{"id":87,"date":"2018-07-11T03:36:51","date_gmt":"2018-07-11T03:36:51","guid":{"rendered":"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/?page_id=87"},"modified":"2024-01-24T00:08:51","modified_gmt":"2024-01-24T00:08:51","slug":"expriment-4-first-order-circuits","status":"publish","type":"page","link":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/experiments\/expriment-4-first-order-circuits\/","title":{"rendered":"#4: First and Second Order Circuits"},"content":{"rendered":"<h2>Objectives<\/h2>\n<ul>\n<li>To study the step response of first order circuits.<\/li>\n<li>To understand the concept of the time constant.<\/li>\n<li>To study the step response of second order circuits.<\/li>\n<li>To understand the difference between overdamped, critically damped and underdamped responses.<\/li>\n<li>To determine theoretically and experimentally the damped natural frequency in the under-damped case.<\/li>\n<\/ul>\n<h2>Equipment<\/h2>\n<ul>\n<li>Breadboard<\/li>\n<li>Function generator<\/li>\n<li>Oscilloscope<\/li>\n<li>Digital multimeter (DMM)<\/li>\n<\/ul>\n<h2>Background<\/h2>\n<h3>A.    First Order Circuits<\/h3>\n<p>First-order transient circuits are described by a first order differential equation. First-order circuits contain a resistor and only one type of storage element, either an inductor or a capacitor, i.e. RL or RC circuits.<\/p>\n<p>For a step voltage\/current source input, the output can be expressed as<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-80215ae2a1a4d513185a69df0995ee06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;&#40;&#116;&#41;&#61;&#88;&#40;&#92;&#105;&#110;&#102;&#116;&#121;&#41;&#43;&#091;&#88;&#40;&#48;&#41;&#45;&#88;&#40;&#92;&#105;&#110;&#102;&#116;&#121;&#41;&#093;&#32;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#101;&#94;&#123;&#45;&#123;&#116;&#32;&#92;&#111;&#118;&#101;&#114;&#32;&#92;&#116;&#97;&#117;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"297\" style=\"vertical-align: -5px;\"\/><\/p>\n<p>Where, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-6df8f6cd344cd92c97127cda8d84049b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;&#40;&#48;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\"\/> is the circuit response at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-095ec4e668d44957abf895379d06cf66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\"\/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-20690a2f38eb2578b12f56f20f20f54c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;&#40;&#92;&#105;&#110;&#102;&#116;&#121;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\"\/> is the response at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-d8d05cc90555da54b121a358326c8460_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#32;&#61;&#92;&#105;&#110;&#102;&#116;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"\/>. The parameter <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-13197f4653c1fd428a291609eb1e3b87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>\u00a0is called time constant of the circuit and gives the time required for the response (i) to rise from zero to 63% (or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-86ca5d90f41654557457b4864ceb494d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#32;&#123;&#49;&#32;&#92;&#111;&#118;&#101;&#114;&#32;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"38\" style=\"vertical-align: -6px;\"\/>) of its final steady value as shown in Figure 4 &#8211; 1 (a), or (ii) to fall to 37% (or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-699ddcd8b87c93caed35a478ac5e47e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#32;&#92;&#111;&#118;&#101;&#114;&#32;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\"\/>) of its initial value as shown in Figure 4 &#8211; 1 (b). Therefore, the smaller the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-13197f4653c1fd428a291609eb1e3b87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, the faster the circuit response is.<\/p>\n<p>For a RC circuit<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{black}\\tau = R \\times C<\/span><\/p>\n<p>For a RL circuit<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{black}\\tau = {L \\over R}<\/span><\/p>\n<p>Applying the equations above, the voltage responses across the capacitor and the resistor in Figure 4-1 can be written as:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-91404ffec16e36e029354927566fae0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#67;&#40;&#116;&#41;&#61;&#69;&#40;&#49;&#45;&#101;&#94;&#123;&#45;&#123;&#116;&#32;&#92;&#111;&#118;&#101;&#114;&#32;&#82;&#67;&#125;&#125;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#32;&#102;&#111;&#114;&#32;&#125;&#116;&#32;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"243\" style=\"vertical-align: -4px;\"\/><\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-0f46b5b3ae685e1c1a58746820e05b14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#82;&#40;&#116;&#41;&#61;&#69;&#101;&#94;&#123;&#45;&#123;&#116;&#32;&#92;&#111;&#118;&#101;&#114;&#32;&#82;&#67;&#125;&#125;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#32;&#102;&#111;&#114;&#32;&#125;&#116;&#32;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"198\" style=\"vertical-align: -4px;\"\/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-243 aligncenter\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1-300x164.png\" alt=\"\" width=\"500\" height=\"273\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1-300x164.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1-768x420.png 768w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1-1024x560.png 1024w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1.png 1968w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-244 aligncenter\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1a-300x206.png\" alt=\"\" width=\"438\" height=\"301\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1a-300x206.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1a-768x526.png 768w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1a-1024x701.png 1024w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1a.png 1886w\" sizes=\"auto, (max-width: 438px) 100vw, 438px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-245 aligncenter\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1b-300x208.png\" alt=\"\" width=\"437\" height=\"303\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1b-300x208.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1b-768x534.png 768w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1b-1024x712.png 1024w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-1b.png 1862w\" sizes=\"auto, (max-width: 437px) 100vw, 437px\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 4 &#8211; 1 A first order circuit and its responses. (a) voltage over the capacitor; (b) voltage over the resistor.<\/strong><\/p>\n<h3>B.    Second Order Circuits<\/h3>\n<p>Second-order circuits are RLC circuits that contain two energy storage elements. They can be represented by a second-order differential equation. A characteristic equation, which is derived from the governing differential equation, is often used to determine the natural response of the circuit. The characteristic equation usually takes the form of a quadratic equation, and it has two roots s<sub>1<\/sub> and s<sub>2<\/sub>.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-172ed151e9ed97cf2b34204148af48c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#50;&#115;&#94;&#50;&#43;&#97;&#95;&#49;&#115;&#43;&#97;&#95;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"151\" style=\"vertical-align: -4px;\"\/><\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-5b1a856762bf5f3674014172c1bdd9ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#95;&#49;&#61;&#123;&#45;&#97;&#95;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#95;&#49;&#94;&#50;&#45;&#52;&#97;&#95;&#50;&#97;&#95;&#48;&#125;&#32;&#92;&#111;&#118;&#101;&#114;&#32;&#50;&#97;&#95;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"152\" style=\"vertical-align: -8px;\"\/><\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-63e63332dbc72eee1e8c96a72fb13623_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#95;&#49;&#61;&#123;&#45;&#97;&#95;&#49;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#95;&#49;&#94;&#50;&#45;&#52;&#97;&#95;&#50;&#97;&#95;&#48;&#125;&#32;&#92;&#111;&#118;&#101;&#114;&#32;&#50;&#97;&#95;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"152\" style=\"vertical-align: -8px;\"\/><\/p>\n<p>When these roots are rewritten as,<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-9c715bfe926b46d0fe56cc8ec0a5dbce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#95;&#49;&#61;&#45;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#94;&#50;&#45;&#92;&#111;&#109;&#101;&#103;&#97;&#95;&#48;&#94;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"165\" style=\"vertical-align: -6px;\"\/><\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-d250157251193a2763edf46bc79f9f67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#95;&#50;&#61;&#45;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#94;&#50;&#45;&#92;&#111;&#109;&#101;&#103;&#97;&#95;&#48;&#94;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"165\" style=\"vertical-align: -6px;\"\/><\/p>\n<p>then the natural response of the circuit is determined by:<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-afe21b5b1722b5b150bc3346e86ed640_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#94;&#50;&#32;&#62;&#32;&#119;&#95;&#48;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"63\" style=\"vertical-align: -5px;\"\/>, there are two real and distinct roots <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-3998a7ead38819fd05930ac99ab5ca9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#111;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -1px;\"\/> Overdamped, as shown in Figure 4 &#8211; 2 (a) and (d).<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-554b89f38272786e769c973ffffa8930_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#94;&#50;&#32;&#61;&#32;&#119;&#95;&#48;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"63\" style=\"vertical-align: -5px;\"\/>, there are two real equal roots <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-3998a7ead38819fd05930ac99ab5ca9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#111;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -1px;\"\/> Critically damped, as shown in Figure 4 &#8211; 2 (b) and (e).<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-f735988ddb7fa9fc568960165b27ee2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#94;&#50;&#32;&#60;&#32;&#119;&#95;&#48;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"63\" style=\"vertical-align: -5px;\"\/>, there are two complex roots <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-3998a7ead38819fd05930ac99ab5ca9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#111;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -1px;\"\/> Underdamped, as shown in Figure 4 &#8211; 2 (c) and (f).<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-251\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1ad-300x97.png\" alt=\"\" width=\"649\" height=\"210\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1ad-300x97.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1ad-768x248.png 768w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1ad-1024x331.png 1024w\" sizes=\"auto, (max-width: 649px) 100vw, 649px\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-252\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1be-300x99.png\" alt=\"\" width=\"652\" height=\"215\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1be-300x99.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1be-768x253.png 768w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1be-1024x338.png 1024w\" sizes=\"auto, (max-width: 652px) 100vw, 652px\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-253\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1cf-300x100.png\" alt=\"\" width=\"651\" height=\"217\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1cf-300x100.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1cf-768x256.png 768w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-1cf-1024x341.png 1024w\" sizes=\"auto, (max-width: 651px) 100vw, 651px\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 4 &#8211; 2 Second order circuits natural responses<\/strong><\/p>\n<h2 style=\"text-align: left;\">Preparation<\/h2>\n<h3>A.    First Order Circuits<\/h3>\n<p>Figure 4 &#8211; 3 and Figure 4 &#8211; 4 show various RC and RL circuits. For all circuits, R = 1 k\u2126, C = 0.1 uF, and L = 100 mH.<\/p>\n<ol>\n<li>For the circuits in Figure 4 &#8211; 3, derive the analytical expression for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-4e24d74ee3ebf96630fac7858f027548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#111;&#117;&#116;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\"\/> given that <span class=\"katex-eq\" data-katex-display=\"false\"> \\color{black}E <\/span> is a constant. Sketch or plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-4e24d74ee3ebf96630fac7858f027548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#111;&#117;&#116;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\"\/> for each circuit.<\/li>\n<li>For the circuits in Figure 4 &#8211; 4, <span class=\"katex-eq\" data-katex-display=\"false\">\\color{black}V_{in}(t)<\/span> is a symmetric square wave input with amplitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-5c7812fc5ceb9be76dffea670cc9215e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#109;&#32;&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\"\/> and period <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-2a16b84e92a3361965bf90060e2f1336_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#95;&#83;&#61;&#49;&#48;&#32;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#92;&#116;&#97;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"93\" style=\"vertical-align: -3px;\"\/>. Sketch or plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-4e24d74ee3ebf96630fac7858f027548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#111;&#117;&#116;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\"\/> for each circuit using superposition and the hint below. Show at least three cycles.<\/li>\n<li>For the circuit in Figure 4 &#8211; 4 (d), with <span class=\"katex-eq\" data-katex-display=\"false\">\\color{black}V_{in}(t)<\/span> remains unchanged, sketch or plot the voltage across the inductor. Show at least three cycles. Compare this voltage waveform to the one obtained from the circuit in Figure 4 &#8211; 4 (c). What do you observe? Why?<\/li>\n<\/ol>\n<p><strong>Hint:<\/strong> the square wave can be broken up into a series of step voltage waveforms with alternating polarities. Each of these step voltage inputs generates an output, and the total output response is the summation of all the individual outputs.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-246 aligncenter\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-2-300x183.png\" alt=\"\" width=\"653\" height=\"398\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-2-300x183.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-2-768x469.png 768w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-2-1024x625.png 1024w\" sizes=\"auto, (max-width: 653px) 100vw, 653px\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 4 &#8211; 3  First order circuits with step input voltage<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-247 aligncenter\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-3-300x184.png\" alt=\"\" width=\"651\" height=\"400\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-3-300x184.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure4-3-1024x627.png 1024w\" sizes=\"auto, (max-width: 651px) 100vw, 651px\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 4 &#8211; 4  First order circuits with square wave input<\/strong><\/p>\n<h3>B.    Second Order Circuits<\/h3>\n<p>Figure 4 &#8211; 5 and Figure 4 &#8211; 6 show various 2nd order circuits. For all circuits, C = 0.01 uF and L = 100 mH.<\/p>\n<ol>\n<li>For both circuits in Figure 4 &#8211; 5, <span class=\"katex-eq\" data-katex-display=\"false\"> \\color{black}E <\/span> is a constant. Write down the characteristic equation. Also, calculate the resistance range for R for the following cases.\n<ol>\n<li>Over-damped response<\/li>\n<li>Critically damped response<\/li>\n<li>Under-damped response<\/li>\n<\/ol>\n<li>For the circuit in Figure 4 &#8211; 5 (a), plot or sketch the response <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-4e24d74ee3ebf96630fac7858f027548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#111;&#117;&#116;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\"\/> when the resistor has the following resistance values.\n<ol>\n<li>R = 22 k\u2126<\/li>\n<li>R = 6.3 k\u2126<\/li>\n<li>R = 2.2 k\u2126<\/li>\n<\/ol>\n<li>For the circuit in Figure 4 &#8211; 5 (b), plot or sketch the response <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-4e24d74ee3ebf96630fac7858f027548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#111;&#117;&#116;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\"\/> when the resistor has the following resistance values.\n<ol>\n<li>R = 680 \u2126<\/li>\n<li>R = 1.6 k\u2126<\/li>\n<li>R = 4.7 k\u2126<\/li>\n<\/ol>\n<li>For both circuits in Figure 4 &#8211; 6, <span class=\"katex-eq\" data-katex-display=\"false\">\\color{black}V_{in}(t)<\/span> is a symmetric square wave input with a frequency of 400 Hz and an amplitude of 4 V. Set R = 470 \u2126 for the circuit in Figure 4 &#8211; 6 (a) and R = 22 k\u2126 for the circuit in Figure 4 &#8211; 6 (b).\n<ol>\n<li>Calculate \u03b1, \u03c9<sub>o<\/sub>, and \u03c9<sub>d<\/sub>.<\/li>\n<li>Plot or sketch the output voltage, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-4e24d74ee3ebf96630fac7858f027548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#111;&#117;&#116;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\"\/>.<\/li>\n<\/ol>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-254 aligncenter\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-2-300x78.png\" alt=\"\" width=\"700\" height=\"182\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-2-300x78.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-2-768x201.png 768w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-2-1024x268.png 1024w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 4 &#8211; 5 Second order circuits with step input voltage<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-255 aligncenter\" src=\"http:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-3-300x74.png\" alt=\"\" width=\"702\" height=\"173\" srcset=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-3-300x74.png 300w, https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/uploads\/2018\/07\/figure5-3-1024x254.png 1024w\" sizes=\"auto, (max-width: 702px) 100vw, 702px\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 4 &#8211; 6 Second order circuits with square wave input<\/strong><\/p>\n<h2>Simulation<\/h2>\n<p>Perform the following circuit simulations using a circuit simulator.<\/p>\n<ol>\n<li>Build and simulate the circuits in Figure 4 &#8211; 4. Set the input voltage to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-b33ccfd383bee644e515d219b2d9bd12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#32;&#92;&#112;&#109;&#32;&#53;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"37\" style=\"vertical-align: 0px;\"\/> with a frequency of 1 kHz. Display both <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-ede73a07f893cbca19785539e16d78a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#105;&#110;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-4e24d74ee3ebf96630fac7858f027548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#111;&#117;&#116;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\"\/> on the oscilloscope. Compare the results with the plots from PREPARATION.<\/li>\n<li>For the circuit in Figure 4 &#8211; 4 (d), with the input voltage remains unchanged, display the voltage across the inductor. Compare this voltage waveform to the one obtained from the circuit in Figure 4 &#8211; 4 (c). What do you observe? Why?<\/li>\n<li>Build and simulate the circuits in Figure 4 &#8211; 6. Set the input voltage to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-99b9b0dc445869059edf87c2a07d9464_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#109;&#32;&#52;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"37\" style=\"vertical-align: -1px;\"\/> with a frequency of 400 Hz. Display both <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-ede73a07f893cbca19785539e16d78a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#105;&#110;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-4e24d74ee3ebf96630fac7858f027548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#111;&#117;&#116;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\"\/> on the oscilloscope. Compare the results with the plots from PREPARATION.<\/li>\n<\/ol>\n<h2>First Order Circuits Experiment<\/h2>\n<p>Construct all four circuits in Figure 4 &#8211; 4. Use the same component values as in the PREPARATION and the same input settings used in the SIMULATION. Complete the measurements described below. Refer to Experiment #3 for instructions on how to use function generator and oscilloscope.<\/p>\n<h3>A. Response to square wave input<\/h3>\n<ol>\n<li>Measure both the input and the output using the oscilloscope. Connect Ch1 to the input and Ch2 to the output so that both waveforms are displayed on the screen. Compare these waveforms to the results from PREPARATION and SIMULATION. <\/li>\n<li>For the circuit in Figure 4 &#8211; 4 (d), measure the voltage across the inductor using the oscilloscope. Since any voltage measured by the probe used in the lab is with respect to the ground, the voltage across the inductor has to be measured indirectly. This measurement can be obtained using the \u2018Math\u2019 button on the oscilloscope. Compare this voltage waveform to the one obtained from the circuit in Figure 4 &#8211; 4 (c). What do you observe? Why?<\/li>\n<li>Save the screen images to a USB drive using the \u2018Save Load\u2019 button.<\/li>\n<\/ol>\n<h3>B. Time constant measurement<\/h3>\n<ol>\n<li>Turn off the input (Channel 1) by pressing the channel number button. Now only the output is displayed on the screen.<\/li>\n<li>Zoom in on the output curve on the oscilloscope such that a large portion (at least half a cycle) of the rise\/drop of a cycle is displayed on the screen. Consider the rise and drop over one half cycle only for each circuit. Use cursors to determine the maximum voltage difference E for the output. The time it takes for the output to rise from the minimum value to 63% of E or to drop from the maximum to 37% of E is the time constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-13197f4653c1fd428a291609eb1e3b87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> of the circuit. Record this time constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-13197f4653c1fd428a291609eb1e3b87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> for each circuit.<\/li>\n<\/ol>\n<h2>Second Order Circuits Experiment<\/h2>\n<p>On the function generator use the same square wave input settings as in SIMULATION. Build both circuits shown in Figure 4 &#8211; 6.<\/p>\n<h3>A. Natural responses<\/h3>\n<ol>\n<li>Measure the output voltage <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-4e24d74ee3ebf96630fac7858f027548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#95;&#123;&#111;&#117;&#116;&#125;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\"\/> for the following six cases.<\/li>\n<ol>\n<li>R = 22 k\u2126 in the circuit of Figure 4 &#8211; 6 (a)<\/li>\n<li>R = 6.3 k\u2126 in the circuit of Figure 4 &#8211; 6 (a)<\/li>\n<li>R = 2.2 k\u2126 in the circuit of Figure 4 &#8211; 6 (a)<\/li>\n<li>R = 680 \u2126 in the circuit of Figure 4 &#8211; 6 (b)<\/li>\n<li>R = 1.6 k\u2126 in the circuit of Figure 4 &#8211; 6 (b)<\/li>\n<li>R = 4.7 k\u2126 in the circuit of Figure 4 &#8211; 6 (b)<\/li>\n<\/ol>\n<li>On the oscilloscope, connect Ch1 to the input and Ch2 to the output so that both the input and the output are displayed on the screen.<\/li>\n<li>For each case, save the screen image with the associated measurements for both the input and the output on to a USB drive.<\/li>\n<li>For each case, indicate if the output response is overdamped, critically damped or underdamped.<\/li>\n<\/ol>\n<\/ol>\n<h3>B. Damped natural frequency measurement<\/h3>\n<ol>\n<li>Set R = 470 \u2126 for the circuit in Figure 4 &#8211; 6 (a) and R = 22 k\u2126 for the circuit in Figure 4 &#8211; 6 (b). Measure and save the screen image for both the input and the output, and compare them with the results from PREPARATION and SIMULATION.<\/li>\n<li>Zoom in on the output curve so that at least two whole oscillations (ripples) of the output from the beginning of an output cycle are displayed. Use the cursors to measure the time period T<sub>d<\/sub> between the first two peaks (or between two zero phases). \u03c9<sub>d<\/sub> is calculated using:<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-content\/ql-cache\/quicklatex.com-e67a4e3a2826b57f3174e6bdc40eaee5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#95;&#100;&#61;&#123;&#50;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#112;&#105;&#32;&#92;&#111;&#118;&#101;&#114;&#32;&#84;&#95;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"65\" style=\"vertical-align: -8px;\"\/><\/p>\n<p><!-- \n\n<h2>Report<\/h2>\n\n\nPrepare your report as per the guidelines given in APPENDIX III. --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Objectives To study the step response of first order circuits. To understand the concept of the time constant. To study &hellip; <a href=\"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/experiments\/expriment-4-first-order-circuits\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">#4: First and Second Order Circuits<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"parent":77,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-87","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-json\/wp\/v2\/pages\/87","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-json\/wp\/v2\/comments?post=87"}],"version-history":[{"count":28,"href":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-json\/wp\/v2\/pages\/87\/revisions"}],"predecessor-version":[{"id":1276,"href":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-json\/wp\/v2\/pages\/87\/revisions\/1276"}],"up":[{"embeddable":true,"href":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-json\/wp\/v2\/pages\/77"}],"wp:attachment":[{"href":"https:\/\/www.ece.ucf.edu\/labs\/EEL3123\/wp-json\/wp\/v2\/media?parent=87"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}