Power Quality & Harmonics

Harmonic Filter Design

A harmonic filter is not simply a capacitor tuned to a harmonic. It is a frequency-dependent power-system component that supplies reactive power at the fundamental, diverts harmonic current at a chosen order, and reshapes the network impedance. This part works through the design of a single-tuned shunt filter — tuning, sizing, capacitor and reactor duty, the frequency scan, tolerances and the full design procedure.

Reading time ≈ 18 min · Part Two of the series

Harmonic filters reduce distortion and stop harmonic currents from flowing into parts of the system where they cause overheating, resonance, voltage distortion or interference. A passive shunt filter is designed to give a low-impedance path for selected harmonic currents while also supplying reactive power at the fundamental frequency. A filter is therefore at once a harmonic-mitigation device, a reactive-power device, a frequency-dependent impedance, and an item that must be checked for voltage, current, thermal and protection duty.

Key idea

A series reactor turns a plain capacitor into a frequency-selective branch tuned to order \(h_n=\sqrt{X_{C1}/X_{L1}}\).
Filters are usually tuned slightly below the target harmonic, for tolerance and ageing margin.
The RLC calculation is only the first stage — the design must be verified against the network frequency response.
Capacitor and reactor duty (voltage, current, thermal) must be checked, not just the fundamental MVAr.

Section 1

Why a filter, and why a reactor

A passive shunt filter provides a low impedance at selected harmonic frequencies, so the harmonic current flows mainly into the filter instead of into the upstream network.

For a harmonic source at a bus, the harmonic voltage is approximately:

\[ V_h = Z_h I_h \]

\(V_h\): harmonic voltage at the bus
\(I_h\): injected harmonic current at order \(h\)
\(Z_h\): network impedance seen at harmonic order \(h\)

Reducing \(Z_h\) at a problematic order reduces the voltage distortion — which is why filter design must always be linked to system impedance, not treated as a standalone RLC circuit. A plain capacitor is the wrong starting point because its reactance falls with frequency:

\[ X_C(h)=\frac{X_{C1}}{h} \]

\(X_C(h)\): capacitive reactance at harmonic order \(h\)
\(X_{C1}\): capacitive reactance at the fundamental frequency

so a plain bank attracts harmonic current and may resonate with the upstream inductance. Adding a series reactor makes the branch a tuned circuit: at the tuning order \(h_n\) the inductive and capacitive reactances are equal, \(h_n X_{L1}=\dfrac{X_{C1}}{h_n}\), which 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 frequency
\(X_{C1}\): capacitor reactance at the fundamental frequency

and the tuned frequency is \(f_n=h_n f_0=\dfrac{1}{2\pi\sqrt{LC}}\), where \(L\) and \(C\) are the reactor and capacitor values. The reactor converts a plain bank into a frequency-selective branch that can be designed either to absorb a chosen harmonic or to avoid resonance with one.

Section 2

Exact tuning and practical detuning

A filter may be tuned to a specific order — the 5th, 7th, 11th or 13th — but in practice it is usually tuned slightly below that order:

Table 1 — Typical practical tuning, set slightly below the target harmonic.

Target Harmonic	Practical Tuning Order
5th	4.7 or 4.8
7th	6.6 or 6.7
11th	10.5 or 10.6
13th	12.4 or 12.6

This is because component tolerances, capacitor ageing, temperature and system-frequency variation shift the actual tuning point. If a filter is tuned exactly at the harmonic, a small change in \(L\) or \(C\) can push the tuning above the harmonic, reducing performance or creating an unwanted resonance. Tuning just below the target gives a practical margin. The tuning order is:

\[ h_n=\frac{f_n}{f_0}=\frac{1}{\omega_0\sqrt{LC}}, \qquad \omega_0=2\pi f_0 \]

\(f_n\): tuned frequency of the filter
\(f_0\): fundamental frequency
\(\omega_0\): fundamental angular frequency
\(L,C\): reactor inductance and capacitor capacitance

Section 3

The single-tuned shunt filter

The most common passive filter is the single-tuned shunt filter: a capacitor and reactor in series, the branch connected in shunt to the bus. Its impedance at harmonic order \(h\) is:

\[ Z_F(h)=R+j\left(hX_{L1}-\frac{X_{C1}}{h}\right) \]

\(Z_F(h)\): filter branch impedance at order \(h\)
\(R\): effective resistance of the reactor and branch
\(X_{L1},X_{C1}\): reactor and capacitor reactance at the fundamental

At the tuning order the reactances cancel, so the branch impedance collapses to the resistance, \(Z_F(h_n)\approx R\) — a low-impedance path that diverts harmonic current near the tuned frequency into the filter. The filter is capacitive below the tuning frequency and inductive above it, so it changes the frequency response of the whole network — the reason a system study is always needed.

Section 4

Sizing the capacitor and reactor

Design usually starts from the required fundamental reactive power. For a three-phase bank, the fundamental capacitive reactance is:

\[ X_{C1}=\frac{kV^{2}}{Q_C} \]

\(X_{C1}\): capacitor reactance at the fundamental
\(kV\): line-to-line voltage (kV)
\(Q_C\): capacitor reactive power (MVAr)

Once the tuning order \(h_n\) is chosen, the reactor reactance and the component values follow:

\[ X_{L1}=\frac{X_{C1}}{h_n^{2}} \qquad L=\frac{X_{L1}}{\omega_0} \qquad C=\frac{1}{\omega_0 X_{C1}} \]

\(X_{L1}\): required reactor reactance at the fundamental
\(L\): reactor inductance
\(C\): capacitor capacitance
\(\omega_0\): fundamental angular frequency

These define the basic filter components — the starting point, not the finished design.

Section 5

Filter reactive power is not the capacitor MVAr

With a reactor in series, the filter's fundamental reactive output is not exactly the capacitor MVAr, because the reactor absorbs some reactive power. The net fundamental reactance is \(X_F=X_{C1}-X_{L1}\), so the filter reactive power is:

\[ Q_F=\frac{kV^{2}}{X_{C1}-X_{L1}}=\frac{h_n^{2}}{h_n^{2}-1}\,Q_C \]

\(Q_F\): net filter reactive power at the fundamental
\(Q_C\): capacitor reactive power alone
\(h_n\): tuning order

so the filter delivers slightly more fundamental reactive power than the capacitor alone. The difference is small for high tuning orders, but more noticeable for low-order detuned filters below the 5th harmonic.

Section 6

Characteristic reactance and quality factor

At the tuned frequency the equal inductive and capacitive reactances define the characteristic reactance, and its ratio to the branch resistance gives the quality factor:

\[ X_n=\sqrt{X_{L1}X_{C1}} \qquad Q=\frac{X_n}{R} \]

\(X_n\): characteristic reactance (\(=X_L(h_n)=X_C(h_n)\))
\(Q\): quality factor of the tuned branch
\(R\): effective resistance of the branch

The quality factor sets how sharp the tuning is:

Table 2 — Effect of quality factor on a single-tuned filter.

Quality Factor	Interpretation
Low \(Q\)	Wider bandwidth, more damping, lower peak current at exact tuning
High \(Q\)	Sharper tuning, lower impedance at the tuned frequency, narrower bandwidth

A typical single-tuned filter has \(30 < Q < 100\), but the value should follow the required harmonic absorption, damping, equipment duty and system frequency response. A very high \(Q\) is effective at one harmonic but sensitive to detuning; a lower \(Q\) is less selective but better damped.

Section 7

Capacitor and reactor duty

The capacitor sees not only the fundamental voltage but the harmonic voltage developed across its reactance. The harmonic voltage, RMS voltage and (conservative) peak voltage are:

\[ V_{C,h}=\frac{X_{C1}}{h}I_h \qquad V_{C,\text{rms}}=\sqrt{\sum_{h=1}^{\infty}V_{C,h}^{2}} \qquad V_{C,\text{peak}}\approx\sum_{h=1}^{\infty}\sqrt{2}\,V_{C,h} \]

\(V_{C,h}\): capacitor voltage at harmonic order \(h\)
\(I_h\): harmonic current at order \(h\)
\(V_{C,\text{rms}},V_{C,\text{peak}}\): total RMS and (worst-case) peak capacitor voltage

The peak expression is conservative — it assumes all harmonic peaks coincide — but capacitors are sensitive to overvoltage, so a conservative check is useful. The capacitor current and reactive loading, and the reactor reactive loading, are:

\[ I_{C,\text{rms}}=\sqrt{\sum_{h=1}^{\infty}I_{C,h}^{2}} \qquad Q_C=\sum_{h=1}^{\infty} h\left(\frac{V_{C,h}}{V_{C,1}}\right)^{2}Q_{C1} \qquad Q_L=\sum_{h=1}^{\infty}hX_{L1}I_h^{2} \]

\(I_{C,\text{rms}}\): total RMS capacitor current
\(Q_{C1},V_{C,1}\): capacitor reactive power and voltage at the fundamental
\(Q_L\): reactor harmonic reactive loading

Harmonic voltage adds to capacitor loading with a multiplier of \(h\), so high-order harmonics can stress the capacitor even when their magnitude is modest. The capacitor and reactor must both be rated for the expected harmonic currents:

Table 3 — Capacitor voltage duties to consider.

Duty	Reason
Fundamental voltage	Normal operating voltage
Harmonic RMS voltage	Dielectric heating
Peak voltage	Insulation stress
Overvoltage margin	System voltage variation
Switching transients	Energisation and back-to-back switching
Ageing and tolerance	Capacitance changes over time

The reactor must also be checked for thermal duty, insulation class, magnetic saturation, audible noise and tolerance.

Section 8

Frequency scan and design procedure

A filter should be checked with a frequency scan of the complete network — the driving-point impedance at the bus over a range of orders:

\[ Z(h)=\frac{V_h}{I_h} \]

\(Z(h)\): driving-point impedance at harmonic order \(h\)
\(V_h,I_h\): harmonic voltage and current at the bus

Before installation the scan finds existing resonances; after installation it shows how the filter has reshaped the response. The study should confirm the filter:

Table 4 — What the frequency scan must confirm.

Requirement	Explanation
Reduces impedance near the target harmonic	Harmonic current is diverted into the filter
Does not create a new dangerous resonance	Avoid high impedance near dominant harmonics
Does not overload the capacitor or reactor	Check RMS and peak duty
Works under different network configurations	Minimum and maximum fault levels, switched banks
Remains acceptable with component tolerance	Check detuned conditions
Does not attract excessive background harmonics	Consider pre-existing voltage distortion

A practical single-tuned filter design procedure is:

Table 5 — Single-tuned filter design procedure.

Step	Action
1	Identify the harmonic problem: dominant order, distortion level and source spectrum
2	Define the objective: PCC compliance, equipment protection, capacitor duty or resonance mitigation
3	Select the filter type: single-tuned, high-pass, C-type, detuned bank, multi-branch or active
4	Select the fundamental reactive power \(Q_C\)
5	Select the tuning order \(h_n\), normally slightly below the harmonic to be filtered
6	Calculate \(X_{C1}=kV^2/Q_C\)
7	Calculate \(X_{L1}=X_{C1}/h_n^2\)
8	Calculate \(C=1/(\omega_0 X_{C1})\) and \(L=X_{L1}/\omega_0\)
9	Select the quality factor \(Q=X_n/R\) and find the branch resistance
10	Check capacitor RMS voltage, peak voltage, RMS current and reactive loading
11	Check reactor current, voltage, thermal duty, saturation and insulation
12	Run a frequency scan before and after installation
13	Run harmonic load-flow for normal, outage and minimum short-circuit conditions
14	Verify compliance with applicable harmonic limits
15	Confirm protection, switching and control requirements

The RLC calculation is only the first stage; the final design must be verified in the complete system.

Section 9

Multiple-branch filters

A single branch tunes to one order. Where several orders matter, branches are connected in parallel — for example near the 5th, 7th, 11th and 13th. Each branch has impedance:

\[ Z_{F,k}(h)=R_k+j\left(hX_{L1,k}-\frac{X_{C1,k}}{h}\right) \]

\(Z_{F,k}(h)\): impedance of branch \(k\) at order \(h\)
\(R_k,X_{L1,k},X_{C1,k}\): resistance and fundamental reactances of branch \(k\)

and the total filter impedance is the parallel combination:

\[ \frac{1}{Z_{F,total}(h)}=\sum_{k=1}^{N}\frac{1}{Z_{F,k}(h)} \]

\(Z_{F,total}(h)\): combined impedance of all branches
\(N\): number of filter branches

A multi-branch bank gives low impedance at several orders but a more complex response, so the complete bank must be checked to ensure no parallel resonance is created between branches or with the external network. Multi-branch filters are common in large industrial systems, HVDC schemes, arc-furnace installations and converter-dominated networks.

Section 10

Detuned banks and filter types

A detuned capacitor bank is not meant to absorb a harmonic strongly; it is designed to avoid resonance with dominant harmonics and to reduce the harmonic current the capacitor absorbs. The detuning reactor shifts the bank's resonant frequency below the lowest dominant harmonic — commonly below the 5th. The reactor percentage and resulting tuning order are:

\[ p=\frac{X_{L1}}{X_{C1}}\times100\% \qquad h_n=\frac{1}{\sqrt{p}} \]

\(p\): reactor percentage (here in per unit inside the root)
\(h_n\): resulting tuning order of the detuned bank

For a 7% reactor, \(h_n=\dfrac{1}{\sqrt{0.07}}=3.78\): the branch resonates below the 5th harmonic and is inductive at the 5th, so it does not act as a strong sink for 5th-harmonic current. This is why detuned banks are preferred over plain banks where nonlinear loads are present. Where broad absorption is needed instead of single-order selectivity, high-pass or damped filters are used:

Table 6 — Common passive filter types.

Filter Type	Main Purpose
Single-tuned filter	Absorb one dominant harmonic
Double-tuned filter	Absorb two selected harmonics in one branch arrangement
High-pass filter	Absorb a broad range of higher-order harmonics
C-type filter	Provide damping with reduced fundamental-frequency losses
Detuned capacitor bank	Avoid resonance and provide reactive power
Multi-branch filter bank	Absorb several characteristic harmonics

The choice depends on the harmonic spectrum, reactive-power requirement, loss evaluation, filter duty, system impedance and operational flexibility.

Section 11

Tolerances, limits and protection

Tuning depends on both \(L\) and \(C\), so component variation shifts the tuned frequency. The sensitivity is:

\[ \frac{\Delta f_n}{f_n}\approx-\frac{1}{2}\left(\frac{\Delta L}{L}+\frac{\Delta C}{C}\right) \]

\(\Delta f_n/f_n\): per-unit shift in tuned frequency
\(\Delta L/L,\ \Delta C/C\): per-unit change in reactor and capacitor value

An increase in capacitance lowers the tuned frequency; a decrease raises it. Design should therefore include tolerance cases (minimum/maximum capacitance, reactor tolerance, frequency variation, failed elements, and minimum/maximum system strength) — another reason to tune slightly below the target. Filters are normally designed to satisfy applicable limits, with IEEE 519 for control at the user PCC, IEC TR 61000-3-6 for emission allocation on MV/HV/EHV systems, and CIGRE guidance for frequency scans and modelling. Meeting a PCC limit does not by itself prove the filter is correctly rated — a filter can improve PCC distortion while being internally overloaded, so both external compliance and internal duty must be checked.

Filters also need suitable protection and switching, since the branch can see inrush, harmonic overload, capacitor unbalance, reactor overheating, switching transients and overvoltage:

Table 7 — Typical harmonic-filter protection functions.

Protection Function	Purpose
Capacitor unbalance protection	Detect failed capacitor elements
Overcurrent protection	Protect against faults and overloads
Overvoltage protection	Protect the capacitor dielectric
Reactor thermal protection	Prevent overheating
Neutral or residual protection	Detect abnormal branch conditions
Fuse protection	Isolate failed capacitor units
Switching control	Avoid excessive transients or wrong sequence
Discharge resistors	Safely discharge capacitors after disconnection

Energisation and back-to-back switching can produce high inrush, so circuit-breaker duty, pre-insertion resistors, controlled switching or damping may be needed for larger installations. Before acceptance, the design should be confirmed against the full check-list — fundamental reactive power, tuning order, frequency scan, harmonic load-flow, capacitor and reactor duty, losses, tolerance and switching cases, protection settings and PCC compliance — across minimum and maximum short-circuit capacity, different transformer arrangements, capacitor switching, load variation, filter outage and background distortion.

Section 12

Key message

A harmonic filter is a frequency-dependent power-system component, not just a capacitor tuned to a harmonic. At order \(h\) its impedance is \(Z_F(h)=R+j\left(hX_{L1}-\dfrac{X_{C1}}{h}\right)\); at the tuning order \(h_n=\sqrt{X_{C1}/X_{L1}}\) the reactances cancel and \(Z_F(h_n)\approx R\), which is how it traps harmonic current. But the same filter supplies fundamental reactive power, reshapes the network response and creates new equipment duties — capacitor voltage and current, reactor current and thermal duty, tolerance, resonance, switching transients and protection all have to be checked.

Filter design = RLC tuning + network frequency response + equipment duty.

Key message

The RLC tuning calculation is only the first stage. A good harmonic-filter design must satisfy harmonic performance, reactive-power requirements, equipment ratings, protection requirements and the full range of system operating conditions at the same time — verified by frequency scans and harmonic load-flow across minimum and maximum fault levels, switching states and component tolerances, and checked for both PCC compliance and internal equipment duty.

Part Two Reading now

Harmonic Filter Design

Designing a single-tuned filter — tuning order, capacitor and reactor sizing, equipment duty, frequency scans and the full procedure.

Series progress 2 of 4

Part Three

Second-Order Damped Harmonic Filters

The second-order damped filter — the capacitor–reactor–resistor branch, quality factor and damping, impedance and equipment duty for broad-spectrum mitigation.