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Before diving into ferroresonance, it is worth reviewing general circuit resonance concepts.
When a system is excited in a periodic way, we say that it tends to vibrate at its natural frequency. If an excitation source drives the system at its natural frequency, we observe a dramatic response, which is when we say that the system is in resonance. For example, if we kept plucking a guitar string at its natural frequency, we would be driving a resonant vibration in that string.
In this article, 'the system' will refer to an electrical circuit, but almost every system and object has a natural frequency, which is determined by the properties of the system/object. In the image below, the dramatic system response would be characterized by the sharp increase in amplitude.
Phase and amplitude responses over a range of driving frequencies for an arbitrary system. At resonance, amplitude peaks, and the phase difference between the driving source and system becomes 90 degrees.
In AC circuits, current oscillates at the source's driving frequency, for example, 60Hz in North American power systems. The natural frequency of an AC circuit is defined by its equivalent inductance and capacitance at the source terminals, L and C.
We can observe circuit resonance more efficiently in the frequency domain, where resonance occurs when the inductive reactance (XL) is equal to the capacitive reactance (XC). Below, f (or w, since w = 2𝝅f ) is the frequency at which an RLC circuit is sourced.
Inductive reactance is given by:
Capacitive reactance is given by:
Inductor.
Capacitor.
Setting XL = XC and solving for f or w, we yield a formula for the natural frequency of a circuit.
In the time domain:
In the frequency domain:
Why do these formulas mean that a high or low-amplitude voltage/current response is observed during resonance?
Impedance is a circuit's total opposition to AC current flow. It is determined by the norm of the circuit's equivalent resistive and reactive impedances at the source terminals.
Total impedance is given by:
Series RLC circuit.
Parallel RLC circuit.
During resonance (when XL = XC), energy oscillates between the circuit capacitance and inductance, effectively removing the reactive part of the circuit impedance. During series resonance, impedance dips to a minimum and a high-amplitude response is observed. During parallel resonance, impedance reaches a maximum and a low-amplitude response is observed.
Example plot for series impedance as a function of driving frequency. Global minimum achieved at resonant frequency.
Example plot for parallel impedance as a function of driving frequency. Global maximum achieved at resonant frequency.
Electrical resonance where all of the above is true is called linear resonance, and it can easily be modeled with a differential equation. In other words, it's predictable.
With the background out of the way, the main topic of the article can be introduced. Ferroresonance is a special case of resonance where the capacitance interacts with a non-linear inductance, resulting in an unpredictable amplitude and distorted oscillatory response.
The term "ferroresonance was introduced in the 1920's by Paul Boucherot, a French electrical engineer whose work helped to separate ferroresonance from general electrical resonance.
Four elements are required to drive a ferroresonant circuit response:
Capacitance
Non-linear inductance
Driving source
Low losses
Before exploring how these elements interact, it is important to define what a "non-linear inductance" means.
We say that an iron or ferrite-core inductor has become saturated when the core has become fully magnetized and there cannot be additional flux through it. At this point, the value of the inductance begins to taper off and becomes non-linear with respect to the current passing though it. These kinds of inductors can be found in general electronics, but they're most commonly found in power system transformers, so the rest of the article will focus on ferroresonance in power systems.
The image on the right below shows how flux density and field strength for an inductor are linearly related until the saturation limits are reached and the curve becomes nonlinear.
1P & 3P transformer core configurations.
Flux density through a non-linear transformer core as a function of magnetic field strength.
When a non-linear inductance interacts with circuit capacitance under the right conditions, the inductor core is driven in and out of saturation at semi-random intervals, which is the driving mechanism for ferroresonance. Examples of power system capacitance include shunt capacitors (for power factor correction), series capacitors (for line impedance reduction), L-L capacitive coupling and L-G capacitive coupling. The system's non-linearity also allows ferroresonance to occur for a range of capacitance values, which means that the resonant frequency changes while the system is operating, comparable to a moving target. However, capacitor-inductor interaction isn't the only factor that can lead to ferroresonance.
Low losses favor the onset and persistence of ferroresonance since minimal damping allows resonant energy to circulate in the power system rather than dissipating. Light-load or no-load conditions are particularly susceptible. For example, an oil field customer might use a large fraction of their transformer capacity during the day, but draw very little current overnight. Similarly, a transformer that services a pump motor may experience abrupt changes in loading as the pump is switched on and off, introducing transient loading conditions to the power system. Under these conditions, transformer core losses and load damping are insufficient to suppress oscillations, allowing the non-linear magnetizing inductance to interact with system capacitance in an unstable manner.
In short, the right (or wrong) combination of capacitance, non-linear inductance (transformer core characteristics), driving source characteristics and transformer loading level can create an environment where ferroresonance is a risk.
Diagram of the capacitive coupling within a 3-core distribution cable. The capacitive coupling between three 1-core cables is similar.
Ferroresonance can be mathematically understood as a non-linear circuit phenomenon in which multiple steady-state operating points exist for the same system parameters. Under normal conditions, the system settles into a predictable, low-voltage operating state. Under ferroresonant conditions, however, the system may transition to an alternative steady state characterized by abnormally high and distorted voltages.
The figure on the right illustrates this behavior for a simplified series ferroresonant circuit. The blue curve represents the non-linear voltage–current characteristic of the transformer magnetizing inductance, which changes slope as the core enters and exits saturation. The red line represents the linear voltage–current relationship imposed by the source and series impedance. Steady-state solutions occur at the intersection points of these curves. Because the inductance is non-linear, multiple intersections (and therefore multiple valid steady-state operating points) may exist for the same source voltage and frequency. Small disturbances or switching events can cause the system to jump from the normal operating point to a ferroresonant state, resulting in sustained overvoltage. The ferroresonant operating point is generally that which rests at a higher voltage level.
Ferroresonance can manifest in several distinct forms depending on the frequency content and stability of the resulting voltage oscillation. The waveforms shown below illustrate typical ferroresonant responses; in practice, any of these behaviors may appear on either the transformer primary or secondary voltage.
Fundamental Mode:
In the fundamental mode, the voltage oscillates at the system’s fundamental source frequency (f0). Although the frequency is unchanged, the waveform is often highly distorted due to repeated saturation of the transformer core, resulting in strong harmonic content.
Subharmonic Mode:
In the subharmonic mode, voltage oscillations occur at integer submultiples of the fundamental frequency (f0/2, f0/3, ...). These responses are typically accompanied by significant waveform distortion and can produce unexpectedly high voltages.
Quasiperiodic Mode:
In the quasi-periodic mode, the voltage waveform contains two or more incommensurate frequencies simultaneously, resulting in a response that never exactly repeats in time. As system parameters change, the waveform may transition between different periodic or quasi-periodic states (this transition is known as a bifurcation).
Chaotic Mode:
In the chaotic mode, the voltage waveform becomes irregular and aperiodic. Although governed by deterministic system equations, the response is extremely sensitive to initial conditions and exhibits a broadband frequency spectrum. Chaotic ferroresonance is particularly damaging due to its unpredictability and potential for sustained overvoltages.
Ferroresonance is typically initiated by a disturbance that pushes the power system away from its normal operating state. These disturbances may be external, such as lightning strikes or insulation failures, or internal, such as asynchronous switching events. Common asynchronous switching scenarios include single-phase or two-phase fuse cutout operations, banked recloser tripping, and conductor breakage.
However, the presence of a triggering event alone does not guarantee that ferroresonance will occur. Transitions between the normal operating state and a ferroresonant state depend strongly on the system’s initial conditions at the instant the disturbance takes place. Factors such as the charge on system capacitances, residual flux in transformer cores, and the instantaneous source voltage determine whether the system settles back into normal operation or transitions into ferroresonance.
Because these initial conditions are difficult to measure and vary continuously during normal operation, ferroresonance is inherently unpredictable. As such, relatively small disturbances can occasionally produce severe and sustained overvoltage conditions.
Lightning strike to distribution poles
Fused cutout (one phase open)
Several real-world power system configurations are known to be susceptible to ferroresonance. The following examples represent some of the most common and practically relevant scenarios encountered in distribution and substation systems
Lightly loaded pad-mount transformer fed through a sufficiently long underground cable:
One of the most common and well-documented ferroresonant configurations occurs when a lightly loaded or unloaded transformer is connected in series with significant system capacitance. In distribution systems, this frequently arises when a three-phase pad-mounted transformer is supplied through a sufficiently long underground cable.
In this configuration, the effective series capacitance is formed by phase-to-phase and phase-to-ground capacitive coupling within the cable. Underground cables exhibit substantially higher capacitance than overhead lines due to their close conductor spacing and the presence of high-permittivity dielectric materials.
If an asynchronous switching event occurs while the transformer is lightly loaded, the resulting transient may drive the system into a ferroresonant state. For this configuration, fundamental-frequency or quasi-periodic ferroresonant responses are most commonly observed.
This scenario is considered the most prevalent ferroresonant configuration in distribution systems and is therefore a primary focus during system planning and risk assessment.
Substation voltage transformer and breaker grading capacitance:
A second, less common ferroresonant configuration can occur in substations where breaker grading capacitance is effectively placed in series with a phase-to-ground voltage transformer.
Voltage transformers are used to provide voltage measurements while electrically isolating the primary system. When a breaker equipped with grading capacitors opens, the grading capacitance may remain connected in series with the VT primary winding, which can form a ferroresonant circuit.
The resulting oscillations may occur at the fundamental frequency or at subharmonic frequencies, and sustained overvoltages can result. Reported overvoltages are typically in the range of 2–4 volts per unit.
It is important to note that not all circuit breakers employ grading capacitors, and the presence of this ferroresonant configuration depends on specific breaker and VT designs. As a result, this scenario is less common in modern installations but remains a recognized risk in certain substations.
Substation grading capacitance.
Voltage transformers on ungrounded or impedance-grounded systems:
Ferroresonance may also occur on the secondary side of transformers, particularly when phase-to-ground voltage transformers are connected to an ungrounded or weakly grounded neutral system. In these cases, capacitive coupling between one or more phases and ground can interact with the transformer magnetizing inductance to form a ferroresonant circuit.
This situation may arise intentionally, such as in systems employing phase-to-ground fault detection schemes, or unintentionally due to a floating or broken neutral. A neutral grounded through an inductive impedance can create a similar susceptibility by placing the inductance in series with system capacitance.
These configurations are most often associated with subharmonic or quasi-periodic ferroresonant responses. Although the phenomenon originates on the secondary side, the resulting voltage distortion and overvoltage can propagate and affect upstream distribution equipment.
Example of an inductive grounding impedance.
There are, of course other examples of situations where a ferroresonant circuit can be formed on distribution systems. For more insight into these scenarios, I recommend reading this paper by P. Ferracci, since the examples above are drawn directly from his work.
There are a few notable effects of ferroresonant overvoltages on the primary distribution system (and secondary network):
Surge arrestor and insulator flashover/damage
Transformer overheating or failure
Recloser nuisance tripping and miscoordination
Conductor breakdown
Overvoltage to customer equipment (motors, power electronics, lighting, etc.)
Severe waveform distortions and harmonic content
Several observable signs can indicate that a power system is experiencing, or is susceptible to, ferroresonance. One of the most common indicators is abnormal noise and vibration from transformers, caused by magnetostriction when the transformer core is driven into saturation. Magnetostriction occurs as magnetic domains within the core realign, producing mechanical stress and vibration. During ferroresonant conditions, this noise is often described as unusually loud humming, buzzing, or rattling—sometimes compared to loose hardware tumbling inside the transformer. In practice, this is symptomusually detected by a customer or lineman.
Excessive transformer heating is another important warning sign. Sustained overvoltage and elevated magnetizing current during ferroresonance can lead to rapid temperature rise, even when the connected load is light or unchanged.
In addition, voltage waveform distortion associated with ferroresonance may propagate into the secondary network, resulting in visible light flicker, erratic operation of sensitive equipment, or abnormal meter and relay behavior. Customers may report flicker, but ferroresonance may not be detected until a power quality investigation is completed (phase recording, acoustic monitoring, etc.).
Note that a ferroresonant state does not always need to be persistent for these indicators to be present.
Effective mitigation is essential to minimize the damaging effects of ferroresonance. Mitigation strategies generally fall into two categories: engineering controls, which reduce the likelihood of ferroresonance occurring, and operational practices, which limit exposure during system switching and abnormal conditions.
Engineering Controls:
One common practice is to limit the length of underground cable feeding pad-mounted transformers, ensuring that it remains below established ferroresonance risk thresholds. Cable length directly influences the amount of L-L capacitive coupling, which plays a critical role in ferroresonant behavior. Transformer loading is also evaluated during planning, with particular attention given to pad-mount installations that may experience frequent light-load or no-load conditions.
Reducing effective system capacitance further decreases the number of configurations in which ferroresonance can occur. This may include phasing out unnecessary shunt capacitors and applying improved voltage and power factor control practices to achieve acceptable performance without excessive capacitive compensation.
Another important mitigation technique is the use of gang-operated switches and reclosers, which ensure the simultaneous operation of all three phases, reducing the likelihood of asynchronous switching conditions. Further, fault interruption technology and microcontrollers allow reclosers to rapidly detect abnormal current behavior associated with ferroresonance and isolate affected sections of the system to limit equipment damage. See the photo to the right.
All of the techniques above are implemented by system-planning and operational engineering groups, who have visibility into distribution models and can determine if a given site poses a ferroresonance risk.
Operational Practices:
From an operational standpoint, asynchronous energization or de-energization of pad-mount transformers and their associated underground feeding lines should be avoided, as these actions introduce transients that may trigger ferroresonant responses.
Care is also taken to avoid energizing lightly loaded or unloaded transformers whenever possible. When light-load operation cannot be avoided, temporary or permanent resistive loading may be applied at the transformer secondary terminals. These load resistors provide additional damping, suppressing sustained oscillations and reducing the likelihood of ferroresonance.
Linemen should use their intuition and experience to help determine if there is a risk of ferroresonance, and communicate with customers and engineering groups to develop a viable solution.
Underground Cable Bury
Power Capacitors
G&W Diamondback Gang-Switch
Damping Resistor
References:
[1] D. A. N. Jacobson, "Examples of Ferroresonance in a High Voltage Power System,", 2003. [Online]. Available: https://pages.mtu.edu/~bamork/FR_WG/Panel/paper03gm0984.pdf. [Accessed: Mar. 15, 2025].
[2] P. Ferracci, n°190, "Ferroresonance,", March 1998. [Online]. Available: https://www.studiecd.dk/cahiers_techniques/Ferroresonance.pdf. [Accessed: Mar. 15, 2025].
[3] M. V Escudero, I. Dudurych, M. Redfern, "Understanding Ferroresonance,", October 2004. [Online]. https://www.researchgate.net/publication/4164306_Understanding_ferroresonance. [Accessed: Mar. 15, 2025].
[4] B. Baldwin, S. Sabade, S. Joshi, "A Study of Ferroresonance and Mitigation Techniques,", April 28, 2013. [Online]. https://pages.mtu.edu/~bamork/EE5223/EE5223TermProj_Ex1.pdf. [Accessed: Mar. 15, 2025].
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