Cable Systems - Postive/Negative Seq Impedance

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1 Cable Systems - Postive/Negative Seq Impedance Nomenclature: GMD GMR - geometrical mead distance between conductors; depends on construction of the T-line or cable feeder - geometric mean raduius of conductor (single conductor, not bundle); values are given in tables with conductor properties GMR and GMD calculations and Estimations Conductor Reactance z primitive z aa z ab z aa z ba z ca z na z na z n3a z ac z ab z bb z cb z nb z nb z n3b z ac z bc z cc z nc z nc z n3c z z ij ba z bb z bc Z in z ca z cb z cc z an z bn z cn z nn z nn z n3n z an z bn z cn z an z bn z cn z an z bn z cn z nn z nn z n3n z an3 z bn3 z cn3 z an3 z bn3 z cn3 z nn3 z nn3 z n3n3 z na Z z nj na z nb z nc z nn z n3a z nb z n3b z nc z n3c z nn z nn z n3n z nn z nn z n3n z nn3 z nn3 z n3n3 z primitive z ij z nj z in z nn File: Cable_Tline_Calcs_Rev.xmcd Page: of 3

2 3PH, 4W, grounded WYE line segement Vs ag Vs bg Vs cg Vs ng Vr ag Vr bg Vr cg Vr ng + z aa z ba z ca z na z ab z bb z cb z nb z ac z bc z cc z nc z an z bn z cn z nn I a I b I c I n Vs abc Vs ng Vr abc Vr ng + z ij z nj z in z nn I abc I n Because Vs ng and Vr ng are grounded, then Vs ng and Vr ng Vs abc Vr abc + z ij I abc + z in I n + z nj I abc + z nn I n I n z nn z nj I abc ( ) Vs abc Vr abc + z ij I abc + z in z nn z nj I abc File: Cable_Tline_Calcs_Rev.xmcd Page: of 3

3 ( ) I abc Vs abc Vr abc + z ij z in z nn z nj Z abc Z ij z in z nn z nj Vs abc Vr abc + Z abc I abc Sequence values V ag V bg V cg Vlg Vlg Vlg : e jdeg A s :.5.866j j j.5.866j Vlg abc A s Vlg I abc A s I Vlg A s Vlgabc A s 3 A s j.67.89j j j File: Cable_Tline_Calcs_Rev.xmcd Page: 3 of 3

4 Vs abc Vr abc + Z abc I abc A s Vs abc A s Vr abc + A s Z abc A s I Vlg s_ Vlg r_ + Z I Z A s Z abc A s Z Z Z Z Z Z Z Z Z Z For symetrical, transposed line: z s 3 z aa + z bb + z cc ( ) z m 3 z ab + z bc + z ca ( ) Consequently: Z abc z s z m z m z m z s z m z m z m z s z ( z s + z m ) z z ( z s z m ) Or by applying GMD and GMR calcualtions for system of n cond n cond : Between phases: D ij GMD ij 3 Dab D bc D ca ft Phase and neutral: D in GMD in 3 Dan D bn D cn ft z ii r i j.34 GMR i File: Cable_Tline_Calcs_Rev.xmcd Page: 4 of 3

5 z nn r i j.34 z ij j.34 z in j.34 z in z z ii + z ij 3 z nn ( ) z z z ii z ij D ij D in z z r i + j.34 GMR n D ij GMR i File: Cable_Tline_Calcs_Rev.xmcd Page: 5 of 3

6 Series Impedance for Underground Line Where: d c - phase conductor diameter [in] d od - nominal diameter over the concentric neutrals of the cable [in] d s - diameter of a concentric neutral strand [in] GMR c - geometric mean radius of the phase conductor [ft] GMR s - geometric mean radius of a neutral strand [ft] k GMR cn GMRs kr k ft r c - resitance of the phase conductor [ ] r s - resistance of olid neutral strand [ ] k - number of concentric neutral strands File: Cable_Tline_Calcs_Rev.xmcd Page: 6 of 3

7 Example: Phase: () () (3) Neutral: (4) (5) (6) Cable is 5kcmil, 5kV, Al with (3)-#4 annealed, coated copper wires (/3 neutral), OD cab :.9in, k : 3 f : 6Hz For cable: 5 kcmil AA: GMR abc :.7 OD cond :.567in r c :.4 For neutral strand: #4 Copper GMR s :.8ft d s :.64in r cs : 4.87 R : OD cab d s.5ft 3 GMR cn GMRs 3R :.4864 ft r cn : r cs k.44 File: Cable_Tline_Calcs_Rev.xmcd Page: 7 of 3

8 D D D 45 D 54.5ft D :.5 D 45 :.5 D 3 D 3 D 56 D 65.5ft D 3 D 3 D 64 D 46.ft D 3 :. D 46 : D 4 D 5 D 36 R.5ft D 4 :.5 D 5 D 5.5ft D 5 :.5 D 6 D 6.5ft D 6 D 6.ft D 6 : z r c.953 : + + j.34 GMR ( j) abc z : z z 33 : z z 44 r cn.953 : + + j.34 GMR ( j) cn z 55 : z 44 z 66 : z 44 z.953 : + j.34 + ( j) D z : z z 3 : z z 3 : z z : + j.34 + ( j) D 3 z 3 : z z : + j.34 + ( j) D z 4 : z 4 z 5 : z 4 z 5 : z 5 z 36 : z 4 z 63 : z 36 File: Cable_Tline_Calcs_Rev.xmcd Page: 8 of 3

9 z : + j.34 D ( j) 5 z 5 : z 5 z 4 : z 5 z 4 : z 4 z 35 : z 5 z 53 : z 35 z 6 : z 5 z : + j.34 + ( j) D 6 z 6 : z 6 z 34 : z 6 z 43 : z z : + j.34 + ( j) D 45 z 54 : z 45 z 56 : z 45 z 65 : z 56 z : + j.34 + ( j) D 46 z 64 : z z 6 : z 6 z ij : z z z 3 z z z 3 z 3 z 3 z j j j j j j j j j z in : z 4 z 4 z 34 z 5 z 5 z 35 z 6 z 6 z j j j j j j j j j z nj : z 4 z 5 z 6 z 4 z 5 z 6 z 43 z 53 z j j j j j j j j j z nn : z 44 z 54 z 64 z 45 z 55 z 65 z 46 z 56 z j j j j j j j j j File: Cable_Tline_Calcs_Rev.xmcd Page: 9 of 3

10 z abc : z ij z in z nn z nj j z abc j.85.4j z : A s zabc A s j j j.85.4j j j i z i.8.8i.8.8i i i i i i File: Cable_Tline_Calcs_Rev.xmcd Page: of 3

11 Shunt capacitance Primitive matrices are used for UG cables and OH lines. Sometimes, if cable system is simple (3/c or 3-/c only), capacitance of cables are calculated directly from physical properties. p primitive p aa p aa p ba p ca p na p na p n3a p ab p ac p ab p bb p cb p nb p nb p n3b p ac p bc p cc p nc p nc p n3c p an p bn p cn p nn p nn p n3n p an p bn p cn p nn p nn p n3n p an3 p bn3 p cn3 p nn3 p nn3 p n3n3 p p ij ba p bb p bc p in p ca p cb p cc p an p bn p cn p an p bn p cn p an3 p bn3 p cn3 p na p nb p nc p p nj na p nb p nc p nn p n3a p n3b p n3c p nn p nn p n3n p nn p nn p n3n p nn3 p nn3 p n3n3 p primitive p ij p nj p in p nn Potential coefficients in cable is: S ii p ii.7689 RD i μf S ij p ij.7689 D ij μf where: S ii - distance from conductor i to its image i' [ft] S ij RD i - distance from conductor i to the image of conductor j [ft] - radius of conductor i in [ft] File: Cable_Tline_Calcs_Rev.xmcd Page: of 3

12 D ij - distance from conducor i to conductor j [ft] p abc p ij p in p nn p nj c abc p abc ( ) y abc jω μs + c abc ω πf Calculation cables capacitance from physical properties Phase to ground capacitance C pg R b RD c πε krd s k R b where: ε ε r ε o ε r - is the relative permittivity of the material (see table below) ε o 8.85 F : is the vacuum permittivity, or m ε o.443 μf File: Cable_Tline_Calcs_Rev.xmcd Page: of 3

13 Example with previous example data: R b :.5ft R : R b.63in ε r :.3 k 3 OD cond RD c :.84 in d s RD s :.3 in y ag + jπfc pg y ag : + πfj R b RD c πε r ε o krd s k R b j μs y bg : y ag y cg : y ag y abc : y ag y bg y cg j j j μs y : A s yabc A s j y j j μs File: Cable_Tline_Calcs_Rev.xmcd Page: 3 of 3