{"id":3173,"date":"2024-08-26T22:31:49","date_gmt":"2024-08-26T22:31:49","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=3173"},"modified":"2024-08-26T22:31:49","modified_gmt":"2024-08-26T22:31:49","slug":"complex-equations","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/08\/26\/complex-equations\/","title":{"rendered":"Complex Equations"},"content":{"rendered":"\n

If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal. Hence, if a<\/em>+jb<\/em>=c<\/em>+jd<\/em>, then a<\/em>=c<\/em> and b<\/em>=d<\/em>. This is a useful property, since equations having two unknown quantities can be solved from one equation. Complex equations are used when deriving balance equations with AC bridges.<\/p>\n\n\n\n

Example 7.3<\/h5>\n\n\n\n

Solve the following complex equations:<\/p>\n\n\n\n

(a) <\/a>3(a<\/em>+jb<\/em>)=9 \u2013j<\/em>2<\/p>\n\n\n\n

(b) <\/a>(2 1 j<\/em>)(\u20132+j<\/em>)=x<\/em>+jy<\/em><\/p>\n\n\n\n

(c) <\/a>(a<\/em> 2 j<\/em>2b<\/em>)+(b<\/em> \u2013j<\/em>3a<\/em>)=5+j<\/em>2<\/p>\n\n\n\n

Solution<\/h5>\n\n\n\n

.<\/p>\n\n\n\n

(a) <\/a>3(a<\/em>+jb<\/em>)=9 \u2013j<\/em>2. Thus, 3a<\/em>+j<\/em>3b<\/em>=9 \u2013j<\/em>2<\/p>\n\n\n\n

Equating real parts gives: 3a<\/em>=9, i.e., a<\/strong><\/em> =3<\/strong><\/p>\n\n\n\n

Equating imaginary parts gives:<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

(b) <\/a>(2 1 j<\/em>)(\u20132+j<\/em>)=x<\/em>+jy<\/em><\/p>\n\n\n\n

Thus, \"image\"<\/p>\n\n\n\n

Equating real and imaginary parts gives: x<\/strong><\/em> =\u20135<\/strong>, y<\/strong><\/em> =0<\/strong><\/p>\n\n\n\n

(c) <\/a>(a<\/em> 2 j<\/em>2b<\/em>)+(b<\/em> \u2013j<\/em>3a<\/em>)=5+j<\/em>2<\/p>\n\n\n\n

Thus, (a<\/em>+b<\/em>)+j<\/em>(\u20132b<\/em> \u2013 3a<\/em>)=5 +j<\/em>2<\/p>\n\n\n\n

\"image\"(7.1)<\/strong><\/p>\n\n\n\n

\"image\"(7.2)<\/strong><\/p>\n\n\n\n

We have two simultaneous equations to solve. Multiplying\u00a0equation (7.1)\u00a0by 2 gives:<\/p>\n\n\n\n

\"image\"(7.3)<\/strong><\/p>\n\n\n\n

Adding\u00a0equations (7.2)\u00a0and\u00a0(7.3)\u00a0gives \u2013a<\/em>=12, i.e.,\u00a0a<\/strong><\/em>\u00a0=\u201312<\/strong><\/p>\n\n\n\n

From\u00a0equation (7.1),\u00a0b<\/strong><\/em>\u00a0=17<\/strong><\/p>\n\n\n\n

Example 7.4<\/h5>\n\n\n\n

An equation derived from an AC bridge network is given by:<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

R<\/em>1<\/sub>, R<\/em>3<\/sub>, R<\/em>4<\/sub> and C<\/em>4<\/sub> are known values. Determine expressions for R<\/em>2<\/sub> and L<\/em>2<\/sub> in terms of the known components.<\/p>\n\n\n\n

Solution<\/h5>\n\n\n\n

Multiplying both sides of the equation by (1\/R<\/em>4<\/sub>+j<\/em>\u03c9C<\/em>4<\/sub>) gives:<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

i.e., \"image\"<\/p>\n\n\n\n

Equating the real parts gives: R<\/strong><\/em>2<\/sub><\/strong>=R<\/strong><\/em>1<\/sub><\/strong>R<\/strong><\/em>3<\/sub><\/strong>\/R<\/strong><\/em>4<\/sub><\/strong><\/a><\/p>\n\n\n\n

Equating the imaginary parts gives:<\/p>\n\n\n\n

\u03c9L<\/em>2<\/sub>=R<\/em>1<\/sub>R<\/em>3<\/sub>\u03c9C<\/em>4<\/sub>, from which, L<\/strong><\/em>2<\/sub><\/strong>=R<\/strong><\/em>1<\/sub><\/strong>R<\/strong><\/em>3<\/sub><\/strong>C<\/strong><\/em>4<\/sub><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal. Hence, if a+jb=c+jd, then a=c and b=d. This is a useful property, since equations having two unknown quantities can be solved from one equation. Complex equations are used when deriving balance equations with AC bridges. Example 7.3 Solve the following complex […]<\/p>\n","protected":false},"author":1,"featured_media":3168,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[428],"tags":[],"class_list":["post-3173","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-complex-numbers"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/08\/mathematics.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=3173"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3173\/revisions"}],"predecessor-version":[{"id":3174,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3173\/revisions\/3174"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/3168"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=3173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}