Ferroresonance in Instrument Transformers: Analysis and Mitigation

Introduction and Physical Fundamentals of Ferroresonance

Ferroresonance represents one of the most complex, destructive, and difficult-to-predict transient and steady-state phenomena in medium- and high-voltage electric power systems. Unlike classical linear resonance, where a linear RLC circuit exhibits minimum or maximum impedance at a specific tuning frequency, ferroresonance is a non-linear phenomenon characterized by the presence of at least one inductive element featuring a ferromagnetic iron core operating in magnetic saturation regions, coupled with distributed or lumped system capacitances (typically underground cable capacitances, long overhead lines, coupling capacitors, or open circuit breakers).

The non-linear behavior of the magnetic core is modeled mathematically through the constitutive relationship between the linked magnetic flux \lambda and the excitation current $i$ , which ceases to be a linear function and is instead described by a hysteresis and saturation curve that can be analytically approximated using high-order polynomial or power functions:

\lambda(i) = L_0 i + \sum_{k=1}^{n} a_k i^{2k+1}

Where $L_0$ represents the inductance in the linear (non-saturated) region, and the coefficients $a_k$ determine the curvature and severity of the non-linearity within the deep saturation zone. When the core of the instrument transformer (whether an inductive voltage transformer - IT, or a capacitor voltage transformer - CVT) enters saturation, its equivalent inductance experiences a drastic reduction, dropping to values close to the winding leakage inductance. This abrupt collapse in inductance radically transforms the impedance of the resonant circuit, shifting the natural oscillation frequency of the system toward subharmonic ( $f_0 < fsystem$ ), harmonic ( $f_0 > fsystem$ ), or quasi-periodic values.

In instrument transformers, this phenomenon is typically triggered under abnormal yet highly probable operating conditions in modern electrical networks: single-phase breaker openings in systems operating with isolated or Petersen-coil-compensated neutrals, intermittent ground faults, no-load transformer energization through lines with significant capacitance, or sustained resonances following the interruption of capacitive fault currents.

Classification and Dynamic Oscillation Modes in Transformers

The highly non-linear nature of the ferroresonant circuit implies the existence of multiple possible stable states for the same circuit parameters (ferro-multistability phenomenon). The transition from a normal operating state to a destructive ferroresonant regime critically depends on initial conditions, the magnitude of the excitation voltage, series-parallel capacitance, and total system losses. Oscillation modes are formally classified into four fundamental categories according to their spectral content and transient signature:

Fundamental Mode (System Frequency Resonance)

Characterized by oscillations at the fundamental network frequency ( $50$ or $60 Hz$ ). The current flowing through the transformer winding contains a massively amplified fundamental component (up to $10$ to $20$ times the rated excitation current) and moderate harmonic distortion. The associated overvoltages can reach values from $2.0$ to $3.5$ per unit (p.u.), causing progressive thermal breakdown of the dielectric insulation due to the exponential increase in core losses caused by hysteresis and eddy currents.

Subharmonic Mode (Oscillations at Fractions of the System Frequency)

Oscillations occur at exact submultiples of the fundamental frequency, typically $f_0 / 2$ , $f_0 / 3$ , or $f_0 / n$ . This mode is extremely dangerous and unsettling for operators, as it generates chaotic-looking current and voltage waveforms on the oscillogram. Induced voltages can exceed $4.0 p.u.$ , and the low-frequency alternating currents force the core into deep bidirectional saturation during every extended cycle, generating severe mechanical stress in the windings due to low-frequency electrodynamic forces.

Harmonic Mode (Oscillations at Integer Multiples of the System Frequency)

Dominated by frequencies higher than the fundamental ( $2f_0, 3f_0, 5f_0$ ). It frequently manifests during no-load transformer energization processes connected to networks with high distributed capacitance. The frequency spectrum displays prominent peaks at even and odd harmonics, generating short-duration transient overvoltages with extremely high dv/dt gradients, which severely stress the turn-to-turn insulation of the instrument transformer windings.

Quasi-Periodic (Chaotic) Mode

Represents a non-linear dynamic regime where voltage and current contain multiple incommensurable frequencies (not related by integer numbers). The system oscillates aperiodically, erratically jumping between different levels of magnetic energy. This mode is visualized in the phase plane through strange attractors and represents the most severe scenario for the structural and thermal integrity of potential transformers, leading to catastrophic failure within minutes.

Equivalent Circuit Analysis and Non-Linear Differential Equations

To rigorously model the phenomenon, consider an equivalent single-phase system composed of an ideal AC voltage source e(t) = E_m \sin(\omega t), a series resistance $R$ representing conductor and network losses, a linear line inductance $L_s$ , a shunt capacitance $C$ (representing line or cable-to-ground capacitance), and an inductive voltage transformer modeled by a non-linear branch relating flux \lambda(i) to a core loss resistance $R_c$ .

Applying Kirchhoff's laws for the loops and nodes of the equivalent circuit, the system of non-linear integro-differential equations governing system dynamics is obtained:

L_s \frac{di_L}{dt} + R i_L + v_c(t) = E_m \sin(\omega t)

C \frac{dv_c}{dt} = i_L - i_\mu(v_c) - \frac{v_c}{R_c}

Where $i_L$ is the current through the series inductance, $v_c$ is the voltage across the capacitor and transformer terminals, and i_\mu(v_c) is the non-linear magnetization characteristic of the ferromagnetic core. Expressing the inverse magnetic flux relationship as a function of voltage v_c = \frac{d\lambda}{dt}, the system can be reduced to a non-linear second-order ordinary differential equation:

\frac{d^2\lambda}{dt^2} + \left(\frac{R}{L_s} + \frac{1}{R_c C}\right) \frac{d\lambda}{dt} + \frac{1}{L_s C} f(\lambda) = \frac{E_m}{L_s} \sin(\omega t)

Where the non-linear function f(\lambda) represents the magnetizing current as a function of linked flux. Analytical solution of this equation is impossible via classical methods due to the non-linear term, requiring advanced numerical time-domain methods (such as higher-order Runge-Kutta algorithms) or approximate analytical methods like harmonic balance and perturbation theory.

Lyapunov stability analysis and the study of Lyapunov exponents allow the determination of critical bifurcation conditions (Hopf and saddle-node bifurcations) that precipitate the system from a stable operating state into chaotic ferroresonance.

Parameterization of Critical Factors and IEEE/IEC Normative Frameworks

International standardization bodies (IEEE and IEC) establish strict guidelines for the evaluation, design, and testing of instrument transformers against ferroresonant risk. The following comparative table details critical electrical parameters, normative thresholds, and associated operational consequences:

Electrical Parameter / Condition	Normative Limit (IEC 61869 / IEEE C57.13)	Critical Failure Condition	Operational and Dielectric Consequence
Capacitance / Inductance Ratio ( $C/L$ )	$C < Ccritical$ (Safety margin > 300%)	Elevated $C$ due to disconnection of long lines or underground cables	Shift of resonance frequency toward the subharmonic band.
Core Saturation Factor ( $n_s = Bsat / Bop$ )	n_s \geq 2.5 for linear nominal operation	Operation with $V > 1.5 p.u.$ or elevated magnetic remanence	Collapse of magnetizing inductance and entry into deep saturation regime.
Excitation Current Density	I_\mu \leq 2\% at 1.0 p.u. ; \leq 10\% at 1.2 p.u. \	Distorted magnetizing currents with peaks exceeding 500\%	Severe thermal overload in primary winding and destruction of solid insulation.
Temporary Transient Overvoltage (TOV)	Withstand $1.9 p.u.$ for $30 seconds$ (IEC 61869-3)	Sustained ferroresonant oscillations exceeding $3.0 p.u.$	Uncontrolled partial discharges, oil-impregnated paper perforation, and explosion.
External Damping Resistance	Calculated nominal thermal dissipation for continuous fault regime	Absence of damping resistance in open secondary winding (broken delta)	Inability to dissipate accumulated reactive energy, perpetuating ferroresonance.

Forensic Engineering of Instrument Transformer Failures

The occurrence of ferroresonance in electrical substations leaves unmistakable physical and metallurgical traces that allow forensic engineers to reconstruct the sequence of events leading to catastrophic equipment failure. Forensic analysis ranges from external visual inspection to analytical metallurgy of the windings.

Electrodynamic and Thermal Deformation of Windings

Under ferroresonant conditions, internal short-circuit and massive excitation currents generate extreme Lorentz forces between the concentric turns of the voltage transformer. Primary windings, constructed with reduced-cross-section copper conductors to withstand high voltages, suffer radial compression and axial stresses exceeding the material's yield strength, causing turn-to-turn short circuits between adjacent coils. Thermally, leakage magnetic flux and eddy currents induced in electrostatic shields and internal metallic structures generate hot spots with temperatures exceeding $400 °C$ , carbonizing cellulosic paper insulation and decomposing insulating oil.

Post-Failure Dissolved Gas Analysis (DGA)

In oil-immersed potential transformers, chromatographic analysis of gases dissolved in the dielectric fluid reveals characteristic failure patterns. Prolonged ferroresonance generates severe thermal decomposition of mineral oil, characterized by massive concentrations of:

Hydrogen ( $H _2$ ): Indicator of partial discharges and severe electrical stress.
Methane ( $CH _4$ ) and Ethane ( $C _2 H _6$ ): Evidence of moderate to severe local overheating ( $300 °C$ to $700 °C$ ).
Acetylene ( $C _2 H _2$ ): Indisputable marker of high-power electric arcs and disruptive discharges between turns or to ground.

Behavior of Primary Protection Elements

Current-limiting fuses installed on the primary side of voltage transformers play a critical and often complex role during ferroresonance. Due to high harmonic content and effective current values that do not reach the fast-melting threshold of the fuse but exceed the thermal dissipation capacity, fuse elements suffer premature aging ("thermal fatigue"). This causes asymmetric phase tripping, perpetuating the single-phase opening condition that feeds and sustains the ferroresonant phenomenon in the remaining healthy phases.

Practical Design Strategies and Advanced Mitigation

Effective mitigation of ferroresonance requires a multidisciplinary approach combining judicious selection of construction parameters, implementation of damping circuit topologies, and adoption of advanced operational philosophies in switch control.

Calculation and Selection of Damping Resistors (Anti-Ferroresonance Dampers)

Inserting a damping resistor into the secondary circuit connected in an open delta ("broken delta") is the most economical and reliable technique in inductive voltage transformers. Disipated power and optimal resistance $Rd$ are calculated using the following analytical formulation based on characteristic system impedance:

Rd = \frac{3 Vsec^2}{Scrit} \left( 1 + \omega^2 Leq Ceq \right)

Where $Vsec$ is the rated secondary voltage, $Scrit$ is the critical apparent dissipation power required to absorb oscillating reactive energy, and $Leq$ and $Ceq$ represent equivalent saturation inductance and system shunt capacitance, respectively. Thermal sizing of this resistor must guarantee that it can withstand the resulting zero-sequence current indefinitely without losing its ohmic properties.

Design Criteria in Capacitor Voltage Transformers (CVTs)

In CVTs, the presence of the capacitive divider ( $C_1, C_2$ ) coupled to the inductive electromagnetic compensation circuit ( $L_t$ ) creates a circuit inherently prone to ferroresonance. To neutralize this risk, a ferroresonance suppression circuit (suction circuit or ferro-detector) is mandatorily incorporated into the secondary or an auxiliary tertiary winding of the intermediate transformer. This circuit consists of a non-linear element (zinc oxide varistor - MOV or a tuned RLC circuit) that introduces high electrical conductance exclusively when subharmonic voltage components or overvoltages exceeding $1.2 p.u.$ are detected, instantly damping oscillation without interfering with the CVT's metrological accuracy under normal operation.

Operational Switching Strategies (Sequential Tripping & Controlled Switching)

From the perspective of substation automation, prevention must focus on eliminating the root causes of ferroresonance:

Use of Synchronized Three-Phase Circuit Breakers: Coordinated opening and closing of breaker poles via controlled switching controllers eliminates the possibility of prolonged single-phase openings in isolated systems.
Elimination of Critical No-Load Networks: Avoid energizing instrument transformers through excessively long, unloaded underground cable sections.
Effective Neutral Grounding: Maintain a sufficiently low grounding impedance in main power transformers to avoid severe neutral point displacements that overexcite adjacent voltage transformers.

Practical Application and Computational Analysis via Vexten Suite

To illustrate the analytical rigor demanded in high-end engineering design, a case study solved via the advanced calculation engine of the Vexten Suite platform is presented, integrating international standards for short-circuit analysis, cable sizing, and harmonic resonance mitigation.

Definition of the Network Scenario in Vexten Suite

A high-to-medium voltage substation ( $138 kV / 13.8 kV$ ) feeding a system with a symmetrical three-phase short-circuit power of $Ssc = 2500 MVA$ is analyzed. The medium-voltage network features a long polymer underground cable system providing a total phase-to-ground capacitance of C = 1.2\,\mu F . A bank of inductive voltage transformers with linear magnetizing inductance $L_0 = 45 H$ and deep saturation inductance $Lsat = 0.8 H$ is installed.

Step 1: Short-Circuit Validation and Excitation Levels (IEC 60909 / IEEE 141)

Using the Vexten Suite short-circuit module, symmetrical and asymmetrical fault current components and the thermal shock factor ( $k_t$ ) are calculated to verify the mechanical strength of instrument transformers against transient events:

I_{k1''} = \frac{1.05 \, V_n}{\sqrt{3} \, \sqrt{R_1^2 + X_1^2}} = \frac{1.05 \times 138000}{\sqrt{3} \times \sqrt{0.12^2 + 2.45^2}} = 33.85 kA

The network short-circuit power factor (\cos\varphi = R_1 / Z_1) determines the decay of the DC unidirectional current component, which interacts with the iron core and can induce a critical magnetic remanence initial state evaluated by Vexten as \psirem = 0.85\,\psisat.

Step 2: Cable Sizing and Harmonic Derating (IEC 60287 / NEC 310)

During a ferroresonant event, the current spectrum is severely contaminated by low- and medium-order harmonics. The Vexten Suite cable sizing module calculates the harmonic content reduction factor ( $Fharm$ ) applicable to control and power cables associated with instrument transformers, ensuring that XLPE insulation operating temperature does not exceed nominal $90 °C$ :

Fharm = \sqrt{ \frac{1}{1 + \sum_{h=2}^{n} (\mu_h)^2 (h)^2} }

Where \mu_h is the harmonic ratio of order $h$ current relative to the fundamental. For a typical spectrum obtained in ferroresonant simulation where the third harmonic reaches 35\% and the fifth reaches 20\%, Vexten Suite determines a strict thermal derating factor of $Fharm = 0.74$ , demanding immediate oversizing of instrumentation secondary conductor cross-sections to prevent thermal degradation of insulation.

Step 3: Resonance Analysis and Damping Tuning

The Vexten Suite non-linear dynamic analysis engine executes time-domain numerical integration of the circuit system with inputted parameters. Results show that without the damping network, the voltage across the voltage transformer terminals reaches a sustained peak of $3.42 p.u.$ in a $1/3$ order subharmonic mode, with an excitation current exceeding $45 A$ rms.

Applying Vexten Suite optimization routines, the optimal damping resistance for the open delta secondary is calculated:

R_{d,opt} = 125.4\,\Omega \quad ( with nominal thermal dissipation Ptherm = 1.8 kW )

Virtual incorporation of this value into the simulation model instantly collapses the ferroresonant oscillation within less than $2.5$ fundamental frequency cycles, returning the system to its stable operating state, keeping voltages within metrological margins and absolutely protecting the dielectric and thermal integrity of the instrument transformer.

Engineering Conclusions

Ferroresonance in instrument transformers constitutes a critical multi-physics phenomenon that challenges traditional linear analysis methods. Its understanding demands a deep mastery of non-linear circuit theory, power system dynamics, and the thermoductive behavior of ferromagnetic materials. Through the rigorous application of international IEC and IEEE standards, preventive design based on secondary damping, and the use of advanced simulation tools such as Vexten Suite, electrical engineers can guarantee the reliability, safety, and operational continuity of high- and medium-voltage electrical installations against this complex transient regime.