Resonance is one of the most important subjects in harmonic studies because it can significantly increase harmonic voltages or currents in a power system. Even if the harmonic source is relatively small, resonance can amplify its effect and create unacceptable distortion, overheating or equipment stress.
Key idea
- Resonance occurs where inductive and capacitive reactances are equal; in harmonic studies it is expressed as a harmonic order.
- Series resonance → low impedance → high harmonic current.
- Parallel resonance → high impedance → high harmonic voltage.
- It is a system-level issue — capacitor banks, network strength and switching state all move the resonant point.
Section 1
Resonance fundamentals
In a power system, inductance and capacitance are always present. Transformers, lines, cables, reactors and machines contribute inductance; capacitor banks, cable capacitance, filters and some power-electronic equipment contribute capacitance. When the inductive and capacitive reactances become equal at a particular frequency, resonance can occur:
\[ X_L=\omega L \qquad X_C=\frac{1}{\omega C} \]
- \(X_L\)
- inductive reactance
- \(X_C\)
- capacitive reactance
- \(\omega\)
- angular frequency
- \(L,C\)
- inductance and capacitance
At resonance \(X_L=X_C\), so \(\omega_r L=\dfrac{1}{\omega_r C}\), which gives the angular resonant frequency and its value in hertz:
\[ \omega_r=\frac{1}{\sqrt{LC}} \qquad f_r=\frac{1}{2\pi\sqrt{LC}} \]
- \(\omega_r\)
- angular resonant frequency
- \(f_r\)
- resonant frequency (Hz)
- \(L,C\)
- circuit inductance and capacitance
In harmonic studies, resonance is usually expressed in terms of harmonic order rather than frequency:
\[ h_r=\frac{f_r}{f_0} \]
- \(f_0\)
- fundamental frequency of the system
So if a 50 Hz system has a resonant frequency of 250 Hz, the resonant harmonic order is \(h_r=\tfrac{250}{50}=5\) — the system is resonant near the 5th harmonic. This matters because many nonlinear loads inject harmonic currents at specific orders: six-pulse converters commonly produce 5th, 7th, 11th and 13th harmonics. If the network resonance falls close to one of these orders, the distortion can become much more severe.
There are two main types of resonance, with opposite effects:
Table 1 — The two types of resonance.
| Resonance Type | Main Effect |
|---|
| Series resonance | Low impedance and high current |
| Parallel resonance | High impedance and high voltage |
Section 2
Series resonance
Series resonance occurs when resistance, inductance and capacitance are connected in series and the inductive reactance equals the capacitive reactance. For a series RLC circuit:
\[ Z=R+j\left(X_L-X_C\right) \]
- \(Z\)
- series-circuit impedance
- \(R\)
- resistance
- \(X_L,X_C\)
- inductive and capacitive reactance
- \(j\)
- imaginary unit
At resonance \(X_L=X_C\), so the reactances cancel and the impedance collapses to the resistance, \(Z=R\):
Series resonance produces a low-impedance path at the resonant frequency.
Because the impedance is low, even a small harmonic voltage at the resonant frequency can drive a large harmonic current. For harmonic order \(h\) the reactances scale as \(X_L(h)=hX_L\) and \(X_C(h)=\dfrac{X_C}{h}\), so series resonance occurs when \(hX_L=\dfrac{X_C}{h}\), giving:
\[ h_r=\sqrt{\frac{X_C}{X_L}} \qquad X_r=\sqrt{X_L X_C} \]
- \(X_L,\ X_C\)
- fundamental-frequency reactances
- \(X_r\)
- reactance at resonance
The quality factor of a series resonant circuit is:
\[ Q=\frac{X_r}{R} \]
- \(Q\)
- quality factor of the series resonant circuit
- \(X_r\)
- reactance at resonance
- \(R\)
- series resistance
The quality factor indicates how sharp the resonance is. A high \(Q\) means the resistance is small compared with the reactance, so the resonance is sharp and the current amplification can be large. A low \(Q\) means the circuit is more damped, the resonance is less severe and the amplification is smaller.
Worked example. For a series RLC circuit with \(X_C=1.6\ \Omega\) and \(X_L=0.064\ \Omega\), the resonant harmonic order is
\[ h_r=\sqrt{\frac{1.6}{0.064}}=5 \]
so the circuit resonates at the 5th harmonic. The reactance at resonance is \(X_r=\sqrt{0.064\times1.6}=0.32\ \Omega\). With a quality factor \(Q=100\), the resistance is
\[ R=\frac{X_r}{Q}=\frac{0.32}{100}=0.0032\ \Omega \]
This very small resistance means the circuit is lightly damped: at the 5th harmonic the impedance becomes very low, and a relatively small 5th-harmonic voltage can drive a large 5th-harmonic current. This is why series resonance is often associated with excessive harmonic current, which can overload capacitors, reactors, cables, transformers and harmonic filters.
Section 3
Parallel resonance
Parallel resonance occurs when resistance, inductance and capacitance form a parallel path and the inductive and capacitive effects cancel in terms of admittance. The total admittance becomes small at the resonant frequency, and since impedance is the inverse of admittance, the equivalent impedance becomes high:
Parallel resonance produces a high impedance at the resonant frequency.
Because the impedance is high, even a small harmonic current injected at the resonant frequency can produce a large harmonic voltage. This is why parallel resonance is so important: nonlinear loads inject harmonic currents into the network, and if the network impedance is high at one of those frequencies, the resulting voltage distortion can be large. The same reactance relationships apply, \(X_L(h)=hX_L\) and \(X_C(h)=\dfrac{X_C}{h}\), so the resonant order is found in the same way as the series case:
\[ h_r=\sqrt{\frac{X_C}{X_L}} \qquad X_r=\sqrt{X_L X_C} \]
- \(h_r\)
- resonant harmonic order
- \(X_L,X_C\)
- fundamental-frequency reactances
- \(X_r\)
- reactance at resonance
The resonant order is calculated identically, but the effect is opposite: series resonance makes the impedance low, parallel resonance makes it high. For a parallel resonant circuit the quality factor is written as:
\[ Q=\frac{R}{X_r} \]
- \(Q\)
- quality factor of the parallel resonant circuit
- \(R\)
- parallel resistance
- \(X_r\)
- reactance at resonance
Here a larger resistance gives a higher quality factor and a sharper resonance peak; a lower resistance gives more damping and reduces the peak. Critical damping — where the peak is no longer severe — corresponds in this simplified formulation to \(Q=\tfrac{1}{2}\), or equivalently \(R=0.5\,X_r\).
Worked example. For a parallel RLC circuit with \(X_C=60\ \Omega\) and \(X_L=0.495\ \Omega\), the resonant harmonic order is
\[ h_r=\sqrt{\frac{60}{0.495}}=11.01 \]
so the circuit resonates close to the 11th harmonic. The reactance at resonance is \(X_r=\sqrt{0.495\times60}=5.45\ \Omega\). With a quality factor \(Q=3\), the resistance is
\[ R=Q X_r=3\times5.45=16.35\ \Omega \]
The circuit therefore has a resonance peak around the 11th harmonic. If an 11th-harmonic current is injected into this network, the 11th-harmonic voltage may become significantly amplified.
Section 4
Series and parallel resonance compared
Series and parallel resonance arise from the same basic condition — equal inductive and capacitive reactances — but their practical effects are opposite:
Table 2 — Series versus parallel resonance.
| Item | Series Resonance | Parallel Resonance |
|---|
| Circuit behaviour at resonance | Low impedance | High impedance |
| Main risk | High harmonic current | High harmonic voltage |
| Indication in impedance scan | Sharp dip | Sharp peak |
| Excitation causing severe response | Harmonic voltage | Harmonic current |
| Practical concern | Overcurrent in resonant path | Voltage distortion at bus |
Series resonance attracts current.
Parallel resonance amplifies voltage.
In a harmonic study, a series resonance point appears as a dip in the impedance–frequency curve, and a parallel resonance point appears as a peak.
Section 5
Why resonance matters in power systems
Power systems contain both inductance and capacitance, so resonance is not only a theoretical circuit concept — it is a practical network issue. Typical contributors are:
Table 3 — Typical inductive elements.
| Element | Contribution |
|---|
| Transformers | Leakage inductance |
| Overhead lines | Series inductance |
| Cables | Series inductance and capacitance |
| Reactors | Designed inductance |
| Motors and generators | Inductive behaviour |
| Utility source | Short-circuit impedance |
Table 4 — Typical capacitive elements.
| Element | Contribution |
|---|
| Power-factor correction capacitor banks | Shunt capacitance |
| Harmonic filters | Tuned capacitance |
| Long cables | Shunt capacitance |
| GIS and busbar systems | Stray capacitance |
| Converter filters | Filter capacitance |
When capacitor banks are added to a network, they can shift the resonant frequency of the system — one of the most common reasons to carry out a harmonic study before installing power-factor-correction capacitors. A bank installed to improve displacement power factor may also create or move a resonance point close to a characteristic harmonic order; if it does, the bank may carry excessive harmonic current, or the bus voltage distortion may increase.
For example, if a system resonates close to the 5th harmonic and a nonlinear load injects 5th-harmonic current, the 5th-harmonic voltage may be amplified — causing voltage distortion, capacitor overloading, transformer heating, nuisance tripping or failure of sensitive equipment. Capacitor banks should therefore not be assessed only at the fundamental frequency; their behaviour at harmonic frequencies must also be checked.
Section 6
Resonance and harmonic sources
Resonance becomes a problem when the resonant frequency is close to a harmonic frequency that is actually present in the system. A resonance near the 17th harmonic with no significant 17th-harmonic source may pose limited practical risk; but a resonance near the 5th, 7th, 11th or 13th harmonic on a site with six-pulse drives or rectifiers may be much more serious. The severity depends on several factors:
Table 5 — Factors affecting resonance severity.
| Factor | Effect |
|---|
| Harmonic source magnitude | Larger harmonic current or voltage creates larger distortion |
| Resonant harmonic order | Resonance near a strong harmonic is more critical |
| System damping | More damping reduces amplification |
| Network strength | Weak systems have higher impedance and may show higher distortion |
| Capacitor size | Capacitors can shift the resonance frequency |
| Transformer impedance | Affects the system inductance and resonant point |
| Cable length | Long cables add capacitance and can affect resonance |
| Filter tuning | Tuned filters intentionally create controlled resonance paths |
Resonance therefore cannot be assessed from one component alone — it is a system-level issue.
Section 7
Resonance in frequency scans
A frequency scan is one of the most useful tools for identifying resonance. It calculates the driving-point impedance at a selected bus over a range of harmonic orders, normally plotted as \(|Z(h)|\) versus harmonic order \(h\). A peak indicates parallel resonance; a dip indicates series resonance.
A frequency scan does not directly show the harmonic distortion level — it shows where the network is likely to amplify harmonic voltages or currents. To calculate the actual distortion, the harmonic source spectrum must also be included:
Frequency scan identifies resonance risk.
Harmonic power flow calculates the actual distortion.
Section 8
Practical engineering interpretation
The most important practical message is that resonance can make a harmonic problem much worse than expected. A nonlinear load may inject a known harmonic current spectrum: if the system impedance is low at those frequencies, the voltage distortion may be acceptable; but if the impedance is high due to parallel resonance, the same current may cause excessive voltage distortion. Similarly, a harmonic voltage source may cause little current under normal impedance, but a series resonance path can make the harmonic current excessive. In harmonic studies, engineers therefore normally check:
Table 6 — Typical checks in a harmonic study.
| Check | Purpose |
|---|
| Harmonic source spectrum | Identify which harmonic orders are injected |
| Frequency scan | Identify resonance points |
| Harmonic power flow | Calculate harmonic voltages and currents |
| Capacitor loading | Check overcurrent and overheating risk |
| Transformer and cable loading | Check additional harmonic losses |
| Voltage THD | Check compliance and power quality |
| Individual harmonic voltages | Identify the dominant harmonic orders |
| Effect of switching conditions | See how resonance changes with network configuration |
Resonance is also sensitive to operating conditions: switching a capacitor bank on or off can move the resonance point, and changing transformer tap position, network configuration, cable connection or filter status can all change the frequency response. For this reason, harmonic studies often consider several operating scenarios rather than a single normal operating case.
Key message
Resonance occurs where inductive and capacitive reactances are equal at a particular frequency — expressed in harmonic studies as a harmonic order. Series resonance creates a low-impedance path and can produce high harmonic current; parallel resonance creates a high-impedance point and can produce high harmonic voltage. Both can be damaging, especially when the resonant order is close to a harmonic produced by nonlinear loads. The purpose of resonance analysis is to identify whether the network will amplify harmonic currents or voltages, and whether damping, filtering or design changes are required.
