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
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
( ) 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.5 +.866j.5 +.866j.5.866j Vlg abc A s Vlg I abc A s I Vlg A s Vlgabc A s 3 A s.333.333.333.333.67 +.89j.67.89j.333.67.89j.67 +.89j File: Cable_Tline_Calcs_Rev.xmcd Page: 3 of 3
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 +.953 + j.34 GMR i + 7.934 File: Cable_Tline_Calcs_Rev.xmcd Page: 4 of 3
z nn r i +.953 + j.34 z ij.953 + j.34 z in.953 + 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 + 7.934 D ij GMR i + 7.934 + 7.934 File: Cable_Tline_Calcs_Rev.xmcd Page: 5 of 3
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
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
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 + 7.934 (.55 +.456j) abc z : z z 33 : z z 44 r cn.953 : + + j.34 GMR + 7.934 (.39 +.33j) cn z 55 : z 44 z 66 : z 44 z.953 : + j.34 + (.95 +.47j) D z : z z 3 : z z 3 : z 3 7.934 z 3.953 : + j.34 + (.95 +.963j) D 3 z 3 : z 3 7.934 z 4.953 : + j.34 + (.95 +.34j) D 4 7.934 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
z 5.953 : + j.34 D + 7.934 (.95 +.47j) 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 6.953 : + j.34 + (.95 +.963j) D 6 z 6 : z 6 z 34 : z 6 z 43 : z 34 7.934 7.934 z 45.953 : + j.34 + (.95 +.47j) D 45 z 54 : z 45 z 56 : z 45 z 65 : z 56 z 46.953 : + j.34 + (.95 +.963j) D 46 z 64 : z 46 7.934 z 6 : z 6 z ij : z z z 3 z z z 3 z 3 z 3 z 33.55 +.456j.95 +.47j.95 +.963j.95 +.47j.55 +.456j.95 +.47j.95 +.963j.95 +.47j.55 +.456j z in : z 4 z 4 z 34 z 5 z 5 z 35 z 6 z 6 z 36.95 +.34j.95 +.47j.95 +.963j.95 +.47j.95 +.34j.95 +.47j.95 +.963j.95 +.47j.95 +.34j z nj : z 4 z 5 z 6 z 4 z 5 z 6 z 43 z 53 z 63.95 +.34j.95 +.47j.95 +.963j.95 +.47j.95 +.34j.95 +.47j.95 +.963j.95 +.47j.95 +.34j z nn : z 44 z 54 z 64 z 45 z 55 z 65 z 46 z 56 z 66.39 +.33j.95 +.47j.95 +.963j.95 +.47j.39 +.33j.95 +.47j.95 +.963j.95 +.47j.39 +.33j File: Cable_Tline_Calcs_Rev.xmcd Page: 9 of 3
z abc : z ij z in z nn z nj.798 +.446j z abc.39 +.33j.85.4j z : A s zabc A s.39 +.33j.789 +.44j.39 +.33j.85.4j.39 +.33j.798 +.446j.46 +.4666i z.56 +.65i.8.8i.8.8i.4874 +.45i.53 +.3i.56 +.65i.64 +.45i.4874 +.45i File: Cable_Tline_Calcs_Rev.xmcd Page: of 3
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
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
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 96.847j μs y bg : y ag y cg : y ag y abc : y ag y bg y cg 96.847j 96.847j 96.847j μs y : A s yabc A s 96.84793j y 96.84793j 96.84793j μs File: Cable_Tline_Calcs_Rev.xmcd Page: 3 of 3