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Qualitative FEM Study of Proximity Loss Reduction by Various Winding Configurations – Part I

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Qualitative FEM Study of Proximity Loss Reduction by Various Winding Configurations – Part I EVENTSLOSSES Qualitative FEM study of proximity loss reduction by various winding configurations – Part I ABSTRACT How to make a 1 MHz transformer without Skin depth and proximity effects in expensive Litz wire transformer windings are important phenomena influencing the design even 1. Introduction power transformers (e.g. as opposed to at power frequencies (50-60 Hz). How- signal transformers). Electrical resistance ever, they become critically important Efficiency of energy transformation is of a given winding has an immediate im- at elevated frequencies, especially for affected by multiple loss components oc- pact on the total efficiency at full load due high-frequency transformers, operating curring in transformers. These compo- to Joule‘s heating. Conductors with grea- in switched-mode power supplies for nents depend on the given application, ter effective cross-sectional area must be example, at any power level. This article topology of conversion (conventional or used to reduce the losses and improve the presents a numerical study of optimum switched mode), frequency of operation, efficiency. winding configurations which can dras- energy required for forced cooling, etc. tically reduce the proximity effects. It However, transformer designers also have is possible to make transformers ope- Loss in the windings (copper loss) is a to take into account other, more complex rating even at 1 MHz without the use of significant part of the total loss in most phenomena, such as the skin effect. Elec- very expensive Litz wire. KEYWORDS With an appropriate core-winding configuration it copper loss, proximity effect, proximity loss, skin effect, eddy currents, wind- is possible to use solid wire instead of expensive ings multi-strand Litz wire, even at 1 MHz 70 TRANSFORMERS MAGAZINE | Volume 3, Issue 1 Stan ZUREK Efficiency of energy transformation is affected by multiple loss components occurring in transform- ers, which depend on the given application, top- ology of conversion, frequency of operation, etc. Even at 50 Hz or 60 Hz the skin depth in 2. Proximity loss copper is around 9 mm (Fig. 1a, b). This is partly the reason for the high-current bus- There is also a phenomenon calledprox­ bars in low-voltage switchgear to be made imity effect which can have even more with copper bars not thicker than 10 mm severe consequences than the skin depth. (Fig. 1c). The proximity loss depends strongly not only on the frequency, but also on a num- For higher frequencies the skin depth re- ber of layers in a given set of conductors. duces proportionally to the square root of The name “proximity” comes from the fact frequency. This becomes an important li- that the effect is caused by the presence of mitation for High-Frequency (HF) trans- other conductors with currents in the im- formers, because the conductors cannot mediate vicinity of each other [3, 4]. be thicker than around double the thick- ness of the skin depth in order to use the Any electric current generates magnetic copper efficiently. For this reason many field around itself and this field penetrates HF transformers use thin but wide cop- any neighbouring conductor. According per tape in order to maintain the effective to the Faraday’s law, if the magnetic field cross-sectional area of the conductor wit- varies then a voltage will be induced in a hout increasing its thickness. conducting loop directly, proportionally to the amplitude and frequency of the field tromagnetic wave penetrating a conduc- HF transformers are widely used in penetrating it. Such induced voltage gives ting material (copper) induces surface s witched mode power supplies (SMPS) rise to eddy current, whose position and eddy currents which oppose the penet- from a sub-W to hundreds of kW. The direction is such as to oppose the mag- rating field, thus attenuating the internal currents usually have very high harmonic netic field generating it (as defined by the electromagnetic field. The attenuation fol- content (triangular, trapezoidal or rectan- Lenz’s law). lows an exponential function and at a dis- gular pulses) which further exacerbates tance of one skin depth the field is attenua- the losses due to the fact that skin depth For neighbouring conductors the effect ted to 37 % of the surface value. Therefore, becomes shallower for higher frequencies. leads to “side” eddy currents (Fig. 2). The the material deeper inside the conductor The problem is described in more detail amplitudes of the eddy currents add/sub- carries less current and the copper is used elsewhere [1, 2]. This paper focuses only tract accordingly with the direction of less inefficiently. on sinusoidal currents. the main current and hence the localised a) b) c) Figure 1. (a) Skin depth for various conductors and frequencies; (b) Hypothetical current distribution in 40 mm diameter copper rod at DC and 50 Hz; (c) LV distribution board with 10 mm thick copper busbars www.transformers-magazine.com 71 EVENTSADVERTORIALMAGNETICLOSSES CORES The proximity loss strongly depends not only on ductors so the power loss is at similar level as it would be for DC current (P = 100 %). the frequency, but also on the number of layers But the inner conductors are exposed to in a given set of conductors much greater amplitudes of H and thus P can grow disproportionately large (over 600 % peak in Fig. 3). The situation shown in Fig. 3 is for con- ductor thickness equal to the skin depth, which is usually sufficient for minimising problems with just the skin effect. How- ever, as shown in Fig. 3, the proximity ef- fect can produce significantly elevated los- ses even for such conditions. This is a real problem for high-frequency applications like HF transformers for SMPS (Fig. 4). The problem occurs also in small electro- nic transformers [5], medium-frequency transformers [6], rotating machines [7], as well as three-phase busbars whose thick- ness is usually limited to 10 mm for this reason [3]. Usually, proximity loss calculations are Figure 2. Illustration of the proximity effect [2] too complex to be solved analytically [8]. Therefore, many studies are carried out numerically, by employing FEM or F inite c urrent density in the conductor is not ratio of 1:1 the currents flowing in prima- Element Modelling [9, 10]. This is not al- uniform. As a result, the effective resistance ry and secondary can be assumed equal. ways practical, because 3D problems are of the wire is increased for alternating cur- very computationally intensive and multi- rents (so-called AC resistance) and thus the The magnetic field strengthH builds up strand wires cannot be fully modelled, copper losses can be elevated significantly. with each layer of the primary winding, especially for higher frequencies or larger to reach the maximum between the wind- components, in which each turn of the The distribution of current density de- ings. The current in the secondary wind- Litz wire could be composed from hund- pends on many factors, for instance, on ing flows in the opposite direction and reds of strands (Fig. 5). the number of adjacent conductors. The thus H is reduced with each layer of sec- effect is schematically illustrated in Fig. ondary. There are many technical and scientific 3. A hypothetical simplified single-phase papers describing various ways of calcu- transformer has the tape windings made However, the localised eddy currents are lating the losses due to proximity effect. as three layers of primary and three layers stronger in the presence of higher H, and A well-known design principle is to fol- of secondary (each winding is connected the power P lost in the windings is pro- low the so-called Dowell’s curves, which in series so that there are three turns in portional to the square of the eddy cur- are generalised and normalised curves each coil). At full load and with the turn rents. H is much lower in the outer con- helping in the assessment of the apparent increase in AC resistance over the DC re- sistance (coefficient k in Fig. 4) [11]. As seen from Fig. 4 the proximity loss can be reduced in two ways. The first route is The localised eddy cur- rents are stronger in the presence of a higher H, and the power P lost in the windings is propor- Figure 3. Illustration of proximity loss in single-phase transformer windings with 3-layer primary tional to the square of (dark grey) and 3-layer secondary (light grey); each winding is connected in series the eddy currents 72 TRANSFORMERS MAGAZINE | Volume 3, Issue 1 The layer thickness is the most important param- eter for proximity effects equal to skin depth) will suffer from k = 2, so the proximity loss would be responsible for additional loss of 200 %. The Q value can be reduced by using thinner conductors (like in Litz wire). If the conductors in the hypothetical trans- former had half the thickness, then the Q value would be reduced to 0.5. But then Figure 4. Dowell’s curves for tape conductors [11] each turn would be constructed from two sub-layers so that the total number of layers would be 6. For Q = 0.5 this would produce lower AC resistance than the The proximity loss can Reducing the number of layers can also original configuration (Fig. 4), but it still produce improvement of performance might be an insufficient improvement be reduced by reducing (Fig. 4). For instance, the above men- and the “strand” thickness would have to the wire thickness and tioned hypothetical three-layer trans- be reduced further. This carries a penalty the number of layers former with Q = 1 (conductor thickness in terms of the associated costs of thinner to reduce the wire thickness (which is pro- portional to the ratio of layer thickness to the skin depth).
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