Ferroresonance in Current Transformers: Impact on Protection Accuracy

Introduction and Conceptual Framework of Ferroresonance in High-Voltage Systems

Ferroresonance in Current Transformers (CTs) constitutes one of the most complex, destructive, and difficult-to-predict electromagnetic phenomena in modern electrical power systems. Unlike classical linear resonance, where capacitive reactance exactly balances inductive reactance at a fixed fundamental frequency, ferroresonance is a non-linear phenomenon characterized by the presence of a magnetic-flux-dependent ferromagnetic inductance operating under deep saturation conditions.

In the context of high and extra-high-voltage substations (Gas Insulated Substations [GIS] and Air Insulated Substations [AIS] from 132 kV up to 765 kV), this phenomenon is frequently triggered during external faults or circuit breaker opening and closing operations. During an external short circuit, the asymmetric fault current flowing through the primary of the CT contains a decaying direct current (DC) component. This component shifts the operating point on the magnetic core's hysteresis curve into the deep saturation zone. In this state, the magnetizing inductance of the CT drops drastically, oscillating between extremely high values (in the linear region) and values close to zero (in the deep saturation region).

The resulting equivalent circuit couples this non-linear inductance with the distributed and concentrated capacitances of the system, such as the inter-turn capacitance, bushing capacitance, busbar and supporting structure capacitances, and the capacitances of control or power cables connected to the secondary. The result is the appearance of sustained overvoltages and overcurrents with severe harmonic content, capable of destroying solid and liquid insulation, damaging magnetic cores through mechanical deformation, and causing the erratic operation or catastrophic blocking of numerical protection systems.

Physical Fundamentals and Non-Linear Mathematical Modeling of the Magnetic Core

To analytically understand ferroresonance, it is imperative to model the non-linear behavior of the ferromagnetic core of the CT. The relationship between the concatenated magnetic flux \lambda(t) and the magnetizing current i_\mu(t) is not single-valued, but is governed by the hysteresis loop and eddy current effects (iron losses). Mathematically, the magnetization characteristic can be approximated using strict analytical functions, such as the generalized arctangent function or models based on Jiles-Atherton theory:

i_\mu(\lambda) = a \cdot \lambda + b \cdot \sinh(c \cdot \lambda)

Where $a$ , $b$ , and $c$ are constants of the core's ferromagnetic material (generally grain-oriented silicon steel alloys or amorphous metals). The incremental or differential inductance of the core is defined as the derivative of the flux with respect to the current:

Ldiff(\lambda) = \frac{d\lambda}{di_\mu} = \frac{1}{a + b \cdot c \cdot \cosh(c \cdot \lambda)}

When the CT is operating in the linear zone, $Ldiff$ exhibits a maximum value $Lmax$ . However, upon a severe external fault, the inrush of a primary current i_1(t) with an angular phase shift \theta and a damping time constant \taudc induces a transient magnetic flux that exceeds the saturation induction $Bsat$ :

i_1(t) = Imax \left[ \sin(\omega t + \alpha - \theta) - e^{-\frac{t}{\taudc}} \sin(\alpha - \theta) \right]

This asymmetric flux saturates the core. At the exact instant when the core enters deep saturation, the incremental inductance $Ldiff$ drops by several orders of magnitude. The system's equivalent circuit viewed from the current transformer terminals reduces to a series or parallel Inductor-Capacitor-Resistor (RLC) resonant circuit, where the capacitance associated with the distributed capacitive elements of the substation environment interacts with the collapsed inductance of the CT:

\frac{d^2\lambda}{dt^2} + \frac{R}{Ldiff(\lambda)} \frac{d\lambda}{dt} + \frac{1}{C \cdot Ldiff(\lambda)} \lambda = \frac{v(t)}{C \cdot Ldiff(\lambda)}

This non-linear second-order ordinary differential equation governs ferroresonant modes, enabling the existence of multiple stable states for the same excitation condition (fundamental resonance, subharmonic of order $1/2, 1/3$ , and harmonic of order $2, 3, 5$ ).

Triggering Mechanism During External Faults

Forensic analysis of multiple fault events in transmission networks reveals that the triggering mechanism of ferroresonance in CTs responds to a chained sequence of transient electromagnetic events:

Occurrence of External Fault and Aperiodic Components

A three-phase, phase-to-phase, or single-phase-to-ground short circuit occurs on a line adjacent to the substation. The system impedance determines a high $X/R$ ratio, generating an exponentially decaying direct current component of large magnitude in the primary short-circuit current.

Unidirectional Saturation and Operating Point Shift

The asymmetric primary current forces the CT core to magnetize heavily in a single direction during the first cycles of the fault. The remanent magnetic flux accumulated from previous cycles adds to the transient fault flux, exceeding the saturation knee point. At this point, the reluctance of the magnetic circuit increases exponentially.

Backup Circuit Breaker Opening and Load Disconnection

The automatic circuit breaker clears the external fault. At the exact moment of electric arc interruption (zero-crossing of the modified primary current), the magnetic energy stored in the core and the electrostatic energy in the system's stray capacitances become trapped in a closed loop without sufficient dissipation.

Excitation of Non-Linear Oscillatory Modes

The residual energy finds a path through the busbar system shunt capacitance and the CT windings. By dynamically varying the core inductance along the hysteresis cycle, energy is transferred from the fundamental frequency to subharmonic and elevated harmonic frequencies, establishing a sustained ferroresonant regime.

Forensic Failure Analysis and Consequences on Substation Equipment

The consequences of an unattended or poorly mitigated ferroresonant event are devastating for substation infrastructure. The following table details the correlation between critical electrical parameters, regulatory limits, and operational and dielectric consequences:

Electrical / Mechanical Parameter	Standard Limit (IEEE / IEC)	Critical Ferroresonance Condition	Operational and Dielectric Consequence
Core Saturation Factor ( $n = Val/Vsec$ )	IEC 61869-2: n \ge 10 to $20$ for protection	Collapse of $Ldiff$ to < 5\% of nominal value	Secondary signal deformation, remanent saturation, and differential protection blocking.
Transient Overvoltage in Secondary Terminals	IEC 60044-1 / IEEE C57.13: Max. 2.5 kV peak in open circuit	Amplitudes reaching 5 kV to 15 kV sustained peak	Secondary insulation puncture, disruptive discharge at terminals, and destruction of control cabinets.
Total Harmonic Distortion (THD) in Current/Voltage	IEEE 519: Voltage THD \le 5\% at point of common coupling	THD exceeding 45% with predominance of 2nd, 3rd, and 5th harmonics	Severe winding overheating due to eddy currents and supplementary iron losses.
Dissipated Thermal Energy ( $I^2t$ )	Short-circuit thermal withstand: $100$ kA for 1s	Sustained resonant currents for several minutes	Thermal degradation of oil-impregnated paper insulation, combustible gas generation, and tank explosion.

Design Strategies, Mitigation, and Normative Criteria

Mitigating ferroresonance in Current Transformers requires a multidisciplinary approach ranging from the selection of magnetic materials to the implementation of external damping networks (snubbers) and advanced secondary topological configurations.

Selection of High-Permeability Magnetic Alloys

The use of cores manufactured from nickel-iron alloys (Mu-Metal or Permalloy) or amorphous metals significantly reduces magnetizing current and exhibits extremely narrow hysteresis loops. This minimizes post-fault remanent flux ( $B_r$ ), making it difficult for the core to enter the deep saturation region:

B_r \le 0.05 \cdot Bsat

Implementation of Secondary Damping Networks (Snubbers)

To prevent energy from oscillating between the non-linear inductance of the CT and stray capacitances, a resistor-capacitor (RC) network or a metal-oxide surge arrester (MOV) is installed directly at the secondary terminals of the CT (terminals S1-S2). The optimal design of the damping resistance $R_d$ is calculated using the characteristic impedance of the equivalent resonant circuit:

R_d = 2 \sqrt{\frac{Lmax}{Ceq}}

Where $Lmax$ is the inductance in the linear region of the core and $Ceq$ is the equivalent capacitance seen from the secondary, considering interconnection cables and the input impedance of protection relays.

Modification of Constructive Design Parameters

Reduction of the transformation ratio and increase of the magnetic core cross-sectional area to ensure a safety margin greater than 300% against the maximum unidirectional component of the short-circuit current calculated according to IEC 60909.
Physical arrangement of windings with inter-layer electrostatic shields to minimize stray capacitance $C12$ between primary and secondary.

Practical Application and Analysis with Vexten Suite

To illustrate the analytical rigor required in high-voltage substation engineering, the calculation and simulation methodology implemented in the Vexten Suite platform is presented for validating the behavior of a 230 kV CT during an external-fault-induced ferroresonance event.

Step 1: Short-Circuit Current and DC Component Calculation (IEC 60909 / IEEE 141 Standard)

Considering a 230 kV substation with a symmetrical three-phase short-circuit power $S'_{k3} = 25 GVA$ and an $X/R = 35$ ratio, the initial symmetrical short-circuit current $I'_{k}$ and peak current $i_p$ are calculated as:

I'_{k} = \frac{c \cdot U_n}{\sqrt{3} \cdot Z_k} = \frac{1.1 \times 230 kV }{\sqrt{3} \times 2.115\,\Omega} = 69.08 kA

i_p = \kappa \cdot \sqrt{2} \cdot I'_{k} = \left( 1.02 + 0.98 \cdot e^{-3/35} \right) \cdot \sqrt{2} \cdot 69.08 kA = 1.785 \times 1.414 \times 69.08 = 174.4 kA peak

Step 2: Evaluation of Magnetic Overload Factor and Saturation Risk

Vexten Suite processes the constructive data of the CT (Accuracy class 0.2S/PS, accuracy limit factor $FLS = 30$ , secondary winding resistance Rct = 1.2\,\Omega, rated burden Z_n = 5\,\Omega). The maximum induced magnetic flux is evaluated by comparing the limiting secondary electromotive force $Val$ with the induced transient voltage $Vst$ :

Vst = Imax \cdot \left( Rct + Z_n \right) \cdot \left( 1 + \frac{X/R}{\omega \cdot \taudc} \right)

The software determines that for a fault clearance time of $120 ms$ , the core reaches a magnetic flux density B(t) = 2.4 T , widely exceeding the saturation limit of the grain-oriented silicon steel alloy ( $Bsat = 1.95 T$ ).

Step 3: Control Cable Sizing Verification (IEC 60287 / NEC 310)

The elevated harmonic currents generated during the ferroresonant state cause a substantial increase in Joule losses and eddy current losses in the copper conductors connecting the CT to the control room. Vexten Suite applies the Harmonic Derating Factor (HDF) based on the IEC 60287 standard:

HDF = \sqrt{ \frac{1 + \sum_{h=2}^{n} I_h^2}{1 + \sum_{h=2}^{n} I_h^2 \cdot (1 + \lambdah)} }

Where $I_h$ is the normalized harmonic component and \lambda_h represents supplementary losses in the conductor due to skin effect and proximity effect at harmonic frequency $h$ . For a $4 mm ^2$ copper control cable with 50% current THD, the allowable current-carrying capacity is reduced by 32%, forcing the engineer to recalibrate the cable cross-section to $10 mm ^2$ to prevent premature thermal degradation of the XLPE insulation.

Step 4: Dynamic Mitigation and Stability Verification

Finally, Vexten Suite's transient analysis module simulates the insertion of a damping network composed of a capacitor bank C_s = 0.5\,\mu F and a non-inductive resistor R_s = 400\,\Omega in parallel with the secondary burden. Simulation results confirm that the residual oscillatory energy is dissipated in less than $1.5$ power system cycles ( $30 ms$ ), completely suppressing the ferroresonant overvoltage and guaranteeing the integrity of primary and backup protection systems.

Conclusions and Advanced Engineering Recommendations

Ferroresonance in Current Transformers during external faults represents a critical engineering challenge that cannot be mitigated using traditional design methods based solely on steady-state operation. Comprehensive analysis demonstrates that the non-linear interaction between the magnetic core saturation inductance and the stray capacitances of the high-voltage system generates severe overvoltages and extreme harmonic distortion.

Electrical engineers and design professionals must adopt the following guidelines:

Always incorporate electromagnetic transient studies (EMTP/ATP or Vexten Suite) in the basic engineering phase for substations of $132 kV$ and above, explicitly modeling the non-linear saturation of CT cores.
Specify current transformers with high-permeability cores or sizing factors at least 50% higher than standard protection requirements when short-circuit levels present $X/R$ ratios exceeding $30$ .
Install standardized secondary $RC$ damping networks in all critical installations exposed to severe switching maneuvers or high-asymmetry three-phase faults.