doc/examples/notebooks/superposition1.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {
      "image/png": {
       "height": 150,
       "width": 485
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from lcapy import Circuit, t\n",
    "cct = Circuit(\"\"\"\n",
    "V1 1 0_1 8; down\n",
    "R1 1 x 3; right=1.5, i=I\n",
    "R2 x 2 1; right=1.5\n",
    "V2 2 0_2 4; down\n",
    "W 0_1 0; right\n",
    "W 0 0_2; right\n",
    "Ox x 0; down, v=V_x\n",
    "\"\"\")\n",
    "cct.draw(label_ids=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's determine the current through R1.  There are many ways to solve this; the easiest is to combine the sources, combine the resistances, and then use Ohm's law. You can also use the principle of superposition."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPCAYAAADZCo4zAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAZUlEQVQYGWP8//8/Az7AgizZ0NAQAuTPBmJFIPsDSI4FyBAA0iDBe0CsBMQgPhwwIlsBVJwGlJkJxIIwE5jgSnEwRqgClFAEBQ04JIGhBgo9ISB2AWKQonNAfAaIz6IENVAAAwAAzp8eRPTSpxEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct.R1.i"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Not surprisingly due to KCL, this is the same as the current through R2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPCAYAAADZCo4zAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAZUlEQVQYGWP8//8/Az7AgizZ0NAQAuTPBmJFIPsDSI4FyBAA0iDBe0CsBMQgPhwwIlsBVJwGlJkJxIIwE5jgSnEwRqgClFAEBQ04JIGhBgo9ISB2AWKQonNAfAaIz6IENVAAAwAAzp8eRPTSpxEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct.R2.i"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now given the current, we can use Ohm's law to determine the voltage drop across R1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVQoFXWR4Q2CQAyFwTiAjiAjOAMjyAqygcR//DOOgCvICIyAbsAMsgF+r1zJmWCTXl9f3/Wud+k0TYmsruuMUFoyL0dCA98qTSUMooq4CMEn6k+8BD82UmMiz6FoBNg6kVQiXPgGj8HFxyY+2Wphd0fYC7tF3RtxdkcvekS0A7/wFmxHW8dI4JPrzjrl5rXVjirS6UK44gW4+ysM4o8idrCp2ZHJZ+5n7cl039yfZyAZEItcNR9Gb9UjVIzNT+lcaE8QK9ikySXUF47LMCQ5ZBGJJbrD65mSL0Q8VMJ3oR2mAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$3$$"
      ],
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cct.R1.Z * cct.R1.I).time()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lcapy can determine this directly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVQoFXWR4Q2CQAyFwTiAjiAjOAMjyAqygcR//DOOgCvICIyAbsAMsgF+r1zJmWCTXl9f3/Wud+k0TYmsruuMUFoyL0dCA98qTSUMooq4CMEn6k+8BD82UmMiz6FoBNg6kVQiXPgGj8HFxyY+2Wphd0fYC7tF3RtxdkcvekS0A7/wFmxHW8dI4JPrzjrl5rXVjirS6UK44gW4+ysM4o8idrCp2ZHJZ+5n7cl039yfZyAZEItcNR9Gb9UjVIzNT+lcaE8QK9ikySXUF47LMCQ5ZBGJJbrD65mSL0Q8VMJ3oR2mAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$3$$"
      ],
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct.R1.v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using KVL, the unknown voltage is thus the voltage generated by the source V1 minus the voltage drop across R1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOCAYAAAD9lDaoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVQoFX2RsRGCQBBFTzQwdCzh7EAbILAFW7AFQy6FDrAEsQMNyBVCMyxBCc3wfeFmGGbwZj67/++/veVu2jSNYRVhGBpQ5Xn+ga/JHZjDHxOZnHNvCgu5eytBP4jPOvFJvAILbuCMQVq71CmKokxxDIE3/4v+OM2174yabQVStFKa76TCCfEIEvIYFORbohmdgzkvoNKcvtNvw+Cjv7N0swGfDBQDg2jdaVad2nM7pRf85d5l0sCbXtGn2lxSq2WKSVJfUYTrOpZgJ+7fzpLrnTSH8pe4uhDNFzKnevV2JG/QAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct.V1.v - cct.R1.v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, Lcapy can determine this directly.  Here Ox is the name of the open circuit over which we wish to determine the voltage difference."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct.Ox.V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, we can query Lcapy for the voltage at node 'x' with respect to ground."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOCAYAAAD9lDaoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVQoFX2RsRGCQBBFTzQwdCzh7EAbILAFW7AFQy6FDrAEsQMNyBVCMyxBCc3wfeFmGGbwZj67/++/veVu2jSNYRVhGBpQ5Xn+ga/JHZjDHxOZnHNvCgu5eytBP4jPOvFJvAILbuCMQVq71CmKokxxDIE3/4v+OM2174yabQVStFKa76TCCfEIEvIYFORbohmdgzkvoNKcvtNvw+Cjv7N0swGfDBQDg2jdaVad2nM7pRf85d5l0sCbXtGn2lxSq2WKSVJfUYTrOpZgJ+7fzpLrnTSH8pe4uhDNFzKnevV2JG/QAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct['x'].v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we should check that the answer makes sense.  It is between 4 and 8 V and so is plausible.  Oh, we should also specify the units for each of our answers.  This helps to find errors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now try using superposition.  We start by analysing the circuit with all the sources except V1 killed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {
      "image/png": {
       "height": 150,
       "width": 386
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "cct2 = cct.kill_except('V1')\n",
    "cct2.draw(label_ids=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a simpler circuit to analyse.  We can either recognise the circuit as a voltage divider or calculate the current and find the voltage drops.  Let's use the latter approach.  Combining the resistances gives 4 ohms and using Ohm's law, the current through R1 and R2 is given by:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOCAYAAAD9lDaoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVQoFW2QMQ6CUBBEF/UA6BGgtSKxp/AIxtLOeAMpodUbGDstpbTTgsZSbyBn4Ab4Roz5CpMMu7M7u//zvbquLcsy38w2UAiaYAn1h/I+lGFPYRHH8Rkei6IYUzsQb+iyh9CGJfyCgQRRwZOKMs3hXeIPV7TPQCBTCQOEju2C7+niXWDoSV0/MdSmFjBMKcqwJa86TTR3MMegH7DWcTRkMOJKUfjZRGOtomuQHugj0JgRQtdArns1mxAR+cQ1qAk0aF6apnJfYK7CB3qzEYwYDHWcnl7G932ILvTQ9gKAV0kgBsj0zgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct2.V1.v / 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lcapy can determine this directly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOCAYAAAD9lDaoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVQoFW2QMQ6CUBBEF/UA6BGgtSKxp/AIxtLOeAMpodUbGDstpbTTgsZSbyBn4Ab4Roz5CpMMu7M7u//zvbquLcsy38w2UAiaYAn1h/I+lGFPYRHH8Rkei6IYUzsQb+iyh9CGJfyCgQRRwZOKMs3hXeIPV7TPQCBTCQOEju2C7+niXWDoSV0/MdSmFjBMKcqwJa86TTR3MMegH7DWcTRkMOJKUfjZRGOtomuQHugj0JgRQtdArns1mxAR+cQ1qAk0aF6apnJfYK7CB3qzEYwYDHWcnl7G932ILvTQ9gKAV0kgBsj0zgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct2.R2.i"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The voltage drop across R2 can then be found from Ohm's law."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPCAYAAADZCo4zAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAZUlEQVQYGWP8//8/Az7AgizZ0NAQAuTPBmJFIPsDSI4FyBAA0iDBe0CsBMQgPhwwIlsBVJwGlJkJxIIwE5jgSnEwRqgClFAEBQ04JIGhBgo9ISB2AWKQonNAfAaIz6IENVAAAwAAzp8eRPTSpxEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct2.R2.Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOCAYAAAD9lDaoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVQoFW2QMQ6CUBBEF/UA6BGgtSKxp/AIxtLOeAMpodUbGDstpbTTgsZSbyBn4Ab4Roz5CpMMu7M7u//zvbquLcsy38w2UAiaYAn1h/I+lGFPYRHH8Rkei6IYUzsQb+iyh9CGJfyCgQRRwZOKMs3hXeIPV7TPQCBTCQOEju2C7+niXWDoSV0/MdSmFjBMKcqwJa86TTR3MMegH7DWcTRkMOJKUfjZRGOtomuQHugj0JgRQtdArns1mxAR+cQ1qAk0aF6apnJfYK7CB3qzEYwYDHWcnl7G932ILvTQ9gKAV0kgBsj0zgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cct2.R2.I * cct2.R2.Z)(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, Lcapy can determine this directly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOCAYAAAD9lDaoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVQoFW2QMQ6CUBBEF/UA6BGgtSKxp/AIxtLOeAMpodUbGDstpbTTgsZSbyBn4Ab4Roz5CpMMu7M7u//zvbquLcsy38w2UAiaYAn1h/I+lGFPYRHH8Rkei6IYUzsQb+iyh9CGJfyCgQRRwZOKMs3hXeIPV7TPQCBTCQOEju2C7+niXWDoSV0/MdSmFjBMKcqwJa86TTR3MMegH7DWcTRkMOJKUfjZRGOtomuQHugj0JgRQtdArns1mxAR+cQ1qAk0aF6apnJfYK7CB3qzEYwYDHWcnl7G932ILvTQ9gKAV0kgBsj0zgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct2['x'].v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we have determine the voltage Vx due to source V1.  Let's now kill just V2 and determine the voltage Vx due to V2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {
      "image/png": {
       "height": 150,
       "width": 482
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "cct3 = cct.kill_except('V2')\n",
    "cct3.draw(label_ids=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this circuit the current flows in the opposite direction to the labelled current I.  Thus we will get a negative value for the current.  If you do not like negative currents, just relabel the current direction."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPCAYAAAD+pA/bAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAi0lEQVQ4EWP8//8/Ay0BCzUNb2hoCAGaNxuIFYHsDyCzKbYAaJAA1NB7QFoJiEF8OGCkZhABLUsDmjwTiAVhPmCCW0UjxqgFBAOWBZoK9hJUiaogFajvHKoQdh7IAlB6NcYuTbnoaCSjhyFKLgZJUiUnA+MRlHuFgNgFiEGWgBLAGSA+SxULgAbhBAARyCXrnE17rwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$-1$$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct3.R1.i"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The unknown voltage Vx can be found as before, either by using the principle of the voltage divider or by calculating the current through R1 and R2 and then determining the voltage dropped across R2 (taking into account that the current flows in the opposite direction to that shown):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVQoFXWR4Q2CQAyFwTiAjiAjOAMjyAqygcR//DOOgCvICIyAbsAMsgF+r1zJmWCTXl9f3/Wud+k0TYmsruuMUFoyL0dCA98qTSUMooq4CMEn6k+8BD82UmMiz6FoBNg6kVQiXPgGj8HFxyY+2Wphd0fYC7tF3RtxdkcvekS0A7/wFmxHW8dI4JPrzjrl5rXVjirS6UK44gW4+ysM4o8idrCp2ZHJZ+5n7cl039yfZyAZEItcNR9Gb9UjVIzNT+lcaE8QK9ikySXUF47LMCQ5ZBGJJbrD65mSL0Q8VMJ3oR2mAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$3$$"
      ],
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cct3.V2.V + cct3.R2.I * cct3.R2.Z)(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, the voltage drop across R1 (taking into account the current direction) is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVQoFXWR4Q2CQAyFwTiAjiAjOAMjyAqygcR//DOOgCvICIyAbsAMsgF+r1zJmWCTXl9f3/Wud+k0TYmsruuMUFoyL0dCA98qTSUMooq4CMEn6k+8BD82UmMiz6FoBNg6kVQiXPgGj8HFxyY+2Wphd0fYC7tF3RtxdkcvekS0A7/wFmxHW8dI4JPrzjrl5rXVjirS6UK44gW4+ysM4o8idrCp2ZHJZ+5n7cl039yfZyAZEItcNR9Gb9UjVIzNT+lcaE8QK9ikySXUF47LMCQ5ZBGJJbrD65mSL0Q8VMJ3oR2mAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$3$$"
      ],
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-cct3.R1.I * cct3.R1.Z)(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now know Vx due to V1 and Vx due to V2.  Due to linearity, the total voltage Vx is simply the sum of the two contributions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOCAYAAAD9lDaoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVQoFX2RsRGCQBBFTzQwdCzh7EAbILAFW7AFQy6FDrAEsQMNyBVCMyxBCc3wfeFmGGbwZj67/++/veVu2jSNYRVhGBpQ5Xn+ga/JHZjDHxOZnHNvCgu5eytBP4jPOvFJvAILbuCMQVq71CmKokxxDIE3/4v+OM2174yabQVStFKa76TCCfEIEvIYFORbohmdgzkvoNKcvtNvw+Cjv7N0swGfDBQDg2jdaVad2nM7pRf85d5l0sCbXtGn2lxSq2WKSVJfUYTrOpZgJ+7fzpLrnTSH8pe4uhDNFzKnevV2JG/QAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cct2['x'].v + cct3['x'].v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the same as the result determined earlier."
   ]
  }
 ],
 "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": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}