A second-order damped filter is used when harmonic mitigation is required over a wider frequency range than a single-tuned filter can normally provide. The capacitor supplies reactive power at the fundamental; the parallel reactor and resistor shape the filter impedance at harmonic frequencies. Compared with a high-\(Q\) single-tuned filter, a damped filter is less selective but more robust.
Key idea
- A capacitor in series with a parallel reactor–resistor branch gives a broad, damped low-impedance region.
- The resistor deliberately damps the resonance — flatter response, lower detuning sensitivity, but real power loss.
- Its quality factor is defined the other way round, \(Q=R/X_n\), with low typical values (\(0.5\) to \(5\)).
- It must be thermally and system-dynamically rated — capacitor, reactor and resistor duty all checked.
Section 1
Why damping is added
A second-order damped filter provides low impedance near and above the selected tuning frequency, while adding damping so that sharp resonance is avoided.
A single-tuned filter gives a very low impedance around one harmonic order — useful when one dominant harmonic (5th or 7th) is known, but sensitive to component tolerance, capacitor ageing and configuration changes. A damped filter deliberately includes resistance: the resistor absorbs part of the harmonic energy and reduces the sharpness of the resonance, giving a wider response.
Table 1 — Filter behaviour by type.
| Filter Type | Behaviour |
|---|
| Single-tuned filter | Sharp low impedance at one harmonic order |
| Second-order damped filter | Wider low-impedance region with more damping |
| High-pass damped filter | Useful for higher-order harmonic absorption |
The resistor improves damping but introduces real power loss, so it must be thermally rated for the expected harmonic current and duty cycle. This makes a damped filter useful where the spectrum is broad, where higher-order harmonics are present, or where the system frequency response is uncertain.
Section 2
Basic tuning principle
For a filter tuned to order \(h_n\), the capacitor and reactor are chosen so their reactances are equal at the tuning frequency. With \(X_L(h)=hX_{L1}\) and \(X_C(h)=\dfrac{X_{C1}}{h}\), the condition \(h_n X_{L1}=\dfrac{X_{C1}}{h_n}\) gives:
\[ h_n=\sqrt{\frac{X_{C1}}{X_{L1}}} \qquad X_{L1}=\frac{X_{C1}}{h_n^{2}} \]
- \(h_n\)
- tuning order of the filter
- \(X_{L1}\)
- reactor reactance at the fundamental
- \(X_{C1}\)
- capacitor reactance at the fundamental
and the tuned frequency is \(f_n=h_n f_0=\dfrac{1}{2\pi\sqrt{LC}}\), where \(f_0\) is the fundamental frequency and \(L,C\) are the reactor and capacitor values.
Section 3
Sizing the components
Design starts from the required capacitor reactive power. For a three-phase bank:
\[ X_{C1}=\frac{kV^{2}}{Q_C} \]
- \(X_{C1}\)
- capacitor reactance at the fundamental
- \(kV\)
- line-to-line voltage (kV)
- \(Q_C\)
- capacitor bank rating (MVAr)
and with the tuning order chosen, the reactor reactance and component values follow:
\[ X_{L1}=\frac{X_{C1}}{h_n^{2}} \qquad C=\frac{1}{\omega_0 X_{C1}} \qquad L=\frac{X_{L1}}{\omega_0} \]
- \(L,C\)
- reactor inductance and capacitor capacitance
- \(\omega_0\)
- fundamental angular frequency, \(\omega_0=2\pi f_0\)
Section 4
Characteristic reactance and quality factor
At the tuning frequency the equal reactances define the characteristic reactance, equivalently:
\[ X_n=h_n X_{L1}=\frac{X_{C1}}{h_n}=\sqrt{X_{L1}X_{C1}} \]
- \(X_n\)
- characteristic reactance (\(=X_L(h_n)=X_C(h_n)\))
It is used to size the damping resistor. For a second-order damped filter the quality factor is defined as the ratio of resistance to characteristic reactance — the inverse of the single-tuned convention:
\[ Q=\frac{R}{X_n} \quad\Longrightarrow\quad R=Q X_n \qquad (\text{single-tuned uses } Q=X_n/R) \]
- \(Q\)
- quality factor of the damped filter
- \(R\)
- damping resistor value
Typical values are low, \(0.5 < Q < 5\): a low \(Q\) gives stronger damping and a flatter response; a higher \(Q\) gives less damping and a sharper response.
Table 2 — Practical meaning of the damped-filter quality factor.
| Quality Factor | Practical Meaning |
|---|
| \(Q\approx 0.5\) | Strong damping, wider response, higher resistor duty |
| \(Q\approx 2\) | Moderate damping, common practical compromise |
| \(Q\approx 5\) | Weaker damping, more selective, lower resistor current at some frequencies |
The choice of \(Q\) is not only mathematical — it affects performance, resistor losses, damping, harmonic current sharing and system resonance.
Section 5
Filter reactive power
Because the reactor is in series with the capacitor, the net fundamental reactive power is not exactly the capacitor MVAr. With net reactance \(X_F=X_{C1}-X_{L1}\):
\[ Q_F=\frac{kV^{2}}{X_{C1}-X_{L1}}=\frac{h_n^{2}}{h_n^{2}-1}\,Q_C \]
- \(Q_F\)
- net filter reactive power at the fundamental
- \(Q_C\)
- capacitor bank rating
so the filter output is slightly higher than the capacitor MVAr — small for high tuning orders, more noticeable for low ones.
Section 6
Impedance of the damped filter
The filter is a capacitor in series with a parallel reactor–resistor branch. The parallel branch impedance at order \(h\) is:
\[ Z_{LR}(h)=\frac{j hX_{L1}R}{R+jhX_{L1}} \]
- \(Z_{LR}(h)\)
- parallel reactor–resistor impedance at order \(h\)
- \(R\)
- damping resistance
so the total filter impedance, separated into real and imaginary parts, is:
\[ Z_F(h)=-j\frac{X_{C1}}{h}+\frac{j hX_{L1}R}{R+jhX_{L1}}=\frac{R\,(hX_{L1})^{2}}{R^{2}+(hX_{L1})^{2}}+j\left[\frac{hX_{L1}R^{2}}{R^{2}+(hX_{L1})^{2}}-\frac{X_{C1}}{h}\right] \]
- \(Z_F(h)\)
- total filter impedance at harmonic order \(h\)
The real and imaginary parts are therefore:
\[ R_F(h)=\frac{R\,(hX_{L1})^{2}}{R^{2}+(hX_{L1})^{2}} \qquad X_F(h)=\frac{hX_{L1}R^{2}}{R^{2}+(hX_{L1})^{2}}-\frac{X_{C1}}{h} \]
- \(R_F(h)\)
- resistive (real) part of the filter impedance
- \(X_F(h)\)
- reactive (imaginary) part of the filter impedance
At low frequencies the capacitor dominates; at high frequencies the resistor provides damping and stops the filter becoming an uncontrolled sharp resonant branch.
Section 7
Reactor and resistor duty
Because the reactor is in parallel with the resistor, the reactor current is not the full branch current. By current division, the reactor and resistor currents (for branch current \(I_h\)) are:
\[ |I_{L,h}|=|I_h|\frac{R}{\sqrt{R^{2}+(hX_{L1})^{2}}} \qquad |I_{R,h}|=|I_h|\frac{hX_{L1}}{\sqrt{R^{2}+(hX_{L1})^{2}}} \]
- \(I_{L,h},I_{R,h}\)
- reactor and resistor current at order \(h\)
- \(I_h\)
- total current into the parallel branch
As the order rises, \(hX_{L1}\) grows and current shifts from the reactor towards the resistor — which is why resistor thermal duty is central to the design. The resistor power loss is:
\[ P_R=\sum_{h=1}^{\infty}R\,|I_{R,h}|^{2}=\sum_{h=1}^{\infty}R\,|I_h|^{2}\frac{(hX_{L1})^{2}}{R^{2}+(hX_{L1})^{2}} \]
- \(P_R\)
- total harmonic power loss in the resistor
The resistor must be rated for continuous loss, short-time overload and ventilation, checked against background distortion, worst-case source operation, filter-outage and resonance cases. The reactor must be checked for:
Table 3 — Reactor duties to verify.
| Duty | Reason |
|---|
| RMS current | Thermal design |
| Harmonic current spectrum | Frequency-dependent heating |
| Peak current | Magnetic and insulation stress |
| Saturation | Avoid nonlinear behaviour |
| Voltage across reactor | Insulation rating |
| Audible noise | Harmonic forces and vibration |
Section 8
Practical design procedure
Table 4 — Second-order damped filter design procedure.
| Step | Action |
|---|
| 1 | Identify the dominant harmonic orders and background distortion |
| 2 | Define the purpose: PCC compliance, equipment protection, or broad high-frequency damping |
| 3 | Select the capacitor reactive power \(Q_C\) |
| 4 | Calculate \(X_{C1}=kV^2/Q_C\) |
| 5 | Select the tuning order \(h_n\) |
| 6 | Calculate \(X_{L1}=X_{C1}/h_n^2\) |
| 7–8 | Calculate \(C=1/(\omega_0 X_{C1})\) and \(L=X_{L1}/\omega_0\) |
| 9 | Calculate characteristic reactance \(X_n=\sqrt{X_{L1}X_{C1}}\) |
| 10 | Select quality factor \(Q\), normally 0.5 to 5 |
| 11 | Calculate resistor value \(R=Q X_n\) |
| 12 | Calculate the filter impedance \(Z_F(h)\) over the relevant range |
| 13 | Check capacitor voltage, current and reactive loading |
| 14 | Check reactor RMS / peak current, thermal duty and saturation |
| 15 | Check resistor RMS current and power loss |
| 16 | Run a network frequency scan with and without the filter |
| 17 | Run harmonic load-flow for normal and contingency configurations |
| 18 | Verify applicable harmonic limits and internal equipment duty |
The core design formulas, on a consistent per-phase or three-phase base, are:
\[ X_{C1}=\frac{kV^{2}}{Q_C},\quad X_{L1}=\frac{X_{C1}}{h_n^{2}},\quad X_n=\sqrt{X_{L1}X_{C1}},\quad R=Q X_n,\quad Q_F=\frac{h_n^{2}}{h_n^{2}-1}Q_C \]
The RLC calculation gives the initial component values; it does not complete the design, which must be validated in the full network model. The voltage and MVAr convention must be checked carefully to avoid errors in \(X_C\), \(L\), \(C\) and the currents.
Section 9
Single-tuned versus damped, and tuning above an order
Table 5 — Single-tuned versus second-order damped filter.
| Feature | Single-Tuned Filter | Second-Order Damped Filter |
|---|
| Main purpose | Absorb one selected harmonic | Provide wider damping over a range |
| Quality factor | Usually high | Usually low |
| Typical \(Q\) | \(30 < Q < 100\) | \(0.5 < Q < 5\) |
| Impedance shape | Sharp minimum near tuning | Broader, more damped |
| Resistor duty | Usually small / only reactor resistance | Intentional damping element |
| Detuning sensitivity | Higher | Lower |
| Losses | Lower | Higher (damping resistor) |
| Typical use | Dominant 5th, 7th, 11th, 13th | Higher orders, broad-spectrum damping, resonance control |
A damped filter may be tuned above a harmonic order — for example \(h_n > 17\) — when the aim is not to trap a low-order harmonic sharply but to damp higher orders. The resistor then prevents a sharp, potentially dangerous impedance peak. The tuning order should be chosen from:
Table 6 — Choosing the tuning order of a damped filter.
| Consideration | Explanation |
|---|
| Harmonic source spectrum | Which orders are present and dominant |
| Existing resonance points | Avoid placing resonance near dominant harmonics |
| Background distortion | Filter may absorb harmonics already present |
| Reactive power requirement | Capacitor size affects tuning and system voltage |
| Equipment duty | Resistor, reactor and capacitor must be rated |
| System configurations | Minimum and maximum fault levels change the response |
| Compliance objective | PCC limits and internal equipment limits may differ |
Section 10
Frequency response and compliance
A damped filter must be checked with a frequency scan that includes the upstream network, transformers, cables, capacitor banks, filters, machines and the relevant configurations. The driving-point impedance is:
\[ Z_{\text{bus}}(h)=\frac{V_h}{I_h} \]
- \(Z_{\text{bus}}(h)\)
- driving-point impedance at the bus, order \(h\)
- \(V_h,I_h\)
- harmonic voltage and current at the bus
performed before and after the filter is connected. A successful design should:
Table 7 — What the frequency scan must show.
| Requirement | Meaning |
|---|
| Reduce impedance near problematic harmonics | Lower harmonic voltage distortion |
| Avoid high parallel-resonance peaks | Prevent harmonic amplification |
| Provide adequate damping | Reduce sensitivity to operating-condition changes |
| Avoid excessive filter current | Protect capacitor, reactor and resistor |
| Remain acceptable under contingencies | Robust under network changes |
IEEE 519 is used for distortion at the user PCC and IEC TR 61000-3-6 for emission allocation on MV/HV/EHV systems, with CIGRE guidance emphasising frequency-domain modelling. But PCC compliance is not enough on its own — a damped filter can meet the PCC limit while suffering high resistor losses or capacitor current, so internal duty must be checked separately:
Table 8 — External compliance and internal duty checks.
| Check | Purpose |
|---|
| PCC voltage distortion | Compliance with network distortion limits |
| PCC current distortion | Compliance with emission limits |
| Capacitor RMS voltage | Dielectric stress |
| Capacitor peak voltage | Insulation duty |
| Capacitor RMS current | Thermal duty |
| Reactor RMS current | Thermal duty |
| Reactor peak current | Saturation and mechanical duty |
| Resistor power loss | Continuous and overload thermal rating |
| Switching transients | Energisation and de-energisation stress |
| Protection coordination | Safe isolation under abnormal conditions |
Section 11
Worked example
For a 33 kV, 6.8 MVAr capacitor bank used as a second-order damped filter tuned to \(h_n=4\):
\[ X_{C1}=\frac{33^{2}}{6.8}=160.1\ \Omega \qquad X_{L1}=\frac{160.1}{4^{2}}=10.0\ \Omega \qquad X_n=\sqrt{160.1\times10.0}=40.0\ \Omega \]
The damping resistor follows \(R=Q X_n\) for the chosen quality factor:
Table 9 — Resistor value versus quality factor (\(X_n=40\ \Omega\)).
| \(Q\) | \(R\) |
|---|
| 0.5 | \(20\ \Omega\) |
| 2 | \(80\ \Omega\) |
| 3 | \(120\ \Omega\) |
| 5 | \(200\ \Omega\) |
and the filter reactive rating is:
\[ Q_F=\frac{h_n^{2}}{h_n^{2}-1}Q_C=\frac{16}{15}\times6.8=7.25\ \text{MVAr} \]
so the filter output is slightly higher than the capacitor MVAr, because the series reactor reduces the net capacitive reactance at the fundamental.
Section 12
Key message
A second-order damped filter is used when mitigation needs damping over a wider frequency range than a single-tuned filter — especially for higher-order harmonics, broad-spectrum distortion and resonance damping. It is tuned by \(h_n=\sqrt{X_{C1}/X_{L1}}\); its resistor is sized by \(R=Q X_n\) with \(X_n=\sqrt{X_{L1}X_{C1}}\) and a low \(0.5 < Q < 5\); and its impedance is \(Z_F(h)=-j\dfrac{X_{C1}}{h}+\dfrac{j hX_{L1}R}{R+jhX_{L1}}\). The most important checks are capacitor duty, reactor duty, resistor power loss, frequency response, harmonic load-flow and PCC compliance.
A damped filter is not only tuned — it is also thermally and system-dynamically rated.
Key message
A damped filter must reduce harmonic distortion without creating new resonance problems or overloading its own capacitor, reactor or resistor. The damping resistor is an intentional design element — it broadens and stabilises the response but dissipates real power, so it must be thermally rated for the full range of harmonic and operating conditions. As always, the RLC tuning is only the first stage: the finished design has to satisfy harmonic performance, equipment ratings, protection and the network frequency response together, verified for both PCC compliance and internal equipment duty.
