Short answer
Harmonic amplification transformer removed cases arise because the parallel resonance between a capacitor bank and the supply depends on bus short-circuit power. The resonance order is approximately the square root of short-circuit power divided by capacitor reactive power. When one of two parallel transformers is taken out, short-circuit power at the bus roughly halves and the resonance moves to a lower order. In an illustrative bus fed by two 1,000 kVA, 6 % impedance transformers from a 250 MVA upstream network, with 600 kvar of plain capacitors, resonance sits near the 7th harmonic with both transformers in service and near the 5.1th with one. The 5th harmonic current from drives and rectifiers can then be amplified; the amount depends on damping and the complete network. Frequency scans must cover each transformer configuration with the capacitor bank unchanged, because the bank usually stays connected.
Resonance and short-circuit power
| Configuration | Bus S_sc | h_r |
|---|---|---|
| Both transformers | ≈ 29.4 MVA | ≈ 7.0 |
| One transformer | ≈ 15.6 MVA | ≈ 5.1 |
h_r ≈ sqrt(Ssc / Q_c)
S_sc is the short-circuit power at the bus, set by upstream network and transformer impedances in series.
Illustrative configurations
Illustrative example. Two 1,000 kVA, 6 % transformers in parallel feeding an LV bus; upstream network 250 MVA; 600 kvar plain capacitor bank on the bus. Values chosen for demonstration.
Each transformer alone: 1 / 0.06 ≈ 16.7 MVA.
both: Ssc = 1 / (1/33.3 + 1/250) ≈ 29.4 MVA; h_r ≈ sqrt(29.4 / 0.6) ≈ 7.0
one: Ssc = 1 / (1/16.7 + 1/250) ≈ 15.6 MVA; h_r ≈ sqrt(15.6 / 0.6) ≈ 5.1
With both transformers, resonance near the 7th may already be a concern; with one, it sits almost on the 5th.
Why the bank stays in
Automatic power-factor controllers keep stages in service to meet power factor, even when a transformer is out for maintenance. Unless the design forces capacitor stages out with the transformer, the resonance shift happens unnoticed until capacitors fail or voltage distortion rises; see capacitor overload from background harmonics.
Frequency scans per configuration
| Configuration | Scan |
|---|---|
| Both transformers, all stages | Base case |
| One transformer, all stages | N-1 |
| One transformer, partial stages | Controller states |
| Upstream network weak | Minimum short-circuit |
The scan should use realistic damping; see resonance scans with frequency-dependent transformer damping.
Mitigations
- Detuned banks, which avoid the simple source–bank parallel resonance above the branch tuning order; validate the complete network response; see detuning factor in transformer-fed capacitor banks.
- Interlocks that reduce capacitor stages when a transformer is out.
- Filters designed for all configurations; see single-tuned and damped filters on transformer buses.
- Monitoring of voltage distortion with alarms.
Step voltage changes also grow in the weak configuration; see capacitor-bank step size and voltage steps.
What this article does not cover
How load sharing changes between parallel transformers is not covered here.
Where this reasoning stops. The resonance formula ignores load damping and cable capacitance; scans give the real picture.
Practical takeaway
Recalculate capacitor-bank resonance for each transformer configuration, expect the order to fall when a transformer is removed, and use detuning, interlocks or filters so that no configuration resonates on a significant harmonic.
Reference framework
- IEEE 519, Standard for harmonic control in electric power systems.
- IEEE 1036, Guide for the application of shunt power capacitors.