Eddy Current Losses.pdf

Eddy Current Losses in Transformer

Windings and Circuit Wiring

Lloyd H. Dixon, Jr.

Introduction

As switching power supply operating fre-

quencies increase, eddy current losses and

parasitic inductances can greatly impair circuit

performance. These high frequency effects are

caused by the magnetic field resulting from

current flow in transformer windings and circuit

wiring.

This paper is intended to provide insight into

these phenomena so that improved high fre-

quency. performance can be achieved. Among

other things, it explains (I) why eddy current

losses increase so dramatically with more wind-

ing layers, (2) why parallelling thin strips does-

n't work, (3) how passive conductors (Faraday

shields and C.T .windings) have high losses,

and (4) why increasing conductor surface area

will actually worsen losses and parasitic induc-

tance if the configuration is not correct.

energy is stored in air gaps, insulation between

conductors, and within the conductors, where

relative permeability JLr is essentially 1.0 and

constant. The energy density then becomes:

w = \BH = \'tJ.(j/I2 J/m3

where ILOis the absolute permeability of free

space ( =47r .10"7 in S.I. units). Total energy W

(Joules) is obtained by integrating the energy

density over the entire volume, v, of the field:

W = \ILO f H2dv Joules

Within typical transformers and inductors, the

magnetic energy is almost always confined to

regions where the field intensity H is relatively

constant and quite predictable. This often oc-

curs in circuit wiring, as well. In these cases:

W = \ILO H2A. e Joules (2)

and from (1), He = NI. Substituting for H:

W = \ILO ~fA/e

Basic Principles

The following principles are used in the

development of this topic and are presented

here as a review of basic magnetics.

1. Ampere's Law: The total magneto-motive

force along any closed path is equal to the total

current enclosed by that path:

F = iHde = I, = NI Amps

Joules

(3)

(1)

whereA is the cross-section area (m1 of the

region normal to the flux, and e is the length

of the region in meters (and the effective

length of the field).

4. Circuit inductance: Inductance is a meas-

ure of an electrical circuit's ability to store

magnetic energy. Equating the energy stored in

the field from (3) with the same energy in

circuit terms:

W = \Lf = \ILO N2fA/e

L = ILON2A/e

where F is the total magneto-motive force

(in Amperes) along a path of length e (m), H

is field intensity (Aim), and I, is the total

current through all turns enclosed by the path.

2. Conservation or energy: At any moment

of time, the current within the conductors and

the magnetic field are distributed so as to

minimize the energy taken from the source.

3. Energy content or the field: The magnetic

field is energy. The energy density at any point

in the field is:

w = f HdB Jouleslm3

(4)

Skin Effect

Figure 1 shows the magnetic field (flux lines)

in and around a conductor carrying dc or low

frequency current I. The field is radially sym-

metrical, as shown, only if the return current

with its associated field is at a great distance.

At low frequency, the energy in the magnetic

field is trivial compared to the energy loss in

the resistance of the wire. Hence the current

where B is the flux density (Tesla). In

switching power supplies, almost all magnetic

R2-1

reduced, so the resistance at high frequency

(and resulting losses) can be many times

gJ:eater than at low freQuencv.

Fig. 1 -IsQ/ated CQnductQrat LQW Frequency

distributes itself uniformly throughout the wire

so as to minimize the resistance loss and the

total energy expended.

Around any closed path outside the wire

(and inside the return current), magneto-motive

force F is constant and equal to total current I.

But field intensity H varies inversely with the

r,adial distance, because constant F is applied

across an increasing t ( =2m).

Within the conductor, F at any radius must

equal the enclosed current at that radius,

therefore F is proportional to r2.

Fig. 3 -High Frequency Cu"ent Distribution

Penetration depth: Penetration or skin

depth, OPEN, is defmed as the distance from

the surface to where the current density is l/e

times the surface current density (e is the

naturallog base)[l]:

OPEN = [ p/(1TIJ.£)]112 m

where p is resistivity. For copper at 100°C,

p = 2.3.10-6 n-cm, IJ.= IJ.o= 41T.10°7,and:

OPEN = 7.5/(£)112 cm

(6)

Fig. 2 -Eddy Current at High Frequency

At high frequency: Figure 2 is a super-

position model that explains what happens

when the frequency rises. The dash lines repre-

sent the uniform low frequency current distri-

bution, as seen in Fig. I. When this current

changes rapidly, as it will at high frequency, the

flux within the wire must also change rapidly.

The changing flux induces a voltage loop, or

eddy, as shown by the solid lines near the wire

surface. Since this induced voltage is within a

conductor, it causes an eddy current coincident

with the voltage. Note how this eddy current

reinforces the main current flow at the surface,

but opposes it toward the center of the wire.

The result is that as frequency rises, current

density increases at the conductor surface and

decreases toward zero at the center. as shown

in Fig. 3. The current tails off exponentially

within the conductor. The portion of the

conductor that is actually carrying current is

From (6), OPEN= .024 cm at 100 kHz, or

.0075 cm at 1 MHz.

Eqs. (5) and (6) are accurate for a flat con-

ductor surface, or when the radius of curvature

is much greater than the penetration depth.

Although the current density tails off expo-

nentially from the surface, the high frequency

resistance (and loss) is the same as if the cur-

rent density were constant from the surface to

the penetration depth, then went abruptly to

zero as shown on the right hand side of Fig. 3.

This equivalent rectangular distribution is

easier to apply.

Equivalent circuit model: Another way of

looking at the high frequency effects in trans-

former windings and circuit wiring is through

the use of an equivalent electrical circuit

model. This approach is probably easier for a

circuit designer to relate to.

Figure 4 is the equivalent circuit of the

isolated conductor of Figs. 1 to 3. With current

I flowin, through the wire, L% accounts for the

energy ~L%I2 stored in the external magnetic

field. L% is the inductance of the wire at high

frequencies.

Point A represents the outer surface of the

conductor, while B is at the center. R; is the

R2-2

(5)

Fig. 4 -Conductor Equivalent Circuit

resistance, distributed through the wire from

surface to center. Think of the wire as divided

into many concentric cylinders of equal cross

section area. The Ri elements shown in the

drawing would correspond to the equal resis-

tance of each of the cylinders. Likewise, the

internal inductance L; accounts for the magnet-

ic energy distributed through the cylindrical

sections. The energy stored in each section

depends on the cumulative current flowing

through the elements to the right of that

section in the equivalent circuit.

Note that external inductance Lx of the wire

(or the leakage inductance of a winding) limits

the maximum di/dt through the wire, depend-

ing on the source compliance voltage, no

matter haw fast the switch turns on.

The time domain: If a rapidly rising current

is applied to the wire, the voltage across the

wire is quite large, mostly across Lx. Internal

inductance Li blocks the current from the wire

interior, forcing it to flow at the surface

through the left-most resistance element, even

after the current has reached its final value and

the voltage across Lx collapses. Although the

energy demand of Lx is satisfied, the voltage

across the wire is still quite large because

current has not penetrated significantly into the

wire and must flow through the high resistance

of the limited cross-section area at the surface.

Additional source energy is then mostly dissi-

pated in the resistance of this surface layer.

The voltage across this Ri element at the

surface is impressed across the adjacent Li ele-

ments toward the center of the wire, causing

the current in L; near the surface to rise.

Current cannot penetrate without a field being

generated within the conductor, and this

requires energy. As time goes on, conduction

propagates from the surface toward the center

(at B in the equivalent circuit), storing energy

in L;. More of the resistive elements conduct,

lowering the total resistance and reducing the

energy going into losses. Finally, conduction is

uniform throughout the wire, no further energy

goes into the magnetic fields external or

internal to the wire, and a small amount of

energy continues to be dissipated in Ri over

time.

Note that the concept of skin depth has no

meaning in the time domain.

The frequency domain: Referring again to

Fig. 4 with a sine wave current applied to the

terminals, it is apparent that at low frequencies,

the reactance of internal inductance Li is neg-

ligible compared to Ri. Current flow is uniform

throughout the wire and resistance is minimum.

But at high frequency, current flow will be

greatest at the surface (A), tailing off exponen-

tially toward the center (B).

Penetration depth (skin depth) is clearly

relevant in the frequency domain. At any

frequency, the penetration depth from Eq. (5)

or (6) reveals the percentage of the wire area

that is effectively conducting, and thus the ratio

of dc resistance to ac resistance at that

frequency.

Although the current waveforms encountered

in most switching power supplies are not

sinusoidal, most papers dealing with the design

of high frequency transformer windings use a

sinusoidal approach based on work done by

Dowell in 1966.[2] Some authors use Fourier

analysis to extend the sinusoidal method to

non-sinusoidal waveforms.

Proximity Effect

Up to this point a single isolated conductor

has been considered. Its magnetic field extends

radially in all directions, and conduction occurs

across the entire surface.

When another conductor is brought into

close proximity to the first, their fields add

vectorially. Field intensity is no longer uniform

around the conductor surfaces, so high frequen-

cy current flow will not be uniform.

For example, if the round wire of Fig. 1 is

close to another wire carrying an equal current

in the opposite direction (the return current

path?), the fields will be additive between the

two wires and oppose and cancel on the out-

sides. As a result, high frequency current flow

is concentrated on the wire surfaces facing each

other, where the field intensity is greatest, with

R2-3

little or no current on the outside surfaces

where the field is low. This pattern arranges

itself so as to minimize the energy utilized,

hence inductance is minimized. As the wires

are brought closer together, cancellation is

more complete. The concentrated field volume

decreases, so the inductance is reduced.

Circuit wiring: The field and current distri-

bution with round wires is not easy to compute.

A sinlpler and more practical example is given

in Figure 5. The two flat parallel strips shown

are actually the best way to implement high

frequency wiring, minimizing the wiring induc-

tance and eddy current losses. These strips

could be two wide traces on opposite sides of

a printed circuit board. (Don't use point-to-

point wiring. It is much more important to

collapse the loop and keep the outgoing and

return conductors as intimate as possible, even

if the wiring distances are greater .)

length, divided equally between each of the two

conductors. If one conductor is much wider

than the other, such as a strip vs. a ground

plane, most of the inductance calculated in (6)

is in series with the narrower conductor. This

is good for keeping down noise in the ground

returns.

Note that current penetration is from one side

only --the side where the field is. This means

that a strip thicker than the penetration depth

is not fully utilized. The equivalent circuit

model of Fig. 4 still applies, with A at the

surface adjacent to the field. But B becomes

the opposite side of the strip, not the center,

since there is no penetration from the side with

no field.

Bad practice: Figures 6 and 7 show what not

to do for circuit wiring (unless you want high

inductance and eddy current losses). Although

these strips have large surface areas, proximity

effects in these configurations result in very

little surface actually utilized. Remember that

the field is concentrated directly between the

two conductors so as to minimize the stored

energy.

In Fig. 6 this results in current flow only at

the edges facing each other. Also, because the

concentrated field region is short, the energy

density is very high, and the inductance is

several times greater than in Fig. 5.

The Fig. 7 configuration is not quite as bad

as Fig. 6 because the current does spread out

somewhat in one of the two conductors, but it

is still many times worse than the proper con-

figuration in Fig. 5. The message is: large

I---t--"i

1+ + + + +11-

Fig. 5 -Circuit Wiring -Flat Parallel Strip

The + signs indicate current flow into the

upper strip, the. indicates current out of the

lower strip. Between the strips, the magnetic

field is high and uniform, so the current is

spread out uniformly on the inner surfaces. On

the outside of the strips the field is very low, so

the current is almost zero. This results in the

minimum possible energy storage (and wiring

inductance) for this configuration. If the

breadth of the strip, t, is much greater than

the separation, w, the energy is almost entirely

contained between the two strips. Then t and

ware the length and width of the field, and can

be used to calculate the inductance. Convertin~

Eq. (4) to cm and with N = 1 turn, the induc-

tance per centimeter length of the 2-conductor

strip is:

!1 11

Fig. 6 -Bad Wiring Practice -Side by Side

L = 12.5 wit

oHlcm

(7)

If the strips have a breadth of 1 cm and are

separated by 0.1 cm, the combined inductance

of the pair is only 1.25 oH for each centimeter

Fig. 7 -Bad Wiring Practice -Right Angled

R2-4

surface area doesn't improve high frequency

performance if the configuration is wrong.

Inductor Windings: Figure 8 is a simple

inductor. The winding consists of 4 turns in a

single layer. Assuming a current of 1 Ampere

through the winding, the total magneto-motive

force F = NI along any path linking the 4 turns

is 4 Ampere-turns. The field is quite linear

across the length of the window because of the

addition of the fields from the individual wires

in this linear array. The winding could have

been a flat strip carrying 4 Amps with the same

result.

Fig. 9 -Transfonner Windings

is neglected). So if the secondary load current

is 4 A through 1 turn, the primary current

through 4 turns must be 1 Amp. The fields

tend to cancel not only outside both windings,

but in the center of the two windings as well.

Whatever field might remain is shorted out by

the high permeability core which has no gap.

Thus the field generated by the current in the

windings, F = 4 A, exists only between the

windings: So at high frequency, current flow is

on the outside of the inner winding and on the

inside of the outer winding, adjacent to the

field.

Fig. 8 -Inductor Winding

Multiple Layer Windings

Figure 10 shows a transformer with multi-

layer windings and its associated low frequency

mmf (F) and energy density diagrams. One half

of the core and windings are depicted. At low

frequency, current (not shown) is uniformly

distributed through all conductors, because they

are much thinner than the penetration depth.

Without the ferrite core, the field outside of

the winding would have been weak because of

cancellation, but with the high permeability

core, the external field is completely shorted

out. This means the entire field, F = NI is

contained across the window inside the wind-

ing. Field intensity, H equals NI/t. In the

center, the entire field is compressed across the

small air gap. Field intensity (H, = NI/t.) is

therefore much greater within the gap, so the

energy stored in the gap (using Eq. 2 or 3) is

much greater than the energy in the much larg-

er window.

At high frequency, current flow is concentra-

ted on the inner surface of the coil, adjacent to

the magnetic field. The field outside the coil is

negligible, so no current flows on the outer

surface.

Transformer Windings: Figure 9 shows a

transformer with a four turn single layer

primary and a 1 turn single layer copper strip

secondary. In any transformer, the sum of the

Ampere-turns in all windings must equal zero

(except for a small magnetizing current which

F"=Ni~L

JL~L

Fig. 10- Mu/tip/e Layer Winding

R2-5

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