{"id":8734,"date":"2024-05-28T22:01:01","date_gmt":"2024-05-28T14:01:01","guid":{"rendered":""},"modified":"2024-05-28T22:01:01","modified_gmt":"2024-05-28T14:01:01","slug":"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206","status":"publish","type":"post","link":"https:\/\/mushiming.com\/8734.html","title":{"rendered":"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206"},"content":{"rendered":"<p><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"display: none;\"> \n <path stroke-linecap=\"round\" d=\"M5,0 0,2.5 5,5z\" id=\"raphael-marker-block\" style=\"-webkit-tap-highlight-color: rgba(0, 0, 0, 0);\"><\/path> \n<\/svg> <\/p>\n<div class=\"toc\">\n<h4>First-Order Circuits<\/h4>\n<ul>\n<li>7.1 Introduction<\/li>\n<li>\n<ul>\n<li>RC circuits(\u52a8\u6001\u7535\u8def\u53ca\u5176\u9636\u6570)<\/li>\n<li>RL circuits<\/li>\n<li>\u603b\u7ed3<\/li>\n<li>\u4f8b\u9898<\/li>\n<\/ul>\n<\/li>\n<li>7.2 The Source-Free(\u65e0\u6e90) RC Circuit(RC\u7535\u8def\u7684\u96f6\u8f93\u5165\u54cd\u5e94)<\/li>\n<li>7.3 The Source-Free RL Circuit(RL\u7535\u8def\u7684\u65e0\u6e90\u54cd\u5e94)<\/li>\n<li>7.4 Singularity Functions<\/li>\n<li>\n<ul>\n<li>unit step functions<\/li>\n<li>unit impulse functions<\/li>\n<li>unit ramp functions<\/li>\n<\/ul>\n<\/li>\n<li>7.5 Step Response(\u9636\u8dc3) of RC Circuit<\/li>\n<li>\n<ul>\n<li>Step response<\/li>\n<li>Forced response (Zero-state response)<\/li>\n<\/ul>\n<\/li>\n<li>7.6 Step Response of RL Circuit<\/li>\n<li>7.7 Complete Response<\/li>\n<li>\n<ul>\n<li>1. Decomposition of complete response (\u5168\u54cd\u5e94\u7684\u5206\u89e3)<\/li>\n<li>2.Determining response with 3 items(\u4e09\u8981\u7d20\u6cd5)<\/li>\n<li>Applications<\/li>\n<li>\n<ul>\n<li>1.Delay Circuits<\/li>\n<li>2.Photoflash Unit<\/li>\n<li>3.Relay Circuits<\/li>\n<li>4.Automobile Ignition Circuit<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>\u603b\u7ed3<\/li>\n<\/ul>\n<\/div>\n<h2>7.1 Introduction<\/h2>\n<p>Two types of simple circuits:a circuit comprising a resistor and capactior and a circuit comprising a resistor and an inductor.<br \/> These are called <strong>RC<\/strong> and <strong>RL<\/strong> circuits,respectively.<br \/> Resistive circuits<br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/994b7b9c3ef846c1b0e751c59bd44171.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c1\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c1\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/3b6c397def24495498a61f254ec308b9.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c2\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c2\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> \u6ce8\u91ca:\u635f\u53d8<br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/0fc65a73a7004c7eb2ac28d2a677a631.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c3\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c3\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/7bd11de20971456dbae4d35c3638708c.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c4\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c4\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><\/p>\n<h3>RC circuits(\u52a8\u6001\u7535\u8def\u53ca\u5176\u9636\u6570)<\/h3>\n<p>\u52a8\u6001\u7535\u8def:\u542b\u6709\u52a8\u6001\u5143\u4ef6(\u7535\u5bb9\u6216\u7535\u611f)\u7684\u7535\u8def<br \/> \u7279\u5f81:\u5f53\u7535\u8def\u7684\u7ed3\u6784\u6216\u53c2\u6570\u6539\u53d8\u65f6,\u7535\u8def\u53ef\u80fd\u4ece\u4e00\u79cd\u5de5\u4f5c\u72b6\u6001\u8f6c\u53d8\u5230\u53e6\u4e00\u79cd\u5de5\u4f5c\u72b6\u6001<br \/> \u6362\u8def:\u7535\u8def\u7684\u7ed3\u6784\u6216\u53c2\u6570\u6539\u53d8<br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/add5d26c34544562a4194d01c8ffa846.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c5\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c5\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> Before switching k,circuit has reached steady state:<br \/> <span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> i = 0 , u c = 0 i=0,u_c=0 <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.65952em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.83888em; vertical-align: -0.19444em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.64444em; vertical-align: 0em;\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span><\/span><br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/14b838709bbb4c5fa0350475451c3cab.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c6\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c6\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> Since k has been closed for a long time,the capacitor voltage has reached steady state again:<br \/> <span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> i = 0 , u c = U s i=0,u_c=U_s <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.65952em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.83888em; vertical-align: -0.19444em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.83333em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.10903em;\">U<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.10903em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br \/> \u8fc7\u6e21\u8fc7\u7a0b:\u7535\u8def\u7531\u4e00\u4e2a\u7a33\u6001\u8fc7\u6e21\u5230\u53e6\u4e00\u4e2a\u7a33\u6001\u7684\u8fc7\u7a0b<br \/> \u8fc7\u6e21\u72b6\u6001(\u77ac\u6001,\u6682\u6001,\u52a8\u6001)<br \/> \u8fc7\u6e21\u8fc7\u7a0b\u4ea7\u751f\u539f\u56e0:\u7535\u8def\u5185\u90e8\u542b\u6709\u50a8\u80fd\u5143\u4ef6 ,\u7535\u8def\u5728\u6362\u65f6\u80fd\u91cf \u7535\u8def\u5728\u6362\u65f6\u80fd\u91cf \u7535\u8def\u5728\u6362\u65f6\u80fd\u91cf \u53d1\u751f\u53d8\u5316\uff0c\u800c\u80fd\u91cf\u7684\u50a8\u5b58\u548c\u91ca\u653e\u90fd\u9700\u8981\u65f6\u95f4<br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/d25237454aad41ac85864283f50229f6.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c7\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c7\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> \u7814\u7a76\u8fc7\u6e21\u8fc7\u7a0b\u5177\u6709\u5b9e\u9645\u610f\u4e49<br \/> \u5229\u7528:\u4ea7\u751f\u5404\u79cd\u6ce2\u5f62<br \/> \u63d0\u9632:\u6682\u6001\u8fc7\u7a0b\u77ac\u95f4\u53ef\u80fd\u51fa\u73b0\u9ad8\u7535\u538b,\u5927\u7535\u6d41,\u4f7f\u4eea\u5668\u8bbe\u5907\u635f\u574f<\/p>\n<p>\u52a8\u6001\u7535\u8def\u7684\u9636\u6570<br \/> \u63cf\u8ff0\u52a8\u6001\u7535\u8def\u7684\u65b9\u7a0b\u662f\u5fae\u5206\u65b9\u7a0b<br \/> \u65b9\u7a0b\u9636\u6570=\u7535\u8def\u9636\u6570<br \/> \u4e00\u9636\u7535\u8def:\u63cf\u8ff0\u7535\u8def\u7684\u65b9\u7a0b\u662f\u4e00\u9636\u5fae\u5206\u65b9\u7a0b,\u4e00\u9636\u7535\u8def\u4e2d\u4e00\u822c\u53ea\u6709\u4e00\u4e2a\u52a8\u6001\u5143\u4ef6<br \/> \u4e8c\u9636\u7535\u8def:\u63cf\u8ff0\u7535\u8def\u7684\u65b9\u7a0b\u662f\u4e8c\u9636\u5fae\u5206\u65b9\u7a0b,\u4e00\u822c\u6709\u4e8c\u4e2a\u52a8\u6001\u5143\u4ef6<\/p>\n<p>;<img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/917601ab272147329a4a68571335b684.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c8\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c8\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> \u5e94 \u7528 K V L : R i + u c = u s \u7535 \u5bb9 V C R : i = C d u c d t R C d u c d t + u c = u s \u4e00 \u9636 \u5fae \u5206 \u65b9 \u7a0b \u4e5f \u5373 \u4e00 \u9636 \u7535 \u8def \u65b9 \u7a0b \u7684 \u9636 \u6570 \u901a \u5e38 \u7b49 \u4e8e \u7535 \u8def \u4e2d \u52a8 \u6001 \u5143 \u4ef6 \u7684 \u4e2a \u6570 \u5e94\u7528KVL:Ri+u_c=u_s\\\\ \u7535\u5bb9VCR:i=C\\frac{du_c}{dt}\\\\ RC\\frac{du_c}{dt}+u_c=u_s\\\\ \u4e00\u9636\u5fae\u5206\u65b9\u7a0b\u4e5f\u5373\u4e00\u9636\u7535\u8def\\\\ \u65b9\u7a0b\u7684\u9636\u6570\u901a\u5e38\u7b49\u4e8e\u7535\u8def\u4e2d\u52a8\u6001\u5143\u4ef6\u7684\u4e2a\u6570 <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.68333em; vertical-align: 0em;\"><\/span><span class=\"mord cjk_fallback\">\u5e94<\/span><span class=\"mord cjk_fallback\">\u7528<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">K<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.22222em;\">V<\/span><span class=\"mord mathdefault\">L<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.76666em; vertical-align: -0.08333em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.68333em; vertical-align: 0em;\"><\/span><span class=\"mord cjk_fallback\">\u7535<\/span><span class=\"mord cjk_fallback\">\u5bb9<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.22222em;\">V<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">C<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.65952em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.24121em; vertical-align: -0.345em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">C<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.896208em;\"><span class=\"\" style=\"top: -2.655em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">d<\/span><span class=\"mord mathdefault mtight\">t<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.4101em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">d<\/span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.164543em;\"><span class=\"\" style=\"top: -2.357em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.24121em; vertical-align: -0.345em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">C<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.896208em;\"><span class=\"\" style=\"top: -2.655em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">d<\/span><span class=\"mord mathdefault mtight\">t<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.4101em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">d<\/span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.164543em;\"><span class=\"\" style=\"top: -2.357em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0em; vertical-align: 0em;\"><\/span><span class=\"mord cjk_fallback\">\u4e00<\/span><span class=\"mord cjk_fallback\">\u9636<\/span><span class=\"mord cjk_fallback\">\u5fae<\/span><span class=\"mord cjk_fallback\">\u5206<\/span><span class=\"mord cjk_fallback\">\u65b9<\/span><span class=\"mord cjk_fallback\">\u7a0b<\/span><span class=\"mord cjk_fallback\">\u4e5f<\/span><span class=\"mord cjk_fallback\">\u5373<\/span><span class=\"mord cjk_fallback\">\u4e00<\/span><span class=\"mord cjk_fallback\">\u9636<\/span><span class=\"mord cjk_fallback\">\u7535<\/span><span class=\"mord cjk_fallback\">\u8def<\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0em; vertical-align: 0em;\"><\/span><span class=\"mord cjk_fallback\">\u65b9<\/span><span class=\"mord cjk_fallback\">\u7a0b<\/span><span class=\"mord cjk_fallback\">\u7684<\/span><span class=\"mord cjk_fallback\">\u9636<\/span><span class=\"mord cjk_fallback\">\u6570<\/span><span class=\"mord cjk_fallback\">\u901a<\/span><span class=\"mord cjk_fallback\">\u5e38<\/span><span class=\"mord cjk_fallback\">\u7b49<\/span><span class=\"mord cjk_fallback\">\u4e8e<\/span><span class=\"mord cjk_fallback\">\u7535<\/span><span class=\"mord cjk_fallback\">\u8def<\/span><span class=\"mord cjk_fallback\">\u4e2d<\/span><span class=\"mord cjk_fallback\">\u52a8<\/span><span class=\"mord cjk_fallback\">\u6001<\/span><span class=\"mord cjk_fallback\">\u5143<\/span><span class=\"mord cjk_fallback\">\u4ef6<\/span><span class=\"mord cjk_fallback\">\u7684<\/span><span class=\"mord cjk_fallback\">\u4e2a<\/span><span class=\"mord cjk_fallback\">\u6570<\/span><\/span><\/span><\/span><\/span><br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/8627a36803294c45ae149726da5a420d.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c9\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c9\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> R ( C 1 + C 2 ) d u c d t + u c = u s R(C_1+C_2)\\frac{du_c}{dt}+u_c=u_s <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.301108em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.24121em; vertical-align: -0.345em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.301108em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.896208em;\"><span class=\"\" style=\"top: -2.655em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">d<\/span><span class=\"mord mathdefault mtight\">t<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.4101em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">d<\/span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.164543em;\"><span class=\"\" style=\"top: -2.357em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br \/> \u4e00\u9636\u7535\u8def<\/p>\n<p>\u52a8\u6001\u7535\u8def\u5206\u6790\u65b9\u6cd5:<\/p>\n<ol>\n<li>\u65f6\u57df\u5206\u6790\u6cd5<br \/> \u7ecf\u5178\u6cd5:\u76f4\u63a5\u6c42\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b<br \/> \u72b6\u6001\u53d8\u91cf\u6cd5<br \/> \u6570\u503c\u6cd5<\/li>\n<li>\u53d8\u6362\u6cd5<br \/> \u5085\u91cc\u53f6\u53d8\u6362\u6cd5:\u9891\u57df\u5206\u6790\u6cd5<br \/> \u62c9\u666e\u62c9\u65af\u53d8\u6362\u6cd5:\u590d\u9891\u57df\u5206\u6790\u6cd5<\/li>\n<\/ol>\n<h3>RL circuits<\/h3>\n<p>Before switching k,circuit has reached steady state:<br \/> <span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> i = 0 , u L = 0 i=0,u_L=0 <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.65952em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.83888em; vertical-align: -0.19444em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.328331em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">L<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.64444em; vertical-align: 0em;\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span><\/span><br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/db55d5cc8c3e428ca1c4c75c594cbd6a.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c10\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c10\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> Since k has been closed for a long time,the capacitor voltage has reached steady state again<br \/> <span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> u L = 0 , i = U s R u_L=0,i=\\frac{U_s}{R} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.328331em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">L<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.85396em; vertical-align: -0.19444em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.04633em; vertical-align: -0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.36033em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.10903em;\">U<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.10903em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/70ccc71da69d48dcb1dc51afa7a91f8b.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c11\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c11\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/98bfa8a175e64ae3bad36c160580ec7b.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c12\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c12\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> Applying Kirchhoff\u2019s laws to purely resistive circuits results in algebraic equations<\/p>\n<p>While applying Kirchhoff\u2019s laws to RC and RL circuits produces differential equations(\u5fae\u5206\u65b9\u7a0b)<\/p>\n<p>The differential equations resulting from analyzing RC and RL circuits are of the first order.Hence.the circuits are collectively known as first-order circuits.<\/p>\n<p>A first-order circuit is characterized by a first-order differential equation.<\/p>\n<p>RC circuits<br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/0efb7c76745340309e5fabb7039f3e0c.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c13\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c13\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> Applying KVL:<br \/> <span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> R i + u c = u s ( t ) Ri+u_c=u_s(t) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.76666em; vertical-align: -0.08333em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><br \/> By definition(\u5b9a\u4e49)(VCR):<br \/> <span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> i = C d u c d t &nbsp; \u2192 R C d u c d t + u c = u s ( t ) &nbsp; \u2192 R d i d t + i C = d u s ( t ) d t i=C\\frac{du_c}{dt}\\\\ ~\\\\ \u2192RC\\frac{du_c}{dt}+u_c=u_s(t)\\\\ ~\\\\ \u2192R\\frac{di}{dt}+\\frac{i}{C}=\\frac{du_s(t)}{dt} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.65952em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.05744em; vertical-align: -0.686em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">C<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.37144em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord mathdefault\">t<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0em; vertical-align: 0em;\"><\/span><span class=\"mspace nobreak\">&nbsp;<\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.36687em; vertical-align: 0em;\"><\/span><span class=\"mrel\">\u2192<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.05744em; vertical-align: -0.686em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">C<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.37144em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord mathdefault\">t<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0em; vertical-align: 0em;\"><\/span><span class=\"mspace nobreak\">&nbsp;<\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.36687em; vertical-align: 0em;\"><\/span><span class=\"mrel\">\u2192<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.05744em; vertical-align: -0.686em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.37144em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord mathdefault\">t<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord mathdefault\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.02252em; vertical-align: -0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.33652em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">C<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em; vertical-align: -0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord mathdefault\">t<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>RL circuits<br \/> <img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/238675ab85f04e11a942c423637f1077.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206 (https:\/\/mushiming.com\/)  \u7b2c14\u5f20\" title=\"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206  \u7b2c14\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> Applying KVL:<br \/> <span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> R i + u L = u s ( t ) Ri+u_L=u_s(t) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.76666em; vertical-align: -0.08333em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.328331em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">L<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><br \/> By definition(\u5b9a\u4e49)(VCR):<br \/> <span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> u L = L d i d t &nbsp; \u2192 R i + L d i d t = u s ( t ) &nbsp; \u2192 R L u l + d u L d t = d u s ( t ) d t u_L=L\\frac{di}{dt}\\\\ ~\\\\ \u2192Ri+L\\frac{di}{dt}=u_s(t)\\\\ ~\\\\ \u2192\\frac{R}{L}u_l+\\frac{du_L}{dt}=\\frac{du_s(t)}{dt} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.328331em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">L<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.05744em; vertical-align: -0.686em;\"><\/span><span class=\"mord mathdefault\">L<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.37144em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord mathdefault\">t<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord mathdefault\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0em; vertical-align: 0em;\"><\/span><span class=\"mspace nobreak\">&nbsp;<\/span><\/span><span class=\"mspace newline\"><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.36687em; vertical-align: 0em;\"><\/span><span class=\"mrel\">\u2192<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.76666em; vertical-align: -0.08333em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.00773em;\">R<\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.05744em; vertical-align: -0.686em;\"><\/span><span class=\"mord mathdefault\">L<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.37144em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord mathdefault\">t<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">d<\/span><span class=\"mord mathdefault\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"\u7535\u8def\u539f\u7406\u4e00\u9636\u7535\u8def_\u4e00\u9636\u7535\u8def\u548c\u4e8c\u9636\u7535\u8def\u600e\u4e48\u533a\u5206First-OrderCircuits7.1Introduction\u603b\u7ed3\u4f8b\u98987.2TheSource-Free(\u65e0\u6e90)RCCircuit...","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"_links":{"self":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/8734"}],"collection":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/comments?post=8734"}],"version-history":[{"count":0,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/8734\/revisions"}],"wp:attachment":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/media?parent=8734"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/categories?post=8734"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/tags?post=8734"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}