Harmonic Studies & Modelling

Transformer Modelling for Harmonic Studies

A transformer is mainly inductive across the harmonic range, so it interacts with capacitive cables, capacitor banks and filters to form resonance — and its losses set the damping at that resonance. This page covers frequency-dependent leakage reactance and resistance; winding connections, vector group and zero-sequence paths; tap position; high-frequency terminal capacitances; the magnetising branch and saturation; and transformer harmonic heating through the K-factor, Factor-K and the harmonic loss factor.

Reading time ≈ 26 min · Part Four of the series

Power transformers are important in harmonic studies because they influence both harmonic propagation and network resonance. A transformer is mainly inductive over the normal harmonic frequency range, so it interacts with capacitive elements such as cables, capacitor banks and filters. Depending on the network configuration and frequency, this interaction can create parallel resonance or series resonance.

The governing relationship is the same as for any network component, and because a transformer affects \(Z_h\), an inaccurate transformer model can lead to incorrect prediction of harmonic voltage distortion, resonance amplification, filter loading or equipment duty.

Key idea

Leakage reactance scales roughly with order (\(X_T(h)\approx hX_{T1}\)); resistance needs a frequency-dependent model because it sets damping.
Winding connection and vector group drive harmonic phase shift, cancellation, triplen circulation and zero-sequence paths.
High-frequency capacitances matter only above the normal harmonic range (roughly above 4 kHz).
For transformer heating, harmonic eddy losses scale with \(I_h^2h^2\) — captured by the K-factor, Factor-K and \(F_{HL}\).

Key terms used on this page

01Leakage reactance, \(X_L\)

The inductive reactance associated with leakage flux between transformer windings.

02Frequency-dependent resistance, \(R(h)\)

Winding resistance adjusted for harmonic frequency, including skin, proximity and eddy-current effects.

03Damping

The loss mechanism that limits the height of harmonic resonance peaks.

04Vector group

The winding connection and phase-shift arrangement, such as Dyn11 or YNd1.

05Zero-sequence path

The path available for zero-sequence currents, depending on winding connection and grounding.

06Triplen harmonics

Harmonics that are multiples of three — the 3rd, 9th, 15th and so on.

07Magnetising branch

The transformer branch representing core magnetisation (\(X_M\) and \(R_{Fe}\)).

08Saturation

Nonlinear core behaviour when the magnetic flux is high.

09K-factor / Factor-K / \(F_{HL}\)

Indices used to assess transformer heating under harmonic current loading.

Section 1

Why transformer modelling matters

A transformer is not only a voltage-changing device in harmonic studies. Its leakage inductance can combine with cable capacitance, capacitor banks or filters to create resonance, while its winding and stray losses provide damping. So transformer modelling affects both the location of resonance and the height of the resonance peak.

At fundamental frequency, a transformer is commonly represented using leakage impedance, winding resistance, a magnetising branch, tap position and winding connection — usually sufficient for load-flow and short-circuit studies. The harmonic voltage is still governed by:

\[ V_h=Z_h\,I_h \]

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

Notation: harmonic order \(h\) versus sequence index

On this page the subscript \(h\) means harmonic order — so \(I_h\) is the current at harmonic order \(h\). The subscripts 0, 1 and 2 mean sequence components: \(I_0\) (zero-sequence), \(I_1\) (positive-sequence) and \(I_2\) (negative-sequence) current, with \(V_0,\ V_1,\ V_2\) the matching voltages. Where both are needed, they are combined — for example \(I_{1,h}\) means the positive-sequence current at harmonic order \(h\). Note that a bare \(I_1\) can mean either “fundamental current” or “positive-sequence current” depending on context, so this page states which is meant wherever it is used.

For harmonic studies, additional issues become important: frequency-dependent resistance and leakage reactance, winding connection and vector group, the zero-sequence path, tap-changer position, losses and damping, harmonic heating, and high-frequency capacitances where relevant. The model must be chosen for the purpose: a simple series impedance may suffice for normal harmonic penetration, while a more detailed model is needed where high-frequency response, terminal overvoltages, resonance, saturation or harmonic heating matter.

Section 2

Basic transformer representation

For most harmonic studies, a transformer may be represented by a series impedance between its two terminals:

\[ Z_T(h)=R_T(h)+jX_T(h) \]

\(Z_T(h)\): transformer impedance at harmonic order \(h\)
\(R_T(h)\): transformer resistance at harmonic order \(h\)
\(X_T(h)\): transformer leakage reactance at harmonic order \(h\)
\(h\): harmonic order
\(j\): imaginary operator

The leakage reactance is normally assumed to increase approximately linearly with harmonic order, which is generally acceptable over the frequency range used for harmonic studies:

\[ X_T(h)=h\,X_{T1} \]

\(X_T(h)\): leakage reactance at harmonic order \(h\)
\(X_{T1}\): leakage reactance at fundamental frequency
\(h\): harmonic order

This is a simplified inductive scaling. It is normally acceptable for many frequency-domain harmonic studies, but detailed models may require manufacturer-specific frequency-dependent impedance.

The resistance is more difficult. Transformer resistance increases with frequency because of skin effect, proximity effect and stray losses, and it is important because it provides damping at resonance. Both quantities rise with order — \(h\uparrow \Rightarrow R_T(h)\uparrow\) and \(h\uparrow \Rightarrow X_T(h)\uparrow\) — but \(X_T(h)\) is often treated as approximately linear, while \(R_T(h)\) requires more careful modelling.

Section 3

Transformer resistance and harmonic damping

Transformer losses are not only an equipment-rating issue — they also affect network harmonic damping. At a parallel resonance, the harmonic impedance peak depends strongly on the available damping, and transformer winding resistance and stray losses can reduce its height: higher harmonic resistance → more damping → lower resonance peak.

If transformer resistance is underestimated, the calculated peak may be too high, leading to unnecessary filtering or excessive mitigation. If it is overestimated, the calculated peak may be too low, hiding a real harmonic risk. The resistance model should therefore be selected carefully. The reactance largely determines where the resonance occurs; the resistance determines how high the peak becomes:

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

\(Z_{peak}\): approximate impedance magnitude at resonance
\(R_{\text{damping}}\): effective damping resistance in the network
\(\propto\): “is proportional to”

This is a conceptual relationship, not a detailed design formula. It is included to show why transformer resistance and loss modelling can change the harmonic result.

Practical message

Transformer harmonic resistance controls resonance damping — especially where the harmonic impedance at the studied node is dominated by transformer impedance.

Section 4

Frequency-dependent transformer impedance

In harmonic analysis software, frequency-dependent transformer impedance may be defined for positive and zero sequence. The positive-sequence short-circuit impedance is used for balanced harmonic propagation; the zero-sequence impedance is used where zero-sequence or unbalanced propagation is relevant:

\[ Z_1(h)=R_1(h)+jX_1(h) \qquad\qquad Z_0(h)=R_0(h)+jX_0(h) \]

\(Z_1(h)\): positive-sequence transformer impedance at order \(h\)
\(Z_0(h)\): zero-sequence transformer impedance at order \(h\)

In many practical studies \(X_1(h)\approx hX_1(1)\) and \(X_0(h)\approx hX_0(1)\), but the resistance should be represented using a suitable frequency-dependent characteristic.

Manufacturer and measurement-based data

The best transformer harmonic model is normally based on manufacturer-provided frequency-dependent data or a measured frequency response. Useful data includes \(R(f)\), \(X(f)\), \(L/R(f)\), \(Z_1(f)\), \(Z_0(f)\) and, where relevant, terminal-capacitance equivalents. Factory acceptance test data — especially the short-circuit impedance and X/R ratio — can be very useful. Where frequency-dependent data is not available, recognised empirical models may be used, but sensitivity studies should be performed because different models can produce significantly different damping at resonance.

Order of preference

Use measured or manufacturer data first; use empirical models only when measured data is unavailable; and perform sensitivity checks where resonance damping is important.

Impedance input data

For harmonic network studies, the required transformer data may include the positive-sequence short-circuit impedance \(Z_1(h)\), the zero-sequence short-circuit impedance \(Z_0(h)\) and the magnetising admittance \(Y_m(h)\). Frequency-dependent impedance may be entered as relative values or as absolute per-unit values, depending on the software and data source. If values already exist in a test report, take care to avoid overwriting or double-scaling. As a rule, define frequency-dependent impedance relative to rated values when it is linked to type data, unless the software explicitly requires absolute values.

Section 5

High-frequency capacitances

For normal harmonic studies, transformer stray capacitances are often neglected, because their effect is usually small within the lower harmonic range used for most power-quality assessments. At higher frequencies, additional capacitances become important. These represent equivalent terminal capacitances and winding-to-earth or winding-to-winding effects; they do not reproduce every physical winding capacitance, but they reproduce the external frequency response seen at the transformer terminals:

\[ C_{W(HV)-E}\,,\qquad C_{W(LV)-E}\,,\qquad C_{W(HV)-W(LV)} \]

\(C_{W(HV)-E}\): capacitance between the high-voltage winding and earth
\(C_{W(LV)-E}\): capacitance between the low-voltage winding and earth
\(C_{W(HV)-W(LV)}\): capacitance between the high-voltage and low-voltage windings

These capacitances may improve the external terminal response at higher frequencies, but they do not represent detailed internal winding voltage stress.

Transformer high-frequency equivalent circuit. (a) positive-sequence: HV and LV series resistance and leakage reactance, a magnetising branch of X_M and R_Fe, an ideal w1:w2 transformer, with winding-to-earth capacitances C_W(HV)-E and C_W(LV)-E and a winding-to-winding capacitance C_W(HV)-W(LV),1. (b) zero-sequence: similar series elements with X_M0 and R_M0 and an earthing branch 3 X_E,HV and 3 R_E,HV to earth, and capacitance C_W(HV)-W(LV),0. — **Figure 1 —** Conceptual high-frequency transformer model including winding-to-earth and winding-to-winding equivalent capacitances: **(a)** positive-sequence and **(b)** zero-sequence systems. The capacitances improve the calculated terminal frequency response but do not represent internal winding voltage stress.

The purpose of this model is to improve the calculated frequency response at the transformer terminals; it is useful when high-frequency terminal behaviour is important. It does not calculate internal voltage stress inside the winding — that requires a specialised winding model from the manufacturer. In short, the external capacitance model gives terminal voltage and current response, but not internal winding stress.

When transformer capacitances should be considered

Capacitances are normally not required for low-order harmonic studies, such as assessment up to the 50th harmonic in many 50 Hz systems. A practical threshold is around \(f\approx 4\ \text{kHz}\); above this range, transformer capacitances may begin to influence the frequency response, depending on transformer size, design and voltage level.

Table 1 — Study conditions where transformer capacitance may matter.

Study Condition	Why Capacitance May Matter
High-frequency resonance	Capacitances affect resonance frequency
Fast-front or high-frequency power quality	Terminal response depends on capacitance
Converter switching-frequency effects	Higher-frequency components may be present
Transformer terminal overvoltage studies	Capacitance influences voltage distribution
EMI or supraharmonic studies	Frequency range may exceed normal harmonics

For most conventional harmonic compliance studies, the capacitances may be ignored; for high-frequency power-quality studies they should be considered. The equivalent network also depends on the sequence system and winding connection: for positive-sequence studies it represents the external coupling between HV terminal, LV terminal and earth; for zero-sequence studies it must also reflect neutral grounding, winding connection and the available zero-sequence paths (Figure 1b). The zero-sequence model is especially important for triplen harmonics, grounded-neutral currents, zero-sequence voltage propagation, unbalanced sources and earth-referenced terminal voltages. Because the winding connection changes the equivalent capacitance network, transformer capacitances cannot be applied blindly — delta or star arrangements must be considered.

Section 6

Magnetising branch

The magnetising branch normally consists of \(X_M\) (magnetising reactance) and \(R_{Fe}\) (core loss). In normal harmonic studies it is often ignored, because the core is assumed to operate in its linear region during steady state, and the model reduces to \(Z_T(h)=R_T(h)+jX_T(h)\). If transformer saturation is important, however, the magnetising branch cannot be ignored.

When is the magnetising branch required?

Usually not required for normal linear harmonic penetration studies.
Required for transformer energisation and inrush studies.
Required for ferroresonance studies.
Required when overvoltage or saturation may distort the magnetising current.
Required for voltage-transformer studies where core saturation may dominate behaviour.

Table 2 — Conditions where the nonlinear magnetising branch matters.

Condition	Relevance
Elevated voltage operation	Core may move closer to saturation
Transformer energisation	Inrush and saturation produce harmonics
Ferroresonance studies	Nonlinear magnetising branch is essential
Voltage transformer studies	Saturation may dominate behaviour
Long-duration overvoltage	Magnetising current may become distorted

If saturation is the main concern, a time-domain model or a nonlinear magnetising characteristic should be used. For frequency-domain studies, saturation may sometimes be approximated using a harmonic current source derived from the flux-current curve at the operating voltage. In short: ignore the magnetising branch for normal linear harmonic studies, but include nonlinear magnetisation for saturation and ferroresonance studies.

Section 7

Transformer winding connections

Winding connections are very important in harmonic studies. The vector group affects the phase shift of harmonic voltages and currents, harmonic cancellation, zero-sequence current paths, triplen harmonic propagation and neutral current.

The vector group determines two things in harmonic studies. First, it determines the phase shift between the two sides of the transformer, which can cause harmonic currents from different sources to cancel or add. Second, it determines whether zero-sequence and triplen harmonic currents can pass through the transformer or instead circulate within a delta winding.

A delta winding can trap triplen harmonic currents under balanced conditions. Triplen harmonics — orders \(h=3,\ 9,\ 15,\ \ldots\) — are zero-sequence harmonics in balanced three-phase systems. They are important because they can add in the neutral conductor or circulate inside delta windings rather than cancelling between phases. The practical rules are: a delta winding → circulating triplen currents can be trapped; a grounded star winding → zero-sequence current may flow through the neutral. This can lead to neutral-current heating or protection operation if not represented correctly.

Table 3 — Effect of transformer winding connection on harmonic propagation.

Winding Connection	Harmonic Modelling Relevance
Delta winding	Can trap balanced triplen harmonic currents as circulating currents
Grounded star winding	Can allow zero-sequence current to flow through the neutral
Ungrounded star winding	Usually blocks the zero-sequence current path
Zig-zag winding	May provide a zero-sequence path depending on grounding and design
Phase-shifting transformer	Can shift harmonic phase angles and affect cancellation

Section 8

Phase shift and harmonic order

Transformer phase shift depends on vector group and harmonic order. A 30-degree fundamental phase shift does not have the same effect at every harmonic. If the transformer has a fundamental phase shift \(\theta\), the harmonic phase shift at order \(h\) is \(h\theta\).

This is the basis of multi-pulse converter cancellation. Combining star and delta secondary windings creates a 30-degree phase shift, letting two six-pulse rectifiers behave as a twelve-pulse arrangement and cancelling some characteristic harmonics. For a six-pulse converter the characteristic harmonics are \(h=6n\pm1\); for a twelve-pulse arrangement the dominant characteristic harmonics become \(h=12n\pm1\). The 5th and 7th harmonics can therefore be significantly reduced compared with a six-pulse arrangement.

Practical message

Winding connection is part of the harmonic mitigation mechanism — not only a transformer construction detail.

Section 9

Tap changer position

In plain terms, the tap position changes the effective turns ratio and can also change the impedance seen from each side of the transformer. In harmonic studies this changes how harmonic voltage and current transfer between voltage levels, so the tap position used in the model should match the study operating case — or sensitivity cases should be tested.

Tap changer position affects the transformer turns ratio and leakage reactance. The turns ratio affects the transfer of harmonic voltages and currents between voltage levels, with the transfer impedance following the square of the turns ratio. Tap position can therefore influence the resonance frequency, harmonic transfer between voltage levels, damping between networks, filter performance and the compliance margin.

Practical recommendation

Include the tap range and tap-position sensitivity in harmonic studies — especially where the result is close to the applicable limit or where resonance is near an important harmonic order.

Section 10

Transformer impedance models

Several models may be used to represent frequency-dependent impedance. The most common use a series impedance with frequency-dependent resistance and approximately linear reactance, \(Z_T(h)=R(h)+jX(h)\) with \(X(h)\approx hX_1\). The main difference is how \(R(h)\) is calculated, and because different models produce different damping, the model choice can significantly affect the impedance peak at resonance.

Table 4 — Common transformer resistance models.

Model Type	Main Concept	Practical Use
Constant resistance	\(R(h)=R_1\)	Simple; may underestimate high-frequency losses
Square-root resistance	\(R(h)=R_1\sqrt{h}\)	Simple approximation
IEEE-type correction	\(R(h)=R_{dc}(1+Ah^B)\)	Includes skin-effect-type increase
CIGRE / Electra-type	Empirical resistance functions	More representative for some transformers
Manufacturer model	Measured frequency-dependent data	Preferred where available

The most important point is not the name of the model, but whether it gives a realistic value of transformer damping over the frequency range of interest.

Section 11

Transformer losses and harmonic heating

Transformers experience additional losses when supplying nonlinear loads — ohmic winding losses, winding eddy-current losses, stray magnetic losses and core losses. Harmonic currents increase RMS current and may increase losses significantly. Winding eddy-current losses are especially important because they increase approximately with the square of frequency. For harmonic order \(h\):

\[ P_{e,h}=P_{e,1}\,I_h^2\,h^2 \qquad\qquad P_{e,total}=P_{e,1}\sum_{h=1}^{h_{max}} I_h^2\,h^2 \]

\(P_{e,h}\): eddy-current loss component associated with harmonic order \(h\)
\(P_{e,1}\): rated winding eddy-current loss at fundamental frequency
\(I_h\): RMS current at harmonic order \(h\) (as a fraction of rated or total RMS current, per the chosen method)
\(h\): harmonic order

This shows why high-order harmonic currents may contribute significantly to transformer heating even when their current magnitude is small.

Because of the \(h^2\) weighting, even a small high-order current can produce a significant eddy-current loss.

Heating indices in plain language

K-factor — an index used to express the heating effect of harmonic currents in transformers, weighting higher harmonic orders more heavily. Factor-K — a transformer rating concept used in some standards and specifications to indicate suitability for nonlinear loads. Harmonic loss factor, \(F_{HL}\) — a factor used to estimate the increase in winding eddy-current losses due to harmonic currents.

These indices describe heating duty. They are not the same as harmonic voltage distortion or harmonic compliance limits.

K-factor

The K-factor represents the heating effect of harmonic currents on winding eddy-current losses. It is defined as the harmonic current spectrum weighted by \(h^2\):

\[ K=\sum_{h=1}^{h_{max}} I_h^2\,h^2 \]

where \(I_h\) is expressed in per unit of total RMS current (per the adopted convention). The K-factor is the ratio of eddy-current losses under nonsinusoidal load current to those under sinusoidal current of the same RMS value; a higher K-factor means the transformer must tolerate greater harmonic heating. A K-rated transformer is designed to withstand this additional heating without exceeding its thermal limits.

Factor-K

Factor-K is another method, commonly associated with European practice. It considers the harmonic spectrum, total RMS current and a winding-dependent exponent, effectively derating transformer capability according to the harmonic heating contribution. A general form uses a frequency exponent \(h^q\), where \(q\) depends on winding construction — typically \(q=1.7\) for round or rectangular conductors and \(q=1.5\) for foil-type LV windings. The exact value should be obtained from the manufacturer; Factor-K requires manufacturer-dependent winding information and should not be applied blindly.

Harmonic loss factor F_HL

The harmonic loss factor links the harmonic current spectrum to the additional winding eddy-current losses:

\[ F_{HL}=\frac{\displaystyle\sum_{h=1}^{h_{max}}\left(\frac{I_h}{I_1}\right)^2 h^2}{\displaystyle\sum_{h=1}^{h_{max}}\left(\frac{I_h}{I_1}\right)^2} \]

\(I_1\): fundamental current (here \(I_1\) denotes the fundamental, not the positive sequence)
\(I_h\): harmonic current at order \(h\)

The numerator is the harmonic spectrum weighted by \(h^2\); the denominator normalises by the total current spectrum. \(F_{HL}\) acts as an eddy-current loss multiplier due to the harmonic spectrum.

K-factor, Factor-K and F_HL compared

All three describe transformer duty under harmonic current, but they are not identical and not interchangeable without checking definitions.

Table 5 — Harmonic-heating quantities compared.

Quantity	Main Purpose	Typical Use
K-factor	Select transformer for harmonic heating duty	Common in North America
Factor-K	Derate / select transformer under harmonic loading	Common in European practice
F_HL	Evaluate harmonic loss impact	IEEE-style transformer loading assessment

A report should clearly state which factor is used and how the harmonic current spectrum was defined. Calculating transformer harmonic duty typically requires the data below.

Table 6 — Input data for transformer harmonic-duty assessment.

Data	Purpose
Harmonic current spectrum	Determines harmonic heating
Fundamental current	Reference for harmonic ratios
Total RMS current	Determines loading
Rated copper losses	Base winding loss
Eddy-current loss ratio	Determines sensitivity to harmonics
Winding construction	Needed for the Factor-K exponent
Transformer rating	Thermal capability
Cooling mode	Affects permissible loading
Ambient temperature	Affects thermal margin
Existing load profile	Determines sustained heating

For software calculation, a typical required input is the ratio of winding eddy-current losses to copper losses. If manufacturer data is unavailable a default assumption may be used, but this introduces uncertainty — use manufacturer loss data wherever possible, especially for transformers supplying large nonlinear loads.

Section 12

Effect of the transformer model on network impedance

The selected transformer model may not affect every part of the network equally. If the studied node is electrically far from large transformers, transformer damping may have little impact on the harmonic impedance. If the node is close to transformers, or if the resonance is formed by transformer inductance and cable or capacitor capacitance, the transformer model can have a major effect: a node dominated by lines/cables → minor impact; a node dominated by transformer resonance → major impact.

Different resistance models can produce very different resonance-peak magnitudes. If the leakage inductance is assumed constant, the resonance frequency may stay similar while the peak amplitude changes — a useful diagnostic: same resonance frequency but different peak magnitude → the damping model is controlling the result.

Two-winding and three-winding transformers

Many harmonic transformer models are developed for two-winding transformers and can often be extended to three-winding units — but carefully. Three-winding transformers may have tertiary-winding effects, different leakage paths, additional harmonic circulation, different zero-sequence behaviour and higher harmonic losses. A delta tertiary may trap triplen harmonics and provide a path for circulating zero-sequence harmonic currents. Do not blindly apply a two-winding model to a three-winding transformer without checking winding connections, leakage impedance distribution and zero-sequence paths.

Section 13

Workflow and sensitivity

A practical workflow for transformer modelling in harmonic studies is as follows:

Define the study objective — resonance screening, harmonic penetration, filter design, transformer heating, high-frequency terminal response, or saturation/ferroresonance.
Select the model complexity — \(Z_T(h)=R(h)+jX(h)\) for normal penetration; add equivalent capacitances for high-frequency terminal response; use a nonlinear magnetising characteristic or time-domain model for saturation.
Define the winding connection and vector group explicitly.
Include tap position and tap-range sensitivity.
Define frequency-dependent resistance and reactance.
Include zero-sequence impedance where relevant.
Calculate harmonic-heating indices (K-factor, Factor-K or \(F_{HL}\)) if transformer duty is being assessed.
Perform sensitivity studies where manufacturer data is uncertain.

Table 7 — Relative sensitivity of harmonic results to transformer modelling choices.

Parameter	Typical Impact	Comment
Frequency-dependent resistance	High near resonance	Controls damping and peak magnitude
Leakage reactance	High	Controls resonance location
Tap position	Medium to high	Changes leakage reactance and transfer impedance
Vector group	High	Affects phase shift and harmonic transfer
Delta winding	High for triplens	Can trap zero-sequence harmonics
Zero-sequence impedance	High for unbalanced / triplen	Must match winding and grounding
Magnetising branch	Low normally, high in saturation	Needed for ferroresonance and overvoltage
Stray capacitance	Low below normal harmonic range	Important at higher frequencies
Eddy-current loss ratio	High for heating assessment	Needed for K-factor and \(F_{HL}\)

In short: for resonance studies, focus on \(R(h)\), \(X(h)\), tap position and connections; for heating studies, focus on the harmonic current spectrum and eddy-current losses; for high-frequency studies, include the capacitances.

Minimum data for a defensible transformer model

A transformer harmonic model should state:

transformer rating and voltage ratio;
winding connection and vector group;
tap position or tap range used in the study;
leakage impedance at rated frequency;
resistance and loss data;
frequency-dependent resistance assumption;
reactance scaling assumption;
zero-sequence impedance and grounding arrangement;
whether the magnetising branch is included or neglected;
whether saturation is represented;
terminal-capacitance assumptions for high-frequency studies;
harmonic-heating data, if duty is assessed;
frequency range and highest harmonic order considered.

Section 14

Reporting and summary

A harmonic study report should clearly state how transformers were represented. A weak statement — “transformers were included in the model” — tells the reader nothing.

Examples of a clear modelling statement

“Transformers were represented using frequency-dependent series impedance, with leakage reactance scaled linearly with harmonic order and resistance modelled using a frequency-dependent loss characteristic.”

“Winding connections, grounding and zero-sequence impedance paths were explicitly represented.”

“Equivalent winding-to-earth and winding-to-winding capacitances were included to reproduce transformer terminal frequency response.”

“Transformer harmonic loading was assessed using the harmonic current spectrum and winding eddy-current loss factor.”

Common modelling mistakes

Using only the 50 Hz impedance without considering frequency dependency.
Ignoring transformer resistance when resonance damping is important.
Modelling the transformer as a simple inductor in a compliance study.
Ignoring vector group and phase shift when harmonic cancellation is relevant.
Ignoring delta-winding behaviour for triplen harmonics.
Ignoring zero-sequence paths in unbalanced studies.
Using the wrong tap position.
Applying K-factor or \(F_{HL}\) without defining the current spectrum and base current.
Adding capacitances without stating the frequency range where they are valid.

To summarise: transformers contribute inductance, losses, phase shift, zero-sequence paths and damping, and can interact with capacitive elements to create resonance. For most conventional studies a transformer can be represented by \(Z_T(h)=R_T(h)+jX_T(h)\) with \(X_T(h)\approx hX_T(1)\) and frequency-dependent resistance \(R_T(h)>R_T(1)\) — the resistance model being important because it controls damping at resonance. Winding connections must be represented because they affect harmonic phase shift, cancellation, triplen circulation and zero-sequence paths, and tap position should be considered because it affects leakage reactance and harmonic transfer between voltage levels.

For high-frequency studies, equivalent capacitances \(C_{W(HV)-E}\), \(C_{W(LV)-E}\) and \(C_{W(HV)-W(LV)}\) may be added to improve the external terminal frequency response — though they do not represent internal winding stress. For harmonic duty, the main concern is additional heating, with winding eddy-current losses increasing approximately as \(I_h^2 h^2\); useful indices are the K-factor, Factor-K and \(F_{HL}\).

In short: transformer modelling affects harmonic propagation, resonance frequency, damping and equipment heating. Leakage reactance mainly influences where resonance occurs, while frequency-dependent resistance and losses influence how high the resonance peaks become. For normal steady-state studies the transformer is usually represented by frequency-dependent leakage impedance; for unbalanced or triplen studies, winding connection, grounding and zero-sequence paths must also be represented; for saturation, inrush or ferroresonance studies, a nonlinear magnetising branch or time-domain model may be required. A clear report should then state:

the transformer model type;
the impedance assumptions;
the resistance and damping assumptions;
the vector group;
the tap position;
the zero-sequence path;
the capacitance treatment;
whether harmonic heating has been assessed.

Key message

Use frequency-dependent transformer impedance for resonance and compliance studies; use winding-connection and zero-sequence modelling for unbalanced and triplen harmonics; use harmonic loss factors for heating assessment; and add equivalent capacitances only when the frequency range makes them relevant. A robust study should state the transformer model type, frequency-dependent resistance, reactance scaling, winding connection, tap position, zero-sequence path, capacitance treatment and harmonic-heating assessment — only then can transformer effects on propagation, resonance, damping and equipment duty be interpreted correctly.

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