Frequency response analysis is the starting point for obtaining a methodology for determining the equivalent circuit for transformers and motors. The analysis enables the diagnosis of mechanical faults in electrical machines and the determination of parameters for simulating voltage gradients.
Frequency response in linear systems analysis
Frequency response is a technique employed in the analysis of linear systems. The technique involves presenting the behavior of the analyzed system across a wide range of frequencies. This technique is widely used in Materials Engineering and Electronics to evaluate the quality of components and circuits.
Importance of frequency response analysis for transformers
In nominal operation, distribution and power transformers operate at a single frequency (50 or 60 Hertz) determined by the electrical system to which they are connected. Therefore, their usual approach in engineering courses considers that their equivalent circuit consists of resistive and inductive components, which are predominant at this frequency.
However, like any other electrical equipment, the transformer has capacitive components, as previously discussed. The effect of capacitive components is very small at the nominal operating frequency and is disregarded in the traditional model.
The reactive components (inductances and capacitances) of the transformer are strongly dependent on the winding geometry. Therefore, by measuring these quantities accurately, an indication of mechanical faults in the active part or degradation of insulation can be obtained.
Thus, frequency response analysis is the starting point for obtaining a methodology for determining the equivalent circuit for transformers and motors. The analysis enables the diagnosis of mechanical faults in electrical machines and the determination of parameters for simulating voltage gradients.
Methodology for determining the equivalent circuit
Frequency response analysis is the starting point for obtaining a methodology for determining the equivalent circuit for transformers and motors. Furthermore, this analysis enables the diagnosis of mechanical faults in electrical machines and the determination of parameters for simulating voltage gradients.
Measurement techniques of frequency response
Measurement techniques of frequency response can be classified into two major groups, according to the philosophy of data acquisition. They are: time-domain techniques and frequency-domain techniques, which are presented below.
Time-domain:
(Vaessen, P. T. M.; Hanique, E., 1992) proposed the first work to introduce this type of technique. Time-domain measurement techniques apply a wide spectral range electrical signal to the transformer and measure the filtered signal at a second point. We record the signals through analog/digital conversion and then transform these signals into the frequency domain using the Fast Fourier Transform (FFT). The relation between the applied and measured signals in the transformer is called frequency signature.
The main advantages of this technique are:
Obtaining the signature in a few seconds: The spectrum of the applied signal (usually from 10 Hz to 10 MHz (Feser, K.; Christian, J.; Kachler, A.; Neumann, C.; Sundermann, U.; Loppacher, M., 2000) results in a short duration, less than a second. The processing of mathematical calculations (FFT and convolution) is also carried out very quickly due to the high capacity of today's computers.
Linearly spaced samples: Samples are taken at regular time intervals (sampling rate of the A/D converter). This allows, in a later stage of diagnosis, the use of estimation tools based on the Method of Least Squares (MLS), which are efficient, well-refined, and popular methods. Thus, it eliminates the need for specialists for analysis.
Possibility of "online" use: This measurement technique can be applied without disconnecting or even turning off the transformer. By choosing a spectral signal amplitude much smaller than the nominal operating voltage of the transformer, simply overlay the signal on the equipment's power supply voltage, without affecting the integrity of the electrical system.
However, there are some limitations of measurement techniques in the time domain, including:
Frequency domain undersampling: A fixed sampling rate allows the samples to be equally spaced in time. However, in the frequency domain, it causes the frequency lower than the sampling rate to be over-sampled and the higher frequencies to be under-sampled (Kim, J. W.; Park, B.; Joeng, S.C.; Kim, S. W.; Park, P., 2005).
Over-sampling requires more memory demand in the acquisition system, and under-sampling causes errors in the construction of the signature. As the frequency range for studying transformer signatures is large (from 10 Hz to 10 MHz), choosing a single sampling rate is impractical. It is also difficult to use more than one rate, or a dynamic rate, since the spectral signal is short-lived and the acquisition system would not find time to change the rate during measurement.
Restriction on measurement range: Similarly to the frequency range, the measurement range (amplitude of measurements) for studying transformer signatures is quite wide. As there is no time available to change the range during measurement, the acquisition system would require a high resolution, or more precisely, an A/D converter with a high number of bits. Such system is difficult to find in the market, as it has a high cost and would also limit its use only to laboratories where electrical and environmental conditions are very rigorously controlled.
Susceptibility to noise: At the time of measurement, it is impossible to limit or filter any noise in the acquired signal, as the noise may have the same fundamental characteristics as the applied signal. Interference can be reduced in the signal processing stage after acquisition; however, there may be loss of information in the transformer signature structure.
High voltage variation rates: The broad spectrum characteristic of the applied signal means, in practical terms, that this signal subjects the transformer to high voltage variation rates. This implies high induced voltages at the transformer terminals. Therefore, precautions must be taken to ensure the physical integrity of the equipment involved in the measurement, including the transformer under analysis, and also for the safety of people present during the test.
Frequency domain:
Measurement in the frequency domain consists of applying a known electrical signal of pure frequency (sinusoidal signal) to the transformer and measuring the signal filtered by the transformer, which, by hypothesis, is also sinusoidal of the same frequency. The process is repeated "n" times, changing the frequency of the applied signal. In the end, a set of values that represent the transformer signature is obtained.
The equipment available on the market has high resolution and a wide frequency range in generating sinusoidal signals, from 0.1 Hz to 100 MHz. This more than adequately covers the range for studying transformer signatures (in technical literature, studies conducted on transformers used frequency ranges with a maximum frequency of up to 10 MHz (Oliveira, O. B.; Cerqueira, W. R.; Rocha, A. C. O.; Mendes, J. C., 1997) (Ryder, S. A., 2003) (Sweetser, C.; McGrail, T., 2003). Furthermore, the frequency sampling interval can be easily changed during the measurement runtime, allowing optimization of data without under- or over-sampling.
The main advantages of this technique are:
Wide measurement range: As expected for measuring a sinusoidal signal, the acquisition system does not need high resolution for capture, requiring, for most cases, only an 8-bit converter. It is also possible to change the range and measurement sensitivity during runtime.
Robustness to noise: At the time of measurement, the acquisition system expects a sinusoidal signal of the same frequency as the applied signal, rejection filters can be implemented for other frequencies, provided they do not significantly alter the amplitude and phase of the measured signal. Average-based filters are perfectly suited for this application, with the advantage of not depending on the signal frequency.
Low cost and possibility of field use: The characteristics of this measurement technique result in a low-cost system, where the equipment is commonly found on the market. As the system is robust, it can be applied in the field without the need for strict control over the electrical and environmental conditions at the measurement site.
The main disadvantages of this technique are:
Deficiency in parameter estimation and identification techniques: Parameter estimation is a subsequent step to the measurement and of great importance to establish objective criteria in transformer diagnosis. In section 3.2.5, a technique with good accuracy was presented; however, it is necessary to know the initial values, or at least estimate them, for rapid algorithm convergence. Furthermore, the work (Zambrano, G. M. V.; Ferreira, A. C.; Calôba, L. P., 2006) did not reveal the computational effort time spent on this operation.
Because the samples were taken directly in the frequency domain and they are not linearly spaced, there are not many efficient estimation techniques for this approach. Thus, the diagnosis would be based on subjective criteria, usually on the experience of specialists.
It cannot be implemented "online": As it uses sinusoidal signals, measurement in the transformer would be masked by its operating voltage, which is also sinusoidal. It is not a strong disadvantage, as even the time-domain techniques, seen earlier, still do not have functional implementation in online systems.