AC Design of 2-Stage Miller Amplifier Explainer.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "2a4712c0",
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext autoreload\n",
    "%autoreload 2\n",
    "\n",
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b50ce458",
   "metadata": {},
   "outputs": [],
   "source": [
    "gm1, go1, CM, gm2, YL, s = sp.symbols(r\"g_{m1} g_{o1} C_M g_{m2} Y_L s\")\n",
    "ro1, RL, GL, CL = sp.symbols(r\"r_{o1} R_L G_L C_L\")\n",
    "w, f = sp.symbols(r'\\omega f')\n",
    "\n",
    "G_exact = sp.Matrix(\n",
    "    [\n",
    "        [0, 0, 0, 1],\n",
    "        [-gm1, go1 + s * CM, -s * CM, 0],\n",
    "        [0, gm2 - s * CM, YL + s * CM, 0],\n",
    "        [1, 0, 0, 0],\n",
    "    ]\n",
    ")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "501986c0",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "\\begin{bmatrix}\n",
    "i_i \\\\\n",
    "i_x \\\\\n",
    "i_o \\\\\n",
    "v_s \n",
    "\\end{bmatrix}\n",
    "&=\n",
    "\\begin{bmatrix}\n",
    "      0 &             0 &        0 & 1 \\\\\n",
    "-g_{m1} & g_{o1} + sC_M &    -sC_M & 0 \\\\\n",
    "      0 & g_{m2} - sC_M & Y_L+sC_M & 0 \\\\\n",
    "      1 &             0 &        0 & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "v_i \\\\\n",
    "v_x \\\\\n",
    "v_o \\\\\n",
    "i_s\n",
    "\\end{bmatrix}\n",
    "\\\\\n",
    "\\begin{bmatrix}\n",
    "0 \\\\\n",
    "0 \\\\\n",
    "0 \\\\\n",
    "1 \n",
    "\\end{bmatrix}\n",
    "&=\n",
    "\\begin{bmatrix}\n",
    "      0 &             0 &        0 & 1 \\\\\n",
    "-g_{m1} & g_{o1} + sC_M &    -sC_M & 0 \\\\\n",
    "      0 & g_{m2} - sC_M & Y_L+sC_M & 0 \\\\\n",
    "      1 &             0 &        0 & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "v_i \\\\\n",
    "v_x \\\\\n",
    "v_o \\\\\n",
    "i_s\n",
    "\\end{bmatrix}\n",
    "\\\\\n",
    "\\vec{\\mathbf{s}} &= \\mathbf{G}\\vec{\\mathbf{r}}\\\\\n",
    "\\mathbf{G}^{-1} \\vec{\\mathbf{s}} &= \\vec{\\mathbf{r}}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "4ed2ce3e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\\\frac{C_{M} g_{m1} s + Y_{L} g_{m1}}{C_{M} Y_{L} s + C_{M} g_{m2} s + C_{M} g_{o1} s + Y_{L} g_{o1}}\\\\\\frac{C_{M} g_{m1} s - g_{m1} g_{m2}}{C_{M} Y_{L} s + C_{M} g_{m2} s + C_{M} g_{o1} s + Y_{L} g_{o1}}\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                                                                                    1],\n",
       "[   (C_M*g_{m1}*s + Y_L*g_{m1})/(C_M*Y_L*s + C_M*g_{m2}*s + C_M*g_{o1}*s + Y_L*g_{o1})],\n",
       "[(C_M*g_{m1}*s - g_{m1}*g_{m2})/(C_M*Y_L*s + C_M*g_{m2}*s + C_M*g_{o1}*s + Y_L*g_{o1})],\n",
       "[                                                                                    0]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_vector = sp.Matrix([0, 0, 0, 1])\n",
    "R_vector = G_exact.inv()*S_vector\n",
    "R_vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "690579a8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2.0 i \\pi C_{M} f g_{m1} - g_{m1} g_{m2}}{2.0 i \\pi C_{M} f g_{m2} + 2.0 i \\pi C_{M} f \\left(2.0 i \\pi C_{L} f + \\frac{1}{R_{L}}\\right) + \\frac{2.0 i \\pi C_{M} f}{r_{o1}} + \\frac{2.0 i \\pi C_{L} f + \\frac{1}{R_{L}}}{r_{o1}}}$"
      ],
      "text/plain": [
       "(2.0*I*pi*C_M*f*g_{m1} - g_{m1}*g_{m2})/(2.0*I*pi*C_M*f*g_{m2} + 2.0*I*pi*C_M*f*(2.0*I*pi*C_L*f + 1/R_L) + 2.0*I*pi*C_M*f/r_{o1} + (2.0*I*pi*C_L*f + 1/R_L)/r_{o1})"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gain = R_vector[2]\n",
    "gain = gain.subs({\n",
    "    go1: 1/ro1,\n",
    "    YL: GL + s*CL\n",
    "})\n",
    "gain = gain.subs({\n",
    "    GL: 1/RL,\n",
    "    s: 2*sp.pi*f*1j\n",
    "})\n",
    "gain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "46cddfcc",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "fn = sp.lambdify(f, gain.subs({\n",
    "    CM: 5e-12,\n",
    "    ro1: 500e3,\n",
    "    RL: 2e3,\n",
    "    CL: 15e-12,\n",
    "    gm1: 10e-3,\n",
    "    gm2: 10e-3,\n",
    "}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "398dcd0e",
   "metadata": {},
   "outputs": [],
   "source": [
    "q = np.logspace(-3, 12, 100)\n",
    "mag = 20*np.log10(np.abs(fn(q)))\n",
    "pha = np.angle(fn(q), deg=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "d20eb71f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "\n",
    "ax = fig.add_subplot(2, 1, 1)\n",
    "ax.plot(q, mag)\n",
    "\n",
    "ax = fig.add_subplot(2, 1, 2, sharex=ax)\n",
    "ax.plot(q, pha)\n",
    "ax.set_xscale('log')"
   ]
  }
 ],
 "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": "3.8.8"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}