The harmonic impedance of an AC network is one of the most important inputs in harmonic performance studies and AC filter design — it converts harmonic currents into harmonic voltages, amplifies or attenuates pre-existing distortion, and decides how the network interacts with filters, converters, capacitor banks, cables and transformers. It is not a single number: seen from a point of connection it changes with frequency and with operating condition, appearing inductive at one frequency, capacitive at another and strongly resonant at a frequency that need not be an integer harmonic. This page is the practical, specification-focused companion to Harmonic Impedance Loci and Envelopes: how to turn many credible network conditions into a technically justified frequency-band envelope, why minimum resistance and damping angle so often dominate filter design, how performance and rating envelopes differ, and what a sound impedance specification and report should contain.
Reading time ≈ 30 min · APS technical note
The harmonic impedance of an AC network is one of the most important inputs in harmonic performance studies and AC filter design. It governs how harmonic currents are converted into harmonic voltages, how pre-existing harmonic voltages are amplified or attenuated, and how the network interacts with filters, converters, capacitor banks, cables, transformers and other frequency-dependent equipment. Because so many downstream decisions depend on it, the impedance should rarely be represented by a single fixed value.
In real transmission and distribution networks the impedance seen from a point of connection changes with frequency and with operating condition. The network can appear inductive at one harmonic frequency, capacitive at another, and strongly resonant at a frequency that is not an exact integer harmonic. A harmonic impedance envelope is therefore not merely a conservative boundary drawn around a few calculated points; it is an engineering representation of the credible range of network impedance at the relevant harmonic frequencies. This page is a practical companion to the more theoretical note on harmonic impedance loci and envelopes — here the emphasis is on building, justifying and specifying an envelope for harmonic assessment and AC filter rating.
One message frames everything that follows: the harmonic impedance envelope defines the electrical environment against which the filter is designed. If the envelope is wrong, the filter design basis is wrong.
Main takeaway
Network harmonic impedance is frequency- and operating-condition-dependent — capture its credible range as an envelope, not one scan.
The aim is not the largest envelope or the smallest, but a technically justified one: too narrow is non-conservative, too wide drives unnecessary filter cost.
For filter design, minimum resistance and damping angle often matter as much as maximum impedance magnitude — the worst case usually sits on the boundary.
Treat performance and rating envelopes separately, divide the range into frequency bands, and apply realistic band-edge tolerance.
Main output
The main output of this study is a set of technically justified R–X impedance envelopes, divided by harmonic frequency band and by assessment condition, that can be used directly for harmonic performance and AC filter rating studies. The output should not be only a frequency-scan plot: it should state the envelope boundaries, the frequency bands, the tolerance assumptions, the performance/rating distinction, and the critical parameters that drive the design.
Assessment condition means whether the envelope is being used for harmonic-voltage performance, equipment rating, contingency assessment, a future network condition, or another defined study case.
What to remember
Network harmonic impedance changes with frequency and with operating condition.
Integer-harmonic calculations alone can miss resonances.
The worst harmonic-performance or filter-rating condition often lies on the envelope boundary.
Minimum resistance and damping angle can be more important than maximum impedance magnitude.
Performance envelopes and rating envelopes may not be the same.
The final envelope must be technically justified, not simply conservative.
Performance envelopes are used to check harmonic-voltage compliance; rating envelopes are used to check equipment thermal and electrical duty. They may be different, because the worst voltage-distortion case is not always the worst filter-current or component-rating case.
An envelope is not a compliance limit
The impedance envelope is not itself a harmonic-distortion limit. It is the network representation used to calculate harmonic voltage distortion, filter current and equipment rating. The compliance limits come from the grid code, connection agreement, IEC framework, IEEE framework or project specification.
Key terms used on this page
01PCC / PoC
Point of Common Coupling / Point of Connection — the busbar where harmonic compliance is normally assessed.
02Harmonic impedance, \(Z_h\)
The complex network impedance \(R_h+jX_h\) seen from the point of connection at harmonic order \(h\).
03Impedance envelope
A boundary in the R–X plane enclosing the credible impedance points for a harmonic order or frequency band.
04Minimum resistance, \(R_{\text{min}}\)
The lowest credible network resistance in a band — controls damping and the height of resonant peaks.
05Damping angle
The impedance angle \(\theta=\tan^{-1}(X/R)\); an angle near \(90^\circ\) means a lightly damped, strongly reactive network.
06Performance envelope
The impedance range over which harmonic voltage-distortion limits must be met.
07Rating envelope
The impedance range used to determine equipment thermal and electrical duty (capacitor, reactor, resistor, arrester).
08Pre-existing distortion
Background harmonic voltage already present on the network, separate from the new plant’s emission.
09Detuning
A shift in a filter’s tuning frequency caused by component tolerance, temperature, ageing and system-frequency variation.
Abbreviations and symbols
01AC
Alternating current.
02BESS
Battery energy storage system.
03FACTS
Flexible AC transmission system.
04HVDC
High-voltage direct current.
05LCC
Line-commutated converter.
06VSC
Voltage-source converter.
07PCC
Point of common coupling.
08R–X plane
Resistance–reactance plane.
09TR
Technical Report (an IEC document type).
10RMS
Root mean square.
11THD
Total harmonic distortion.
12TDD
Total demand distortion.
13EMT
Electromagnetic transient.
14\(R_{\text{min}}\)
Minimum-resistance boundary of the envelope.
15\(R_{\text{max}}\)
Maximum-resistance boundary of the envelope.
16\(X_{L,\text{max}}\)
Maximum inductive-reactance boundary.
17\(X_{C,\text{max}}\)
Maximum capacitive-reactance boundary.
18\(\theta\)
Impedance angle.
19\(\epsilon\)
Frequency tolerance.
What not to do
Do not use one short-circuit impedance as the harmonic impedance.
Do not scan integer harmonics only.
Do not use one large envelope for all harmonics unless it is justified.
Do not ignore minimum resistance and damping angle.
Do not mix performance and rating cases.
Do not apply IEC, IEEE, CIGRÉ and grid-code requirements without defining the hierarchy.
Section 1
What is network harmonic impedance?
Network harmonic impedance is the equivalent impedance of the AC network seen from a selected busbar or point of connection at a particular harmonic frequency. It is a complex quantity:
\[ Z_h = R_h + jX_h \]
\(Z_h\)
network harmonic impedance at harmonic order \(h\)
\(R_h\)
equivalent network resistance at order \(h\)
\(X_h\)
equivalent network reactance at order \(h\)
\(j\)
imaginary operator
\(h\)
harmonic order
The harmonic frequency follows from the harmonic order and the fundamental frequency:
\[ f_h = h\,f_1 \]
\(f_h\)
harmonic frequency
\(f_1\)
fundamental frequency, normally 50 Hz or 60 Hz
On a 50 Hz system \(h=5\) is 250 Hz and \(h=35\) is 1750 Hz. Resonances, however, need not fall on integer orders.
The magnitude and angle of the harmonic impedance are:
magnitude of the harmonic impedance at order \(h\)
\(\theta_h\)
impedance (damping) angle at order \(h\)
\(X_h\) is the net reactance, evaluated at each frequency; \(X_h\gt 0\) means the network looks inductive there and \(X_h\lt 0\) capacitive. Because \(R_h\gt 0\) for a passive network the angle stays unambiguous in the right half-plane. When the net reactance passes through zero or changes sign, the network is close to resonance and the impedance is mainly resistive.
Why this matters — the harmonic relationship
Harmonic voltage is produced from harmonic current by the same relationship as for any component, \(V_h=Z_h\,I_h\). A higher \(|Z_h|\) at the frequency of an injected harmonic current produces a higher harmonic voltage; a resonant, lightly damped impedance amplifies both the plant’s own emission and any pre-existing background voltage. Filter performance depends on magnitude, damping, angle and resonance behaviour — so both resistance and reactance matter, not magnitude alone.
Section 2
Why harmonic impedance is not short-circuit impedance
For short-circuit studies, engineers often use the network impedance at fundamental frequency. This is not sufficient for harmonic studies. At harmonic frequencies the network behaves differently because so much of it is frequency-dependent:
line and cable reactance change with frequency;
shunt capacitance becomes far more significant at higher frequency;
transformers have frequency-dependent behaviour and stray capacitance;
loads provide damping, but the damping changes with frequency and load composition;
capacitor banks and filters create series and parallel resonances;
cables and overhead lines can create their own resonances;
power-electronic converters interact with the network impedance;
the equivalent impedance changes whenever circuits, generators, transformers, filters or capacitor banks are switched in or out.
The harmonic impedance at the point of connection may therefore change rapidly with frequency: a small frequency change can produce a large change in magnitude or angle, especially near resonance. This is why harmonic studies require a frequency scan rather than a single impedance value.
Takeaway
A short-circuit impedance is not a harmonic impedance. A network that appears strong at fundamental frequency can still have severe harmonic resonances at higher frequencies — a high fault level is no guarantee of low harmonic impedance.
Section 3
Why it changes with operating condition
The harmonic impedance of the AC network changes because the network itself changes. The conditions that can significantly affect the impedance seen from a converter station, renewable plant, BESS, industrial facility or customer connection include:
transmission elements switched in or out for planned outages, protection operation or maintenance;
generation dispatch changing with demand and system operation;
load level and load composition changing through the day and season;
reactive-compensation switching and control actions;
shunt capacitor banks, harmonic filters, shunt reactors, SVCs or STATCOMs connected or disconnected;
transformer tap positions changing;
cable and overhead-line temperature changing conductor resistance and parameters;
future network development changing system topology;
contingency conditions creating weak-grid or unusual resonance conditions.
A harmonic impedance study should therefore include credible network operating conditions, not only the normal intact condition. One operating condition gives a single impedance locus; many credible conditions give the family of loci from which an envelope is drawn — see the companion note on impedance loci and envelopes for how those loci are plotted in the R–X plane.
Takeaway
An impedance envelope must represent credible network variation. A single intact-network condition is not enough for filter specification or compliance assessment.
Section 4
Major and minor resonance
A resonance occurs when the inductive and capacitive behaviour of the network interact at a particular frequency. A parallel resonance creates a high impedance: if a harmonic current is injected near that frequency, the resulting harmonic voltage can be high. A series resonance creates a low-impedance path that may increase harmonic current through filters, capacitor banks or network elements.
Major resonance and minor loops
A major resonance is usually associated with a change in the sign of reactance — the impedance locus moves from inductive to capacitive or back. A minor loop is a resonant feature where the impedance changes rapidly but the reactance does not change sign. Both matter: a minor loop can still affect harmonic voltage, filter current, damping and sensitivity to model uncertainty.
A reactance sign change (an \(X=0\) crossing) marks a resonance, but it does not by itself say which kind: it is a parallel resonance where \(|Z|\) is at a maximum and a series resonance where \(|Z|\) is at a minimum. The study should not only locate the largest impedance peak. It should also identify rapid impedance changes, minor loops and resonant regions that may affect filter duty or distortion compliance. For voltage distortion at the PCC, parallel resonance is usually the dominant concern; for branch current and component duty, series resonance can be decisive.
Section 5
Why integer-harmonic scans are not enough
A common error is to calculate network impedance only at integer harmonic orders such as the 2nd, 3rd, 5th, 7th, 11th and 13th. This can miss important resonances, because a resonance may occur between integer orders. If the study checks only the integer points, a sharp resonance between two harmonics may be invisible.
This matters because real systems are not exact. Network-frequency variation, transformer tap position, line and cable temperature, data uncertainty, equipment tolerance and modelling simplification can all shift a resonant frequency. A resonance that appears between two harmonics in the study may move closer to an integer harmonic in service. Harmonic impedance should therefore be calculated on a quasi-continuous frequency basis.
Choosing the frequency step
The familiar rule \(\Delta f\leq f_0/10\) (so \(\le 5\) Hz near 50 Hz) is only a loose upper ceiling for not missing low-order behaviour — it is not a recommended resolution. The binding requirement is resolving sharp series and parallel resonance peaks, whose bandwidth can be a few hertz or less for high-Q conditions. For transmission, HVDC and filter-design studies a \(1\) Hz step is a sensible default, with adaptive refinement (\(0.1\)–\(0.5\) Hz) around high-Q resonances. The step must be fine enough to reveal resonances that affect performance or rating, not merely convenient for computation. See impedance loci and envelopes for the resolution rule in context.
Takeaway
Integer-harmonic points alone can hide critical resonances. The study output should include frequency scans with a small enough step to reveal non-integer resonant behaviour.
Section 6
From R–X scatter to a usable envelope
The R–X plane is simply a map of resistance on the horizontal axis and reactance on the vertical axis: each point shows how the network looks electrically at one frequency and one operating condition.
When the impedance is swept across frequency, each complex value \(Z_h=R_h+jX_h\) is a point in the R–X plane, with resistance on the horizontal axis and reactance on the vertical axis. One operating condition traces a path — the impedance locus; many credible conditions produce a family of loci, or a cloud of points. An R–X envelope is a boundary drawn around that cloud, representing the range of impedances that should be considered in harmonic performance and filter-rating studies.
The purpose of an envelope is to simplify the design process without losing the important impedance variations. Instead of asking the filter designer to evaluate every scanned point for every condition, the customer specifies envelopes for defined harmonic bands, and the designer checks performance and rating against the relevant boundaries. A good envelope provides a common basis for bidders and contractors, a conservative-but-not-excessive representation of the network, a practical input for both performance and rating calculations, coverage of outages and future uncertainty, and a transparent record of the assumptions used.
The detailed taxonomy of envelope shapes — circle, sector, polygon and convex-hull polygon — and how each trades simplicity against realism is covered in the companion note on harmonic impedance loci and envelopes. In short: a circle or sector is simple but encloses impedance regions the network never presents; a polygon or enhanced polygon follows the credible point cloud more closely and is usually preferred for detailed compliance and filter design.
Takeaway
An impedance envelope is a design input, not a decorative plot. It directly affects harmonic compliance, filter size, filter damping, component rating and project cost.
Section 7
Why the envelope boundary matters
In many harmonic performance and filter-rating calculations, the worst resonant condition is not found inside the envelope — it is found on the boundary. The boundary represents the limiting values of resistance, reactance, impedance angle and magnitude that create the maximum harmonic voltage or filter current. Critical points often occur:
close to the origin of the R–X plane;
along the minimum-resistance boundary;
along the maximum or minimum damping-angle boundary;
near the transition between inductive and capacitive regions;
near resonant points where the impedance is mainly resistive;
at the maximum-impedance-magnitude boundary, where pre-existing voltage distortion is amplified.
This means the boundary must be defined carefully — it is not enough to draw a convenient large shape around the data. If the boundary includes unrealistic low-resistance or low-damping regions, the design becomes unnecessarily conservative; if it excludes credible low-resistance or high-resonance regions, the design becomes non-conservative. A robust assessment therefore tests sufficient points on and near the boundary, not only a base case at the centre of the cloud.
When a filter designer searches for the worst case, the critical result often occurs on the envelope boundary. This is why the boundary must represent realistic combinations of resistance and reactance, not only a conservative outer shape.
Conservatism should be realistic. An envelope that includes impossible combinations of low resistance, high impedance magnitude and extreme angle can force unnecessary filter size, damping, losses and cost without improving the real system performance.
Section 8
One large envelope versus frequency-band envelopes
A simple approach is to draw one large circle or sector around all impedance points from all harmonic orders and all conditions. This appears safe but is usually too pessimistic. Different parts of a large envelope are controlled by different harmonic orders and different operating conditions, so a low harmonic may be assessed against an impedance point that can only occur at a much higher harmonic. The result is artificial conservatism that can drive:
unnecessary filter branches;
larger Mvar ratings;
excessive damping and higher losses;
a larger footprint and higher cost;
more complex compliance demonstration and operational restrictions.
In most cases it is better to divide the impedance data into harmonic frequency bands and define a separate envelope for each band, with the bands chosen from the actual behaviour of the network rather than a fixed template.
Low-order bands
At lower harmonic orders the impedance is often more stable and easier to define (though this depends on the network). Where the variation is smooth and resonances are few, narrow envelopes can be used with confidence. Low-order harmonics frequently control the design because converter and load emissions are concentrated there, filters may be sharply tuned to low orders, and voltage-distortion limits are often more restrictive. Low-order envelopes should not automatically inherit the worst high-order damping angles or impedance radii.
Mid-range and high-order bands
At higher orders the number of resonances usually increases and the impedance can change rapidly with small frequency changes — especially in networks with long cable circuits, multiple capacitor banks, filters at several substations, HVDC converters, FACTS devices, renewables, BESS or weak-grid conditions. Modelling accuracy also reduces at higher frequencies because equipment data is less certain and frequency-dependent parameters become more important. Broader bands may then be needed, but each band must still be selected carefully so it covers credible resonance movement without introducing unrealistic impedance regions.
Table 1 — Band width as a function of harmonic order (indicative).
Wider bands, but still bounded to credible regions
Takeaway
The objective is to be realistically conservative. A single all-harmonics envelope can drive filter over-design by allowing impedance combinations that cannot occur at the harmonic order being assessed.
Section 9
Tolerance at the edges of a band
When an envelope is defined for a band, impedance data immediately outside the nominal band should also be considered, because resonances can shift due to fundamental-frequency variation, modelling uncertainty, equipment-parameter tolerance, transformer tap position, line or cable temperature, future network development, unstudied operating conditions, simplified load models and filter-tuning tolerance.
The amount of tolerance should not be a fixed rule for all orders. A \(\pm 1\) harmonic-order band is a constant \(\pm f_1\) in absolute terms (\(\pm 50\) Hz on a 50 Hz system) at every order, so its relative width scales as \(1/h\): at \(h=2\) it is \(\pm 50\%\) of the harmonic frequency, whereas at \(h=40\) it is only \(\pm 2.5\%\). A fixed \(\pm 1\) order may therefore be reasonable at high orders but is far too severe at low orders, and a percentage-based tolerance gives a more balanced relative allowance across the range.
The tolerance range is defined by applying a percentage frequency allowance around the nominal harmonic frequency:
lower and upper frequencies included in the tolerance range
\(f_h\)
nominal harmonic frequency
\(\epsilon\)
selected frequency tolerance, expressed as a fraction
The tolerance need not be the same at every frequency. Where a resonance is dominated by equipment with well-known parameters, a smaller \(\epsilon\) may be justified; where it depends on uncertain network data, a larger \(\epsilon\) may be required. The underlying causes are partly absolute (system-frequency drift is roughly constant in hertz) and partly relative (\(L\) and \(C\) tolerances scale the tuning frequency), so the most physical model combines an absolute floor with a percentage term; a pure percentage is a reasonable simplification.
Frequency-band envelopes should normally overlap slightly for the same reason — a resonance may shift from one band into the next because of small parameter changes. The band-edge tolerance and the band overlap together provide margin against frequency uncertainty.
Section 10
Multiple envelopes in the same band
Sometimes the scatter plot for one band forms two or more separate clusters. A typical case is a particular outage or operating condition that produces impedance points far from the normal operating cloud. If one large envelope is drawn around all clusters, it may enclose a large region of impedance values that never occurs in service.
In that situation it is often better to define two or more separate envelopes for the same band — for instance one for the intact network and one for the outage condition. This reduces unnecessary conservatism while still covering the credible conditions. The specification must then state clearly when each envelope applies.
Takeaway
One cloud of points → one envelope; two distinct clouds → two envelopes. Separated clusters should not be forced into one oversized boundary.
Section 11
Performance and rating envelopes
Harmonic studies usually distinguish between performance conditions and rating conditions, and the impedance envelopes for the two may differ.
Table 2 — Performance versus rating envelopes.
Aspect
Performance Envelope
Rating Envelope
Purpose
Demonstrate that voltage-distortion limits are met at the PCC and specified buses
Determine equipment thermal and electrical duty
Governs
Harmonic voltage distortion, compliance margin
Capacitor RMS current and peak voltage; reactor and resistor duty; arrester duty
Conditions
Credible continuous operating states
May include severe or less-probable configurations
Limit basis
Must meet continuous performance limits
Need not meet continuous limits, but must not overstress equipment
At low harmonic orders the performance and rating envelopes may be similar. At higher orders they often differ, because outages, tolerances and future-network uncertainty can shift resonances into the rating cases. A design can satisfy voltage-distortion limits yet still overload a filter component, and a filter can be thermally adequate yet fail to deliver the required harmonic performance — so both must be checked.
A condition that is acceptable for short-duration equipment rating may not be acceptable for continuous harmonic-voltage performance.
Takeaway
Performance and rating envelopes should be treated separately where the applicable network conditions differ. Harmonic compliance and equipment rating are related but not identical.
Section 12
Critical envelope parameters: minimum resistance and damping angle
Several envelope parameters are especially important for filter design: minimum resistance \(R_{\text{min}}\); maximum resistance \(R_{\text{max}}\); maximum inductive reactance \(X_{L,\text{max}}\); maximum capacitive reactance \(X_{C,\text{max}}\); minimum and maximum damping angle; maximum impedance magnitude; the frequency band; and the tolerance applied to that band. Two of these — minimum resistance and damping angle — frequently dominate the result.
Why minimum resistance matters
Minimum resistance controls damping. A lower \(R_{\text{min}}\) generally means a higher resonant peak and larger harmonic-voltage amplification. Conceptually, at a parallel resonance the peak impedance rises as the series loss in the resonant loop falls:
approximate impedance magnitude at the parallel resonance
\(R_{\text{series}}\)
effective series loss resistance in the resonant loop (network plus filter-branch losses)
\(L,\ C\)
effective inductance and capacitance of the resonant loop
\(\propto\)
“is proportional to”
A conceptual relationship, not a design formula. The inverse dependence holds for the series loss in the loop; if the damping is instead modelled as a shunt resistance across the tank, the peak is directly proportional to that resistance. The design-critical quantity is the maximum \(|Z|\) at the parallel resonance that a low \(R_{\text{min}}\) produces — not \(R_{\text{min}}\) itself. This relationship is only used to explain the effect of damping; it must not be used as a substitute for a frequency-domain harmonic scan or a detailed filter-rating calculation.
Low \(R_{\text{min}}\) is especially important for single-tuned filters, high-Q filters, capacitor banks, lightly damped network resonances, pre-existing distortion, filter-rating calculations and converter stations with strict harmonic limits. A small reduction in \(R_{\text{min}}\) can produce a significant increase in filter current and harmonic-voltage distortion, so it should be chosen from the scatter plot, study assumptions and measurement evidence where available, with a reasonable allowance for uncertainty — not set arbitrarily.
Maximum impedance magnitude is not always the design-driving parameter. For filter rating and resonance amplification, the minimum-resistance boundary can be more critical because it controls damping; a high impedance with reasonable damping may be less severe than a lower-resistance resonance close to the filter tuning frequency.
Why the damping angle matters
The impedance angle \(\theta=\tan^{-1}(X/R)\) indicates the balance between resistance and reactance. A high positive angle means a strongly inductive, weakly damped network; a high negative angle means a strongly capacitive, weakly damped network. If the envelope applies one conservative limiting angle across all harmonics, the calculated distortion can be much higher than necessary — particularly at low orders, where the real impedance may be better damped than a broad envelope suggests. An angle near \(90^\circ\) represents a lightly damped condition that can substantially increase the calculated distortion.
In plain terms, the damping angle is not a separate physical component — it describes how resistive or reactive the equivalent network impedance is at a harmonic frequency. An angle close to \(\pm 90^\circ\) means the impedance is mainly reactive and weakly damped; a smaller angle means the resistance contribution is larger and the resonance is more damped.
Design consequence
The most important envelope outputs are not only maximum impedance magnitude. \(R_{\text{min}}\) and the damping angle can be equally important, and sometimes more important, for sharply tuned or lightly damped designs. If \(R_{\text{min}}\) is set too low or the limiting angle too close to \(90^\circ\), the filter may be oversized; if they are too optimistic, the filter may be under-rated.
Section 13
Constructing a harmonic impedance envelope
A practical construction process runs as follows:
Define the study point — the point of connection, converter busbar, filter busbar, substation busbar, offshore-platform busbar or industrial connection point.
Define the harmonic frequency range from equipment, standards, project requirements and possible resonances.
Define credible network cases — normal operation, planned outages, contingencies, load levels, generation dispatch, reactive-compensation states and future configurations.
Run frequency scans for each case at a sufficiently fine step.
Plot the calculated impedance points in the R–X plane.
Identify resonances, rapid transitions, low-resistance regions, high-impedance regions and isolated clusters.
Select harmonic bands based on the actual impedance behaviour.
Apply appropriate frequency tolerance at the band edges.
Define the envelope shape for each band.
Check whether the envelope includes unrealistic impedance areas.
Identify the critical boundary points for performance and rating.
Document all assumptions, network cases, tolerance values and excluded conditions.
Recommended study inputs
The quality of the envelope depends on the quality of the input data. A robust study should include:
line data — positive- and zero-sequence parameters, capacitance, frequency dependence where required, mutual coupling and overhead/cable distinction;
cable data — capacitance, sheath bonding, screen return path and frequency-dependent resistance;
transformer data — leakage reactance, winding connection, tap range, earthing, impedance tolerance and stray capacitance where needed;
load data — harmonic-frequency damping representation;
existing filters and capacitor banks — tuning frequency, component tolerances, losses, damping resistors, switching status and detuning range;
power-electronic equipment — represented to suit the study objective (harmonic current/voltage source for passive frequency-domain studies; detailed impedance/admittance models for stability and interaction studies).
Takeaway
The main deliverable is not only an R–X plot. It is a defensible set of frequency-band envelopes, traceable to the studied network conditions, that can be used directly for performance and rating studies.
Section 14
Load damping and frequency-dependent modelling
Load damping is one of the most important uncertainties in harmonic impedance studies. Loads provide resistance at harmonic frequencies and so reduce resonance peaks, but the amount of damping depends on the load type — motor, resistive, rectifier, power-electronic and lightly loaded feeders all behave differently. If load damping is ignored, the study becomes overly conservative; if it is overestimated, the study becomes non-conservative.
Crucially, load does not only damp — it can also detune. Rotating machines present significant reactance (locked-rotor / sub-transient inductance) that shifts the parallel-resonance frequency as well as adding some damping, so a motor-heavy feeder can move a resonance onto a harmonic rather than simply suppressing it. Power-electronic and constant-power loads can present low, or even non-conservative, damping at harmonic frequencies. So “loads add resistance and reduce peaks” is an over-simplification: the truly conservative case is the load scenario with the least net damping, which is not necessarily the minimum-MW case.
For planning studies, credible minimum-load (and minimum-damping) conditions should usually be considered, because damping is often lowest when the network is lightly loaded — and that is often when resonance peaks are highest. Load models should be tested with sensitivity analysis wherever the result is important to the filter design.
Frequency dependence
At harmonic frequencies, frequency dependence can be significant: overhead-line series resistance increases with frequency through skin effect and earth-return (Carson) effects; cable series resistance likewise increases with frequency while its internal inductance decreases modestly; transformer behaviour is not accurately represented by fundamental-frequency impedance alone; load damping changes with frequency; shunt capacitive susceptance scales with frequency, so capacitance dominates behaviour at higher orders and must be represented; and filters and capacitor banks are strongly frequency-dependent. A model adequate for load-flow or short-circuit studies may not be adequate for harmonic resonance studies.
Takeaway
Load damping can decide whether a resonance is severe or acceptable, and frequency dependence can move and sharpen resonances. Both should be treated as explicit study assumptions, not hidden defaults.
Section 15
Measuring the network harmonic impedance
Network harmonic impedance can be measured as well as calculated. Measurement is valuable, but it has important limitations, and the literature (notably CIGRÉ WG B4.47) is consistent on how it should be used: as a way to confirm and improve the calculated impedance, not as a substitute for it.
Why measurement alone cannot define the envelope
A measurement is generally only relevant for the particular operating condition(s) present at the time it is taken. It is not feasible to extrapolate a measurement from one operating condition to another, nor to predict how the impedance will change with load pattern, outages or future network development. An impedance envelope, by contrast, must span many credible conditions — so measurement informs and validates the envelope, but the envelope itself still has to be built from calculation across the full range of network states.
Pre-existing distortion and accuracy
For several measurement techniques, accuracy is greatest at the harmonic orders where the magnitude of pre-existing distortion is low. Many of these techniques also assume that the pre-existing distortion stays constant for the duration of the measurement, which in reality may not always hold. Both effects limit confidence at exactly the low orders that often dominate filter design, so the operating-state context of any measurement must be recorded alongside the result.
Measurement versus calculation — what correlates, and what does not
Comparisons generally show good correlation in where the network resonances occur and in the reactive component of impedance, but poorer correlation in the resistive (damping) component — measured results typically show more damping than calculated. This points to limitations in how the frequency variation of resistance is modelled for the various network elements, and it reinforces why the minimum-resistance and damping-angle assumptions (Section 12) deserve careful, evidence-based treatment.
Measurement techniques
Several families of technique are used in practice, each with its own intrusion, accuracy and frequency-coverage trade-offs:
Monitor the network response to energisation / de-energisation of mechanically switched capacitors, shunt reactors or transformers and derive impedance by algorithm.
Minimal intrusion — suitable transducers often already exist, but must be calibrated for accuracy at harmonic frequencies.
Large disturbing-load distortion
Measure the voltage (and current) distortion produced by a large pulsed AC-DC converter load.
Accurate at the load’s characteristic harmonic orders; less reliable at low-order non-characteristic harmonics where the disturbing harmonic is small or pre-existing distortion is significant; limited to the node of connection.
Interharmonic noise injection
Inject a noise signal into the control loop of a large disturbing load (converter or SVC) to raise the interharmonic spectrum, where pre-existing distortion is low, then interpolate to the integer harmonics.
Pre-existing distortion is usually very low between harmonics; autocorrelation and large sample counts allow very low, essentially imperceptible injection levels.
Dedicated injection device
Inject a disturbance from a small electronically commutated single- or three-phase load coupled to the HV system through a coupling capacitor.
Reliable where pre-existing distortion is low and there is no adjacent strong harmonic source; drawback is the intrusive coupling-capacitor connection into the HV system.
Measured impedance is a snapshot of the system condition at the time of measurement. It can support model validation, but it cannot replace the envelope study unless all relevant operating conditions, outages, load levels, generation patterns and future network states are represented. Measurement should therefore be treated as validation evidence, not as the only design basis for long-term filter specification.
Takeaway
Where it is practicable, measurement of network harmonic impedance should be encouraged — but viewed as a means to confirm the calculated impedance, not to replace it. Measurement is much needed to validate, and potentially improve, the harmonic network models. Any available measurement results should be taken into account by the customer when choosing the network impedance loci and envelopes to be used for AC harmonic filter design.
Section 16
Existing filters, capacitor banks and pre-existing distortion
Existing filters and capacitor banks can dominate the harmonic impedance at certain frequencies. If a nearby filter is accurately known, it may reduce uncertainty in the impedance around its tuning frequency, and a smaller frequency tolerance may then be justified in that region. If the filter condition is uncertain, or the filter may be switched out, the study should include the relevant switching states. The same applies to shunt capacitor banks — their switching status moves resonant frequencies and changes network impedance significantly. The study should not assume all filters or capacitor banks are always in service unless the operating philosophy guarantees it.
Pre-existing harmonic distortion
Harmonic performance studies often need to consider both new harmonic emissions and pre-existing background distortion, which may be represented as a background harmonic voltage at the connection point or behind an equivalent network impedance. The impedance envelope governs how this background distortion is amplified or attenuated at the filter busbar. For aggregate-distortion studies the critical impedance may occur away from the origin of the R–X plane — not only near \(R_{\text{min}}\) — so the envelope definition should also consider maximum impedance magnitude, \(R_{\text{max}}\), \(X_{L,\text{max}}\), \(X_{C,\text{max}}\), resonant points away from the origin, and the phase relationship between harmonic sources where relevant.
Pre-existing distortion should not be ignored, especially in networks with existing converters, industrial loads, electric-arc furnaces, railway supplies, offshore networks or multiple renewable plants.
Takeaway
Filters and capacitor banks are not passive background details — their switching state can dominate the envelope. And because pre-existing distortion can make a different part of the envelope critical, the envelope must support both new-emission and aggregate-distortion assessment.
Section 17
Filter rating and detuning
Harmonic performance and filter rating are related but not the same. Performance concerns voltage-distortion limits at the PCC or other specified buses; rating concerns the thermal and electrical duty of filter components. Filter-rating checks should consider harmonic current through capacitors and reactors, harmonic voltage across capacitors, resistor and reactor thermal duty, capacitor RMS current and peak voltage, filter detuning, fundamental-frequency voltage variation, pre-existing distortion, background harmonic current, multiple harmonic sources, and both performance and rating network conditions.
Detuning and tolerance
AC filters are affected by component tolerance and ageing. The tuning frequency follows \(f_{tune}=1/\!\left(2\pi\sqrt{LC}\right)\), so a fractional detuning of about \(-\tfrac12\!\left(\Delta L/L+\Delta C/C\right)\) arises from capacitor manufacturing tolerance, temperature and ageing, from reactor inductance tolerance, and from system-frequency variation. For a tuned filter, even small detuning can significantly change harmonic current and busbar voltage distortion — especially when the network minimum resistance is low. Single-tuned filters are therefore often deliberately tuned slightly below the target harmonic (typically near \(h-0.05\)), so that as the filter detunes and ages the network-side parallel resonance is kept away from the harmonic being filtered. Filter detuning should be assessed together with the network impedance envelope and the pre-existing-distortion assumptions, not in isolation.
Takeaway
Filter tuning is not fixed for the life of the project. Detuning and network-impedance uncertainty must be assessed together, because their effects combine at exactly the lightly damped conditions that drive filter current.
Section 18
Customer and contractor responsibilities
For projects involving AC filters, HVDC, FACTS, large renewables, BESS or major industrial harmonic sources, the customer (system operator or developer) should normally provide the network-impedance specification. The customer is best placed to define credible network conditions, future developments, operating restrictions, outage conditions and planning assumptions. The contractor or filter designer designs the filters against the specified envelopes and advises if an envelope appears excessive, insufficient, inconsistent or unclear.
If a small part of an envelope drives a large increase in filter cost or complexity, the customer and contractor should jointly review the original network condition that created that part of the envelope. The key question is whether that impedance region is credible and relevant for performance or rating. If it is credible and important, the filter should be designed for it. If it is extremely rare or not relevant for continuous performance, the parties may agree a different treatment — such as rating-only assessment, an operational restriction, or a higher permitted distortion for that exceptional condition. Any such treatment should be agreed and documented.
Table 4 — Customer and contractor outputs.
Party
Output
Customer
Defines the network conditions, planning assumptions, future developments, operating restrictions and impedance envelopes.
Contractor
Uses the specified envelopes to design the filters, check performance, check rating, and identify any envelope region that creates disproportionate cost or complexity.
Takeaway
Envelope definition is not only a calculation task. It is also a specification and risk-allocation task between customer and contractor.
Section 19
What a good specification should include
Ambiguity in the impedance specification leads to large differences in filter design. A clear harmonic-impedance specification should state:
Specification checklist
point of connection or busbar where impedance is calculated;
fundamental frequency and harmonic frequency range;
frequency-scan resolution;
network operating conditions, contingencies and outage assumptions;
load levels and generation-dispatch assumptions;
reactive-compensation switching states and existing filter / capacitor-bank states;
future-network developments included;
performance conditions and rating conditions;
harmonic-band definitions and the frequency tolerance applied to each band;
envelope shapes and parameters — \(R_{\text{min}}\), \(R_{\text{max}}\), \(X_{L,\text{max}}\), \(X_{C,\text{max}}\) and limiting impedance angles;
whether pre-existing distortion is included;
whether values are in ohms or per unit, with base voltage and base power;
whether positive-sequence, phase-domain or sequence-domain impedance is used;
whether envelopes are for performance, rating or both;
required design margin and reporting format.
Suggested methodology for project studies
A practical end-to-end methodology: define the study scope and harmonic limits; identify the PCC and filter busbars; collect present and future network data; select credible operating conditions; build frequency-dependent network models; run frequency scans at adequate resolution; generate R–X scatter plots per condition; identify major resonances, minor loops and rapid transitions; group orders into justified bands; apply edge tolerances; define envelopes per band; separate performance and rating envelopes where required; check for unrealistic regions; perform performance calculations using envelope boundaries; perform rating calculations using rating conditions; review the conditions that drive filter size, damping or cost; refine the envelope where justified by evidence; and document assumptions, limitations and responsibilities.
Suggested report wording
Suggested report narrative
“The AC network harmonic impedance was assessed using frequency-domain scans over the specified harmonic range, at the point of connection, for credible variations in topology, loading, generation dispatch, reactive compensation and outage conditions. Because the impedance varies significantly with both frequency and network condition — changing between inductive and capacitive and exhibiting series and parallel resonances at non-integer frequencies — it was assessed using a quasi-continuous frequency scan rather than integer harmonic points only. The resulting points were represented in the R–X plane and grouped into harmonic-frequency bands; for each band an envelope was defined to include the credible scatter while avoiding unnecessary inclusion of non-realistic regions, with frequency tolerance applied at the band edges. Separate envelopes were used for performance and rating where the applicable network configurations differ. Critical parameters — minimum resistance, damping angle, maximum impedance magnitude and the inductive/capacitive boundaries — were reviewed because they significantly influence harmonic-voltage distortion and filter current. The envelopes were used as the basis for harmonic performance and AC-filter rating, with the filter design checked against the relevant boundaries including detuning, pre-existing distortion and component tolerances.”
Suggested specification wording
Suggested specification clause
“The Contractor shall design the AC harmonic filters and demonstrate harmonic performance using the AC network harmonic impedance envelopes provided by the Customer. The envelopes are defined in the R–X plane at the specified point of connection and apply over the stated harmonic frequency bands. The Contractor shall consider the full boundary of each relevant envelope when assessing harmonic-voltage distortion, filter current, component rating and performance compliance, and shall not replace the specified envelopes with a single short-circuit impedance or with impedance values calculated only at integer harmonic orders. The study shall account for resonance between harmonic orders, frequency tolerance, filter detuning, pre-existing distortion, and the distinction between performance and rating conditions. Where the Contractor identifies that a small part of an envelope has a disproportionate effect on filter cost, rating or complexity, the Contractor shall notify the Customer with supporting evidence; any modification, exclusion or reclassification of an envelope region shall be subject to Customer approval and documented in the study report.”
Section 20
Relationship with IEC, IEEE and CIGRÉ guidance
Harmonic impedance, emission limits and filter design sit across several standards and guidance documents. For project work the applicable basis should be stated clearly, because IEC, IEEE, CIGRÉ and utility specifications may all be relevant but should not be mixed without care over which document controls the harmonic limits, the impedance modelling and the acceptance criteria.
The principal references are:
IEC TR 61000-3-6 (Edition 2.0, 2008) — assessment of emission limits for connecting distorting installations to MV, HV and EHV systems. It provides a coordination framework, via compatibility and planning levels and emission allocation, rather than mandatory limits.
IEEE Std 519-2022 — voltage and current distortion limits and the separate responsibilities of the system owner/operator and the user, applied at the point of common coupling. The 2022 edition is a full Standard, superseding the 2014 Recommended Practice; the companion application guide is IEEE Std 519.1-2015.
CIGRÉ Technical Brochures — TB 766 (Network Modelling for Harmonic Studies, JWG C4/B4.38, 2019) is the canonical reference for network and component modelling, scatter plots and impedance-envelope construction; TB 553 (Special Aspects of AC Filter Design for HVDC Systems, WG B4.47, 2013) and the foundational TB 139 (1999) cover AC network harmonic impedance, pre-existing harmonics and AC-DC interaction for HVDC AC filters.
IEC TR 62001 series — guidance to the specification and design evaluation of AC filters for HVDC systems: Part 1 Overview (2016), Part 2 Performance (2016), Part 3 Modelling (IEC TR 62001-3:2016) and Part 4 Equipment (2021). Part 3 addresses AC network impedance modelling, pre-existing harmonics, harmonic interaction and filter-performance simulation. The series is a Technical Report set, scoped to conventional AC filters and line-commutated (LCC) HVDC; for VSC HVDC harmonic performance the better references are the relevant CIGRÉ B4/C4 brochures and the project specification.
Table 5 — Reference standards and guidance for harmonic impedance and AC filter studies.
Document
What It Covers
IEC TR 61000-3-6 (Ed. 2.0, 2008)
Harmonic emission/voltage coordination on MV/HV/EHV via compatibility and planning levels and emission allocation — guidance, not mandatory limits.
IEC TR 61000-3-13 (2008)
Companion to 61000-3-6 for the unbalance (negative-sequence) coordination side, where unbalance is in scope.
IEC 61000-4-7 (Ed. 2.0, 2002 + A1:2008)
Reference harmonic/interharmonic measurement method underpinning the characterisation of pre-existing harmonics.
IEEE Std 519-2022 (+ 519.1-2015 guide)
Voltage and current distortion limits and owner/user responsibilities at the PCC; the 2022 edition is a Standard, superseding 519-2014.
CIGRÉ TB 766 (JWG C4/B4.38, 2019)
Network modelling for harmonic studies — impedance-envelope construction, scatter plots and component models. Canonical envelope reference.
CIGRÉ TB 553 (2013) / TB 139 (1999)
AC network harmonic impedance, pre-existing harmonics, AC-DC interaction and AC filter design for HVDC.
IEC TR 62001-1/-2/-3/-4 (2016–2021)
HVDC AC-filter guidance — overview, performance, modelling (Part 3) and equipment; scoped to conventional AC filters and LCC HVDC.
ENA EREC G5/5 (2020) / national grid code
Project-specific harmonic planning and emission limits for GB connections, or the equivalent national code in other jurisdictions.
Which document controls?
IEC, IEEE, CIGRÉ and the applicable grid code may all be relevant on one project, but they serve different roles: a grid code or IEC TR 61000-3-6 / IEEE 519 framework usually sets the limits; CIGRÉ TB 766 and the IEC TR 62001 series inform the impedance modelling and envelope practice; and IEC 61000-4-7 governs the measurement of any harmonics used to validate the study. The specification should name the single controlling document for limits and avoid blending criteria from different sources without stating the hierarchy. Where a document type is known it should be cited (Technical Report, Technical Specification or Standard) rather than a bare number — for instance the IEC TR 62001 series, not “IEC 62001” generically. Where a project has a grid code, connection agreement or employer’s requirement, that project document normally defines the controlling limits, compliance points, assessment conditions and reporting requirements; IEC, IEEE and CIGRÉ guidance should then be used consistently within that project hierarchy.
Takeaway
Standards and guidance documents should be applied consistently. The project should state which document controls harmonic limits, impedance modelling and acceptance criteria, and should not blend differing IEC, IEEE, CIGRÉ and utility criteria without explaining the hierarchy.
Section 21
Practical checks, main outputs and common mistakes
An impedance envelope should not be accepted merely because it contains all the points. Before accepting an envelope for design, check that it is realistic, traceable and suitable for both performance and rating.
Checks before accepting an envelope
Does it include all credible operating conditions, relevant outages and contingencies, and future development where required?
Was the frequency step small enough to capture resonances, and were non-integer resonances considered?
Were frequency tolerances applied at the band edges?
Are the bands too wide at low orders or too narrow at high orders?
Does the envelope include large unrealistic regions, and are separated clusters better represented by more than one envelope?
Are \(R_{\text{min}}\) and the damping angle technically justified?
Are performance and rating conditions separated correctly?
Are pre-existing harmonic voltages considered, and are existing filters and capacitor banks represented correctly?
Are load-damping assumptions credible, and are the results sensitive to modelling uncertainty?
Are all per-unit bases stated, and are the final results traceable to the original network conditions?
Main outputs of the study
The study should provide: R–X scatter plots for all relevant conditions; frequency scans at a small enough step to find non-integer resonances; harmonic bands chosen from the actual impedance behaviour; separate envelopes per band where required; a clear distinction between performance and rating envelopes; defined boundaries for \(R_{\text{min}}\), \(R_{\text{max}}\), inductive and capacitive reactance, damping angle and maximum impedance magnitude; the frequency tolerance at each band edge; identification of the critical boundaries that drive distortion or rating; identification of rare conditions needing separate treatment; a clear statement of assumptions and exclusions; and the final envelopes for the filter designer.
Quality-control checks
The final envelope is checked against the original scatter plots.
Each boundary is traceable to a harmonic band and a network condition.
No envelope includes a large unrealistic region without justification.
The frequency tolerance is stated explicitly.
The performance/rating classification is stated explicitly.
Any excluded operating condition is documented.
The per-unit base is stated wherever per-unit impedance is used.
The final plots and data tables are reproducible from the study model.
Common mistakes
Using only integer harmonics in the frequency scan and missing non-integer resonances.
Using one large envelope for all harmonic orders, creating an unnecessarily conservative and expensive design.
Applying a fixed \(\pm 1\) harmonic tolerance at all frequencies — too severe at low order.
Ignoring the data immediately outside a band, when a resonance just outside can shift in.
Assuming the worst condition is always inside the envelope, when it usually lies on the boundary.
Focusing only on maximum impedance magnitude and ignoring minimum resistance or damping angle.
Assuming performance and rating envelopes are the same.
Ignoring pre-existing harmonic distortion.
Overestimating load damping, or using simplified load models without sensitivity studies.
Using a single network condition for a project that will operate under many topologies.
Excluding rare but credible outage conditions without documenting why — or including extremely rare, irrelevant conditions in the continuous performance envelope.
Using per-unit values without clearly stating base voltage and base power.
Mixing IEC, IEEE, CIGRÉ and utility criteria without stating which document controls the specification.
Final takeaway
A harmonic impedance envelope is a design basis. It controls the calculated harmonic distortion, filter current, component rating, damping requirement and cost. The envelope must include credible network conditions and reasonable uncertainty, but it must not include unrealistic impedance regions that drive artificial over-design. The final output should be a traceable set of frequency-band-specific envelopes for performance and rating studies.
Final deliverable
The final deliverable should be a data package, not only a report: envelope plots, boundary coordinates, frequency bands, tolerance assumptions, performance/rating classification and the list of network cases used to create each envelope.
Two companion notes on the harmonic impedance of an AC network. The first builds the impedance loci and envelope in the R-X plane and explains the envelope types; the second — this note — turns that into a specification and AC-filter design basis. Together they cover the theory, construction, design drivers and specification.
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02Part TwoReading now
Network Harmonic Impedance Envelopes
The practical, specification-focused note — frequency-band envelopes and tolerance, minimum resistance and damping angle as design drivers, performance versus rating, measurement, and specification wording.
