**Time-Frequency Analysis of Complex Space-Phasor in Power Electronics**^{ }^{} Zbigniew Leonowicz, Tadeusz Lobos, Tomasz Sikorski

*Abstract* - This paper considers applying non-parametric and parametric methods for calculation of time-frequency representation of non-stationary signals in power electronics. Space-phasor is proposed as a complex representation of 3-phase signal in order to calculate spectrum of positive and negative-sequence components. The developed methods are tested with non-stationary multiple-component signals occurring during the fault operation of inverter-fed drives. Firstly, the main differences between parametric and non-parametric methods are underlined. Then some aspects of additional kernel functions are introduced on the basis of Wigner and Choi-Williams distributions. Additional comparison of uncertainty of measurements using described methods, as well as a widely used Fourier technique can be useful for practitioners. Proposed methods allows tracking instantaneous frequency as well as magnitude that can lead to applications in diagnosis and power quality area. *Index Terms* – Time-Frequency analysis, power system harmonics, power electronics, Wigner distribution, subspace method, measurement uncertainty## I.INTRODUCTION

Representation of signals in time and frequency domain has been of interest in signal processing areas for many years, especially when analysing time-varying non-stationary signals. This kind of representation attracts nowadays more attention also in electrical engineering. The main motivations which incline to joint time-frequency analysis originate from character of the signals which appear in power systems and also from constantly increasing capabilities of signal processing methods.

Modern frequency power converters generate a wide spectrum of harmonics components, which can deteriorate the quality of the delivered energy, increase the energy losses and decrease the reliability of a power system. In some cases, large converters systems generate not only characteristic harmonics typical for the ideal converter operation, but also considerable amount of non-characteristic harmonics and interharmonics which may strongly deteriorate the quality of the power supply voltage. Interharmonics are defined as non-integer harmonics of the main fundamental under consideration. The estimation of the harmonic components is very important for control and protection tasks [10,11].

The standard method for study time-varying signals is short-time Fourier transform (STFT) which is based on the assumption that for a short-time period the signal can be considered as stationary. The spectrogram utilizes short-time windows whose length is chosen so that over the length of the window the signal is stationary. Then, the Fourier transform of this windowed signal is calculated to obtain the distribution of the frequency components over the frequency spectrum at the time corresponding to the centre of the window. The crucial drawback of this method is that the length of the window is related to the frequency resolution. Increasing the window length allows improving frequency resolution but it causes that the nonstationarities occurring during this interval will be smeared in time and frequency. This inherent relationship between time and frequency resolution becomes more important when one is dealing with signals whose frequency content is changing rapidly.

The time-frequency characterization of signals that can overcome the above-mentioned drawback became a major goal for signal processing areas. Observing recent approaches to the time-frequency representations we can distinguish two main groups, namely, non-parametric and parametric methods. Further, due to different structure of definition of the equation the non-parametric methods can be divided into groups, which carry out the linear or non-linear operation on the signal. If there is a need to scale the time or frequency argument we treat the representations as a scalogram or spectrogram respectively [7,12,13].

The first suggestions for designing non-parametric, bilinear transformations were introduced by Wigner, Ville and Moyal at the beginning of nineteen-forties in the context of quantum mechanics area. Next two decades beard fruit of significant works by Page, Rihaczek, Levin, Mark, Choi and Williams, Born and Jordan, who provided unique ideas for time-frequency representations, reintroduced to signal analysis [6,13]. Then in nineteen-eighties Cohen employed concept of kernel function and operator theory to derive a general class of joint time-frequency representation. It was shown that many bilinear representations can be written in one general form that is traditionally named Cohen’s class [5].

According to the quoted above development of non-parametric two-dimensional transformations it is worth discussing details about Wigner and Wigner-Ville distribution, which can be treated as a basic equations of one family. Time-varying spectra obtained using the Wigner Distribution (WD) shows better frequency concentration and less phase dependence than Fourier spectra [4,5]. Phase dependence can be defined as the frequency estimation error when the initial phase of many harmonic components varies. However, for a multi-component signal, which can be represented as a sum of mono-components, the time-frequency representation is composed of distributions of each component (auto-terms) and the interactions of each pair of different components (cross-terms). Cross-terms are located between the auto-terms and have an oscillating nature. They reduce auto-terms resolution, obscures the true signal features and makes interpretation of the distribution difficult. One way of lowering the cross-term interferences bases on the convolution of WD with a smoothing kernel [1,2,4]. Such approach is used in the case of Choi-Williams Distribution (CWD) when exponential weighted function is applied to suppress cross-term components [3].

Considering parametric methods of spectrum estimation, which are based on the linear algebraic concepts of subspaces, leads to so-called ''subspace methods''. Its resolution is theoretically independent of the signal-to-noise ratio. The model of a signal is a sum of random sinusoids in the background of noise of a known covariance function. It was shown that the eigenvectors of the correlation matrix of the signal may be divided into two groups, namely, the eigenvectors spanning the signal space and eigenvectors spanning the orthogonal noise space. The eigenvectors spanning the noise space are, in general, the ones whose eigenvalues are the smallest and equal to the noise power. One of the most important technique is the min-norm method [7,12]. In order to adapt this high-resolution method for analysis of non-stationary signals we use a similar approach as in short-time Fourier transform. The time-varying signal is broken up into minor segments with the help of the temporal window function and each segment is analyzed independently.

In authors’ opinion there is urgent need for better parameter estimation of distorted electrical signals that can be achieved by applying the time-frequency analysis. Mentioned goal is strongly supported by an issue of energy quality and its wide-understood influence on energy consumers as well as producers. The proper estimation of signal components is therefore very important for control and protection tasks.

In the paper we present results of investigations of a converter-fed induction motor drive under transient working conditions. Proposed approach includes representation of 3-phase system by complex space-phasor and its subsequent time-frequency analysis using Min-Norm method and Wigner as well as Choi-Williams Distributions. General purpose of the work is to emphasize the advantages and disadvantages of proposed methods in point of their application for time-varying spectral estimation in electrical engineering. Especially the significance of the calculations in case of space-phasor analysis is underlined which allows to analyze separately the respective nonstationarities occurring in negative and positive sequence components.

## II.PROPOSED APPROACH

*Complex Space-Phasor*

Having a 3-phase voltage system

we can define complex space-phasor

given by [8,9]:

It describes, in addition to the positive-sequence component, an existing negative-sequence component, harmonic and non-harmonic frequency components of the signal.

Introducing the zero sequence component

, gives complete and unique representation of 3-phase system. Figs. 1a and 1b illustrate an example of 3-phase signal containing 1-st and 5-th harmonics (THD=45%) as its trajectory in the complex space-phasor plane.

*Min-Norm Method *

It is assumed that the data can be modeled as a sum of

*M* complex sinusoids in complex white Gaussian noise.

for

*n = *0,1

*,...,N*-1

z[n] is complex white Gaussian noise with zero- mean and variance

.

*A*_{i} – amplitudes

*f*_{i} – frequencies and

- phases.

The

*N×N* autocorrelation matrix [7] for

* N > M* is:

where

**P**_{i} stands for powers of each complex sinusoid,

**e**_{i} for eigenvectors of the autocorrelation matrix and

** I** for identity matrix (* means complex conjugate and

^{T} matrix transposition).

is the sum of a signal autocorrelation matrix

and a noise autocorrelation matrix

.

The frequency information is contained within the matrix

. The matrix

can be decomposed into its eigenvectors and eigenvalues. The eigenvectors corresponding to the

*M* largest eigenvalues contain information about signal parameters. To extract the information it is also possible to use the property of the orthogonality of eigenvectors. It is said that eigenvectors containing the information about signal span the signal subspace and the remaining span the noise subspace. The signal vectors are all orthogonal to all vectors in the noise subspace. The earliest application of this property is the Pisarenko Harmonic Decomposition (PHD). Because of difficulties with practical applications of the PHD method other subspace methods are in use, such as Min-Norm method.

The matrix of eigenvectors can be defined using noise eigenvectors

as:

*N-M* smallest eigenvalues of the correlation matrix (matrix dimension is

*N*>

*M*+1) correspond to the noise subspace and

*M* largest (all greater than the noise variance

) corresponds to the signal subspace.

Min-norm method uses only one optimal vector

**d** for frequency estimation. This vector, belonging to the noise subspace, has minimum Euclidean norm and his first element equal to one. We can present

in the form:

where

is the upper row of the matrix. Hence,

, where

(

is an auxiliary vector which satisfies the condition

). The above conditions are expressed by the following equation:

Pseudospectrum (not a true spectrum, because it does not contain any information about the true energy of the signal) is defined with the help of

**d** as:

where

.

Since each of the elements of the signal vector is orthogonal to the noise subspace, the quantity (Eqn. ) exhibits sharp peaks at the signal component frequencies.

In order to adapt this high-resolution method for analysis of non-stationary signals we use similar approach as in short-time Fourier transform. The time varying signal is broken up into minor segments (with the help of the temporal window function) and each segment (possibly overlapping) is analyzed. The denominator of (Eqn. ) is estimated for each time instant. Instantaneous estimates of

can be used as estimates of the instantaneous frequency of the signal.

*Wigner Distribution and Choi-Williams Distribution*

The Wigner (WD) and Wigner-Ville (WVD) distributions are the time-frequency representations given by [1,2,4,6,13]:

where:

*t* - time,

*ω* - angular frequency,

*τ* - time shift,

- analysed real signal,

analytic signal,

, with orthogonal imaginary part obtained using Hilbert transform.

The most prominent component of quoted equations is so called instantaneous autocorrelation function

. Fourier transform executed along time-shift variable brings a two-dimensional plane which represents the distribution of the frequency components, here called auto-terms (a-t), over the frequency spectrum at particular time. Unfortunately, bilinear nature of discussed equation manifests itself in existence of undesirable components, called cross-terms (c-t). Cross-terms are located between the auto-terms and have an oscillating nature. When the real signals are investigated, which are characterised by smooth spectrum, the problem of undesirable cross-terms appear to be interactions between components localized in negative and positive part of frequency axis. Thus, in case of real signal analysis it is proposed to preprocess the signal into its complex analytic form which spectrum has zero values in negative part of the frequency axis. Such approach is characteristic for Wigner-Ville Distribution. It is worth emphasizing that discussed approach decreases the number of cross-terms, however it can not be applied when the complex signals are investigated.

This paper provides an idea to apply time-frequency representations for analysis of complex space-phasor. The sense of proposed approach is based on the knowledge that spectrum of the space-phasor contains information about positive and negative-sequence components simultaneously, along the positive and negative part of frequency axis, respectively. Thus, time-frequency analysis would track the changes of positive and negative sequence components simultaneously. Precedent discussion is illustrated in Fig. 1c which illustrates the Wigner Distribution of complex space-phasor representing a 3-phase signal with positive sequence 1-st harmonic and negative sequence 5-th harmonic.

Discrete-time signal x(

*m*), with the sampling period normalized to unity, leads to the discrete-time Wigner distribution [2].

where variables

*m* and

*n* correspond to the discrete-time and discrete time-shift respectively. If we further digitize the frequency, where the variable

*k* corresponds to the discrete frequency, the discrete WD becomes:

Observing the exponential operator it is worth underlining that Wigner Distribution allows to track correctly time-frequency plane up to a quarter of sampling frequency. It is the second feature of non-parametric bilinear transformations which distinguish the method from classical spectrogram or parametric techniques.

As previuously mentioned, Wigner Distribution can be treated as a basic transformation within the non-parametric family which allows for further modifications. These additional modifications are oriented to suppress the cross-terms and finally to adapt particular transformation to the analyzed signal.

One representative of Cohen’s family is the Choi-Williams Distribution (CWD). Lowering the cross-term interferences is achieved here by the convolution the integrant of WD with Gaussian smoothing kernel in form

[1,3,6,13]:

where:

*t* – time,

*ω* – angular frequency,

*τ* – time lag,

*θ* – angular frequency lag,

*u* – additional integral time variable.

One crucial property of kernel function is smoothing effect of the cross-terms with preservation of useful distribution’s properties. Introduced by Choi and Williams, Gaussian kernel belongs to a subclass characterized by a very specific structure of the kernel function that can be treated as a one-dimensional function of product of

. From the mathematical point of few such kernel functions allow suppressing the effect of undesirable cross-terms.

## III.INVESTIGATIONS AND RESULTS

*Fault operation of the inverter drive*

In the paper we show investigation results of a 3kVA-PWM-converter drive with a modulation frequency of 1 kHz supplying a 2-pole, 1 kW asynchronous motor (supply voltage 220 V, nominal power 1,1 kW, slip 6 %, cos

*φ*=0.81). Characteristic RC-damping components at the rectifier bridge and at the converter valves are considered. To design the intermediate circuit, the L, C values for a typical 3 kVA converter are chosen. Fault operation of the converter drive is considered as a short-circuit between motor leads (R and S) which occurs at the time 19ms. Main frequency of the converter is 60 Hz. Sampling frequency is 5 kHz. 3-phase current signal at the converter output during the short-circuit is illustrated in Fig. 2. Complex space-phasor of such signal was calculated on basis of Eqn. and its trajectory under transient condition is presented in Fig. 3.

Time-frequency representations of the complex space-phasor were investigated using parametric Min-Norm method with a sliding temporal window and Wigner as well as Choi-Williams distributions from non-parametric group.

When the Min-Norm method was applied the time-frequency representation was calculated from the time interval of 40 samples. Sliding the window along the signal we obtain instantaneous spectrum of calculated space-phasor, as shown in Fig. 4a. After the short-circuit the Min-Norm method enabled to detect two intermodulation frequencies (880 Hz and 1120 Hz), and two additional components (1920 Hz and 2070 Hz). Details of tracked instantaneous spectrum for particular time after the short-circuit can be observed in Fig. 4b. This figure can be treated as a cross-section of obtained representation. Additionally, the comparison with classical power spectral density (PSD) is delivered. Proposed method is characterized by more sharp detection of investigated components than classical Fourier algorithm.

The results obtained when the Wigner distribution was applied are shown in Fig.5. Time-frequency plane shown in Fig. 5a underline the problem, discussed in previous section, of cross-terms which strongly deteriorate the image of tracked auto-term components. Observing the details with the use of cross-sections. Figs. 5b and 5c, it can be found that before and after the fault the basic component 60Hz and the modulation component 1000Hz exist in positive-sequence. Additionally, 880Hz component is discovered in positive-sequence and –1120Hz component in negative-sequence. After the fault 1120Hz component appears in positive-sequence. Arising asymmetry brings also a group of negative-sequence harmonics (-60Hz, -880Hz, -1000Hz, -1120Hz) which can be observed in cross-section illustrated in Fig. 5b.

The influence of additional smoothing kernel function is presented on the basis of Choi-Williams Distribution with Gaussian kernel. Comparing Fig.5a, illustrating time-frequency plane obtained when Wigner Distribution was applied, with Fig. 6a, which show time-frequency plane obtained using Choi-Williams Distribution, we can notice suppression effect of cross-terms. Attached cross-sections after the short-circuit, Fig. 6b, and before the event, Fig. 6c, underline influence of the Gaussian kernel on lowering the cross-terms participation. Simultaneously, observation of the appearing components confirms the results obtained using Wigner Distribution and Min-Norm method.

Additional comment requires the range of observed frequency. Min-Norm method similarly as Fourier-based technique allow to track the instantaneous spectrum up to the half of sampling frequency i.e. to 2500Hz. Bilinear transformation calculate s instantaneous spectrum up to the quarter of sampling frequency. This is the reason why applying Wigner and Choi-Williams Distribution could not detect higher harmonics 1920 Hz and 2070 Hz, recognized by other methods.

*Uncertainty of measurements*

The comparison of uncertainty of measurements using all described methods, as well as a widely used power spectrum estimator (FFT-based method) can be useful for practitioners, helping to choose a method which provides most accurate results under special requirements of a given measurement setup. Many publications are devoted to the theoretical assessment of performance of the above-mentioned methods. Most of the complicated formulas are derived under restrictive assumptions, such as e.g. equal power of harmonic components, very limited number of components, and so on. From the engineering point of view, in the area of interest of the authors, it is more important to know what are the limits of each method and how the accuracy is affected in usual experimental setup.

In power systems, the analyzed waveforms usually consist of many harmonic components, sometimes with low-amplitude, sub- or interharmonics added [11]. Such signals are not difficult to analyze using FFT-based methods, provided that a long recording of a stationary signal is available. Such assumption is often not fulfilled, since many fault-mode or transient state records contain highly nonstationary components of relatively short duration.

The following experiment is designed to compare the uncertainty in time and frequency of parameter estimation (amplitude and frequency of each signal component). Testing signals are designed to belong to a class of waveforms often present in power systems. The signals have following parameters:

one 50 Hz main harmonic with unit amplitude,

random number(from 0 to 8) of higher odd harmonic components with random amplitude (lower than 0.5) and random initial phase,

sampling frequency 5000 Hz,

each signal generation repeated 1000 times with re-initialization of random number generator,

SNR=20 dB if not otherwise specified,

size of the correlation matrix = 50 for Min-Norm method,

signal length 200 ms if not otherwise specified.

Several experiments with simulated stochastic signals were performed, in order to compare different performance aspects of parametric (Min-Norm) and non-parametric methods (WVD, CWD and power spectrum ). Each run of frequency and amplitude estimation is repeated many times (Monte Carlo approach) and the mean-square error (MSE) of parameter estimation is computed. Selected results are presented in Figs. 7,8,9. From the analysis of results presented in Fig. 7 it follows that parametric Min-Norm method shows very high accuracy in frequency estimation but relatively low in amplitude estimation (this is most likely caused by the inaccuracies in estimation of the autocorrelation matrix when its size is limited to 50). In Fig. 8 the problem of “masking” of harmonics with low amplitude by high-amplitude components is addressed. The error of estimation is plotted, when gradually increasing partially random the amplitude of higher harmonics.

Additional kernel function has an influence on uncertainty of measurements. A relatively high error in the case of Choi-Williams algorithm depending on noise level, amplitude of higher harmonics as well as measuring window length can be observed in Figs.7,8,9. Comparing non-parametric family with Min-Norm method we can notice better accuracy of frequency estimation in case of parametric family. On the other hand, non-parametric methods allow estimating amplitude of tracked components with better accuracy than Min-Norm technique.

## IV.CONCLUSIONS

Proposed methods can be treated as a comprehensive diagnostic tests of signal distortion in electrical engineering area. Additional degree of freedom which bring two-dimensional time-frequency representations allow to track distribution of frequency components over the frequency spectrum parallel with dynamics of investigated phenomena over the time. It acquires significance when transient and non-stationary phenomena are investigated.

Proposed in the paper idea of initial transformation of investigated 3-phase signal into its complex space-phasor form gives the possibilities of simultaneous observation of positive and negative-sequence, represented along positive and negative part of frequency axis, respectively.

Non-parametric bilinear transformations, such as Wigner or Choi-Williams distribution, belong to first group of possible solution which enable to track instantaneous spectrum. Formula of the equation contribute to the time-frequency representation oscillating components which has no physical meaning and obscure desirable view of tracked real components. The most prominent contribution of undesirable cross-terms is observed in case of Wigner Distribution. Applying additional kernel function into definition brings smoothing effect of the cross-terms. Example of mentioned direction is Choi-Williams Distribution where Gaussian kernel function is responsible for lowering interferences between tracked components. However, additional kernel function has an influence on uncertainty of measurements. Practical comparison of Wigner and Choi-Williams distributions allow to notice quite sensitivity to influence of noise, amplitude of higher harmonics and measuring window length on accuracy of frequency and amplitude estimation when Choi-Williams equation was adapted.

Comparing non-parametric family with Min-Norm method we can notice better accuracy of frequency estimation in case of parametric family. On the other hand, non-parametric methods allow to estimate amplitude of tracked components with better accuracy than Min-Norm technique.

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**FIGURES:** Fig. 1. An example of 3-phase signal with the 5-th harmonic (a); trajectory of complex space-phasor (b) and its Wigner Distribution (c)

Fig. 2. 3-phase current signal at the converter output during a two-phase short-circuit

Fig. 3. Trajectory of complex-space-phasor of signal from Fig. 2

Fig. 4. Time-frequency representation of the complex-space phasor from Fig. 3, obtained using Min-Nom method with sliding window (a) and its cross-section for t=22ms (b).

Fig. 5. Time-frequency representation of the complex-space phasor from Fig. 3, obtained using Wigner Distribution (a) and its cross-sections for t=22ms (b) and t=10ms (c).

Fig. 6. Time-frequency representation of the complex-space phasor from Fig. 3, obtained using Choi-Williams Distribution (a) and its cross-sections for t=22ms (b) and t=10ms (c).

Fig. 7. Influence of noise level on accuracy of frequency (a) and amplitude (b) estimation.

Fig. 8. Influence of amplitude of higher harmonics on accuracy of frequency (a) and amplitude (b) estimation.

Fig. 9. Influence of measuring window length on accuracy of frequency (a) and amplitude (b) estimation.

**LIST OF ALL FIGURES’ CAPTIONS:** Fig. 1. An example of 3-phase signal with the 5-th harmonic (a); trajectory of complex space-phasor (b) and its Wigner Distribution (c)

Fig. 2. 3-phase current signal at the converter output during a two-phase short-circuit

Fig. 3. Trajectory of complex-space-phasor of signal from Fig. 2

Fig. 4. Time-frequency representation of the complex-space phasor from Fig. 3, obtained using Min-Nom method with sliding window (a) and its cross-section for t=22ms (b).

Fig. 5. Time-frequency representation of the complex-space phasor from Fig. 3, obtained using Wigner Distribution (a) and its cross-sections for t=22ms (b) and t=10ms (c).

Fig. 6. Time-frequency representation of the complex-space phasor from Fig. 3, obtained using Choi-Williams Distribution (a) and its cross-sections for t=22ms (b) and t=10ms (c).

Fig. 7. Influence of noise level on accuracy of frequency (a) and amplitude (b) estimation.

Fig. 8. Influence of amplitude of higher harmonics on accuracy of frequency (a) and amplitude (b) estimation.

Fig. 9. Influence of measuring window length on accuracy of frequency (a) and amplitude (b) estimation.