{ "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": "iVBORw0KGgoAAAANSUhEUgAAAgMAAACeCAYAAACvvAnjAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4Ae2dTdLdtrGGP6lcGdybgaJU3XnsHcjSCiLvwJZXYHsHdmmUzFT2DmyvQLZ3YGcFkr0D5c5TZVmD5A4y0X0ffiAD/h3+gSTI86KKhyQINBov0OhGA+S5efv27Y2P42Pwl7/85Z7b8fjtuHYbup9s10eM9XZYry03a9LPpZ+8c+NweAT++te/PlYl3tXxzeErEyqgOj3Q5cc6Pj9LnfauhzC9Ix7u6/xUxxd783Pm8oWvZfLMDZyobjnJ5B0sHofjIqDOhBHwpc4fHbcWbc5Vnx8V+4XOv7SfOmYJAsL0Q+V/o/NPS+g4bzcCwtUy2Q2NY3sQyEEm7/bw5ujjIPC1WP3kOOwOcyrBwCtwo7MNgWG4JqcQrj8o02c635uc2RnGIGCZHIOS01QI5CCTXiaomuN4F+pAzPB+0fnN8bi/yPFTPf3yYopMHgp73MEoVWaD7+n4XnFHmHGzTPCtjlN5lFSfXYNlclf4i8Itk/PawJ6Bebjlkgul+SwXZlLwIUFGqb6bQqEyMAd6KVjro4Hy/0HHV0qAgmV5Y7cgPkbtsVC6v4vJezqDt0M6BCyTF7BUf7NM9uCzt0zaGOhpmNyj1XFwpb/U+WxeARTq4s1twuVT0WFdHKW3Zng/Io5i3bU9VN+vdOCmHhNItxjrMQVdQxrhbpm80NDCxzJ5AZ/waDeZ9DLBcOPkmoIZyNhBP9c61PjSYIG7/aHOn9UeRDd61pz5ouw/iPMEOsR1usDH0KBIpWOpgvJKhflHXcMjGxsLpa9zbGyQvrNMxScJEU94Ip7rvmtfBd4KNpWWfHeWred4NEiHh2BXI6aTweNFWiZv28wy2e672cukjYF2ox0l5rEG8FUVzw5AoEw7lz1UV2bdGD9sfKsUsK7ZN/FQRxyg0zKUJtKAHi5/cEbxFkHXGAettXbFU16ytx9Ej7I/0rmppInH1dqr6PXsJx1jlTz7G57oOM1rqarLXsEyKeTV9yyTjR54BJn0MkGj0Y5wG4Tt5RF4Hcuj6sSMm8GUne61EJ6hBFG2lSEQEqHMmhv2oFOLm0ED8rh9a3R0/4GOFzqqINoYH4UhoOum56JKN/ZCNHCnYvx0hS6eutLBN5sbhwK4UieHBQiozVCAlslbDOl7TbmxTN5ikq1M2jOwYADYMSuDd5d7eEeWFheNixWl2hVQjrw10aqz4pg5V7Nk3aNEm7NpaI6mQeIQHun8Ogz093XN2wJf674yWHT9s+JQ0J/rWqcbjJXKk0DElBDKYiCt6tTIT9t/34jrusVgIW3Fa1cixVHWGHo92R0dELBMBiDUhy2T3WKRtUzaGOhutNxjsS77lEV2vAcFN/SRG1zfKNuuwEDLDHZMwBh43ZFwCo0yOzj/icFNB3TBvLb8oPh4A2GZb9ZZtDAqik2Puu6jwZJI756KKBMDcp93oUqmciivqJ/OTa9Llc4XgwhYJvshskzeYpO1TL7T335+kjECCFdrlpwjv1IwuM2Z9aNoOhVnSFNTso26IEQcraC8zY1y95SoK+0UGjeiCx3OBS2d/86hKBT2aKWpPKURAbmuUHyXQOkoj82TvWv3gdY9nWvl6x6vRNMb0WUQdZVP3OR69RG64njLZGh89UXL5AFl0sbAwUYvCRqDzo3ONYWQYzXEI4oT5cYufBTWAx1dRgybAvu8AlSNTYXf6qgpSuVhnfa5jjiAS6HI40hdT6FBVmZ6zTVgZuXFxjwSqPzCUOC6LygN/IyZyd9XOhR9ueeAduY/BFiWKNsaPGtrsSF9LU5pCOQv8xURF35IV/SrC2n8qAcBtYFlMmAjLCyTtzJ8OJm0MdAj4BlHM/AMKqEc+NfAUCh+nVHEpYeg9gaEnrGWf8krcAMdHWzQI90rHdQfhc+rcTWFF9K2FFuIH0uDAQ1vxo3yxQbMd4piuYG45kyc5LOD6FEPvhFAvcCkFhQf81QaDPDCB5q6eAEDsBoTKPuSMTaGxjWnsUxaJg8vk/6jooMNYaVS0LnT5Z5rdcQvihwl956uKwWuaxTW+zonM3BEi/0FtVcQdX9VYQoGSstAxncZMC4cJiIg3ApDTWfLZA92wsYyOQED4bW5TN7taTtH54sA7uQjhvJNgWrjYxhEmd0nMwQCMJQ1xjV/RBwHeRaeeBdudK6MrsFM+mvjEWmcpBuBo2Jnmexuz+SxR5BJLxMkb/bVCTLQT9kcNsiQOiqDApZoqahZ46cc3OpJFDWKSQfraJ/qXNLFFf9nHUkD5ejggz240KcoxKR87EhsqjGUpI13rO/eRVsmB1rAMlm8Nj1lgrK5TNozMNCJ93gclBhu9a7w34r8r64HC+Jw4bEuz5o1B0YB7ns27aUM5UwEgwDX6mr/rSD6CN7V/U2v6s1SDG8oTDGCMC7/R4dDDwLCE8Mya5kUf6V89dSiM7rMY5nshGd5pNrlEDJpY2B5W69BgZ3sYzd/pSi/tUtdRFk/5iMZyYKEAs8AmwrxCHCUA5Eu0weVV3o60hPPl+J3AecpHP5+SuIrTXsEmcRDMSlYJifBNTfxIWTSywRzm3elfBJO3PXManF18yGe5hfk/qVn/5e4+MEv7SUs75lo8cW78r39hKTbpITf5u62Nhfbxcys7z/F4b+34/JYJQlTy2TCJpvZRxNysC2pmfXdXCbtGdi2XwyWpo5TvCamMy7upiEwmH9mAmY9rONTHrN3XPjFa4E6Jw2hDGhf46w9KZYmtg0C6rOWyW2gdik7ImDPwI7gdxWtgQdX3+zZrPK/q/yXFG3xxbuy7FDejc5FmTqXM3aWDqp1Z8VPpfu2LKPn/Eo0ex45OjUCwvpOaprXQk/Y5SqTLLPBWxkei9d4X0NT1i2TJVIZnHOTSRsDGXSKBgvM0l8Sp87Cpp7aV/caaVu3So8Cn7JrtSovIvZQ17Uv7U2lq/SHVT7iHcOHQbX2rQLF4zHhq4CHep9cPDssQ6CSEbV9TjJZM/rFG1+r7JV9PTukTIpvy+Oy/jsq991RqZxoSwQK97wEgB2oXZ+0TMaLyiiUmwje1zWegDJ8pws2EDLwzfZSlMSOdFZ9mWnxdgXLJpVnJNSB9li1TUI5PuWFgGVyp/awPG4HvD0D22E9qqSggHqtexFJ9mqhymKPQGtfguIvlT+qHgdOhBE26u+SD1zHJuu/V8TvmpG+v0XAMrlrT7hGeQTwzWXSnoFd+/mswtd4m2AWIyfNlPyVygPg9E/x+I8D8Jkri7nI5Bm9eNcoj/TzzWXSxkCuw0s/Xwj8UT9/2l+rfJ6Ab+egqhniqt9F2BEC96dl4Gchk+qftT0Ey6qUTe5rlEfA31wmbQxk0+fNSCYIPBMfrWUSDbTsr3ieCY+p2bgngq9TEzU9I5AAgWuUR2DbXCa9ZyBBb92YBJva6CgOKyAgpT/675JXKH4vkvwXBTMwh3kIWCbn4TaY60rlEVw2l0kbA4PdMbsEzODezY6rEzGkAeja3hrAuGy+OXGiFl29KpbJFSG+QnkEzc1l0ssEK3bilUgXg7YExN6BlQC+QrIYl1v+F8bZILZMnq1F96/P5jJpY2D/Rp/EgYwA3LkcdBYHI5ACgYciYs/ATCQtkzOBc7ZLCGwukzYGLjVHvs9eijUbA/m2z2E4Cx4mvEz0KYf5CFgm52PnnBECe8mkjYGoEQ50yRfyeP/WwQgsRQCjkv+jwNvkMB8By+R87JyzjsAuMmljoN4IR7nj86i4kRyMwFIE6Ef+xPJSFG//5dMyuRxHU7gd2zeXSRsDB+x6msXRUbAeHYzAUgQ+EoH4n+6W0rvK/JbJq2z2tSq9i0zaGFirOden+40GoA/XL8YlnBUB9R/2CryrM54mh+UIWCaXY3jVFPaUSRsDx+16fA2v9aW841bHnO+AwBOV+cMO5Z61SMvkWVt2u3rtJpM2BrZr5KQlhdkcfz3s5YKkyF4VMYxJLxEkanLLZCIgr5vMbjJpY+DYHY8/Jjnjn5Mcu1UOwL0U12OxyVsE/r5A2vayTKbF82qo7S2TSYwBVWLU7HRsuqtp/YUVFZ7FRkLjuhDI68yOR+CT66z6erW2TK6H7RVQ3lUmFxsD6vyfq5EejGwoNiuR3iEdArvsPE3HviltjYBkkL9i/lJnf1tgHfAtk+vgelqqOcjknbdv396IEXYVf6+D92Rf6/69MagrHbvZH+k82lWttBgDb3T+ZkwZW6cRX2DxqQ4+6hMPlj/Csw68IJ/pPLrOSr9qEC+0A4bWV6sWZOKHRyD03y90PtTmU/F7KLkUv5bJw0vLNhXIRSYLz4CYQTmj/Fg/HPWxA6VHOJ/q3KkUea7jlY7aEoLuUVgoU/JnFcQThsr/BqY+0n11KA4jCTcOXxrL6k9dxBc7wrM0rsSXQ14I0I+PZggcTi4tk3l1+sy5yUImm39hjLv/2UjgcDVe2onMBiUMAf7esxnIR/4sBiUJLoYJnhH4/bPuW+9dI9w6MJhIM8pgUrpJQfTfTsrQSKz8jRjfGoE2AlP6idLeaVMYjlG+Qv51nm2kKu/ucikeLJPDze0UCxFQPxtFQelmyeMY4u+UiVQIwksYq+ieKM8lZV54GpTmzS3Z//wqDnf7bzpwV7ae/yflZld/U0ko+T8N8IPBQL1X2YEtuqs19GZIZlKQsMTYZEaJJ+q57lsGXiasnpUNFDnHkrC7XFomlzRfPa9lso5HbneVMSDGepV3k2k1KuthQwoR4+LSB03I/0TH7JmD8i4OoYPiEWFJYMgwwcvx3eJCTWARAmon+haKBgOO/S3fK65pxLKc86HiO5ex9MwhYwTUbhhzlsuM2yhmzTIZo3HM69gYYID9SY2Kor+vg0EWYWR9v6n4MRyag+9N6BB4C8qB+oHimE2/0Lm5uY380NnNGBBPKBNmj7xvfclwUZIigMOlpZGQrH4SbfB4quPX8ARsc/GKBJYOdUL5/wGOA7a/6bLpVaHvtvooeRzyRkBtarnMu4m6uLNMdqFyoLjCGAgDKoMngVeOCuWvM4ocC51XZeLwUDctpaj0DL6lQfFY9yj7vsAmPAyQwSA6lDUqbUTsF+Vr8h09Li7LZY5WXZoJuRc9PAeT3M3KA9/Qx/NQ5NUZrJ/q8KxVIMwI70d5UBy0SzPQ9+i/DsdDwHJ5vDazTB6vzWocF8aAYkpF+4kUVdMLwKy2GYjDZd4XGIiHlCb5GcgHg3gqB4fBtBMT4AUhjPEK3Kac8Cu+qR/uarwrMR7gTbzDDAQafbTLWIUqButa/WYG184yAQHL5QSwckhqmcyhFZbxUBoDhfJuKCwoM4PtUpT3Fd81GyMPAWXXla94GH4wOroMjTjN2tfU46bRkXvLVDpel6zqzb0Ss+8BZR9bxiWN0uNAPpYjCH/UwTcL7MIu4Jj/IwzBl+WW2NC60T1GGJjXDFvdf66juVw1nwHnvBGetEE5mYgRKWWryyAb8tqVeWvtFxOPr8VDJZdc6xleN0I52WCSU8mt4i2XBTzpf0J/sEymh3Y0xSADbL5FFsaGj0pjoKW8RRBB4ng+lhrpAiPkG5r5DhkUU4qdmxbvxCjAVC9mKwxOheLR/QNdl4NgH40CV6W1AhJYKYMwxSNQ7LvQdVPJ0zY1Y4s0zbiU/FwrLeHapexvFI+8zP0Q1my5VJksc1Y86RrF/7OO+ENqlksBkjoIa8tkalBn0FM7YPh2TU4vUntHGVFkXcobgXqj57VZV6B2SVgLBal81WBMGTpgMA6UC53BoLx9s49LeYdmH+SFx09Fn0FraBbySGkqb4euwYUySpcm9LrCi65Ix81HQJgzuKPwMQIgRNsVBldoj2JmqOvSG4Pna65igr7DtgjMlkuxiTzHb5egoIhjM3M8llkuE7apsLVMJsRzD1LvqFDc3DdqzEp5B0b4JG+x01/PuGZjYKkwOWNAdIXafgHlqc2oowx4Bkp6UXT7UjQqS7/9dFFMMVCIAjz2zt5VPumezSiJ+rEs0ArgoqMyLloJHNGLgHDrtXoDpk1ce9u2txA/2BOBJXLJWPFygHnL5QBAUx9bJqcill96jAGUd9MQgFNm7s+5UHhPjR2/AoiF/ah40v6plLzyFF6HMEA3UzKgx5Z68/nq9+KLVwp544CZBF6QuI43usfLAT7PeD6DIQa1miEjOmBCPIeDETACDQQkI7PlUnlrMizSyB/04rHGctnA3LdGAGMA5dT1ChavvX0sIXqso/kK3POePIouXpf7WnnwJtzo3DcrQ9HWFCXptw7ij88M8846a43scyiV/mtdYyQ06z6aReXlS4sskTD48Colgfvd633Lin+NQJ4ISEYWy6VosJSE16/mSVK85TLPZjdXOyJQ/GvhnPIlUCi36t35KTSUt9ijoHO8qWcKiWzSqg4MNhgSh69LNqCakcMjEORit30aKp8xhr1GY74seni8XQEjsBQBPANzQ+lqmzPLZbZNfgcjYATOiQAettLLtmkNgyHAmyYs8d2Ee86j9ihtyqwLMwKZIDDbMwD/Eq7ygzqjhUx5Cotd50JQM8FhNhuqhz0Ds9FzRiOQFoFyfBHVeHmPCUvxGmra0kzNCJwHgSWeAVAoNt/pPEWx47qb402gvGxCGHSoB3sfcIfi6fhV5749EtnwbkaMwIkR4BU39kFxroLk8vBjTlUZXxiBFRBY5BmAHwkZM31ekxtUgkrDe99sDBrtSaAMByNgBIyAETACRmA9BBYbA+uxZspGwAgYASNgBIzAFgjc3aIQl2EEjIARMAJGwAjki4CNgXzbxpwZASNgBIyAEdgEgaUbCGcxqT0DXf9VMItWLplUJz5wUtu0lAtvR+RDeN45Yz85YltM4fmMbWbZntIDhtNatocx2iPF5saAOkKx+16VbX42dI/6pyyTtwneV/3iz56mpH+NtO4Lz6c64tfErhGHQ9TZsn2IZsqFSct2Li0R+Nh0mUCDBW8efKbzqQwB1QevwI3ONgRCx0pxEp68dfJCZwxIh4wRUBtZtjNun9xYs2zn1iI3N5saA6o+3xj4JD8YFnPEX+b6i4qLYWwT0KDBPxBiQPLuuEO+CFi2822bLDmzbOfVLJstE6jh+VLfLzrv8onStWBXfZgR8dGhrn9+HF1shM9q32BQGcywUarwzH8pxP/7rtttg/j5XMfg9ynEFcsE3+rgI1cOmSEQ9V3LdkfbRPhYttv4WLbbmOwSs6VngNnzs11quW6hdOZFa9oaLPiHR/5CebXBIkCA8uejTyhgeOZz0rsF+NDBjPJiUBpwYdMpRoxDfghYtnvaRH3Wst2DDdGW7QvgbPxoE8+AGpw19Zc6n23mwCz7oerV+alTxfPFxTig1D6I0+saGsR1znrH0KAApWOZgvJKw+SPuoZ2/E32+K9cUayrtkfEE8bHc9137anAQOFfH0u+lbQzYDSQphPrzhyOXB0BtZtl+xZly3a7t1m225hkG7OJMaDaM3MYnAFmi1I/YyjglrdDAySKlvqy1l3N9nXNUslDHXGARgubiTSgxyz/sfJVbnddYxxU7nXdV7wonnI7DRDFTwqiS9ldfxVLPJ+q7lX0evaTDoyBi6+b6jkejcF0kxh34hQIWLaFovqmZbvRm4SJZbuBSc63dzdiDiW1aE19Iz5HF6P6MOumXmxwq0KIRwkyI4+VL2nAoIlDC5sZNKDNDK1Jmz+QesHDOIg+xgf8NdPHyUZdiwZuUIyfrtDFU1c6+GA/w1Ag3ZOhRH6+KQKt/rtp6SsUFuTPsm3ZXqF35Utydc+ABAuL+WW+EMzmjBkRs+tmQDmyUbLlElccbvlqlqx7lGiXq340jajwR7p+HfC+r2s2CH6t+6axAs/F0oGejd3ApyztEMpCQVd1aqTCGPm+Edd1i8FC2hqvHQkxsj7WcapXUzvqeYio0P6WbbWWsLBsd/day3Y3LtnFbuEZYJBvKcbskIgYYpDTMTRTJU2XUuqcjUfk40uMgddxRLieQqPMDr8oeRQqChojrIa7nvGFRJYOftP1W51nr78rP7P+oU2PLInAy1BgIO3zLsR5y3rFcb7eDwHLdj/2lu1bbCzb/X0kqyerewZU20JJZVXrC8xIyaEsmfXj4o833FW5Qhpc7V2Bzs/RCsrX3Ch3T4m60k6hcSO60OFc0NL57xyKQmFXSxWK66wPeQl6zgDWN8snSfEqotJRHhsnu4wh0pW02AdQlR/iu7wRXQZRQSf+ES2MDw5e5azRjdP5ejMELNsBavVHy3a3p9GyvZk4LitoC2MABVOboS5jeb3cEmiUJwqOnfgorQc6unhnYyBu+K7AhsJvddQUpdIzU3/eyIBCKxR5I34KDbIyKDfdtczKiw13JFD5haHAdV9QGvgZ4y1gGQJFj+FEoI35bDDLEqWSBsuaVyCkr8WRWYH8Zb4i4sIP6WpGzoW0frQuApZt4at+bdm+HQss2+vK26rU77x9i7d4nSAhYbB4pfOddUpYh6r4RUH/poMd7LUd97pnPR9FWO3ab3KhZyhn8r3SgRKGHrRaCk9x4NMyLMbSUDoGIjwZhE90XxgvOuO5oOxfdd3LK5nmBtGlXuDBPgTKKIwBxcc8lQYQLmVm9F11LYyKMXwqDXsQXoxJq3QOKyEg/C3blm3L9krytQfZtY0BlCKu5T/sUbklZYpnlCmK7j1dV0pc1yh4/pAIRbs4iA6b4mqvIC4mejACUzBQWgwPjLExHoyDIXEcdoW/ZXuguab06wFSh308BQOltWzv2NJ3Vy6bmWOlSFcuKzV5OiahWkNXZ2XGyww/iSFQUL+dVV+tYhOW9JEbncf2k1+VnFmpw74IWLaH8WcMsWxbtod7SgYp1jYGWFs+ZAjKiTWwT0uFpWvc8c9SVki0KYOZ7rUquDkD5mH7Vcq+szOtw7aBZXuznmPZ3gzq5QW9s5zERQrMHkbtJr1IJXooQaaDscZcztjZ7Ec5xbvzOqcMlIU7FIOAmetLnd+kLABaoskyAZv9nulITj81v6noqa4sw7CMNNYrQNFXg08qnFeig8xZtgfAVd+2bA9gFD22bEdgbH252DOgzs6mMNbXu8J/K/K/uh4siGONnY/68Cc3HBgFrOOzgz9pEG1m7WzIwyPAgXGwSgj1WIV2xkS/CxhPYREF9D9TMjjtPATUNtnLtnicJZOh31m253WNMbks22NQyijNYmNAdWHmjDLeKvBaWfMVljkf6RnLL8sCzILKd/fH5pucTgPUVVnGM+v7+8nAOsNcBI4g28jm3GDZnovcQD7L9gBAGT5etEygBsddzwYZ/pCCL/Lx9bs4/Es3/xdHJLge9dndBOUUJKiTDmYQ5bJEKtKmMw+Bfyrbv+dlda6xCKjPW7bHguV0qRCwbKdCcgadpcYAbnrWxDAItgrMVv6kMt/oYNMdSrpvmWI0T6I19MEFvgcwmp4TLkNAWB/q2xTLaptfbuFv2c6vWU7BkWU7z2Zcagzgopvt2lanKJV5HzrF52/Lh0pfuAR1LsrUuXTds3RQbUKbShf6ynNo5SP+mcnFATw+UPyWhlpcvq8PjID6Ta6yzR6BYhwI8D4Wr/FkoDZmkEbPLdsBLJ+MQB8Ci4wBEWWWXnwGVwLHjvvaJ3j7Ci3jlR6FNUVZVeWVNHR+qKP22d0ZdCNyx7pUXTGoGAzx0MQGEd9EABsHIzAHgUrW1K9yku3acp1446uXU8aQOVjskkf1smzvgvx1Fnp3YbVZS79Rp+UVseamvoWk69lVBsqNHf33dY0noAzf6YINhAxYhcegfHD2s+rLDIm3K3itsjIEQr1pj1XbJJTj0zkRsGzv2K6W7R3Bv9KiF3kGggK6ZJUne7VQZbE5sblB8Ubxl8o/e7NihPGaZTFwx5VVHIZRbRYVPz/w9e/F++8OzP8hWFf/GfLaWbbXbUnL9rr4mnoDgaWegQa51u0abxO0CrniiDVfqcwV1n+KsX/kytwV8ZWLbJ/VG2jZviJhyqGqaxsDCOphP1uaQwMN8AC+nYOhZnazPsYyUF4Oj92fcmiF2363e1uon5/R+0ULW7bz6OdXw8XaxsDVALlTRZ+p3NYyiQZI9lc834mntYu9pwJer12I6RuBnRGwbO/cANdW/KI9AyPAYt2RwdthBQSk9NkvwOZB3ibgK5DMJsCbDyU1NxQq+hSB/6Kgng77ImDZXhF/y/aK4Jp0JwJrGwPM4N7tLNmRSRDQoHFtbw1g7JzV0EnSJzYiYtleGWjL9soAm3wNgbWXCYpBW52aAdzBCKRAAONyy//CSMHzGWlYts/YqvvWybK9I/6rGgMyAnDnctDIDkYgBQJ8SMmegRRILqBh2V4AnrP2IWDZ7kNmg/hVjYHAP18otDGwQWOevYjgYcLLRJ9y2B8By/b+bXAKDizb+zfjFsYAX8jjnVkHI7AUAYxK/o8Cb5PD/ghYtvdvg7NwYNneuSW3MAb4Oh7uHwcjsBQB+pE/sbwUxXT5LdvpsLx2SpbtnXvA6saAZnEM3lh9DkZgKQIfiUD8D3VL6Tn/AgQs2wvAc9YmApbtJiIb369uDIT6fKOB48ON6+biToSA+g97Bd7VmdmoQz4IWLbzaYtDcmLZzqPZtjIG+Bpe60t5eUBgLg6CwBPx+cNBeL0mNi3b19Ta69TVsr0OrpOobmIMhNkcfz3s5YJJzePEEQIYk14iiADJ4dKynUMrHJ4Hy3YGTbiJMRDqyR+KnPVPRTJoyvOyIIXzWLXjLYJdvi8w1ogdm+6ELWXZPmGjblEly/YWKI8rYzNjQI1ebCS84gFzXIs4VRcCeAQ+6Xqwdpz66+cq48HIctjTQPqrCpbtq2ru1JW1bKdGdCa9O2/fvp2ZdXo2DRpsAvteZ393YDp8V5lDfYW/Yn6l8zdLAFB+lDR/ckR4rvvBjYhKw6bXRzqP9miFct7ovIjfgssD/ai+lu0DtVcOrKrPWLZzaIjAw2aeAcpT4/OxmK91vrrZU8DbpwkIqJ+wx+SezosUq/LzcWJCPu8AAA6SSURBVBwUNEqdv4b9XsfFoLQot6chTystz3VgpMBjFXT/lW4+43kVeQUXqq9l+wraOVUVg9xYtlMBmoDO7H8tVGMucikoP1ahgxEYREB95dPBREqgdHea6cp+pnNhUOiMUYA7/4GOS94B+uelDYvsY8AQeN0sM+Qj/yHfoBEulu2ORnVUegTU1wZlW2lacg0nii90iM6W7QRNs8QY6GygBDxdHYnQqfGWMKsc5cK+OpBmVFi4stYPrl3LUijyS8bAE+W/pMyh2flpZAYnHb/p+EIHM+ZDBfFs2U7YYsITpWX5ToupZTshnpCabQwk5uO05DQQMIPEZYzyeU8Heyaan9TFjf2h4kevTSu9wzACT5UEhV3hrevSfX+/L7vSsFdg6M0F2vXSdw/I/0THoiUO5XfIGAH1Fcv3Pu1j2U6Me8sYCIMlQP8aykKBHXKGkxirueRQ/n8gc8D2N102Z15YuZXCIq3DMgQC1ij1poH1MFC+pOyZ9bfaQzQZ+PEWlMYdSw3sP3ihM16dOJAfOlkZAwEXy3fcUsuuLd/L8JucO/Rhy/Zk5C5nqBkDApnB7msdH+m6cKHqjKJi8GgOqopyGIHA+1EavANdbmOUxuCmtoiOL4cRKNcim0qd/ky4ZAxgMCAHtSBZgNZPOjMQPdaZdusLr/QAecomiF/Ld/rWsHynx3SIomV7CKEZzytjQAMFigp3NTuh47VUBhDiHWYgICxjpcPaIX/I0Qwon0vr0830vh9G4OOQhDcC4tQocjYRxu0SP+eamX/XxsAyHUZALCNlfHwmPzKVRVB9Ld8rtESjH1m+V8C4g6RluwOUpVGVMSBC5UyI1z3Y7ELgvewfdd+cXRUP/TMeAWEIviy31JSI7hmkwbymnHT/uY6m63l8gU5ZLL0Iw8r40jVKHmPg2QA89/W8y4NTZsNAvrRfgHS0J+XlEizfK7aE+pble0V8G6Qt2w1AUtzGxkAxwFkBpYC1TkOYMmMo9l3ouqnki44d5yCN7m2AxaBMuBZ+pRJuerTY0EeYvY4faJez7Ftq3b9DBkV3rvViLd8rYas+YfleCdsmWct2E5F097ExANUX6UibEgio8/6sEwofI0CnYsZYzPh1zyz1KZG6Lr0xuKB5D95eAYBZFmpeGJFi3wuv/V2a9VMiLv7SoOA+DihV2qsy1nSNZ6dJk/zQySlYvhO3htrd8p0Y05HkLNsjgRqbLDYGcGuWn2ut5VeH57W3IbdoLY9vbhEQbvEGoxosAdMmrjYCaihNvxGu7AmoZdQ9hhez9TEbYZGFvvX+2n6BQJf0zcGJsojPJVi+V2gJtb/lewVc+0gKb8t2HzgL42NjAFdXbRObgGd2QzyHgxE4EgIYWcXsPurHf2YwGVEJFPujnnSVkg908eI0DTqyoiSaBkIPyU2iLd+bwOxCNkDAsr0CyJUxoAEN9ynuTgYNXosicF8zEG6j/WsEskfgE3H4pfovM3y+lfGBrsfO1J8rfd+rnngW+H+N4vUmnfs8OY+VLhvZEZ+WbzWIwykQsGyv0Iyb/mvhCvybpBFYBQEpTwzi6nsbUwpR3mKDoc4YIQ5GwAhkhIBlu7sx7nZHO9YIXD0CLbf6BETwHpDfwQgYgfwQsGx3tImNgQ5QHGUENHvg9UP2AzDLHx1CevLNfn2xWZho8bXD8qtrzce+NwJGYAICQTazkO0JbK+e1MbA6hC7gAMjwAeLyo/1jK0G6VPvFWAjZLEZciwTTmcEjMBFBHKR7YtMbvnQxsCWaLusQyGgGQRvHvB57vIbEBf5D+lIP3aj4kV6fmgEjMA6CFi227h6A2EbE8cYgawQ0MDFNxJwa/a9uZAVv2bGCBiB4yFgz8Dx2swcGwEjYASMgBFIioCNgaRwmpgRMAJGwAgYgeMhUCwTyP349nism2MjcCwEJGd3LnGs52w+5GNFzcBXDwmvb0+131+Ur/pnxtqTcGP57kLFcUYgHQJDsp2upPUoec/AetiashFIgoAGGu8ZSIKkiRgBI9CHgJcJ+pBxvBEwAkbACBiBK0HAxsCVNLSraQSMgBEwAkagD4Hqj4r6Ehw9Xi5W/nAGN6vDlSJwhvW8K226i9W2bF+E5yoeWrbTNfOpjQF1lAdA5Q6TrsOYkhHIAQHLdg6tYB7OhMCpjQE1FH9IwZ/GOBgBI3AuBCzb52pP12ZnBE5rDEQzh1+2wlhl8pW4LD5FKz74lv3TUPd3w/kTxfOJ3VOFnHBfCVja7HTtNhcrtXfp8bNs68uUAUfL9twO5XwFAqc1BlQ7FOFmXgENUHy/HkMgC2NAfHwpnqo/zNE177D/rOM9HWcLGGEf6jjl53pVr5/O1mAL62PZtmwv7ELO3kTglN8Z0OCJtfy9zu83Kxzf6zmzZ/4a9gMd8czrRz37JtDhj2cuGhV6zgbFRyPSzSpPdDE04PGxDgLKAX6hV4avle6H8kbXfEjqA50LRaIzmLzS8b6uN5tRlfw0z+KB2R3YVgZLnEbx1PmRjnLzJzxjaBVto3MthPRvdE7218G1AnyTBQJqX8u2ZTuLvng2Js7qGRhcTwzKgxnGMx0f6b4yBnTNLJOZNMoXWr1B6VDIT3UeMjxQbrPKE21mvF/pjIL/SWcMgyrovlD0Ov+go/waHUr2ZZUovwve8ug1SlSPYpavM3X+u85D+ILPzzq+01G1ZX7VNkcLEbBs3/5FtmV7YUdy9joCpzMGpAhQjLiNO12rikd5o4hI92fdtxSS4lCqKFzSdNJRfBkYnDAcOkOq8kSnWCdVIT82C9IzlCV8YsQU+xZ0bs6QMQ5I16pvk97a9+IBw2gwKF1Z58rjMZCJdqA9Or0NA3n9OHME1B8s27eybtnOvK8ekb27R2R6gGdc+sz2+8Lf9OChjiF3OQYDbuehPQBPlKYpnHHZqcqLlwhi+uX1/fKieQ5KFXf7xdl1M98a94EXZu5jZu9lnVsGUBdvoR1oDww+h/MhYNlutKlluwGIb2cjcCpjICiBhzp3ziQVz6yR2eaYnbevle67S8iKHgq211hIXN7H8CKarZm94qgTB0sINX50z2yKemP8jFHASrpq+Fh8XDKe4sLLOg95Z+I81P9JHOHr4yOgPoOBZ9mOmtKyHYHhy8UInG2ZoFyTbwETBAf3NK7yTmOhkQml0uv+D2lZSuhUVCuUVyj7Bo83oRy8GPF+gSJZePaFzsUeg3B/o3PNYGjSXOte5YL/EKZx8dS5ZfzECTquaQ/qO9bg6CDhqAwRsGxHjRJk2bIdYeLLZQicxhiQcDBzYM0cV2JXKNeRRykj0WEWPaSIWG7oo5esPPFSustvdF2ut/9RZTPrx4PBBsgar7rnGbwxYKBUCfDUh0+RYK2fwM+YZZeCBaUv69xpbF3gkzcmyrwXkvnRURBQX7BsR40VZMmyHWHiy+UIZG8MhIHgRmeU86XAK4K4w/sCLn3CGK/AbcrhXwYplHFXSFleMbNXIbyKV8zqAy7fKg6l3zXT55sC8Me5CspXGilV3EYXg69oNvgo6zxqv0CUl/YAE4fMEQh9+EZny7Zle0xvtWyPQWlmmqyNAQ0SzGjZgMfa/ZASQ9lc+qDOfTBSmi7FyaNaULp7OoYGKWj2pUlZHjPd2qw68IZHgFfvnuqozfgV/wfFTQ7KhwEB5pzHhpZnIs4omhhqfR6UOGl8XczulbfXMwCvOpr4075TeI/L9PVGCKjdLNu3WFu2O/qcZbsDlJWjsjUG1BnKmfVLYYAy6TUGlHbMWjRW5SglEcpGqfyiY25IWR4DZ8ujgcAE5pLNhEUT5ZrsrQPRgzeU9igjLNSHE3XuxV/06BMYCvAbB4ywZlz83Nc7I6C2s2z/pw0s2//BoriybDcA2ej27kblTC5GHYINcSjAwvUfOkgfHbwCxUdq+hIovphhKt0YxflI6XoVUVTGJYWfpDzxUcyQVWaXu7x8Bh+5BvAGT74IWR2KYxB8HOJqyztRnZ9fqBRfV+wyMDCQcsbjQpWu45HazbKtpo76uWW73vUt23U8NrnL1jNQ1l4CU74uhxu8tUNcz5khtmbNZf7ojMIhLbOSXsNB9Ej3TMeYgDLqMy5SlVd+UbDrNUc+10tg01wRxP8DHWMMmTLLqmfxglFUGEZxQYr/jXidy/rFj8v9Aq18JFIePEFdAyiP8Qx0GQk8c8gIAbWjZfu2PSzboV9atvcT0Lv7FT2pZBQrX9YrZ8JxZoyEQeWtvCgIFA9/4INRUAvQ1lEYAjqPdTOjdEuF3KS3uDzxgaHxpEa4flMZAUQrPfj0GSf1nPvfMYPn6AqFG1n1aRk1iqONOLoGUGixxNHKxwOHLBGwbHc3i2W7jYtlu41JspjD/FGRlEDru/yKQ2nggq5tnruEjtKigIoBSOdS6b/WNS7szploHz2lx9VNvt6Ni3PLUz54bIZXiq95R3TPxryHOp7rYJNh7XmTwN73gV8MltKww6vzQgd8f6uDZ+BKiD0+tXjR6fIo3CieQZRlo0ltWZTmn10QUFtZtuXdEw412dW9ZTvqkcLDsh3hkfrySMYAgsGM/j11isINrDOvzfH/AqVST43PIL3QQS/uph8k4gRJEFBbYDDwr4a9xlmSgkwkKQJqL8t2UkTPR8yyvX6b3l2/iGQllDPlwgugzsHM8qXOuxkCoWbw1fumQ7Lam9AYBOgbZT8Zk95p8kCgbDPLdh7tkSMXlu2VW+UwxoCUPt4AXL/lej8DSDmIrAxTP3nxhWuP/QzMSh12QiDgTzvUXK07seNiJyCgNrNsT8Dr2pJatrdp8cMYAwGOQvmrc/Atfv5joFgu2Aaqi6Wwfo2r02E/BMDfHpr98F9asmV7KYLnzW/Z3qBtD2UMSPnjGcAAYONg4VLcAKPBIsQXSxVsWiv/N2AwjxOkQyDgDv65GIfpKncllNR2lu0raesp1bRsT0FrWdrDbCAsq6nOwTIBH6Xo3E1epvPZCBiBYyFg2T5We5nbcyHw/0VfjBLP46TiAAAAAElFTkSuQmCC\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": { "application/pdf": "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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2021-04-27T16:02:58.463706\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.3.4, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "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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\n", "text/latex": [ "$\\displaystyle \\frac{5 i \\left(\\frac{i \\left(2 + i\\right) e^{t \\left(-1 - 2 i\\right)}}{5} - \\frac{i \\left(2 + i\\right)}{5}\\right) \\theta\\left(t\\right)}{4} - \\frac{5 i \\left(\\left(- \\frac{1}{5} - \\frac{2 i}{5}\\right) e^{t \\left(-1 + 2 i\\right)} + \\frac{1}{5} + \\frac{2 i}{5}\\right) \\theta\\left(t\\right)}{4}$" ], "text/plain": [ " ⎛ t⋅(-1 - 2⋅ⅈ) ⎞ \n", " ⎜ⅈ⋅(2 + ⅈ)⋅ℯ ⅈ⋅(2 + ⅈ)⎟ ⎛⎛ 1 2⋅ⅈ⎞ t⋅(-1 + 2⋅ⅈ\n", "5⋅ⅈ⋅⎜─────────────────────── - ─────────⎟⋅θ(t) 5⋅ⅈ⋅⎜⎜- ─ - ───⎟⋅ℯ \n", " ⎝ 5 5 ⎠ ⎝⎝ 5 5 ⎠ \n", "────────────────────────────────────────────── - ─────────────────────────────\n", " 4 4 \n", "\n", " \n", ") 1 2⋅ⅈ⎞ \n", " + ─ + ───⎟⋅θ(t)\n", " 5 5 ⎠ \n", "─────────────────\n", " " ] }, "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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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2021-04-27T16:03:00.490226\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.3.4, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "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": [ "$\\displaystyle \\frac{1}{C L s^{2} + C R s + 1}$" ], "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": "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\n", "text/latex": [ "$\\displaystyle \\frac{1}{- C L \\omega^{2} + i C R \\omega + 1}$" ], "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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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2021-04-27T16:03:02.128799\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.3.4, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "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": [ "$\\displaystyle \\frac{5}{s^{2} + 2 s + 5}$" ], "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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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2021-04-27T16:03:09.847820\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.3.4, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "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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\n", "image/svg+xml": [ "\n", "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2021-04-27T16:03:10.255413\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.3.4, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "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", "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own 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: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*." ] } ], "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": "3.7.10" } }, "nbformat": 4, "nbformat_minor": 1 }