{"id":543,"date":"2018-12-30T05:38:23","date_gmt":"2018-12-30T04:38:23","guid":{"rendered":"https:\/\/lab.fawno.com\/?p=543"},"modified":"2021-01-14T14:01:07","modified_gmt":"2021-01-14T13:01:07","slug":"","status":"publish","type":"post","link":"https:\/\/lab.fawno.com\/en\/2018\/12\/30\/apuntes-de-electronica-circuito-rc-en-continua\/","title":{"rendered":"","raw":""},"content":{"rendered":"","protected":false,"raw":""},"excerpt":{"rendered":"","protected":false,"raw":""},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","_editorskit_title_hidden":false,"_editorskit_reading_time":0,"_editorskit_typography_data":[],"_editorskit_blocks_typography":"","_editorskit_is_block_options_detached":false,"_editorskit_block_options_position":"{}","_es_post_content":"\n

En la entrada anterior<\/a> no analic\u00e9 por completo el circuito RC en continua. Dedicar\u00e9 esta entrada al an\u00e1lisis de la carga y descarga del condensador en un circuito RC.<\/p>\n\n\n\n

Analizar\u00e9 el circuito de descarga, en este caso contamos con el condensador C<\/em> cargado a una tensi\u00f3n Vc0<\/sub><\/em> y la resistencia R<\/em> conectada en paralelo al condensador. En el instante inicial la corriente es:<\/p>\n\n\n\n

I_0=\\frac{V_{c0}}{R}<\/pre><\/div>\n\n\n\n
La definici\u00f3n de intensidad de corriente<\/a> el\u00e9ctrica es:<\/p>\n\n\n\n
 Cantidad de carga por unidad de tiempo.<\/p>Wikipedia: Corriente el\u00e9ctrica<\/a><\/cite><\/blockquote>\n\n\n\n
i = \\frac{q}{t}<\/pre><\/div>\n\n\n\n
As\u00ed pues para saber la cantidad de carga el\u00e9ctrica en funci\u00f3n del tiempo:<\/p>\n\n\n\n
\\tag{1} q(t)=\\int{i\u00b7dt}<\/pre><\/div>\n\n\n\n
Efectivamente, tenemos que integrar debido a que tenemos que considerar no la cantidad de carga en un instante sino tambi\u00e9n la suma de todos los instantes previos.<\/p>\n\n\n\n
La carga el\u00e9ctrica que tiene un condensador se defin\u00eda como:<\/p>\n\n\n\n
\\tag{2} Q=C\u00b7V_c\\\\\n\\vphantom{}\\\\\nV_c=\\frac{Q}{C}<\/pre><\/div>\n\n\n\n
Sustituyendo en (2) Q<\/em> por su valor q(t)<\/em> en (1)<\/p>\n\n\n\n
\\tag{3} V_c=\\frac{1}{C}\\int{i\u00b7dt}<\/pre><\/div>\n\n\n\n
Podemos considerar que el circuito de descarga es un circuito de carga con E = 0:<\/p>\n\n\n\n
\\tag{4} E=V_r+V_c=0\\\\\n\\vphantom{}\\\\\nV_c=-V_r=-R\u00b7i<\/pre><\/div>\n\n\n\n
Sustituimos en (3) Vc<\/sub> por su valor en (4):<\/p>\n\n\n\n
\\tag{5} \\frac{1}{C}\\int{i\u00b7dt}=-Ri<\/pre><\/div>\n\n\n\n
Derivando la expresi\u00f3n respecto del tiempo:<\/p>\n\n\n\n
\\tag{6} \\frac{1}{C}i=-R\\frac{di}{dt}\\\\\n\\vphantom{}\\\\\n-\\frac{1}{RC}dt=\\frac{1}{i}di<\/pre><\/div>\n\n\n\n
Integramos:<\/p>\n\n\n\n
\\tag{7} \\int{-\\frac{1}{RC}dt}=\\int{\\frac{1}{i}di}\\\\\n\\vphantom{}\\\\\n-\\frac{1}{RC}t+K=\\ln{i}<\/pre><\/div>\n\n\n\n
Hallamos el valor de K:<\/p>\n\n\n\n
K=\\frac{t}{RC}+\\ln{i}<\/pre><\/div>\n\n\n\n
En el instante inicial t = 0 e i = I0<\/sub>:<\/p>\n\n\n\n
\\tag{8} K=\\ln{I_0}<\/pre><\/div>\n\n\n\n
Sustituimos en (7) el valor de K:<\/p>\n\n\n\n
\\tag{9} -\\frac{1}{RC}t+\\ln{I_0}=\\ln{i}\\\\\n\\vphantom{}\\\\\n-\\frac{1}{RC}t=\\ln{i}-\\ln{I_0}\\\\\n\\vphantom{}\\\\\n-\\frac{1}{RC}t=\\ln{\\frac{i}{I_0}}\\\\\n\\vphantom{}\\\\\ne^{-t\/RC}=\\frac{i}{I_0}<\/pre><\/div>\n\n\n\n
<\/a>Y finalmente despejamos la intensidad:<\/p>\n\n\n\n
\\Large \\boxed{i=I_0\u00b7e^{-t\/RC}}<\/pre><\/div>\n\n\n\n
Como he dicho anteriormente el circuito de descarga es un circuito de carga en cortocircuito (E = 0). Lo interesante de la ecuaci\u00f3n expresada en funci\u00f3n de la intensidad inicial es:<\/p>\n\n\n\n
  • No importa si el circuito es de carga o descarga<\/li>
  • No importa el estado inicial del condensador<\/li>
  • S\u00f3lo necesitamos saber o calcular la corriente inicial.<\/li><\/ul>\n\n\n\n
    Hoy en d\u00eda resulta f\u00e1cil encontrar esta demostraci\u00f3n, pero hace unos 25 a\u00f1os no era tan f\u00e1cil y lo que encontr\u00e9 en libros era la formula final expresada en tensiones para circuito de carga. En aquel entonces apenas hab\u00eda aprendido a integrar y derivar uno o dos cursos antes y se me ocurri\u00f3 intentar demostrar la formula.<\/p>\n","_es_post_name":"apuntes-de-electronica-circuito-rc-en-continua","_es_post_excerpt":"","_es_post_title":"Apuntes de electr\u00f3nica: Circuito RC en continua","_en_post_content":"","_en_post_name":"","_en_post_excerpt":"","_en_post_title":"","edit_language":"en","footnotes":""},"categories":[65,62],"tags":[63,66,43,59],"class_list":["post-543","post","type-post","status-publish","format-standard","hentry","category-apuntes","category-electronica","tag-apuntes","tag-condensadores","tag-electricidad","tag-electronica"],"ninja_gutenberg_blocks_featured_media_urls":{"thumbnail":"","ninja_gutenberg_blocks_landscape_large":"","ninja_gutenberg_blocks_portrait_large":"","ninja_gutenberg_blocks_square_large":"","ninja_gutenberg_blocks_landscape":"","ninja_gutenberg_blocks_portrait":"","ninja_gutenberg_blocks_square":"","full":""},"_links":{"self":[{"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/posts\/543","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/comments?post=543"}],"version-history":[{"count":15,"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/posts\/543\/revisions"}],"predecessor-version":[{"id":1582,"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/posts\/543\/revisions\/1582"}],"wp:attachment":[{"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/media?parent=543"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/categories?post=543"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lab.fawno.com\/en\/wp-json\/wp\/v2\/tags?post=543"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}