31:T482,md` ## Z-transform Z-transform corresponds to ${tex` \{ x (t) \} `} is determined by series of samples in discrete time ${tex` t_k = k\ h `}, where ${tex` k = 0, 1,\dots`} and ${tex` h `} is the sampling period: ${tex.block` \underline{x} = (^\sim x_0\!^\sim, x_1\!^\sim x_2\!^\sim \dots ^\sim) = \left.(x_0 \,z^0 + x_1 \,z^{-1} + x_2 \,z^{-2} + \dots)\right|_{z\,=\,(^\sim 1 ^\sim 0 ^\sim)} `} Replacing continuous time function ${tex`\{1\}`} by sequence of samples ${tex` (^\sim 0.5^\sim, 1^\sim 1^\sim1^\sim \dots^\sim) `} and replacing Heaviside operator ${tex` p `} utilizing algorithm of numerical integration ${tex.block` \underline{p} = \frac{2}{h} \left.\frac{1 - z^{-1}}{1 + z^{-1}}\right|_{z = (^\sim 1 ^\sim 0 ^\sim)} = \frac{2}{h} \frac{(^\sim 1 ^\sim , -1 ^\sim)}{(^\sim 1 ^\sim , 1 ^\sim)} = \frac{1}{h}\ (^\sim 2^\sim, -4^\sim 4^\sim -4^\sim \dots^\sim) `} we obtain expression for approximate sequence of samples of function ${tex`\{ x(t) \}`}: ${tex.block` \underline{x} = {\color{magenta}X (\underline{p})} {\color{brown}\ \cdot \ \underline{p} \cdot (^\sim0.5^\sim, 1^\sim1^\sim1^\sim \dots ^\sim) } `} `32:T4a0,outPNxy = {RLC_js ; //wait for new values pyPN` h = 0.01 #sampling period p = PolyNum('const:(~2~,-4~4~-4~4~...~)') / h # = 2/h * (~1~-1~)/(~1~1~) E_0 = 10 [R, L, C] = window.RLC # print(R, L, C) E = PolyNum('const:(~1~,2~2~2~2~...~)') * (E_0 / 2) # E_0 for t > 0 Z_C = 1 / ( p*C ) Z = p*L + R + Z_C U_R = E * R / Z U_C = E * Z_C / Z # data for d3chartPN() window._h = h window._E_asPointsList = E.asPointsList(h) window._U_R_asPointsList = U_R.asPointsList(h) window._U_C_asList = U_C.asList() ` return ([ //data for d3chartPN() {'xy':window._E_asPointsList, 'nmHtml':'‒‒‒ e(t)', 'color':`${colors_[0]}`} , {'xy':window._U_R_asPointsList, 'nmHtml':'‒○‒ u_R(t)', 'color':`${colors_[2]}` ,'pathAttrs':{"stroke-width": 1} ,'symbAttrs':{'symb': "circle", 'r':3}} , {'yh':window._U_C_asList, 'h':window._h, 'nmHtml':'‒○‒ u_C(t)', 'color':`${colors_[1]}` //can be ovr. in ..Attrs as "stroke": "..." ,'pathAttrs':{"stroke-width": 2.5, "stroke": "DarkBlue"} ,'symbAttrs':{'symb': "circle", 'r':3}} ]); // var obj=window._ans; return obj.__class__.__repr__(obj) }33:[] 2f:{"title":"02. Laplace transform and Z-transform (pyPN)","cells":[{"id":861,"value":"md`\n# 02. Laplace transform and Z-transform (pyPN)\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1899,"value":"md`like https://observablehq.com/@andrzejq/02-laplace-transform-and-z-transform, \nbut very slow!`","pinned":false,"hidden":false,"mode":"ojs"},{"id":938,"value":"html`

(css)

\n\n $\"RLC-circuit\"$ `","pinned":false,"hidden":false,"mode":"ojs"},{"id":954,"value":"md`\n### Example - RLC circuit:\n\n${tex.block`\nZ_C(p) = \\cfrac{1}{p C} \\ \\ \\ Z(p) = R + p L + Z_C(p)\n`}\n${tex.block`\n\\{i(t)\\} = \\cfrac{E_0\\, p^{-1} + L\\, i(0) - u_C(0)\\, p^{-1}}{Z(p)} { \\color{brown}\\ \\cdot \\ p \\cdot \\{1\\} }\n`}\n${tex.block`\n\\text{assuming} \\ i(0) = 0, u_C(0) = 0:\n`}\n${tex.block`\n\\{u_C(t)\\} = (\\ E_0\\, p^{-1}\\ /\\ Z(p)\\ ) \\ Z_C(p)\\ { \\color{brown}\\cdot \\ p \\cdot \\{1\\} } \n = (\\ E_0 \\cdot \\{1\\}\\ )\\ Z_C(p)\\ /\\ Z(p)\n`}\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1038,"value":"RLC_js = { //global variables JS -> py\n window.RLC = RLC; //sliders_\n return window.RLC //not important\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1029,"value":"$32","pinned":true,"hidden":false,"mode":"ojs"},{"id":1249,"value":"chart_RLC = d3chartPN_(this, outPNxy, {xMinMax_:[0, 0.65], yMinMax_:[-2, 18]})","pinned":true,"hidden":false,"mode":"ojs"},{"id":1183,"value":"viewof RLC = sliders_( \n [ /*params_ - (only name is required) */\n {name:'R', attrStr:`step=\"0.1\" min=\"5\" max=\"200\" value=\"20\"`, htmlLabel:'R'}\n ,{name:'L', attrStr:`step=\"0.01\" min=\"0.5\" max=\"7\" value=\"2\"`, htmlLabel:'L'}\n ,{name:'C', attrStr:`step=\"0.00001\" min=\"0.0001\" max=\"0.005\" value=\"0.001\"`,htmlLabel:'C'}\n ]\n ,/*d3format_*/ \"<.3g\"\n);","pinned":false,"hidden":false,"mode":"ojs"},{"id":1632,"value":"colors_ = [\"Crimson\",\"Blue\",\"Violet\",\"Purple\"]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1630,"value":"styles = html`\n(styles for sliders)\n