Equivalent Electrician Circuit.pdf

Ocl 94

Lloyd Dixon

Purpose:

I. To define the electrical circuit equivalents of

magnetic device structures to enable improved

analysis of circuit performance.

2. To define the magnitude and location of

relevant parasitic magnetic elements to enable

prediction of performance effects

3. To manipulate parasitic elements to obtain

improved or enhanced circuit performance

4. To encourage the circuit designer to be more

involved with magnetic circuit design.

provide a simple electrical equivalent circuit that

lends itself to circuit analysis.

The magnetic structure shown in Figure 1 will be

the first demonstration of this technique. This

simple inductor is built upon a ferrite core, with air

gaps to store the required inductive energy created

by simply shimming the two core halves apart (not

usually a good practice, but inexpensive).

Magnetic Definitions:

Systeme International (SI) Units and Equations

are used throughout this paper .

Simplifying the Magnetic Structure:

The fIrst task is to take a magnetic device

structure and reduce it to a few lumped elements -

as few as possible for the sake of simplicity.

because tllC equivalent electrical circuit will have

just as many elements. This is not an easy task -

just about everything is distributed. not lumped: the

magnetic force from current in the windings.

distributed flux. fringing flux adjacent to gaps. and

stray fields. Boiling this down to a few elements

with reasonable accuracy requires a little insight

and intuition and experience -but it can be done.

Finite Element Analysis softwarc on the other

hand is extremely accurate because it does just tlle

opposite -it chops the structure up unto a huge

number of tiny elements.[!] It is a useful tool which

can provide a great deal of knowlcdge about what

goes on inside the magnetic device. but it does not

Fig ,. -Magnetic Structure -Inductor

The first thing to do is to simplify -judiciously.

Looking at the inherent symmetry of the structure,

the two outer legs can be combined into a single

leg with twice the area. Fringing fields around the

gaps will be ignored, except the effective gap area

might be increased to take the fringing field into

account.

The number of flux paths will be minimized,

eliminating those that are trivial. For example, flux

in the non-magnetic mateial adjacent to the core

will be ignored, because it is trivial compared to

the flux in the neighboring ferrite. On the other

hand, if there were two windings, the small amount

of flux between the windings must not be ignored

-it constitutes the small but important leakage

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inductance between the two windings.

The core will be divided into regions of similar

cross-section and flux density. The distributed

magnetic force from the winding will be lumped

and assigned a specific location in the physical

structure.

combining the two centerpost halves into a single

element, likewise combining the outer leg portions

on both sides of the gap. The magnetic field source,

the ampere-turns of the winding, which is really a

circulatory field is assigned to any discrete point

where the nux is not divided. It would be incorrect

to locate this source in series with the outer leg,

becauseit would drive the nux in the stray field in

the wrong direction.

ASSUNE TYPICAL VALUES :

A.=1cmz:

te=10cm

011

Aw=2cmz

(ETO3~)

NAX Ht=Nlm=JmAw=~OO.2=800

RELUCTANCE

~ MMfl-

R -HliBA

= 'I ~A

r -

-FULL UTILIZ:

WmEHt.BmA.!Z

=800..2S..0001!2-10mJ

GAP LENGTH-FULL UTILIZ:

tg=NI!H-NII"!B

=600...'..10-7j.25=.4cm

-. !

APPORTION ,. TO Re, Ro

.13

APPTN 19 TO Rcg, Rog

~=~o~r,

~0=47T.10-7

~r= 1 (o;r), =3000 (f.rrlt.)

Ro=(l.12)1(JJ.°~rA.)

=.051(JJ.oJJ.r..0001)

Ro=.133.101 (omit 10')

Ra IS AN ESTIMATE

Fig 2. -Core Parameters and Utilization

For the purpose of illustration. the parameters of

the core (ETD34) are given in Figure 2. The

maximum ampere-turns obtainable is calculated

from the window area times the max. current

density in copper of 400Ncm2. The maximum flux

at saturation equals the saturation flux density

(0.25T) times the core area. From this. the maxi-

mum possible energy storage (in an appropriate

gap) equals I/2BH. for a total of IOmJ. The gap

length required to achieve this full utilization is

calculated as shown in Figure 2. These calculations

are not relevant to the modeling process. but they

help indicate the suitability of this core for .the

inlended application.

The Reluctance Diagram: Next, a reluctance

diagram will be created. modeling the physical

structure. Reluctance is essentially magnetic imped-

ance. It is a measure of the opposition to flux

within any region of the magnetic device.

Fig 3. -The Simplified Reluctance Model

So the reluctance diagram includes the reluctance

of the combined centerpost ferrite and also the

centerleg gap, the combined outer leg ferrite and

the outer leg gap, and the reluctance of the stray

field outside the core (the calculation is an educated

guess). This "magnetic circuit" can be analyzed just

as though it were an electrical circuit. Remember

that it is not an elecrical circuit -reluctance is

definitely not the same as resistance -it stores

energy rather than dissipate it. But the reluctance

diagram does follow the same rules as an electrical

circuit, and the amount of flux in each path can be

calculated based on the magnetic force and the

reluctance.

Much can be learned by examining the reluc-

tance model and playing .'what if'. games. Note that

in each leg, the calculated gap reluctances are more

than 100 times greater than the adjacent ferrite core

legs. This indicates that the core reluctances could

be eliminated, impairing accuracy by less than I %.

Note also that the flux in the suay field is actully

large than the flux in the outer leg of the core,

because the suay field reluctance is less than the

gap reluctance. This means that much noise is

propageted outside the core, and the inductance

---

R=-=~--'~

cI>

fL4

The reluctance of each significant region of the

device is calculated from its area, length and

permeability, and inserted with its specific value

into the appropriate location in the reluctance

model, as ShO\\'ll in Figure 3. Again, because of

inherent symmelry, the model can be simplified by

R3-2

~AX ENERGY

value obtained depends heavily on the stray field,

which is difficult to calculate. If the centerleg gap

is eliminated and the outer leg gap correspondingly

increase, the amount of s.l:rayfield increases further.

On the other hand, if the outer leg gap is closed

and the centerleg gap is widened, the stray field is

almost eliminated, because the reluctance of the

centerleg without gap is much less than the stray

field reluctance. In this manner, the reluctance

model is useful without even converting it to the

equivalent electrical circuit.

to the dual.

A planar circuit is defined as one that can be

drawn on a plane surface with no crossovers. The

duality process fails if there are crossovers, which

can occur with complex core structures.ISJActually,

with only three windings on a simple core, if the

reluctance model includes every theoretically

possible flux linkage combination between wind-

ings, there will be crossovers. But most of these

theoretical linkages are trivial, and should be

ignored, for the sake of simplicity if nothing else.

This is where common sense comes in.

Creating the Dual: The process for creating the

electrical dual from the reluctance model is actually

quite simple, as illustrated in Figure 4.

Magnetic-Electrical Duality:

Fifty years ago, E. Colin Cherry published a

paper showing the duality between magnetic circuits

and electrical circuits.[3] It is well know that two

electrical circuits can be duals of each other -the

Cuk converter is the dual of the flyback (buck-

boost), for example. Electrical circuits that are duals

are not therefore equivalent. Magnetic circuits and

electrical circuits are in a different realm, and yet

in this case the duals are truly equivalent.

A dual is created by essentially turning the

circuit inside out and upside down. Some of the

magnetic-electrical duality relationships and rules

are:

~ ,

.~111/

-'¥+

: !

-51

, , l ' ""

r ' I"

' ,

1 \

: I

-c ';: -0

\ RS

I \

~--

..-,

..' -RCG

..,

" \\

Nodes Meshes (loops)

Open Short

Series Elements Parallel Elements

Magn. Force

Fig 4. -The Magnetic / Electrical Dual

Ampere- Turns

First, identify each mesh, or loop in the reluc-

tance model. In this case there are 3 loops. (The

outside is always considered a loop. Topologically

a simple circle has two loops -the inside and the

outside.) Put a dot in tile center of each loop (any

convenient location on the outside). These dots will

be the nodes of the electrical circuit. Draw a dash

line from electrical node to node though every

intervening element. The dash lines are branches in

the electrical circuit. The intervening elements

become elements of the new circuit, but they are

transformed: Reluctances become their reciprocal-

permeances, the magnetic winding becomes the

electrical terminals, with d(j)l/dt translating into

Vl/Nl (Faraday's Law), and magnetic force trans-

lating into NlIl in series with the terminals. Note

that the 5 nodes in the original reluctance model

are automatically converted into 5 loops in the

electrical dual. (Don't forget the outside is a loop!)

dcl>/dt

Volts/turn

Reluctance

Permeance

Polarity Orientation: Rotate in same direction

Circuits must be planar

Permeance is the reciprocal of reluctance:

p=2.= R

Pemleance is actually the inductance for 1 turn.

Multiply pemleance by N2 to obtain the inductance

value referred to all N-turn winding.

Polarity Orientation means that for elements

such as the windings that have polarity, to assign

the proper polarity in the dual, rotate all polarity

indications in the same direction from the original

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, ,

AIl of the values in the equivalent electrical circuit

at this stage pertain to a one-turn winding. The

electrical equivalent circuit is redrawn ~ shown in

Figure 5:

inductance. But if the outer leg gap was closed up,

its inductance would become infinite, making the

stray field inductance irrelevant. The overall induc-

tance would then equal the 14~ of the centerleg

gap.

A simple transformer: A transformer with two

windings is shown in Figure 7. The transformer bas

no gap -energy storage is undesirable. The flux

between the two windings, although small, is very

important because the energy contained between the

windings constitutes leakage inductance.

UNITS:

Henry./Turn

p = 1/R

= ~A/1

= L/N2

1/Ro

= 7.5~H/N2

V1/N1

= d-l/dl

N111

= Hi

Fig 5. -The Equivalent Electrical Circuit

If the winding has fifteen turns, penneances are

converted to inductance values by multiplying by

152. Likewise, tenninal V/N is multiplied by 15 to

become tenninal voltage, and NI is divided by N to

become tenninal current. The final electrical equiv-

alent circuit is shown below:

WITH N = 15 TURNS

Fig 6. -Final Electrical Equivalent Circuit

This simple example obviously produces a trivial

result. There are five inductive elements which

represent the five reluctances in the reluctance

model. The five inductances combine to form a

single inductance value of IO~. But each of the

five elements is clearly identifiable back to the

reluctance it represents. Note that the series reluc-

tances of the centerleg ferrite and centerleg gap

show up as parallel inductances in Fig. 6. and the

stray field reluctance which was in parallel with the

outer leg are now series inductances. The circuit

shows that the high inductance values contributed

by the centerleg ferrite and outer leg ferrite are

irrelevant in parallel with the much lower gap

inductances. It shows that the stray field inductance

makes a very significant contribution to the overall

Figure 8 is the reluctance model with specific

values calculated for the same Em34 core used in

the previous example, but ungapped. The ampere

turns in the two windings cancel except through the

region between the windings (RI2). Thus, when the

transformer is loaded, this is the only place the

fields don't cancel, storing considerable leakage

inductance energy as a function of load current.

R3-4

transfonner is added to one tenninal pair to allow

for a turns ratio other than 1:1, and to provide

galvanic isolation.

Coupled Inductor: Topic M7 in the Design

Reference Section of the Seminar Manual describes

the benefits of coupled filter inductors in multi-

output buck regulators. Figure 12 shows the physi-

cal structure and resulting equivalent circuit which

can provide insight into the design of this device.

L12

NI, N2 = 15 TURNS

L IN J.LH

--r-rv;'"'-

Log,

14.(

( rlt

1;li

4 LOOPS, 4 NODES IN BOTH

I -N

1 --

'"":

R.f.

Fig 9. -The

Transformer

Electrical

Dual

~ -

Leg L12

Topic.

..7.

WITH Np .20

TURNS, N. -2

TURNS

+L.

Fig 12. -Coupled Inductor

Fractional Tur~: Transfonners with fractional

turns have been featured in previous Seminars.

Figure 13 provides insight into the design and

behavior of this device.

Fig 10. -Transformer Equivalent Circuit

A Three-Winding Transformer: Figure 11 is

a three-winding transfonner showing the physical

structure and the equivalent electrical circuit. The

reluctance model is not shown. The equivalent

circuit model shows how the leakage inductances

are distributed between windings and their magni-

tudes.

~ Vln ,

;Hp

V."I~]

IN.

"""

ErrECTIVE

LEAKAGE

--0

'L'

HOW TO STIrrEN

-1

Fig

13. -Transformer

with

Fractional

Turns

PER~EANCE

IN ~H/turn2

The Gyrator-Capacitor Approach:

Another method for defining an equivalent

electrical circuit of a magnetic device structure

completely avoids the Reluctance/Dualityapproach

that has been discussed up to this point.l4.5J The

Fig 11. -Three-Winding Transformer

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