Short answer
A transformer capacitor bank detuning factor, usually written p, is the ratio of the series reactor’s reactance to the capacitor’s reactance, both at fundamental frequency. It sets the series resonance of the reactor-capacitor branch at harmonic order 1/sqrt(p). In an illustrative case, 7 % detuning gives a tuning order of 3.78, about 189 Hz on a 50 Hz system, and 14 % gives 2.67, about 134 Hz. Both reactances must be referred to the same fundamental frequency and the same capacitor rating; mixing 50 and 60 Hz data or nominal and actual capacitance shifts the result. The reactor also raises the capacitor’s fundamental voltage above bus voltage by a factor of 1/(1 − p), about 7.5 % for p = 7 %. Capacitor voltage ratings must allow for that rise plus harmonic voltage.
Definition
| p | h_r | f_r | U_C / U_bus |
|---|---|---|---|
| 5.67 % | 4.2 | 210 Hz | 1.060 |
| 7 % | 3.78 | 189 Hz | 1.075 |
| 14 % | 2.67 | 134 Hz | 1.163 |
p = X_L / X_C (at f_1)
h_r = 1 / sqrt(p)
f_r = f_1 / sqrt(p)
Below f_r the branch is capacitive and supplies reactive power; above it the branch is inductive. In the simple inductive-source/single-bank model, that removes the source–bank parallel resonance above the branch tuning frequency; other capacitances and network branches still require a full scan.
Common values
Illustrative example. 50 Hz system; detuned banks at p = 5.67 %, 7 % and 14 %. Values chosen for demonstration.
| p | h_r | f_r at 50 Hz | Typical intent |
|---|---|---|---|
| 5.67 % | 4.2 | 210 Hz | Detune below 5th |
| 7 % | 3.78 | 189 Hz | Detune below 5th, more margin |
| 14 % | 2.67 | 134 Hz | Detune below 3rd |
The choice depends on which harmonics are present. A 7 % bank still exposes the system to 3rd harmonic resonance if large 3rd harmonic currents exist and the bank is connected so that they can flow.
Same fundamental base
- Frequency: with the same ideal L and C, changing from 60 Hz to 50 Hz makes X_L 5/6 as large and X_C 6/5 as large; p therefore becomes (50/60)² of its 60 Hz value.
- Capacitance: the capacitor’s actual capacitance, within tolerance, sets X_C.
- Voltage: p is a ratio and is independent of voltage, but the reactor must be designed for the capacitor’s rated reactive power at the capacitor’s voltage.
Tolerance and drift effects are discussed in tolerance drift in tuned harmonic filters.
Capacitor voltage rise
U_C = Ubus / (1 − p)
For p = 7 %, U_C ≈ 1.075 U_bus. On a 400 V bus the capacitor sees about 430 V at fundamental, plus harmonic voltage. A 400 V-rated capacitor would operate above its rated fundamental voltage in this example. Select a suitable higher rating from the actual continuous voltage, harmonic duty and manufacturer limits; 440 V is not an unconditional selection rule. Harmonic contributions are covered in capacitor terminal voltage with series detuning reactors.
Reactive output
The branch’s reactive output at bus voltage is:
Q = Ubus^2 / (X_C − X_L) = QC,bus / (1 − p)
where Q_C,bus is what the capacitor alone would give at bus voltage. Acceptance testing must use this relationship; see capacitor bank acceptance: measured versus nameplate kvar.
What this article does not cover
The input list for capacitor resonance studies is not repeated here. System resonance with transformers is in harmonic amplification after one transformer is removed.
Where this reasoning stops. Ideal components are assumed; real reactors have losses and saturation.
Practical takeaway
Define p from fundamental reactances on one base, choose it to place the tuning order below the lowest significant harmonic, and rate capacitors for the 1/(1 − p) voltage rise plus harmonics.
Reference framework
- IEC 60831-1, Shunt power capacitors of the self-healing type for a.c. systems up to 1 kV.
- IEC 61642, Industrial a.c. networks affected by harmonics: application of filters and shunt capacitors.