Important Formula of Symmetrical Fault Analysis

( 1 ) Percentage Resistance

%R = (  IR / V ) × 100

OR

%R = (  R )(  kVA ) / 10 ( kV )²

Where

I = Full load current

V = Rated Voltage

R = Resistance in ohm

            Resistance R = ( %R ) ×  V / I × 100  

(  2 ) Percentage Reactance

%X = (  IX / V ) × 100

                            OR

%X = (  X )(  kVA ) / 10 ( kV )²

Where

I = Full load current

V = Rated Voltage

X = Resistance in ohm 

           Reactance X = ( %X ) ×  V / I × 100

( 3 ) Percentage reactance at Base kVA

        = ( Base kVA / Rated kVA ) × Percentage reactance at rated kVA

( 4 ) Reactance at two voltage level

X2 = ( E2 / E1 ) ² × X1

Where

X1 = Reactance at voltage level E1

X2 = Reactance at voltage level E2

·       Resistance and reactance refer to phase values not line values

( 5 ) Per unit

Per unit value of any quantity = Actual value of quantity / Base value of quantity

Per unit kV = Actual kV / Base kV

Per unit current I_pu = Actual Current / Base current

                                = ( Actual Current × kV_B ) / kVA_B

Per unit impedance Z_pu 

 = Actual impedance / Base impedance                                                              = Actual impedance × kVA_B / [( kV_B )² ×1000 ]

( 6 ) Base value

Base current = Base kVA / Base kV

Base impedance = Base voltage / Base current

                           = ( kV_B )² × 1000 / kVA_B

                           = ( kV_B )² / MVA_B  Ohms

Base impedance Z_B = V_B × I_B

( 7 ) Three phase System Star connection

Base current = kVA_B  / ( √ 3 ×  kV_B )

Where kV_B = Base Line to line voltages

        kVA_B = Base kVA

Base Impedance Z_B = ( kV_B ) × 1000 / ( √ 3 × I_B )

                                = ( kV_B )² × 1000 / kVA_B

Per unit impedance Z_PU = Actual impedance × kVAB / ( kVB )² × 1000

( 8 ) Per unit impedance in two cases

Z_{PU NEW}  = ( kVA_NEW  ) / ( kV_NEW )²

Z_{PU OLD} = ( kVA_OLD  ) / ( kV_OLD )²

Z_{PU NEW} 

    = Z_{PU OLD} × [( kVA_NEW  ) / ( kV_NEW )² ] / [( kVA_OLD  ) / ( kV_OLD )² ]

   = Z_{PU OLD} × [ kVA_NEW / kVA_OLD ] × [ ( kV_OLD )² / ( kV_NEW )² ]

( 9 ) Per unit impedance in the two-winding transformer

Z_O1 = Impedance of transformer as referred to primary side

Z_O2 = Impedance of transformer as referred to secondary side

kVA_B = Rated kVA of transformer

kV_B1 = Rated voltage of the transformer primary winding

kV_B2 = Rated voltage of the transformer secondary winding

N1 = Primary winding turns

N2 = Secondary winding turns

         Z_O2 = Z_O1 × ( N2 / N1 ) ²

                = Z_O1 × (  kV_B2 / kV_B1 ) ²…………… ( 1 )

Z_{O2 ( PU )} = Z_O2 × kVA_B / [( kV_B2 )² × 1000 ] ……….. ( 2 )

From equation ( 1 ) and ( 2 )

Z_{O2 ( PU )} = Z_O1 ×  [ (  kV_B2 / kV_B1 ) ² ]  × kVA_B / [( kV_B2 )² × 1000 ]

                 = Z_O1 × kVA_B  / [ ( kV_B1 ) ² × 1000 ]

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