{"id":15002,"date":"2023-01-01T22:33:23","date_gmt":"2023-01-02T03:33:23","guid":{"rendered":"https:\/\/panamahitek.com\/?p=15002"},"modified":"2023-01-05T14:53:36","modified_gmt":"2023-01-05T19:53:36","slug":"methods-for-the-analysis-of-alternating-current-ac-circuits","status":"publish","type":"post","link":"https:\/\/panamahitek.com\/en\/methods-for-the-analysis-of-alternating-current-ac-circuits\/","title":{"rendered":"Methods for the analysis of Alternating Current (AC) Circuits"},"content":{"rendered":"<p style=\"text-align: justify;\">This is a post that I have wanted to write for a long time, but it is only now that I have the time, the disposition, and the appropriate mental state to share my ideas and knowledge on this subject. In this publication, I will try to explain what I consider to be the two main methods for analyzing circuits with alternating current (AC) power sources: <strong>Laplace Transform<\/strong> and <strong>Complex Frequency<\/strong>.<\/p>\n<p style=\"text-align: justify;\">This is a somewhat lengthy post given the number of concepts that I must define before getting into the substance. Afterwards, in separate publications, I will focus on AC analysis techniques, specifically mesh analysis, nodal analysis, and possibly matrix analysis.<\/p>\n<p style=\"text-align: justify;\">Before getting into the substance, I would like to clarify that this post will present important mathematical complexity. As an engineer, my strength has never been algebra or mathematics in general, but rather the application of different types of mathematical models to describe the behavior of all kinds of physical phenomena.<\/p>\n<p style=\"text-align: justify;\">To solve the equations, I will use the TI-Nspire CX CAS simulator, one of the Texas Instruments calculators. This machine accompanied me during my undergraduate studies, in my master&#8217;s degree, and now in my doctoral degree. My faithful companion.<\/p>\n<p>That said, let&#8217;s begin.<\/p>\n<h5><strong>The excitation function<\/strong><\/h5>\n<p style=\"text-align: justify;\">The times that I have had to teach the Circuit II course at the Technological University of Panama, I have started by explaining the concept of the excitation function. This is a mathematical model that allows us to describe any type of power source, whether voltage or current, in an electric circuit. For me, understanding this concept is fundamental.<\/p>\n<p>The mathematical model is as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14981\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-120.png\" alt=\"alternating current\" width=\"348\" height=\"29\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-120.png 761w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-120-300x25.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-120-696x58.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-120-741x63.png 741w\" sizes=\"auto, (max-width: 348px) 100vw, 348px\" \/><\/p>\n<p>Where:<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><strong>v(t)<\/strong> is the value of the excitation function at any time<\/li>\n<li><strong>V<sub>m<\/sub><\/strong> is the maximum amplitude of the function<\/li>\n<li><strong>t<\/strong> is time in seconds<\/li>\n<li><strong>\u03c3<\/strong> is the neperian frequency, given in nepers per second<\/li>\n<li><strong>\u03c9<\/strong> is the angular frequency, given in radians per second<\/li>\n<li><strong>\u03c6<\/strong> is the phase angle of the function, given in degrees<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">By modifying these parameters, it is possible to construct any waveform used in electrical circuits, either as voltage sources or current sources. For example:<\/p>\n<p style=\"text-align: justify;\">If we consider that <strong>\u03c3=0<\/strong>, <strong>\u03c9=0<\/strong> and<strong> \u03c6=0<\/strong>, we will have a constant function, such as that of direct current (DC) power sources<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14834\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-61.png\" alt=\"alternating current\" width=\"303\" height=\"24\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-61.png 871w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-61-300x24.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-61-768x61.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-61-696x55.png 696w\" sizes=\"auto, (max-width: 303px) 100vw, 303px\" \/><\/p>\n<p>If we consider that <strong>\u03c3\u22600<\/strong>, <strong>\u03c9=0<\/strong> and <strong>\u03c6=0<\/strong>, we will have an exponential function:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14983\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-121.png\" alt=\"alternating current\" width=\"364\" height=\"22\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-121.png 1016w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-121-300x18.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-121-768x47.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-121-696x42.png 696w\" sizes=\"auto, (max-width: 364px) 100vw, 364px\" \/><\/p>\n<p style=\"text-align: justify;\">If we consider that <strong>\u03c3=0<\/strong>, <strong>\u03c9\u22600<\/strong> and <strong>\u03c6\u22600<\/strong>, we will have a cosine function:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14985\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-122.png\" alt=\"alternating current\" width=\"452\" height=\"24\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-122.png 1236w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-122-300x16.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-122-1024x54.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-122-768x40.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-122-696x37.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-122-1068x56.png 1068w\" sizes=\"auto, (max-width: 452px) 100vw, 452px\" \/><\/p>\n<p style=\"text-align: justify;\">The excitation function is a sinusoidal function that can be damped by having non-zero values for both frequencies of the function. It&#8217;s worth noting that a cosine function can be transformed into a sine function by modifying the value of \u03c6 (specifically with a phase shift of 90\u00ba).<\/p>\n<p style=\"text-align: justify;\">When I talk about this topic with my students, I like to ask them: how do we model square wave functions, sawtooth waves or similar functions? Well, for that in engineering we study a procedure that allows us to model periodic functions as a sum of sinusoidal functions: the <strong>Fourier Series<\/strong>.<\/p>\n<p style=\"text-align: justify;\">For instance, let&#8217;s say we want to create a mathematical model that reproduces the following signal:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14845\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-64.png\" alt=\"alternating current\" width=\"600\" height=\"404\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-64.png 1300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-64-300x202.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-64-1024x690.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-64-768x518.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-64-696x469.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-64-1068x720.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-64-623x420.png 623w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/p>\n<p style=\"text-align: justify;\">It is a sawtooth signal. The <strong>Fourier Series<\/strong> allows us to construct a signal like that from sines and cosines, using the following mathematical model:<\/p>\n<p><a href=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-13157 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-2.jpg\" alt=\"\" width=\"424\" height=\"62\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-2.jpg 472w, https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-2-300x44.jpg 300w\" sizes=\"auto, (max-width: 424px) 100vw, 424px\" \/><\/a>Where:<\/p>\n<p><a href=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-3.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-13159 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-3.jpg\" alt=\"\" width=\"642\" height=\"48\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-3.jpg 904w, https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-3-300x23.jpg 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-3-768x58.jpg 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2021\/08\/series_de_fourier-3-696x52.jpg 696w\" sizes=\"auto, (max-width: 642px) 100vw, 642px\" \/><\/a><\/p>\n<p style=\"text-align: justify;\">If we use this model to model the sawtooth signal shown above, the process would be:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14855 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-66.png\" width=\"601\" height=\"348\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-66.png 1117w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-66-300x174.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-66-1024x592.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-66-768x444.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-66-696x403.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-66-1068x618.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-66-726x420.png 726w\" sizes=\"auto, (max-width: 601px) 100vw, 601px\" \/><\/p>\n<p style=\"text-align: justify;\">If we assign a value to the &#8220;m&#8221; of the sum, we are indicating how many harmonics we want to produce. This is known as the expansion of the <strong>Fourier Series<\/strong>. When we expand the series, what we get is a sum of sines and cosines:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14857 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-67.png\" width=\"619\" height=\"178\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-67.png 1117w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-67-300x86.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-67-1024x294.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-67-768x221.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-67-696x200.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-67-1068x307.png 1068w\" sizes=\"auto, (max-width: 619px) 100vw, 619px\" \/><\/p>\n<p style=\"text-align: justify;\">As we see, we have specified that we want an expansion of 10 harmonics. The result of the expansion is 10 sinusoidal functions, which fit the general model of the excitation function that we have presented here. If we plot the expansion of the <strong>Fourier Series<\/strong>, the result is:<\/p>\n<p style=\"padding-left: 40px;\"><a href=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/fourier_series.gif\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14859 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/fourier_series.gif\" alt=\"\" width=\"502\" height=\"336\" \/><\/a><\/p>\n<p style=\"text-align: justify;\">As we can see, adding more harmonics to the expansion of the <strong>Fourier Series<\/strong> allows the mathematical model to gradually become more accurate in representing the sawtooth wave we wanted to model initially. The more harmonics that are used, the closer the waveform will be to the desired shape.<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/fourier_series_2.gif\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14861 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/fourier_series_2.gif\" alt=\"\" width=\"450\" height=\"301\" \/><\/a><\/p>\n<p>Regardless of the number of harmonics needed, with the excitation function (or the sum of functions) we can describe any periodic wave. We can also model non-periodic waves, using unit step functions or playing with the neperian frequency of the mathematical model.<\/p>\n<p>Regardless, it is imperative to understand the excitation function and its importance in the analysis of electrical circuits.<\/p>\n<h5><strong>Integro-differential equation system<\/strong><\/h5>\n<p style=\"text-align: justify;\">What makes the analysis of AC electrical circuits difficult is not the excitation function, but the presence of reactive elements, specifically inductors and capacitors.<\/p>\n<p>For example, suppose we have the following electrical circuit:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14866 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-69.png\" width=\"498\" height=\"141\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-69.png 1204w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-69-300x85.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-69-1024x290.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-69-768x218.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-69-696x197.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-69-1068x302.png 1068w\" sizes=\"auto, (max-width: 498px) 100vw, 498px\" \/><\/p>\n<p style=\"text-align: justify;\">Suppose we want to know the current delivered by the power source <strong>v<sub>s<\/sub>(t)<\/strong> in the time domain. According to <a href=\"https:\/\/panamahitek.com\/ley-de-las-corrientes-de-kirchhoff-metodo-de-nodos\/\"><strong>Kirchhoff&#8217;s Current Law<\/strong><\/a>, this current will be the sum of the 3 currents, the one from the resistor, the inductor, and the capacitor. This can be expressed as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14870 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-71.png\" width=\"536\" height=\"181\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-71.png 1248w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-71-300x101.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-71-1024x345.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-71-768x259.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-71-696x235.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-71-1068x360.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-71-1245x420.png 1245w\" sizes=\"auto, (max-width: 536px) 100vw, 536px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14868 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-70.png\" width=\"292\" height=\"31\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-70.png 688w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-70-300x32.png 300w\" sizes=\"auto, (max-width: 292px) 100vw, 292px\" \/><\/p>\n<p style=\"text-align: justify;\">In the case of the resistor, the current can be easily calculated using Ohm&#8217;s Law. In the case of the inductor and the capacitor, each has a mathematical model for calculating current as a function of time.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14872 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-72.png\" width=\"621\" height=\"58\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-72.png 1335w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-72-300x28.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-72-1024x95.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-72-768x71.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-72-696x65.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-72-1068x99.png 1068w\" sizes=\"auto, (max-width: 621px) 100vw, 621px\" \/><\/p>\n<p style=\"text-align: justify;\">Since the three elements are connected in parallel to the power source, the voltage at each element will be equal to the voltage of the power source. With that said, the current delivered by the power source is defined as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14874\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-73.png\" alt=\"transformada de laplace\" width=\"369\" height=\"60\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-73.png 750w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-73-300x49.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-73-696x113.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-73-741x122.png 741w\" sizes=\"auto, (max-width: 369px) 100vw, 369px\" \/><\/p>\n<p style=\"text-align: justify;\">This is an integro-differential equation, that is, with integrals and derivatives in the same expression. For a mathematician, it probably isn&#8217;t very difficult to solve the derivative and integral in this expression and get the expression that represents the current delivered by the source. In engineering, you are taught to work with this type of expressions in a course called Ordinary and Differential Equations. From this point of view, the problem doesn&#8217;t seem so complicated.<\/p>\n<p style=\"text-align: justify;\">However, what happens when we need to find the answer to multiple unknowns? A system of simultaneous integro-differential equations? That is precisely the problem we face when analyzing electrical circuits with multiple meshes, multiple nodes, and multiple power sources.<\/p>\n<h5 style=\"text-align: justify;\"><strong>Transformation of mathematical models from the time domain to the frequency domain<\/strong><\/h5>\n<p style=\"text-align: justify;\">When we have mathematical expressions that are difficult to solve in the time domain, it is very convenient to work in the frequency domain. Essentially, what we do is apply the Laplace Transform to the mathematical models in terms of time for each of the elements. Then we move on to solve the algebra in the frequency domain, and finally return to the time domain, either with an <strong>Inverse Laplace Transform<\/strong> or with an approximate method such as <strong>Complex Frequency<\/strong>.<\/p>\n<p style=\"text-align: justify;\">There are other options for AC circuit analysis, such as the <strong>Fourier Transform<\/strong>, but in this post we will focus on the previously mentioned methods: <strong>Complex Frequency<\/strong> and <strong>Laplace Transform<\/strong>.<\/p>\n<p style=\"text-align: justify;\">When working with the <strong>Laplace Transform<\/strong>, we often use a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Laplace_transform#Properties_and_theorems\">table of transforms<\/a> to convert different types of mathematical models in the time domain to their equivalent in the frequency domain. In circuit analysis, the transforms we use most often are:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-14879 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-75.png\" alt=\"transformada de laplace\" width=\"881\" height=\"166\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-75.png 881w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-75-300x57.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-75-768x145.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-75-696x131.png 696w\" sizes=\"auto, (max-width: 881px) 100vw, 881px\" \/><\/p>\n<p style=\"text-align: justify;\">As we can see, derivatives are transformed into &#8220;s&#8221; and integrals into &#8220;1\/s&#8221;. In this way, it is very easy to do circuit analysis, since it is no longer necessary to deal with integrals or derivatives. In both cases there are terms associated with the initial conditions of the circuit, but in this post we will consider that the inductors and capacitors are discharged. Later I hope to make a post in which I will explain the procedure used to analyze circuits with initial load conditions.<\/p>\n<p style=\"text-align: justify;\">In the following table I am sharing with you the mathematical models for the resistance, inductor and capacitor in the time domain and their equivalent in the frequency domain.<\/p>\n<figure id=\"attachment_14876\" aria-describedby=\"caption-attachment-14876\" style=\"width: 601px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14876 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-74.png\" alt=\"transformada de laplace\" width=\"601\" height=\"697\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-74.png 601w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-74-259x300.png 259w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-74-362x420.png 362w\" sizes=\"auto, (max-width: 601px) 100vw, 601px\" \/><figcaption id=\"caption-attachment-14876\" class=\"wp-caption-text\">Table for converting from the time domain(dominio del tiempo) to the frequency domain (frequency domain).<\/figcaption><\/figure>\n<p style=\"text-align: justify;\">I think the best way to illustrate the use of this table is by solving an example. Let&#8217;s go.<\/p>\n<h5><strong>Circuit analysis in the frequency domain<\/strong><\/h5>\n<p>An\u00e1lisis de circuitos en el dominio de la frecuencia con el siguiente circuito:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14882\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76.png\" alt=\"transformada de laplace\" width=\"404\" height=\"136\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76.png 1604w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-300x101.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-1024x345.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-768x259.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-1536x517.png 1536w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-696x234.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-1068x360.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-1248x420.png 1248w\" sizes=\"auto, (max-width: 404px) 100vw, 404px\" \/><\/p>\n<p style=\"text-align: justify;\">A simple circuit with two power sources, in which we have to find the current <strong>i<sub>o<\/sub>(t)<\/strong>. Next, I will describe the steps that in my opinion should be followed to analyze this circuit.<\/p>\n<p style=\"text-align: justify;\">The first step for me is to convert the impedances and power sources to the frequency domain. To convert the impedances we use the table from the previous section. We do not apply <strong>Laplace Transform<\/strong> to the power sources, but instead we convert them to the frequency domain in the following way:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14884\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-77.png\" alt=\"transformada de laplace\" width=\"203\" height=\"36\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-77.png 489w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-77-300x53.png 300w\" sizes=\"auto, (max-width: 203px) 100vw, 203px\" \/><\/p>\n<p style=\"text-align: justify;\">Note that in the time domain we use v (lowercase) and in the frequency domain we use V (uppercase). The reason for keeping the power sources as variables will be explained later. That said, our circuit converted to the frequency domain would look like this:<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14886 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-78.png\" width=\"392\" height=\"155\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-78.png 1387w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-78-300x119.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-78-1024x405.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-78-768x303.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-78-696x275.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-78-1068x422.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-78-1063x420.png 1063w\" sizes=\"auto, (max-width: 392px) 100vw, 392px\" \/><\/p>\n<p style=\"text-align: justify;\">The resistance remains at 4. The inductance becomes an impedance represented by 2s and the capacitor becomes an impedance represented by 1\/0.25s. This transformation has been made according to what is shown in the table in the previous section. The power sources are expressed as variables in the frequency domain.<\/p>\n<p style=\"text-align: justify;\">Once the circuit is converted to the frequency domain, the techniques for analyzing circuits are used, either mesh analysis, node analysis, or any other analysis method based on Ohm&#8217;s Law. In my case, I will use node analysis to construct the following system of equations:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14888 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-79.png\" width=\"451\" height=\"184\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-79.png 1382w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-79-300x122.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-79-1024x417.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-79-768x313.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-79-696x284.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-79-1068x435.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-79-1031x420.png 1031w\" sizes=\"auto, (max-width: 451px) 100vw, 451px\" \/><\/p>\n<p style=\"text-align: justify;\">Now we sum the two currents that enter the node I have named<strong> V<sub>x<\/sub>(s)<\/strong> and equate it to the current<strong> I<sub>o<\/sub>(s)<\/strong>. Then define the current <strong>I<sub>o<\/sub>(s)<\/strong> as the voltage<strong> V<sub>x<\/sub>(s)<\/strong> divided by the impedance of the capacitor.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-14892 \" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-81.png\" width=\"402\" height=\"170\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-81.png 752w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-81-300x127.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-81-696x294.png 696w\" sizes=\"auto, (max-width: 402px) 100vw, 402px\" \/><\/p>\n<p style=\"text-align: justify;\">This system of equations has been defined based on <strong>Ohm&#8217;s Law<\/strong> and <strong>Kirchhoff&#8217;s Current Law<\/strong>.<\/p>\n<p style=\"text-align: justify;\">This system of equations has two unknowns:<strong> I<sub>o<\/sub>(s)<\/strong> and <strong>V<sub>x<\/sub>(s)<\/strong>. Using the calculator, I will solve the system of equations:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-14894 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-82.png\" alt=\"laplace transform\" width=\"1078\" height=\"327\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-82.png 1078w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-82-300x91.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-82-1024x311.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-82-768x233.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-82-696x211.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-82-1068x324.png 1068w\" sizes=\"auto, (max-width: 1078px) 100vw, 1078px\" \/><\/p>\n<p>This way, the expression for<strong> I<sub>o<\/sub>(s)<\/strong> in the frequency domain is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14896 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83.png\" width=\"416\" height=\"61\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83.png 986w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83-300x44.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83-768x113.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83-696x102.png 696w\" sizes=\"auto, (max-width: 416px) 100vw, 416px\" \/><\/p>\n<p style=\"text-align: justify;\">At this point, we need the expression for <strong>i<sub>o<\/sub>(t)<\/strong>. So far, we have <strong>I<sub>o<\/sub>(s)<\/strong>. At this point, we have to decide on the type of procedure we will use to express the response, either the <strong>Inverse Laplace Transform<\/strong> or the <strong>Complex Frequency<\/strong> method.<\/p>\n<p style=\"text-align: justify;\">Here is the importance of leaving the sources expressed as V<sub>s1<\/sub>(s) and V<sub>s2<\/sub>(s) because, as we will see below, we will be able to use the two methods we are discussing to find the solution to the problem without having to recalculate everything from the beginning.<\/p>\n<h5><strong>Inverse Laplace Transform<\/strong><\/h5>\n<p style=\"text-align: justify;\">To apply the <strong>Inverse Laplace Transform<\/strong> it is necessary to assign a value to the power sources<strong> V<sub>s1<\/sub>(s)<\/strong> and <strong>V<sub>s2<\/sub>(s)<\/strong> using the Laplace Transform of V<sub>s1<\/sub>(t) and V<sub>s2<\/sub>(t).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-14898\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-84.png\" alt=\"laplace transform\" width=\"451\" height=\"125\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-84.png 972w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-84-300x83.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-84-768x213.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-84-696x193.png 696w\" sizes=\"auto, (max-width: 451px) 100vw, 451px\" \/><\/p>\n<p>These operations can be done with the Ti-Nspire CX CAS calculator:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14900\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-85.png\" alt=\"laplace transform\" width=\"405\" height=\"164\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-85.png 954w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-85-300x122.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-85-768x312.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-85-696x282.png 696w\" sizes=\"auto, (max-width: 405px) 100vw, 405px\" \/><\/p>\n<p style=\"text-align: justify;\">Now we replace these expressions in the expression<strong> I<sub>o<\/sub>(s)<\/strong> that we defined in the previous section:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-14902 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-86.png\" alt=\"laplace transform\" width=\"1107\" height=\"301\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-86.png 1107w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-86-300x82.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-86-1024x278.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-86-768x209.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-86-696x189.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-86-1068x290.png 1068w\" sizes=\"auto, (max-width: 1107px) 100vw, 1107px\" \/><\/p>\n<p style=\"text-align: justify;\">This polynomial can be broken down into smaller polynomials through the method of partial fractions. This should allow for the formation of polynomials that can be converted to the frequency domain through a table of <strong>Inverse Laplace Transforms<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-14904 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-87.png\" alt=\"transformada de laplace\" width=\"1104\" height=\"93\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-87.png 1104w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-87-300x25.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-87-1024x86.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-87-768x65.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-87-696x59.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-87-1068x90.png 1068w\" sizes=\"auto, (max-width: 1104px) 100vw, 1104px\" \/><\/p>\n<p style=\"text-align: justify;\">This step is not necessary for me, because I have a calculator that allows me to directly do the inverse <strong>Laplace transform<\/strong>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-14906 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-88.png\" alt=\"laplace transform\" width=\"1103\" height=\"149\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-88.png 1103w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-88-300x41.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-88-1024x138.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-88-768x104.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-88-696x94.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-88-1068x144.png 1068w\" sizes=\"auto, (max-width: 1103px) 100vw, 1103px\" \/><\/p>\n<p>This would be the final expression for<strong> i<sub>o<\/sub>(t)<\/strong>, which I present adjusted below:<img loading=\"lazy\" decoding=\"async\" width=\"1189\" height=\"65\" class=\"alignnone wp-image-14908 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89.png\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89.png 1189w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-300x16.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-1024x56.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-768x42.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-696x38.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-1068x58.png 1068w\" sizes=\"auto, (max-width: 1189px) 100vw, 1189px\" \/><\/p>\n<p style=\"text-align: justify;\">This would be the answer to the problem. This mathematical model describes the current <strong>i<sub>o<\/sub>(t)<\/strong> at any time greater than t=0.<\/p>\n<p style=\"text-align: justify;\">I must emphasize that the answer obtained with the <strong>Laplace Transform<\/strong> is an <span style=\"text-decoration: underline;\">exact answer<\/span>. As I will explain later, with the <strong>Complex Frequency<\/strong> method we will obtain an <span style=\"text-decoration: underline;\">approximate response<\/span> that will converge with the response of the<strong> Laplace Transform<\/strong> once the terms with exponential functions have been extinguished.<\/p>\n<h5><strong>Complex frequency<\/strong><\/h5>\n<p style=\"text-align: justify;\">The <strong>Complex Frequency<\/strong> method makes it possible to obtain an approximate answer to a circuit analysis problem in alternating current, without the need to use the <strong>Inverse Laplace Transform<\/strong>.<\/p>\n<p style=\"text-align: justify;\">This method is the one normally used in Steady State Sinusoidal Analysis or phasor analysis. It is also the basis for the analysis of electrical power circuits. To apply this method, it is necessary to understand the concept of complex frequency variable.<\/p>\n<p style=\"text-align: justify;\">Here it will not be necessary to do an <strong>Inverse Laplace Transform<\/strong>. We will simply replace the variable &#8220;s&#8221; in the polynomial<strong> I<sub>o<\/sub>(s)<\/strong> by a complex number, whose value will depend on the angular frequency (<strong>\u03c3<\/strong>) and the angular frequency (<strong>\u03c9<\/strong>) of the excitatory function used to power the electrical circuit.<\/p>\n<p style=\"text-align: justify;\">The following table presents the form that the complex frequency variable will have as a function of the frequency(s) of the excitatory function:<\/p>\n<figure id=\"attachment_15004\" aria-describedby=\"caption-attachment-15004\" style=\"width: 1218px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-15004 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted.png\" alt=\"complex frequency\" width=\"1218\" height=\"310\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted.png 1218w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-300x76.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-1024x261.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-768x195.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-696x177.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-1068x272.png 1068w\" sizes=\"auto, (max-width: 1218px) 100vw, 1218px\" \/><figcaption id=\"caption-attachment-15004\" class=\"wp-caption-text\">Types of voltage sources for which complex frequency analysis can be used<\/figcaption><\/figure>\n<p style=\"text-align: justify;\">At the beginning of this post we invested a section in explaining the characteristics of this function.<\/p>\n<p style=\"text-align: justify;\">Analyzing the original circuit, we can identify the type of exciter function of the power sources:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" id=\"thepasted-2\" class=\"wp-image-14882 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76.png\" width=\"410\" height=\"138\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76.png 1604w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-300x101.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-1024x345.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-768x259.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-1536x517.png 1536w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-696x234.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-1068x360.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-76-1248x420.png 1248w\" sizes=\"auto, (max-width: 410px) 100vw, 410px\" \/><\/p>\n<p style=\"text-align: justify;\">Both are cosine power sources. Even though the power source on the right uses the sine function, it can be converted to a cosine function with a 90\u00ba phase shift. The two sources have the same angular frequency: 2 rad\/s (the number inside the parentheses that accompanies the &#8220;t&#8221;).<\/p>\n<p style=\"text-align: justify;\">Taking this into account we can define our complex frequency variable:<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14915 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-91.png\" width=\"137\" height=\"37\" \/><\/p>\n<p style=\"text-align: justify;\">In this expression, the &#8220;j&#8221; represents the imaginary operator, that is, square root of -1:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14917 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-92.png\" width=\"103\" height=\"33\" \/><\/p>\n<p style=\"text-align: justify;\">For mathematicians, the imaginary operator is normally represented by the letter &#8220;i&#8221;, but in Electrical Engineering it is customary to use &#8220;j&#8221; to avoid confusion with the i used to represent the intensity of electric current.<\/p>\n<p style=\"text-align: justify;\">We say that &#8220;s&#8221; is the variable of complex frequency because depending on the type of exciter function this variable can be a real number (exponential source), imaginary (sinusoidal source), complex (damped sinusoidal) or zero (DC case).<\/p>\n<p style=\"text-align: justify;\">Note that in order to use this method it is necessary that all power sources use the same angular frequency and natural frequency. If we had power sources with different frequencies, it would be necessary to use the superposition method, which requires special considerations for its implementation. This is a disadvantage of this method with respect to that of the<strong> Laplace Transform<\/strong>, which is independent of the frequency of the excitatory function(s).<\/p>\n<figure id=\"attachment_14919\" aria-describedby=\"caption-attachment-14919\" style=\"width: 502px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14919\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-93.png\" alt=\"frecuencia compleja\" width=\"502\" height=\"212\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-93.png 1439w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-93-300x127.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-93-1024x432.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-93-768x324.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-93-696x294.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-93-1068x451.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-93-996x420.png 996w\" sizes=\"auto, (max-width: 502px) 100vw, 502px\" \/><figcaption id=\"caption-attachment-14919\" class=\"wp-caption-text\">Same angular frequency in both power sources<\/figcaption><\/figure>\n<p style=\"text-align: justify;\">Another requirement of this method is to define the power sources in their phasor form. We can see this in the table that we shared above with the types of sources, specifically in the column on the right. We need to define the power sources V<sub>S1<\/sub> and V<sub>S2<\/sub> as phasors:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-14946 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106.png\" alt=\"complex frequency\" width=\"1094\" height=\"429\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106.png 1094w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-300x118.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-1024x402.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-768x301.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-696x273.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-1068x419.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-1071x420.png 1071w\" sizes=\"auto, (max-width: 1094px) 100vw, 1094px\" \/><\/p>\n<p style=\"text-align: justify;\">Basically what we do is take the amplitude and express it as a complex number in its polar form, whose phase angle will be the phase difference \u03c6 of the cosine function. When we have a sine function, we convert it to cosine by adding a 90\u00ba offset, which causes the phasor form to have a 90\u00ba lag.<\/p>\n<p style=\"text-align: justify;\">Once the power sources have been defined in their phasor form and a value has been assigned to the complex frequency variable, we replace these values \u200b\u200bin the<strong> I<sub>o<\/sub>(s)<\/strong> expression that we left defined a while ago:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" id=\"thepasted-9\" class=\"aligncenter wp-image-14896\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83.png\" alt=\"frecuencia compleja\" width=\"416\" height=\"61\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83.png 986w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83-300x44.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83-768x113.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-83-696x102.png 696w\" sizes=\"auto, (max-width: 416px) 100vw, 416px\" \/><\/p>\n<p>Substituting, we get the following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14946\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106.png\" alt=\"complex frequency\" width=\"553\" height=\"217\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106.png 1094w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-300x118.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-1024x402.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-768x301.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-696x273.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-1068x419.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-106-1071x420.png 1071w\" sizes=\"auto, (max-width: 553px) 100vw, 553px\" \/><\/p>\n<p>The answer we got is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14948 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-107.png\" width=\"275\" height=\"25\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-107.png 473w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-107-300x27.png 300w\" sizes=\"auto, (max-width: 275px) 100vw, 275px\" \/><\/p>\n<p>Here we have the answer expressed in its rectangular form and in its polar form. From this result we can obtain an answer in the time domain:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-14950\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-108.png\" alt=\"complex frequency\" width=\"501\" height=\"31\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-108.png 801w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-108-300x19.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-108-768x48.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-108-696x43.png 696w\" sizes=\"auto, (max-width: 501px) 100vw, 501px\" \/><\/p>\n<p style=\"text-align: justify;\">Both answers are valid, one from the rectangular answer and the other from the polar answer. Both are equivalent, but with different notations.<\/p>\n<p style=\"text-align: justify;\">If you ask me, I preferred the first answer, the rectangular answer, to avoid using offset angles with the polar answer. Using offset angles always causes inconvenience when entering data into the calculator.<\/p>\n<h5><strong>Comparison of responses obtained with each method<\/strong><\/h5>\n<p>The answer obtained with the <strong>Inverse Laplace Transform<\/strong> was:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" id=\"thepasted-11\" class=\"aligncenter wp-image-14908 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89.png\" alt=\"transformada de laplace\" width=\"1189\" height=\"65\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89.png 1189w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-300x16.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-1024x56.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-768x42.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-696x38.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-89-1068x58.png 1068w\" sizes=\"auto, (max-width: 1189px) 100vw, 1189px\" \/><\/p>\n<p>The response obtained with the <strong>Complex Frequency<\/strong> method was:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14952 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-109.png\" width=\"260\" height=\"29\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-109.png 440w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-109-300x33.png 300w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/p>\n<p style=\"text-align: justify;\">At first it seems that the answers are not the same, but in reality they are the same answer. As I mentioned before, the <strong>Inverse Laplace Transform<\/strong> allows us to calculate the exact answer to the problem. This response is made up of the <span style=\"text-decoration: underline;\">natural response<\/span> and the <span style=\"text-decoration: underline;\">forced response<\/span>, which in turn is the sinusoidal response in steady state.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15011 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-3.png\" alt=\"laplace transform\" width=\"1513\" height=\"195\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-3.png 1513w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-3-300x39.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-3-1024x132.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-3-768x99.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-3-696x90.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-3-1068x138.png 1068w\" sizes=\"auto, (max-width: 1513px) 100vw, 1513px\" \/><\/p>\n<p style=\"text-align: justify;\">In the case of the phasor method (<strong>Complex Frequency<\/strong>), the result will be the forced response, without the natural response.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-15006 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-1.png\" width=\"232\" height=\"51\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-1.png 527w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-1-300x66.png 300w\" sizes=\"auto, (max-width: 232px) 100vw, 232px\" \/><\/p>\n<p style=\"text-align: justify;\">The natural response occurs during circuit energization, but this will die down after some time. This is because the natural response is multiplied by an exponential function with a negative exponent. This type of function complies with the form:<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-14958 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-112.png\" width=\"110\" height=\"42\" \/><\/p>\n<p style=\"text-align: justify;\">In this type of functions we have a time constant, denoted by \u03c4, which allows us to calculate the discharge time of the function. In a function of this type, the discharge time is equal to 5\u03c4. Let&#8217;s see an example:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-15015 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-5.png\" width=\"550\" height=\"369\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-5.png 1195w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-5-300x201.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-5-1024x686.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-5-768x515.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-5-696x467.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-5-1068x716.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-5-627x420.png 627w\" sizes=\"auto, (max-width: 550px) 100vw, 550px\" \/><\/p>\n<p style=\"text-align: justify;\">In this graph the value of \u03c4 is 0.8. The discharge time of this function will be 5&#215;0.8=4.0. That is precisely what the graph shows.<\/p>\n<p style=\"text-align: justify;\">That said, when we factor the answer obtained with the<strong> Inverse Laplace Transform<\/strong> we have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1368\" height=\"111\" class=\"alignnone wp-image-14970 size-full\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-118.png\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-118.png 1368w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-118-300x24.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-118-1024x83.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-118-768x62.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-118-696x56.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2022\/12\/Pasted-118-1068x87.png 1068w\" sizes=\"auto, (max-width: 1368px) 100vw, 1368px\" \/><\/p>\n<p style=\"text-align: justify;\">The term circled in red is an exponential function with a negative exponent, such as the one shown above. The tau is 2 (\u03c4=2), so the natural response is expected to die out after 10 seconds (5\u03c4=10). After this time, the response obtained with the <strong>Complex Frequency<\/strong> method and the response obtained with the <strong>Inverse Laplace Transform<\/strong> will converge to the same value:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-15021 aligncenter\" src=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-8.png\" width=\"550\" height=\"369\" srcset=\"https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-8.png 1179w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-8-300x201.png 300w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-8-1024x687.png 1024w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-8-768x515.png 768w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-8-696x467.png 696w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-8-1068x717.png 1068w, https:\/\/panamahitek.com\/wp-content\/uploads\/2023\/01\/Pasted-8-626x420.png 626w\" sizes=\"auto, (max-width: 550px) 100vw, 550px\" \/><\/p>\n<p style=\"text-align: justify;\">As we can see, in the initial moments of the energization of the circuit, the natural response that we calculate with the <strong>Inverse Laplace Transform<\/strong> can bee seen. But after the time 5\u03c4 of the exponential function of the natural response is exceeded, the responses obtained with each method of analysis converge. This is the reason why the <strong>Complex Frequency<\/strong> method is considered a valid analysis method, because even though it does not allow us to calculate the natural response of the circuit, it does allow us to know the response in steady state.<\/p>\n<h5><strong>Advantages and disadvantages of the methods used<\/strong><\/h5>\n<p style=\"text-align: justify;\">The <strong>Complex Frequency<\/strong> method allows us to obtain the steady state response of an alternating current circuit without having to deal with the <strong>Inverse Laplace Transform<\/strong>. Said method requires significant algebraic management, which is impractical when the necessary tools are not available to solve the mathematical problem. By representing the power sources as phasors and replacing the complex frequency variable by the corresponding complex number, it is not necessary to solve a single algebraic operation.<\/p>\n<p style=\"text-align: justify;\">On the other hand, when working with <strong>Complex Frequency<\/strong> it is not possible to consider power sources with multiple frequencies and it is necessary to use the superposition method. This makes it especially difficult to use this technique when working with systems with harmonic content or DC components, such as in current electrical systems. The same applies to electrical circuits in which the power source must be modeled by means of a <strong>Fourier Series<\/strong>. Even so, the <strong>Complex Frequency<\/strong> method or Steady State Sinusoidal Analysis (Phasors) is the preferred method for the analysis of power systems worldwide.<\/p>\n<p style=\"text-align: justify;\">The <strong>Laplace Transform<\/strong> is used in cases where it is imperative to know the natural response of a system, in instances where there are power sources with multiple frequencies or in systems with non-periodic disturbances. Remember that the <strong>Complex Frequency<\/strong> method gives us the stable response in the long term and ignoring the initial moments after a change in the excitatory function of the system.<\/p>\n<p style=\"text-align: justify;\">Finally, I must highlight that in this post we have analyzed systems that are energized from a zero energy state, that is, without energy stored in the inductors and capacitors. When we have an initial charge on inductors and capacitors the <strong>Complex Frequency<\/strong> method becomes especially difficult to use because of all the mathematical procedures that are required to calculate the natural response. But that is a topic that I will deal with in another post later.<\/p>\n<h5><strong>Conclusion<\/strong><\/h5>\n<p style=\"text-align: justify;\">In this post I have presented a detailed explanation on the analysis of electrical circuits with time dependent power sources (and\/or alternating current). Two main methods have been mentioned: the <strong>Laplace Transform<\/strong> and the <strong>Complex Frequency<\/strong>. An example of how to use these methods to analyze a circuit and obtain the response of the circuit in the time domain has been presented. The excitatory function has also been mentioned as a mathematical model that allows describing any type of power source in an electrical circuit.<\/p>\n<p style=\"text-align: justify;\">I hope this post has been helpful in understanding how circuit analysis is done with time-dependent power supplies. If you have any questions or comments, I&#8217;ll be happy to answer! Thank you for reading!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a post that I have wanted to write for a long time, but it is only now that I have the time, the disposition, and the appropriate mental state to share my ideas and knowledge on this subject. In this publication, I will try to explain what I consider to be the two [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":14977,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[2074],"tags":[2469,2472,2462,2473,2468,2476,2470,2481,2477,2480,2478,2463,2461,2464,2474,2465,2053,2466,2467,2479,2471,2475],"class_list":{"0":"post-15002","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-electricity","8":"tag-alternating-current","9":"tag-angular-frequency","10":"tag-circuit-analysis","11":"tag-complex-frequency","12":"tag-electric-circuits","13":"tag-excitation-function","14":"tag-frequency-domain","15":"tag-inverse-laplace-transform","16":"tag-kirchhoffs-current-law","17":"tag-laplace-transform","18":"tag-mathematical-models","19":"tag-matrix-analysis","20":"tag-maximum-amplitude","21":"tag-mesh-analysis","22":"tag-neperian-frequency","23":"tag-node-analysis","24":"tag-ohms-law-en","25":"tag-phase-angle","26":"tag-texas-instruments-calculator","27":"tag-ti-nspire-cx-cas-simulator","28":"tag-time-domain","29":"tag-time-dependent-power-sources"},"yoast_head":"<!-- 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