Thermal noise calculation for QUAK/ESR

Thermal noise.ipynb

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   "source": [
    "# Thermal noise\n",
    "\n",
    "## Thermal noise in a Ohmic resistor\n",
    "\n",
    "One sided power spectral density (mean square voltage variance per Hertz of bandwidth) for Ohmic resistor $R$:\n",
    "\n",
    "$\n",
    "\\bar{v_n^2} = 4 k_B T R\n",
    "$\n",
    "\n",
    "This means for a bandwidth of $f_{\\text{bw}}$:\n",
    "\n",
    "$\n",
    "v_n = \\sqrt{\\bar{v_n^2}} * \\sqrt{f_{\\text{bw}}}\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "37f73329",
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "kb = 1.38e-23\n",
    "\n",
    "def oneSidedPowerSpectralDensity(R, *, T = 300):\n",
    "    return 4 * kb * T * R\n",
    "\n",
    "def thermalNoiseRMS(R, bandwidthHz, *, T = 300):\n",
    "    return math.sqrt(oneSidedPowerSpectralDensity(R, T=T)) * math.sqrt(bandwidthHz)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0fa28502",
   "metadata": {},
   "source": [
    "Example: A $50 \\Omega$ resistor at $300 K$ would have a one sided spectral power density of $0.9 \\frac{\\text{nV}}{\\sqrt{\\text{Hz}}}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "0a191199",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.099450532861861e-10"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "math.sqrt(oneSidedPowerSpectralDensity(50))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e9e17a13",
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   "source": [
    "def valueWithSiPrefix(value):\n",
    "    if value > 1e15:\n",
    "        return (value / 1.0e15, \"P\")\n",
    "    if value > 1e12:\n",
    "        return (value / 1.0e12, \"T\")\n",
    "    if value > 1e9:\n",
    "        return (value / 1.0e9, \"G\")\n",
    "    if value > 1e6:\n",
    "        return (value / 1.0e6, \"M\")\n",
    "    if value > 1e3:\n",
    "        return (value / 1.0e3, \"k\")\n",
    "    if value < 1e-15:\n",
    "        return (value * 1.0e18, \"a\")\n",
    "    if value < 1e-12:\n",
    "        return (value * 1.0e15, \"f\")\n",
    "    if value < 1e-9:\n",
    "        return (value * 1.0e12, \"p\")\n",
    "    if value < 1e-6:\n",
    "        return (value * 1.0e9, \"n\")\n",
    "    if value < 1e-3:\n",
    "        return (value * 1.0e6, \"u\")\n",
    "    if value < 1:\n",
    "        return (value * 1.0e3, \"m\")\n",
    "    return (value, \"\")\n",
    "\n",
    "def printValueWithSiPrefix(value, baseunit):\n",
    "    val, prefix = valueWithSiPrefix(value)\n",
    "    print(\"{} {}{}\".format(val, prefix, baseunit))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "91a63af0",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f3dbBandwidth(timeConstant):\n",
    "    return 1.0 / (2.0 * math.pi * timeConstant)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebb9a40b",
   "metadata": {},
   "source": [
    "## Time constant of filters\n",
    "\n",
    "### RC-Lowpass\n",
    "\n",
    "A simple filter with R in series, C as shunt. This has a time constant of $\\tau = R * C$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "36847c65",
   "metadata": {},
   "outputs": [],
   "source": [
    "def rcLowpassTimeConstant(R, C):\n",
    "    return R * C\n",
    "def rcLowpassBandwidth(R, C):\n",
    "    return f3dbBandwidth(R * C)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f762009b",
   "metadata": {},
   "source": [
    "## Our QUAK/ESR setup"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5eb6edb",
   "metadata": {},
   "source": [
    "Our lock in amplifier at QUAK/ESR has a time constant of $100 ms = 0.1s$ and thus a bandwidth of $10 Hz$.\n",
    "\n",
    "We have to account for the factor of $2 \\pi$ since the bandwidth specification convention for a system is the point where the power drops about to half ($- 3 dB$):\n",
    "\n",
    "$\n",
    "f_{3dB} = \\frac{1}{2 \\pi \\tau}\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "316b675f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time constant of quak/esr lock in: 0.1\n",
      "Bandwidth of quak/esr lock in: 1.5915494309189535\n",
      "Expected thermal noise: 1.147955978598872 nV\n"
     ]
    }
   ],
   "source": [
    "quakEsrLockInTimeConstant = 100e-3\n",
    "\n",
    "quakEsrThermalNoise = thermalNoiseRMS(50, f3dbBandwidth(quakEsrLockInTimeConstant))\n",
    "quakEsrThermalNoiseSI, quakEsrThermalNoiseUnit = valueWithSiPrefix(quakEsrThermalNoise)\n",
    "print(\"Time constant of quak/esr lock in: {}\".format(quakEsrLockInTimeConstant))\n",
    "print(\"Bandwidth of quak/esr lock in: {}\".format(f3dbBandwidth(quakEsrLockInTimeConstant)))\n",
    "print(\"Expected thermal noise: {} {}V\".format(quakEsrThermalNoiseSI, quakEsrThermalNoiseUnit))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14847562",
   "metadata": {},
   "source": [
    "We have two or three amplifiers.\n",
    "\n",
    "### Two amplifiers (2x ZFL-500LN+)\n",
    "\n",
    "In case we're using 2x ZFL-500LN+ each has $20 dB$ gain, thus a voltage factor of $10$ per amplifier, i.e. voltage gain of $100$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "beb941bd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "229.5911957197744 nV\n"
     ]
    }
   ],
   "source": [
    "printValueWithSiPrefix(2 * 100 * quakEsrThermalNoise, \"V\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db7bc874",
   "metadata": {},
   "source": [
    "### Three amplifiers (2x ZFL-500LN+, 1x ZX60-P103LN+)\n",
    "\n",
    "When we're using three amplifiers we add a third ```ZX60-P103LN+``` to our setup (at the output stage) - this has a gain of $24 dB$ thus a voltage gain of around $16.85$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "e31e5f2f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.8686116478781987 uV\n"
     ]
    }
   ],
   "source": [
    "printValueWithSiPrefix(2 * 100 * 16.85 * quakEsrThermalNoise, \"V\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9c314bd",
   "metadata": {},
   "source": [
    "### Three amplifiers plus 3dB attenuator (2x ZFL-500LN+, -3dB attenuator,  1x ZX60-P103LN+)\n",
    "\n",
    "Adding a 3 dB attenuator in front of the ```ZX60-P103LN+``` would induce a voltage gain factor of $0.70$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "c5bd1cfa",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.7080281535147392 uV\n"
     ]
    }
   ],
   "source": [
    "printValueWithSiPrefix(2 * 100 * 16.85 * 0.7 * quakEsrThermalNoise, \"V\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "23aa8ef9",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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