Power System Fundamentals · Training Guide

Resonance and Ferroresonance in Power Systems Part Three — Ferroresonance in VTs, Transformers and Network Models

Reading time ≈ 30 min

Start here

Ferroresonance in voltage transformers, power transformers and network models

Ferroresonance can occur when capacitance interacts with a nonlinear magnetic device such as a voltage transformer, power transformer, or reactor. In power networks, this condition is often linked to switching operations, open breakers, grading capacitors, long lines, cables, weak sources, or lightly loaded transformers.

Unlike normal linear resonance, ferroresonance is difficult to predict because the magnetic core does not behave with a constant inductance. Once the core saturates, the inductance changes, and the circuit can move into a different operating state. This may lead to overvoltages, distorted waveforms, transformer overfluxing, insulation stress, overheating, and equipment failure.

What this page teaches

why VTs are sensitive to ferroresonance;
why CVTs and electromagnetic VTs behave differently;
how grading capacitors and breaker capacitances provide the capacitive path;
why ungrounded or weakly grounded systems increase the risk;
why the transformer magnetising curve must be represented correctly;
how damping affects the final operating state;
why ferroresonance may be fundamental, subharmonic, quasi-periodic or chaotic.

A stability problem, not just a magnitude problem

Ferroresonance is not only an overvoltage-magnitude problem; it is a nonlinear stability problem — the same circuit can settle into several different operating states depending on initial conditions and damping.

Table 1 — Equipment types prone to ferroresonance with their risk mechanisms and modelling needs.

Equipment	Ferroresonance Risk Mechanism	Important Modelling Requirement
Electromagnetic VT	Nonlinear magnetising branch with system capacitance	Saturation curve and core losses
CVT	Capacitive divider and tuning-circuit interactions	Detailed CVT equivalent circuit
Power transformer	Saturating core fed through capacitance or a weak source	Magnetising curve and residual flux
Breaker grading capacitor	Provides the capacitive feed path	Accurate capacitance representation
Ungrounded system	Neutral displacement and weak damping	Zero-sequence network detail

Voltage transformers

Ferroresonance in voltage transformers

Voltage transformers are important because they are directly connected to the network and contain magnetic cores. If the surrounding circuit provides enough capacitance and low damping, a voltage transformer can become part of a ferroresonant circuit.

Two broad VT technologies are commonly considered: capacitive voltage transformers and electromagnetic voltage transformers.

A capacitive voltage transformer, or CVT, includes a capacitor divider and an intermediate electromagnetic transformer. CVTs have their own possible internal ferroresonance mode, involving the capacitor divider and the intermediate transformer. This is why ferroresonance-suppression circuits are normally fitted.

These suppression circuits are designed to damp abnormal oscillations. They may include tuned filters, saturable reactors, or surge suppressors. Because of this built-in suppression, CVTs are generally less prone to external network-driven ferroresonance than electromagnetic VTs, but they should not be considered immune.

Electromagnetic voltage transformers are more sensitive to network ferroresonance because their magnetic core can saturate. When they are energised through grading capacitors, stray capacitance, cable capacitance, or other weak capacitive paths, they may form a nonlinear LC circuit.

Open breakers

Voltage transformers and circuit-breaker grading capacitors

One important ferroresonance mechanism involves open EHV circuit breakers with grading capacitors.

A grading capacitor is connected across the open breaker contacts to distribute voltage more evenly. Although the capacitance is small, it can still allow a small current to pass through the open breaker. This means that equipment on the apparently isolated side of the breaker may still receive energy.

If a VT is connected on that isolated side, the grading capacitor can excite the VT. The VT magnetising inductance is nonlinear, and the grading capacitor provides the capacitance needed for ferroresonance. Stray capacitance and distributed capacitance in the surrounding substation can also contribute.

This is why open-breaker capacitance is important. A circuit may look isolated on the single-line diagram, but electrically it may still be weakly energised through grading capacitors.

Substation configurations

Busbar and line voltage transformers

Busbar voltage transformers

Busbar VTs can be exposed to ferroresonance when a busbar is isolated but connected to feeders whose circuit breakers have grading capacitors.

In this arrangement, the isolated busbar is not directly connected to the source, but it can still be energised through the grading capacitors across the open feeder breakers. If a VT is connected to the busbar, the grading capacitors and the VT magnetising branch can form a nonlinear resonant circuit.

The equivalent circuit can be understood as a nonlinear inductor, represented by the VT core, connected with distributed capacitance from grading capacitors and stray capacitances.

The practical lesson is simple: isolated busbars with connected electromagnetic VTs should not automatically be treated as electrically inactive if nearby breakers include grading capacitors.

Line voltage transformers

Line VTs can also be exposed to ferroresonance during line de-energisation or incomplete isolation.

A typical risk condition can occur when the line circuit breaker is open and fitted with grading capacitors, the bus disconnector remains closed, and the line disconnector is open. In this condition, the VT may be trapped between switching devices and may be energised through the breaker grading capacitor.

This type of arrangement can allow ferroresonance to develop if the VT core saturates and the surrounding capacitance provides the required energy exchange path.

This risk is particularly relevant during line de-energisation sequences, switching operations, commissioning, or abnormal isolation arrangements.

System configurations

Double-circuit configurations and ungrounded neutral systems

Voltage transformers in double-circuit configurations

In a double-circuit overhead line, one circuit may be out of service while the other circuit remains energised. Even if the out-of-service circuit is open, it may still be capacitively coupled to the energised circuit.

This coupling can excite a VT connected to the out-of-service circuit. If the VT is electromagnetic and the circuit has low damping, ferroresonance may occur.

The important mechanism is capacitive coupling between the two circuits. The energised circuit supplies energy through mutual capacitance, while the VT provides the nonlinear inductance.

This condition links directly with the resonance behaviour discussed for multiple-circuit rights of way. The same coupling that can produce line resonance can also supply energy to a VT ferroresonant circuit.

Voltage transformers in ungrounded neutral systems

Ferroresonance is also common in medium-voltage distribution systems with ungrounded or isolated neutrals.

In these systems, VTs may be connected from phase to ground. Because the system neutral is not directly grounded, switching transients, earth faults, or core saturation can cause neutral displacement. The displaced neutral can create abnormal phase-to-ground voltages across the VTs.

This condition may look similar to a single-line-to-ground fault, but the mechanism can be ferroresonance rather than a simple fault.

The risk is influenced by the VT saturation characteristic, network capacitance, damping, and grounding arrangement. VTs with a lower saturation point may be more vulnerable because they enter the nonlinear region more easily.

HV/MV transformer capacitance coupling

Ferroresonance may also occur through capacitance coupling between transformer windings and the surrounding system.

One risk condition can occur when the HV side is grounded at a remote point, the MV side has VTs installed, and there is little or no load connected to the MV side. During an HV-side earth fault, the HV neutral potential can rise. This disturbance can be transferred to the MV side through transformer capacitances.

The coupling path can include transformer capacitance, zero-sequence capacitance, and saturated VTs. If the MV-side VTs enter saturation, a ferroresonant condition may be sustained even after the original HV fault is cleared.

This condition is important because the MV side may appear lightly energised or weakly connected, but capacitive coupling can still provide enough energy to excite ferroresonance.

Power transformers

Ferroresonance in power transformers

Transformer-terminated transmission lines

A transformer-terminated transmission line can experience ferroresonance when the line is out of service while a nearby parallel circuit remains energised.

The transformer provides nonlinear magnetising inductance. The line-to-ground capacitance and inter-circuit capacitance provide the capacitance. The energy required to sustain the oscillation can be supplied through capacitive coupling from the energised parallel circuit.

This condition may occur even when circuit breakers exist on both sides of the transformer if uneven tripping, incomplete switching, or breaker failure leaves part of the circuit in an abnormal condition.

Line length is important. Very short lines may not provide enough capacitance to support resonance. Medium-length lines may support fundamental-frequency ferroresonance. Longer lines may support sub-harmonic ferroresonance.

The possible consequences include transformer overfluxing, insulation stress, and accelerated ageing.

Transformer connected to a series-compensated line

A transformer connected to a series-compensated line can also be exposed to ferroresonance.

The risk may appear during load rejection or when a low-voltage circuit breaker opens. In this condition, the transformer inductance can interact with the series capacitor.

The resulting series ferroresonance may cause dielectric stress, overheating, and equipment damage.

This condition is different from shunt-compensated line resonance because the capacitance is intentionally installed in series with the line. However, the same general principle applies: capacitance and nonlinear inductance can exchange energy in a way that produces abnormal voltage and current.

Energisation risk

Lightly loaded and partially energised transformers

Lightly loaded transformer energised through a cable or long line

A ferroresonance condition can occur when a transformer is energised through a long cable or overhead line from a weak source.

A weak source has low short-circuit capacity. This means the source impedance is relatively high, and the system may not provide strong damping. If the transformer is lightly loaded, there may also be little load loss to suppress oscillations.

The transformer magnetising inductance and the cable or line capacitance can then interact. If the transformer core saturates, the circuit may enter a ferroresonant state.

This condition is important during black-start restoration, remote transformer energisation, weak-grid operation, and long radial feeder energisation.

Single-phase or two-phase transformer energisation

Ferroresonance can also occur when a transformer is energised on only one or two phases.

This is more common in distribution networks where fuses, single-phase switching, or incomplete switching may leave one or two phases energised while the remaining phase is open.

The mechanism can involve series resonance between transformer limb inductance and capacitance from the line, capacitor banks, or stray winding capacitance. If at least one phase remains energised, sufficient capacitance exists between the transformer and the open point, and the loading or losses are low, ferroresonance may develop.

The result can be significant overvoltage, often in the range of several per unit, depending on the circuit configuration and transformer saturation behaviour.

This condition is important because the transformer may not look fully energised, but the partial energisation can create a dangerous nonlinear resonance.

Analysis approaches

Modelling and studying ferroresonance

Ferroresonance is nonlinear and highly sensitive to initial conditions, system topology, component nonlinearities, switching instant, trapped flux, and damping.

This means that a single simulation or a single hand calculation is rarely enough. The same network may behave normally in one switching condition and enter ferroresonance in another.

The main purpose of a ferroresonance study is to identify whether dangerous operating states are possible, what conditions trigger them, how severe the overvoltages or overcurrents may be, and what damping or mitigation is required.

Analytical solution methods

Analytical methods can be useful for understanding ferroresonance, especially for periodic solutions.

Galerkin and harmonic balance methods represent network equations using Fourier series. These methods are suitable for periodic ferroresonance and fundamental-frequency solutions. However, they cannot reliably detect chaotic or aperiodic behaviour.

Pseudo-arc-length continuation extends harmonic balance methods and can include sub-harmonic solutions. However, it still has limitations for chaotic or pseudo-periodic ferroresonance.

Incremental describing function methods are based on nonlinear control theory. They can be useful for predicting critical jump values in VT ferroresonance, but they require accurate equipment data.

Energy-transfer approaches can also be used to estimate whether switching events provide enough energy to trigger ferroresonance in VTs.

These analytical approaches are valuable for insight, but they are normally not sufficient for final engineering assessment of complex substations or transmission networks.

Digital simulation methods

Digital simulation is widely used for ferroresonance studies because it can represent nonlinear components and time-domain switching behaviour.

EMT-type programs are particularly useful because they can model periodic, sub-harmonic, quasi-periodic, and chaotic waveforms. They can also include transformer saturation, breaker operation, trapped charge, grading capacitors, line capacitance, and damping.

Nonlinear dynamics tools can also be used to understand ferroresonance behaviour. These include Lyapunov exponents, bifurcation diagrams, phase-space trajectories, and Poincaré maps.

The main challenge is sensitivity. Ferroresonance depends strongly on initial conditions. This means that many simulation runs may be required to identify the range of switching conditions that can lead to ferroresonance.

Phase space, Poincaré sections and bifurcation diagrams

Some mathematical tools are useful for understanding ferroresonance behaviour.

Phase space represents the system state as a trajectory. A periodic oscillation appears as a closed loop. A chaotic oscillation appears as a more complex pattern that does not close in the same way.

A Poincaré section samples the phase trajectory at fixed time intervals. This can simplify complex time-domain behaviour into a pattern of discrete points.

A bifurcation diagram shows how the system behaviour changes when a parameter is varied. It can reveal where the system moves from normal operation to periodic ferroresonance, sub-harmonic ferroresonance, or chaotic behaviour.

These tools are mainly used for detailed analysis and research, but the engineering message is simple: ferroresonance can change suddenly when parameters or initial conditions change.

The advanced terms in plain language

Poincaré map — a way of checking whether the waveform repeats regularly or moves between different states.
Bifurcation — a sudden change from one operating pattern to another when a parameter changes.
Chaotic response — an irregular nonlinear oscillation that does not settle into a simple repeating waveform.

Model scope and lines

Network extent and line modelling

Network extent for ferroresonance studies

For many ferroresonance studies, it is not necessary to model the entire power system in detail.

A minimal network model may be sufficient if it correctly represents the ferroresonant loop, the source strength, the relevant capacitances, the nonlinear magnetic element, and the damping.

The external network can often be represented by a Thevenin equivalent at the point of connection. However, the local circuit around the ferroresonant loop must be represented carefully.

The aim is not to build the largest model. The aim is to build the right model for the mechanism being studied.

Overhead line modelling

Accurate capacitance modelling is important for ferroresonance studies involving overhead lines.

The line geometry, tower structure, conductor spacing, height above ground, phase arrangement, and transpositions influence the capacitance matrix. These details can affect the coupling between phases and between adjacent circuits.

Phase transpositions should be explicitly represented when they are relevant to the study. If the line is not transposed, the model should preserve the asymmetry.

For many power-frequency ferroresonance studies, a Bergeron line model may be sufficient because it can represent multi-phase distributed parameters at the frequency of interest.

Frequency-dependent models, such as the J. Marti line model, may be needed when the switching transient itself initiates the ferroresonance or when higher-frequency behaviour is important.

Corona effects are often neglected in ferroresonance studies, but if they are neglected, suitable engineering margin may be required.

Passivity of line models

The passivity of line models is important in EMT studies.

A non-passive model can artificially create energy in the simulation. This may produce false resonances or exaggerate the severity of an oscillation.

In practical terms, the line model should not behave as an artificial energy source. The real part of the terminal admittance should remain non-negative over the frequency range relevant to the study. If the model violates this behaviour, the simulated resonance may be partly created by the model rather than by the physical network.

For ferroresonance studies, this is especially important because the phenomenon itself is sensitive to energy balance. A modelling error that injects energy can create a false ferroresonant response.

Component representation

Transformer, reactor and substation equipment modelling

Transformer modelling

Transformer modelling is one of the most important parts of a ferroresonance study.

For power transformers, three-phase models should normally be used because phase coupling can affect the response. For voltage transformers, single-phase models may be sufficient, but the external winding connection and grounding arrangement should be represented correctly.

The nonlinear magnetic core is the most critical part of the model. Saturation and losses should be represented with care.

When detailed data is missing, a single-valued nonlinear inductor with an appropriate damping resistor may be used as a practical approximation. More advanced hysteresis models, such as Preisach-type models, can be more accurate but are rarely used in routine studies because the required data is often unavailable.

The saturation curve should be connected in the part of the transformer model that best represents the physical core location, usually in parallel with the winding closest to the core.

Core losses may be represented in different ways. A detailed hysteresis loop can be accurate but requires special test data. A nonlinear resistor may be limited because real core losses depend on flux rather than only voltage. A calibrated linear resistor is commonly used when it is matched to known core losses.

The magnetising curve is the main nonlinear element

The magnetising curve is not a detail — it is the main nonlinear element. Generic transformer models may not be sufficient for ferroresonance studies if the magnetising curve, residual flux, core losses or damping path is not represented correctly.

Shunt reactor modelling

Shunt reactors must also be modelled carefully when they are part of the resonant loop.

Different core types have different magnetic coupling behaviour. Single-phase reactors have no interphase magnetic coupling. Shell-type, four-leg, or five-leg reactors usually have minimal coupling. Three-leg core reactors can have strong interphase coupling and may need a full three-phase representation.

Many shunt reactors remain approximately linear up to a defined overvoltage level, often above rated voltage. However, near resonance, saturation may become important and should be represented.

Shunt reactors may have a very high quality factor. Therefore, losses should be included using an appropriate series resistance or other damping representation. Without realistic losses, the model may overstate the persistence or severity of resonance.

Substation equipment modelling

All capacitances in the ferroresonant loop should be considered.

This may include busbar capacitance, disconnector capacitance, VT capacitance, surge arrester capacitance, circuit-breaker capacitance, transformer bushing capacitance, and shunt capacitor banks.

Busbars can usually be represented using lumped π-equivalent models unless they are very long. Long busbar arrangements may require a distributed representation.

Circuit breakers should be represented as ideal switches with grading capacitors in parallel where applicable. This is especially important in VT ferroresonance studies, where grading capacitors can supply the energy path across an open breaker.

Surge arresters may be included if energy absorption or arrester duty is part of the study.

Current transformers and line traps are often not required unless they are directly involved in the ferroresonant loop.

Capacitor banks should be represented as lumped capacitors, with the correct wye or delta connection.

Model confidence

A ferroresonance EMT model should not be trusted only because it runs successfully. It should be checked for realistic capacitance paths, the transformer saturation curve, residual flux, losses, damping and the switching sequence.

Critical parameters

Sensitivity to key parameters and the role of damping

Sensitivity to magnetising curve representation

The magnetising curve strongly affects ferroresonance behaviour.

A VT or transformer that saturates easily is more prone to ferroresonance. A device with a higher knee point may be less likely to enter fundamental-frequency ferroresonance, although this depends on the surrounding circuit.

The saturated inductance is an important parameter. It is related to the air-core inductance and stray flux paths. It may be estimated analytically or by electromagnetic simulation when detailed data is required.

Different curve representations can produce different study outcomes. A piecewise-linear representation may predict normal behaviour or overvoltage depending on the case. Polynomial representations may trigger or suppress ferroresonance depending on how the curve is fitted.

A limited polynomial model may not have enough flexibility for detailed studies. Therefore, magnetising curve representation should not be treated as a minor modelling detail.

Sensitivity to circuit-breaker closing times

Circuit-breaker pole closing times can influence ferroresonance.

Asynchronous pole closing may initiate or suppress ferroresonance depending on the switching sequence and the initial flux condition of the transformer or VT core.

This is important because the same network may behave normally for one closing instant and become ferroresonant for another. For this reason, switching sensitivity studies may be required.

Influence of damping

Damping is one of the most important factors in ferroresonance.

Damping represents dissipative mechanisms that remove energy from the oscillation. It controls both the amplitude and the persistence of resonance.

Damping may come from winding resistance, core losses, connected load, damping resistors, reactor losses, neutral resistors, or secondary burden.

If damping is too low, ferroresonance is more likely to start and continue. If sufficient damping is present, the oscillation may decay quickly or may not develop at all.

This is why damping resistors, VT burden, neutral resistors, and realistic loss modelling are important in ferroresonance studies.

Summary

Final learning points

Key takeaways

Ferroresonance occurs when capacitance interacts with a nonlinear magnetic device — such as a VT, transformer, or reactor — in a circuit with low damping and a continuous energy source.
Electromagnetic VTs are more vulnerable to external network ferroresonance than CVTs — because CVTs normally include suppression circuits. However, CVTs also have their own internal ferroresonance mode, which is why suppression circuits are required.
Open circuit breakers with grading capacitors can energise isolated VTs — and create a nonlinear LC circuit even when the circuit appears isolated on the single-line diagram.
Busbar VTs, line VTs, VTs in double-circuit configurations, and VTs in ungrounded neutral systems — can all be exposed to ferroresonance under suitable switching or fault conditions.
Transformer ferroresonance can occur — in transformer-terminated lines, weak-source energisation, single-phase or two-phase energisation, and series-compensated line arrangements.
Ferroresonance is highly sensitive to initial conditions, switching instant, trapped flux, topology, saturation, capacitance, and damping — a single simulation is rarely sufficient. Many switching scenarios must be evaluated.
Analytical methods are useful for insight, but EMT simulation is normally required — for practical engineering studies that must assess time-domain behaviour, trapped charge, saturation, and damping.
The ferroresonant loop must be modelled correctly — including the source equivalent, line and cable capacitances, grading capacitors, transformer or VT saturation, reactor behaviour, damping, and relevant substation capacitances.
Line-model passivity should be checked — because a non-passive model may create artificial energy and produce false resonance behaviour in EMT simulation.
Magnetising curve representation is critical — different saturation models can produce different ferroresonance results. This is not a minor modelling detail.
Damping can prevent ferroresonance from starting or continuing — low-loss systems are more susceptible, while damping resistors, realistic losses, VT burden, or connected load can suppress the oscillation.
For junior engineers, the key message is simple — ferroresonance is not caused by one component alone. It is caused by the interaction of capacitance, nonlinear inductance, low damping, and an energy source. The most dangerous cases often occur when equipment looks isolated or lightly energised but is still connected through capacitance.

Part Three Reading now

Ferroresonance in VTs, Transformers and Network Models

VT ferroresonance scenarios, transformer core saturation mechanisms, grading capacitor effects, busbar and line configurations, and network topologies that sustain or trigger ferroresonant states — with sensitivity analysis and damping approaches.

Series progress 3 of 5

Part Four

Mitigation of Ferroresonance

Practical mitigation methods for VTs and power transformers — passive and active damping, open-delta resistors, and black-start energisation strategies.

Previous Resonance in Shunt-Compensated Transmission Circuits

EHV line charging capacitance, shunt reactor sizing, transformer leakage reactance resonance conditions, open-phase ferroresonance, and double-circuit resonance in shared corridor configurations.