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:[] 2f:{"title":"02. Laplace transform and Z-transform","cells":[{"id":861,"value":"md`\n# 02. Laplace transform and Z-transform\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":938,"value":"html`

(css)

\n\n`\n// regexp for math:\n// ^\\$\\$([^$]+?)^\\$\\$$\n// €{tex.block`\\1 `}\n\n// \\$([^$]+?)\\$\n// ${tex`\\1`}\n\n// €{tex.block`\n// \\${tex.block`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":872,"value":"md`\n## Laplace transform\n\nThe functions of continuous time defined for ${tex`t \\geq 0`}, can be expressed in **Mikusinski's operational calculus** in form of functions of **Heaviside's operator** ${tex` \\ p = \\cfrac{1}{\\int_0^t} \\ `} where ${tex`\\color{magenta}{X(p)}`} denote ${tex`\\color{magenta}{\\text{Laplace transform}}`} formulas: \n\n ${tex.block`\n\\{ x (t) \\} = \\color{magenta}{X(p)} { \\color{brown}\\ \\cdot \\ p \\cdot \\{1\\} }\n `}\n\n ${tex` \\{1\\} `} denotes function, with value ${tex` 1 `} for all ${tex` t \\geq 0 `}.\n\nFor example ${tex` \\{ 200\\ \\sin(2t) \\} = { \\color{magenta}200\\ \\cfrac{2}{p^2 + 4}} {\\color{brown} \\ \\cdot \\ p \\cdot \\{1\\} } `}.\n\n

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":0,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":1752,"value":"html`

(img \"RLC\")

\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":"h = 0.01; //sampling period","pinned":false,"hidden":false,"mode":"ojs"},{"id":1029,"value":"outPNxy = {\nlet p = new PN('c_P_TRAP', 1/h);\nlet E_0 = 10\nlet [R, L, C] = RLC;\nlet E = new PN('c_05111',E_0);// const for t > 0\nlet Z_C = new PN(1)\n .div(p.mul(C));\nlet Z = Z_C.add(p.mul(L)).add(R);\nlet U_R = E.mul(R)\n .div(Z);\nlet U_C = E.mul(Z_C)\n .div(Z);\n\nreturn ([ //data for d3chartPN()\n {xy:E.asPointsList(h), nmHtml:'‒‒‒ e(t)', color:`${colors_[0]}`}\n , {xy:U_R.asPointsList(h), nmHtml:'‒○‒ u_R(t)', color:`${colors_[2]}` \n ,pathAttrs:{\"stroke-width\": 1}\n ,symbAttrs:{symb: \"circle\", r:3}}\n , {yh:U_C.asList(), h:h, nmHtml:'‒○‒ u_C(t)', color:`${colors_[1]}` \n //can be ovr. in ..Attrs as \"stroke\": \"...\"\n ,pathAttrs:{\"stroke-width\": 2.5, \"stroke\": \"DarkBlue\"} \n ,symbAttrs:{symb: \"circle\", r:3}} \n]);\n}","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