\\n",
    "                nearest_branch_points[1],\\\n",
    "                z_branch_points,\n",
    "                z_zero_points))\n",
    "    return w_branch_points, w_zero_points"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are ready to see how the conformal transform works. For $z_F'$ and $z_H'$ we shall for now take some random values between $z_F$ and $z_G$, and $z_H$ and $z_I$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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PPfecfPPNN0Hr8sBES1sBTawA3AvgCAAHgOMAPiz12nMlCRWHWyjh6ic1YsQI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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x841a4e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z_F1 = z_F*0.8+z_G*0.2\n",
    "z_H1 = z_H*0.8+z_I*0.2\n",
    "z_zero_points = [z_F1, z_H1]\n",
    "z_branch_points = z_branch_points_cpw(s1,s2,w1,w2,w3)\n",
    "w_branch_points, w_zero_points = w_special_points(z_branch_points, z_zero_points)\n",
    "z_B,z_C,z_D,z_E,z_F,z_G,z_H,z_I = z_branch_points\n",
    "w_B,w_C,w_D,w_E,w_F,w_G,w_H,w_I = w_branch_points\n",
    "z_F1,z_H1 = z_zero_points\n",
    "w_F1,w_H1 = w_zero_points\n",
    "\n",
    "x = np.linspace(-(2*s2+2*s1+w1+w2+w3)*0.5,(2*s2+2*s1+w1+w2+w3)*1.5,251)\n",
    "y = np.linspace(0, (np.max(x)-np.min(x))*0.4, 201)\n",
    "\n",
    "plt.figure(figsize=(10,5))\n",
    "plt.clf()\n",
    "axes=plt.subplot(2,1,1)\n",
    "vertices = [i for i in z_zero_points] + [i for i in z_branch_points]\n",
    "lines = [(z_B-100,z_B,'-'),(z_B,z_C,':'),(z_C,z_D,'-'),(z_D,z_E,':'),(z_E,z_F,'-'),\\\n",
    "         (z_F,z_F1,':'),(z_F1,z_G,':'),(z_G,z_H,'-'),(z_H,z_H1,':'),(z_H1,z_I,':'),(z_I,z_I+100,'-')]\n",
    "for z1,z2,style in lines:\n",
    "    plt.plot([z1.real, z2.real], [z1.imag, z2.imag], color='black', linestyle=style, linewidth=(3 if style=='-' else 1))\n",
    "\n",
    "plt.xlim(np.min(x), np.max(x))\n",
    "plt.ylim(-0.5*np.max(y), np.max(y))\n",
    "\n",
    "plt.scatter([j.real for j in vertices], [j.imag for j in vertices])\n",
    "\n",
    "annotations = ['B', 'C', 'D', 'E', 'F', 'F\\'', 'G', 'H', 'H\\'', 'I']\n",
    "annotation_positions = [(a*0.10+0.05, 0.05) for a in range(10)]\n",
    "annotation_points = [z_B,z_C,z_D,z_E,z_F,z_F1,z_G,z_H,z_H1,z_I]\n",
    "\n",
    "for annotation, annotation_position, annotation_point in zip(annotations, annotation_positions, annotation_points):\n",
    "    axes.annotate(annotation, xy=(annotation_point.real, annotation_point.imag), xytext=annotation_position, \n",
    "              xycoords='data',  textcoords='axes fraction', arrowprops={'facecolor':'black', 'shrink':0.05})\n",
    "    \n",
    "axes=plt.subplot(2,1,2)\n",
    "vertices = [i for i in w_zero_points] + [i for i in w_branch_points]\n",
    "lines = [(w_B,w_C,':'),(w_C,w_D,'-'),(w_D,w_E,':'),(w_E,w_F,'-'),\\\n",
    "         (w_F,w_F1,':'),(w_F1,w_G,':'),(w_G,w_H,'-'),(w_H,w_H1,':'),(w_H1,w_I,':'),(w_I,w_B,'-')]\n",
    "for z1,z2,style in lines:\n",
    "    plt.plot([z1.real, z2.real], [z1.imag, z2.imag], color='black', linestyle=style, linewidth=(3 if style=='-' else 1))\n",
    "    \n",
    "plt.scatter([j.real for j in vertices], [j.imag for j in vertices])\n",
    "\n",
    "plt.xlim([w_E.real-(w_B.real-w_E.real)*0.2, w_B.real+(w_B.real-w_E.real)*0.2])\n",
    "plt.ylim([w_B.imag-(w_C.imag-w_B.imag)*0.4, w_C.imag+(w_C.imag-w_B.imag)*0.3])\n",
    "\n",
    "annotations = ['B', 'C', 'D', 'E', 'F', 'F\\'', 'G', 'H', 'H\\'', 'I']\n",
    "annotation_positions = [(0.95, 0.05), (0.95, 0.75), (0.05, 0.75), (0.05, 0.05), (0.200, 0.50), \\\n",
    "                        (0.30, 0.50), (0.30, 0.05), (0.325, 0.05), (0.35, 0.50), (0.4, 0.05),]\n",
    "annotation_points = [w_B,w_C,w_D,w_E,w_F,w_F1,w_G,w_H,w_H1,w_I]\n",
    "\n",
    "for annotation, annotation_position, annotation_point in zip(annotations, annotation_positions, annotation_points):\n",
    "    axes.annotate(annotation, xy=(annotation_point.real, annotation_point.imag), xytext=annotation_position, \n",
    "              xycoords='data',  textcoords='axes fraction', arrowprops={'facecolor':'black', 'shrink':0.05})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On the branch points, the transform of the real axis turns by a right angle counter-clockwise. On the zero points, in rotates by $2\\pi$ and goes in the reverse direction.\n",
    "\n",
    "The transform $w(z)$ is already half of the solution of the problem: If we look at its imaginary part, $\\operatorname{Im}w(z)$, it is constant on the conductors. This is make it a good candidate for $\\phi(x,y)$.\n",
    "However we wanted $\\phi(x,y)=0$ on $z_E<x<z_F, y=0$, so that the central conductor is grounded. Apart from that, it would be nice if we could scale $\\frac{\\phi_1}{\\phi_2}$. For these purposes, we can vary $z_F'$ and $z_H'$ (they need to remain real, however)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Find the zero points of the transform derivative from the potential of the ports\n",
    "\n",
    "If we want, for example, $\\phi_2=0$ (which is useful for determining the capacitance matrix), we would need to find $z_F'$ and $z_H'$ from the equations\n",
    "\\begin{equation}\n",
    "\\operatorname{Im}w(z_E)=0\n",
    "\\end{equation}\n",
    "\\begin{equation}\n",
    "\\operatorname{Im}w(z_G)=0\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def find_zero_points(z_branch_points, z_zero_points_initial, z_zero_point_types, constraint_point_ids, constraint_types):\n",
    "    def constraint(t_zero_points):\n",
    "        w_branch_points, w_zero_points = w_special_points(z_branch_points, \\\n",
    "        [t if point_type=='real' else 1j*t for t,point_type in zip(t_zero_points, z_zero_point_types)]) \n",
    "        z_B,z_C,z_D,z_E,z_F,z_G,z_H,z_I = z_branch_points           \n",
    "        w_B,w_C,w_D,w_E,w_F,w_G,w_H,w_I = w_branch_points\n",
    "        constraint_values = \\\n",
    "        [w_branch_points[constraint_point_id[0]].real-w_branch_points[constraint_point_id[1]].real \\\n",
    "         if constraint_types[constraint_id] == 'real' else \\\n",
    "         w_branch_points[constraint_point_id[0]].imag-w_branch_points[constraint_point_id[1]].imag \\\n",
    "         for constraint_id, constraint_point_id in enumerate(constraint_point_ids)]\n",
    "        return constraint_values\n",
    "        \n",
    "    return root(constraint, z_zero_points_initial).x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Derivative of the conformal transform (defines the electric and magnetic fields)\n",
    "It turns out that the real part of $w(z)$ makes a good candidate for $\\psi(x,y)$, too. The corresponding electric and magnetic fields can be calculated as their derivatives, which is even easier than the potentials, since we don't need to evaulate the integral for them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def conformal_transform_derivative(z,z_branch_points,z_zero_points):\n",
    "    result = 1\n",
    "    for branch_point_id, branch_point in enumerate(z_branch_points):\n",
    "        result /= np.sqrt(z-branch_point)\n",
    "    for zero_point_id, zero_point in enumerate(z_zero_points):\n",
    "        result *= (z-zero_point)\n",
    "    return result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Defining geometric properties of the coplanar waveguide coupler\n",
    "Let's for example take two 50-Ohm lines with different cross-section scales."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "w = 10\n",
    "s = 6\n",
    "s1 = 6\n",
    "s2 = 12\n",
    "w1 = 10\n",
    "w2 = 20\n",
    "w3 = 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Evaluating the conformal transform special points (zeros and branch points) if when of the nodes has a potential and all others are grounded (z-space & w-space)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "z_branch_points = z_branch_points_cpw(s1,s2,w1,w2,w3)\n",
    "z_zero_points_initial = [(z_branch_points[4]+z_branch_points[5]).real/2, \\\n",
    "                         (z_branch_points[6]+z_branch_points[7]).real/2]\n",
    "z_zero_points = find_zero_points(z_branch_points, z_zero_points_initial, ['real', 'real'], \\\n",
    "                                        [(3,0),(5,0)], ['imag', 'imag'])\n",
    "w_branch_points, w_zero_points = w_special_points(z_branch_points, z_zero_points)\n",
    "z_B,z_C,z_D,z_E,z_F,z_G,z_H,z_I = z_branch_points\n",
    "w_B,w_C,w_D,w_E,w_F,w_G,w_H,w_I = w_branch_points\n",
    "z_F1,z_H1 = z_zero_points\n",
    "w_F1,w_H1 = w_zero_points"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# plotting conformal transform & fields"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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dWPkeeylCQ8HNmsljNfPfmjTRD4za7bKrr6JCtjZx9Xn1sbAQlhUt4MXlfHr2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QAAAg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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x843ea58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-(2*s2+2*s1+w1+w2+w3)*0.5,(2*s2+2*s1+w1+w2+w3)*1.5,251)\n",
    "y = np.linspace(0, (np.max(x)-np.min(x))*0.4, 201)\n",
    "\n",
    "plt.figure(figsize=(10,5))\n",
    "plt.clf()\n",
    "axes=plt.subplot(2,1,1)\n",
    "vertices = [i for i in z_zero_points] + [i for i in z_branch_points]\n",
    "lines = [(z_B-100,z_B,'-'),(z_B,z_C,':'),(z_C,z_D,'-'),(z_D,z_E,':'),(z_E,z_F,'-'),\\\n",
    "         (z_F,z_F1,':'),(z_F1,z_G,':'),(z_G,z_H,'-'),(z_H,z_H1,':'),(z_H1,z_I,':'),(z_I,z_I+100,'-')]\n",
    "for z1,z2,style in lines:\n",
    "    plt.plot([z1.real, z2.real], [z1.imag, z2.imag], color='black', linestyle=style, linewidth=(3 if style=='-' else 1))\n",
    "\n",
    "plt.xlim(np.min(x), np.max(x))\n",
    "plt.ylim(-0.5*np.max(y), np.max(y))\n",
    "\n",
    "plt.scatter([j.real for j in vertices], [j.imag for j in vertices])\n",
    "\n",
    "[xm,ym] = np.meshgrid(x,y)\n",
    "um = conformal_transform_derivative(xm+1j*ym,z_branch_points,z_zero_points).real\n",
    "vm = conformal_transform_derivative(xm+1j*ym,z_branch_points,z_zero_points).imag\n",
    "wm = np.zeros(um.shape, dtype=np.complex)\n",
    "plt.streamplot(x,y,vm,um)\n",
    "\n",
    "annotations = ['B', 'C', 'D', 'E', 'F', 'F\\'', 'G', 'H', 'H\\'', 'I']\n",
    "annotation_positions = [(a*0.10+0.05, 0.05) for a in range(10)]\n",
    "annotation_points = [z_B,z_C,z_D,z_E,z_F,z_F1,z_G,z_H,z_H1,z_I]\n",
    "\n",
    "for annotation, annotation_position, annotation_point in zip(annotations, annotation_positions, annotation_points):\n",
    "    axes.annotate(annotation, xy=(annotation_point.real, annotation_point.imag), xytext=annotation_position, \n",
    "              xycoords='data',  textcoords='axes fraction', arrowprops={'facecolor':'black', 'shrink':0.05})\n",
    "    \n",
    "    \n",
    "axes=plt.subplot(2,1,2)\n",
    "vertices = [i for i in w_zero_points] + [i for i in w_branch_points]\n",
    "lines = [(w_B,w_C,':'),(w_C,w_D,'-'),(w_D,w_E,':'),(w_E,w_F,'-'),\\\n",
    "         (w_F,w_F1,':'),(w_F1,w_G,':'),(w_G,w_H,'-'),(w_H,w_H1,':'),(w_H1,w_I,':'),(w_I,w_B,'-')]\n",
    "for z1,z2,style in lines:\n",
    "    plt.plot([z1.real, z2.real], [z1.imag, z2.imag], color='black', linestyle=style, linewidth=(3 if style=='-' else 1))\n",
    "    \n",
    "plt.scatter([j.real for j in vertices], [j.imag for j in vertices])\n",
    "\n",
    "wx = np.linspace(w_E.real, w_B.real, 251)\n",
    "wy = np.linspace(w_B.imag, w_C.imag, 101)\n",
    "wxm,wym = np.meshgrid(wx,wy)\n",
    "wvm = 0*np.ones(wxm.shape)\n",
    "wum = 1*np.ones(wxm.shape)\n",
    "plt.streamplot(wxm,wym,wvm,wum)\n",
    "\n",
    "plt.xlim([w_E.real-(w_B.real-w_E.real)*0.2, w_B.real+(w_B.real-w_E.real)*0.2])\n",
    "plt.ylim([w_B.imag-(w_C.imag-w_B.imag)*0.4, w_C.imag+(w_C.imag-w_B.imag)*0.3])\n",
    "\n",
    "annotations = ['B', 'C', 'D', 'E', 'F', 'F\\'', 'G', 'H', 'H\\'', 'I']\n",
    "annotation_positions = [(0.95, 0.05), (0.95, 0.75), (0.05, 0.75), (0.05, 0.05), (0.125, 0.05), \\\n",
    "                        (0.20, 0.05), (0.275, 0.05), (0.35, 0.05), (0.425, 0.05), (0.5, 0.05),]\n",
    "annotation_points = [w_B,w_C,w_D,w_E,w_F,w_F1,w_G,w_H,w_H1,w_I]\n",
    "\n",
    "for annotation, annotation_position, annotation_point in zip(annotations, annotation_positions, annotation_points):\n",
    "    axes.annotate(annotation, xy=(annotation_point.real, annotation_point.imag), xytext=annotation_position, \n",
    "              xycoords='data',  textcoords='axes fraction', arrowprops={'facecolor':'black', 'shrink':0.05})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Putting the potential on each of the waveguides and evaluating the capacitance matrix vector\n",
    "The elements of the capacitance matrix $C_{ij}$ are defined as the charge induced on on the conductor $i$ if the conductor $j$ is given a unit potential while all other conductors are grounded:\n",
    "\\begin{equation}\n",
    "q_i=C_{ij}\\phi_j\n",
    "\\end{equation}\n",
    "Similarly, the inductance matrix elements $L_{ij}$ are defined by the flux through the loop $i$ when a unit current is circulating in the loop $j$:\n",
    "\\begin{equation}\n",
    "\\Phi_i=L_{ij}I_j\n",
    "\\end{equation}\n",
    "Since we are dealing with a 2D cross-section, we have:\n",
    "* conductor cross-section instead of loops and conductors (1D cross-section of 2D surfaces),\n",
    "* linear charge density instead of charge\n",
    "* linear flux density instead of flux\n",
    "* linear capactance density instead of capacitance\n",
    "* linear inductance density instead of inductance\n",
    "\n",
    "In order to get the \"real thing\" we should multiply the linear densities by the length of the CPW.\n",
    "\n",
    "The following code solves the equations of $z_F'$ and $z_H'$ and fills in the first column-vectors of $C_{ij}$ with $(\\frac{\\operatorname{Re}w(z_C)-\\operatorname{Re}w(z_D)}{\\operatorname{Im}w(z_C)}, \\frac{\\operatorname{Re}w(z_G)-\\operatorname{Re}w(z_H)}{\\operatorname{Im}w(z_C)})$. After that, it exchanges the first CPW with the second one, solves again and fills in the second column-vectors. The dimensionless capacitance and inductance matrices are inverses of each other: $L_{ij} = C^{-1}_{ij}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "z_branch_points = z_branch_points_cpw(s1,s2,w1,w2,w3)\n",
    "z_zero_points_initial = [(z_branch_points[4]+z_branch_points[5]).real/2, \\\n",
    "                         (z_branch_points[6]+z_branch_points[7]).real/2]\n",
    "z_zero_points = find_zero_points(z_branch_points, z_zero_points_initial, ['real', 'real'], \\\n",
    "                                        [(3,0),(5,0)], ['imag', 'imag'])\n",
    "w_branch_points, w_zero_points = w_special_points(z_branch_points, z_zero_points)\n",
    "w_B,w_C,w_D,w_E,w_F,w_G,w_H,w_I = w_branch_points\n",
    "c1_ = [(w_C.real-w_D.real)/(w_C.imag-w_B.imag), (w_G.real-w_H.real)/(w_C.imag-w_B.imag)]\n",
    "\n",
    "z_branch_points = z_branch_points_cpw(s2,s1,w2,w1,w3)\n",
    "z_zero_points_initial = [(z_branch_points[4]+z_branch_points[5]).real/2, \\\n",
    "                         (z_branch_points[6]+z_branch_points[7]).real/2]\n",
    "z_zero_points = find_zero_points(z_branch_points, z_zero_points_initial, ['real', 'real'], \\\n",
    "                                        [(3,0),(5,0)], ['imag', 'imag'])\n",
    "w_branch_points, w_zero_points = w_special_points(z_branch_points, z_zero_points)\n",
    "w_B,w_C,w_D,w_E,w_F,w_G,w_H,w_I = w_branch_points\n",
    "c2_ = [(w_G.real-w_H.real)/(w_C.imag-w_B.imag), (w_C.real-w_D.real)/(w_C.imag-w_B.imag)]\n",
    "\n",
    "C_dimensionless = np.asarray([c1_, c2_])\n",
    "L_dimensionless = np.linalg.inv(np.asarray(C_dimensionless))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Converting dimensionless capacitance and inductance to SI and multipling by relative permittivities\n",
    "The effective relative permittivity of a dielectric substrate is given by\n",
    "\\begin{equation}\n",
    "\\epsilon_{\\mathrm{eff}}=\\frac{\\sqrt{\\epsilon_{xx}\\epsilon_{yy}}+1}{2}\n",
    "\\end{equation}\n",
    "Since we have only solved Laplace's equation in the upper half-plane, we have overestimated the inductance and underestimated the actual capacitance by a factor of two. The resulting capacitance and inductance matrices in SI units are caluclated below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "epsilon_eff = 6.45\n",
    "C_l = 2*C_dimensionless*epsilon_0*epsilon_eff\n",
    "L_l = L_dimensionless*mu_0/2\n",
    "coupling_matrix = L_dimensionless*mu_0/2/np.sqrt(epsilon_0*mu_0*epsilon_eff)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mutual linear capacitance density matrix, F/m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  1.68853488e-10,  -1.69303012e-11],\n",
       "       [ -1.69303011e-11,   1.68187296e-10]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C_l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mutual linear inductance density matrix, Hn/m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  4.29352441e-07,   4.32200666e-08],\n",
       "       [  4.32200665e-08,   4.31053112e-07]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L_l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mutual \"impedance\" matrix, Ω\n",
    "This matrix is essentially the linear inductance matrix in resistance units."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 50.68212109,   5.10183345],\n",
       "       [  5.10183344,  50.88287366]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coupling_matrix"
   ]
  }
 ],
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