Windings.pdf

Section

Understanding the rules governing magnetic field

behavior is fundamentally important in designing and

optimizing magnetic devices used in high frequency

switching power supply applications. Paralleled

windings can easily fail in their intended purpose,

eddy current losses and leakage inductances can eas-

ily be excessive. These are some of the problems that

are addressedin this Section.

Even if you never participate in transformer or

inductor design, these magnetic principles apply in

optimizing circuit layout and wiring practices, and

minimizing EM!..

Reference paper (R2): "Eddy Current Losses in

Transformer Windings and Circuit Wiring," included

in this Manual, is a useful supplement.

Conservation of Energy

Like water running downhill, electrical current

always takes the easiestpath available. The path taken

at dc and low frequencies can be quite different from

the path taken by the high frequency current compo-

nents.

The basic rule governing the current path: Cur-

rent flows in the path(s) that resu/t in the /owest ex-

penditure of energy. At low frequency, this is ac-

complished by minimizing PR losses. At high fre-

quency, current flows in the path(s) that minimize

inductive energy -energy transfer to and from the

magnetic field generated by the current flow. Energy

conservation causes high frequency current to flow

near the surface of a thick conductor, and only certain

surfaces, even though this may results in much higher

I2R losses. If there are several available paths, HF

current will take the path(s) that minimize inductance.

This may have undesirable side effects as shown in

one of the examples below.

Examples are given later which demonstrate how

to manipulate the field and current path to advantage.

Figure 3-1 Skin Effect Model

happens when a high frequency ac current is put

through?

Figure 3-1 happens to be a high-frequency model

of a single wire. A represents the surface of the wire,

B is the center. Lx represents the inductance per unit

length external to the wire (what would be the meas-

ured inductance of the wire). Li is inductance distrib-

uted within the wire, from the surface to the center.

(Copper is non-magnetic, just like air, and stores

magnetic energy in the same way.) Ri is the distrib-

uted longitudinal resistance from the surface to the

center. Collectively, Ri is the dc resistance of the

wire. All of the above values are per unit length of

wire.

At dc or low frequency ac, energy transfer to in-

ductance Li, over time, is trivial compared to energy

dissipated in the resistance. The current distributes

itself uniformly through the wire from the surface to

the center, to minimize the rR.t loss. But at high fre-

quency, over the short time spans involved, rR .t loss

is less than the energy transfer to and from Li. Current

flow then concentrates near the surface, even though

the net resistance is much greater, in order to mini-

mize energy transfer to Li. If we look at this strictly

from a circuit point of view, at high frequency, the

impedance of Li near the surface blocks the current

from flowing in the center of the wire.

Penetration depth (or skin depth), Dpen, is de-

fined as the distance from the conductor surface to

where the current density (and the field, which termi-

Skin Effect

The circuit of Figure 3-1 shows an inductor in se-

ries with an L-R transmission line. What happens

when a dc current is put through this circuit? What

3-1

nates on the current flow) is lie times the surface CUf-

rent density:

intensity, H, by 5 times. Energy density, proportional

to H2, is 25 times weaker.

meters

(W/cmJ = Y2BH= Y2,uH1

In copper at 100°C, resistivity,

p = 2.3-10-8 Q-m,

Therefore, total energy (volume times energy)

and inductance, are 5 times less in Fig. 3-2 above than

with the more concentrated field in (R2) Fig. 6.

An important principle is demonstrated here: If

the field (and the current that produces it) is given the

opportunity to spread out, it will do so in order to

minimize energy transfer. The stored energy (and in-

ductance) between the conductors varies inversely

with the length of the field.

Visualize the magnetic field equipotential sur-

faces stretched across the space between the two con-

ductors, terminating on the current flow at each sur-

face. Visualize the flux lines, all passing horizontally

between the two conductors, normal to the equipo-

tential surfaces. The flux return paths encircle the

conductors in wide loops spread out over a distance -

the field here is very weak.

and~r=

-7.6 7

(copper)

.J7

At 100 kHz in copper, DpEN= .024 cm.

Proximity Effect

When two conductors, thicker than DpEN,are in

proximity and carry opposing currents, the high fre-

quency current components spread across the surfaces

facing each other in order to minimize magnetic field

energy transfer (minimizing inductance) This high

frequency conduction pattern is shown in reference

(R2) Figs 5,6, and 7. (R2) Fig. 5 is reproduced below

as Fig. 3-2.)

T w

Examples

In the examples of winding structures given

below:

.Each + represents 1 Ampere into the page

.Each. represents 1 Ampere out of the page

.Fine lines connecting + and. represent edge view

of the field equipotential surfaces.

.Conductor size is much greater than penetration

(skin) depth.

Simple Transformer Windings:

If the two flat conductors of Fig. 3-2 are placed

within a transformer core, the only change in the field

pattern is that the fringing field at the conductor ends

is reduced. The conductors are now called "wind-

ings", and the inductance representing the energy

stored between the conductors is called "leakage in-

ductance". In Figure 3-3, one of the flat strips is re-

placed by 4 wires. This could be a 4-turn winding

carrying 3A, opposed by a single turn secondary car-

rying l2A. Or, the four primary wires could be in

parallel, giving a l-turn primary carrying l2A. In ei-

ther case, the field pattern spreads itself across the

entire window, and the same minimized energy is

Fig. 3-2 Proximity Effect

In all three configurations, current does not flow

on any other surfaces because that would increase the

volume of the magnetic field and require greater en-

ergy transfer. Inductance is thus minimized, but ac

resistance is made higher, especially in (R2) Figs 6

and 7. Note how in Fig. 3-2, the preferred configura-

tion, the current distributes itself fairly uniformly

across the two opposing surfaces. This results in sig-

nificantly less stored energy than R2 Fig. 6, even

though the length and volume of the field in Fig. 3-2

is 5 times greater. This is because spreading the cur-

rent over 5 times longer distance reduces the field

3-2

PEN -

Figure 3-3 Single Layer Windings

depth, the field cannot penetrate the conductor, and

current flow is confined to near the conductor surface.

A strange thing happens --since the field cannot

penetrate the conductor, the entire 8 Ampere field

tenninates at this inner surface of the inner layer. This

requires a total of 8 Ampere-turns at this surface --

2A per wire --since the field can be tenninated only

by current flow. Inside the outer layer, there is a 4A

field, from the 4 A-t flowing in the outer layer. This

field must tenninate on the outside of the inner layer,

because it cannot penetrate. This requires 4 A-t in the

opposite direction of the current in the wire! !

Thus the inner layer has 8 A-t on its inner surface,

and 4 A-t in the opposite direction on its outer sur-

face. Each inner wire has 2A on its inner surface and

lA in opposite direction on its outer surface. The net

current remains lA in all series wires in both layers.

But since loss is proportional to r, the loss in the in-

ner layer is 12+ 22= 5 times larger than the loss in the

outer layer, where only the net lA flows on its inner

surface!!

Not only is the rR loss larger becausethe current

is confined to the surface, it also increases exponen-

tially as the number of layers increases. This is be-

cause the field intensity increases progressively to-

ward the inside of the winding. Since the field cannot

penetrate the conductors, surface currents must also

increase progressively in the inner layers. For exam-

ple, if there were 6 wire layers, all wires in series car-

rying lA, then each wire in the inner layer will have

6A flowing on its inner surface (facing the secondary

winding) and 5A in the opposite direction on its outer

surface. The loss in the inner layer is 62 + 52 = 61

times larger than in the outer layer which has only the

net IA flowing on its inner surface!!

If the wire diameter is reduced, approaching the

penetration depth, the + and -currents on the inner

and outer surfaces of each wire start to merge, par-

tially canceling. The field partially penetrates through

the conductor. When the wire diameter is much less

than the penetration depth, the field penetrates com-

pletely, the opposing currents at the surfaces com-

pletely merge and cancel, and the IA current flow is

distributed throughout each wire.

stored between the windings. The conductors are

thicker than DPEN,so the high frequency currents flow

near the surfaces in closest proximity, thus terminat-

ing the field.

Figure3:4Two l~er- windings :::Series

Multiple Layer Windings:

In Figure 3-4, an 8-turn primary carrying lA is

opposed by a 2-turn secondary carrying 4A. The 8

turns of wire, sized for the required nns current, can-

not fit into the window breadth, so it is configured in

two layers. As expected, there is an 8 Arnpere-turn

field stretched across the entire window. But since the

conductor thickness is much greater than the skin

3-3

Calculation of the rR loss when the conductor

size (layer thickness) is similar to the penetration

depth is very complex. A method of calculating the ac

resistance was published by Dowell(l), and is dis-

cussed extensively in Reference (R2). Figure 3-5

(from R2, Fig. 15), based on Dowell's work, shows

the ratio of RAc/RDCvs. layer thickness/DPENand the

number of layers in each winding section. Read (R2)

for a detailed discussion.

2. Carstenl2]has applied Dowell's sine wave so-

lution to a variety of non-sinusoidal waveforms en-

countered in SMPS applications, providing curves

that include hannonic effects.

3. Computerize Dowell's work, in order to ap-

ply it to any non-sinusoidal waveshape. O'Meara[3]

can be helpful.

4. A computer program "PROXY" (proximity

effect analysis) is available from KO Systerns[4].

100

ParaUeled Layers:

The transformer of Fig. 3-6 is the same as Fig. 3-

4 with the winding layers reconfigured in parallel,

resulting in a 4-turn, 2 Amp primary, and a l-turn, SA

secondary. The intention is to have lA in each pri-

mary wire and 4A in each secondary strip, with the

same field pattern as in Fig. 3-4, but it doesn't happen

that way.

I ~ 20

1171

~ 10

1 2

Q = LayerThickness/Dpen

Figure. 3-5 Eddy Current Losses --RAdRDc

The curves of Fig. 3-5 graphically show the high

ac resistance that results when the layer thickness

equals or exceeds the penetration depth, especially

with multiple layers. With the large ac currents in a

transformer, RAc/Roc of 1.5 is generally considered

optimum. A lower RAc/Roc ratio requires finer wire,

and the wire insulation and voids between wires re-

duce the amount of copper, resulting in higher dc

losses. In a filter inductor with small ac ripple current

component, a much larger RAcIRoc can be tolerated.

Although the curves of Fig. 3-5 are quite useful,

keep in mind that an accurate solution requires har-

monic (Fourier) analysis of the current waveform.

Loss must then be calculated independently for each

harmonic, since DpENdiffers for each harmonic fre-

quency, and these losses added to obtain the total loss.

Alternative methods of calculating eddy current

losses include:

I. Calculate based on the fundamental only, ig-

nore the harmonics and add 50% to the calculated

loss.

Figure 3-6 Paralleled Two-Layer Windings

Whenever windings are paralleled, alternative

current paths are provided. In Fig. 3-6, resistance in

each of the paralleled windings causes the current to

divide nearly equally at dc and low frequencies. But

at high frequency, stored energy becomes more im-

portant than rR "t. All of the high frequency current

components will flow on the inside surfaces of the

inner layers directly facing each other. The high fre-

quency current in the outer layers is zero. Any current

in the outer layers contributes to an additional field,

between inner and outer layers, requiring additional

energy. With series-connected layers the current has

3-4

no alternative -it must flow in all layers, resulting in

additional energy stored between layers as shown in

Fig. 3-4. When there is an alternative, as with paral-

leled layers, the high frequency current will flow so

as to minimize the stored energy.

The leakage inductance between the windings in

Fig 3-6 is slightly smaller than it would have been if

the current divided equally -a small benefit. But only

a tiny fraction of the available copper is utilized,

making fR loss prohibitively large.

Another example: A low voltage, high current

secondary might use a single turn of copper strap, but

the thickness required to carry the rms current is 5

times the skin depth. It might seem logical to parallel

5 thin strips, each one skin depth in thickness. Result:

All the HP current will flow in the one strip closest to

the primary. If ac current were to flow in the other

strips further from the primary, PR loss would be

less, but more stored energy is required because the

field is bigger (increased separation).

Rule: If you provide alternative current paths, be

sure you know what the rules are.

Window Shape-Maximize Breadth

The shape of the winding window has a great im-

pact on the eddy current problem. Modern cores in-

tended for high frequency SMPS applications have a

window shape with a winding breadth (width) several

times greater than its height. For the same number of

turns, the number of layers required is thereby mini-

mized. As shown in Figure 3-7, the window has twice

the breadth as the core in Fig. 3-4, so that only one

layer is required. This results in a very significant re-

duction in eddy current losses, as can be seen from

Fig 3-5.

Another major benefit of the wider window is that

the stored energy (leakage inductance) is minimized.

With the 8 turns at lA of Fig 3-4 fit into a single

layer, (and the opposing 2-turn strap also in a single

layer), the total magnetic force, 1.,remains 8 Ampere-

turns. However, the field intensity H, stretched over

twice the distance, is half as much. Flux density B is

also halved (B = J1H), therefore energy density Y2BH

is one-fourth as much. However, the volume of the

field is increased, perhaps doubling, therefore total

energy-and leakage inductance-is halved.

Figure. 3-7 Wide Window Breadth

The penalty for stretching out the winding is in-

creased capacitance between windings.

Interleaving:

If the stretched out winding of Fig. 3-7 were

folded in half, it could then fit into the original win-

dow, as shown in Figure 3-8. This "interleaved"

winding has the same low eddy current loss, low field

intensity, and low inductance as the winding of Fig.

3-7.

Figure. 3-8 Interleaved Windings

3-5

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