{
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   "source": [
    "Jochen Kieninger<br>\n",
    "**Electrochemical Methods for the Micro- and Nanoscale**<br>\n",
    "www.electrochemical-methods.org\n",
    "\n",
    "## Task 7.2 (Electrochemical impedance spectroscopy - Randles circuit)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "4b447e3f-71dd-4cf7-ae78-ef5ab6313e4e",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import re, im, I, simplify, symbols, sqrt, atan"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "97a63ce4-b23b-4018-954c-aea4275bd724",
   "metadata": {},
   "outputs": [],
   "source": [
    "Ru, Rct, Cdl, omega = symbols('R_u R_ct C_dl, omega', real=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "218acdc4-ed87-4611-9fba-e9fcb6f7421e",
   "metadata": {},
   "source": [
    "The complex impedance of a Randles circuit with uncompensated resistance $R_u$ in series with a parallel circuit of the charge transfer resistance $R_{ct}$ and the double layer capacity $C_{dl}$ is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ac6d4468-d746-491f-8971-8b5d7a46802b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle R_{u} + \\frac{1}{i C_{dl} \\omega + \\frac{1}{R_{ct}}}$"
      ],
      "text/plain": [
       "R_u + 1/(I*C_dl*omega + 1/R_ct)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z = Ru+1/(I*omega*Cdl + 1/Rct)\n",
    "Z"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "357b74ea-e3d4-4e0e-b7ff-109f8894ad89",
   "metadata": {},
   "source": [
    "Because of the uncompensated resistance, the equations are not as handsome as one would wish,\n",
    "\n",
    "- Real part:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "07b1a16d-40bb-4e27-ab4a-261a4a12421b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{R_{ct} + R_{u} \\left(C_{dl}^{2} R_{ct}^{2} \\omega^{2} + 1\\right)}{C_{dl}^{2} R_{ct}^{2} \\omega^{2} + 1}$"
      ],
      "text/plain": [
       "(R_ct + R_u*(C_dl**2*R_ct**2*omega**2 + 1))/(C_dl**2*R_ct**2*omega**2 + 1)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(re(Z))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a730f8a-1a5d-4aec-a80b-4af10dd2157b",
   "metadata": {},
   "source": [
    "- Imaginary part:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "b27d29a3-4f4f-4b0d-8198-5a6f03efce1c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{C_{dl} R_{ct}^{2} \\omega}{C_{dl}^{2} R_{ct}^{2} \\omega^{2} + 1}$"
      ],
      "text/plain": [
       "-C_dl*R_ct**2*omega/(C_dl**2*R_ct**2*omega**2 + 1)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(im(Z))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73438eb6-4795-43d2-9991-eee8d279d60c",
   "metadata": {},
   "source": [
    "- Absolute value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8b066e3a-d528-4dbe-86df-3d68732235ff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{C_{dl}^{2} R_{ct}^{4} \\omega^{2} + \\left(R_{ct} + R_{u} \\left(C_{dl}^{2} R_{ct}^{2} \\omega^{2} + 1\\right)\\right)^{2}}}{C_{dl}^{2} R_{ct}^{2} \\omega^{2} + 1}$"
      ],
      "text/plain": [
       "sqrt(C_dl**2*R_ct**4*omega**2 + (R_ct + R_u*(C_dl**2*R_ct**2*omega**2 + 1))**2)/(C_dl**2*R_ct**2*omega**2 + 1)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(sqrt(re(Z)**2 + im(Z)**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b846b6a-bb51-4eda-9e09-6e59c53b66b3",
   "metadata": {},
   "source": [
    "- Phase:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "64112451-8015-4f55-a549-de4ddd4bca85",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\operatorname{atan}{\\left(\\frac{C_{dl} R_{ct}^{2} \\omega}{R_{ct} + R_{u} \\left(C_{dl}^{2} R_{ct}^{2} \\omega^{2} + 1\\right)} \\right)}$"
      ],
      "text/plain": [
       "-atan(C_dl*R_ct**2*omega/(R_ct + R_u*(C_dl**2*R_ct**2*omega**2 + 1)))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(atan(im(Z)/re(Z)))"
   ]
  }
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