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| In the [[power systems|power systems analysis]] field of [[electrical engineering]], a '''per-unit system''' is the expression of system quantities as fractions of a defined base unit quantity. Calculations are simplified because quantities expressed as per-unit do not change when they are referred from one side of transformer to the other. This can be a pronounced advantage in power system analysis where large numbers of transformers may be encountered. Moreover, similar types of apparatus will have the impedances lying within a narrow numerical range when expressed as a per-unit fraction of the equipment rating, even if the unit size varies widely. Conversion of per-unit quantities to volts, ohms, or amperes requires a knowledge of the base that the per-unit quantities were referenced to.
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| The main idea of a per unit system is to absorb large difference in absolute values into base relationships. Thus, representations of elements in the system with per unit values become more uniform.
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| A per-unit system provides units for; [[electric power|power]], [[voltage]], [[Current (electricity)|current]], [[Electrical impedance|impedance]], and [[admittance]]. Except impedance and admittance, any two of these are independent and can be arbitrarily selected as base values, usually power and voltage. All quantities are specified as multiples of selected base values. For example, the base power might be the rated power of a [[transformer]], or perhaps an arbitrarily selected power which makes power quantities in the system more convenient. The base voltage might be the nominal voltage of a [[Electrical bus|bus]]. Different types of quantities are labeled with the same symbol ('''pu'''); it should be clear from context whether the quantity is a voltage, current, etc.
| | Apart from the Clash of Clans hack tool; there are unquestionably also hack tools to other games. Those can check out those hacks and obtain hundreds of which they need. It is sure that will have lost of fun once they feature the hack tool at their disposal.<br><br>To appreciate coins and gems, you must obtain the Clash behind Clans hack equipment all by clicking on the purchase button. Contingent towards the operating framework that you are utilizing, you will operate the downloaded document as the admin. Furnish our log in Id and choose the gadget. When this, you are get into the quantity of necklaces or coins that [http://En.Wiktionary.org/wiki/individuals individuals] and start off my Clash of Clans hack instrument.<br><br>In case you have any kind of questions about where by as well as the way to utilize clash of clans hack android ([http://circuspartypanama.com clicking here]), you are able to e mail us in our site. For those who have little ones who delight in video games, then you know how challenging it really is always to pull them out of the t. v.. Their eye can prove stuck towards the monitor for hours as these kinds of products play their preferred games. If you want aid regulating your your child's clash of clans Hack time, then your pursuing article has some recommendations for you.<br><br>It's possible, but the volume of absence one day would abatement by 59 one. 5% through 260 treasures to hundred or so gems. Or, as long as you capital to generate up the 1 business day bulk at 260 gems, the band would require to acceleration added considerably and also 1 holiday would turn into additional expensive.<br><br>Hold out for game of the season editions of a lot of titles. These usually come out per time of year or higher after some of the initial headline, but include a lot of the particular down-loadable and extra page content which was released all the way through steps once the first headline. These sport titles supply a excellent deal more bang for currently the buck.<br><br>To allow them to defeat higher-level villages, this task aids you to make use of a mixture of troops that Barbarians plus Archers nicely those suicide wall bombers to bust down filters. Goblins can also be a useful accent the combo simply considering that they attack different buildings. You should understand if you would like to begin worrying more or less higher troops when your family can''t win battles now with Barbarians.<br><br>Obviously individuals who produced these Crack Clash of Tourists are true fans related with the sport themselves, and as well this is exactly the activities ensures the potency of all our alternative, because we will needed to do the game ourselves. |
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| Per-unit being used in [[Power flow study|power flow]], [[short-circuit evaluation|short-circuit]] and [[motor starting methods|motor starting]] studies, it is important for all [[Power engineering|power engineers]] to be familiar with the concept.
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| ==Purpose==
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| There are several reasons for using a per-unit system:
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| * Similar apparatus (generators, transformers, lines) will have similar per-unit impedances and losses expressed on their own rating, regardless of their absolute size. Because of this, per-unit data can be checked rapidly for gross errors. A per unit value out of normal range is worth looking into for potential errors.
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| * Use of the constant <math> \scriptstyle \sqrt{3} </math> is reduced in three-phase calculations.
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| * Per-unit quantities are the same on either side of a transformer, independent of voltage level
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| * By normalizing quantities to a common base, both hand and automatic calculations are simplified.
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| * It improves numerical stability of automatic calculation methods
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| The per-unit system was developed to make manual analysis of power systems easier. Although power-system analysis is now done by computer, results are often expressed as per-unit values on a convenient system-wide base.
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| == Base quantities ==
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| Generally base values of power and voltage are chosen. The base power may be the rating of a single piece of apparatus such as a motor or generator. If a system is being studied, the base power is usually chosen as a convenient round number such as 10 MVA or 100 MVA. The base voltage is chosen as the nominal rated voltage of the system. All other base quantities are derived from these two base quantities. Once the base power and the base voltage are chosen, the base current and the base impedance are determined by the natural laws of electrical circuits. Note the base value should only be magnitudes, while the per-unit values are phasors. The phase angles of complex power, voltage, current, impedance etc. are not affected by the conversion to per unit values.
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| By convention, we adopt the following two rules for base quantities:
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| * The value of base power is the same for the entire power system of concern.
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| * The ratio of the voltage bases on either side of a transformer is selected to be the same as the ratio of the transformer voltage ratings.
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| With these two rules, a per-unit impedance remains unchanged when referred from one side of a transformer to the other. This allows us to eliminate ideal transformer from a transformer model.
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| ==Relationship between units==
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| The relationship between units in a per-unit system depends on whether the system is [[single-phase]] or [[three-phase]].
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| ===Single-phase===
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| Assuming that the independent base values are power and voltage, we have:
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| : <math>P_{base} = 1 pu</math>
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| : <math>V_{base} = 1 pu</math>
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| Alternatively, the base value for power may be given in terms of [[Reactive power|reactive]] or [[apparent power]], in which case we have, respectively,
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| : <math>Q_{base} = 1 pu</math>
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| or
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| : <math>S_{base} = 1 pu</math> | |
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| The rest of the units can be derived from power and voltage using the equations <math>S = IV</math>, <math>P = S cos(\phi)</math>, <math>Q = S sin(\phi)</math> and <math> \underline{V} = \underline{I} \underline{Z}</math> ([[Ohm's law]]), <math>Z</math> being represented by <math> \underline{Z} = R + j X = Z cos(\phi) + j Zsin(\phi)</math>. We have:
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| : <math>I_{base} = \frac{S_{base}}{V_{base}} = 1 pu</math> | |
| : <math>Z_{base} = \frac{V_{base}}{I_{base}} = \frac{V_{base}^{2}}{I_{base}V_{base}} = \frac{V_{base}^{2}}{S_{base}} = 1 pu</math>
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| : <math>Y_{base} = \frac{1}{Z_{base}} = 1 pu</math>
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| ===Three-phase===
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| Power and voltage are specified in the same way as single-phase systems. However, due to differences in what these terms usually represent in three-phase systems, the relationships for the derived units are different. Specifically, power is given as total (not per-phase) power, and voltage is line-to-line voltage.
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| In three-phase systems the equations <math>P = S cos(\phi)</math> and <math>Q = S sin(\phi)</math> also hold. The apparent power S now equals <math>S_{base}= \sqrt{3}V_{base} I_{base}</math>
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| : <math>I_{base} = \frac{S_{base}}{V_{base} \times \sqrt{3}} = 1 pu</math>
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| : <math>Z_{base} = \frac{V_{base}}{I_{base} \times \sqrt{3}} = \frac{{V_{base}^2}}{S_{base}} = 1 pu</math>
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| : <math>Y_{base} = \frac{1}{Z_{base}} = 1 pu</math>
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| ==Example of per-unit==
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| As an example of how per-unit is used, consider a three-phase power transmission system that deals with powers of the order of 500 MW and uses a nominal voltage of 138 kV for transmission. We arbitrarily select <math>S_{base} = 500</math> MVA, and use the nominal voltage 138 kV as the base voltage <math>V_{base}</math>. We then have:
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| : <math>I_{base} = \frac{S_{base}}{V_{base} \times \sqrt{3}} = 2.09 \, \mathrm{kA}</math>
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| : <math>Z_{base} = \frac{V_{base}}{I_{base} \times \sqrt{3}} = \frac{V_{base}^{2}}{S_{base}} = 38.1 \, \Omega</math>
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| : <math>Y_{base} = \frac{1}{Z_{base}} = 26.3 \, \mathrm{mS}</math>
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| If, for example, the actual voltage at one of the buses is measured to be 136 kV, we have:
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| : <math>V_{pu} = \frac{V}{V_{base}} = \frac{136 \, \mathrm{kV}}{138 \, \mathrm{kV}} = 0.9855 \, \mathrm{pu}</math>
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| ==Per-unit system formulas==
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| The following tabulation of per-unit system formulas is adapted from Beeman's ''Industrial Power Systems Handbook''.
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| {| class="wikitable"
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| ! Equation
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| | align="left" style="background:##ffdead;" colspan="2"|<math>\text{Base number selection}</math>
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| | ||<math>\text{Arbitrarily selecting from ohm's law the two base numbers: base voltage and base current}</math>
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| |align="center"|<math>1</math>||<math>\text{We have, Z}=\frac{E}{I}</math>
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| |align="center"|<math>2</math>||<math>\text{Base ohms}=\frac{\text{base volts}}{\text{base amperes}}</math>
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| |align="center"|<math>3</math>||<math>\text{Per-unit volts}=\frac{\text{volts}}{\text{base volts}}</math>
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| |align="center"|<math>4</math>||<math>\text{Per-unit amperes}=\frac{\text{amperes}}{\text{base amperes}}</math>
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| |align="center"|<math>5</math>||<math>\text{Per-unit ohms}=\frac{\text{ohms}}{\text{base ohms}}</math>
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| | ||<math>\text{Alternatively, choosing base volts and base kva values, we have,}</math>
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| | ||<math>\text{in single-phase systems:}</math>
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| |align="center"|<math>6</math>||<math>\text{Base amperes }=\frac{\text{base kva * 1000}}{\text{base volts}}</math>
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| |align="center"|<math>7</math>||<math>\text{Base amperes }=\frac{\text{base kva}}{\text{base kv}_{L-L}}</math>
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| |align="center"|<math>8</math>||<math>\text{Base ohms }=\frac{\text{base volts}}{\text{base amperes}}</math>
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| | ||<math>\text{and in three-phase systems:}</math>
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| |align="center"|<math>9</math>||<math>\text{Base amperes }=\frac{\text{base kva * 1000}}{\sqrt{3 } * \text{base volts}}</math>
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| |align="center"|<math>10</math>||<math>\text{Base amperes }=\frac{\text{base kva}}{\sqrt{3 } * \text{base kv}_{L-L}}</math>
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| |align="center"|<math>11</math>||<math>\text{Base ohms }=\frac{\text{base volts}}{\sqrt{3 } * \text{base amperes}}</math>
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| | ||<math>\text{Working out for convenience per-unit ohms directly, we have}</math>
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| | ||<math>\text{for single-phase and three-phase systems:}</math>
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| |align="center"|<math>12</math>||<math>\text{Base ohms }=\frac{\text{ohms * base kva}}{kv_{L-L}^2 * 1000}</math>
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| | align="left" style="background:##ffdead;" colspan="2"|<math>\text{Short-Circuit Calculation Formulas}</math>
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| | ||align="left" style="background:##ffdead;" colspan="2"|<math>\text{Ohms conversions:}</math>
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| |align="center"|<math>13</math>||<math>\text{Per-unit ohms reactance} =\frac{\text{ohms reactance * }\text{kva base}}{kv_{L-L}^2*1000}</math>
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| |align="center"|<math>14</math>||<math>\text{Ohms reactance} =\frac{%\text{ reactance}*kv_{L-L}^2 * 10}{\text{kva base}}</math>
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| |align="center"|<math>15</math>||<math>\text{Per-unit ohms reactance} = \frac{\text{per cent ohms reactance}}{100}</math>
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| | ||align="left" style="background:##ffdead;" colspan="2"|<math>\text{Changing ohms from one kva base to another:}</math>
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| |align="center"|<math>16</math>||<math>%\text{ ohms reactance on kva base}_2=\frac{\text{kva base}_2}{\text{kva base}_1} * %\text{ ohms reactance on base}_1</math>
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| |align="center"|<math>17</math>||<math>\text{0/1 ohms reactance on kva base}_2=\frac{\text{kva base}_2}{\text{kva base}_1}\text{ * 0/1 ohms reactance on base}_1</math>
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| | ||align="left" style="background:##ffdead;" colspan="2"|<math>\text{Changing incoming system reactance:}</math>
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| | ||<math>\text{a. If system reactance is given in percent, use Eq. 16 to change from one kva base to another.}</math>
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| | ||<math>\text{b. If system reactance is given in short-circuit symmetrical rms kva or current, convert to per-unit as follows:}</math>
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| |align="center"|<math>18</math>||<math>\text{0/1 reactance} = \frac{\text{kva base used in reactance in studied calculation}}{\text{system short-circuit kva}}</math>
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| |align="center"|<math>19</math>||<math>\text{0/1 reactance} = \frac{\text{kva base used in reactance in studied calculation}}{\text{system short-circuit current * }\sqrt{3} \text{ * system kv}_{L-L}}</math>
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| |-
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| | ||align="left" style="background:##ffdead;" colspan="2"|<math>\text{Calculating approximate motor kva base:}</math>
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| | ||<math>\text{a. For induction motors and 0.8 power factor synchronous motors}</math>
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| |align="center"|<math>20</math>||<math>\text{kva base} \approx\text{ horsepower rating}</math>
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| | ||<math>\text{b. For unity power factor synchronous motors}</math>
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| |align="center"|<math>21</math>||<math>\text{kva base} \approx\text{ 0.8 * horsepower rating}</math>
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| | ||align="left" style="background:##ffdead;" colspan="2"|<math>\text{Converting ohms from one voltage to another:}</math>
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| |align="center"|<math>22</math>||<math>\text{Ohms on basis of voltage}_1 =(\frac{\text{voltage}_1}{\text{voltage}_2})^2 \text{ * ohms on basis of voltage}_2</math>
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| |-
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| | ||align="left" style="background:##ffdead;" colspan="2"|<math>\text{Short-circuit kva and current calculations}</math>
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| | ||<math>\text{Symmetrical short circuit kva:}</math>
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| |align="center"|<math>23</math>||<math>=\frac{\text{100 * kva base}}{%\text{ X}}</math>
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| |align="center"|<math>24</math>||<math>=\frac{\text{kva base}}{\text{0/1 X}}</math>
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| |align="center"|<math>25</math>||<math>=3 * \frac{\text{Voltage}_{L-N}^2}{\text{ohms reactance}\text{ * 1000}}</math>
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| |align="center"|<math>26</math>||<math>=\frac{\text{kv}_{L-L}^2 \text{ * 1000}}{\text{ohms reactance}}</math>
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| |-
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| | ||<math>\text{Symmetrical short circuit current:}</math>
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| |align="center"|<math>27</math>||<math>=\frac{\text{100 * kva base}}{%\text{ X}* \sqrt{3} *\text{kv}_{L-L}}</math>
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| |-
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| |align="center"|<math>28</math>||<math>=\frac{\text{kva base}}{\text{0/1 X}* \sqrt{3} *\text{kv}_{L-L}}</math>
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| |-
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| |align="center"|<math>29</math>||<math>=\frac{\text{kv}_{L-L} \text{ * 1000}}{\sqrt{3} * \text{ohms reactance}}</math>
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| |-
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| | ||<math>\text{Asymmetrical short-circuit current and kva:}</math>
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| |align="center"|<math>30</math>||<math>\text{Asymmetrical short-circuit current = symmetrical current * X/R factor }</math>
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| |-
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| |align="center"|<math>31</math>||<math>\text{Asymmetrical short-circuit kva = symmetrical kva * X/R factor }</math>
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| |}
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| ==References==
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| *{{cite conference|last=Beeman|first=Donald|title=Short-Circuit-Current Calculating Procedures|booktitle=Industrial Power Systems Handbook|editor-last=Beeman|editor-first=Donald (ed.)|publisher= McGraw-Hill|year=1955|pages=see esp. 38–41, 52–55|url=http://books.google.ca/books/about/Industrial_power_systems_handbook.html?id=zOgiAAAAMAAJ}}
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| *{{cite conference|last=Elgerd|first=Olle I.|title=§2.5 Per-Unit Representation of Impedances, Currents, Voltages and Powers|booktitle=Electric Energy Systems Theory: An Introduction|year=2007|edition=1971 1st|publisher=Tata McGraw-Hill|pages=35–39|url=http://books.google.ca/books/about/Electric_Energy_Systems_Theory.html?id=AKTi3UxfhlgC&redir_esc=y|isbn=978-0070192300}}
| |
| *{{cite journal|last=Yuen|first=Moon H.|title=Short Circuit ABC--Learn It in an Hour, Use It Anywhere, Memorize No Formula|journal=IEEE Trans. on Industry Applications|date=Mar–Apr 1974|volume=IA-10|issue=2|pages=261–272|doi=10.1109/TIA.1974.349143}}
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| *{{cite book|last=William D., Jr|first=Stevenson,|title=Elements of power system analysis|year=1975|publisher=McGraw-Hill|location=New York|isbn=0-07-061285-4|edition=3rd}}
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| | |
| *{{cite book|last=Weedy|first=B.M.|title=Electric power systems|year=1972|publisher=J. Wiley|location=London ; Toronto|isbn=0-471-92445-8|edition=2nd}}
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| *{{cite book|last1=Glover|first1=J. Duncan|last2=Sarma|first2=Mulukutla|last3=Overbye|first3=Thomas J.|title=Power System Analysis and Design|year=2011|publisher=Cengage Learning|isbn=1111425779|pages=108-116|url=http://books.google.ca/books?id=U77A2C37QesC&dq=power+system+analysis+and+design&source=gbs_navlinks_s}}
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| [[Category:Electrical engineering]]
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| [[Category:Electric power]]
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| [[Category:Power engineering]]
| |
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