# Relevance of flux leakage

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### Introduction:

The task presented was to analyse changes in flux levels through a transformer, concentrating mainly on the flux leakage through an air gap.

The line LL' is the reference line along which the data was collected, it is 2.5mm below the upper section of the core. These results were to be represented in graph form, for all possible combinations of the following:

- Core materials: MU3 and CR10 (cold-rolled 1010 steel)
- Coil materials: Copper and Aluminium
- Current in coil 1: 2A, 4A, 6A, 8A, 10A

This report provides details of the use of the MagNet package to perform Finite Element Analysis of the transformer flux. The main purpose of the investigation was to interpret the change in the level of flux leakage in the air gap.

Before presenting the data acquired, it is important to understand the purpose of a transformer, the relevance of flux leakage and the operation of the software being used for the analysis.

### Transformers:

Transformers enable electricity to be distributed from the generating power station, through a series of substations and into homes, commercial and industrial premises at the appropriate voltage.

A basic transformer consists of two conductive coils (or windings) wound round a core, consisting of thin sheets (laminations) of a material such as silicon steel or iron. If a voltage is applied to the primary winding an alternating current will flow. When current flows through a conductor a rotating magnetic field is created, as the coil is wound round a core which has low magnetic resistivity, these fields induce alternating flux within the core. Those flux lines which do not link the primary and secondary windings are called leakage flux, and are usually due to losses through air gaps or material losses. Those which do link both coils are referred to as mutual flux. The reason for using high permeability material in the core is to ensure that as many flux lines as possible do link the two windings. Flux (B) varies in direct proportionality with magnetic field intensity (H) and material permeability (µr) relative to permeability of a vacuum (µ0 = 4p x 10-7). The relationship is defined as:

B = µ0 µr H

The rotating flux induces an alternating voltage in the secondary winding, the magnitude of which depends on the number of turns on each coil.

The flux path which links the windings and induces the secondary voltage Vs. In an ideal transformer (i.e. no losses and 100% power transfer) the relationship between the turns on each winding and the voltages is called the turns ratio and is given: **Vp/Vs = Np/Ns = a** (a is the turns ratio). Using this ratio it is possible to induce a secondary voltage, by stepping the primary voltage up or down as required.

In the real world transformers are not ideal, therefore it is necessary to take into account losses which occur. These losses can be subdivided into two categories, no load or load losses. Load losses encapsulates those relating to the windings, including eddy current and power lost as heat (or I2R losses), they severity is determined by design of the coils and the associated load. The no-load losses are those which the core is responsible for, these are always present and are independent of the transformer load. This type of loss can be further subdivided into the following:

- Hysteresis & eddy currents in core(responsible for over 99% of no-load losses)
- Stray eddy currents in various components
- I2R losses caused by no-load current
- Dielectric losses

Clearly the losses due to hysteresis and eddy currents in the core are the main considerations for designing a transformer core. In order to reduce the effects of eddy currents the laminations used as the core material are very thin, meaning a greater number of laminations are required. The effect of this is that the magnitude of the eddy currents in each lamination is reduced by the factor of the number of laminations, meaning the heat generated is greatly reduced.

In terms of severity, hysteresis losses in the core are the main cause for concern. They occur because the flux is constantly alternating in both magnitude and direction at a rate determined by the frequency. This means that a force has to be created to produce this effect, called coercive force (Hc). Each time this occurs, energy is lost as heat, therefore if the frequency is 50Hz then this heating effect will happen every 50th of a second. The diagram below shows the hysteresis loop, the area of the loop for a material is equal to the heat loss per cycle.

### Finite element analysis:

Finite element analysis is a technique which originated in the late 1950s and was generally exclusive to the aircraft manufacturers, as these companies were the only ones with access to sophisticated enough computers. The finite element method is basically a mathematical system which solves complex problems relating actual physical properties of materials (in this case, flux density). Basically if a problem has a differential solution, using FEA allows it to be solved algebraically, as the software solves all the relevant complex equations. This technique can be applied to a variety of engineering, scientific or mathematical problems. The theory works by dividing model being analysed into sections known as 'elements', which are connected at points called 'nodes'. Each element will have a unique solution to the relevant equations allowing the problem to be solved at any point in the system.

### MagNet

The Infolytica Corporation's MagNet software provides a means of quickly analysing electromagnetic fields using the finite element method. With the full version it is possible to view the results using either 2D or 3D modelling, however, the package available for this design only offered 2D modelling, which was sufficient. The software also allows a designer to view data in graphical form, allowing easier comparisons of results.

For this investigation, one of the key features of the software was its ability to produce shaded plot diagrams. These enable designers to easily view the performance of the design under certain conditions. Since the focus here was on the flux variations, the shaded plots formed showed the different levels of flux density through the core, coils and air gaps.

Another feature offered by this software that was utilised for this investigation was the 'Field Sampler', which was used to form the graphs required for analysis. Using this function allowed the necessary data to be calculated, namely, the change in flux leakage in the air gap for changing values of X, at a level of 2.5mm below the upper surface of the core.

### Method of Investigation:

Using the MagNet software, the first stage was to construct a 2D model of the transformer, with the dimensions: X = 100mm; Y = 150mm; Z = 10mm; XX = 400mm; a = 380mm; and b = 215mm. The sizes of 'a' and 'b' were used to define the specific area of interest and were unique to this particular investigation. An air box of suitable dimensions was created around the transformer.

There were a number of variables to be considered as part of the investigation. The core materials to be considered were MU3 and CR10 (cold-rolled steel), the coil materials were copper and aluminium. This gave four combinations to be analysed for different values of current through coil 1 (2A, 4A, 6A, 8A, 10A). The initial combination was a CR10 core with cooper coils.

Before the model became a useful tool for analysis it had to be broken down into 'elements', which the software does automatically. The accuracy of the results depends on the number of elements, they are directly proportional. Selecting the option 'Initial 2D Mesh' form the 'Solve' menu performs this function.

The area of particular importance to this investigation was the flux (B). Viewing the 'shaded plot' of the transformer provided a means of visualising the sections of greater or lower flux. A selection of these diagrams is contained in Appendix 2. The next step was to use the 'field Sampler' tool to create the graphs which were to be used for analysis. First, the range of measurement had to be defined for X and Y. Since the measurements were all to be taken at 2.5mm below the upper face of the core, the Y value did not change but remained constant at 397.5 (XX - 2.5 = 400 - 2.5 = 397.5). Therefore all the readings were taken along the X axis, the range of values were selected by inputting a start point and an end point.

The relevant results were obtained for each material combination at the different coil 1 currents. These results were used to create a set of graphs for each combination showing how the flux varied with the value of X. For this investigation, the most important results were those obtained in the air gap between the upper surface of the coil and the core. This allows the level of flux leakage in the air gap to be obtained.

### Analysis of Results:

The next section deals with the results obtained and how these were interpreted. The two graphs shown below illustrate the differences between the core materials, MU3 and CR10 (cold-rolled steel). There are a number of obvious differences between the two. Firstly, the flux density values reach a far greater level passing through the MU3 than steel, in fact the peak value is roughly ten times that achieved by the steel core. Also, the graph obtained using MU3 is very smooth in comparison with the steel core, indicating that there are fewer barriers to passage of flux in the MU3 core. These results are not surprising, as they prove that MU3 in a much more permeable material than steel, which was already known. This shows that an MU3 core would provide better flux linkage than a steel core and subsequently, fewer power losses.

Analysing the diagram below, which is of the MU3 core with aluminium coils, it is possible to show the points through the core and the air gap where the magnitude and direction of the flux changes. The relevant points are:

- -90 to 10: At the edge of the core the magnitude of flux is almost zero, increasing steadily in the negative direction before reaching a peak at the start of the air gap. The flow is negative because it opposes the flow into the secondary winding.
- 10 to 320: At this point the flux level drops to almost zero and remains relatively constant through the air gap, indicating that most of the flux is contained within the core. The flow direction is still against the flow towards the secondary coil and is therefore negative.
- 320 to 427.5: Again here it is the flux in the core which is being analysed, as soon as the air gap ends the flux hits its peak value, decreasing steadily as it nears the centre point of the core, where it crosses the zero line becoming positive.
- 427.5 to 535: The flux is now flowing towards the secondary coil and is therefore positive, the density increases steadily as the distance from centre increases. The greatest flux density is reached just before entering the air gap, this is because the sharp corner at this point is restricting the area through which the flux can move, causing a kind of 'piling up' effect.
- 535 to 845: Similar to the observations regarding 10 to 320, except here the flux is flowing toward the secondary coil, therefore remains positive.
- 845 to 945: Here the flux density starts at peak value and decreases towards zero as the end of the core is reached, remaining positive.

The following four graphs show the change in flux density through the first air gap. These were obtained for each of the material combinations and the five values of current in coil 1. The first pair of graphs show the results obtained using a steel core, varying the current and the coil material.

From these results further conclusions can be drawn. Firstly, the two graphs appear to be identical and further analysis of the calculated results proved that this was the case. This suggested that the coil material had no effect on the level of flux. Furthermore, the graphs show a clear relationship between the current through coil 1, and the levels of flux leakage. This was to be expected, since the flux is induced by the flow of current through the primary coil and is therefore proportional to the amount of current applied to the coil.

The next set of graphs display the change in flux in the air gap when using the MU3 core. As before the variables were the coil 1 current and the coil material.

These results confirm the accuracy of some of the previous observations. Namely that the material in the coil has no effect on the flux in the air gap, only the core material and applied current are factors in this. Furthermore, analysing the data from all the graphs displayed above, it can be seen that the flux present in the air gap is fairly small for both core materials and of similar values. However, proportionally there is a big difference between the two, as was shown before the peak flux in the MU3 core is around ten times that of the steel. This means that the mutual flux present in MU3 is much higher than in CR10, so the rate of power transfer from primary to secondary will be far better using MU3.

### Conclusions

From the results obtained there are a number of conclusions which can be made. The graphs relating to the air gap show that:

- The coil material does not affect the level of flux in the air gap, only the core material, therefore the losses are no-load losses.
- MU3 has better magnetic properties than CR-10, higher permeability and better flux linkage.
- The MU3 core loses proportionally much less flux in the air gap than the steel, meaning better power transfer between the windings.
- The steel has more imperfections than the MU3, giving it higher resistivity to magnetising/de-magnetising, therefore the response shows much greater fluctuation and is less easy to predict.
- The other main factor affecting flux in the air gap is applied current, since the field strength is measured in ampere/metre, increasing the current will also increase the field strength. Therefore, using the equation B = µ0H (for air µr = 1) this must consequently increase the flux present in the air gap.

### References

- http://images4.wikia.nocookie.net/engineering/images/0/09/400px-Single-phase_transformer.png (21/10/09)
- http://www.tpub.com/content/doe/h1011v1/img/h1011v1_65_1.jpg (16/11/09)
- http://encyclopedia2.thefreedictionary.com/finite+element+method (25/10/09)
- http://www.acoustics.salford.ac.uk/student_area/bsc3/computer_simulation/Fem.pdf (25/10/09)
- http://www.infolytica.com/en/products/magnet/ (27/10/09)

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