High-voltage calculators.Fault levels, cables, lines, earthing, arc flash and protection.

Fault level

Fault levelThree-phase symmetrical fault current I″k (IEC 60909)

Z_{k} = \sqrt{{R_{k}}^{2} + {X_{k}}^{2}}

Fault levelNetwork feeder (source) impedance from fault level

Z_{Q} = \frac{c {U_{n}}^{2}}{S_{k Q}^{″}}

Fault levelTransformer impedance and IEC 60909 correction factor KT

Z_{T} = \frac{u_{k}}{100} \times \frac{{U_{rT}}^{2}}{S_{rT}}

Fault levelFault current at the end of a cable or overhead line

Z_{Q} = \frac{c {U_{n}}^{2}}{S_{k Q}^{″}}

Fault levelMVA method: source plus transformer plus local infeed

S_{T} = \frac{S_{rT}}{u_{k} / 100}

Fault levelLV-side fault through a transformer and the current seen on the HV side

Z_{Qt} = \frac{c {U_{nLV}}^{2}}{S_{k Q}^{″}}

Fault levelX/R, R and X from fault level and impedance angle

Z = \frac{{U_{n}}^{2}}{S_{k}^{″}}

Asymmetry and duration

Asymmetry and durationPeak short-circuit current ip = κ√2·I″k

κ = 1.02 + 0.98 e^{- \frac{3}{(X / R)}}

Asymmetry and durationDC component and asymmetrical current at time t

i_{dc} = \sqrt{2} I_{k}^{″} e^{- \frac{2 π f t}{(X / R)}}

Asymmetry and durationThermal equivalent short-time current Ith (IEC 60909 m and n)

m = \frac{1}{2 f T_{k} \ln (κ - 1)} [e^{4 f T_{k} \ln (κ - 1)} - 1]

Unbalanced faults

Unbalanced faultsSingle-phase-to-earth fault current I″k1

I_{k 1}^{″} = \frac{\sqrt{3} c U_{n}}{| 2 (R_{1} + j X_{1}) + (R_{0} + 3 R_{N} + 3 R_{f}) + j X_{0} |}

Unbalanced faultsPhase-to-phase fault current I″k2

I_{k 2}^{″} = \frac{c U_{n}}{| 2 (R_{1} + j X_{1}) |}

Per-unit

Per-unitPer-unit base quantities and conversion

I_{b} = \frac{S_{b}}{\sqrt{3} U_{b}}

Per-unitPer-unit impedance change of base

z_{new} = z_{old} {(\frac{U_{old}}{U_{new}})}^{2} (\frac{S_{new}}{S_{old}})

HV cables

HV cablesElectric stress in a screened single-core cable

E_{x} = \frac{U_{0}}{x \cdot \ln (\frac{R_{i}}{r_{c}})}

HV cablesOptimum conductor radius for minimum stress (Ri/rc = e)

r_{c} = \frac{R_{i}}{e}

HV cablesCapacitance of a screened cable (IEC 60287-1-1)

C = \frac{2 π ε_{0} ε_{r}}{\ln (\frac{D_{i}}{d_{c}})}

HV cablesCable charging current and three-phase charging kvar

I_{c} = 2 π f C l U_{0} \times 10^{−3}

HV cablesConductor a.c. resistance at operating temperature (IEC 60287-1-1)

R_{θ} = R_{20} [1 + α_{20} (θ - 20)]

HV cablesInductance and reactance of single-core cables, trefoil and flat

L_{t} = K + 0.2 \ln (\frac{2 s}{d})

HV cablesThree-phase voltage drop of an HV cable

Δ U = \sqrt{3} I l (R \cos φ + X \sqrt{1 - \cos^{2} φ})

Insulation

InsulationDielectric loss per unit length (IEC 60287-1-1)

W_{d} = 2 π f C U_{0}^{2} \tan δ \times 10^{−3}

InsulationInsulation resistance of a cable

R_{ins} = \frac{ρ \ln (\frac{D_{i}}{d_{c}})}{2 π l} \times 10^{−6}

InsulationThermal resistance of insulation T1 (IEC 60287-2-1)

T_{1} = \frac{ρ_{T}}{2 π} \ln (1 + \frac{2 t_{1}}{d_{c}})

Short-circuit rating

Short-circuit ratingAdiabatic short-circuit rating of a conductor (IEC 60949)

I_{ad} = K S \sqrt{\frac{\ln (\frac{θ_{f} + β}{θ_{i} + β})}{t}} \times 10^{−3}

Short-circuit ratingWire screen short-circuit rating (IEC 60949 adiabatic)

S = \frac{n π d_{w}^{2}}{4}

Sheath bonding

Sheath bondingStanding sheath voltage, single-point bonded cables

X_{s} = 2 \cdot 2 π f \times 10^{−7} \ln (\frac{2 s}{d_{m}})

Testing

TestingAfter-installation test voltages for MV extruded cables

U_{ac,5 min} = \sqrt{3} U_{0}

Overhead lines

Overhead linesLine inductance and reactance per phase (GMD/GMR)

D_{m} = \sqrt[3]{D_{ab} D_{bc} D_{ca}}

Overhead linesLine capacitance and charging current per phase

D_{m} = \sqrt[3]{D_{ab} D_{bc} D_{ca}}

Overhead linesSurge impedance and propagation velocity

Z_{c} = \sqrt{\frac{L \times 10^{- 3}}{C \times 10^{- 9}}}

Overhead linesSurge impedance loading (SIL)

SIL = \frac{V^{2}}{Z_{c}}

Overhead linesFerranti voltage rise at the open receiving end

β l = 2 π f \sqrt{L \times 10^{- 3} \times C \times 10^{- 9}} \times l

Overhead linesShort line: sending-end voltage, regulation and losses

\bar{V_{s}} = \bar{V_{r}} + \bar{I} (R + j X) l

Overhead linesMedium line nominal-π: sending-end voltage and current

Z = (R + j X) l

Overhead linesConductor sag: parabola and catenary

w = m \times 9.80665

Transients and insulation

Transients and insulationTravelling wave reflection and transmission at a junction

ρ = \frac{Z_{2} - Z_{1}}{Z_{2} + Z_{1}}

Transients and insulationCorona: disruptive critical and visual inception voltage (Peek)

δ = \frac{3.92 \times 0.750062 \times p}{273 + t}

Transients and insulationUniform-field air breakdown (Paschen, Kuffel, 3 kV/mm)

p_{20} = \frac{p \times 293}{273 + t}

Transients and insulationField between coaxial cylinders and utilisation factor

E_{max} = \frac{V}{r \ln (\frac{R}{r})}

Transients and insulationField between concentric spheres and utilisation factor

E_{max} = \frac{V R}{r (R - r)}

Transients and insulationLightning-induced overvoltage on a line (Rusck, simplified)

V_{max} = \frac{Z_{0} I_{0} h}{y} (1 + \frac{β}{\sqrt{2 - β^{2}}})

Transients and insulationAltitude correction factor for external insulation (IEC 60071-2)

K_{a} = e^{m H / 8150}

Power quality and networks

Power quality and networksThree-phase power and line current

S = \frac{P}{\cos φ}

Power quality and networksCapacitor bank rating for power-factor correction

Q_{c} = P (\tan φ_{1} - \tan φ_{2})

Power quality and networksCapacitor bank voltage rise and parallel resonance

Δ V = \frac{Q_{c}}{S_{sc}} \times 100

Power quality and networksMotor starting voltage dip

S_{m} = k S_{r}

Power quality and networksEmbedded generation voltage rise

Δ V = \frac{P R - Q X}{U}

Power quality and networksRate of change of frequency after a loss of infeed

RoCoF = \frac{Δ P \cdot f_{0}}{2 H S}

Transformers

TransformersTransformer full-load currents and voltage ratio

I_{1} = \frac{S}{\sqrt{3} U_{1}}

TransformersTransformer voltage regulation at a given load

ε_{r} = n \frac{P_{k}}{S} \times 100

TransformersTransformer losses and efficiency at a load

P_{out} = n S \cos φ

TransformersLoad sharing between two transformers in parallel

Y_{1} = \frac{S_{1}}{z_{1}}; Y_{2} = \frac{S_{2}}{z_{2}}

Protection

ProtectionIDMT relay operating time, IEC 60255-151 curves

M = \frac{I_{f}}{I_{s}}

ProtectionInverse-time relay operating time, IEEE C37.112 curves

M = \frac{I_{f}}{I_{s}}

ProtectionGrading margin check between two inverse-time relays

t_{1} = f (c_{1}, \frac{I_{f}}{I_{s1}}, T_{1})

ProtectionRequired grading margin from relay and CT errors (NPAG)

t_{m} = \frac{2 E_{r} + E_{ct}}{100} t + t_{cb} + t_{o} + t_{s}

ProtectionCT knee-point voltage requirement

I_{sf} = \frac{I_{f} I_{sn}}{I_{pn}}

ProtectionCT effective accuracy-limit factor with actual burden

R_{n} = \frac{S_{n}}{I_{sn}^{2}}

ProtectionNeutral earthing resistor value and rating

R = \frac{U}{\sqrt{3} I}

Substation earthing

Substation earthingEarth potential rise from earth fault current (split / reduction factor)

I_{E} = r I_{F}

Substation earthingHOT / COLD and high EPR site classification (ENA TS 41-24, EREC S34)

HOT if EPR > U_{ITU}, U_{ITU} = {\begin{matrix} 650 V & t \leq 0.2 s \\ 430 V & otherwise \end{matrix}

Substation earthingSubstation grid resistance (Sverak, IEEE 80)

R_{g} = ρ (\frac{1}{L_{T}} + \frac{1}{\sqrt{20 A}} (1 + \frac{1}{1 + h \sqrt{\frac{20}{A}}}))

Substation earthingTransferred potential to a remote point (hemispherical approximation)

I_{E} = \frac{EPR}{R_{E}}

Electrode resistance

Electrode resistanceSingle vertical rod resistance (Dwight)

R = \frac{ρ}{2 π L} (\ln (\frac{8 L}{d / 1000}) - 1)

Electrode resistanceRods in parallel (BS 7430 λ factor)

α = \frac{ρ}{2 π R_{1} s}

Electrode resistanceHorizontal buried conductor or strip (Dwight / Sunde)

a = \frac{d}{2000}

Electrode resistancePlate electrode resistance

r = \sqrt{\frac{A}{π}}

Touch and step

Touch and stepTolerable touch and step voltages (IEEE 80, 50 kg or 70 kg body)

C_{s} = 1 - \frac{0.09 (1 - \frac{ρ}{ρ_{s}})}{2 h_{s} + 0.09}

Touch and stepPermissible touch voltage UTp vs fault duration (BS EN 50522)

U_{Tp} = f (t_{f}) (BS EN 50522 Table B.3, log–log interpolation)

Touch and stepBody current from touch voltage (1000 Ω body)

I_{B} = \frac{U_{T}}{Z_{B} + R_{a}}

Soil resistivity

Soil resistivityApparent soil resistivity from Wenner readings (two spacings)

ρ_{1} = \frac{4 π a_{1} R_{1}}{1 + \frac{2 a_{1}}{\sqrt{a_{1}^{2} + 4 b^{2}}} - \frac{a_{1}}{\sqrt{a_{1}^{2} + b^{2}}}}

Arc flash: IEEE 1584

Arc flash: IEEE 1584IEEE 1584-2018 arcing current, incident energy and arc flash boundary (600 V–15 kV, end to end)

I_{arc, V} = 10^{k_{1} + k_{2} \lg I_{bf} + k_{3} \lg G} (k_{4} {I_{bf}}^{6} + k_{5} {I_{bf}}^{5} + \dots + k_{9} I_{bf} + k_{10})

Arc flash: IEEE 1584IEEE 1584-2018 arcing current and reduced arcing current (for protection timing)

I_{arc, V} = 10^{k_{1} + k_{2} \lg I_{bf} + k_{3} \lg G} (k_{4} {I_{bf}}^{6} + k_{5} {I_{bf}}^{5} + k_{6} {I_{bf}}^{4} + k_{7} {I_{bf}}^{3} + k_{8} {I_{bf}}^{2} + k_{9} I_{bf} + k_{10})

Arc flash: IEEE 1584IEEE 1584-2018 enclosure size correction factor CF

W_{1}, H_{1} = \frac{660.4 + (d - 660.4) \frac{V_{oc} + A}{B}}{25.4}

Arc flash: other methods

Arc flash: other methodsIEEE 1584-2002 empirical model (superseded, for comparison with older studies)

\lg I_{a} = K + 0.662 \lg I_{bf} + 0.0966 V + 0.000526 G + 0.5588 V (\lg I_{bf}) - 0.00304 G (\lg I_{bf}) (V < 1 kV)

Arc flash: other methodsRalph Lee theoretical method (above 15 kV, or outside the empirical models)

E = \frac{2.142 \times 10^{6} V I_{bf} t}{D^{2}}

Arc flash: other methodsDoughty–Neal incident energy, 600 V and below (open air and 508 mm box)

E_{MA} = 5271 {D_{A}}^{- 1.9593} t_{A} (0.0016 F^{2} - 0.0076 F + 0.8938) (open air)

Arc flash: other methodsDC arc flash: maximum power method (Ammerman et al. / NFPA 70E Annex D)

I_{arc} = 0.5 I_{bf}

Arc flash: other methodsDC arc flash: Stokes & Oppenlander arc resistance (iterative arcing current)

R_{arc} = \frac{20 + 0.534 G}{{I_{arc}}^{0.88}} \times 10^{3}

Arc flash: risk and PPE

Arc flash: risk and PPEArc blast pressure at a distance (Lee 1987)

P = \frac{11.58 I_{arc}}{{D_{ft}}^{0.9}} (lb/ft²)

Arc flash: risk and PPEArc flash boundary from a known incident energy and distance exponent

D_{B} = D {(\frac{E}{E_{B}})}^{1 / x}

Arc flash: risk and PPEIncident energy at a different working distance or arc duration

E_{2} = E_{1} (\frac{t_{2}}{t_{1}}) {(\frac{D_{1}}{D_{2}})}^{x}

Arc flash: risk and PPEIncident energy reduction from faster clearing or greater distance (maintenance switch, remote operation)

E_{2} = E_{1} (\frac{t_{2}}{t_{1}}) {(\frac{D_{1}}{D_{2}})}^{x}

Arc flash: risk and PPEStoll curve: second-degree burn threshold for an exposure time

E_{b} = 1.2 t^{0.3}

Arc flash: risk and PPEArc-rated PPE check: incident energy against garment arc rating and NFPA 70E categories

margin = \frac{ATPV}{E}

Arc flash: risk and PPEIEC 61482-1-2 box test class check (APC 1 / APC 2) by the DGUV 203-077 energy method

W_{arc} = k_{P} \sqrt{3} U_{n} I_{k} t_{k}

High-voltage calculators.Fault levels, cables, lines, earthing, arc flash and protection.

Fault level

Asymmetry and duration

Unbalanced faults

Per-unit

HV cables

Insulation

Short-circuit rating

Sheath bonding

Testing

Overhead lines

Transients and insulation

Power quality and networks

Transformers

Protection

Substation earthing

Electrode resistance

Touch and step

Soil resistivity

Arc flash: IEEE 1584

Arc flash: other methods

Arc flash: risk and PPE

High-voltage calculators

Apps

Product

For Organisations

Campaigns

Support

About us

Social