{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Circuit Models 101\n",
    "\n",
    "In this notebook, we will explore the basics of circuit models in AutoEIS. We will start by importing the necessary libraries."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:03.780573Z",
     "iopub.status.busy": "2025-08-25T23:17:03.780464Z",
     "iopub.status.idle": "2025-08-25T23:17:05.529629Z",
     "shell.execute_reply": "2025-08-25T23:17:05.528940Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import autoeis as ae\n",
    "ae.visualization.set_plot_style()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Circuit representation\n",
    "\n",
    "In AutoEIS, circuits are represented as strings. Please refer to [circuit notation](../circuit.md) for the syntax of the circuit string. Now, let's create a sample circuit string:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:05.533147Z",
     "iopub.status.busy": "2025-08-25T23:17:05.532743Z",
     "iopub.status.idle": "2025-08-25T23:17:05.539588Z",
     "shell.execute_reply": "2025-08-25T23:17:05.538739Z"
    }
   },
   "outputs": [],
   "source": [
    "circuit = \"R1-[P2,R3]-C4-[[R5,C6],[L7,R8]]-R2\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can visualize the circuit model using the `draw_circuit` function, which requires the `lcapy` package to be installed, and a working LaTeX installation. See [here](https://lcapy.readthedocs.io/en/latest/install.html) for more details."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:05.542223Z",
     "iopub.status.busy": "2025-08-25T23:17:05.542031Z",
     "iopub.status.idle": "2025-08-25T23:17:11.687589Z",
     "shell.execute_reply": "2025-08-25T23:17:11.686722Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/runner/work/AutoEIS/AutoEIS/.venv/lib/python3.10/site-packages/lcapy/__init__.py:76: UserWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html. The pkg_resources package is slated for removal as early as 2025-11-30. Refrain from using this package or pin to Setuptools<81.\n",
      "  import pkg_resources\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Suggestion: add a constraint between nodes (25, 27) and (28, 26) for vertical graph\n"
     ]
    },
    {
     "data": {
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     },
     "metadata": {
      "image/png": {
       "height": 688,
       "width": 1787
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = ae.visualization.draw_circuit(circuit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Querying circuit strings\n",
    "\n",
    "Once you have the circuit string, you can run different queries on it. We're not going to explore all of available queries here, but we'll show you a few of the most common ones. To see the full list of available queries, check out the [API reference](../modules.rst).\n",
    "\n",
    "To get the list of components in the circuit, you can use the `get_component_labels` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:11.694364Z",
     "iopub.status.busy": "2025-08-25T23:17:11.693405Z",
     "iopub.status.idle": "2025-08-25T23:17:11.698471Z",
     "shell.execute_reply": "2025-08-25T23:17:11.698051Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['R1', 'P2', 'R3', 'C4', 'R5', 'C6', 'L7', 'R8', 'R2']"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ae.parser.get_component_labels(circuit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(The impedance of) Each component is represented by one or more parameters. To get the list of parameters that fully describe the circuit, use the `get_parameter_labels` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:11.700804Z",
     "iopub.status.busy": "2025-08-25T23:17:11.700645Z",
     "iopub.status.idle": "2025-08-25T23:17:11.704158Z",
     "shell.execute_reply": "2025-08-25T23:17:11.703408Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['R1', 'P2w', 'P2n', 'R3', 'C4', 'R5', 'C6', 'L7', 'R8', 'R2']"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ae.parser.get_parameter_labels(circuit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that components and parameters are not the same, despite the fact that for single-parameter components they're represented by the same string. For instance, `R1` is both a parameter and a component, but `P2` is a component, which is described by the parameters `P2w` and `P2n`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also validate the circuit using the `validate_circuit` function in case you're not sure if the circuit is valid:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:11.706247Z",
     "iopub.status.busy": "2025-08-25T23:17:11.706132Z",
     "iopub.status.idle": "2025-08-25T23:17:11.709790Z",
     "shell.execute_reply": "2025-08-25T23:17:11.708985Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ae.parser.validate_circuit(circuit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try to validate an invalid circuit string:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:11.720279Z",
     "iopub.status.busy": "2025-08-25T23:17:11.720056Z",
     "iopub.status.idle": "2025-08-25T23:17:12.054314Z",
     "shell.execute_reply": "2025-08-25T23:17:12.053001Z"
    },
    "tags": [
     "raises-exception"
    ]
   },
   "outputs": [
    {
     "ename": "AssertionError",
     "evalue": "Duplicate elements found: {'R1'}",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[7], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[43mae\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mparser\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mvalidate_circuit\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mR1-[R2,P3]-R1\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m~/work/AutoEIS/AutoEIS/src/autoeis/parser.py:58\u001b[0m, in \u001b[0;36mvalidate_circuit\u001b[0;34m(circuit)\u001b[0m\n\u001b[1;32m     56\u001b[0m components \u001b[38;5;241m=\u001b[39m get_component_labels(circuit)\n\u001b[1;32m     57\u001b[0m duplicates \u001b[38;5;241m=\u001b[39m [e \u001b[38;5;28;01mfor\u001b[39;00m e \u001b[38;5;129;01min\u001b[39;00m components \u001b[38;5;28;01mif\u001b[39;00m components\u001b[38;5;241m.\u001b[39mcount(e) \u001b[38;5;241m>\u001b[39m \u001b[38;5;241m1\u001b[39m]\n\u001b[0;32m---> 58\u001b[0m \u001b[38;5;28;01massert\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m duplicates, \u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mDuplicate elements found: \u001b[39m\u001b[38;5;132;01m{\u001b[39;00m\u001b[38;5;28mset\u001b[39m(duplicates)\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m     59\u001b[0m \u001b[38;5;66;03m# Test circuit is not empty\u001b[39;00m\n\u001b[1;32m     60\u001b[0m \u001b[38;5;28;01massert\u001b[39;00m \u001b[38;5;28mlen\u001b[39m(circuit) \u001b[38;5;241m>\u001b[39m \u001b[38;5;241m0\u001b[39m, \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mCircuit string is empty.\u001b[39m\u001b[38;5;124m\"\u001b[39m\n",
      "\u001b[0;31mAssertionError\u001b[0m: Duplicate elements found: {'R1'}"
     ]
    }
   ],
   "source": [
    "ae.parser.validate_circuit(\"R1-[R2,P3]-R1\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another useful query is to compare two circuits to see if they are structurally equivalent. For instance, one would expect that `R1-R2` and `R2-R1` and `R5-R0` are equivalent, i.e., neither the order of appearance of the components nor the labels matter. This is useful for filtering out duplicate circuits, which may (and will) arise during circuit generation using evolutionary algorithms. You can do this using the `are_circuits_equivalent` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:12.059008Z",
     "iopub.status.busy": "2025-08-25T23:17:12.058838Z",
     "iopub.status.idle": "2025-08-25T23:17:12.069806Z",
     "shell.execute_reply": "2025-08-25T23:17:12.069198Z"
    }
   },
   "outputs": [],
   "source": [
    "circuit1 = \"R1-[P2,R3]-C4\"\n",
    "circuit2 = \"C4-R1-[R3,P2]\"\n",
    "circuit3 = \"C0-R5-[R9,P0]\"\n",
    "\n",
    "assert ae.utils.are_circuits_equivalent(circuit1, circuit2)\n",
    "assert ae.utils.are_circuits_equivalent(circuit1, circuit3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Evaluating circuit strings\n",
    "\n",
    "Once you have a valid circuit string, you can calculate the EIS spectra of the circuit model by evaluating it at the frequency range of interest. To do this, you need to convert the circuit string to a function using the `generate_circuit_fn` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:12.071939Z",
     "iopub.status.busy": "2025-08-25T23:17:12.071817Z",
     "iopub.status.idle": "2025-08-25T23:17:12.077977Z",
     "shell.execute_reply": "2025-08-25T23:17:12.077351Z"
    }
   },
   "outputs": [],
   "source": [
    "circuit_fn = ae.utils.generate_circuit_fn(circuit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's calculate the EIS spectra for a few frequencies. For this, we also need to pass the parameters of the circuit model, for which we can use random values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-08-25T23:17:12.080831Z",
     "iopub.status.busy": "2025-08-25T23:17:12.080468Z",
     "iopub.status.idle": "2025-08-25T23:17:12.088948Z",
     "shell.execute_reply": "2025-08-25T23:17:12.088003Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2.47101063-1.90324214e+02j, 2.45090467-4.10119181e+01j,\n",
       "       2.45102511-8.82122947e+00j, 2.47344142-1.90984378e+00j,\n",
       "       2.45028677-4.44860703e-01j, 2.40214456-1.54854868e-01j,\n",
       "       2.31473269-8.41415766e-02j, 2.26697117-4.92385197e-02j,\n",
       "       2.22847868-4.17354044e-02j, 2.18939631-4.07371620e-02j])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "freq = np.logspace(-3, 3, 10)\n",
    "num_params = ae.parser.count_parameters(circuit)\n",
    "p = np.random.rand(num_params)\n",
    "Z = circuit_fn(freq, p)\n",
    "Z"
   ]
  }
 ],
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