Case study
Capacitor Bank Harmonic Resonance Overvoltage Case Study
Electrical engineering case study on diagnosing capacitor-bank harmonic resonance, with short-circuit strength, resonance order, voltage distortion, capacitor current, detuning, protection, and validation evidence.
Power-factor correction can reduce current, losses, and utility penalties, but a capacitor bank also changes the frequency response of the electrical network. If the capacitor bank resonates with upstream system inductance near a dominant harmonic, voltage distortion and capacitor current can rise instead of falling.
This case study follows an industrial 13.8 kV bus where a switched capacitor bank was installed to improve displacement power factor. After energization, variable-speed drives began reporting DC-link overvoltage alarms, capacitor fuses ran hot, and a power-quality logger recorded high fifth-harmonic voltage. The case is realistic but simplified for engineering education.
The engineering question is:
Did the capacitor bank solve a reactive-power problem, or did it create a harmonic resonance that made the electrical system less reliable?
The answer depends on evidence, not on the fact that the bank was correctly sized for fundamental-frequency kvar.
System Context
The plant has medium-voltage motors, low-voltage motor drives, welding loads, and process controls. A 5.0 MVAr capacitor bank is connected to the 13.8 kV bus. It is switched during high production load to keep power factor above the site operating target.
The simplified review data are:
| Item | Value |
|---|---|
| bus voltage | 13.8\ \text{kV} line-to-line |
| system frequency | 60\ \text{Hz} |
| estimated short-circuit strength at the bus | 125\ \text{MVA} |
| capacitor-bank rating | 5.0\ \text{MVAr} |
| dominant nonlinear loads | six-pulse drives and rectifiers |
| pre-event power factor at high load | 0.86 lagging |
| measured bus voltage with bank energized | 1.04\ \text{pu} |
| project action limit for voltage THD | 5\% |
| capacitor-current action limit | 110\% of reviewed rating |
These are project criteria for the case study. Real installations must use the applicable utility requirements, equipment ratings, harmonic limits, insulation coordination, protection settings, and site safety procedures.
Initial Symptom
The capacitor bank worked at the fundamental frequency. The site power factor improved from 0.86 lagging to approximately 0.97 lagging. The problem appeared only after the bank switched in:
- several drives logged intermittent DC-link overvoltage alarms;
- the capacitor-bank compartment temperature rose faster than expected;
- one phase fuse showed discoloration during inspection;
- a power-quality logger recorded voltage distortion near the fifth harmonic;
- the symptom was worst at high drive loading and light utility-source strength.
This pattern suggests a frequency-response problem. A normal 60 Hz load-flow calculation is not enough.
Step 1: Fundamental Capacitor Current
The fundamental line current of a balanced three-phase capacitor bank is:
With:
and:
the current at rated voltage is:
The measured fundamental voltage is 1.04\ \text{pu}, so the fundamental capacitor current is approximately:
Engineering Comment
The bank current is not suspicious at the fundamental frequency. A technician checking only 60 Hz current might conclude that the bank is healthy. The failure mechanism is harmonic current superimposed on the fundamental current.
Step 2: Parallel Resonance Screen
A common screening estimate for the parallel-resonance harmonic order at a bus with a capacitor bank is:
where S_{SC} is the short-circuit strength at the bus and Q_C is the capacitor-bank rating, using the same apparent-power units.
Substitute:
The estimated resonance is near the fifth harmonic:
Engineering Comment
This is the key diagnostic result. Six-pulse rectifiers commonly inject fifth and seventh harmonic currents. A bus resonance near the fifth harmonic is a credible mechanism for amplified fifth-harmonic voltage and excessive capacitor current.
Step 3: Measured Harmonic Evidence
The power-quality logger records the following RMS voltage components while the capacitor bank is energized:
| Component | Frequency | Voltage as percent of fundamental |
|---|---|---|
| fundamental | 60\ \text{Hz} | 100\% |
| 5th harmonic | 300\ \text{Hz} | 9.5\% |
| 7th harmonic | 420\ \text{Hz} | 2.4\% |
| 11th harmonic | 660\ \text{Hz} | 1.3\% |
Voltage THD is:
This exceeds the project action limit of 5\%.
Engineering Comment
Total THD is useful, but the individual harmonic tells the stronger story. The fifth harmonic dominates, and the resonance screen predicted a fifth-harmonic problem. The symptom, calculation, and measurement point to the same mechanism.
Step 4: Capacitor Harmonic Current
Capacitive reactance decreases as frequency increases. For the same harmonic voltage fraction, capacitor current rises roughly in proportion to harmonic order:
Fifth-harmonic capacitor current:
Seventh-harmonic capacitor current:
Eleventh-harmonic capacitor current:
Estimated total RMS capacitor current is:
Relative to the rated fundamental current:
or:
This exceeds the reviewed 110\% action limit.
Engineering Comment
The capacitor bank is not overloaded because it supplies too much 60 Hz reactive power. It is overloaded because harmonic voltage near resonance drives additional current through the capacitors. This explains hot fuses and elevated compartment temperature.
Step 5: Why the Drives Trip
The drives see a distorted bus voltage. A diode or thyristor rectifier front end charges its DC link near voltage peaks, so increased peak distortion and overvoltage can raise DC-link stress even when RMS voltage looks acceptable.
The logger also shows that the fifth-harmonic voltage rises only when both conditions are true:
- the capacitor bank is energized;
- enough nonlinear load is operating to inject harmonic current.
When the capacitor bank is off, voltage THD falls to 3.1\%. When the bank is on during high nonlinear load, voltage THD rises to 9.9\%. That operating-state dependence is strong evidence against a random drive defect.
Step 6: Mitigation Options
The engineering team reviews four options.
| Option | Effect | Limitation |
|---|---|---|
| leave the bank off | removes the resonance condition | power factor penalty and higher feeder current |
| reduce capacitor size | shifts resonance but may move it near another harmonic | does not control the harmonic source |
| add a detuned reactor | moves the bank’s series tuning below the fifth harmonic | requires component rating, protection and thermal review |
| install active or passive filtering | reduces harmonic current or provides controlled impedance | higher cost and commissioning complexity |
The selected correction is a detuned capacitor bank with a reactor chosen so that the bank is not an attractive low-impedance path at the fifth harmonic.
Step 7: Detuned Reactor Check
For the existing bank, approximate per-phase capacitance for a wye-equivalent three-phase capacitor bank is:
where:
Substitute:
The fundamental capacitive reactance is:
A common detuning target is below the fifth harmonic. In this case, the design review uses a tuning order of:
The reactor reactance fraction at the fundamental is:
So the required fundamental reactor reactance is:
The corresponding inductance is:
Engineering Comment
This calculation is a screening calculation, not a procurement specification. A real detuned bank must be designed for capacitor voltage rise, reactor thermal loading, harmonic current spectrum, insulation level, tolerances, protection, ventilation, discharge, switching duty, and manufacturer data.
Step 8: Post-Correction Validation
After the detuned bank is commissioned, the team repeats the same operating case: high nonlinear load, capacitor bank energized, normal utility source, and identical logger configuration.
| Quantity | Before correction | After correction | Review criterion |
|---|---|---|---|
| power factor | 0.97 lagging | 0.96 lagging | at least 0.95 |
| voltage THD | 9.9\% | 3.8\% | at most 5\% |
| fifth-harmonic voltage | 9.5\% | 2.6\% | no dominant resonance |
| capacitor RMS current | 245\ \text{A} | 224\ \text{A} | at most 110\% reviewed rating |
| drive DC-link alarms | repeated | none during test window | no recurrence |
| compartment temperature rise | abnormal trend | stable trend | maintenance acceptable |
The corrected current ratio is:
or:
This satisfies the reviewed current action limit.
Release Decision
The capacitor bank can be released only for the validated operating envelope:
- same bus configuration and source strength;
- same capacitor-bank step and detuned reactor status;
- nonlinear load below the reviewed operating case;
- power-quality logging enabled during the trial period;
- alarm thresholds and trip records reviewed after commissioning;
- maintenance inspection scheduled for fuses, reactors, ventilation and connections.
The release is conditional because resonance depends on system impedance. A transformer change, utility fault-level change, generator mode, added drives, added filters, or capacitor-step change can move the resonance and require a new study.
Engineering Lessons
This case shows why power-factor correction is not just a kvar calculation. At 60 Hz, the capacitor bank did what it was supposed to do. At the fifth harmonic, it created a frequency-response problem.
Good engineering practice connects:
- fundamental reactive-power need;
- short-circuit strength and system impedance;
- harmonic sources and spectrum;
- resonance order and measured voltage distortion;
- capacitor RMS current and thermal evidence;
- mitigation choice and component ratings;
- validation in the same operating state that produced the symptom.
The transferable lesson is simple: never approve a capacitor bank in a nonlinear-load environment from power factor alone. Verify the frequency response, measure the harmonic spectrum, and validate the installed correction under the operating case that matters.