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Formulas of Power Engineering

Cross-section
for direct - and single phase alternating current
(current given)
q = 1 x I x l (mm²)
κ x u
     
for three-phase current
q = 1.732 x I x cosφ x l (mm²)
κ x u

for direct - and single phase alternating current
(capacity given)
q = 2 x I x P (mm²)
κ x u x U
     
for three-phase current
q =     l x P     (mm²)
κ x u x U

Voltage drop
For low voltage cable network of normal operation, a voltage drop of 3-5% is adviseable.
Exeption: higher values (7%) can be permitted in case of network extension or in short circuit.

for direct current (current given)
u = 2 x I x l (V)
κ x q

single phase alternating current (current given)
u = 2 x I x cosφ x l (V)
κ x q

for three-phase current (current given)
u = 1.732 x I x cosφ x l (V)
κ x q


for direct current (capacity given)
u =   2 x l x P   (V)
κ x q x U

single phase alternating current (capacity given)
u =   2 x l x P   (V)
κ x q x U

for three-phase current (capacity given)
u = l x P   (V)
κ x q x U

u = Voltage drop (V)
U = Operating voltage (V)
P = Power (W)
RW = Effective resistance (Ω/km)
L = Inductance (mH/km)
ωL = Inductive resistance (Ω/km)
    ω = 2 π f at 50 Hz = 314
q = Cross-section (mm²)
I = Working current (A)
l = Length of line (m)
κ = Electrical conductivity of conductors
    (m/Ω x mm²)
    κ Copper: 58
    κ Aluminum: 33
Nominal voltage
The nominal voltage is to be expressed with two values of alternating current U0/U (in V).
U0 = voltage between conducter and ground or metallic covering (shield, armoring, concentric conductor)
U = voltage between two outer conductors
U0 = U/√3 for three-phase current systems
U0 = U/2 for single phase and direct current systems
U0/U0 = 1 outer conductor is grounded for single phase and direct current systems
Nominal current
I in A

Active current
IW = I x cosφ

Reactive current
I0 = I x sin φ

Active power
S = U x I                             for single phase current
S = 1.732 x U x I                 for three-phase current

Apparent power
P = U x I x cosφ                   for single phase current
P = 1.732 x U x I x cos φ      for three-phase current
P = U x I                             for direct current

Reactive power
Q = U x I x sin φ                 for single phase current
Q = 1.732 x U x I x sin φ     for three-phase current
(Voltampere reactive)           Q = P x tan φ

Phase angle
φ is a phase angle between voltage and current
cos φ = 1.0 0.9 0.8 0.7 0.6 0.5  
sin φ = 0 0.44 0.6 0.71 0.8 0.87  
Insulation resistance
RInsu = SInsu x In Da x 10-8 (MΩ x km)
l d


Specific insulation resistance
RS = R x 2π x l x 108
In Da
di
Da = Outer diameter over insulation
d = Conductor diameter
di = Inner diameter of insulation
l = Length of the line
SInsu = Spec. resistance of insulation materials (Ω x cm)
Mutual capacity
for single conductor, three-cond. and H-cable
CB =     ξr x 10³     (nF/km)
18 In Da
d

Inductance
single-phase
0.4 x ( In Da  + 0.25)    mH/km
r

three-phase
0.2 x ( In Da  + 0.25)    mH/km
r
Da = Distance mid to mid of both conductors
r = Conductor radius
ξr = Dielectric constant
0.25 = Factor for low frequency
Ground capacitance
EC = 0.6 x CB

Charging current
(only for three-phase)
ILad = U x 2 π f x CB x 10-6 (A/km per conductor at 50Hz)

Charging power
PLad = ILad x U

Leakage and loss factor
G = tanδ x ωC(S)
tanδ = G/ωC

ω = 2 π f
C = Capacity
tanδ = Loss factor
S = Siemens = 1/1Ω
Dielectric loss
DV = U² U x 2 π f x CB x tan x 10-6 (W/km)

It should be noted that for the current load of the insulated cables and wires of selected cross-section, the power ratings are to be considered.
To estimate the voltage drop of insulated cables and wires with large cross-sections of single and three-phase overhead line, the active resistance as well as indictive resistance must be considered.

Formula for single phase:
U = 2 x l x I x (RW x cosφ + ωLx sinφ) x 10-3 (V)

Formula for three-phase
U = 1.732 x l x I x (RW x cosφ + ωLx sinφ) x 10-3 (V)
f at 50Hz
tanδ PE/VPE cables ~ 0.0005
EPR ~ 0.005
Paper single conductor, three-conductor, H-cable ~ 0.003
Oil-filled and pressure cable ~ 0.003
PVC-cable ~ 0.05
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