FILTER INDUCTOR DESIGN

Filter Inductor Design

A variety of factors constrain the design of a magnetic device. The peak flux density must not saturate the core. The peak ac flux density should also be sufficiently small, such that core losses are acceptably low. The wire size should be sufficiently small, to fit the required number of turns in the core window. Subject to this constraint, the wire crosssectional area should be as large as possible, to minimize the winding dc resistance and copper loss. But if the wire is too thick, then unacceptable copper losses occur due to the proximity effect. effect. An air gap is needed when when the device device stores significant energy. e nergy. But an air gap is undesirable in transformer applications. It should be apparent that, for a given magnetic device, some of these constraints are active while others are not significant. The objective of this handout is to derive a basic procedure for designing a filter inductor. In Section 1, several types of magnetic devices commonly encountered in converters are investigated, and the relevant design constraints are discussed. The design of  filter inductors is then covered in Sections 2 and 3. In the filter inductor application, it is necessary to obtain the required inductance, avoid saturation, and obtain an acceptably low dc winding resistance and copper loss. The geometrical constant K g is a measure of the effective magnetic size of a core, when dc copper loss and winding resistance are the dominant constraints [1,2]. The discussion here closely follows [3]. Design of a filter inductor involves selection of a core having a K g sufficiently large for the application, then computing the required air gap, turns, and wire size. Design of transformers and ac inductors, where core loss is significant, is covered in a later handout. 1.

Seve everal types of magn agnetic devi device cess,

their  B-H  loops, and core core v s .

copper loss Filter inductor 

A filter inductor employed in a CCM buck converter is illustrated in Fig. Fi g. 1(a). 1(a) . In this application, the value of inductance  L is usually chosen such that the inductor current ripple peak magnitude ∆i is a small fraction of the full-load inductor current dc component

Filter Inductor Design

a)

b)  L

i(t) i(t)

∆i L

 I 

+  – 0

T s

 DT s

t 

Filter inductor employed in a CCM buck converter: (a) circuit schematic, (b) inductor current waveform.

Fig. 1

a)

b) core reluctance R  c

Φ +

i(t) + v(t)  –

F c

–

R  c

n turns

Fig. 2

air gap reluctance R  g

n i(t)

+  –

Φ(t)

R  g

Filter inductor: (a) structure, (b) magnetic circuit model.

 I , as illustrated in Fig. 1(b). As illustrated in Fig. 2, an air gap is employed which is

sufficiently large to prevent saturation of the core by the peak current  I + ∆i. The core magnetic field strength H c(t) is related to the winding current i(t) according to  H c(t ) =

R  c n i (t ) l c R  c + R  g

(1)

where lc is the magnetic path length of the core. Since  H c(t) is proportional to i(t),  H c(t) can be expressed as a large dc component  H c0 and a small superimposed ac ripple ∆ H c, where R  c  H c0 = nI  l c R  c + R  g R  c ∆ H c = n ∆i l c R  c + R  g

(2)

A sketch of  B(t) vs. H c(t) for this application is given in Fig.

 B

3. This device operates with the minor  B-H  loop illustrated.

 Bsat 

The size of the minor loop, and hence the core loss, depends on the magnitude of the inductor current ripple

∆i .

minor B-H loop,  filter inductor 

The

∆ H c

copper loss depends on the rms inductor current ripple,

 H c0

 H c

essentially equal to the dc component  I . Typically, the core  B-H loop, large excitation

loss can be ignored, and the design is driven by the copper loss. The maximum flux density is limited by saturation of  Fig. 3

Filter inductor minor  B-H  loop.

Filter Inductor Design

the core. Proximity losses are negligible. Although a high-frequency ferrite material can be employed in this application, other materials having higher core losses and greater saturation flux density lead to a physically smaller device.  Ac inductor 

An ac inductor employed in a resonant circuit is illustrated in Fig. 4. In this application, the high-frequency current variations are large. In consequence, the  B(t)-H(t) loop illustrated in Fig. 5

 B

a)  L

is large. Core loss and proximity usually this

loss

 Bsat 

are

significant

application.

in

 B-H loop, for  operation as ac inductor 

i(t)

b) i(t)

–∆ H c

The

∆ H c

∆i

maximum flux density is limited

by

core

rather than saturation. Both

core

loss

∆i

loss and

copper loss must be

t  core B-H loop

Fig. 4 Ac inductor, resonant converter example: (a) resonant tank  circuit, (b) inductor current waveform.

Fig. 5 Operational B-H  loop of ac inductor.

accounted for in the design of this element. A high-frequency material having low core loss, such as ferrite, is normally employed. Transformer 

Figure 6 illustrates a conventional transformer employed in a switching converter. Magnetization of the core is modeled by the magnetizing inductance  L mp . The magnetizing current imp(t) is related to the core magnetic field H(t) according to Ampere’s law  H (t ) =

n i mp(t ) lm

(3)

However, imp(t) is not a direct function of the winding currents i1(t) or i2(t). Rather, the a)

b) v1(t)

i1(t)

n1 : n2 imp(t)

+

v1(t)

 Lmp

v2(t)

 –

area λ1

i2(t)

+

 –

imp(t) ∆imp

t 

Fig. 6

 H c

Conventional transformer: (a) equivalent circuit, (b) typical primary voltage and magnetizing current waveforms.

Filter Inductor Design

 B

magnetizing current is dependent on the applied winding voltage waveform v1(t) . Specifically, the maximum ac flux density is directly proportional to the applied volt-seconds

λ 1. A typical  B-H  loop for this application is illustrated in

λ1 2n 1 A c

 B-H loop, for  operation as conventional transformer 

n 1∆i mp lm

Fig. 7.

 H c

In the transformer application, core loss and core B-H loop

proximity losses are usually significant. Typically the maximum flux density is limited by core loss rather than by Fig. 7 saturation. A high-frequency material having low core loss

Operational  B-H loop of a conventional transformer.

is employed. Both core and copper losses must be accounted for in the design of the transformer. Coupled inductor 

a)

A coupled inductor is a filter

inductor

having

n1

multiple

+ v1

windings. Figure 8(a) illustrates coupled inductors in a two-output

i1

vg

+  –

–

forward converter. The inductors can be wound on the same core, because

the

winding

i2 n2 turns

voltage

v2

waveforms are proportional. The inductors of the SEPIC and Cuk 

+

–

b)

converters, as well as of multiple-

i1(t)

output buck-derived converters and

 I 1

∆i1

some other converters, can be coupled.

The

inductor

current

i2(t)

ripples can be controlled by control

∆i2

 I 2

of the winding leakage inductances [4,5]. Dc currents flow in each winding as illustrated in Fig. 8(b), and the net magnetization of the core is proportional to the sum of 

t 

Fig. 8

Coupling the output filter inductors of a twooutput forward converter: (a) schematic, (b) typical inductor current waveforms.

the winding ampere-turns:  H c(t ) =

R  c n 1i 1(t ) + n 2i 2(t ) lc R  c + R  g

(4)

Filter Inductor Design

 B

As in the case of the single-winding filter inductor, the size of the minor  B-H  loop is proportional to the total

minor B-H loop, coupled inductor 

current ripple, Fig. 9. Small ripple implies small core

∆ H c

loss, as well as small proximity loss. An air gap is  H c0

employed, and the maximum flux density is limited by saturation.

 H c

 B-H loop, large excitation

Fig. 9 Coupled inductor minor  B-H  loop.

Flyback transformer 

As discussed the Experiment 7

a) n1 : n2

handout, the flyback transformer functions

i1

as an inductor with two windings. The primary winding is used during the

+

i2

imp  Lmp

v

+  –

vg

–

transistor conduction interval, and the secondary

is

used

during

the

diode

conduction interval. A flyback converter is

b) i1(t)

illustrated in Fig. 10(a), with the flyback 

i1,pk 

transformer modeled as a magnetizing inductance

in

parallel

with

an

ideal i2(t)

transformer. The magnetizing current imp(t) is proportional to the core magnetic field strength  H c(t) . Typical DCM waveforms are given in Fig. 10(b).

t 

imp(t) i1,pk 

Since the flyback transformer stores energy, an air gap is needed. Core loss depends on the magnitude of the ac component of the magnetizing current. The  B-H  loop for discontinuous conduction

t 

Fig. 10 Flyback transformer: (a) converter schematic, with transformer equivalent circuit, (b) DCM current waveforms.

mode operation is illustrated in Fig. 11. When the

 B

converter is designed to operate in DCM, the core loss is significant. The maximum flux density is then chosen to maintain an acceptably low core loss. For CCM operation, core loss is less significant, and the maximum flux density is limited only by saturation of the core. In either case, winding proximity losses may be of consequence.

 B-H loop, for  operation in  DCM flyback  converter 

 H c n 1 1, pk  R  c l m R   +R   c g

core B-H loop

Operational  B-H loop Fig. 11 of a DCM flyback transformer.

Filter Inductor Design

2.

Filter inductor design constraints

Let us consider the design of the filter inductor illustrated in  L

Figs. 1 and 2. It is assumed that the core and proximity losses are negligible, so that the inductor losses are dominated by the low-

i(t)

frequency copper losses. The inductor can therefore be modeled by

 R

the equivalent circuit of Fig. 12, in which  R represents the dc resistance of the winding. It is desired to obtain a given inductance  L and given winding resistance  R . The inductor should not saturate when a given worst-case peak current  I max is applied. Note that

Fig. 12 Filter inductor equivalent circuit.

specification of  R is equivalent to specification of the copper loss Pcu, since 2

Pcu = I rms  R

(5)

The inductor winding resistance influences both converter efficiency and output voltage. Hence in design of a converter, it is necessary to construct an inductor whose winding resistance is sufficiently small. It is assumed that the inductor geometry is topologically equivalent to Fig. 13(a). An equivalent magnetic circuit is illustrated in Fig. 13(b). The core reluctance R c and air gap reluctance R g are lc µc Ac lg R  g = µ0 Ac R  c =

(6)

where lc is the core magnetic path length,  Ac is the core cross-sectional area, µc is the core permeability, and lg is the air gap length. It is assumed that the core and air gap have the same cross-sectional areas. Solution of Fig. 13(b) yields ni = Φ R  c + R  g

(7)

Usually, R c