Understanding Frequency Response of Induction Motor Winding through Electromagnetic Wave Equations

. Frequency response analysis offers an insight about the integrity of machine windings, when employed as a tool for condition monitoring. To ensure that, an electromagnetic wave is injected from one terminal of winding, and the power of the wave at the receiving terminal is measured. The power at the terminals is measured in terms of either voltage or current. This difference in power at the two terminals can be attributed to the medium’s permittivity, permeability and conductivity, through which the signal is being transmitted. This paper offers an explanation for the behavior of the voltage gain frequency response of induction motor winding and propagating medium parameters by employing the fundamental electromagnetic wave equations. Their explanation illustrates how these parameters can affect the response. The correlation established using Maxwell’s equation and these parameters with frequency response analysis is evident while identifying open winding fault and issue with machine core inductance. The results are analyzed and interpreted with the new correlation.


Introduction
An electromagnetic wave (Eq. 1) is a physical phenomenon that propagates in both space and time through a medium.These waves transfers energy from one end to another, with or without a conductor.One such example is the transmission line.This electrical energy is consumed globally, supplied via transmission lines and is utilised to power a variety of industries.
A(x, t) = A max cos(k x − ω t) Approximately 85% of the industry's electrical machines are induction motors, primarily squirrel cage motors.Their robust structure and ease of control using power electronic converters are the primary reasons for their high demand in low, medium and high power industrial applications [1].
These motors are used in various mechanical applications, such as driving draught pumps in power plants, pumping water to cooling towers, driving conveyor belts, etc. .The absence of the brush & slipring arrangement in squirrel cage machines results in spark-free operation; this makes them ideal for use in explosion prone environments, such as the oil industry, petroleum industry, and coal mining, among others [2].
For such important operations, well-conceived maintenance strategy will guarantee a safe and uninterrupted operation in the industry.Maintenance engineers most frequently employ three maintenance strategies viz.Breakdown maintenance, Time-based maintenance & Condition-based maintenance.The maintenance method is selected and implemented based on the machine's size, redundancy, importance to the system and it's financial impact on the industry.
Researchers and scientists have developed condition monitoring techniques to detect mechanical anomalies in an induction motor over the past few decades.These methods compare the effect caused by a specific parameter during normal and abnormal operations to ascertain the nature and cause of a fault [3,4,5].
Motor current signature analysis and the vibration analysis are the methods widely studied in the domain of condition monitoring of induction motor as they are non-invasive and online methods.Observing the partial power spectrum of the induction motor [6] and measuring axial emf with a search coil [7] are among a few of the many techniques described in the literature.
Motor current signature analysis monitors the frequency spectrum of stator current for a fully loaded induction motor [8,9,10].The presence of specific frequency components aids in the diagnosis of machine rotor broken bar, bearing, and eccentricity faults.
Vibration sensors attached to the machine's body are used to monitor the mechanical vibrations in the machine [11].The recorded vibration spectrum is processed via wavelet transform or short-time Fourier transform in order to identify mechanical defects.
The current waveform or vibration in a machine is dependent on a number of other parameters, such as supply voltage, machine design parameters, loading condition, etc.This implies that the presence of faultdetection frequency components in the current signal or vibration signal may result from the aforementioned uncertainties, thereby creating a false impression of a mechanical fault.This shortcoming of current signature and vibration analysis prompted authors to investigate and implement frequency response-based diagnosis in induction motor.
The frequency response is an offline, non -invasive condition monitoring technique; employed widely to detect mechanical deformations in a transformer [12,13,14].Multiple sinusoidal low voltage signals are used to energize the winding at one terminal and the responses to these signals are recorded at another terminal (based on the circuit connections).The principle behind this technique is electrical resonance [15], in which the R, L and C parameters of the winding, leads to the presence of peaks and troughs, based on the connection arrangements used for the measurment.
Frequency response analysis is a comparison-based technique in which the traces are compared to a reference trace, also known as baseline data, which is typically provided by the manufacturer.When baseline data is unavailable, a comparison is conducted between the trace of an identical unit or among the phases of the same unit, assuming the phase winding is congruent.This test is performed to verify the winding's electrical and mechanical integrity after various incidents, such as lightning strikes, seismic events, post-transportation assembly, etc.; used for routine diagnostics, etc. Diagnostics for critical induction motors can also be conducted using an equivalent approach.
The authors here are listing some of the work carried out in the domain of transformer diagnosis using freqeuncy response which the readers can work upon in the case of induction motor diagnosis.
-The comparison of frequency with baseline data can be conducted visually, or with the aid of statistical indicators such as the Arithmetic Sum of Logarithmic error (ASLE), Variance, Min-Max ratio, Compartitive standard deviation (CSD), ttest, Root Mean Square Error (RMSE), and others [16,17].
-A R-L-C ladder network exhibiting the similar frequency behaviour as the physical system is synthesized and described for driving point impedance connection in [18,19].
-The frequency response spectrum is divided into small sections, which attributes the faulty component of the transformer by varying the circuit parameters of the R-L-C network of the transformer, which helps to identify the fault component [20].
-The analysis of frequency response in terms of poles and zeros of the transfer function is studied and reported for normal and abnormal transformer conditions [21,22].
-The behaviour of frequency response due to other factors such as connection cables, terminating resistances, environmental effects, etc. is studied and reported in the literature [23].
The method of FRA is used for the detection of stator winding faults in induction motors [24,25,26,27,28,29], also, the effect of winding connection arrangement on the frequency response for the winding fault is demonstrated and studied using various statistical indicators [30].The frequency response is also, used to determine the differential-mode, common-mode, and zerosequence impedances of the induction motor to study the effect of shaft voltage and bearing currents in PWM inverter drives.The high-frequency circuit models of induction motor representing the frequency response for the listed impedances is available in [31,32].However, these circuits are exclusively used for the study of shaft voltages and bearing currents and are not suitable for the diagnosis, as the frequency response connection arrangements may differ to the measurement connection used for differential mode, common mode and zero-sequence impedances.The influence of rotor position on the frequency response of a rotating machine is discussed in [33].
In this paper, the authors present the voltage gain frequency response of an induction motor winding and propose a novel explanation for the response's behaviour employing the fundamentals of electromagnetic wave propagation.
In the subsequent sections, the authors explained the experimental setup developed in the laboratory and the summary of the experiments carried out; the authors have illustrated and explained the frequency response of an open-turn fault in an induction motor winding using the proposed explanation; in addition, the effect of core inductance on the frequency response of the machine winding is studied, shown and explained using electromagnetic wave equations.The proposed explanation is used to identify and distinguish the open turn fault in the winding apart from the fault introduced due to magnetic core in the machine.

2.
Frequency response of Induction machine

Experimental setup
The frequency response measurement system is developed in the laboratory using an Arbitrary Function Generator (AFG) and a Digital Signal Oscilloscope (DSO) (Tab.1); both the instruments are programmable and communicated over USB using MAT-LAB [34,35].
The AFG is configured to produce sine wave with frequency ranging from 20 Hz to 1 MHz.The study employs a total of 1200 frequency points, logarithmically spaced over frequency range [36].The voltage sine wave is injected into one of the coil's terminals.The magnitude of the gain is computed by capturing the peak values of the input voltage (V i ) and output voltage (V o ), i.e. 20 * log 10

Vo
Vi .In addition, the phase difference of the system is computed by identifying the peak time instant for V o and V i .Gain and phase difference are plotted with their respective frequency and the resulting trace represents the frequency response of the coil.The discretization effect and measurement noise are reduced by processing captured waveform with a sine wave curve fitting algorithm [37].
Another method for capturing frequency response is described in [38], which provides only gain-magnitude data.This method of measurement is faster than the system described above.

Connection arrangements
The frequency response of three low-powered induction machines are investigated.The studied machines' stator windings are distributed and randomly wound.
The connection configuration for measuring frequency response depends on the accessibility of the winding terminals.
There are two possible connection configurations for measuring the frequency response of an induction motor.
Winding measurement: By injecting the voltage signal at one terminal and observing the output voltage response at the subsequent termination of the winding, the frequency response of the winding is determined.Other windings apart from the test winding are left open (Fig. 1(a)).
Line measurement: In this configuration, the response of series-connected coils is measured.The diagram of the connection is shown in Fig. 1(b).The stator winding contains an inaccessible internal star point.
Machine 1 has an open turn fault in one of the phases, whereas Machines 2 and 3 are used to examine the effect of core inductance on the induction machine frequency response.The specifications of the machine under study is shown in Tab. 2 and the machine's connection configuration and the type of research conducted are detailed in Tab. 3.    The Fig. 2 depicts a typical frequency response of healthy induction motor [27]; due to the presence of one resonance point in the trace (Fig. 2), the trace is divided into two parts , i.e. a segment with ω < ω r and the other segment with ω > ω r , where ω r is the resonating frequency.The response can be further divided into multiple regions, based on the presence of resonance frequencies.
The behaviour observed in the trace (Fig. 2) of the winding can be explained using the transmission line analogy and applying Maxwell's equation for electro-  magnetic wave propogation (Eq. 2 & Eq. 3).
In a transmission line, the frequency of the wave is the same at both ends, but the velocity (Eq.5) of the wave as it progresses through a medium varies depending on the medium's permeability (µ) and permittivity (ε).Consequently, a wave travelling through different mediums will experience a change in wavelength, resulting in a phase difference between the sending end wave and the receiving wave.
The downward slope observed in region 1 (ω < ω r ) of Fig. 2 is due to the dominant conduction current component (Eq.3).The (Eq. 6), indicates that the conduction current is independent of the frequency but, the relative permeability (µ r ) of a ferromagnetic material is frequency sensitive, as reported in [39].This study has reported that µ r is constant up to 1 × 10 3 Hz and then begins to decline, reaching unity for frequencies greater than 1 × 10 5 Hz.This decline results in the reduction of conduction current component, resulting in a downward slope in voltage gain of the induction motor.Here, the consequences of skin effect on conductivity of the winding is neglected.
In the region 2 of the frequency spectrum (ω > ω r ), the displacement current component Eq. 7 dominates, which is dependent on the wave frequency (ω) and the medium parameters; permittivity (ε) and permeability (µ) of the medium.In this range of testing, the medium's permeability is now equal to that of vacuum and remains constant.In addition, the permittivity(ε) of the medium (insulators) is constant over the test frequency range [40].Consequently, the velocity of the traversing wave is constant, and the magnitude of the displacement current is frequency-dependent; hence the rise in output voltage V o , which can be observed as a rising slope in region 2 of frequency response.

Fault identification
The frequency response of machine 1 is depicted in Fig. 3. Since neither the baseline data nor the sister unit is available, the fault is identified by comparing the frequency response of individual windings.In the low-frequency region, winding R − R 1 lacks the initial downward slope, and its output voltage V o is lower than that of the other windings.
Referring Sec.2.3., the conduction current contributes primarily to the output voltage in region 1 of the spectrum.Variation in medium's conductivity (σ) or permeability (µ) may account for the reduced conduction current.
All of the phase windings are wound on a single core, and the presence of a downward slope in the traces of the other two windings indicates that the core is healthy.As the medium's permeability (µ) remains unchanged, the conductivity (σ) of the conductor must have changed.Since the conduction current decreases, and based on the Eq. 6, we can infer that the conductivity (σ) of the medium is reduced, indicating an open winding.A rise in the voltage gain in region 1 of R−R 1 trace, is due to the dominant contribution of the displacement current over conduction current component.
A visual examination of the machine revealed that a few turns of the phase R winding have burnt (Fig. 4).Also, the measured resistance values for all the windings are presented in Tab. 4.

Effect of rotor core
In order to examine the effect of the rotor core on the frequency response, the frequency response of ma- chine 2 is measured under two conditions: (i) with the rotor assembly and (ii) without the rotor assembly.Measurement is performed using the winding connection configuration.The frequency response for both cases is illustrated in the Fig. 5. Gain variation is observed among cases in the low-frequency region of the frequency response spectrum, as anticipated.
In case (i) the medium permeability µ med1 is the result of all three cores, i.e., stator core (µ st ), airgap(µ ag ) & rotor core (µ rt ), whereas in case (ii), the medium permeability µ med2 is the result of µ st & µ ag .
As, the traversing medium involving the rotor has a greater permeability (µ med1 > µ med2 ), lower current is required to generate the same amount of flux, compared to case (ii).Hence, the lesser output voltage V o .This anomaly is observed in the Fig. 5, wherein the response for case (i) is lower compared to case (ii).For the high-frequency region, the traces are identical in both cases.As discussed previously, medium permeability is µ med1 = µ med2 = µ 0 .In addition, ε is constant; consequently, the wave velocity for a given frequency is identical for both cases, making the responses identical.
To confirm the same, a similar study is conducted on machine 3, which is a slip-ring machine.In this study, the frequency response of both the stator and the rotor is measured via the line connection.The frequency response of the stator (or rotor) winding is measured with the following conditions: (i) rotor (or stator) terminals kept open, and (ii) rotor (or stator) terminals shorted.The connection configurations are depicted in Fig. 6.
The frequency response for the stator winding is depicted in Fig. 7(a), while the frequency response for the rotor winding is depicted in Fig. 7(b).The results demonstrate the similar behaviour mentioned in the earlier paragraph.

Conclusion
The frequency response for an induction motor winding is presented and analyzed with a novel approach to distinguish and identify the winding fault(s).A very different approach to study the frequency response of induction machine winding using concepts of electromagnetic wave propagation equations is adopted.This way suggests that the conduction current drawn by the circuit is responsible for variation in low frequency region of the frequency response, leading to highlighting an open turn fault in the machine winding.This approach offered an explanation to the deviation in frequency response due to core inductance variation and hence established relationship with the conduction current.
The high frequency region in the responses indicated a match.The physics of wave propagation supported it due to the fact that displacement current remains unchanged; as the medium parameters are constant in this frequency range, resulting in a constant wave velocity.
The experimental results presented through cases in this research work indicates that frequency response analysis can be extended well as a diagnostic tool for the three phase induction motor and the electromagnetic wave propagation effectively helps fault identification and interpretations.
Connection for line measurement.

Fig. 1 :Tab. 3 :
Fig. 1: Connection arrangement for FRA of induction machine (a) Connection for winding measurement and (b) Connection for line Measurement.

Fig. 5 :
Fig.5: Effect of core inductance on frequency response of machine 2.
Stator winding frequency response (rotor terminal open).Stator winding frequency response (rotor terminal short).
Rotor winding frequency response (stator terminal open).
Frequency response of machine 3 (stator side).
Frequency response of machine 3 (rotor side).

Fig. 7 :
Fig. 7: Connection arrangement for FRA of induction machine (a) Frequency response of machine 3 (stator side) and (b) Frequency response of machine 3 (rotor side).
Tab. 1: Instruments used for frequency response analysis Tab. 2: Specifications of the 3 -ϕ induction motors tested