SFRA for Transformer Winding Mechanical Deformation Detection

Physical-Mathematical Foundations of Sweep Frequency Response Analysis (SFRA)

Sweep Frequency Response Analysis (SFRA) is the highest-resolution non-destructive methodology for evaluating the mechanical and structural-integrity of the core-winding assembly in power transformers. The fundamental premise of the method lies in the fact that a power transformer constitutes a complex passive distributed-parameter RLC network (resistance, inductance, capacitance, and conductance), whose spectral transfer function is bijective with respect to its physical geometry and dielectric hyperstructure.

At medium and high frequencies, the winding geometry, inter-disk dielectric clearances, insulation distance to the tank, and relative spatial positioning between high-voltage, low-voltage, and tertiary/tap windings dictate a unique spectral fingerprint. Any dimensional alteration imperceptible at a macroscopic scale—whether induced by electrodynamic short-circuit forces, seismic events, or mechanical stresses during transportation—alters the local distribution of inductances and capacitances, producing an unequivocal shift in the poles and zeros of the system transfer function.

Distributed Parameter Model of Transformer Windings

To analytically model the spectral behavior of a winding, it is discretized into infinitesimal or discrete sections forming a ladder network model. Each elemental section $dx$ along the axial axis of the winding possesses self-inductances, mutual inductances with respect to adjacent and non-adjacent sections, series capacitances between turns/disks, shunt capacitances to ground (core and tank), and associated dielectric and ohmic losses.

The partial differential equations governing the propagation of the voltage wave $v(x,t)$ and current wave $i(x,t)$ along the winding correspond to the generalization of the Telegrapher's Equations in the spectral frequency domain:

Where:

• $s = \sigma + j\omega = j2\pi f$ is the complex Laplace frequency.
• $R(x, s)$ is the frequency-dependent distributed resistance due to skin effect and proximity effect in conductors ($R \propto \sqrt{f}$).
• $L(x, s)$ is the distributed self-inductance per unit length.
• $M(x, \xi)$ is the kernel of the mutual inductance integral operator between axial positions $x$ and $\xi$.
• $Cg(x)$ is the shunt distributed capacitance to ground (between the winding and the grounded magnetic core or tank wall).
• $Cs(x)$ is the series distributed capacitance (capacitance between consecutive turns and inter-disk capacitance).
• $G(x, s)$ is the transverse conductance of the cellulose/oil dielectric insulation ($G = \omega Cg \tan\delta$).
• $Lw$ represents the total equivalent length of the winding conductor.

Formulation of the Transfer Function

The SFRA test injects a constant-amplitude sinusoidal signal $Vin(f)$ (typically 10 V peak-to-peak) into one winding terminal across a logarithmic frequency sweep generally ranging from $10 Hz$ to $2 MHz$, while measuring the voltage response $Vout(f)$ at another terminal, terminating both channels with a standard reference impedance of $50\ \Omega$.

The response is commonly expressed on a decibel scale of voltage attenuation or gain:

Alternatively, in terms of the spectral phase response $\phi(f)$:

In the Laplace operational domain, the input impedance $Zin(s)$ or voltage transfer function $H(s)$ can be rigorously expressed through pole ($p_j$) and zero ($z_i$) expansion:

The resonance frequencies (local maxima, poles) and anti-resonance frequencies (local minima, zeros) are governed by the roots of the characteristic polynomial. For a discrete elemental winding section of order $N$, the primary resonant frequencies $\omega_k$ are analytically parameterized by:

This mathematical relationship demonstrates beyond doubt the extreme sensitivity of the method: a millimeter-level geometric alteration modifying the inter-disk spacing alters the series capacitance $C_s$, directly shifting the natural frequency $\omega_k$ across the medium- and high-frequency regions of the spectrum.

Frequency Band Segmentation and Substructural Diagnostics

The complete transfer function spectrum of SFRA ($10 Hz$ to $2 MHz$) does not respond homogeneously to all physical components of the power transformer. Due to the differing behaviors of inductive reactance ($X_L = 2\pi f L$) and capacitive reactance ($X_C = \frac{1}{2\pi f C}$), the frequency response is analytically segmented into four distinct functional bands, each dominated by a specific structural subset of the equipment.

Very Low and Low Frequency Band (< 2 kHz): Core Magnetizing Domain

At frequencies below $2 kHz$, the capacitive reactance of the insulation system is extremely high ($X_C \to \infty$), behaving effectively as an open circuit. Consequently, the injected current flows predominantly through the non-linear magnetizing inductance of the magnetic core ($L_m$) and the DC winding resistance ($Rdc$).

The transfer function in this region is dominated by an initial deep anti-resonance followed by a resonance corresponding to the equivalent circuit formed by the core magnetizing inductance in parallel with the equivalent capacitance to ground:

Any phenomenon that alters the magnetic core permeability $\mu_r$, the continuity of the silicon steel laminations, the core grounding status, or the presence of residual magnetism following a fault event will induce a massive shift in both amplitude and spectral position across this low-frequency band.

Medium-Low Frequency Band (2 kHz to 100 kHz): Inter-Winding Interaction and Global Structures

In this spectral range, the reactance of the magnetizing inductance increases substantially, and the inter-winding capacitance ($Cinter$) begins to provide a low-impedance path. The response signature in this band reflects the mutual interaction between concentric High Voltage (HV), Low Voltage (LV), and Tap/Regulation windings.

The behavior is dictated by the short-circuit leakage inductance ($L_k$) and inter-winding capacitance $Cinter$. Variations in this zone reveal global winding tilting, global axial displacement of the entire assembly, or structural defects in internal lead connections and On-Load Tap Changers (OLTC).

Medium-High Frequency Band (100 kHz to 500 kHz): Main Winding Structure and Axial/Radial Geometry

At these frequencies, the leakage inductance $L_k$ and the series inter-disk or inter-turn capacitances ($C_s$) form a complex multiphase network of poles and zeros. The magnetic core steel acts practically as a magnetic flux shield due to surface eddy currents induced in the laminations, reducing the effective core permeability to $\mu_r \approx 1$.

Therefore, the response depends almost exclusively on the internal geometry of the winding itself. Localized disk displacements, radial buckling deformation, conductor tilting, or axial pressing block collapse directly alter the local inter-disk capacitance $C_s(x)$ and incremental leakage inductance $\Delta L_k$, manifesting as horizontal (frequency) and vertical (attenuation) shifts of multiple resonant peaks.

High Frequency Band (500 kHz to >2 MHz): Output Leads, Internal Connections, and Bushings

At the upper end of the spectrum (> 500 kHz), attenuation is dominated by low-magnitude parasitic capacitances, internal interconnection leads, bushings, and the self-inductance of flexible busbars.

Observable alterations in this region that are unsupported by corresponding shifts at lower frequencies are typically attributed to variations in measurement ground leads, degradation of bushing connections, changes in the physical geometry of transformer drop conductors, or compromised electrostatic shielding screens.

Forensic Analysis of Mechanical Failure Modes and Spectrum Transformations

Electrodynamic or Lorentz forces acting upon transformer conductors during an external short-circuit event are proportional to the square of the instantaneous fault current ($F \propto Isc^2$). These forces resolve into two main vector components: radial force ($F_r$) and axial force ($F_z$). The manifestation of these forces imposes irreversible plastic deformations on the mechanical structure of the winding, which translate into highly specific SFRA spectral signatures.

Radial Buckling

Inwardly directed radial forces act to mechanically compress the inner winding (typically the low-voltage winding situated closest to the core leg). If the radial force exceeds the critical limit for plastic yield or buckling of the copper conductor, the failure phenomenon known as Buckling occurs.

Two primary mechanical typologies exist:

• Free Buckling: The winding deforms into a lobed profile without a fixed anchor point, creating periodic crests and troughs along its circumference.
• Forced Buckling: The winding collapses inward between the axial support sticks or spacers on the core leg, producing squeezed, localized periodic deformations.

Effect on the SFRA Spectrum: Radial buckling deformation decreases the physical distance between the inner winding and the core, substantially increasing the capacitance to ground ($C_g \uparrow$). Furthermore, the symmetry of the inter-winding gap is altered, modifying the mutual inter-winding capacitance ($Cinter$). On the SFRA trace, this is diagnosed via a systematic leftward shift (toward lower frequencies) of resonance peaks within the 100 kHz to 500 kHz band, accompanied by amplitude variations in anti-resonant troughs.

Axial Displacement and Disk Tilting

Axial forces originate from asymmetries in leakage magnetic flux at the upper and lower ends of the winding. If the HV and LV windings are not perfectly aligned magnetically, a destructive net axial force component arises, driving the inner winding downward and the outer winding upward.

When the exerted axial pressure overcomes the mechanical pre-clamping force (clamping rings and pressboard blocks), two failure modes develop:

1. Global Axial Displacement: The entire winding slides axially along the core leg.
2. Disk Tilting: Adjacent winding disks twist tangentially or rotate into a "zigzag" pattern due to the failure and collapse of inter-disk insulating spacers.

Effect on the SFRA Spectrum: Axial displacement drastically alters mutual magnetic coupling and increases short-circuit leakage inductance ($L_k \uparrow$). In the SFRA signature, axial displacement causes a pronounced alteration in the medium-low frequency band (2 kHz - 100 kHz), marked by a prominent leftward shift of the first inter-winding resonance trough and amplitude variations exceeding 3-6 dB in intermediate peaks.

Inter-Turn Short Circuit or Inter-Disk Dielectric Breakdown

Dielectric failure stemming from Kraft paper degradation or transient switching/lightning overvoltages can short-circuit two or more consecutive turns. An inter-turn short circuit voids the magnetic flux inside the shorted loop due to Lenz's law, driving a massive internal circulating current that screens out and cancels the inductance of that winding segment.

Effect on the SFRA Spectrum: The elimination of equivalent inductance causes a dramatic loss in magnetic energy storage capacity. On the SFRA trace, the low- and mid-frequency signatures lose their characteristic resonant peaks, transforming the transfer function curve into a flattened, highly attenuated response across the low- and mid-frequency bands, resembling a heavily damped or permanently shorted circuit.

Statistical Comparison Algorithms and Standardized Diagnostic Criteria

SFRA interpretation relies heavily on comparative analysis of spectral traces using three standardized evaluation methodologies:

• Baseline Comparison (Fingerprint vs. Actual): Direct comparison with factory baseline data or pre-commissioning fingerprints. This represents the gold standard for diagnostic sensitivity and certainty.
• Phase-to-Phase Comparison: Comparison between symmetrical phases of the same machine (e.g., Phase A vs. Phase C) recorded under identical test configurations.
• Sister Transformer Comparison: Comparison with a twin unit of identical mechanical design and manufacturer.

To eliminate subjectivity during visual Bode plot inspection, international standards such as IEEE C57.149, IEC 60076-18, and CIGRE WG A2.26 recommendations mandate the use of advanced mathematical metrics, primarily the Cross-Correlation Coefficient ($Rxy$ or $CC$) and the Mean Squared Error Logarithmic Difference.

Cross-Correlation Coefficient ($Rxy$)

Given two frequency response vectors expressed in decibels, $X_k = Hbase(f_k)$ and $Y_k = Hactual(f_k)$, evaluated discretely over $N$ frequency points within defined frequency sub-bands:

Where $\bar{X}$ and $\bar{Y}$ correspond to the arithmetic means of the response magnitudes within the evaluated sub-band:

Absolute Spectrum Distance Metric ($Dabs$)

To quantify amplitude deviations without over-sensitivity to pure phase shifts or narrow-peak spectral offsets, the Average Absolute Spectral Distance is applied:

Diagnostic Decision Criteria

In accordance with standard DL/T 911-2016 and IEEE/IEC guidelines, structured severity thresholds for the cross-correlation coefficient within the critical winding band ($100 kHz - 500 kHz$) are categorized according to the following analytical scale:

Comparative Matrix of Failure Phenomena, Standardized Criteria, and Diagnostic Impact

The following comprehensive table consolidates spectral signatures, equivalent RLC parametric variations, governing standards, metric thresholds, and critical operational consequences for primary failure modes detectable by SFRA:

Mechanical / Structural Failure Mode	Affected Frequency Band	Equivalent RLC Parametric Variation	SFRA Spectral Signature Behavior	Standardized Metric Thresholds (IEC 60076-18 / IEEE C57.149)	Critical Dielectric or Operational Impact
Radial Buckling	Medium-High Frequency (100 kHz – 500 kHz)	$C_g \uparrow$ (10% - 30%) $Cinter \downarrow$ $L_k variable$	Horizontal leftward shift of resonant peaks toward lower frequencies. Peak amplitude variation.	$Rxy < 0.95$ in medium-high band. $Dabs > 3.5 dB$.	Reduction of oil dielectric clearances. Inter-winding dielectric breakdown driven by partial discharge.
Disk Tilting / Axial Displacement	Medium-Low to Medium-High (20 kHz – 300 kHz)	$C_s \uparrow$ or $\downarrow$ $L_k \uparrow$ (1% - 5%) $M altered$	Severe amplitude variation in peaks and troughs. Appearance of new parasitic pole-zero pairs.	$Rxy < 0.98$ in mid-band. $\Delta L_k > 2\%$ in short-circuit impedance test.	Shearing of critical insulating spacers. Complete loss of axial mechanical clamping pre-load.
Loss of Core Grounding	Low Frequency (10 Hz – 1 kHz)	$C_g floating$ $G \to \infty (parasitic)$	Disappearance of characteristic low-frequency trough. Vertical upward shift of the entire trace.	Extreme deviation with $Rxy < 0.90$ in low sub-band (< 1 kHz).	Destructive partial discharge induced by floating potential. Massive oil gassing (DGA generation).
Core Residual Magnetism	Very Low Frequency (10 Hz – 500 Hz)	$\mu_r \downarrow (local saturation)$ $L_m \downarrow$	Rightward shift of the first resonance peak toward higher frequencies. Signature normalizes post-demagnetization.	$Rxy < 0.95$ in < 1 kHz band, but full normalization achieved after DC demagnetization cycles.	Severe inrush currents during energization. Nuisance tripping of differential protection relays (87T).
Inter-Turn Short Circuit	Low, Medium, and High (10 Hz – 500 kHz)	$L_m \to 0$ $L_k \downarrow dramatic$ $R \uparrow (losses)$	Complete collapse of spectral signature. Curve loses distinct peaks and degrades into a flat attenuation response.	$Rxy < 0.80$ across multiple bands simultaneously. Magnetizing inductance destroyed.	Immediate catastrophic failure. Instantaneous Buchholz relay (63) and overcurrent (50) trip.
Lead Displacement and Connection Tilting	High Frequency (500 kHz – 2 MHz)	$Llead altered$ $Cstray altered$	Peak shifts isolated strictly to the region above 500 kHz. Absolute stability in low- and mid-frequency bands.	$Rxy < 0.95$ localized to $f > 500 kHz$. Lower sub-bands maintained at $Rxy > 0.998$.	Minor risk of immediate dielectric breakdown, unless tank clearance distances are compromised.

Mechanical Mitigation Strategies and Quantitative Modeling of Electrodynamic Forces

Preventing structural damage detected by SFRA requires rigorous mathematical modeling of electrodynamic forces acting upon winding geometries during three-phase and phase-to-ground short-circuit events. Robust engineering design must ensure dynamic stresses do not exceed the yield strength limits of copper or silver-bearing copper alloys (Cu-Ag).

Modeling of Radial Short-Circuit Forces

The radial force density per unit length $F_r$ acting on a circular winding subject to peak asymmetrical short-circuit current $ipeak$ is defined by:

Where:

• $\mu_0 = 4\pi \times 10^{-7} H/m$ is the magnetic permeability of free space.
• $N$ is the total number of turns in the winding.
• $ipeak$ is the peak short-circuit current (including asymmetry DC offset factor $Kpeak \cdot \sqrt{2} Isc$).
• $D_m$ is the mean winding diameter.
• $heq$ is the equivalent magnetic height of the winding taking end-field fringing into account.

The mean mechanical tensile/compressive tangential stress (Hoop Stress, $\sigma_{\theta}$) induced across conductor cross-sectional area $A_c$ is expressed analytically as:

To guarantee protection against permanent buckling, the strict design design boundary dictates that:

Where $\sigma_{0.2}$ is the conventional 0.2% proof stress of the conductor material (typically $120 - 220 MPa$ for hard-drawn copper) and $FS$ is the static/dynamic safety factor ($FS \ge 1.5$).

Calculation of Axial Compression and Clamping Forces

The total axial compressive force $F_z$ acting to collapse the winding toward its magnetic mid-plane is derived by integrating the product of radial leakage flux $B_r(z)$ and total current:

Axial crushing stress imposed upon insulating pressboard spacers must not exceed the hot dynamic compressive strength limit of cured cellulose (typically $\sigma_{\text{comp}} \le 40 - 70 MPa$).

Modern Constructive Mitigation Techniques

• Epoxy-Bonded Continuously Transposed Conductors (Bonded CTC): Application of CTC conductors coated with B-stage epoxy resin. Thermo-curing bonds individual strands into a rigid monolithic beam, dramatically increasing the section modulus and improving radial buckling resistance by a factor of 3 to 5 compared to non-bonded conductors.
• Dynamic Axial Clamping via Spring Elements or Hydraulic Systems: Installation of constant pre-load mechanical clamping systems to automatically compensate for cellulose pressboard creep and aging contraction over the lifespan of the asset, ensuring the winding never drops below its critical minimum axial clamping threshold ($Fpreload > F_{z, dynamic }$).

Advanced Integration with Vexten Suite for SFRA Analysis and Structural Integrity

Within the Vexten Suite analytical framework, the specialized SFRA diagnostic module serves as the primary processing core for forensic evaluation and structural health prediction of power transformation assets. The platform seamlessly marries spectral data processing with deterministic power system dynamic simulations.

Short-Circuit Fault Correlation via IEC 60909 and IEEE 141

The Vexten Suite short-circuit engine (Vexten Fault Engine module) simulates electromagnetic fault transients across transmission and distribution networks in full compliance with IEC 60909 and IEEE 141 standards. When a near-substation fault occurs, Vexten Suite precisely calculates the peak asymmetrical fault current ($i_p$) experienced by the transformer:

Where $\kappa$ is the peak factor calculated as a function of the system $R/X$ ratio at the fault location, and $I_k''$ is the initial symmetrical short-circuit current. Using this calculated current, Vexten Suite synthesizes a 3D electrodynamic force map ($F_r, F_z$) representing the mechanical stress profile imposed on the windings. If cumulative material fatigue thresholds are breached, the platform automatically triggers an alert for asset management personnel, generating a high-priority work order for immediate field SFRA testing.

Vexten Spectral Signature Algorithm and Automatic Sub-Band Decomposition

The Vexten SFRA Analyzer software package automates the ingestion of standardized XML/ASCII trace files (CIGRE / IEEE C57.149 formats). It natively executes the following advanced mathematical procedures:

1. Cubic Spline Spectral Alignment: Logarithmic response interpolation to harmonize non-uniform frequency test points recorded across disparate test instruments (e.g., Omicron, Doble, Megger).
2. Adaptive Four Sub-Band Standard Decomposition: Automated segmentation of spectral vectors into physical sub-regions corresponding to core, inter-winding interaction, main winding structure, and high-frequency lead circuits.
3. Vectorial Cross-Correlation Matrix and Spectrum Distance Metrics (SDM): Parallel execution of multivariate statistical comparisons between baseline fingerprints and field test data.

If the calculated Vexten Suite metric within the medium-high band falls below $0.98$, the software triggers early warning alerts and cross-correlates the response with dielectric insulation power factor/capacitance (C1/C2) data and short-circuit impedance test measurements (IEC 60076-5), isolating the exact mechanical failure mode and estimating the percentage loss of remaining mechanical useful life (RUL) of the transformer.