Elegant procedure to estimate series capacitance of a uniform transformer winding from its measured FRA: implementable on existing FRA instruments

: A simple procedure to estimate series capacitance of a uniform transformer winding from its measured frequency response analysis (FRA) and shunt capacitance is presented. Unlike previously published approaches, this method does not involve any cumbersome and time-consuming curve-fitting nor running optimisation/search algorithms, and neither does it require data of winding geometry. The procedure relies on a property that is observable in the impedance function of a lossless winding, viz., the ratio of the product of squares of open circuit natural frequencies to the product of squares of short circuit natural frequencies bears a special relation to impedance function coefficients. Its feasibility was initially verified by simulation, and then by experiments on small-sized continuous-disk and interleaved-disc windings, followed by a large-sized 33 kV, 3.5 MVA continuous-disc winding, and finally on a 315 kVA 11/13.8 kV transformer. After measuring FRA, the process involves just finding roots of a polynomial, from which the initial impulse voltage distribution constant and series capacitance can directly be determined. Given these attractive features, authors believe that this method is implementable on existing FRA instruments, so that, along with routinely measured FRA, these two important constants of a winding can be displayed.


Introduction
Series capacitance (C s ) of transformer winding is a vital design parameter which along with the shunt capacitance governs the initial impulse voltage distribution when a surge impinges on it. Its value essentially conveys the net capacitive coupling offered by different disks/double-disks of the winding. Designers employ the well-established method of interleaving of individual turns to increase series capacitance and consequently achieve a more uniform initial impulse voltage distribution. Hence, knowledge of series capacitance is paramount for ascertaining or predetermining the winding's transient behaviour [1][2][3][4]. Although shunt capacitance is readily measurable, the same is not the case for series capacitance. Also, there exists no simple method to crosscheck how closely was the intended design value of C s actually realised after manufacture of the winding. Ideally speaking, it would be desirable to perform this cross-check on every manufactured winding, and preferably this check is based on measurements.
The other additional benefits that arise from C s estimation are: (a) Foremost, it acts as a cross-verification of the design data and proves how closely was the design value of C s reproduced by the manufacturing process. Unfortunately, this cannot be determined by any other measurement-based method, other than by doing an initial impulse voltage distribution measurement which would require a sacrificial winding. (b) Permits construction of a lumped-parameter ladder network equivalent circuit.
(c) Estimating C s of existing windings can be now rendered possible by the proposed method when geometry and other winding data are unavailable, which is often the scenario.
(d) And more importantly, based on the recent work from the authors' research group [5], it is expected that C s would also possess diagnostic capabilities similar to the quantity called equivalent air-core inductance of a winding.
These are the important reasons for estimating C s , and in that context, the objective of this contribution is to devise a simple method to indirectly measure the series capacitance of a winding via the frequency response analysis (FRA) measurement.

Literature review
A brief summary of previous efforts directed towards estimation of series capacitance is given as follows: 1. Predetermination of the series capacitance of winding during its design stages was a much-discussed topic in the early 1950s-1960s, and all these efforts essentially pursued an analytical or semi-analytical approach, and required complete data on winding geometry, insulation data, clearances etc. [6][7][8][9]; a requirement not easily available to end-users. Many formulae were developed during that time period, but interestingly each one yielded a different result even for a given winding, and so it was arguable which one of them was more appropriate to use. In other words, there was no consensus amongst the developed methods. Moreover, each of these formulae was specific for a given type or structure of winding, and so needed to be entirely reworked when the design changed; which is not a trivial task. 2. In later years (1990s) researchers in [10][11][12] reported the use of finite element method (FEM) and charge simulation method (CSM) based approaches to predetermine C s , but, as in early efforts, these also required data about winding geometry. Usually, this information is hard to get, and only available with manufacturers/designers. Even though in both these approaches series capacitance can be estimated, but there was no simple way to cross-verify them. 3. In the past decade or so, FRA has become a de facto monitoring tool of the power utilities to assess the mechanical integrity of the winding [13][14][15][16]. Keeping this in mind, the authors' research group successfully demonstrated the possibility of indirectly estimating C s from measured FRA data, for both single and three-phase transformers [17,18]. This was a major step forward. However, fitting an accurate curve for measured FRA was a crucial and time-consuming step that needed special skills requiring some background knowledge and experience. This was its drawback that prevented its immediate deployment in industry. Some simplifications were suggested by authors in [19], but this approach was non-FRA-based, required additional instruments etc. Specifically, it was a time-domain approach that required exciting the winding by a nearly step-like waveform using a recurrent surge generator.
So, analysis of the published literature on the estimation of C s leads to the following observation: 1. A direct measurement to estimate C s is ruled out. 2. Also, there exists no simple approach for determining C s from measured FRA that can be used by unskilled operators. 3. Consequently, it is imperative to explore alternative methods that would possess the following features: • be simple and easily implementable in software; • be implementable on existing FRA instruments as an addon; • be based on FRA and any other terminal measurement; • avoid complexities like optimisation, curve-fitting etc.; • involve minimum post-processing after acquiring FRA data; • be applicable to all types of uniform windings.
Consequently, developing a method that possesses ALL the aforementioned features is the main objective of this paper.

Methodology
The underlying principle is initially presented for a lossless case. The losses in the winding are neglected [20] while formulating the method and later a method is suggested to overcome its effect. It was possible to put together a simple procedure by a deft manipulation and combination of a few well-known properties that correlate the roots of a polynomial to its coefficients. One special case of the Vieta's theorem [21] is exploited in its formulation. Details are discussed under the following subheadings.

Background
Consider an N-section, lossless, lumped-parameter, mutually coupled ladder-network model (as shown in Fig. 1) of a uniform transformer winding [3]. The model consists of per section series capacitance (C s ), per section shunt capacitance (C g ), per section self inductance (L ii ) and mutual inductance (L i j ) between any two sections. For representing frequency response, the driving point impedance (DPI) function is considered, since it possesses some unique properties like physical realisability, the alternating arrangement of poles and zeros etc. which are very useful in this work. In general, the DPI function for a lossless N-section ladder network (with its neutral open) can be written as [22] Let the numerator and denominator of DPI(s) be defined as follows: Q(S) can be further rewritten as Polynomials on the RHS of (2) and (4) are similar in structure but differ only in their coefficients. The following properties can be easily observed from these two polynomials: 1. The powers of 's' are always an even number, hence, they are both even polynomials. The number of sign changes for P(s) and P( − s) is zero. The same is true for Q(s)/s as well. Hence, from Descartes' rule-of-signs [21] there exist no positive or negative real roots for these two expressions. 2. Since losses are neglected, all the roots of P(s) and Q(s)/s will be purely imaginary and shall exist as complex conjugate pairs. Let the roots of P(s) be ± jω i ′ and all the non-zero roots of Q(s) be ± jω i . Obviously from the definition, since ω i ′s are the roots of the numerator polynomial they correspond to the short circuit natural frequencies (scnf), and likewise, ω i s being roots of denominator polynomial correspond to the non-zero open circuit natural frequencies (ocnf) of the circuit under consideration. Furthermore, these ocnf and scnf correspond to the peaks and troughs in the DPI magnitude plot, respectively.

Linking roots of a polynomial to its coefficients
Next, some unique properties of such polynomials are invoked and then used subsequently to formulate the proposed method. Property 1: If a new polynomial is constructed from a given polynomial by reversing the order of all the coefficients, then the roots of the new polynomial will be the multiplicative inverse of the roots of the original polynomial [21].
Property 2: Starting from an even polynomial, if a new polynomial is constructed by halving the powers of its variable without altering the coefficients, then the roots of the new polynomial will be square of the roots of an original polynomial [21].
Using these two properties, from P(s) construct a new polynomial P 1 (s) such that the coefficients of P(s) are reversed and the powers of 's' are halved, and that leads to Due to Properties 1 and 2, the roots of the new polynomial P 1 (s) will be 1/(jω i ′) 2 . Likewise, from (4), a new polynomial Q 1 (s)/s can be constructed as which would have its roots as 1/(jω i ) 2 . Property 3: (Vieta's Theorem): The symmetric sum of the roots of a polynomial is defined as those sums of roots which are unchanged by any permutation of these roots. There will be N symmetric sums for an Nth order polynomial and the pth elementary symmetric sum of a set of N roots is the sum of all products of p of those roots (1 ≤ p ≤ N). The symmetric sum of roots of polynomial is related with to polynomial coefficients, by the well-known Vieta's theorem [21], which states that, if we Here, the value of p ranges from 1 to N. Hence, N such equations can be derived which relates the roots and coefficients of a DPI function. However, here we are considering a special case, wherein p = N. For the polynomial P 1 (s), if we consider p = N, then the Nth symmetric sum gives the product of all the roots of a polynomial P 1 (s). As P 1 (s) is the numerator of DPI (whose roots are the scnfs by definition), let this term be called Π scnf , and can be written as: It is easy to observe that for both odd and even values of N, the negative sign gets cancelled on both the sides of (8) and hence can be simplified as And, similarly, for (6), we can write Employing the same logic as above, it can be simplified as Next, dividing (9) by (11), we get Taking this ratio is a crucial step in the formulation of the method that will become evident next. The LHS of (12) is a term that consists of entirely measurable quantities extractable from DPI magnitude plot. In other words, this quantity is nothing but a ratio of the product of the squares of peak frequencies to the product of the squares of the trough frequencies.
On the other hand, the terms a 0 , b 0 , a N and b N on the RHS of (12) (being functions of the circuit elements) were computed for the network shown in Fig. 1, for different values of N, say, N = 3, 4, 5, …, 20 using symbolic computation facility in MAPLE. It emerges that all these four terms are functions of C g and/or C s alone, for any N. The expressions corresponding to RHS of (12) can be nicely combined into a compact single expression in γ alone (viz., by substituting γ = C g /C s ). For brevity, these are shown in Table 1, only for N = 3-7, as an example. From a study of these individual coefficients and its ratios (for different values of N), the following salient features can be observed: 1. For any value of N, the numerator and denominator of the ratio a 0 /b 0 can be individually represented as a product of two terms; the first term is purely capacitive and the second term is purely inductive. Most importantly, the inductive term is common to both the numerator and denominator, and hence cancels out. 2. Writing DPI in normalised form b N is a function of C g alone and can be generalised as 4NC g , whereas, a N = 4, always. Hence, b N /a N = NC g . Since a ratio of the coefficients are taken, normalisation does not alter the final results. 3. Hence, the ratio (a 0 /b 0 ) ⋅ (b N /a N ) is always a function of C g and C s alone. 4. In the neutral open condition, there will be an equal number of non-zero peaks and troughs. So, LHS of (12) is a dimensionless quantity. 5. Thus, in compact form, we can write: Which in turn can, in general, be written as where δ i and β i are the corresponding coefficients of numerator and denominator polynomials of f (γ), respectively. For brevity, both these coefficients are shown in Tables 2 and 3 up to N = 17. However, these can be computed and stored for any desirable higher value of N if required. Once the numerical value of Π scnf /Π ocnf is computed from measured DPI, solving (14) directly leads to the value of γ. Since the value of C s and C g cannot be negative, so the value of γ should always be positive. So, before solving (14), the existence of a positive real root can be ascertained using Descartes' rule-of-signs. The total shunt capacitance of a transformer winding can be measured by an LCR meter. So, using this measured value and computed γ, the value of C s can be estimated.

Salient features
This procedure for estimation of C s has the following advantages compared to all other previously published methods in the literature: 1. The entire DPI data need not be processed, but, only data points pertinent to the peaks and troughs are required. In addition, the measured value of total shunt capacitance is the only other data required. 2. No curve-fitting or optimisation exercise is required. In practical DPI measurements, it is often observed that a dominant pole tends to mask nearby non-dominant poles and these have to be carefully considered during curve-fitting. Hence, fitting DPI magnitude is invariably a task that calls for mathematical skills and experience. This is completely avoided in the proposed method. 3. From the point of implementation on an FRA instrument, the proposed method is very simple and elegant. It only requires a lookup table for storing the δ i and β i coefficients (a one-time exercise) and a simple algorithm for finding roots. Hence, the proposed method is ideally suited for use in factories, as well as, in laboratories.
4. The initial impulse voltage distribution constant, popularly termed as α of the winding, is the ratio of the square root of the total shunt to the total series capacitance, and can readily be computed from γ using Finally, it is important to highlight here that α can be calculated without measuring either C g or C s explicitly. To the best of authors' knowledge, this is the first time an indirect measurement-based method is proposed for determining α without the explicit knowledge of either C g or C s .

Limitation of method and means to overcome it
The analytical formulation of the proposed method was built around the assumption that losses in a transformer winding are small enough to be ignored. This is far from true, especially at the higher frequencies, which makes its implementation questionable. This issue arises from the fact that this method presumes peaktrough pair values as extracted from the DPI magnitude are the true value of pole-zero; this is true, if and only if, the losses are negligible. However, when losses become significant at higher frequencies, the peak-trough pairs as observed from the magnitude plot WILL NO LONGER COINCIDE with the true value of polezero, and hence they cannot be directly determined by extracting it from the DPI magnitude plot. Thus, the method will begin to yield erroneous results. To overcome this limitation, authors propose to extract and use all peak-trough pairs that lie below a frequency  3  1  6  9  4  1  8  20 16  5  1  10  35  50 25  6  1  12  54  112  105 36  7  1  14  77  210  294  196 49  8  1  16  104  352  660  672  336 64  9  1  18  135  546  1287  1782  1386 540 81  10  1  20  170  800  2275  4004  4290  2640 825 100  11  1  22  209  1122  3740  8008  11,011  9438  4719 1210 121  12  1  24  252  1520  5814   limit of about 2-2.5 Mrad/s, wherein the presumption is still largely applicable. The real part of the pole or zero is very small to be safely ignored in this frequency interval, a fact which was confirmed by curve-fitting exercise. Most importantly, the imposition of this bound permits implementation of the method as it is. Ignoring higher frequency peak-trough pairs does not seriously affect end results (as will become evident later), compared to the scenario of including all peak-trough pairs which leads to unacceptable errors. This modified approach is used.
(Note: The method was initially developed for a lossless case since it is easy to describe, explain and cross-check. Furthermore, losses in a transformer winding are usually very small and can be neglected for practical purposes at lower frequencies. However, when frequency increases, they tend to become significant and cannot be ignored. For this reason, the proposed method had to be modified to consider the defined ratio of Π scnf /Π ocnf ONLY up to a certain frequency limit. The objective was to develop a simple, and easily implementable method which can be deployed on existing FRA measuring instrument. Hence, this simplification was proposed so that the method is applicable without or with losses).

Simulation
Initially, the proposed procedure was checked by simulation studies using an eight-section (N = 8) mutually coupled ladder network, as shown in Fig. 1. Since all the parameters of this network are known, the capability of the proposed steps can be judged. The elements of the network used were C s = 0.444 nF and C g = 0.25 nF, which corresponds to α = 6. The self-inductance of the first section and mutual inductances between 1st and ith section are given in Table 4. Symmetry considerations are invoked for determining the rest of them. Initially, a lossless case is considered, followed by a lossy case modeled by a series resistance of 10 Ω/ section.

Without loss
After plugging in all these values into a circuit simulation software (PSpice), the DPI magnitude was determined by performing ACanalysis, and is shown in Fig. 2. The peak-trough pairs outputted by Matlab program 'findpeaks' are marked on Fig. 2 and the same is also tabulated in Table 5. Steps for computing C s are as follows: 1. Number of peak-trough pairs were found to be eight, so N = 8.

With loss
Loss was considered by inserting a resistance of 10 Ω/section in series with the inductor. The above computations are repeated. The DPI magnitude plot is shown in Fig. 3, along with the peak-trough pairs in Table 6.

Implementation on uniform transformer windings
Triggered by the success of the proposed method in simulation studies, the next step was to examine its applicability on uniform transformer windings. In all these experiments, DPI was measured by manually sweeping the frequency and measuring the response (viz., input current) at each discrete frequency step. The instruments used for this were as follows: During DPI measurement, as sine waves are being measured, elimination of noise was achieved by using 'averaging option' available on the digital oscilloscope. Furthermore, at each measurement the vertical amplitude of the channel measuring current was dynamically adjusted so that the measured waveform always occupies >90% of the full-scale amplitude of the channel; this guarantees a high signal-to-noise ratio. (Note: The achievable accuracy of the proposed method significantly depends on our ability to accurately identify and extract the DPI magnitude peaks and troughs. So, all the above-mentioned efforts were exercised to attain the maximum possible signal-to-noise ratio and hence achieve the highest possible accuracy.) Initially, experiments were performed on single isolated continuous-disc and interleaved-disc windings. Then, the method is examined on an actual two-winding single-phase testing transformer. Details of each case is presented below.

Case A: single, isolated, fully interleaved-disc winding
A uniform fully interleaved-disc winding was chosen which had 16 discs with 10 turns per disc. The paper insulated copper turn has a cross-section area of 30 mm 2 . The winding had a height of 215 mm, and inner and outer diameters of 260 and 350 mm, respectively. The insulation thickness, duct spacing etc. corresponded to an 11 kV rating. An aluminum sheet was concentrically placed to simulate the ground plane. (Note: The DPI of ONLY a linear system can be defined. In the frequency interval of 10 kHz-1 MHz, the winding behaves as a linear element, since the iron core repels almost all the flux due to eddy currents and in turn, the winding offers a constant inductance value (equivalent to an air-core inductance). Keeping this scenario in mind, the iron core which acts as a magnetic shield in this frequency interval was emulated by a grounded aluminium cylinder.) The measured DPI plot is shown in Fig. 4. The peak-trough pairs were found by 'findpeaks' and they are marked on it for clarity. These frequencies are also tabulated in Table 7. The steps for estimating C s are as follows: 1. Peak-trough pairs in DPI magnitude were four, hence N = 4. 2. Picking up the first two peak-trough pairs (which are below 2.5 Mrads/s) in Table 7, compute Π scnf /Π ocnf = 1.6313. Tables 2 and 3 for N = 4, f (γ) is constructed and equated to Π scnf /Π ocnf . The simplified expression is 6.3686γ 4 + 47.6861γ 3 + 104.5329γ 2 + 56.2190γ − 10.1022 = 0

Using coefficients in
4. Since Π scnf /Π ocnf < 2N, there will exist ONLY one positive real root for (20). 5. Using the positive root of γ = 0.1406, the value of α was computed as 1.4996. This low value of α is a characteristic feature of a fully interleaved-disc winding. 6. The measured value of total shunt capacitance was 0.410 nF at 1 kHz, so shunt capacitance per section is C g = 0.1025 nF. Estimated series capacitance per section was C s = 0.7290 nF.

Case B: 2.2 kV continuous-disc winding in presence of a shorted LV winding
One healthy phase of an HV winding (along with an inner LV winding) was scavenged from an old discarded transformer. It was one of the phases of a Δ − Y transformer of rating 70 kVA, 2200/220 V, 25 Hz. (Note: It is well-known that the measured DPI would be modified due to the presence of a shorted secondary winding, in addition to other surrounding conditions, other neighbouring windings, terminal condition of all non-tested windings etc. However, the value of C s of the excited winding, which would be extracted from the measured DPI, will remain unchanged. This fact has been previously examined in [18] and proved.) An aluminium sheet was concentrically placed inside the LV winding which emulates the presence of core. The non-tested LV winding is shorted and connected to the aluminium sheet. The DPI magnitude was measured and the peak-trough pairs are extracted and tabulated in Table 8. The DPI magnitude plot is shown in Fig. 5, and also contains markings of each peak-trough pair.

Case C: 33 kV, 3.5 MVA, continuous-disc winding
Next, experiments were done on another uniform single isolated continuous-disc winding manufactured specifically for this purpose. This was one of the HV windings of a new 3-Φ, 33 kV, 3.5 MVA transformer. The winding had 24 number of double-discs with 22 turns per disc. Its total height was 570 mm. Spacing between each disc is 3 mm, and the inner and outer diameters of each disc is 395 and 486 mm, respectively. An aluminium sheet was concentrically placed to simulate the ground plane. Measured DPI magnitude is shown in Fig. 6. The peak-trough pairs outputted by 'findpeaks' are tabulated in Table 9.
4. Since Π scnf /Π ocnf < 2N, there exist ONLY one positive real root for (22). 5. The only positive root of γ is 0.7029. 6. This corresponds to an estimated value of α = 11.7375. This is a typical value of α that corresponds to continuous disc winding. 7. The measured total shunt capacitance is 1.15 nF at 1 kHz, hence shunt capacitance per section C g = 0.0821 nF. The estimated series capacitance per section is C s = 0.1169 nF. So, the total series capacitance is 0.0083 nF.

Verification by initial impulse voltage distribution:
As it is well-known that there is no direct method to verify the correctness of the estimated C s , the authors measured the initial impulse voltage distribution and compare it with the one computed using the above-estimated value of C s . This measurement was carried out by exciting the winding by a near-step-like (70 V, 0.27/36 µs) impulse voltage waveform produced using Haefely's repetitive surge generator (RS482). The voltage magnitude at each doubledisk junction was measured corresponding to the time instant at which the input excitation is maximum. Plotting these voltages leads to the initial impulse voltage distribution, which is shown in Fig. 7, along with that computed using estimated C s = 0.1169 nF.
The close match of estimated and measured distributions goes to show that the proposed method has satisfactorily estimated the value of C s .

Verification by CSM:
One more way of verifying the estimated C s was explored. Since, in this particular case, the authors had access to design data (physical dimensions of the winding are given in Fig. 8) this gave an opportunity to compute C s using CSM, and it in-turn can be used to cross-verify the results gotten from the proposed method. The computation procedure and algorithm described in [12] was followed. As per this procedure, initially, the turn-to-turn capacitance matrix of the winding is computed. Then, capacitance between neighbouring discs (interdisc capacitances) is obtained by adding the capacitances existing between all the turns in neighbouring discs. Finally, the series combination of all the disc capacitances provides the net series

Implementability on existing FRA instruments
Given the simplicity of the proposed method, the authors foresee its portability on commercial FRA instrument to be straightforward. The expected major steps/aspects in that regard are listed as follows: • The DPI magnitude data is acquired in a normal fashion.
• The peak-trough detection algorithm used in this work can be converted and ported into the instrument software. This algorithm always detected the peak-trough pairs accurately, except in one case, wherein peak-trough 7-7′ in Fig. 10 was not detected. Thus, this algorithm is robust and can be used.  • Once all peak-trough pairs are identified, N is determined.
• Then, Π scnf /Π ocnf can be determined using all peak-trough pairs that are below the upper bound of 2-2.5 Mrads/s. • From the stored lookup table, pertinent δ i and β i for a given N are used to construct the polynomial f (γ). • A check whether Π scnf /Π ocnf < 2N is necessary to ensure the presence of a single positive real root. • To find roots, the bisection method is a good option to use. An initial guess for the root has to be set. For this purpose, as only the positive root is required, the initial interval for γ can be set as 0-5. • The convergence criterion can be set as 0.001. • Note: This bisection algorithm was implemented and run on all the cases discussed in the previous section. In all runs, the positive root was estimated accurately in less than a few seconds. Thus, this method of finding roots is a satisfactory choice. • Authors expect instrument manufacturers to implement this feature as an add-on option. This being a software addition, it can be retro-fitted.

Conclusions
A simple and elegant procedure for estimating series capacitance of a uniform transformer winding from measured frequency response data was presented. The theoretical aspects of the proposed method were derived by exploiting properties that correlate the roots of a polynomial to its coefficients. Invoking these properties on drivingpoint-impedance function of a winding (modelled as a N-section mutually coupled ladder network) led to the establishment of the proposed procedure. The method was implemented on a variety of uniform transformer windings to check its feasibility. Finally, it was also successfully implemented on an actual single-phase, twowinding transformer. All these experimental results prove its feasibility. The proposed method is free from mathematical complexities, is straightforward to implement, is time-efficient and therefore ideally suited to be deployed on existing commercial FRA instruments, as a software add-on option. This feature is expected to add a new dimension to the FRA instruments.