{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "a21c989f",
   "metadata": {},
   "source": [
    "<!--NOTEBOOK_HEADER-->\n",
    "*This notebook contains material from [cbe61622](https://jckantor.github.io/cbe61622);\n",
    "content is available [on Github](https://github.com/jckantor/cbe61622.git).*\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa04270e",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
    "< [3.0 Analog Measurement](https://jckantor.github.io/cbe61622/03.00-Analog-Measurement.html) | [Contents](toc.html) | [3.2 Sampling Continuous Time Signals](https://jckantor.github.io/cbe61622/03.02-Sampling-Continuous-Time-Signals.html) ><p><a href=\"https://colab.research.google.com/github/jckantor/cbe61622/blob/master/docs/03.01-Low-Level-Measurment.ipynb\"> <img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a><p><a href=\"https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.ipynb\"> <img align=\"left\" src=\"https://img.shields.io/badge/Github-Download-blue.svg\" alt=\"Download\" title=\"Download Notebook\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5977003-b449-4c9f-aa41-3a5adbcc1c58",
   "metadata": {
    "nbpages": {
     "level": 1,
     "link": "[3.1 Low-Level Measurement](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1-Low-Level-Measurement)",
     "section": "3.1 Low-Level Measurement"
    }
   },
   "source": [
    "# 3.1 Low-Level Measurement"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6aa18f61-bd85-4726-a9e4-31d9ddfbff75",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.1.1 References](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1.1-References)",
     "section": "3.1.1 References"
    }
   },
   "source": [
    "## 3.1.1 References\n",
    "\n",
    "* [Low Level Measurements Handbook - 7th Edition (Kiethley)](https://download.tek.com/document/LowLevelHandbook_7Ed.pdf)\n",
    "* [Art of Electronics, Chapter 8]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87908ad3-4c63-4e11-8d69-316cb8ea615a",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.1.2 Limits to Measurements](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1.2-Limits-to-Measurements)",
     "section": "3.1.2 Limits to Measurements"
    }
   },
   "source": [
    "## 3.1.2 Limits to Measurements\n",
    "\n",
    "Sources of noise\n",
    "\n",
    "White noise:\n",
    "\n",
    "* *Shot noise* from the discrete nature of charge carriers\n",
    "* *Johnson noise* caused by thermal fluctuations\n",
    "\n",
    "Other noise:\n",
    "\n",
    "* *Flicker 1/f noise* from a variety of sources\n",
    "* .... and many more!\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8ba870cf-5be5-4450-aefc-3dbbb57523bf",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.1.2.1 Shot Noise](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1.2.1-Shot-Noise)",
     "section": "3.1.2.1 Shot Noise"
    }
   },
   "source": [
    "### 3.1.2.1 Shot Noise\n",
    "\n",
    "* Description and theory, Walter Schottky, 1918. \n",
    "* [Poisson Process - Wolfram MathWorld](https://mathworld.wolfram.com/PoissonProcess.html)\n",
    "\n",
    "Shot noise is the noise due to descrete nature of electric charge. Shot noise can be modeled as a Poisson process. A fundamental consideration in electrical and optical devices including photography.\n",
    "\n",
    "In a Poisson process:\n",
    "\n",
    "* Poisson processes model the occurrence of independent events that occur one at a time.\n",
    "* The probability of exactly one event in a sufficiently short interval $h$ is $P = \\nu h$ where the constant $\\nu$ is the average rate of events.\n",
    "* Each interval is a Bernoulli trial. The number of events in a time period is given by\n",
    "\n",
    "$$P(k) = \\frac{(\\nu h)^k}{k!} e^{-\\nu h}$$\n",
    "* $\\mu = \\nu h$ is the expected number of events in the interval $h$\n",
    "* $\\sigma^2 = \\nu h$ is also the variance in the number of events in interval $h$.\n",
    "* The waiting time between events follows an exponential distribution. The probability of waiting longer than $t$ is \n",
    "\n",
    "$$P(t > h) = e^{-\\nu h}$$\n",
    "\n",
    "or less that $t$\n",
    "\n",
    "$$P(t \\leq h) = 1 - e^{-\\nu h}$$\n",
    "\n",
    "This last expression is the cumulative distribution function. The probably density function is the derivative\n",
    "\n",
    "$$f(h) = \\nu e ^ {-\\nu h}$$\n",
    "\n",
    "**Frequency domain**\n",
    "\n",
    "[A frequency-domain derivation of shot noise](https://core.ac.uk/download/pdf/33125817.pdf)\n",
    "\n",
    "$$i_{noise} = \\sqrt{2 q I_{dc}}$$\n",
    "\n",
    "where $q = 1/C$ is the electron charge $1.60\\times 10^{-19}$ C.\n",
    "\n",
    "**Signal to Noise Ratio**\n",
    "\n",
    "Signal to noise ratio is normally expressed as the power of the signal divided by the power of the noise. \n",
    "\n",
    "$$\\text{SNR} = \\frac{\\mu^2}{\\sigma^2}$$\n",
    "\n",
    "If the interval is $\\delta t$ the current is $I$, rate of charge carriers is $C I h$ where $C$ is Coulomb constant. The SNR is then\n",
    "\n",
    "$$\\text{SNR} = \\frac{(C I \\delta t)^2}{C I \\delta t} = C I \\delta t$$\n",
    "\n",
    "Question: Suppose you need 5 sigma accuracy (i.e, $\\frac{\\mu}{\\sigma} > 5$) and are measuring a 1 nA signal, what is the fastest sampling rate you can expect?\n",
    "\n",
    "**Simulated Photon Noise**\n",
    "\n",
    "A sequence of images in which the average number of photons captured per pixel increases by factors of 10x between images. [source](https://commons.wikimedia.org/wiki/File:Photon-noise.jpg)\n",
    "\n",
    "![](https://upload.wikimedia.org/wikipedia/commons/4/42/Photon-noise.jpg)\n",
    "\n",
    "\n",
    "Question:\n",
    "\n",
    "1. Which image has the most noise?\n",
    "2. Which image has the lower signal to noise ratio?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "8c8afde4-2fd9-4538-b575-de57f003b82c",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.1.2.1 Shot Noise](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1.2.1-Shot-Noise)",
     "section": "3.1.2.1 Shot Noise"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "nu = 6241509.074 electrons per second\n",
      "expected number of electrons per step nu*h = 6.241509074\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'seconds')"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Demonstration\n",
    "# \n",
    "# Simulate shot noise for a 1 pA current\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "import random\n",
    "import math\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# constants\n",
    "C = 6.241509074e18     # number of electrons in a Coulomb\n",
    "I = 1e-12              # 1 pA of current\n",
    "h = 1e-6               # 1 microsecond\n",
    "\n",
    "nu = C*I\n",
    "print(\"nu =\", nu, \"electrons per second\") \n",
    "print(\"expected number of electrons per step nu*h =\", nu*h)\n",
    "\n",
    "K = 600\n",
    "\n",
    "# simulate arrival of K charge carriers\n",
    "dt = np.random.exponential(1/nu, size=(K))\n",
    "t = np.cumsum(dt)\n",
    "\n",
    "# plot results\n",
    "y = np.array([1]*len(t))\n",
    "\n",
    "fig, ax = plt.subplots(2, 1, figsize=(12, 8))\n",
    "ax[0].plot(t, y, '.')\n",
    "\n",
    "ax[1].step(t, np.cumsum(y)/C, where=\"pre\")\n",
    "ax[1].plot(t, I*t)\n",
    "ax[1].set_ylabel('coulombs')\n",
    "ax[1].set_xlabel('seconds')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4d5b4a6-e328-459b-b1f8-4775a7dc5d6c",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.1.2.1 Shot Noise](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1.2.1-Shot-Noise)",
     "section": "3.1.2.1 Shot Noise"
    }
   },
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "91c9ec54-31f6-4dba-83d7-31fab4377ca4",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.1.2.1 Shot Noise](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1.2.1-Shot-Noise)",
     "section": "3.1.2.1 Shot Noise"
    }
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "9a77b302-ede4-4b33-8dbf-d48bfc33cb0d",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.1.2.2 Johnson-Nyquist Noise](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1.2.2-Johnson-Nyquist-Noise)",
     "section": "3.1.2.2 Johnson-Nyquist Noise"
    }
   },
   "source": [
    "### 3.1.2.2 Johnson-Nyquist Noise\n",
    "\n",
    "* Discovered and measured by John B. Johnson, Bell Labs, 1926\n",
    "* Explained by Harry Nyquist, Bell Labs, 1928. \n",
    "\n",
    "Johnson noise has the same origins as black-body radiation. The average (root mean square) voltage due to thermal noise in a resistor is\n",
    "\n",
    "$$v_{noise} = \\sqrt{4kTRB}$$\n",
    "\n",
    "where $k$ is Boltzmann's constant, T is absolute temperature, R is resistance, and B is bandwidth in Hertz. At 20 deg C \n",
    "\n",
    "\\begin{align*}\n",
    "4kT & = 1.62 \\times 10^{-20} & V^2/Hz-\\Omega \\\\\n",
    "\\sqrt{4ktR} & = 1.27\\times 10^{-10}\\sqrt{R} & V/Hz^{1/2}\n",
    "\\end{align*}\n",
    "\n",
    "The short-circuit current noise is\n",
    "\n",
    "$$i_{noise} = v_{noise}/R = \\sqrt{\\frac{4kTB}{R}}$$\n",
    "\n",
    "Question: What is the RMS voltage of 10k ohm resistor?\n",
    "\n",
    "* High resistances results in noisy voltage signals\n",
    "* Low resistance results in noisy currrent signals\n",
    "\n",
    "$$P_{noise} = v_{noise}i_{noise} = 4kTB$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66e727fd-26d4-4837-8e09-5b3377d5e5a4",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.1.3 Chemical Applications](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1.3-Chemical-Applications)",
     "section": "3.1.3 Chemical Applications"
    }
   },
   "source": [
    "## 3.1.3 Chemical Applications\n",
    "\n",
    "* Electrochemical Measurements\n",
    "    * Ion-selective electrodes\n",
    "    * pH measurements\n",
    "    * Conductivity cells\n",
    "    * Potientiostat/Galvanostat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "01f0b7e8-3952-47dc-9885-7527c251e483",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.1.3 Chemical Applications](https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.html#3.1.3-Chemical-Applications)",
     "section": "3.1.3 Chemical Applications"
    }
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "0417db2c",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
    "< [3.0 Analog Measurement](https://jckantor.github.io/cbe61622/03.00-Analog-Measurement.html) | [Contents](toc.html) | [3.2 Sampling Continuous Time Signals](https://jckantor.github.io/cbe61622/03.02-Sampling-Continuous-Time-Signals.html) ><p><a href=\"https://colab.research.google.com/github/jckantor/cbe61622/blob/master/docs/03.01-Low-Level-Measurment.ipynb\"> <img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a><p><a href=\"https://jckantor.github.io/cbe61622/03.01-Low-Level-Measurment.ipynb\"> <img align=\"left\" src=\"https://img.shields.io/badge/Github-Download-blue.svg\" alt=\"Download\" title=\"Download Notebook\"></a>"
   ]
  }
 ],
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