Fixed parameter calculation, added explanation

13375a02 · Pascal Engeler · 6ed161e7 · 13375a02 · 13375a02
Commit 13375a02 authored 5 years ago by Pascal Engeler
--- a/parameter_calculation/.ipynb_checkpoints/Untitled-checkpoint.ipynb
+++ b/parameter_calculation/.ipynb_checkpoints/Untitled-checkpoint.ipynb
@@ -2,7 +2,7 @@
 "cells": [
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 137,
   "metadata": {},
   "outputs": [],
   "source": [
@@ -20,11 +20,13 @@
    "\n",
    "Mass: ug\n",
    "\n",
-    "Voltage: V\n",
+    "Voltage: uV\n",
+    "\n",
+    "Current: nA\n",
    "\n",
    "Capacitance: mF\n",
    "\n",
-    "Charge: C\n",
+    "Charge: nC\n",
    "\n",
    "Time: ms\n",
    "\n",
@@ -35,7 +37,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 138,
   "metadata": {},
   "outputs": [],
   "source": [
@@ -44,12 +46,14 @@
    "drum_thickness = 0.3 #um\n",
    "electrode_thickness = 0.05 #um\n",
    "electrode_area = 200.*200. #um^2\n",
-    "drum_gap = 1. #um"
+    "drum_gap = 1. #um\n",
+    "drum_resonance = 10. #kHz\n",
+    "drum_Q = 10000. #no units"
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 139,
   "metadata": {},
   "outputs": [],
   "source": [
@@ -57,57 +61,185 @@
    "rho_si3n4 = 3.44e-6 #ug/um^3\n",
    "rho_au = 1.93e-5 #ug/um^3\n",
    "coulomb_k = 9.0e12 #um/mF\n",
-    "permittivity = 8.85e-15 #mF/um"
+    "permittivity = 8.85e-15 #mF/um or nN/(mV)^2"
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 140,
   "metadata": {},
-   "outputs": [],
-   "source": []
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Si3N4 Mass: 0.8105309046261666\n",
+      "Au Mass: 0.1544\n",
+      "Drum Mass: 0.9649309046261666\n"
+     ]
+    }
+   ],
+   "source": [
+    "#Drum mass\n",
+    "m_si3n4 = (0.5*drum_diameter)**2*3.141592653589793*drum_thickness*rho_si3n4\n",
+    "m_au = 4.*electrode_area*electrode_thickness*rho_au\n",
+    "m = m_si3n4 + m_au\n",
+    "print(f'Si3N4 Mass: {m_si3n4}')\n",
+    "print(f'Au Mass: {m_au}')\n",
+    "print(f'Drum Mass: {m}')"
+   ]
  },
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 141,
   "metadata": {},
-   "outputs": [],
-   "source": []
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Capacitance: 3.54e-10 nC/uV\n"
+     ]
+    }
+   ],
+   "source": [
+    "#Drum capacitance\n",
+    "capacitance = permittivity*electrode_area/drum_gap #nC/uV\n",
+    "print(f'Capacitance: {capacitance} nC/uV')"
+   ]
  },
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 142,
   "metadata": {},
   "outputs": [],
-   "source": []
+   "source": [
+    "#Electrode charge\n",
+    "def charge(volt):\n",
+    "    return capacitance * volt*1e6"
+   ]
  },
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 143,
   "metadata": {},
-   "outputs": [],
-   "source": []
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Charge at 5V: 0.00177 pC\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(f'Charge at 5V: {charge(5.)} pC')"
+   ]
  },
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 144,
   "metadata": {},
   "outputs": [],
-   "source": []
+   "source": [
+    "def force(volts):\n",
+    "    nanocouls = charge(volts)\n",
+    "    return coulomb_k * nanocouls*(-nanocouls) / drum_gap**2"
+   ]
  },
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 145,
   "metadata": {},
-   "outputs": [],
-   "source": []
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Force at 1.0 um and 5 V: -28196100.0 nN = -28.196099999999998 mN\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(f'Force at {drum_gap} um and 5 V: {force(5.)} nN = {force(5.)*1e-6} mN')"
+   ]
  },
  {
   "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 146,
   "metadata": {},
   "outputs": [],
-   "source": []
+   "source": [
+    "#Damping coefficient\n",
+    "c = 2.*3.141592653589793*drum_resonance / drum_Q"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 149,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Gap: 1.0 um\n",
+      "1/Gap^2: 1.0 um^2\n",
+      "2/Gap^3: 2.0 um^3\n",
+      "Capacitance: 3.54000000E-10 nC/uV\n",
+      "Coulomb_k * Capacitance^2 / m: 1.16883395E-06 um/mF\n",
+      "Prefactor k_pf:  2.922084873105368e-07\n",
+      "Drum Mass: 9.64930905E-01 ug\n",
+      "[F] = nN\n",
+      "Damping c: 6.28318531E-03 kHz\n",
+      "w^2: 4.09132068E+03 kHz^2\n"
+     ]
+    }
+   ],
+   "source": [
+    "#Print relevant parameters\n",
+    "print(f\"Gap: {drum_gap} um\")\n",
+    "print(f\"1/Gap^2: {1./drum_gap**2} um^2\")\n",
+    "print(f\"2/Gap^3: {2./drum_gap**3} um^3\")\n",
+    "print(f\"Capacitance: {capacitance:.8E} nC/uV\")\n",
+    "print(f\"Coulomb_k * Capacitance^2 / m: {coulomb_k*capacitance**2/m:.8E} um/mF\")\n",
+    "print(f\"Prefactor k_pf:  {coulomb_k * permittivity**2 * electrode_area**2 / (4 * m * drum_gap**4)}\")\n",
+    "print(f\"Drum Mass: {m:.8E} ug\")\n",
+    "print(f\"[F] = nN\")\n",
+    "print(f\"Damping c: {c:.8E} kHz\")\n",
+    "print(f\"w^2: {(2.*3.141592653589793*drum_resonance)**2/m:.8E} kHz^2\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Force per unit mass\n",
+    "\n",
+    "The force per unit mass is now\n",
+    "\n",
+    "$\\text{F} / \\text{m} = \\text{Hooke} + \\text{Damping} + \\text{Drive} + \\text{Coupling}$,\n",
+    "\n",
+    "with\n",
+    "\n",
+    "$\\text{Hooke} = - \\omega^2 \\, x,$\n",
+    "\n",
+    "$\\text{Damping} = - c \\, \\dot{x},$\n",
+    "\n",
+    "$\\text{Drive} = k_{\\text{pf}} \\, V^2 \\, (1. + \\frac{x}{\\text{gap}}),$\n",
+    "\n",
+    "$\\text{Coupling} = \\sum_i k_{\\text{pf}} \\, t_i^2 \\, (1. + \\frac{x + x_i}{\\text{gap}}),$\n",
+    "\n",
+    "where\n",
+    "\n",
+    "$k_\\text{pf} = \\frac{k_c\\,\\varepsilon^2\\,A^2}{4\\,m\\,d^4}$\n",
+    "\n",
+    "and V, t_i are the driving voltage and the coupling electrode i voltage.\n",
+    "\n",
+    "The derivation of the coulomb term goes via assuming that our electrodes form parallel capacitors, which is valid when \n",
+    "$1\\,\\text{um} \\approx \\text{gap} << \\text{linear dim} \\approx 200\\, \\text{um}$.\n",
+    "Then the capacity is given by $C = \\frac{\\varepsilon\\,A}{d^2}$, which depends on the gap between the two electrodes, $d = \\text{gap}$. "
+   ]
  },
  {
   "cell_type": "code",

 %% Cell type:code id: tags:

 ``` python
 import numpy as np
 import matplotlib.pyplot as plt
 ```

 %% Cell type:markdown id: tags:

 # Units

 The units that suit the situation are

 Mass: ug

-Voltage: V
+Voltage: uV
+
+Current: nA

 Capacitance: mF

-Charge: C
+Charge: nC

 Time: ms

 Frequency: kHz

 Length: um

 %% Cell type:code id: tags:

 ``` python
 #Drum parameters:
 drum_diameter = 1000. #um
 drum_thickness = 0.3 #um
 electrode_thickness = 0.05 #um
 electrode_area = 200.*200. #um^2
 drum_gap = 1. #um
+drum_resonance = 10. #kHz
+drum_Q = 10000. #no units
 ```

 %% Cell type:code id: tags:

 ``` python
 #Physical constants
 rho_si3n4 = 3.44e-6 #ug/um^3
 rho_au = 1.93e-5 #ug/um^3
 coulomb_k = 9.0e12 #um/mF
-permittivity = 8.85e-15 #mF/um
+permittivity = 8.85e-15 #mF/um or nN/(mV)^2
 ```

 %% Cell type:code id: tags:

 ``` python
+#Drum mass
+m_si3n4 = (0.5*drum_diameter)**2*3.141592653589793*drum_thickness*rho_si3n4
+m_au = 4.*electrode_area*electrode_thickness*rho_au
+m = m_si3n4 + m_au
+print(f'Si3N4 Mass: {m_si3n4}')
+print(f'Au Mass: {m_au}')
+print(f'Drum Mass: {m}')
 ```

+%% Output
+
+    Si3N4 Mass: 0.8105309046261666
+    Au Mass: 0.1544
+    Drum Mass: 0.9649309046261666
+
 %% Cell type:code id: tags:

 ``` python
+#Drum capacitance
+capacitance = permittivity*electrode_area/drum_gap #nC/uV
+print(f'Capacitance: {capacitance} nC/uV')
 ```

+%% Output
+
+    Capacitance: 3.54e-10 nC/uV
+
 %% Cell type:code id: tags:

 ``` python
+#Electrode charge
+def charge(volt):
+    return capacitance * volt*1e6
 ```

 %% Cell type:code id: tags:

 ``` python
+print(f'Charge at 5V: {charge(5.)} pC')
 ```

+%% Output
+
+    Charge at 5V: 0.00177 pC
+
 %% Cell type:code id: tags:

 ``` python
+def force(volts):
+    nanocouls = charge(volts)
+    return coulomb_k * nanocouls*(-nanocouls) / drum_gap**2
 ```

 %% Cell type:code id: tags:

 ``` python
+print(f'Force at {drum_gap} um and 5 V: {force(5.)} nN = {force(5.)*1e-6} mN')
 ```

+%% Output
+
+    Force at 1.0 um and 5 V: -28196100.0 nN = -28.196099999999998 mN
+
 %% Cell type:code id: tags:

 ``` python
+#Damping coefficient
+c = 2.*3.141592653589793*drum_resonance / drum_Q
 ```

 %% Cell type:code id: tags:

 ``` python
+#Print relevant parameters
+print(f"Gap: {drum_gap} um")
+print(f"1/Gap^2: {1./drum_gap**2} um^2")
+print(f"2/Gap^3: {2./drum_gap**3} um^3")
+print(f"Capacitance: {capacitance:.8E} nC/uV")
+print(f"Coulomb_k * Capacitance^2 / m: {coulomb_k*capacitance**2/m:.8E} um/mF")
+print(f"Prefactor k_pf:  {coulomb_k * permittivity**2 * electrode_area**2 / (4 * m * drum_gap**4)}")
+print(f"Drum Mass: {m:.8E} ug")
+print(f"[F] = nN")
+print(f"Damping c: {c:.8E} kHz")
+print(f"w^2: {(2.*3.141592653589793*drum_resonance)**2/m:.8E} kHz^2")
+```
+
+%% Output
+
+    Gap: 1.0 um
+    1/Gap^2: 1.0 um^2
+    2/Gap^3: 2.0 um^3
+    Capacitance: 3.54000000E-10 nC/uV
+    Coulomb_k * Capacitance^2 / m: 1.16883395E-06 um/mF
+    Prefactor k_pf:  2.922084873105368e-07
+    Drum Mass: 9.64930905E-01 ug
+    [F] = nN
+    Damping c: 6.28318531E-03 kHz
+    w^2: 4.09132068E+03 kHz^2
+
+%% Cell type:markdown id: tags:
+
+# Force per unit mass
+
+The force per unit mass is now
+
+$\text{F} / \text{m} = \text{Hooke} + \text{Damping} + \text{Drive} + \text{Coupling}$,
+
+with
+
+$\text{Hooke} = - \omega^2 \, x,$
+
+$\text{Damping} = - c \, \dot{x},$
+
+$\text{Drive} = k_{\text{pf}} \, V^2 \, (1. + \frac{x}{\text{gap}}),$
+
+$\text{Coupling} = \sum_i k_{\text{pf}} \, t_i^2 \, (1. + \frac{x + x_i}{\text{gap}}),$
+
+where
+
+$k_\text{pf} = \frac{k_c\,\varepsilon^2\,A^2}{4\,m\,d^4}$
+
+and V, t_i are the driving voltage and the coupling electrode i voltage.
+
+The derivation of the coulomb term goes via assuming that our electrodes form parallel capacitors, which is valid when
+$1\,\text{um} \approx \text{gap} << \text{linear dim} \approx 200\, \text{um}$.
+Then the capacity is given by $C = \frac{\varepsilon\,A}{d^2}$, which depends on the gap between the two electrodes, $d = \text{gap}$.
+
+%% Cell type:code id: tags:
+
+``` python
 ```

--- a/parameter_calculation/Untitled.ipynb
+++ b/parameter_calculation/Untitled.ipynb
@@ -2,7 +2,7 @@
 "cells": [
  {
   "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 137,
   "metadata": {},
   "outputs": [],
   "source": [
@@ -37,7 +37,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 127,
+   "execution_count": 138,
   "metadata": {},
   "outputs": [],
   "source": [
@@ -53,7 +53,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 128,
+   "execution_count": 139,
   "metadata": {},
   "outputs": [],
   "source": [
@@ -66,7 +66,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 129,
+   "execution_count": 140,
   "metadata": {},
   "outputs": [
    {
@@ -91,7 +91,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 130,
+   "execution_count": 141,
   "metadata": {},
   "outputs": [
    {
@@ -110,7 +110,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 131,
+   "execution_count": 142,
   "metadata": {},
   "outputs": [],
   "source": [
@@ -121,7 +121,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 132,
+   "execution_count": 143,
   "metadata": {},
   "outputs": [
    {
@@ -138,7 +138,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 133,
+   "execution_count": 144,
   "metadata": {},
   "outputs": [],
   "source": [
@@ -149,7 +149,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 134,
+   "execution_count": 145,
   "metadata": {},
   "outputs": [
    {
@@ -166,7 +166,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 135,
+   "execution_count": 146,
   "metadata": {},
   "outputs": [],
   "source": [
@@ -176,7 +176,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 136,
+   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
@@ -188,6 +188,7 @@
      "2/Gap^3: 2.0 um^3\n",
      "Capacitance: 3.54000000E-10 nC/uV\n",
      "Coulomb_k * Capacitance^2 / m: 1.16883395E-06 um/mF\n",
+      "Prefactor k_pf:  9.17164116E-11 mFum/ug\n",
      "Drum Mass: 9.64930905E-01 ug\n",
      "[F] = nN\n",
      "Damping c: 6.28318531E-03 kHz\n",
@@ -202,12 +203,42 @@
    "print(f\"2/Gap^3: {2./drum_gap**3} um^3\")\n",
    "print(f\"Capacitance: {capacitance:.8E} nC/uV\")\n",
    "print(f\"Coulomb_k * Capacitance^2 / m: {coulomb_k*capacitance**2/m:.8E} um/mF\")\n",
+    "print(f\"Prefactor k_pf:  {permittivity * electrode_area / (4. * m * drum_gap**2):.8E} mFum/ug\")\n",
    "print(f\"Drum Mass: {m:.8E} ug\")\n",
    "print(f\"[F] = nN\")\n",
    "print(f\"Damping c: {c:.8E} kHz\")\n",
    "print(f\"w^2: {(2.*3.141592653589793*drum_resonance)**2/m:.8E} kHz^2\")"
   ]
  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Force per unit mass\n",
+    "\n",
+    "The force per unit mass is now\n",
+    "\n",
+    "$\\text{F} / \\text{m} = \\text{Hooke} + \\text{Damping} + \\text{Drive} + \\text{Coupling}$,\n",
+    "\n",
+    "with\n",
+    "\n",
+    "$\\text{Hooke} = - \\omega^2 \\, x,$\n",
+    "\n",
+    "$\\text{Damping} = - c \\, \\dot{x},$\n",
+    "\n",
+    "$\\text{Drive} = k_{\\text{pf}} \\, V^2 \\, (1. + \\frac{x}{\\text{gap}}),$\n",
+    "\n",
+    "$\\text{Coupling} = \\sum_i k_{\\text{pf}} \\, t_i^2 \\, (1. + \\frac{x + x_i}{\\text{gap}}),$\n",
+    "\n",
+    "where\n",
+    "\n",
+    "$k_\\text{pf} = \\frac{\\varepsilon_0\\,A}{4\\,m\\,\\text{gap}^2}$\n",
+    "\n",
+    "and V, t_i are the driving voltage and the coupling electrode i voltage.\n",
+    "\n",
+    "Note that we choose coordinates where drums from both, sublattice A and B, are displaced in positive direction when they move closer to their respective driving electrode."
+   ]
+  },
  {
   "cell_type": "code",
   "execution_count": null,

 %% Cell type:code id: tags:

 ``` python
 import numpy as np
 import matplotlib.pyplot as plt
 ```

 %% Cell type:markdown id: tags:

 # Units

 The units that suit the situation are

 Mass: ug

 Voltage: uV

 Current: nA

 Capacitance: mF

 Charge: nC

 Time: ms

 Frequency: kHz

 Length: um

 %% Cell type:code id: tags:

 ``` python
 #Drum parameters:
 drum_diameter = 1000. #um
 drum_thickness = 0.3 #um
 electrode_thickness = 0.05 #um
 electrode_area = 200.*200. #um^2
 drum_gap = 1. #um
 drum_resonance = 10. #kHz
 drum_Q = 10000. #no units
 ```

 %% Cell type:code id: tags:

 ``` python
 #Physical constants
 rho_si3n4 = 3.44e-6 #ug/um^3
 rho_au = 1.93e-5 #ug/um^3
 coulomb_k = 9.0e12 #um/mF
 permittivity = 8.85e-15 #mF/um or nN/(mV)^2
 ```

 %% Cell type:code id: tags:

 ``` python
 #Drum mass
 m_si3n4 = (0.5*drum_diameter)**2*3.141592653589793*drum_thickness*rho_si3n4
 m_au = 4.*electrode_area*electrode_thickness*rho_au
 m = m_si3n4 + m_au
 print(f'Si3N4 Mass: {m_si3n4}')
 print(f'Au Mass: {m_au}')
 print(f'Drum Mass: {m}')
 ```

 %% Output

    Si3N4 Mass: 0.8105309046261666
    Au Mass: 0.1544
    Drum Mass: 0.9649309046261666

 %% Cell type:code id: tags:

 ``` python
 #Drum capacitance
 capacitance = permittivity*electrode_area/drum_gap #nC/uV
 print(f'Capacitance: {capacitance} nC/uV')
 ```

 %% Output

    Capacitance: 3.54e-10 nC/uV

 %% Cell type:code id: tags:

 ``` python
 #Electrode charge
 def charge(volt):
    return capacitance * volt*1e6
 ```

 %% Cell type:code id: tags:

 ``` python
 print(f'Charge at 5V: {charge(5.)} pC')
 ```

 %% Output

    Charge at 5V: 0.00177 pC

 %% Cell type:code id: tags:

 ``` python
 def force(volts):
    nanocouls = charge(volts)
    return coulomb_k * nanocouls*(-nanocouls) / drum_gap**2
 ```

 %% Cell type:code id: tags:

 ``` python
 print(f'Force at {drum_gap} um and 5 V: {force(5.)} nN = {force(5.)*1e-6} mN')
 ```

 %% Output

    Force at 1.0 um and 5 V: -28196100.0 nN = -28.196099999999998 mN

 %% Cell type:code id: tags:

 ``` python
 #Damping coefficient
 c = 2.*3.141592653589793*drum_resonance / drum_Q
 ```

 %% Cell type:code id: tags:

 ``` python
 #Print relevant parameters
 print(f"Gap: {drum_gap} um")
 print(f"1/Gap^2: {1./drum_gap**2} um^2")
 print(f"2/Gap^3: {2./drum_gap**3} um^3")
 print(f"Capacitance: {capacitance:.8E} nC/uV")
 print(f"Coulomb_k * Capacitance^2 / m: {coulomb_k*capacitance**2/m:.8E} um/mF")
+print(f"Prefactor k_pf:  {permittivity * electrode_area / (4. * m * drum_gap**2):.8E} mFum/ug")
 print(f"Drum Mass: {m:.8E} ug")
 print(f"[F] = nN")
 print(f"Damping c: {c:.8E} kHz")
 print(f"w^2: {(2.*3.141592653589793*drum_resonance)**2/m:.8E} kHz^2")
 ```

 %% Output

    Gap: 1.0 um
    1/Gap^2: 1.0 um^2
    2/Gap^3: 2.0 um^3
    Capacitance: 3.54000000E-10 nC/uV
    Coulomb_k * Capacitance^2 / m: 1.16883395E-06 um/mF
+    Prefactor k_pf:  9.17164116E-11 mFum/ug
    Drum Mass: 9.64930905E-01 ug
    [F] = nN
    Damping c: 6.28318531E-03 kHz
    w^2: 4.09132068E+03 kHz^2

+%% Cell type:markdown id: tags:
+
+# Force per unit mass
+
+The force per unit mass is now
+
+$\text{F} / \text{m} = \text{Hooke} + \text{Damping} + \text{Drive} + \text{Coupling}$,
+
+with
+
+$\text{Hooke} = - \omega^2 \, x,$
+
+$\text{Damping} = - c \, \dot{x},$
+
+$\text{Drive} = k_{\text{pf}} \, V^2 \, (1. + \frac{x}{\text{gap}}),$
+
+$\text{Coupling} = \sum_i k_{\text{pf}} \, t_i^2 \, (1. + \frac{x + x_i}{\text{gap}}),$
+
+where
+
+$k_\text{pf} = \frac{\varepsilon_0\,A}{4\,m\,\text{gap}^2}$
+
+and V, t_i are the driving voltage and the coupling electrode i voltage.
+
+Note that we choose coordinates where drums from both, sublattice A and B, are displaced in positive direction when they move closer to their respective driving electrode.
+
 %% Cell type:code id: tags:

 ``` python
 ```