Transformer- Find Location of Partial Discharge

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TDOA geometry

Fig 2. TDOA geometry.

Consider an emitter (E in Figure 2) at an unknown location vector

E = (x, y, z)

which we wish to locate. The source is within range of N+1 receivers at known locations

P₀, P₁, ..., P_m, ..., P_N.

The subscript m refers to any one of the receivers:

P_m = (x_m, y_m, z_m)0 ≤ m ≤ N

The distance (R) from the emitter to one of the receivers in terms of the coordinates is

(1)

The math is made easier by placing the origin at one of the receivers (P₀), which makes its distance to the emitter

(2)

Measuring the time difference in a TDOA System[edit]

SignalExamples of measuring time difference with cross correlation.

The distance in equation 1 is the wave speed () times transit time (). A TDOA multilateration system measures the time difference () of a wavefront touching each receiver. The TDOA equation for receivers m and 0 is

(3)

Figure 3a is a simulation of the a pulse waveform recorded by receivers and . The spacing between , and is such that the pulse takes 5 time units longer to reach than . The units of time in Figure 3 are arbitrary. The following table gives approximate time scale units for recording different types of waves.

The red curve in Figure 3a is the cross-correlation function . The cross correlation function slides one curve in time across the other and returns a peak value when the curve shapes match. The peak at time = 5 is a measure of the time shift between the recorded waveforms, which is also the value needed for Equation 3.

Figure 3b is the same type of simulation for a wide-band waveform from the emitter. The time shift is 5 time units because the geometry and wave speed is the same as the Figure 3a example. Again, the peak in the cross correlation occurs at .

Figure 3c is an example of a continuous, narrow-band waveform from the emitter. The cross correlation function shows an important factor when choosing the receiver geometry. There is a peak at Time = 5 plus every increment of the waveform period. To get one solution for the measured time difference, the largest space between any two receivers must be closer than one wavelength of the emitter signal. Some systems, such as the LORAN C and Decca mentioned at earlier (recall the same math works for moving receiver & multiple known transmitters), use spacing larger than 1 wavelength and include equipment, such as a Phase Detector, to count the number of cycles that pass by as the emitter moves. This only works for continuous, narrow-band waveforms because of the relation between phase (), frequency (f) and time (T)

.

The phase detector will see variations in frequency as measured phase noise, which will be anuncertainty that propagates into the calculated location. If the phase noise is large enough, the phase detector can become unstable.

3-D Solution[edit]

Equation 3 is the hyperboloid described in the previous section, where 4 receivers (0 ≤ m ≤ 3) lead to 3 non-linear equations in 3 unknown values (x,y,z). The system must then solve for the unknown emitter location in real time.

Civilian air traffic control multilateration systems use the Mode C SSR transponder return to find the altitude (z). Three or more receivers at known locations are used to find the other 2 dimensions (x, y).

R. Bucher and D. Misra show the detailed algebra to locate 1 transmitter with TDOA between 4 receivers.^[2] Their solution is a set of linear equations to find (x, y) and a quadratic for (z).

Improving accuracy with a large number of receivers can be a problem for devices with small embedded processors because of the time required to solve several simultaneous, non-linear equations (1, 2 & 3). The TDOA problem can be turned into a system of linear equations when there are 3 or more receivers, which can reduce the computation time. Starting with equation 3, solve for R_m, square both sides, collect terms and divide all terms by :

(4)

Removing the 2 R₀ term will eliminate all the square root terms. That is done by subtracting the TDOA equation of receiver m = 1 from each of the others (2 ≤ m ≤ N)

(5)

Focus for a moment on equation 1. Square R_m, group similar terms and use equation 2 to replace some of the terms with R₀.

(6)

Combine equations 5 and 6, and write as a set of linear equations of the unknown emitter location x,y,z

(7)

Use equation 7 to generate the four constants from measured distances and time for each receiver 2 ≤ m ≤ N. This will be a set of N-1inhomogeneous linear equations.

There are many robust linear algebra methods that can solve for the values of (x,y,z), such as Gaussian Elimination. Chapter 15 in Numerical Recipes^[3] describes several methods to solve linear equations and estimate the uncertainty of the resulting values.

2-D Solution

Finding the emitter location in a two dimensional geometry can use any of the methods used for the 3-D geometry. The coordinate frame is typically defined to make the z dimension zero or constant. Examples of 2-D multilateration are short wave radio long distance communications through the Earth's atmosphere, acoustic wave propagation in the sound fixing and ranging channel of the oceans and the LORAN navigation system.

Accuracy

For trilateration or multilateration, calculation is done based on distances, which requires the frequency and the wave count of a received transmission. Fortriangulation or multiangulation, calculation is done based on angles, which requires the phases of received transmission plus the wave count.

For lateration compared to angulation, the numerical problems compare, but the technical problem is more challenging with angular measurements, as angles require two measures per position when using optical or electronic means for measuring phase differences instead of counting wave cycles.

Trilateration in general is calculating with triangles of known distances/sizes, mathematically a very sound system. In a triangle, the angles can be derived if one knows the length of all sides, (see congruence), but the length of the sides cannot be derived based on all of the angles, not without knowing the length of at least one of the sides (a baseline) (see similarity).

In 3D, when four or more angles are in play, locations can be calculated from n + 1 = 4 measured angles plus one known baseline or from just n + 1 = 4 measured sides.

Multilateration is, in general, far more accurate for locating an object than sparse approaches such as trilateration, where with planar problems just three distances are known and computed. Multilateration serves for several aspects:

• over-determination of an n-variable quadratic problem with (n + 1) + m quadratic equations
• stochastic errors prohibiting a deterministic approach to solving the equations
• clustering needs to segregate members of various clusters contributing to various models of solving, i.e. fixed locations, oscillating locations and moving locations

Accuracy of multilateration is a function of several variables, including:

• The antenna or sensor geometry of the receiver(s) and transmitter(s) for electronic or optical transmission.
• The timing accuracy of the receiver system, i.e. thermal stability of the clocking oscillators.
• The accuracy of frequency synchronisation of the transmitter oscillators with the receiver oscillators.
• Phase synchronisation of the transmitted signal with the received signal, as propagation effects as e.g. diffraction or reflection changes the phase of the signal thus indication deviation from line of sight, i.e. multipath reflections.
• The bandwidth of the emitted pulse(s) and thus the rise-time of the pulses with pulse coded signals in transmission.
• Inaccuracies in the locations of the transmitters or receivers when used as a known location

The location of a fault or discharge in a liquid filled transformer is usually associated with the difference in the electrical signal from the discharge compared with the signal picked up from several microphones located inside the transformer. This is based on the difference of propagation of sound through oil. This is normally done when a failure occurs during factory tests but has been done with field testing in the past. It requires very special equipment and may involve several tests that cause the failure to reoccur and is therefore considered destructive testing. It does help in locating the area of the fault and the final procedure is a teardown and partial rebuild. We used it extensively and routinely during factory testing after a failure occurred and were able to find some interesting repairable failures, but most of the time it showed failures which needed extensive work.

A few manufacturers tried to develop equipment that would show locations in the field, but none of the results were satisfactory. Looking for smoke and bubbles after failures was more productive.