# Equations

 Equivalent capacitance of several [n] capacitors in parallel: Ce = ∑Ci where i = 1, …, n Equivalent capacitance of several [n] capacitors in series: Ce = 1 / ∑[1/Ci] where i = 1, …, n Capacitive Reactance: Xce = 1*106 / [2πfCe] Capacitive Susceptance: Bce = 2πfCe / 1*106 Capacitive Admittance: Yce = Res / [Res 2 + Xce2] + jXce / [Res 2 + Xce2] Capacitive Impedance: Zce = Res – jXce Capacitive Reactive Power: Qce = 2*10-9πf CeVnp2 Capacitance: Ce = 1*109Qce / [2πfVnp2] Equivalent inductance of several [n] inductors in parallel: Le = 1 / ∑[1/Li] where i = 1, …, n Equivalent inductance of several [n] inductors in series: Le = ∑Li where i = 1, …, n Inductive Reactance: Xle = 2*10-3πfLe Inductive Susceptance: Ble = 1 / [2*10-3πfLe] Inductive Admittance: Yle = Res / [Res 2 + Xle2] - jXle / [Res 2 + Xle2] Inductive Impedance: Zle = Res + jXle Inductive Reactive Power: Qle = V2 / [2*10-3πf Le] Equivalent resistance of several [n] resistors in parallel: Re       = 1 / ∑[1/Ri] where i = 1, …, n Equivalent resistance of several [n] resistors in series: Re       = ∑Ri where i = 1, …, n Root-Sum-of-Squares Current: IRSS    = [∑I(h)2]1/2 where h = 1, …, n Root-Sum-of-Squares Voltage: VRSS  = [∑V(h)2]1/2 where h = 1, …, n Crest Factor: CF = Vpeak / Vrms Active Power: P1P = IVcosΘ single phase systems P3P = (3)1/2IVcosΘ three phase systems Apparent Power: S1P = I V single phase systems S3P = (3)1/2 I V three phase systems Reactive Power: P1P = IVsinΘ single phase systems P3P = (3)1/2 IVsinΘ three phase systems Power Triangle Equation: S = [P2 + Q2]1/2 Displacement Power Factor: dPf = P (1) / S (1) dPf = cosΘ True Power Factor: Pf = ∑P (h) / ∑S (h) where h = 1, …, n Capacitive Reactive Power Required to Correct a Poor Displacement Power Factor: QC = P(tan-1(Θex) - tan-1(Θc)) System Capacity at the Corrected Power Factor: SC = [1 - Pfex / Pfc]Sex Reduction in Losses at the Corrected Power Factor: Plr = 1 - (Pfex / Pfc)2 Transformer Base Impedance: Zbase = Vnp2 / Snp where Snp is the OA capacity of the transformer Transformer Impedance, %Z Known: Zxfmr = 0.01Zbase%Z Transformer Equivalent Resistance, Efficiency Known: Rxfmr = (1 - Efficiencyxfmr)Zbase Transformer Leakage Reactance: Xxfmr = (Zxfmr2 - Rxfmr2)1/2 Total Harmonic Current Distortion: THDI = (∑I (h)) 1/2 / I (1)1/2 where h = 2, …, n Total Demand Distortion: TDD = (∑I (h)) 1/2 / Id (1)1/2 where h = 1, …, n Total Harmonic Voltage Distortion: THDV = (∑V (h)) 1/2 / V (1)1/2 where h = 2, …, n Resonant Frequency: fr = 1 / [2π(LeCe)1/2] where Ce ≈ 1*109Qce / [2πfVnp2] Le ≈ 1*106Xxfmr / [2πf]
Inrush Current and Frequency for Switching Capacitors Banks
(Reference ANSI Standard C37.0731-1973)
Isolated Power Capacitor Bank:
Imax peak = 1.41[IscI1]1/2, f = f (1)[Isc / I1]1/2
Two Power Capacitor Banks on the Same Bus:
Imax peak = 1747[Vll[I1I2] / (Le[I1 +I2])]1/2, f = [f(1)Vll[I1+I2] / (Le[I1I2])]1/2
 Where: I1, I2 are the currents of the power capacitor banks being switched and bank already energized, respectively. The power capacitor bank being switched is assumed uncharged with closing taking place at the crest of the voltage source. The current used should include the effect of operating the power capacitor bank at a voltage above its nominal rating and the effect of a positive capacitance tolerance. In the absence of specific information, a multiplier of 1.35 times the nominal power capacitor bank current would give conservative results.

## Symbols

C capacitance, Microfards
X reactance, Ohms
π 3.14159
f frequency
B susceptance, mho
R resistance, Ohms
Z impedance, Ohms
J complex operator
Q reactive power, KVAR
L inductance
I current, Amperes RMS
h harmonic number
V voltage, Volts RMS
S apparent power, KVA
P active power, KW
dPf displacement power factor
Pf true power factor

## Subscripts

c corrected
d demand, based on a fifteen (15) or thirty (30) minute demand period
e equivalent
ex existing
es equivalent series
i summation index
ll line-to-line
lr loss reduction
np name plate
xfmr transformer
sc symmetrical short-circuit

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