Power Quality & Harmonics

Harmonic Impedance Loci and Envelopes

The harmonic impedance seen from a point of common coupling is not a single number — it moves with outages, dispatch, load, switching and future network changes. Harmonic impedance loci and envelopes capture that range in the R-X plane, so that a new converter, HVDC link, wind or PV plant or filter can be assessed against every credible network condition rather than one base-case scan. This article explains frequency scans and the R-X locus, the envelope types (circle, sector, polygon and convex hull), frequency bands and resolution, and how an impedance-loci analysis colours each R-X point by harmonic-distortion-limit utilisation to find the critical, worst-case impedance for compliance and filter design.

Reading time ≈ 26 min · APS technical note

Harmonic impedance loci and envelopes represent the range of possible network harmonic impedances seen from a point of assessment, normally the Point of Common Coupling (PCC). They are especially important when assessing new harmonic-emitting plant — HVDC converters, wind farms, PV plants, battery energy storage systems, STATCOMs, SVCs, large variable-speed drives and industrial nonlinear loads.

The main reason for using impedance envelopes is that the network harmonic impedance is not fixed. It changes with operating condition, circuit outages, generation dispatch, load level, capacitor and reactor switching, transformer tap position, network reconfiguration and future system changes. A harmonic study should therefore not rely on only one frequency scan from one network condition: one network condition gives one impedance locus, but many credible network conditions give an impedance envelope.

Key idea

Network harmonic impedance varies with operating state — capture the range as an envelope in the R-X plane, not one scan.
A polygon or convex-hull envelope is usually more realistic than a circle or sector, which include unrealistic empty areas.
Split the frequency range into orders or bands, with fine enough resolution to catch resonances (\(\Delta f\leq f_0/10\)).
Test many R-X points and report the worst case — the critical impedance and the HD-limit utilisation, not just a base case.

Key terms used on this page

01PCC

Point of Common Coupling — the point where harmonic compliance is normally assessed.

02Harmonic impedance, \(Z_h\)

The network impedance seen at harmonic order \(h\).

03R-X plane

A graph with resistance \(R\) on the horizontal axis and reactance \(X\) on the vertical axis.

04Impedance locus

The path followed by harmonic impedance points for one network operating condition.

05Impedance loci

A collection of impedance paths from several network operating conditions.

06Impedance envelope

A boundary drawn around credible impedance points for a harmonic order or frequency band.

07Frequency scan

A calculation of network impedance versus frequency.

08Convex hull

The smallest convex polygon that encloses a set of points.

09HD-limit utilisation

The percentage of the harmonic-distortion limit used by a calculated result.

Section 1

Why harmonic impedance envelopes are used

The harmonic voltage at a bus is governed by the same relationship as for any component:

\[ V_h=Z_h\,I_h \]

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

For a voltage-source converter the relationship may be expressed differently, but the same principle applies: the converter harmonic source interacts with the external network impedance. So the harmonic impedance at the PCC strongly affects the calculated distortion, filter design, equipment duty, background harmonic amplification, connection compliance and the risk of resonance. If the impedance is underestimated, the study may underestimate distortion; if it is overestimated, it may produce unnecessarily onerous mitigation. The purpose of impedance envelopes is to define a realistic but sufficiently conservative range of system impedance values for harmonic assessment.

Section 2

Frequency scan and impedance locus

A frequency scan calculates the network impedance seen from a selected bus as frequency changes. For each frequency the network admittance matrix is solved, a unit harmonic current is injected at the study bus, and the resulting voltage gives the driving-point impedance:

\[ [Y_h][V_h]=[I_h] \qquad\Rightarrow\qquad Z_h=\frac{V_h}{I_h}=R_h+jX_h \]

\(R_h\): resistance component at order \(h\)
\(X_h\): reactance component at order \(h\)

In a conventional frequency-scan plot the magnitude \(|Z_h|\) is plotted against frequency, showing resonance peaks and valleys clearly. In an impedance-locus plot the same complex impedance is plotted in the R-X plane — \(R_h\) on the horizontal axis, \(X_h\) on the vertical axis — and as frequency changes the impedance point traces a path: the harmonic impedance locus. So a frequency scan gives impedance magnitude versus frequency, while an impedance locus gives the complex impedance path in the R-X plane. Both describe the same behaviour, but are useful for different purposes.

Section 3

Interpretation of the R-X plane

Each harmonic impedance point is a complex number plotted in the R-X plane:

\[ Z_h=R_h+jX_h \]

\(Z_h\): harmonic impedance at order \(h\)
\(R_h\): resistance component at order \(h\)
\(X_h\): reactance component at order \(h\)
\(j\): imaginary operator
\(h\): harmonic order, where \(f_h=hf_1\) (\(f_1\) the fundamental, normally 50 Hz)

A positive \(X_h\) normally indicates inductive behaviour, while a negative \(X_h\) normally indicates capacitive behaviour.

Units and sign convention

All impedance points should state the unit — normally ohms or per unit — and the sign convention: \(R\) is the real/resistive part; \(X\) is the imaginary/reactive part; \(X\gt 0\) is inductive and \(X\lt 0\) is capacitive; \(|Z|=\sqrt{R^2+X^2}\) is the magnitude; and \(\theta=\tan^{-1}(X/R)\) is the impedance angle.

\[ |Z_h|=\sqrt{R_h^2+X_h^2} \qquad\qquad \theta_h=\tan^{-1}\!\left(\frac{X_h}{R_h}\right) \]

\(|Z_h|\): magnitude of harmonic impedance at order \(h\)
\(\theta_h\): impedance angle at order \(h\)
\(R_h,\ X_h\): resistance and reactance components at order \(h\)

A harmonic impedance point above the horizontal axis looks inductive at that frequency, below it looks capacitive, and near it is mainly resistive. A crossing near the R-axis may indicate a transition between inductive and capacitive behaviour. This often occurs near a resonance region, but the severity depends on the impedance magnitude, damping and nearby harmonic sources — so the crossing should be treated as an indicator requiring review, not automatically as a compliance failure. A large movement in the R-X plane over a small frequency interval indicates a sensitive resonance region.

Section 4

Series and parallel resonance

In a frequency scan, a parallel resonance normally appears as a peak in impedance magnitude (\(|Z_h|\uparrow\)) and a series resonance as a valley (\(|Z_h|\downarrow\)). In the R-X plane, resonance appears as a loop or sharp movement in the locus, and a major resonance may be associated with a sign change in reactance — \(X_h\gt 0 \rightarrow X_h\lt 0\) or the reverse. The practical message: parallel resonance → high harmonic-voltage risk; series resonance → high harmonic-current risk in a branch. Both matter, but parallel resonance is usually the main concern for voltage distortion at the PCC.

Resonance indication in a frequency scan

\(|Z_h|\uparrow\) indicates a possible parallel-resonance region; \(|Z_h|\downarrow\) indicates a possible series-resonance region, where \(|Z_h|\) is the magnitude of harmonic impedance at order \(h\). Parallel resonance is normally more important for harmonic voltage distortion at the PCC, while series resonance can be important for branch current and component duty.

Section 5

From locus to loci

One frequency scan for one operating condition produces one impedance locus. But a real system operates in many conditions — the intact network, single-circuit and transformer outages, generator dispatch variation, low and high load, capacitor banks in and out of service, reactor switching states and different converter operating points. Each condition gives a different locus, and the collection of these paths is the harmonic impedance loci. So a locus is one operating condition and loci are many; the purpose of creating loci is to understand how much the network harmonic impedance can vary.

Locus or loci?

A locus is the impedance path produced by one network operating condition. Loci are several impedance paths produced by many credible operating conditions. An envelope is the boundary drawn around those credible impedance points.

Section 6

The harmonic impedance envelope

A harmonic impedance envelope is a boundary drawn around the set of possible impedance points in the R-X plane, intended to include all credible impedance values for a selected harmonic order or frequency range. It may be defined for one harmonic order, a narrow band, or a wider range — for example \(h=5\), or \(5\leq h\leq 7\), or \(26\leq h\leq 35\). The envelope is then used by third parties such as manufacturers or developers to assess harmonic emission and filter performance against a credible range of system impedances. In short, the impedance envelope is a usable boundary for harmonic assessment — it converts many operating-case frequency scans into a practical design and compliance input.

Why envelopes are used for connection studies

For a new connection, the developer or manufacturer needs an external AC system impedance to assess distortion. Providing every frequency scan for every condition produces a very large data set; an envelope simplifies this. Instead of thousands of scan points, the system operator provides a boundary in the R-X plane, and the assessing party checks performance for all impedance points inside or on it. The benefits are simpler data exchange, a clear definition of external-network uncertainty, coverage of credible operating states, a usable input for filter design and consistent connection assessment. So the system operator provides the impedance envelope, and the manufacturer or developer designs against it — particularly important for HVDC, wind, PV, BESS, STATCOM and other converter-based connections.

Section 7

Types of impedance envelopes

Different envelope shapes may be used depending on the data quality, intended use and complexity of the network frequency response. The most common are the circle, sector, polygon and convex-hull polygon.

A circular envelope is the simplest, usually defined by a maximum impedance magnitude and a boundary passing through or near the origin. The advantage is simplicity; the disadvantage is that a circle may include a large area of impedance values the network never presents, making the assessment unnecessarily conservative. Acceptable for early screening, but often too conservative for detailed filter design.
A sector envelope limits both magnitude and angle (\(Z_{min}\), \(Z_{max}\), \(\theta_{min}\), \(\theta_{max}\)), reducing unrealistic areas compared with a full circle — simple and more realistic, though it may still include regions not physically produced.
A polygon envelope is defined by a set of vertices \((R_1,X_1),\ldots,(R_n,X_n)\) and can closely follow the calculated points, reducing empty areas and giving a more realistic representation. It needs more effort and points to define, but for detailed compliance and filter design it is often preferred.
A convex-hull polygon tightly encloses all calculated points with the smallest convex boundary — a close fit without manually defining every corner, minimising empty areas, though it may produce many vertices that increase the assessing party’s computation. A simplified enhanced polygon can be a good compromise between accuracy and usability.

Table 1 — Common impedance-envelope types.

Envelope Type	Meaning	Main Risk
Circle	One radius around a reference point	May include unrealistic impedance areas
Sector	Impedance magnitude limited within an angle range	Still may be conservative
Polygon	Boundary drawn around credible impedance points	More realistic but needs careful construction
Convex hull	Smallest convex polygon around the point cloud	Simple and realistic, but may still include empty areas
Multiple envelopes	Separate boundaries for separate operating groups	More accurate, but more data to manage

Empty areas and over-conservatism

An empty area is a region inside the envelope where no calculated impedance point actually exists. Large empty areas make the assessment too conservative, leading to oversized filters, unnecessary damping, higher losses, higher project cost and an unrealistic compliance burden. The envelope should include credible impedance values but avoid large unrealistic areas where practicable — which is why polygon or enhanced-polygon envelopes are often more useful than simple circles.

Section 8

Frequency bands and resolution

A harmonic order can be converted to frequency using \(f_h=hf_1\). For a 50 Hz system, \(h=5\) is 250 Hz and \(h=35\) is 1750 Hz — useful for understanding what the bands physically mean.

A major decision is how to divide the harmonic frequency range into envelopes. A single envelope covering all harmonic orders should generally be avoided, because it can include a very large and unrealistic impedance area. Instead the range should be divided into individual harmonic-order envelopes or frequency-band envelopes. At low orders, narrow envelopes are usually preferred because low-order harmonics are important for compliance and filter design (for example \(h=5\) or \(5\leq h\leq 7\)); at higher orders, wider bands may be acceptable because the network may have many resonances and the impedance may vary rapidly. So low orders → narrow envelopes; higher orders → wider envelopes may be acceptable — though cable-rich networks may have low-order resonances and need narrower or more carefully defined bands.

Frequency resolution

The frequency-scan resolution must be fine enough to capture resonances. If the step is too large, the scan may miss narrow resonances or interharmonic resonance points, giving an envelope that does not include the true maximum impedance. A useful guideline is:

\[ \Delta f \leq \frac{f_0}{10} \]

\(\Delta f\): frequency step used in the frequency scan
\(f_0\): approximate resonance or target frequency

This is a rule of thumb only. Very sharp resonances may require a smaller frequency step.

For a 50 Hz system this gives \(\Delta f\leq 5\) Hz, but in systems with sharp resonances smaller steps such as \(\Delta f = 0.5\) Hz or \(1\) Hz may be needed in sensitive areas. The resolution must be selected based on resonance sharpness, not only calculation convenience.

Overlap between bands

Frequency-band envelopes should normally include some overlap, to account for uncertainty in resonance frequency from component tolerances, system frequency variation, temperature, model uncertainty and future operating changes. At low orders an overlap of about \(\pm 0.5\) harmonic order may be enough if there are no sharp resonances; at higher orders a percentage-based overlap such as \(\pm 10\%\) of the covered range may be more appropriate. The overlap gives margin against frequency uncertainty — important because a resonance may shift from one band into another due to small parameter changes.

Section 9

Phase impedance, critical boundary and distinct groups

In an unbalanced network each phase may have a different harmonic impedance — from untransposed lines, asymmetrical cable layouts, single-phase loads, unequal transformer connections, unbalanced compensation or different phase-conductor geometry. Impedance envelopes should ideally include impedance points from all three phases where phase-domain modelling is relevant, and a single combined envelope may be created to cover all phase impedances. Include all phase impedances where unbalance matters: even a network transposed at fundamental frequency may not be fully balanced at harmonic frequencies.

State the phase basis of the envelope

If phase-domain results are used, the envelope should state whether it includes:

phase A impedance only;
separate phase A, B and C envelopes;
one combined envelope covering all phases;
positive-sequence impedance only;
zero-sequence or negative-sequence impedance where relevant.

Critical boundary of the envelope

The left-hand boundary normally corresponds to lower \(R_h\). Lower resistance means lower damping, and lower damping can increase the impedance peak at resonance. For filter design the low-resistance boundary may therefore be more critical than the maximum \(|Z_h|\) point — so it should be defined carefully: too optimistic and the filter may be under-designed; too conservative and it may be over-designed.

\[ Z_{peak}\propto\frac{1}{R_{\text{damping}}} \]

\(Z_{peak}\): approximate impedance magnitude at resonance
\(R_{\text{damping}}\): effective resistance or loss component providing damping
\(\propto\): “is proportional to”

This is a conceptual relationship, not a detailed design formula. It explains why low-resistance envelope points can be critical.

Multiple envelopes for distinct operating groups

Sometimes different network conditions produce impedance points in separate regions of the R-X plane — one group for the intact network, another for a major outage or different topology. Drawing one envelope around both may include a large empty area between them; in that case it is better to provide two separate envelopes for the same band. So one cloud of points → one envelope; two distinct clouds → two envelopes, improving accuracy and reducing unnecessary conservatism.

For example, if two operating groups create two clearly separated point clouds, one large envelope may include a large unrealistic region between them. In that case separate envelopes should be issued — one for the intact network and one for outage conditions — improving accuracy while still covering credible conditions.

Section 10

Impedance-loci analysis and limit utilisation

Some harmonic analysis tools — such as DIgSILENT PowerFactory — include scripts that test a range of R-X points inside an envelope. The basic concept is: define an external network impedance envelope; generate many R-X points inside it; calculate the harmonic distortion at the PCC for each point; compare each result against the harmonic limits; and identify the worst-case impedance point. This determines not only the maximum harmonic distortion but also where in the R-X plane the limit is approached or exceeded. Each investigated point can then be coloured by its harmonic-distortion-limit utilisation, as in Figure 1, where points run from blue (large margin) through green, yellow and orange to red (limit exceeded).

Harmonic impedance loci analysis in the R-X plane (both axes in Ohm). A black polygon border encloses a grid of impedance test points, each coloured by harmonic-distortion-limit utilisation: blue for 0% to 50%, green for 50% to 75%, yellow for 75% to 90%, orange for 90% to 100% and red for 100% or above. Lower-resistance points on the left have the largest margin; higher-resistance points on the right exceed the limit. — **Figure 1 —** An impedance-loci analysis: each tested R-X point inside the envelope (the black “border point” polygon) is coloured by HD-limit utilisation — blue (large margin) through to red (limit exceeded). Red points identify the external impedance conditions that violate the harmonic limit.

In practical terms, each R-X point represents one possible external network harmonic impedance. The plant harmonic source is connected to that impedance, the resulting PCC harmonic voltage is calculated, and the result is compared with the applicable limit. Repeating this for many points identifies the worst-case external impedance.

Harmonic limit utilisation

HD-limit utilisation means the calculated harmonic distortion expressed as a percentage of the applicable harmonic-distortion limit. When harmonic limits are included, each R-X point can be assessed against the applicable limit through this index:

\[ U_h=\frac{HD_h}{HD_{limit,h}}\times 100\% \]

\(U_h\): harmonic-distortion limit utilisation at order \(h\)
\(HD_h\): calculated harmonic distortion at order \(h\)
\(HD_{limit,h}\): applicable harmonic distortion limit at order \(h\)

If \(U_h\lt 100\%\), the result is below the limit; if \(U_h\geq 100\%\), the limit is reached or exceeded.

Table 2 — Interpretation of harmonic-limit utilisation.

Utilisation	Interpretation
< 50%	Large margin
50% – 75%	Moderate margin
75% – 90%	Small but acceptable margin
90% – 100%	Very close to limit
≥ 100%	Limit exceeded

Red points on a loci plot indicate impedance conditions that violate the harmonic limit, which helps identify whether a harmonic problem is general or only occurs for a narrow part of the envelope.

Worst-case R-X point

A key result of impedance-loci analysis is the worst-case R-X point. For each harmonic order the calculation should identify the critical resistance \(R_{critical}\), the critical reactance \(X_{critical}\) and the maximum distortion \(HD_{max}\) found within the tested envelope. This tells the engineer which external impedance produces the highest distortion — output as the harmonic order + maximum distortion + critical R-X point — usable for filter design, connection compliance, sensitivity analysis and discussion with the system operator or manufacturer.

Section 11

Number and placement of R-X points

The number of R-X points affects accuracy: too few may miss the true worst-case region, while very many increase calculation time. A practical script may let the user define the number of investigated points — for example 1000 or 5000 — distributed uniformly inside the envelope or along the boundary. So more points → better coverage, but more computation time; for detailed compliance studies the number should be high enough to identify the critical region reliably.

Boundary points and internal points

Depending on the objective, the calculation may test boundary points only, or boundary and internal points. For many distortion problems the worst case occurs on or near the boundary, but this is not guaranteed for all converter-source and filter interactions. Testing internal points gives better confidence, especially where the relationship between impedance and distortion is nonlinear. So boundary testing is efficient, but internal-point testing improves confidence — a robust script should allow sufficient coverage of both.

Section 12

Application to compliance and filter design

For harmonic compliance assessment, the impedance envelope represents the external-network uncertainty and the plant model represents the new installation — converter-based generation, an HVDC station, a STATCOM, an SVC, an industrial drive or a harmonic filter. The calculation combines the plant harmonic source with each external impedance point and checks whether the resulting PCC distortion stays within the limit. The practical question is: does the plant comply for all credible external network impedances? If yes, the connection is robust; if no, mitigation or revised filter design may be required.

\[ V_h=Z_{ext,h}\,I_h \]

\(V_h\): calculated harmonic voltage at the PCC
\(Z_{ext,h}\): external network harmonic impedance represented by an R-X point
\(I_h\): harmonic current injected by the plant at order \(h\)

For converter models with Norton or Thévenin equivalents the calculation is more detailed, but the principle is the same: test plant performance against the credible external-impedance range.

Application to filter design

For filter design, envelopes ensure the filter performs across the full credible range of external impedance. A filter designed for one short-circuit level or one condition may fail when the impedance changes, so it should be tested against minimum and maximum system impedance, different impedance angles, resonant network conditions, component tolerances, filter detuning and operating states. The rule: filter design must be robust against the impedance envelope, not only against the base-case frequency scan.

Future-proofing

An envelope may also include future network uncertainty, because systems are changing through renewables, HVDC links, battery storage, EV charging, large cable circuits, converter-based loads and network reinforcements. Future changes can increase or decrease harmonic impedance — added capacity may reduce impedance, while large cable connections can introduce low-order resonances. Because new plant and filters may operate for decades, margin may be added for future-proofing — but larger envelopes increase filter size, damping, losses and cost, so future-proofing is a risk and cost decision that should be agreed between the parties rather than applied blindly.

Section 13

Creating and using impedance envelopes

A practical workflow for creating harmonic impedance envelopes is as follows:

Define the point of assessment — the PCC or connection bus.
Define the harmonic frequency range — \(h=2\) to \(50\), or wider if converter switching harmonics or supraharmonics are relevant.
Define credible network scenarios — intact network, outages, load levels, generation dispatch, reactive-compensation states, converter modes and future states.
Run frequency scans for each scenario at sufficient resolution.
Plot each result in the R-X plane.
Group the results by harmonic order or frequency band.
Create an appropriate envelope shape.
Check whether the envelope includes excessive empty areas.
Introduce overlap between bands where needed.
Provide the envelope definition in tabular form for use by third parties.

The envelope must be traceable to credible network scenarios.

Using an impedance envelope

A practical workflow for using envelopes in an assessment is: import or define the envelope; define the plant harmonic source model; define the applicable harmonic limits; generate R-X test points inside or along the envelope; calculate harmonic distortion for each point; identify the maximum distortion and corresponding R-X point; colour-code the points by limit utilisation; report whether the limit is met for all envelope points; if limits are exceeded, identify whether the issue is localised or widespread; and revise mitigation or request refinement of the envelope if necessary. The envelope is not just a plot — it is a design and compliance input.

Section 14

Sensitivity priorities

The most important sensitivities for impedance-loci and envelope studies are summarised below.

Table 3 — Relative sensitivity of impedance-envelope studies to modelling choices.

Parameter	Typical Impact	Comment
Network outage assumptions	Very high	Can shift resonance significantly
Frequency resolution	Very high	Coarse steps can miss resonance
Envelope frequency-band width	High	Wide bands may be too conservative
Envelope shape	High	Circles may include unrealistic areas
Phase impedance inclusion	High for unbalanced systems	All phases may need inclusion
Compensation switching states	High	Capacitors and reactors strongly affect resonance
Load damping assumptions	High	Changes resonance peak magnitude
Cable-rich network representation	High	Can create low-order resonances
Future network assumptions	Medium to high	May require future-proofing margin
Number of R-X test points	Medium to high	Affects accuracy of worst-case search

The quality of the impedance envelope controls the quality of the harmonic assessment.

Section 15

Reporting and summary

A harmonic study report should clearly state how the impedance envelope was created or used. A weak statement — “harmonic impedance was considered” — tells the reader nothing.

Examples of a clear statement

“Harmonic impedance envelopes were developed in the R-X plane from frequency scans of credible network operating scenarios, including relevant outages, load levels and reactive-compensation states.”

“The plant harmonic model was tested against all investigated R-X points within the relevant impedance envelope, and the maximum harmonic distortion was reported together with the critical R-X point.”

“Filter performance was checked against the external network impedance envelope, including component tolerance and detuning sensitivity. The envelope represents the network scenarios available at the time of the study; future changes may require it to be reviewed.”

A harmonic impedance envelope report should state

point of assessment, normally the PCC;
frequency range and harmonic orders;
frequency resolution;
network scenarios included;
outage cases included;
generation-dispatch assumptions;
load-level assumptions;
capacitor/reactor/filter states;
future-network assumptions, if included;
whether results are positive-sequence or phase-domain;
envelope type: circle, sector, polygon or convex hull;
whether multiple envelopes are used;
units: ohms or per unit;
sign convention for \(X\);
worst-case R-X point for each assessed harmonic or band;
HD-limit utilisation and margin;
limitations and exclusions.

Common modelling mistakes

Using one base-case frequency scan instead of an envelope.
Drawing a very large circular envelope that includes unrealistic impedance points.
Not including credible outage conditions.
Not including capacitor, reactor or filter switching states.
Using too coarse a frequency step and missing a sharp resonance.
Mixing ohms and per-unit values without clear base values.
Not stating whether the envelope is positive-sequence, zero-sequence or phase-domain.
Combining two separate point clouds into one oversized envelope.
Ignoring the low-resistance boundary for filter design.
Reporting only the worst distortion value without identifying the critical R-X point.

To summarise: harmonic impedance loci and envelopes represent the range of possible harmonic impedances seen from a PCC. A frequency scan gives \(|Z_h|\) versus frequency; an impedance locus gives \(Z_h=R_h+jX_h\) plotted in the R-X plane. One operating condition gives one locus; many conditions give many loci; an envelope is a boundary around the credible points, used for connection compliance, filter design, background-amplification studies, data exchange between system operator and manufacturer, and identification of worst-case distortion. The envelope may be a circle, sector, polygon or convex-hull polygon — polygons usually give the most realistic representation by reducing empty areas. The range should normally be divided into orders or bands (avoid a single all-harmonics envelope), with resolution \(\Delta f\leq f_0/10\) as a starting point and finer where resonances are sharp.

In short: harmonic impedance envelopes convert many operating-case frequency scans into a practical design input. Instead of assessing a new plant against one base-case impedance, the plant is tested against a credible range of external network impedances in the R-X plane. For compliance and filter design, the study should identify the critical R-X point, the maximum harmonic distortion and the HD-limit utilisation; and where different operating groups produce separate point clouds, multiple envelopes may be more accurate than one oversized envelope. A clear envelope should state:

the point of assessment and frequency band;
the network scenarios and frequency resolution;
the phase representation;
the envelope type;
the units and sign convention;
the critical R-X point and maximum distortion;
the HD-limit utilisation.

Key message

Harmonic performance should be checked against the impedance envelope, not only against one base-case frequency scan. When using an envelope, test sufficient R-X points and report the maximum harmonic distortion, the critical R-X point, the limit utilisation and whether any envelope points exceed the limit. A robust study should state the PCC, frequency range, network scenarios, frequency resolution, envelope type, frequency bands, phase treatment, number of R-X points, harmonic limits, critical R-X results and future-proofing assumptions — only then can harmonic distortion, resonance risk and filter design be assessed reliably across the credible range of network conditions.

Part One Reading now

Harmonic Impedance Loci and Envelopes

The foundations — the R–X plane, frequency scan versus impedance locus, series and parallel resonance, from one locus to many, and the envelope types (circle, sector, polygon and convex hull).

Series progress 1 of 2

Part Two

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.