Design Optimization of the Power Transformer and Inductor in the Switch-mode Power Supplies

Peter ŠAŠTINSKÝ1, Matúš JENDRUCH1 1 Dept. of Radio and Electronics, Slovak University of Technology, Ilkovičova 3, 812 19 Bratislava, Slovak Republic

peter.sastinsky@stuba.sk, matus.jendruch@stuba.sk

Abstract. This paper discusses the loss in the transformer and the inductor in applications of the DC/DC converters. The analysis of this loss due to eddy currents in windings conductor of these transformers and inductors is made. This eddy current loss is caused particularly by two effects: skin effect and proximity effect, which are also part of this text. The design optimization of these windings to minimize eddy current loss is made.

Keywords Inductor, transformer, skin effect, proximity effect, eddy currents.

1. Introduction Each electronic equipment uses the power supply. In

each modern power supply the power transformer is found. In switch-mode power supplies and DC/DC converters to accumulate of energy the inductor is used. However, these components are not ideal, when even they correct are designed. Therefore different analyses mainly concerning loss are made and their optimal design is found.

2. Eddy current loss in conductor winding of transformers and inductors

2.1 Winding loss due to DC resistance of windings The power PCu,sp dissipated per unit of copper volume

in a copper winding due to its DC resistance is given by

( )2, rmsCuspCu JP ρ= (1)

where Jrms = Irms/ACu is the current density in the conductor and Irms is the RMS current in the winding. However, it is more convenient to express PCu,sp as power dissipated per unit of winding volume Pw,sp. The total volume VCu is given by VCu = kCuVw, where Vw is the total winding volume. Using this result to express Pw,sp yields

( )2, rmsCuCuspw JkP ρ= (2)

where kCu is copper fill factor.

2.2 Skin effect in copper windings Consider the single copper conductor shown in. Fig.

1a, which is carrying a time-varying current i(t). This current generates the magnetic fields shown in Fig. 1a, and they in turn generate the eddy currents illustrated in Fig. 1b. These eddy currents flow in the opposite direction to the applied current i(t) in the interior of the wire and thus tend to shield the interior of the conductor from the applied current and resulting magnetic field. As a result the total current density is largest at the surface of the conductor as shown in Fig. 1c.

a) b) c)

Fig. 1. Isolated copper conductor.

The characteristic decay length in the exponential is termed the skin depth and is given by

ωμσδ 2= (3)

where f = ω/2π is the frequency (in Hz) of the applied magnetic field, μ is the magnetic permeability of the copper and σ is the conductivity of the copper.

f [kHz] 0,05 5 20 500

δ [mm] 10,6 1,06 0,53 0,106

Tab. 1. Skin depth in copper at 100°C for several different frequencies.

Table 1 shows the skin depth in copper at several different frequencies at a temperature of 100°C. If the cross-

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sectional dimensions of the conductor are significantly larger than the skin depth, most of the current carried by the conductor will be constricted to a relatively thin layer at the surface approximately one skin depth in thickness as is illustrated in Fig. 1c. The net result of this is that effective resistance of the conductor will be far larger than the DC resistance because the effective cross-sectional area for current flow is small compared to the geometric cross section of the conductor. This will cause the winding losses to be much larger than if it were a DC current. The solution to this problem is to use conductors with cross-sectional dimensions on the order of the skin depth in size. Such considerations have led to the development of special conductor arrangements for high-frequency applications. These conductor arrangements include Litz wire and thin foil windings. The net effect of these losses is to increase the effective resistance of the winding to a value Rac. This modifies (2) to

( )2, rms

dc

acCuCuspw J

RR

kP ρ= (4)

where Rdc is the DC resistance of the winding.

2.3 Proximity effect Consider the cross-sectional view of an inductor

winding carrying a current I shown in the winding window of ferrite core in Fig. 2a. To simplify our initial considerations, the diameter of the winding conductor or the frequency are assumed to be small enough so that the skin effect can be neglected. The application of Ampere’s law along path A in Fig. 2a encloses several ampere-turns of magnetomotive force (MMF), thus showing that a magnetic field is present in the window. If path B, which located at a greater distance x into the winding window, is taken, a greater amount of MMF is enclosed and the magnetic field is even larger. The approximate distribution of the MMF and thus the magnitude of the magnetic field is illustrated in Fig. 2a. The magnetic field generates eddy currents in the conductor windings in exactly the same manner as was described earlier for the skin effect. In Fig. 2, the magnetic flux is contained in the plane of the winding window and thus is perpendicular to the longitudinal direction (direction of applied current flow) of the conductor windings. Hence the eddy currents flow either parallel or antiparallel to the applied current as diagrammed in Fig. 2b. This generation of eddy currents is termed the proximity effect because the eddy currents in a specific conductor or winding layer are caused by the magnetic fields of the other current-carrying conductors in proximity to the given conductor. These eddy currents will dissipate power Pec and thus contribute to the electrical loss in the winding in addition to those caused by the normal ohmic loss Pdc due to the DC resistance of the windings. Thus the eddy current loss per unit length of conductor winding in Fig. 2 increases dramatically as the position x in the winding window increases, and thus the number of layers contributing to the local magnetic field increases. An

approximate distribution of the eddy current loss with position is shown in Fig. 2a.

a) b)

Fig. 2. Cross-sectional winding view in the core winding window and the associated spatial distribution of the MMF and eddy current loss density. The dots (•) and x’s in the conductor cross sections indicate the relative magnitude spatial distribution of the current density and current direction.

The total power dissipated in a winding is

( ) ( ) =+=+= ecrmsdcrmsecdcw RIRIPPP 22

( ) acrms RI 2= (5)

where Rec is effective eddy current resistance. The net resistance of the winding Rac is given by

dcdc

ecdcRac R

RR

RFR ⎟⎟⎠

⎞⎜⎜⎝

⎛+== 1 (6)

where FR is termed the resistance factor. For the situation illustrated in Fig. 2a where the diameter of the conductor is less than or about equal to the skin depth, the resistance factor will be somewhat greater the unity. If the diameter of the winding conductor is significantly greater than a skin depth, then the eddy currents will flow only near the surface of the conductor. Because of the skin effect, no currents will flow in the interior as is sketched in Fig. 1c. Magnetic flux will be excluded from the interior of the conductor, and the MMF diagram will be modified as shown in Fig. 2a, which shows very little MMF in the interior of the conductor. The confinement of the total current to a thin area on the outside portion of the conductor means that current density will be much larger at the surface of the conductor than in the situation illustrated in Fig. 2b for the low-frequency case. As a consequence, for a high frequencies the resistance factor much greater

(one or two orders) that what would be predicated on the basis of just the proximity effect at low frequencies.

2.4 Sectioning transformer windings to reduce eddy current loss The presence of the secondary winding in a

transformer enables the eddy current loss in a transformer to be minimized. Consider the MMF distribution in the transformer winding window shown in Fig. 3b. The MMF in the secondary section of the transformer has a negative slope and goes back to zero because the induced current in the secondary is opposite that in the primary.

Fig. 3. Winding window in a transformer containing a) a simple

winding arrangement, b) MMF distribution versus position and c) eddy current loss density versus position.

Now consider sectioning the primary winding into two separate parts as shown in Fig. 4a and sandwiching the secondary between the two halves of the primary. The total number of primary and secondary turns remains unchanged. The resulting MMF distribution is also shown in Fig. 4a. The peak value of the MMF with this sandwiched winding is now approximately one-half the peak value of that in Fig. 3. For the simple transformer winding since the number of ampere-turns in each primary half-section is one-half of the value shown in Fig. 3. Since the peak MMF is reduced by a factor of 2, so is the maximum magnetic flux in the winding window. Eddy current loss is proportional to the square of the magnetic flux so the eddy current loss in the transformer of Fig. 4a should be approximately one-fourth that of the transformer of Fig. 3. This approach can be extended by dividing both the primary and secondary windings into more sections as is indicated in Fig. 4b. Now the peak value MMF is one-fourth that of Fig. 3, and so eddy current loss should be one-sixteenth those of simple transformer. In principle this subdividing of the windings into more sections could be

continued until each section consists of one or two winding layers. This approach is not without its disadvantages. Winding a transformer in this manner is complex. The interwinding capacitance is increased in proportion to the number of sections, and the reliability of the insulation and copper fill factor is decreased.

a) b)

Fig. 4. Partitioning of the primary and secondary windings into multiple sections to reduce eddy current loss.

2.5 Optimization of solid conductor windings and minimum winding loss The discussion in previous sections has provided

several quantitative approaches for reducing the eddy current loss in inductor and transformer windings. However, optimizing the design of a winding requires a quantitative procedure for implementing these suggestions and assessing their benefits. Such a procedure has been developed that relates the power dissipated in the winding or section of winding to geometry of the winding (conductor cross-sectional dimensions, number of turns, and number of layers) and the skin depth of the winding conductor. The procedure is based on a fairly general analysis that includes nonuniform magnetic fields across the cross-sectional area of the conductor, skin effect, and eddy current screening. The goal of procedure is to find the combination of optimum conductor diameter or thickness and number of layers so that the total winding loss (DC as well as due to eddy current) is minimized. The procedure is based on the set of curves shown in Fig. 5 of normalized power dissipation in the winding as a function of normalized variable Φ with the number of layers m in the winding section as a parameter. The normalized power dissipation is defined as

( ) δδδ ===

==hdc

dcR

hdc

ac

rmshdc

w

RRF

RR

IRP

,,2

,

(7)

where Rdc,h=δ is the DC resistance of the winding when the conductor diameter or thickness is equal to the skin depth.

The parameter Φ is given by

δhF1=Φ (8)

In this equation, h is the effective conductor height, δ is the skin depth given by Eg. 3, and F1 is copper layer factor. For rectangular conductors the effective conductor height is the equal height h. For round conductors the effective conductor height is (√π).d/2 where d is the conductor diameter. The parameter m in Fig. 5 is the number of layers in the winding section. The copper layer factor is the fraction of the layer width hw (or equivalently the winding window height) that is occupied by copper. For the winding made with rectangular conductor the copper layer factor is b/b0 and for the round conductors the F1=d/d0. The dimensions b0 and d0 include the insulation on the conductor. For a layer composed of a single turn of coil conductor, the layer factor would equal unity.

Fig. 5. Normalized power dissipation in a winding or winding

section as a function of Φ with the number of layers m as a parameter.

A winding or winding portion of m layers has a low-frequency MMF distribution that varies linearly from zero on one side of the section to a maximum at the other side. For the split primary in Fig. 4a, each primary section has Mpri/2 layers where Mpri is the total number of layers in primary. The secondary must also be considered to have two portions each having Msec/2 layers. For the winding arrangement shown in Fig. 4b, the two outside primary

sections have Mpri/4 layers while the central primary section is considered to be two separate portions each also having Mpri/4 layers. The two secondary sections should each be considered to have two portions each with Mpri/4 layers. The number of layers in winding sections for other divisions of the primary and secondary windings would be found in the same manner. The value of Φ in the graph that corresponds to the minimum in the selected curve (selected by the number of layers in the winding section) corresponds to the optimum value of conductor diameter or thickness. When the optimum diameter or thickness is used for conductor winding, the resistance factor has the value

5,1=RF (9)

This means that the power dissipation due to eddy currents is equal to

dcec PP 5,0= (10)

and the total winding loss is

dcw PP 5,1= (11)

3. Summary In this article eddy current loss in the winding

conductors of the power transformer of power supplies due to skin effect and proximity effect was discussed. The design optimization of these windings to minimize eddy current loss was made. Minimal loss in the windings is when the resistance factor is equal 1.5. The availability of the set of curves in this text permits several different winding designs (number of layers per section, number of sections into which the primary and/or the secondary are partitioned, the type of winding conductor, round or rectangular).

References [1] FAKTOR, Z. Transformátory a tlumivky pro spínané napájecí

zdroje. BEN – technická literatura, Praha 2002, ISBN 80-86056-91-0.

[2] MOHAN, N., UNDELAND, T. M., ROBBINS, W. P. Power Electronic. 2nd ed., John Wiley & Sons Inc., Canada 1995, ISBN 0-4715-8408-8.

[3] HRIBIK, J., HRUŠKOVIC, M. Power Supply Source for Class TD Power Amplifiers. In Proceedings of the 17th Internacional Czech-Slovak Scientific Conference “Radioelektronika 2007”. Brno (Czech Republic), 2007, p.105-108.

[4] ERICKSON, R. W., MAKSIMOVIC, D. Fundamentals of Power Electronics. 2nd ed. Springer Science+Business Media Inc., New York 2001, ISBN 0-7923-7270-0.