{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"nbsphinx": "hidden"
},
"source": [
"# Characterization of Systems in the Time Domain\n",
"\n",
"*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Analysis of a Passive Electrical Network\n",
"\n",
"[Electrical networks](https://en.wikipedia.org/wiki/Electrical_network) composed of linear passive elements, like resistors, capacitors and inductors can be described by linear ordinary differential equations (ODEs) with constant coefficients. Hence, in view of the theory of signals and systems they can be interpreted as linear time-invariant (LTI) systems. The different ways to characterize the properties of an LTI system introduced before are illustrated at the example of a second-order analog [low-pass filter](https://en.wikipedia.org/wiki/Low-pass_filter).\n",
"\n",
"![Circuit of a 2nd-order analog low-pass filter](lowpass.png)\n",
"\n",
"It is assumed that no energy is stored in the capacitor and inductor for $t<0$. It is furthermore assumed that $x(t) = 0$ for $t<0$. Hence $y(t) = 0$ and $\\frac{d y(t)}{dt} = 0$ for $t<0$. For illustration, the normalized values $L = 0.5$, $R = 1$, $C = 0.4$ are used for the elements of the electrical network."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Differential Equation\n",
"\n",
"The differential equation describing the input/output relation of the electrical network is derived by applying [Kirchhoff's circuit laws](https://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws) to the network. This results in the following ODE\n",
"\n",
"\\begin{equation}\n",
"C L \\frac{d^2 y(t)}{dt^2} + C R \\frac{d y(t)}{dt} + y(t) = x(t)\n",
"\\end{equation}\n",
"\n",
"This ODE is defined in `SymPy`"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle C L \\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} + C R \\frac{d}{d t} y{\\left(t \\right)} + y{\\left(t \\right)} = x{\\left(t \\right)}$"
],
"text/plain": [
" 2 \n",
" d d \n",
"C⋅L⋅───(y(t)) + C⋅R⋅──(y(t)) + y(t) = x(t)\n",
" 2 dt \n",
" dt "
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import sympy as sym\n",
"sym.init_printing()\n",
"\n",
"t, L, R, C = sym.symbols('t L R C', real=True)\n",
"x = sym.Function('x')(t)\n",
"y = sym.Function('y')(t)\n",
"\n",
"ode = sym.Eq(L*C*y.diff(t, 2) + R*C*y.diff(t) + y, x)\n",
"ode"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The normalized values of the network elements are stored in a dictionary for later substitution"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left\\{ C : \\frac{2}{5}, \\ L : \\frac{1}{2}, \\ R : 1\\right\\}$"
],
"text/plain": [
"{C: 2/5, L: 1/2, R: 1}"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"RLC = {R: 1, L: sym.Rational('.5'), C: sym.Rational('.4')}\n",
"RLC"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Impulse Response\n",
"\n",
"The passive electrical network and the ODE describing its input/output relation can be interpreted as an LTI system. Hence, the system can be characterized by its [impulse response](impulse_response.ipynb) $h(t)$ which is defined as the output of the system for a Dirac Delta impulse $x(t) = \\delta(t)$ at the input. For the given system, the impulse response is calculated by explicit solution of the ODE"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/latex": [
"$\\displaystyle h{\\left(t \\right)} = C_{1} e^{\\frac{t \\left(- R - \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} + C_{2} e^{\\frac{t \\left(- R + \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} - \\frac{e^{\\frac{t \\left(- R - \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}} + \\frac{e^{\\frac{t \\left(- R + \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}$"
],
"text/plain": [
" \n",
" ⎛ ________________⎞ ⎛ ________________⎞ \n",
" ⎜ ╱ ⎛ 2 ⎞ ⎟ ⎜ ╱ ⎛ 2 ⎞ ⎟ \n",
" ⎜ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ⎟ ⎜ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ⎟ \n",
" t⋅⎜-R - ───────────────────⎟ t⋅⎜-R + ───────────────────⎟ \n",
" ⎝ C ⎠ ⎝ C ⎠ \n",
" ──────────────────────────── ──────────────────────────── \n",
" 2⋅L 2⋅L ℯ\n",
"h(t) = C₁⋅ℯ + C₂⋅ℯ - ─\n",
" \n",
" \n",
" \n",
"\n",
" ⎛ ________________⎞ ⎛ ________________⎞ \n",
" ⎜ ╱ ⎛ 2 ⎞ ⎟ ⎜ ╱ ⎛ 2 ⎞ ⎟ \n",
" ⎜ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ⎟ ⎜ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ⎟ \n",
"t⋅⎜-R - ───────────────────⎟ t⋅⎜-R + ───────────────────⎟ \n",
" ⎝ C ⎠ ⎝ C ⎠ \n",
"──────────────────────────── ──────────────────────────── \n",
" 2⋅L 2⋅L \n",
" ⋅θ(t) ℯ ⋅θ(t)\n",
"───────────────────────────────── + ──────────────────────────────────\n",
" ________________ ________________ \n",
" ╱ ⎛ 2 ⎞ ╱ ⎛ 2 ⎞ \n",
" ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ "
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"solution_h = sym.dsolve(\n",
" ode.subs(x, sym.DiracDelta(t)).subs(y, sym.Function('h')(t)))\n",
"solution_h"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The integration constants $C_1$ and $C_2$ have to be determined from the initial conditions $y(t) = 0$ and $\\frac{d y(t)}{dt} = 0$ for $t<0$. "
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left\\{ C_{1} : 0, \\ C_{2} : 0\\right\\}$"
],
"text/plain": [
"{C₁: 0, C₂: 0}"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"integration_constants = sym.solve((solution_h.rhs.limit(\n",
" t, 0, '-'), solution_h.rhs.diff(t).limit(t, 0, '-')), ['C1', 'C2'])\n",
"integration_constants"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Substitution of the values for the integration constants $C_1$ and $C_2$ into the result from above yields the impulse response of the low-pass"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/latex": [
"$\\displaystyle h{\\left(t \\right)} = - \\frac{e^{\\frac{t \\left(- R - \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}} + \\frac{e^{\\frac{t \\left(- R + \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}$"
],
"text/plain": [
" ⎛ ________________⎞ ⎛ ________________⎞ \n",
" ⎜ ╱ ⎛ 2 ⎞ ⎟ ⎜ ╱ ⎛ 2 ⎞ ⎟ \n",
" ⎜ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ⎟ ⎜ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ⎟ \n",
" t⋅⎜-R - ───────────────────⎟ t⋅⎜-R + ───────────────────⎟ \n",
" ⎝ C ⎠ ⎝ C ⎠ \n",
" ──────────────────────────── ──────────────────────────── \n",
" 2⋅L 2⋅L \n",
" ℯ ⋅θ(t) ℯ ⋅θ(\n",
"h(t) = - ────────────────────────────────── + ────────────────────────────────\n",
" ________________ ________________ \n",
" ╱ ⎛ 2 ⎞ ╱ ⎛ 2 ⎞ \n",
" ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ \n",
"\n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
"t)\n",
"──\n",
" \n",
" \n",
" "
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"h = solution_h.subs(integration_constants)\n",
"h"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The impulse response is plotted for the values of $R$, $L$ and $C$ given above"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
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\n",
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