{"id":2851,"date":"2024-08-25T19:37:59","date_gmt":"2024-08-25T19:37:59","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=2851"},"modified":"2024-08-25T19:38:00","modified_gmt":"2024-08-25T19:38:00","slug":"emf-equation","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/08\/25\/emf-equation\/","title":{"rendered":"EMF EQUATION"},"content":{"rendered":"\n

When sinusoidal voltage is applied to the primary winding of a transformer, a sinusoidal flux, as shown in Figure 10.12<\/a> is set up in the iron core which links with primary and secondary winding.<\/p>\n\n\n\n

Let \u0278<\/em>m<\/sub> = Maximum value of flux in Wb<\/p>\n\n\n\n

f <\/em>= supply frequency in Hz (or c\/s)<\/p>\n\n\n\n

N<\/em>1<\/sub> = No. of turns in primary<\/p>\n\n\n\n

N<\/em>2<\/sub> = No. of turns in secondary<\/p>\n\n\n\n

As shown in Figure 10.19<\/a>, flux changes from +\u0278<\/em>m<\/sub> to \u2212\u0278<\/em>m<\/sub> in half cycle, that is, \"img\" second,<\/p>\n\n\n\n

Average rate of change of flux<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

Now, the rate of change of flux per turn is the average induced emf per turn in volt.<\/p>\n\n\n\n

\u2234 Average emf induced per turn = 4f<\/em> \u0278<\/em>m<\/sub> \\volt<\/p>\n\n\n\n

For a sinusoidal wave,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n
\"img\"\/<\/figure>\n\n\n\n

Fig. 10.12<\/strong>  Wave diagram of mutual flux set-up in magnetic core<\/p>\n\n\n\n

\u2234 RMS value of emf induced\/turn, E <\/em>= 1.11 \u00d7 4f<\/em> \u0278<\/em>m<\/sub> = 4.44f<\/em> \u0278<\/em>m<\/sub> volt<\/p>\n\n\n\n

Since primary and secondary have N<\/em>1 <\/sub>and N<\/em>2<\/sub> turns, respectively.<\/p>\n\n\n\n

\u2234 RMS value of emf induced in primary,<\/p>\n\n\n\n

 <\/p>\n\n\n\n

E<\/em>1<\/sub> = (emf induced\/turn) \u00d7 No. of primary turns<\/p>\n\n\n\n

= 4.44N<\/em>1<\/sub>f<\/em> \u0278<\/em>m<\/sub> volt        (10.1)<\/p>\n\n\n\n

Similarly, rms value of emf induced in secondary,<\/p>\n\n\n\n

E<\/em>2<\/sub> = 4.44N<\/em>2<\/sub>f<\/em> \u0278<\/em>m<\/sub> volt        (10.2)<\/p>\n\n\n\n

From eq. (10.1)<\/a>, we get,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

From eq. (10.1)<\/a>, we get,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

<\/a>Equations (10.3)<\/a> and (10.4<\/a>) clearly show that emf induced per turn on both the sides, that is, primary and secondary is the same.<\/p>\n\n\n\n

Again, we can find the voltage ratio,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

Equations (10.1)<\/a> and (10.2)<\/a> can be written in the form of maximum flux density B<\/em>m<\/sub> using relation,<\/p>\n\n\n\n

 <\/p>\n\n\n\n

\u0278<\/em>m<\/sub> = B<\/em>m<\/sub> <\/sup>\u00d7A<\/em>i<\/sub> (where A<\/em>i<\/sub> is iron area)<\/p>\n\n\n\n

\u2234<\/p>\n\n\n\n

 <\/p>\n\n\n\n

E<\/em>1<\/sub> = 4.44 N<\/em>1<\/sub>f B<\/em>m<\/sub>A<\/em>i<\/sub> volts and E<\/em>2<\/sub> = 4.44 N<\/em>2<\/sub>f B<\/em>m<\/sub>A<\/em>i<\/sub> volt<\/p>\n\n\n\n

Example 10.1<\/strong><\/p>\n\n\n\n

The emf per turn for a single-phase 2310\/220 V, 50 Hz transformer is approximately 13 volt. Calculate the number of primary and secondary turns.<\/em><\/p>\n\n\n\n

(P.T.U. Dec. 2009)<\/strong><\/p>\n\n\n\n

Solution:<\/em><\/p>\n\n\n\n

Here,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

\u2234 Primary turns,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

Secondary turns,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

Example 10.2<\/strong><\/p>\n\n\n\n

A power transformer has 1000 primary turns and 100 secondary turns. The cross-sectional area of the core is 6 sq cm and the maximum flux density while in operation is 10,000 Gauss. Calculate turns per volt for the primary and secondary windings.<\/em><\/p>\n\n\n\n

(P.T.U. Dec. 2008)<\/strong><\/p>\n\n\n\n

Solution:<\/em><\/p>\n\n\n\n

Here,<\/p>\n\n\n\n

 <\/p>\n\n\n\n

N<\/em>1<\/sub> = 1000; N<\/em>2 <\/sub>= 100; A<\/em>i<\/sub> = 6 cm2 <\/sup>= 6 \u00d7 10\u22124<\/sup>m2<\/sup><\/p>\n\n\n\n

B<\/em>m<\/sub> = 10000 gauss = 10000 \u00d7 10\u22128 <\/sup>\u00d7 104<\/sup> = 1 tesla<\/p>\n\n\n\n

We know, E<\/em>1 <\/sub>= 4.44 \u00d7 N<\/em>1<\/sub> \u00d7 f <\/em>\u00d7 B<\/em>m<\/sub> \u00d7 A<\/em>i<\/sub> = 4.44 \u00d7 1000 \u00d7 50 \u00d7 1 \u00d7 6 \u00d7 10\u22124<\/sup> = 133.2 V<\/p>\n\n\n\n

E<\/em>2<\/sub> = 4.44 \u00d7 N<\/em>2<\/sub> \u00d7 f <\/em>\u00d7 B<\/em>m<\/sub> \u00d7 A<\/em>i<\/sub> = 4.44 \u00d7 100 \u00d7 50 \u00d7 1 \u00d7 6 \u00d7 10\u22124 <\/sup>= 13.32 V<\/p>\n\n\n\n

On primary side, number of turns\/volt = \"img\"<\/p>\n\n\n\n

On secondary side, number of turns\/volt = \"img\"<\/p>\n\n\n\n

The number of turns per volt or voltage per turn on primary and secondary remains the same.<\/p>\n\n\n\n

Example 10.3<\/strong><\/p>\n\n\n\n

A 25 kVA transformer has 500 turns on the primary and 40 turns on the secondary winding. The primary is connected to 3000 V, 50 Hz mains, calculate (i) primary and secondary currents at full <\/a>load, (ii) the secondary emf, and (iii) the maximum flux in the core. Neglect magnetic leakage, resistance of the winding and the primary no-load current in relation to the full-load current.<\/em><\/p>\n\n\n\n

Solution:<\/em><\/p>\n\n\n\n

    \n
  1. At full load, \"img\"Now,\"img\"Secondary current,\"img\"<\/li>\n\n\n\n
  2. Secondary emf. \"img\"<\/li>\n\n\n\n
  3. Using relation, E<\/em>1<\/sub> = 4.44 \u00d7 N<\/em>1<\/sub> \u00d7 f<\/em> \u00d7 \u0278<\/em>m<\/sub>3300 = 4.44 \u00d7 500 \u00d7 50x<\/em>\u0278<\/em>m<\/sub>or\"img\"<\/li>\n<\/ol>\n\n\n\n

    Example 10.4<\/strong><\/p>\n\n\n\n

    The design requirements of an 11000\/415 V, 50 Hz single-phase core type transformer are approximate emf\/turn 15 V, maximum flux density 1.5 T. Find suitable number of primary and secondary turns and net cross-sectional area of core.<\/p>\n\n\n\n

    Solution:<\/em><\/p>\n\n\n\n

    Here,<\/p>\n\n\n\n

    E<\/em>1<\/sub> = 11000 V; E<\/em>2 <\/sub>= 415 V; f <\/em>= 50 Hz; B<\/em>m<\/sub> = 1.5 T<\/p>\n\n\n\n

    \"img\"\/<\/figure>\n\n\n\n

    No. of primary turns,<\/p>\n\n\n\n

    \"img\"\/<\/figure>\n\n\n\n

    No. of secondary turns,<\/p>\n\n\n\n

    \"img\"\/<\/figure>\n\n\n\n

    Now,<\/p>\n\n\n\n

     <\/p>\n\n\n\n

    E<\/em>1 <\/sub>= 4.44 N<\/em>1<\/sub> \u00d7 f<\/em>\u00b4 \u00d7 A<\/em>i<\/sub> \u00d7 B<\/em>m<\/sub><\/p>\n\n\n\n

    \"img\"\/<\/figure>\n\n\n\n

    \u2234 Net area,<\/p>\n\n\n\n

    \"img\"\/<\/figure>\n\n\n\n

    Example 10.5<\/strong><\/p>\n\n\n\n

    A single phase 50 Hz core-type transformer has rectangular cores 30 \u00d7 20 cm and the maximum allowable density is 1.05 Wb\/sq. m. Find the number of turns per limb on the high- and low- voltage sides for a voltage ratio of 3300\/200 V. Take iron factor as 0.93.<\/p>\n\n\n\n

    <\/a>Solution:<\/em><\/p>\n\n\n\n

    Gross cross-sectional area = 30 \u00d7 20 = 600 cm2<\/sup><\/p>\n\n\n\n

     <\/p>\n\n\n\n

    A<\/em>gc<\/sub> = 600 \u00d7 10\u22124<\/sup> m2<\/sup><\/p>\n\n\n\n

    The iron factor is to be taken into consideration as the laminations are insulated from each other and<\/p>\n\n\n\n

    \"img\"\/<\/figure>\n\n\n\n

    A<\/em>i<\/sub> = K<\/em>i<\/sub> \u00d7 A<\/em>gc<\/sub><\/p>\n\n\n\n

    = 0.93 \u00d7 600 \u00d7 10\u22124 <\/sup>= 558 \u00d7 10\u22124<\/sup> m2<\/sup><\/p>\n\n\n\n

    EMF induced\/turn<\/p>\n\n\n\n

     <\/p>\n\n\n\n

    = 4.44 \u00d7 f <\/em>\u00d7 B<\/em>max<\/sub> \u00d7 A<\/em>i<\/sub><\/p>\n\n\n\n

    = 4.44 \u00d7 50 \u00d7 1.05 \u00d7 558 \u00d7 10\u22124 <\/sup>= 13 V<\/p>\n\n\n\n

    Primary turns<\/p>\n\n\n\n

    \"img\"\/<\/figure>\n\n\n\n

    Secondary turns<\/p>\n\n\n\n

    \"img\"\/<\/figure>\n\n\n\n

    Example 10.6<\/strong><\/p>\n\n\n\n

    The secondary of a 500 kVA, 4400\/500 V, 50 Hz, single-phase transformer has 500 turns. Determine (i) emf per turn, (ii) primary turns, (iii) secondary full-load current, (iv) maximum flux, (v) gross cross-sectional area of the core for flux density of 1.2 tesla and iron factor is 0.92, and (vi) if the core is of square cross-section finds the width of the limb.<\/em><\/p>\n\n\n\n

    Solution:<\/em><\/p>\n\n\n\n

    Here, Rating = 500 kVA; E<\/em>1 <\/sub>= V<\/em>1 <\/sub>= 4400 V; E<\/em>2 <\/sub>= V<\/em>2 <\/sub>= 500 V<\/p>\n\n\n\n

     <\/p>\n\n\n\n

    f <\/em>= 50 Hz; N<\/em>2 <\/sub>= 500; B<\/em>m<\/sub> = 1.2 T; k<\/em>i<\/sub> = 0.92<\/p>\n\n\n\n

      \n
    1. emf per turn = \"img\"<\/li>\n\n\n\n
    2. emf per turn = \"img\"\u2234 Primary turns,\"img\"<\/li>\n\n\n\n
    3. Secondary full-load current, \"img\"<\/li>\n\n\n\n
    4. Maximum flux, \"img\"<\/li>\n\n\n\n
    5. <\/a>Iron area of the core, \"img\"Gross area of the core, \"img\"<\/li>\n\n\n\n
    6. Width of squared limb = \"img\"<\/li>\n<\/ol>\n\n\n\n

      Example 10.7<\/strong><\/p>\n\n\n\n

      A 100 kVA, 3300\/200 V, 50 Hz single-phase transformer has 40 turns on the secondary, calculate (i) the values of primary and secondary currents, (ii) the number of primary turns, and (iii) the maximum value of the flux. If the transformer is to be used on a 25 Hz system, calculate (iv) the primary voltage, assuming that the flux is increased by 10 per cent and (v) the kVA rating of the transformer assuming the current density in the windings to be unaltered.<\/p>\n\n\n\n

      Solution:<\/em><\/p>\n\n\n\n

        \n
      1. Full-load primary current, \"img\"Full-load secondary current, \"img\"<\/li>\n\n\n\n
      2. No. of primary turns, \"img\"<\/li>\n\n\n\n
      3. We know, E<\/em>2<\/sub> =4.44 \u00d7 f<\/em> \u00d7 \u0278<\/em>max<\/sub> \u00d7 N<\/em>2<\/sub>V<\/em>200 = 4.44 \u00d7 50 \u00d7 \u0278<\/em>max<\/sub> \u00d740\u2234\"img\"<\/li>\n\n\n\n
      4. As the flux is increased by 10% at 25 Hz\u2234Flux at 25 Hz, \"img\" = 0.0225 \u00d7 1.1 = 0.02475 Wb\u2234Primary voltage = 4.44 \u00d7 N<\/em>1<\/sub> \u00d7 f<\/em>\u00b4 \u00d7\"img\" volt= 4.44 \u00d7 660 \u00d7 25 \u00d7 0.02475 = 1815 <\/li>\n\n\n\n
      5. For the same current density, the full-load primary and secondary currents remain unaltered.\u2234 kVA rating of the transformer = \"img\"<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

        When sinusoidal voltage is applied to the primary winding of a transformer, a sinusoidal flux, as shown in Figure 10.12 is set up in the iron core which links with primary and secondary winding. Let \u0278m = Maximum value of flux in Wb f = supply frequency in Hz (or c\/s) N1 = No. of turns in primary N2 = No. of […]<\/p>\n","protected":false},"author":1,"featured_media":2841,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[413],"tags":[],"class_list":["post-2851","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-single-phase-transformers"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/08\/power-transformer.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/2851","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=2851"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/2851\/revisions"}],"predecessor-version":[{"id":2852,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/2851\/revisions\/2852"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/2841"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=2851"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=2851"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=2851"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}