4b - Symmetrical Components

Bolted Fault

Non-impedance (zero impedance) fault e.g. dropping metal tool that lands across phases

Equivalent Circuit for Line-to-Ground (L-G) Fault

All (0), (+), and (-) sequence branches Ia0 = Ia1 = Ia2 Ia = 3Ia0 = 3Ia1 = 3Ia2 = 2Ia0 + Ia1 = etc.... |IF| = |Ia| Ib = Ic = 0 Zeq = Z1 + Z2 + Z3

Equivalent Circuit for Double Line to Ground (L-L-G) Fault

All (0), (+), and (-) sequence branches In = Ic + Ib |IF| = |Ib| = |Ic| Ia = 0 = Ia0 + Ia1 + Ia2 Zeq = Z1 + (Z2 || Z0)

Positive Sequence Network Diagram

Draw all EMF sources (gen., motor) as voltage source in series with the machines' impedances (Xg, Xm) Draw remaining device impedances (xfmr, line) in series/parallel with respect to how they connect in the single-line diagram (Xt, XL) Draw reference bus (top or bottom of circuit)

Unbalanced Symmetrical Components

Non-zero-sum current & voltage phasors 0 != Va + Vb + Vc 0 != Ia + Ib + Ic ( = In --> wye-connected only)

Equivalent Circuit for Line-to-line (L-L) Fault

Only (+) and (-) sequence branches Ia = 0 Ib = -Ic |IF| = |Ib| = |Ic| Ia0 = 0 Ia1 = -Ia2 Zeq = Z1+Z2

Equivalent Circuit for Three-phase (3Φ) Fault

Only (+) sequence branch Ia2 = Ia0 = 0 IF = Ia = Ia1 Zeq = Z1

Types of faults

Three-phase (3Φ) = across all 3 phases (same as 3Φ-to-ground) Line-to-line (L-L) = across 2 of 3 total phases Double line to ground (L-L-G) = across 2 of 3 total phases & ground Line-to-ground (L-G) = across 1 of 3 total phases & ground (same as line-to-neutral) All 4 fault types uniquely match an individual 1Φ equivalent circuit

Purpose of symmetrical components

To analyze different faults Calculations are easy w/ a balanced system, but are typically used for approximations Calculations are handled differently in an unbalanced system, which constitutes most real-world situations A set of unbalanced phasors can be represented as the sum of its symmetrical components

Balanced Symmetrical Components

Zero-sum current & voltage phasors 0 = Va + Vb + Vc 0 = Ia + Ib + Ic ( = In --> wye-connected only)

Zero Sequence Network Diagram

Short-circuit all EMF sources (gen., motor) in series with the machines' impedances (Xg, Xm) Draw reference bus (top or bottom of circuit) Modify each transformer connection to the reference bus (depending on connection type on single-line) Modify each EMF source connection to the reference bus (depending on grounding configuration on single-line), which will always be 3-times grounding reactance in series (3*Xgnd) Use Ref. 5.1.5 for this

Negative Sequence Network Diagram

Short-circuit all EMF sources (gen., motor) in series with the machines' impedances (Xg, Xm) Draw remaining device impedances (xfmr, line) in series/parallel with respect to how they connect in the single-line diagram (Xt, XL) Draw reference bus (top or bottom of circuit)

Categories of faults

Symmetrical = 3Φ Asymmetrical = L-L, L-L-G, L-G

Fault

Unintended electrical connection across phases or ground, such as a short circuit Most real-world faults have impedance between the faulted phases & ground Non-impedance (zero impedance) faults are "bolted faults" (e.g. dropping metal tool that lands across phases)

Fault Type Equivalent Circuits

Used to model faults All 4 fault types uniquely match an individual 1Φ equivalent circuit Must always convert 3Φ VL to 1Φ VLn 1Φ VLn will always appear in the (+) sequence branch

3Φ Unsymmetrical Phasors

Va = Va0 + Va1 + Va2 Vb = Va0 + (a^2)Va1 + (a)Va2 Vc = Va0 + (a)Va1 + (a^2)Va2 Ia = Ia0 + Ia1 + Ia2 Ib = Ia0 + (a^2)Ia1 + (a)Ia2 Ic = Ia0 + (a)Ia1 + (a^2)Ia2 Voltage sums are in reference book (matrix format), but current sums are not. Easy to remember, though - just replace V's with I's

3Φ Symmetrical Phasors

Va0 = (Va + Vb + Vc) / 3 Va1 = (Va + (a)Vb + (a^2)Vc) / 3 Va2 = (Va + (a^2)Vb + (a)Vc) / 3 Ia0 = (Ia + Ib + Ic) / 3 Ia1 = (Ia + (a)Ib + (a^2)Ic) / 3 Ia2 = (Ia + (a^2)Ib + (a)Ic) / 3 Voltage sums are in reference book (matrix format), but current sums are not. Easy to remember, though - just replace V's with I's

Symmetrical Component Notation

a = 1<(120°) a^2 = 1<(240°) = 1<(-120°) Va0 = zero sequence Va1 = positive sequence (abC) Va2 = negative sequence (acb)