Shunt, Series and Passive Filter Modelling for Harmonic Studies
Capacitors, reactors and passive filters are installed for fundamental-frequency duties — voltage control, reactive compensation, power transfer or filtering — but at harmonic frequencies the same components reshape the network impedance and can create or shift resonance. This page covers the lumped RLC representation of shunt and series capacitors and reactors and of passive filters; capacitor-bank parallel resonance and detuning; filter tuning and damping; component tolerances and duty; connection and grounding; and why these devices must be modelled as physical RLC networks rather than as reactive-power values.
Reading time ≈ 30 min · Part Seven of the series
Shunt and series compensation devices are among the most influential components in harmonic studies. They are normally installed for fundamental-frequency purposes — voltage control, reactive-power compensation, power-transfer improvement or harmonic filtering — but at harmonic frequencies the same components can significantly change the network impedance and may create or shift resonance. The main devices are shunt capacitors, shunt reactors, series capacitors, series reactors and passive harmonic filters.
All of these can normally be represented as lumped R, L and C elements. The important point is not only to include them, but to represent their frequency-dependent behaviour, losses, connection arrangement, tuning and tolerances correctly. Because these devices can significantly change \(Z_h\), they can change the level of harmonic voltage distortion even when the harmonic current injection is unchanged.
Key idea
Inductor impedance rises with order (\(hX_{L1}\)) and capacitor impedance falls (\(X_{C1}/h\)); their interaction creates resonance.
A shunt capacitor can form parallel resonance with the system at \(h_r=\sqrt{S_{SC}/Q_C}\) — often near a low-order harmonic.
Detuning controls the resonance frequency; damping controls the peak magnitude — both may be needed.
Model the physical RLC circuit, not just the reactive power — and check that every component survives its RMS and harmonic duty.
Key terms used on this page
01Shunt capacitor
A capacitor connected between busbar/phase and earth or neutral to provide reactive-power support.
02Shunt reactor
A reactor connected in shunt to absorb reactive power, often used on long cables or lightly loaded networks.
03Series capacitor
A capacitor connected in series with a line or circuit to reduce effective series reactance.
04Series reactor
A reactor connected in series to limit current or detune a capacitor bank.
05Passive harmonic filter
An RLC circuit designed to provide low impedance at selected harmonic frequencies.
06Detuned capacitor bank
A capacitor bank with a series reactor selected so the bank does not resonate at common harmonic orders.
07Tuning frequency
The frequency where the filter impedance is lowest.
08Damping resistor
A resistor used to reduce the sharpness of resonance and limit amplification.
09Quality factor, \(Q\)
A measure of how sharp or selective a filter response is.
Section 1
Why compensation devices matter
Shunt capacitors, reactors and passive filters can significantly change the harmonic impedance of a network. A capacitor bank may solve a power-factor problem at 50 Hz but create a parallel resonance at a harmonic frequency; a filter may reduce one harmonic order but amplify another if it is not correctly tuned or damped. These components must therefore be modelled carefully in harmonic studies.
The harmonic voltage at a bus is governed by the same relationship as for any component, and compensation devices act directly on \(Z_h\):
\[ V_h=Z_h\,I_h \]
\(V_h\)
harmonic voltage at order \(h\)
\(I_h\)
harmonic current injection at order \(h\)
\(Z_h\)
network harmonic impedance at order \(h\)
At fundamental frequency a capacitor or reactor controls voltage or reactive power; at harmonic frequencies it interacts with the system inductance and capacitance to form resonance. The reactances scale oppositely with order:
inductive and capacitive reactance at fundamental frequency
\(B_C(h),\ B_{C1}\)
capacitive susceptance at order \(h\) and at fundamental frequency
As the harmonic order increases, inductive reactance increases while capacitive reactance decreases — this opposite behaviour is exactly why capacitors and reactors can create resonant conditions at specific frequencies.
So inductors dominate more at higher frequency, while capacitors present lower impedance at higher frequency. When the inductive and capacitive parts of the network interact, resonance can occur — the network impedance becoming very high (parallel) or very low (series). The practical message: compensation devices can improve fundamental-frequency operation but create harmonic-frequency risks.
Section 2
General lumped RLC representation and tuning
Most shunt and series devices are represented using lumped R, L and C elements. The impedance of a series R-L-C branch is:
The tuning condition occurs when the inductive and capacitive reactances are equal, \(h_n X_{L1}=X_{C1}/h_n\), giving the tuning harmonic order (and the equivalent tuning frequency):
At the tuning frequency the inductive and capacitive reactances are approximately equal in magnitude, and the branch presents a low impedance to the targeted harmonic current.
This relationship is central to detuned capacitor banks and passive harmonic filters.
Frequency-dependent parameters
R, L and C may be defined as frequency-dependent characteristics — series resistance \(R_s(f)\), inductance \(L(f)\), capacitance \(C(f)\) and, where applicable, a parallel damping resistance \(R_p(f)\). The harmonic impedance is then evaluated at \(f=h f_0\), where \(f_0\) is the fundamental frequency. If no detailed data is available, the ideal relationships \(X_L(h)=hX_{L1}\) and \(B_C(h)=hB_{C1}\) are normally used — but resistive losses are not always constant; reactor resistance, for example, may increase with frequency through skin and proximity effects. The rule is to use frequency-dependent characteristics where data is available, but not to invent unrealistic damping where it is not.
Section 3
Shunt capacitor modelling
Shunt capacitors are represented as a lumped capacitive susceptance connected to a busbar. At harmonic order \(h\) the susceptance is \(B_C(h)=hB_{C1}\); if the capacitor reactive power at fundamental frequency is \(Q_C\) at nominal voltage \(V\):
So the capacitor draws more harmonic current as frequency increases. The harmonic current through the capacitor is:
\[ I_{C,h}=jB_C(h)\,V_h=jh\,B_{C1}\,V_h \]
\(I_{C,h}\)
capacitor harmonic current at order \(h\)
\(V_h\)
harmonic voltage across the capacitor
The practical meaning is that even a small harmonic voltage can drive a significant capacitor harmonic current, especially at higher orders.
Shunt capacitor losses
Capacitor resistive losses are normally small and often neglected in network impedance models; where manufacturer data exists, they can be represented by a series or parallel resistance. The dielectric loss can be expressed through the loss factor \(\tan\delta\):
\[ P_C \approx \omega C V^2 \tan\delta \qquad\Rightarrow\qquad P_{C,h}\approx h\omega_0 C V_h^2 \tan\delta \]
\(\tan\delta\)
dielectric loss factor
\(\omega_0\)
fundamental angular frequency
Dielectric loss therefore increases with frequency and with the square of harmonic voltage. For most network impedance studies this loss is small compared with other damping, but for capacitor duty assessment the capacitor harmonic current and RMS voltage must still be checked: capacitors may be simple in network models, but their harmonic duty still matters.
Section 4
Capacitor banks and parallel resonance
Parallel resonance occurs when the network inductance and the capacitor-bank capacitance create a high impedance at a particular harmonic frequency. If a harmonic current source is present at or near that frequency, even a small injected current can create a high harmonic voltage. This is why capacitor banks can sometimes increase voltage distortion instead of reducing it. The approximate resonance harmonic order is:
upstream system inductive reactance at fundamental frequency
\(S_{SC}\)
short-circuit power at the capacitor connection point
\(Q_C\)
capacitor bank reactive-power rating
This is a simplified screening equation. Detailed studies should use the frequency-dependent network model, including transformers, cables, lines, loads and other capacitor/filter banks.
This is extremely important: a capacitor bank can shift resonance into low-order harmonics such as \(h=5,\ 7,\ 11,\ 13\). If a harmonic source injects current near the resonance frequency, the voltage distortion can be amplified. The practical rule: never add a capacitor bank without checking harmonic resonance.
Section 5
Detuned and damped capacitor banks
It is worth separating two ideas. A tuned harmonic filter is designed to attract a selected harmonic current, such as the 5th or 7th. A detuned capacitor bank is different: its main purpose is still reactive-power compensation, but a series reactor is added so the bank does not resonate at a dangerous harmonic frequency. Detuning normally shifts the resonance below the lowest significant characteristic harmonic — typically below the 5th. This prevents the bank from acting as a low-impedance sink for a major harmonic and avoids parallel resonance at characteristic frequencies. The tuning order is:
\[ h_n=\sqrt{\frac{X_{C1}}{X_{L1}}} \]
For example, a bank detuned to \(h_n=4.2\) is tuned below the 5th harmonic. Below the tuning frequency the bank behaves mainly capacitive; above it, mainly inductive:
\(h \lt h_n \Rightarrow\) capacitive behaviour
\(h \gt h_n \Rightarrow\) inductive behaviour
This change in behaviour is why detuned banks reduce the risk of low-order harmonic resonance.
Mechanically switched capacitors and damped networks
A mechanically switched capacitor may be a simple bank or a detuned bank; in some systems additional damping is added to form a damped network. A simple bank can shift resonance significantly; a detuned bank moves resonance away from characteristic harmonics; a damped network adds losses to reduce the resonance peak. A damping resistor reduces the sharpness of the impedance peak — useful because a highly tuned circuit can create a narrow but severe resonance.
Detuning vs damping
Detuning controls the resonance frequency; damping controls the resonance peak magnitude. Both may be needed.
Table 1 — Difference between a capacitor bank, a detuned bank and a passive filter.
Device
Main Purpose
Harmonic Behaviour
Plain shunt capacitor bank
Reactive-power compensation
May create parallel resonance
Detuned capacitor bank
Reactive-power compensation with reduced resonance risk
Resonance shifted away from common harmonic orders
Single-tuned filter
Harmonic current absorption
Low impedance near the target harmonic
High-pass filter
Broadband harmonic damping
Reduces higher-order harmonic impedance
C-type filter
Harmonic filtering with reduced fundamental-frequency losses
Useful where losses must be limited
Section 6
Shunt reactor modelling
Shunt reactors are represented as lumped inductive elements connected to a busbar or line terminal. At harmonic order \(h\):
Unlike a shunt capacitor, the reactor provides lower admittance at higher harmonics because its impedance increases with frequency. Shunt reactors are used for fundamental-frequency voltage control, especially in cable-rich or lightly loaded transmission systems, but they still influence the harmonic profile of a line or busbar by changing the network frequency response — so they should be included in harmonic propagation studies even when not intended as harmonic devices.
Air-core and oil-filled reactors
Air-core reactors have no magnetic-core saturation and can usually be modelled as linear inductance with resistance. Oil-filled reactors may include magnetic-core effects, but for most steady-state harmonic studies saturation and magnetic coupling can be neglected if the reactor operates within its linear range. For special studies involving overvoltage, saturation, switching transients or ferroresonance, a more detailed model may be required. So an air-core reactor → a linear R-L model is usually adequate; an oil-filled reactor → a linear model is usually adequate for harmonics, but saturation may matter in special studies.
Shunt reactor losses
Reactor losses should be included where data is available, represented by a series resistance through the quality factor:
The resistance controls damping; if the reactor is part of a tuned or detuned circuit, its resistance may significantly affect the impedance peak. Where no frequency-dependent resistance is available, a constant resistance may be used as an approximation — but actual losses may increase with frequency. Include reactor resistance where it affects filter or detuned-bank damping.
Section 7
Series capacitor modelling
Series capacitors are connected in series with a transmission line to reduce its effective series reactance and increase power transfer. In harmonic studies a series capacitor is a lumped capacitive element inserted between two nodes, with impedance decreasing with order:
Because they sit directly in the current path, series capacitors can strongly affect resonance — forming series resonance with line inductance or interacting with other elements. They must be explicitly represented in harmonic studies, especially where the line is part of a resonance path.
Tolerances and protection equipment
Series capacitor modelling should consider capacitance tolerance, temperature variation, resistive losses, line-section location and any associated damping or bypass circuits. Associated protection equipment may include surge arresters, spark gaps, a bypass circuit breaker, a current-limiting reactor and a damping resistor. For most steady-state harmonic studies this protection equipment need not be represented in detail — ignore it in normal steady-state harmonic studies, but include it when its tuning is within the harmonic frequency range of concern or when background harmonic levels near that tuning are high.
Section 8
Series reactor modelling
Series reactors are lumped inductive elements connected in series with a line, feeder, capacitor bank or filter branch, with impedance:
\[ Z_L(h)=R_L(h)+jhX_{L1} \]
They are commonly used for current limiting, detuning capacitor banks, filter tuning, limiting inrush current, and smoothing or converter interface. For transmission applications they are often air-core and can be represented by a linear R-L model. When the reactor is part of a tuned filter or detuned capacitor bank, however, its inductance and resistance directly determine the tuning frequency and damping — so it should be represented explicitly.
Section 9
Passive harmonic filters
Passive harmonic filters are combinations of resistors, inductors and capacitors arranged to provide a low-impedance path for selected harmonic currents or damping over a wider range. Common types include the single-tuned, double-tuned and triple-tuned filter, the second-order high-pass filter, the C-type filter and the damped high-pass filter. A single-tuned filter gives low impedance near a specific order; a damped filter gives broader damping over a range. The tuning condition is \(h_n=\sqrt{X_{C1}/X_{L1}}\), and the branch impedance of a simple series-tuned filter is:
Common passive filter types
Single-tuned filter — tuned to one dominant harmonic order, such as the 5th. Double-tuned filter — filters two harmonic frequencies using one branch arrangement. High-pass filter — provides damping over a wider high-frequency range. C-type filter — designed to reduce fundamental-frequency resistor losses while still providing harmonic damping.
\[ Z_F(h)=R+j\!\left(hX_{L1}-\frac{X_{C1}}{h}\right) \qquad\Rightarrow\qquad Z_F(h_n)\approx R \]
At the tuning frequency \(h=h_n\), the inductive and capacitive reactances cancel and the branch impedance becomes approximately resistive, providing a low-impedance path for the targeted harmonic current.
Filter damping
Filter damping is controlled by resistance — from a series resistance, a parallel resistance, a damping-resistor branch, or reactor and capacitor losses. For a single-tuned branch, the quality factor is:
\[ Q_f=\frac{X_n}{R} \]
\(Q_f\)
filter quality factor
\(X_n\)
inductive reactance at the tuning frequency
\(R\)
damping resistance
A high-\(Q\) filter is more selective but produces a sharper impedance response; a lower-\(Q\) filter provides more damping but is less selective.
A high quality factor gives a sharp, narrow tuning response with low losses; a low quality factor gives wider damping but higher losses (\(Q_f\uparrow\Rightarrow\) narrower filter, lower losses; \(Q_f\downarrow\Rightarrow\) wider damping, higher losses). The correct value depends on the study objective and the acceptable losses.
Section 10
Component tolerances and fundamental duty
Passive filters and detuned banks are sensitive to component tolerances. Capacitance and inductance vary with manufacturing tolerance, temperature, ageing, operating voltage and frequency. The tuning frequency depends on \(h_n=\sqrt{X_C/X_L}\), or in absolute terms \(f_n=\tfrac{1}{2\pi\sqrt{LC}}\), and a small change in L and C shifts it:
Filter tuning is affected by capacitor tolerance, reactor tolerance, temperature, ageing and system-frequency variation. A filter designed exactly for the 5th harmonic may not remain exactly tuned in service, so harmonic studies should test reasonable detuning cases rather than relying on a single ideal tuning frequency — a filter should not be designed exactly at the problematic harmonic order without considering tolerances and system frequency variation.
Recommended filter sensitivity cases
nominal capacitance and inductance;
maximum capacitance tolerance;
minimum capacitance tolerance;
maximum reactor tolerance;
minimum reactor tolerance;
expected ageing of capacitors;
filter branch out of service;
one capacitor step unavailable;
minimum and maximum short-circuit level;
minimum and maximum load damping.
Fundamental-frequency duty
Shunt capacitor banks and filters also carry fundamental-frequency current and voltage, which may dominate the equipment rating. For a capacitor the fundamental current is \(I_{C,1}=\omega_0 C V_1\); for a filter branch the fundamental current depends on the net capacitive reactance of the branch at fundamental frequency. The total component voltage and current include fundamental and harmonic components:
Filter design is therefore both a harmonic issue and an equipment-duty issue: the engineer must check that capacitors, reactors and resistors can withstand their total RMS voltage, current and losses.
Section 11
RMS quantities and component results
Harmonic results may be reported in different ways. For equipment duty, RMS summation is normally more physically meaningful than arithmetic summation. The harmonic RMS voltage and the arithmetic sum are:
Similarly for current, \(I_{h,rms}=\sqrt{\sum_{k=2}^{n}I_k^2}\) and the total RMS current is:
\[ I_{rms}=\sqrt{I_1^2+I_{h,rms}^2} \]
The harmonic losses in a resistive component sum across orders, \(P_h=\sum_{k=2}^{n}P_k\), and the total loss is \(P_t=P_1+P_h\), where \(P_1\) is the fundamental-frequency loss. RMS values are used for thermal and equipment-duty assessment, while arithmetic values may be useful for conservative voltage-stress checks in some contexts.
Result quantities for filter components
For each filter or compensation component, the study should allow assessment of the voltage across the component, the current through it, the fundamental, harmonic-RMS and total-RMS duty, and the losses at individual orders and in total. For an inductor this includes \(U_L(h)\), \(I_L(h)\) and \(P_L(h)\) with their RMS and total values; for a capacitor, \(U_C(h)\) and \(I_C(h)\) plus dielectric or equivalent loss where applicable; for a damping resistor, \(I_R(h)\), \(P_R(h)\) and \(P_{R,total}\). This matters because a filter may successfully reduce harmonic distortion yet still overload one of its components.
A passive filter is not assessed only by how much distortion it reduces — its components must also be checked for duty. This includes RMS current, harmonic current, capacitor overvoltage, reactor current, resistor thermal loading, fundamental-frequency reactive power, switching transients and overload capability.
Component-duty checklist
A filter or capacitor-bank study should check:
fundamental-frequency voltage across the capacitor;
RMS current through each branch;
individual harmonic current;
total RMS current;
capacitor kvar duty;
reactor current and thermal duty;
damping-resistor energy and thermal rating;
voltage stress under harmonic distortion;
switching and energisation duty, if relevant;
outage conditions and filter overload cases.
Practical rule
Compliance at the busbar does not guarantee filter-component adequacy — every component must be checked against its own duty.
Section 12
Balance, connection and series vs shunt
Capacitor banks, reactors and filters may be represented as balanced or unbalanced. A balanced representation assumes equal phase values (\(C_a=C_b=C_c\), \(L_a=L_b=L_c\), \(R_a=R_b=R_c\)); an unbalanced representation allows different values per phase, e.g.:
An unbalanced model may be required where single-phase components are used, one capacitor unit is out of service, tolerances are phase-specific, unbalanced studies are needed, or zero-sequence flow is important. For a balanced series capacitor the positive- and negative-sequence equations may use the same capacitance, while the zero-sequence value is defined separately. Balanced models are adequate for symmetrical studies; unbalanced models are needed for phase-wise and zero-sequence accuracy.
Connection arrangement
The connection and grounding arrangement affects which harmonic components can flow through the device. A grounded-wye capacitor bank may provide a path for zero-sequence or triplen harmonic components, while an ungrounded bank may block some zero-sequence paths. The model should therefore reflect the actual connection — grounded-wye, ungrounded-wye, delta, single-phase, phase-to-phase or phase-to-earth. This is particularly important for zero-sequence and triplen harmonics: triplen harmonics \(h=3,\ 9,\ 15,\ \ldots\) are zero-sequence in balanced three-phase systems, and their flow depends strongly on neutral and delta paths. A filter or bank connected in delta behaves differently from a grounded-wye bank for triplen harmonics.
Notation: sequence subscripts vs harmonic order
The subscripts 0, 1 and 2 refer to sequence components, not harmonic order: \(I_0,\ I_1,\ I_2\) are the zero-, positive- and negative-sequence currents, \(V_0,\ V_1,\ V_2\) the matching voltages, and \(Z_0,\ Z_1,\ Z_2\) the sequence impedances. Harmonic order is always shown by \(h\) — so \(I_{0,3}\) means the zero-sequence current at the 3rd harmonic.
Series vs shunt compensation
A shunt device is connected between the bus and ground or neutral and changes the bus admittance; a series device is inserted in the line or branch path and changes the branch impedance.
Table 2 — Series and shunt compensation compared.
Device Type
Network Effect
Harmonic Relevance
Shunt capacitor
Adds capacitive admittance
Can create parallel resonance
Shunt reactor
Adds inductive admittance
Changes voltage profile and harmonic response
Series capacitor
Reduces series reactance
Can create series resonance
Series reactor
Increases series reactance
Can detune or limit harmonic current
Passive filter
Adds tuned shunt path
Diverts selected harmonic currents
So shunt devices affect bus harmonic impedance and series devices affect transfer and branch harmonic impedance; both must be represented according to their physical location.
Section 13
Frequency dependency and modelling filters
For many passive components, ideal frequency dependency is sufficient over the normal harmonic range, but some have non-ideal behaviour. Capacitance \(C(f)\approx C\) is often acceptable at lower harmonic frequencies; resistor \(R(f)\approx R\) is acceptable where the resistor has low frequency dependency and temperature coefficient; reactor inductance \(L(f)\approx L\) is acceptable in many studies, but the reactor resistance \(R(f)\) may increase with frequency through skin effect, winding design and core effects. So inductance is often approximately constant, while resistance may be frequency dependent. For converter output filters subject to PWM-related high-frequency components, reactor losses may be significantly higher and frequency dependency more important.
Modelling passive filters
Passive filters should be modelled exactly according to their electrical scheme. A filter is not just a capacitor bank — it may include a series reactor, capacitor, damping resistor, parallel damping branch, multiple tuned branches, a high-pass section or a C-type arrangement. Each component should be represented as an individual lumped element unless the software provides an equivalent validated filter model. For a multi-tuned filter, each tuned branch must be represented because each introduces a different impedance minimum. The filter topology must not be oversimplified, especially where component loading, losses and detuning are being assessed.
Don't model a filter as an ideal short circuit
Do not model a passive filter as an ideal short circuit at the tuned harmonic. Real filters have resistance, losses, tolerances and finite damping. If these are ignored, the model may overestimate filtering performance or underestimate component duty.
Filter performance assessment
A filter model should be assessed using a frequency scan, a harmonic penetration study, a component-duty calculation, a tolerance sensitivity, and outage and operating cases. The frequency scan checks whether the filter creates the intended impedance characteristic and whether it introduces unintended resonance; the penetration study checks whether distortion at the PCC or busbar is reduced to the required level; the duty calculation checks whether capacitor, reactor and resistor ratings are adequate; and the tolerance sensitivity checks whether performance remains acceptable if L and C vary. The workflow is: define filter → frequency scan → harmonic distortion calculation → component duty check → sensitivity assessment.
Section 14
Workflow and sensitivity
For compensation and filter devices, the most important sensitivities are summarised below.
Table 3 — Relative sensitivity of harmonic results to compensation and filter modelling.
Parameter
Typical Impact
Reason
Capacitance value
Very high
Controls tuning and resonance
Reactor inductance
Very high
Controls tuning
Damping resistance
High
Controls resonance peak
Component tolerance
High
Shifts tuning frequency
Connection mode
High for zero sequence
Affects triplen harmonic flow
Neutral grounding
High for zero sequence
Determines return path
Capacitor bank status
High
Changes network resonance
Reactor losses
Medium to high
Affects damping
Temperature and ageing
Medium
Changes capacitance and tuning
Harmonic source location
High
Determines current through the filter
Filter and compensation studies require sensitivity analysis — a single base-case result is not enough where the harmonic margin is small.
Practical modelling workflow
A practical workflow for shunt, series and filter devices is as follows:
Identify all compensation devices in service — capacitors, reactors, filters, series compensation.
Define their connection point and connection arrangement.
Include frequency-dependent characteristics where available.
Include damping resistors and component losses.
Represent detuning reactors or damping networks explicitly.
Include tolerances, ageing and temperature sensitivity where relevant.
Run frequency scans with all credible switching states.
Check component duty under fundamental and harmonic conditions.
The rule: model the physical circuit, not only the intended reactive power.
Minimum data for a defensible model
A robust shunt/series/filter harmonic model should state:
device type: capacitor, reactor, series capacitor, series reactor or filter;
connection arrangement;
grounding arrangement;
rated voltage;
rated reactive power;
capacitance value;
inductance value;
damping resistance, if applicable;
tuning frequency or tuning harmonic order;
component tolerances;
loss assumptions;
filter step configuration;
availability / outage cases;
switching status;
component current and voltage ratings;
frequency range and highest harmonic order assessed.
Section 15
Reporting and summary
A harmonic study report should clearly state how compensation and filter devices were represented. A weak statement — “capacitor banks were included” — tells the reader nothing.
Examples of a clear modelling statement
“Capacitor banks were represented as lumped shunt admittances with susceptance scaled by harmonic order; detuning reactors and damping resistors were explicitly included where installed.”
“Passive filters were represented as individual R, L and C branches according to the manufacturer single-line diagram, including tuning frequency, damping resistance and component tolerances.”
“The phase connection and neutral grounding of capacitor banks, reactors and filters were explicitly represented to capture zero-sequence harmonic paths, and component RMS voltage, RMS current and harmonic losses were calculated for capacitors, reactors and damping resistors.”
Common modelling mistakes
Representing a capacitor bank only at 50 Hz and ignoring harmonic resonance.
Forgetting the series reactor in a detuned bank.
Treating a detuned bank as a harmonic filter.
Ignoring capacitor and reactor tolerances.
Ignoring damping resistance or filter losses.
Modelling a filter as ideal rather than finite impedance.
Ignoring filter outage conditions.
Not checking filter component duty.
Ignoring grounding arrangement and zero-sequence paths.
Assuming the same filter performance for all network short-circuit levels.
To summarise: shunt capacitors and reactors, series capacitors and reactors and passive filters must be represented carefully because they directly affect the network harmonic impedance. For capacitors \(B_C(h)=hB_{C1}\) and \(X_C(h)=X_{C1}/h\); for reactors \(X_L(h)=hX_{L1}\). A capacitor bank can create parallel resonance with the upstream system, \(h_r=\sqrt{X_{C1}/X_{S1}}\) (approximately \(\sqrt{S_{SC}/Q_C}\)), and a detuned bank adds a series reactor to move the tuning away from important orders, \(h_n=\sqrt{X_{C1}/X_{L1}}\). Passive filters are modelled as R-L-C combinations with branch impedance \(Z_F(h)=R+j(hX_{L1}-X_{C1}/h)\), which becomes low at the tuning frequency to divert harmonic current.
The key modelling requirements are the component values, frequency dependency, losses and damping, connection arrangement, neutral grounding, component tolerances, switching states and component duty — in short, model compensation and filters as physical RLC networks, not only as reactive-power values.
In short: shunt capacitors, reactors, series compensation and passive filters can significantly change network harmonic impedance. A capacitor bank can create resonance, while a passive filter can reduce selected harmonic orders but may also introduce new resonance points if it is not correctly damped. For filters, the study should confirm not only compliance improvement but also the thermal and electrical duty of each component. A clear harmonic study should then state:
the device rating and RLC values;
the connection and grounding arrangement;
the tuning frequency;
the component tolerances;
the damping assumptions;
the switching status and outage cases;
the component-duty checks.
Key message
A robust harmonic study should state the device type, connection point, RLC topology, tuning frequency, loss and damping assumptions, frequency-dependent characteristics, connection and grounding, tolerance sensitivity and component-duty results. Only then can the influence of shunt and series compensation on harmonic resonance, distortion and mitigation performance be interpreted correctly — the seventh instalment of this series on harmonic studies in power systems.
Multi-Part Technical Series
Harmonic Studies in Power Systems
A practical series on how harmonic studies are set up and solved — from choosing the modelling domain, through frequency scan, harmonic penetration and balanced versus unbalanced modelling, to time, hybrid and harmonic-domain methods.
07
07Part SevenReading now
Shunt, Series and Passive Filter Modelling for Harmonic Studies
Lumped RLC modelling of shunt and series capacitors and reactors and passive filters; capacitor-bank parallel resonance; detuning; filter tuning, damping and component duty; connection and grounding.
