{"id":3169,"date":"2024-08-26T22:29:12","date_gmt":"2024-08-26T22:29:12","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=3169"},"modified":"2024-08-26T22:29:13","modified_gmt":"2024-08-26T22:29:13","slug":"introduction-34","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/08\/26\/introduction-34\/","title":{"rendered":"Introduction"},"content":{"rendered":"\n

A <\/a>complex number<\/em> is of the form (a<\/em>+jb<\/em>) where a<\/em> is a real number<\/em> and jb<\/em> is an imaginary number<\/em>. Therefore, (1+j<\/em>2) and (5 \u2013j<\/em>7) are examples of complex numbers.<\/p>\n\n\n\n

By definition, \"image\" and j<\/em>2<\/sup>=\u20131<\/p>\n\n\n\n

(Note: In electrical engineering, the letter j is used to represent \"image\" instead of the letter i<\/em>, as commonly used in pure mathematics, because i<\/em> is reserved for current.)<\/p>\n\n\n\n

Complex numbers are widely used in the analysis of series, parallel and series-parallel electrical networks supplied by alternating voltages, in deriving balance equations with AC bridges, in analyzing AC circuits using Kirchhoff\u2019s laws, mesh and nodal analysis, the superposition theorem, with Th\u00e9venin\u2019s and Norton\u2019s theorems, and with delta-star and star-delta transforms, and in many other aspects of higher electrical engineering. The advantage of the use of complex numbers is that the manipulative processes become simply algebraic processes.<\/p>\n\n\n\n

A complex number can be represented pictorially on an\u00a0Argand diagram<\/em>. In\u00a0Figure 7.1, the line 0\u00a0A represents the complex number (2+j<\/em>3), 0B represents (3 \u2013j<\/em>), 0C represents (\u20132+j<\/em>2) and 0D represents (\u20134 \u2013j<\/em>3).<\/p>\n\n\n\n

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

Figure 7.1<\/strong> The Argand diagram<\/p>\n\n\n\n

A complex number of the form\u00a0a<\/em>+jb<\/em>\u00a0is called a\u00a0Cartesian or rectangular complex number<\/em>. The significance of the\u00a0j<\/em>\u00a0operator is shown in\u00a0Figure 7.2. In\u00a0Figure 7.2(a)\u00a0the number 4 (i.e., 4+j<\/em>0) is shown drawn as a phasor horizontally to the right of the origin on the real axis. (Such a phasor could represent, for example, an alternating current,\u00a0i<\/em>=4 sin \u03c9t<\/em>\u00a0amperes, when time\u00a0t<\/em>\u00a0is zero.)<\/p>\n\n\n\n

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

Figure 7.2<\/strong> Significance of the j<\/em> operator<\/p>\n\n\n\n

The number\u00a0j<\/em>4 (that is, 0+j<\/em>4) is shown in\u00a0Figure 7.2(b)\u00a0drawn vertically upwards from the origin on the imaginary axis. Multiplying the number 4 by the operator\u00a0j<\/em>\u00a0results in an anticlockwise phase-shift of 90\u00b0 without altering its magnitude.<\/p>\n\n\n\n

Multiplying\u00a0j<\/em>4 by\u00a0j<\/em>\u00a0gives\u00a0j<\/em>2<\/sup>4, i.e., \u20134, and is shown in\u00a0Figure 7.2(c)\u00a0as a phasor four units long on the horizontal real axis to the left of the origin\u2014an anticlockwise phase-shift of 90 compared with the position shown in\u00a0Figure 7.2(b). Thus, multiplying by\u00a0j<\/em>2<\/sup>\u00a0reverses the original direction of a phasor.<\/p>\n\n\n\n

Multiplying\u00a0j<\/em>2<\/sup>4 by\u00a0j<\/em>\u00a0gives\u00a0j<\/em>3<\/sup>4, i.e., \u2013j<\/em>4, and is shown in\u00a0Figure 7.2(d)\u00a0as a phasor four units long on the vertical, imaginary axis downward from the origin\u2014an anticlockwise phase-shift of 90 compared with the position shown in\u00a0Figure 7.2(c).<\/p>\n\n\n\n

Multiplying\u00a0j<\/em>3<\/sup>4 by\u00a0j<\/em>\u00a0gives\u00a0j<\/em>4<\/sup>4, i.e., 4, which is the original position of the phasor shown in\u00a0Figure 7.2(a).<\/p>\n\n\n\n

By similar reasoning, if a phasor is operated on by \u2013j<\/em> then a phase shift of \u201390\u00b0 (i.e., clockwise direction) occurs, again without change of magnitude.<\/p>\n\n\n\n

In electrical circuits, 90\u00b0 phase shifts occur between voltage and current with pure capacitors and inductors; this is the key as to why j<\/em> notation is used so much in the analysis of electrical networks. This is explained later in this chapter.<\/p>\n","protected":false},"excerpt":{"rendered":"

A complex number is of the form (a+jb) where a is a real number and jb is an imaginary number. Therefore, (1+j2) and (5 \u2013j7) are examples of complex numbers. By definition,  and j2=\u20131 (Note: In electrical engineering, the letter j is used to represent  instead of the letter i, as commonly used in pure mathematics, because i is reserved for current.) Complex numbers are widely used in the analysis […]<\/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-3169","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\/3169","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=3169"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3169\/revisions"}],"predecessor-version":[{"id":3170,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3169\/revisions\/3170"}],"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=3169"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3169"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3169"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

Summarizing<\/em>, application of the operator j<\/em> to any number rotates it 90\u00b0 anticlockwise on the Argand diagram, multiplying a number by j<\/em>2<\/sup> rotates it 180\u00b0 anticlockwise, multiplying a number by j<\/em>3<\/sup> rotates it 270\u00b0 anticlockwise and multiplication by j<\/em>4<\/sup> rotates<\/a> it 360\u00b0 anticlockwise, i.e., back to its original position. In each case, the phasor is unchanged in its magnitude.<\/p>\n\n\n\n