{ "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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"text/latex": [ "$$C L \\frac{d^{2}}{d t^{2}} y{\\left (t \\right )} + C R \\frac{d}{d t} y{\\left (t \\right )} + y{\\left (t \\right )} = x{\\left (t \\right )}$$" ], "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", "%matplotlib inline\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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"text/latex": [ "$$\\left \\{ C : \\frac{2}{5}, \\quad L : \\frac{1}{2}, \\quad R : 1\\right \\}$$" ], "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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ENrAhRD507WKjEP/In7/RgUF7hWfQ27ASY97B3ndwETjQ/ss6cIPGHwyQUKsPGjbsJW77SrWLjUI88OfX69yYqh/vwH0DcMH52DSR1zbKvIPKsntiS4Ynd83CbKrbM/6geORm17AtlOvTLLaY19+a9sP3D63RdQXKLZlUs6bQAsJdg4dgAbAOHJyhtoT7zb7gt/ej031qq37Z+mRzAVi2mlD6fksGPr0ptIBp1+AhEDJfAwenry7hLgV+6T+fT24/ytXBd1bI5fcr5MvGt9KdT7FFhpSu59ttST3hzn1baCmIHYOHHYhG8DI9VodbBCN9SwE3siBqqcbZ6yqdC6iXLTmC0KPEPd9KF0GBV78wI28S7Y7BQ6CYr2SnlsDC/KabXg6EIwgtbsaKoP7d9+ToGb6V7hYTs4qZ3+v6oB7igz4Q7mIjYzla7LBHc1hRM1q7wUMk9iv5KSXwpqpK08sBcwSBj88xvDushKX72egvfCvdLZ47V5VJpjJJZ4e6n+OOZWhNSGdE65bFFXsGCdAhyeblgPnCsMR9KFOgpweiSNarO0KZeQf+SM5gNfXMTt58wWwAtkA6Idp8crtKqlYCT3BBa/NywBxBWOLgPCNs6QYdGLbSXRg/rwcbgedtzs1+3HEECyj7l6JFFkgnRFtWllfpNUiAThxtXg6YIwhb3APY8h+YSbOtdAegCfEOTPNCYmbr6siaMYpNky2QTog2o+CuomqVABz+ZPVyAI4g7N4daMeFaXHAVvq8iA68Mq3ZJhPn2aHZhUaPurLBPSHa7OK7CqxOAh3YiVq9HNDB0RY3rj0+hbindtz+rfRNicpG4FFaArvPDs0uLYoW2SCdD2128V0F1iKBDkxUcHVm8D6leTmAelJHELY4cphv9wYSasBW+kbJZx14kpfAjGVfUjIGWmwuq/iJIg8/C9qSkrzKrkMCnxv3GU59n2leDqCa1I7cFvcAjY8ylkImw8fAFkw807EOrCyBXWeHQoGH/zfR6k4dyCOS0GKX65pLhV+jPSyuq4DqJTAMPXPYwE5h0bwcAAT6StviwPtBF+gEYSE2NawDsyUw+Jss7rXYgtZ0WIEBnwNt9S/fVcEcEhjBohK7XqGlaV4OIJQ6grDE0fzDykdyZ61GfETP+73e3ttMemS25rCr9AUzGQMtskBCZ0HrbIkr8iQSuEPPfTEvJBYvB9wRhCXuCT7TiC7IXE3uyOgGPXai60+6+nacHbpTTnywjhYr1pl5ODN4PxHaePlcOZqTwAsUSez4HGxupns5wJDYDo8e133osviF+8EokTLcYoCdJzZ9najfxG8YyRhobXBPg9bdClfsOSRAepNg7+OzMVQPDhtK6gjCiJu2H9FeP3s6jvqF8sSuC/u5e+M/eDq9/Q95XGeH+ksNTGGgtcA9D9pAoVzJWpYAWQX6LAk0RxAl8H7n7NC/hbZEO11lViaBBeufRp8K6guWBsyCoKx4/hbasrK8Sq9DAngnaTt8x/1THEG4k6bF6meHppXiz/W30PrlcaVoXgIjXop6QSiOILypExLoZ4cmFBGU5W+hDRLJlahxCdypJa8ThuQIwpkuMdI4OzSxHH+2v4XWL48rRfMSePkH4OYxCgCNof0P8JE6A73MN7UAAAAASUVORK5CYII=\n", "text/latex": [ "$$h{\\left (t \\right )} = C_{1} e^{\\frac{t \\left(- R - \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} + C_{2} e^{\\frac{t \\left(- R + \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} - \\frac{e^{\\frac{t \\left(- R - \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}} + \\frac{e^{\\frac{t \\left(- R + \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}$$" ], "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", " ⋅Heaviside(t) ℯ ⋅Heav\n", "───────────────────────────────────────── + ──────────────────────────────────\n", " ________________ ________________ \n", " ╱ ⎛ 2 ⎞ ╱ ⎛ 2 ⎞ \n", " ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ \n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "iside(t)\n", "────────\n", " \n", " \n", " " ] }, "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": [ "$$\\left \\{ C_{1} : 0, \\quad C_{2} : 0\\right \\}$$" ], "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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"text/latex": [ "$$h{\\left (t \\right )} = - \\frac{e^{\\frac{t \\left(- R - \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}} + \\frac{e^{\\frac{t \\left(- R + \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}$$" ], "text/plain": [ " ⎛ ________________⎞ ⎛ ____________\n", " ⎜ ╱ ⎛ 2 ⎞ ⎟ ⎜ ╱ ⎛ 2 \n", " ⎜ ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ⎟ ⎜ ╲╱ C⋅⎝C⋅R - 4\n", " t⋅⎜-R - ───────────────────⎟ t⋅⎜-R + ───────────────\n", " ⎝ C ⎠ ⎝ C \n", " ──────────────────────────── ───────────────────────\n", " 2⋅L 2⋅L \n", " ℯ ⋅Heaviside(t) ℯ \n", "h(t) = - ────────────────────────────────────────── + ────────────────────────\n", " ________________ __________\n", " ╱ ⎛ 2 ⎞ ╱ ⎛ 2 \n", " ╲╱ C⋅⎝C⋅R - 4⋅L⎠ ╲╱ C⋅⎝C⋅R -\n", "\n", "____⎞ \n", " ⎞ ⎟ \n", "⋅L⎠ ⎟ \n", "────⎟ \n", " ⎠ \n", "───── \n", " \n", " ⋅Heaviside(t)\n", "──────────────────\n", "______ \n", " ⎞ \n", " 4⋅L⎠ " ] }, "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": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sym.plot(h.rhs.subs(RLC), (t,-1,10), ylabel=r'h(t)');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Step Response\n", "\n", "The [step response](step_response.ipynb) is derived by integrating over the impulse response $h(t)$. For ease of illustration this is performed for the specific values of the elements given above" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\frac{5 i \\left(- \\frac{e^{t \\left(-1 - 2 i\\right)}}{5} + \\frac{2 i e^{t \\left(-1 - 2 i\\right)}}{5} + \\frac{1}{5} - \\frac{2 i}{5}\\right) \\theta\\left(t\\right)}{4} - \\frac{5 i \\left(- \\frac{e^{t \\left(-1 + 2 i\\right)}}{5} - \\frac{2 i e^{t \\left(-1 + 2 i\\right)}}{5} + \\frac{1}{5} + \\frac{2 i}{5}\\right) \\theta\\left(t\\right)}{4}$$" ], "text/plain": [ " ⎛ t⋅(-1 - 2⋅ⅈ) t⋅(-1 - 2⋅ⅈ) ⎞ ⎛ t⋅(\n", " ⎜ ℯ 2⋅ⅈ⋅ℯ 1 2⋅ⅈ⎟ ⎜ ℯ \n", "5⋅ⅈ⋅⎜- ───────────── + ───────────────── + ─ - ───⎟⋅Heaviside(t) 5⋅ⅈ⋅⎜- ────\n", " ⎝ 5 5 5 5 ⎠ ⎝ \n", "──────────────────────────────────────────────────────────────── - ───────────\n", " 4 \n", "\n", "-1 + 2⋅ⅈ) t⋅(-1 + 2⋅ⅈ) ⎞ \n", " 2⋅ⅈ⋅ℯ 1 2⋅ⅈ⎟ \n", "───────── - ───────────────── + ─ + ───⎟⋅Heaviside(t)\n", " 5 5 5 5 ⎠ \n", "─────────────────────────────────────────────────────\n", " 4 " ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "tau = sym.symbols('tau', real=True)\n", "\n", "he = sym.integrate(h.rhs.subs(RLC).subs(t, tau), (tau, 0, t))\n", "he" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's plot the step response" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sym.plot(he, (t,-1,10), ylabel=r'$h_\\epsilon(t)$');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Transfer Function\n", "\n", "For an exponential input signal $x(t) = e^{s t}$, the [transfer function](eigenfunctions.ipynb#Transfer-Function) $H(s)$ represents the complex weight of the exponential output signal $y(t) = H(s) \\cdot e^{s t}$. The transfer function is derived by introducing $x(t)$ and $y(t)$ into the ODE and solving for $H(s)$" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\frac{1}{C L s^{2} + C R s + 1}$$" ], "text/plain": [ " 1 \n", "──────────────────\n", " 2 \n", "C⋅L⋅s + C⋅R⋅s + 1" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = sym.symbols('s')\n", "H = sym.Function('H')(s)\n", "\n", "H, = sym.solve(ode.subs(x, sym.exp(s*t)).subs(y, H*sym.exp(s*t)).doit(), H)\n", "H" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The transfer characteristic of an LTI system for harmonic exponential signals $e^{j \\omega t} = \\cos(\\omega t) + j \\sin(\\omega t)$ is of special interest for the analysis of electrical circuits. It can be derived from $H(s)$ by substituting the complex frequency $s$ by $s = j \\omega$. The resulting transfer function $H(j \\omega)$ provides the attenuation/amplification and phase the system adds to an harmonic input signal." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAALgAAAAsBAMAAAAk6m+hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMt0Qq5nNdrvvZiKJVETbNVTtAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACbUlEQVRIDe2XP2jUUBzHv3fPXNom155u4nDldNEpQ6FuzXCgY0CouojootPdoKBOV620DoLo0KmQvYOHOAoGdBEOLu5Cu7q02h6tqHj+3iXvT2KkTcHF5gePvD+f3/fy++bleAHyROl0HjofOzO/ky8hF20W4ll+FbZkuYLClv/LlmNPti9kVlRMFg4cZQeG/zCOsq+Hr531ms7hs/fJNGBM74MccHl2sem/Ae4Mb/kiY2rD+ir68dX0qBOj94aLy09T6/Ewdaw0Vxx8Okdr2wqfcP4QLweARL8BdVfhqpc6VrKH5K7xAyj9VAxgT0ejSV/NKnQPGKMUFRJLHhQqLwiZfEa39UuxwIn4gVbbalah5FklIS6xpHg/pOySR/QXJQPrUTyQWTROoPWOhkNiCXFr5DSrAVPXOe1Se1XDa5ziI8is2cvQ0A6sM5LVseT5ySb7otgK6FrldbxtWwvHL0Wz4paCz9DQpbuPa4IdcQJLipfllutvEGZw1A7Hh8PYpDjLCuehoV120REsz5AFRuLsRoNHbXz00cBovcWhTaCDiS7vUhiNxtnnjYYLMOxBobfJxECxCqNtoX+ElEd3+JGUdqmttfmztaknQtRb3YFCB8CcC50VWFKc73AgoN3IzXc9mCHodZUhsiru6GWI0O/AugedFVhS3OKatkONzDf9BdQ9dlNKKzPnwkCiBpXeb5d0Nlsc6z6q70iNF913PrxfHVz1M8S3LF+g7CSJv2xv6uxfxNmV5jUSm2kNmq0Bzu92769o2nIbjD2gpxqhdq/nw1jydVaI5ztWiiz9BzP6B8RSmfIfKTWfGiaw3zwKHDV1Am3VAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\frac{1}{- C L \\omega^{2} + i C R \\omega + 1}$$" ], "text/plain": [ " 1 \n", "──────────────────────\n", " 2 \n", "- C⋅L⋅ω + ⅈ⋅C⋅R⋅ω + 1" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w = sym.symbols('omega', real=True)\n", "\n", "Hjw = H.subs(s, sym.I * w)\n", "Hjw" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The magnitude of the transfer function $|H(j \\omega)|$ is plotted for illustration for the specific values of the elements given above" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "application/pdf": 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[ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sym.plot(abs(Hjw.subs(RLC)), (w, -10, 10),\n", " ylabel=r'$|H(j \\omega)|$', xlabel=r'$\\omega$');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It now becomes obvious, that the low frequencies pass through the system and that high frequencies are attenuated. This motivates the term 'low-pass' for such systems.\n", "\n", "As alternative to the solution of the ODE, the transfer function $H(s)$ is [computed from the impulse response](eigenfunctions.ipynb#Link-between-Transfer-Function-and-Impulse-Response) and plotted for the specific values of the elements given above" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\frac{5}{s^{2} + 2 s + 5}$$" ], "text/plain": [ " 5 \n", "────────────\n", " 2 \n", "s + 2⋅s + 5" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H2 = sym.integrate(h.rhs.subs(RLC)*sym.exp(-s*t), (t, 0, sym.oo), conds='none')\n", "H2.simplify()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "application/pdf": 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"text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sym.plot(abs(H2.subs(s, sym.I*w)), (w, -10, 10),\n", " ylabel=r'$|H(j \\omega)|$', xlabel=r'$\\omega$');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The phase of the transfer function $\\varphi(j \\omega) = \\arg \\{ H(j \\omega) \\}$ provides insight into the phase added to an harmonic signal when passing through the system. It is computed and plotted for the specific values of the elements given above" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "phi = sym.arg(Hjw)\n", "sym.plot(phi.subs(RLC), (w, -10, 10),\n", " ylabel=r'$\\varphi(j \\omega)$', xlabel=r'$\\omega$');" ] }, { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "source": [ "**Copyright**\n", "\n", "The notebooks are provided as [Open Educational Resource](https://de.wikipedia.org/wiki/Open_Educational_Resources). Feel free to use the notebooks for your own educational purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Lecture Notes on Signals and Systems* by Sascha Spors." ] } ], "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": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 1 }