Power Quality & Harmonics

Capacitor Banks, Power Factor Correction, and Harmonic Resonance

Capacitor banks supply reactive power, improve power factor and support voltage — but they also add capacitance that can resonate with the system inductance at a harmonic frequency. This final part links power-factor correction to harmonic resonance, gives quick screening formulas for the resonant order, and shows how transformer magnetising current and winding connection shape harmonic behaviour.

Reading time ≈ 15 min · Part Five of the series

Capacitor banks are widely used to supply reactive power, improve power factor, support voltage and reduce current loading. But they are also important in harmonic studies because they introduce capacitance into the network, which can interact with the system inductance and create resonance at certain harmonic frequencies. A capacitor bank should therefore not be treated only as power-factor-correction equipment — it must also be assessed as part of the harmonic behaviour of the network.

Key idea

A capacitor bank improves power factor at the fundamental, but its capacitance can resonate with system inductance at a harmonic.
The resonant order can be screened quickly from \(\sqrt{\mathrm{SCC}/Q_C}\) or from the capacitor-switching voltage rise.
Larger capacitors lower the resonant order; stronger systems raise it.
Transformer magnetising current is nonlinear, and winding connection governs how triplen harmonics circulate.

Section 1

Capacitor banks and power factor

When a capacitor bank is connected to a bus, it supplies capacitive reactive power locally, reducing the reactive power that the upstream system must supply. As a result, the current drawn from the supply can fall, the bus voltage can rise, and the apparent power demand from the system can decrease. For an inductive load the current lags the voltage; a shunt capacitor injects a leading reactive current that compensates part of the load's lagging current. The main benefits are:

Table 1 — Main benefits of capacitor banks.

Benefit	Explanation
Power factor improvement	The reactive power drawn from the upstream system is reduced
Voltage support	Capacitor current through system reactance causes a voltage rise at the bus
Reduced current	For the same active power, improved power factor reduces supply current
Reduced losses	Lower current reduces \(I^2R\) losses in cables, transformers and lines
Better voltage regulation	Reduced voltage drop improves the voltage profile

However, the same capacitor bank that improves power factor at the fundamental frequency can create harmonic resonance at higher frequencies. This is the main engineering concern of this page.

Section 2

Capacitor switching and bus voltage rise

A power system can often be represented by a Thévenin equivalent source behind an equivalent system impedance, which is usually inductive because transformers, lines, cables and the upstream grid all contribute reactance. With the capacitor bank off, the bus voltage is set by the supply voltage and the drop across the system impedance. When the bank is switched on, capacitor current flows through that (mainly inductive) impedance and raises the bus voltage. The rise is normally small, but it carries useful information about system strength and the resonance condition.

The capacitor current and its fundamental reactance are:

\[ I_C=\frac{V}{X_C} \qquad X_C=\frac{1}{\omega_0 C} \]

\(X_C\): capacitive reactance at the fundamental frequency
\(\omega_0\): fundamental angular frequency

If the upstream system reactance is \(X_S\), the approximate per-unit voltage rise from switching the capacitor is the ratio of system reactance to capacitor reactance:

\[ \Delta V_{bus,pu}\approx \frac{X_S}{X_C} \]

\(\Delta V_{bus,pu}\): per-unit bus voltage rise from switching the capacitor
\(X_S\): upstream system reactance
\(X_C\): capacitor reactance at the fundamental frequency

This relationship is useful because it links the measurable voltage rise directly to the resonance condition, as the next sections show.

Section 3

Resonance caused by capacitor banks

A capacitor bank connected to a power system forms an equivalent LC circuit with the system inductance — the inductance coming from the upstream grid, transformer leakage reactance, lines, cables and reactors. The resonant harmonic order can be estimated from:

\[ h_r=\sqrt{\frac{X_C}{X_S}} \]

\(X_C\): capacitor reactance at the fundamental frequency
\(X_S\): equivalent system reactance at the bus

Resonance therefore depends on both the capacitor size and the strength of the upstream system. A larger capacitor bank has a lower capacitive reactance, so increasing the bank size normally shifts resonance to a lower harmonic order. A stronger upstream system has a lower reactance, so increasing the short-circuit capacity shifts resonance to a higher order:

Table 2 — How capacitor size and system strength move the resonant order.

Change	Effect on Resonance
Larger capacitor bank	Resonance moves to a lower harmonic order
Smaller capacitor bank	Resonance moves to a higher harmonic order
Stronger system / higher short-circuit level	Resonance moves to a higher harmonic order
Weaker system / lower short-circuit level	Resonance moves to a lower harmonic order

This matters because harmonic sources do not produce all orders equally: six-pulse drives and rectifiers usually produce strong 5th, 7th, 11th and 13th harmonics. If the capacitor bank shifts the system resonance close to one of these orders, the distortion can increase significantly.

Section 4

Short-circuit capacity and resonance

The resonant harmonic order can also be expressed using short-circuit capacity and capacitor rating, which are usually available in real projects:

\[ h_r=\sqrt{\frac{\mathrm{SCC}}{Q_C}} \]

\(\mathrm{SCC}\): short-circuit capacity at the bus
\(Q_C\): capacitor bank rating

Both quantities must be on the same MVA base or in compatible units. For example, if \(\mathrm{SCC}=25\,Q_C\) then \(h_r=\sqrt{25}=5\) — the system resonates close to the 5th harmonic, a critical condition if the installation contains 5th-harmonic sources such as six-pulse rectifiers or variable-speed drives. Similarly, if \(\mathrm{SCC}=49\,Q_C\) then \(h_r=\sqrt{49}=7\), a resonance near the 7th harmonic. This simple equation is a quick screening method — it does not replace detailed harmonic analysis, but it flags whether a capacitor bank may create a resonance problem.

Section 5

Bus voltage rise and resonance order

The bus voltage rise from capacitor switching can also be used to estimate the resonant harmonic order:

\[ h_r=\frac{1}{\sqrt{\Delta V_{bus,pu}}} \]

\(\Delta V_{bus,pu}\): per-unit voltage rise caused by switching the capacitor bank

This is practical because voltage rise is often measurable or can be estimated. For example, a rise of \(\Delta V_{bus,pu}=0.04\) gives \(h_r=\tfrac{1}{\sqrt{0.04}}=5\) (resonance near the 5th harmonic), while \(\Delta V_{bus,pu}=0.02\) gives \(h_r=\tfrac{1}{\sqrt{0.02}}=7.07\) (near the 7th). A larger voltage rise corresponds to a lower resonant order — often more concerning, because low orders such as the 5th and 7th are commonly produced by nonlinear loads.

Section 6

Sizing power-factor correction

A low power factor means that, for a given active power, the system current is higher — increasing losses and equipment ratings. For a three-phase system:

\[ I=\frac{P}{\sqrt{3}\,V\cos\phi} \]

\(P\): active power
\(V\): line-to-line voltage
\(\cos\phi\): power factor

so for the same active power and voltage, a lower power factor gives higher current. The disadvantages of low power factor are:

Table 3 — Disadvantages of low power factor.

Disadvantage	Explanation
Higher current	More current is required to deliver the same active power
Higher losses	Line and transformer losses increase with \(I^2R\)
Larger equipment ratings	Transformers, cables and switchgear may need higher ratings
Poor voltage regulation	Higher reactive current causes larger voltage drops
More reactive demand from the grid	The upstream system must supply more reactive power

A capacitor bank improves the power factor by supplying part of the reactive power locally. To move from an original angle \(\phi_1\) to a target angle \(\phi_2\) at active power \(P\), the capacitor reactive power required is:

\[ Q_C=P\left(\tan\phi_1-\tan\phi_2\right) \]

\(Q_C\): capacitor reactive power required
\(P\): active power
\(\phi_1,\phi_2\): original and target power-factor angles

Since \(\phi_1=\cos^{-1}(pf_1)\) and \(\phi_2=\cos^{-1}(pf_2)\), this can be written directly from the original and target power factors:

\[ Q_C=P\left[\tan\!\left(\cos^{-1}(pf_1)\right)-\tan\!\left(\cos^{-1}(pf_2)\right)\right] \]

\(pf_1\): original power factor
\(pf_2\): corrected (target) power factor

This formula is widely used for sizing power-factor-correction capacitors.

Section 7

Why power-factor correction can create harmonic problems

Power-factor correction is beneficial at the fundamental frequency but can create problems at harmonic frequencies. At the fundamental, the capacitor supplies useful reactive power; at harmonic frequencies, its reactance falls as \(X_C(h)=\dfrac{X_C}{h}\), so the bank presents a lower-impedance path to higher-frequency harmonic currents. At the same time the system inductive reactance rises as \(X_S(h)=hX_S\). At a certain order these become equal:

\[ hX_S=\frac{X_C}{h} \]

\(h\): harmonic order at which resonance occurs
\(X_S\): system reactance at the fundamental frequency
\(X_C\): capacitor reactance at the fundamental frequency

which is the resonance condition. After installing a capacitor bank the harmonic impedance of the system can therefore change significantly: a network with acceptable distortion before compensation may become problematic afterwards. This is why capacitor banks and harmonic studies are strongly linked. The practical risks include:

Table 4 — Practical harmonic risks of capacitor banks.

Risk	Explanation
Capacitor overcurrent	The capacitor may carry excessive harmonic current
Voltage distortion	Parallel resonance may amplify harmonic voltages
Transformer heating	Harmonic currents increase losses
Cable heating	RMS current and high-frequency effects increase losses
Nuisance tripping	Protection may operate due to distorted current or voltage
Equipment stress	Capacitors, reactors and insulation may see higher stress
Reduced equipment life	Thermal and dielectric stress may accelerate ageing

Capacitor banks should therefore be checked against harmonic conditions, especially in systems with variable-speed drives, rectifiers, inverters, UPS systems, EV chargers or other nonlinear loads.

Section 8

Transformer harmonics

Transformers also matter in harmonic studies because their magnetising behaviour is nonlinear. If a sinusoidal voltage is applied at no load, the magnetising current is not perfectly sinusoidal, because the magnetising inductance is not constant — it changes with flux density. The no-load magnetising current is often peaked and contains harmonics, the most significant usually being the third, together with other odd harmonics.

A sinusoidal transformer voltage can produce a non-sinusoidal magnetising current.

The opposite is also true: if the magnetising current were forced to be sinusoidal, the flux and induced voltage would become distorted. This behaviour is caused by magnetic saturation and the nonlinear relationship between flux density and magnetising current.

Section 9

Effect of transformer winding connection

In three-phase transformers, harmonic behaviour depends strongly on winding connection, the key issue being the treatment of triplen harmonics — multiples of three (3rd, 6th, 9th, 12th, …). In a balanced three-phase system these are zero-sequence components, in phase in all three phases, so they need a return path. Without one, they cannot flow freely and the voltage or flux waveform may distort.

Delta-connected windings

A delta winding provides a closed path for triplen harmonic currents: the third-harmonic magnetising currents circulate inside the delta and so do not usually appear in the external line currents. This helps suppress triplen harmonics on the line side and reduces voltage distortion at the terminals.

A delta winding traps triplen harmonic currents inside the delta.

Wye-connected windings

A wye winding behaves differently. If the wye is ungrounded, there is no neutral return path for zero-sequence triplen currents, so the third-harmonic magnetising current cannot flow freely. When it is restricted, the flux and phase voltages may distort, and the star point may shift or oscillate because the triplen components affect the phase-to-neutral voltages. However, balanced triplen components are in phase in all three phases and cancel when one phase voltage is subtracted from another:

\[ V_{ab}=V_a-V_b \]

\(V_{ab}\): line-to-line voltage between phases a and b
\(V_a,V_b\): phase-to-neutral voltages of phases a and b

If the third-harmonic components in \(V_a\) and \(V_b\) are equal and in phase, they subtract to zero in the line-to-line voltage. Triplen harmonics may therefore appear in the phase-to-neutral voltage but not in the line-to-line voltage. If the wye neutral is grounded, the neutral provides a return path for triplen currents, allowing the third-harmonic magnetising current to flow and reducing phase-voltage distortion.

Section 10

Practical engineering interpretation

Capacitor banks change the frequency response of the network: they may improve power factor and voltage at the fundamental, but they may also shift resonance close to characteristic harmonic orders. Transformers affect harmonic propagation through their winding connection — delta windings provide a circulating path for triplen harmonics, while wye windings may need a neutral path, otherwise phase-voltage distortion and neutral displacement may occur. For harmonic studies, the following are normally checked:

Table 5 — Items to check in a capacitor / harmonic study.

Item	Why It Matters
Capacitor bank size	Determines reactive compensation and affects resonance order
Short-circuit capacity at the bus	Determines system strength and resonance order
Voltage rise after capacitor switching	Gives an indication of resonance risk
Nonlinear load harmonic spectrum	Identifies which harmonic orders are likely to be injected
Transformer impedance	Contributes to system inductance and resonance
Transformer winding connection	Affects triplen harmonic circulation and propagation
Neutral grounding	Determines whether zero-sequence harmonic currents can flow
Existing or planned capacitor banks	May shift resonance and increase distortion
Operating scenarios	Resonance can change when capacitor banks or feeders are switched

The key point is that power-factor correction should not be designed in isolation. A capacitor size that is correct from a reactive-power point of view may still create harmonic resonance, so capacitor banks should be checked against the harmonic spectrum of the loads and the frequency response of the network.

Key message

Capacitor banks are useful — they supply reactive power, improve power factor, reduce current and support voltage — but they interact with system inductance and can create resonance at harmonic frequencies. Larger capacitors usually move resonance to a lower order; stronger systems move it higher. The resonant order can be screened from \(h_r=\sqrt{\mathrm{SCC}/Q_C}\) or from the switching voltage rise \(h_r=1/\sqrt{\Delta V_{bus,pu}}\). Transformers add their own nonlinear magnetising harmonics, and their winding connection decides whether triplen harmonics circulate, return through the neutral, or appear as voltage distortion. Harmonic problems are rarely caused by one item alone — they result from the interaction between harmonic sources and the frequency-dependent network, so the source, capacitor banks, system strength, transformer connections and network response must be considered together.

Part Five Reading now

Capacitor Banks & Power Factor Correction

How capacitor banks and transformer windings shape harmonics: resonant-order screening, and triplen harmonics in delta and wye windings.

Series progress 5 of 5

Part Four

Resonance in Harmonic Studies

Series and parallel resonance, the resonant harmonic order, quality factor and damping, and reading resonance from a frequency scan.