EEE471Ch8

EEE471Ch8 - Symmetrical Components Chapter 8 1 Symmetrical...

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1 Symmetrical Components Chapter 8
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2 Symmetrical Components Symmetrical Components is often referred to as the language of the Relay Engineer but it is important for all engineers that are involved in power. The terminology is used extensively in the power engineering field and it is important to understand the basic concepts and terminology.
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3 Symmetrical Components Used to be more important as a calculating technique before the advanced computer age. Is still useful and important to make sanity checks and back-of-an-envelope calculation. We will be studying 3-phase systems in general. Previously you have only considered balanced voltage sources, balanced impedance and balanced currents.
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4 Symmetrical Components n a a b c V a V b V c V a V b V c Balanced load supplied by balanced voltages results in balanced currents This is a positive sequence system, In Symmetrical Components we will be studying unbalanced systems with one or more disymmetry. Z Y Z Y Z Y I b I a I c
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5 Symmetrical Components For the General Case of 3 unbalanced voltages V A V B V C 6 degrees of freedom Can define 3 sets of voltages designated as positive sequence, negative sequence and zero sequence
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6 Symmetrical Components Positive Sequence 120 o 120 o 120 o V A1 V B1 V 2 degrees of freedom V A1 = V A1 V B1 = a 2 V A1 A1 a is operator 1/120 o
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7 Symmetrical Components Negative Sequence 120 o 120 o 120 o V A2 V V B2 2 degrees of freedom a is operator 1/120 o V A2 = V A2 V B2 = aV A2 2 A2
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8 Symmetrical Components Zero Sequence 2 degrees of freedom V A0 V B0 V V A0 = V B0 = V
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9 Symmetrical Components Reforming the phase voltages in terms of the symmetrical component voltages: V A = V A0 + V A1 + V A2 V B = V B0 + V B1 + V B2 V = V + V + V What have we gained? We started with 3 phase voltages and now have 9 sequence voltages. The answer is that the 9 sequence voltages are not independent and can be defined in terms of other voltages.
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10 Symmetrical Components Rewritng the sequence voltages in term of the Phase A sequence voltages: V A = V A0 + V A1 +V A2 V B = V A0 + a 2 V A1 + aV A2 V V A = V 0 + V 1 +V 2 V B = V 0 + a 2 V 1 + aV 2 V = V 0 + aV 1 +a 2 V 2 Drop A Suggests matrix notation: V A 1 1 1 V 0 V B 1 a 2 a V 1 V 1 a a 2 V 2 = [V P ] = [A] [V S ]
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11 Symmetrical Components We shall consistently apply: [V P ] = Phase Voltages [V S ] = Sequence Voltages 1 1 1 [A] = 1 a 2 a 1 a a 2 [V P ] = [A][V S ] Pre-multiplying by [A] -1 [A] -1 [V P ] = [A] -1 [A][V S ]= [I][V S ] [V S ] = [A] -1 [V P ]
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12 Operator a a = 1 /120 o = - .5 + j .866 a 2 = 1 / 240 o = - .5 - j.866 a 3 = 1 / 360 o = 1 a 4 = 1 / 480 o = 1 / 120 o = a a 5 = a 2 etc. 1 + a + a 2 = 0 a - a 2 = j 3 1 - a 2 = /30 o 1/a = a 2 3 Relationships of a can greatly expedite calculations -1
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13 Inverse of A [ ] = 2 2 1 1 1 1 1 a a a a A Step 1: Transpose [ ] = 2 2 1 1 1 1 1 a a a a A T Step 2: Replace each element by its minor - - - - - - - - - 1 1 1 1 2 2 2 2 2 2 2 a a a a a a a a a a a a a a 1 1 2 3 2 3
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EEE471Ch8 - Symmetrical Components Chapter 8 1 Symmetrical...

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