An inductor consists of wire wound around a core of ferrite material
that includes an air gap. A subset within the broad inductor category,
power inductors operate as energy-storage devices. They store energy in a
magnetic field during the power supply's switching-cycle on time and
deliver that energy to the load during the off time. To understand power
loss in inductors, you must first understand the basic parameters
associated with inductors. These include magnetomotive force F(t),
magnetic-field strength H(t), magnetic flux Φ(t), magnetic-field density
B(t), permeability µ, and reluctance
R.
To
avoid the complicated physics of electromagnetic fields, we offer only a
brief treatment of these parameters. The magnetic field strength
generated by an inductor is measured in amperes multiplied by turns per
meter. The magnetic field is created when current flows in the turns of
wire that wrap around the magnetic core. For switch-mode power
inductors, we can approximate the magnetic field by assuming it is
completely contained within the core.
Magnetic-field
density, measured in teslas, is equal to the magnetic-field strength,
H(t), multiplied by the magnetic-core permeability, µ:
Magnetic
flux, which is measured in webers, equals the magnetic-field density,
B(t), multiplied by the cross-sectional area of the core, A
C:
Permeability,
measured in henrys/m, expresses the capability of a specific material
to allow the flow of magnetic flux more easily. Thus, higher
permeability enables a material to pass more magnetic flux. Permeability
is a product:
in which µ
0 is the permeability of free space (µ
0 = 4π × 10
-7 H/m) and µ
R is the material's relative permeability (a dimensionless quantity). For example, µ
R for iron is approximately 5000 and µ
R
for air — the other extreme — is 1. The core of a power inductor
contains an air gap and ferrite material, so its effective µ is
somewhere between that of ferrite and air.
Magnetomotive
force, F(t), is approximated in our case as the magnetic-field
strength, H(t), multiplied by the effective length of the core, l
E:
where
the units for F(t) are amperes multiplied by turns. Effective length is
the length of the path followed by the magnetic flux around the core.
In a magnetic circuit, F(t) can be regarded as the generator of magnetic
flux (
Fig. 1). Finally, reluctance, which is measured in amperes
multiplied by turns/weber, is the resistance of a material to magnetic
fields. Reluctance is also the ratio of magnetomotive force, F(t), to
magnetic flux, Φ(t), and therefore depends on the physical construction
of the core. Substitution of the above equations for F(t) and Φ(t)
yields the following equation for reluctance:
Inductors
operate according to the laws of Ampere and Faraday. Ampere's Law
relates current in the windings — or turns of wire — to the magnetic
field in the core of the inductor. As an approximation, one assumes the
magnetic field in the inductor's core is uniform throughout the core
length (l
E). That assumption lets us write Ampere's Law as:
where “n” is the number of wire turns around the inductor core and i(t) is the inductor current.
Faraday's Law relates the voltage applied across the inductor to the magnetic flux contained within the core:
where Φ(t) is the magnetic flux and “n” is the number of wire turns around the core. The functional diagram of
Fig. 1 shows a power inductor and its equivalent magnetic circuit. As shown, the air gap places a high-reluctance element (
RAIR) in series with a low-reluctance ferrite material (
RFe),
thereby locating the bulk of the magnetomotive force, ni(t), at a
desired location — that of the air gap. The inductor value is calculated
as:
Because ferrite materials have
high permeability, they offer an easy path for magnetic flux (low
reluctance). That characteristic helps contain the flux within the
inductor's core, which in turn enables the construction of inductors
with high values and small size. This advantage is evident in the
inductance equation above, in which a core material with high µ value
allows for a smaller cross-sectional area.
Inductor Operation
The power inductor in a buck or boost converter operates as follows. Turning on the primary switch applies a source voltage V
IN across the inductor, causing the current to increase as:
This changing current, di(t)/dt, induces a changing magnetic field in the core material according to Ampere's Law:
In turn, magnetic flux through the inductor's core increases as:
and that increase can be rewritten in terms of magnetic-field density:
The primary switch opens during the off time and removes V
IN,
causing the magnetic field to decrease. In response, a decreasing dΦ/dt
in the inductor's core induces (according to Faraday's Law) a voltage
-n dΦ/dt across the inductor.
A graph of B(t) as a function of H(t) for a sinusoidal input voltage produces the hysteresis loop shown in bold lines on
Fig. 2.
B(t) is measured as H(t) is increased. The response of B(t) versus H(t)
is nonlinear and exhibits hysteresis, hence the name hysteresis loop.
Hysteresis is one of the core-material characteristics that causes power
loss in the inductor core.
Power Loss in the Inductor Core
Energy
loss due to the changing magnetic energy in the core during a switching
cycle equals the difference between magnetic energy put into the core
during the on time and the magnetic energy extracted from the core
during the off time. Total energy (E
T) into the inductor over one switching period is:
Using Ampere's Law:
and Faraday's Law:
the equation for E
T can be rewritten as:
Thus, the total energy put into the core over one switching period is the area of the shaded region within the B-H loop of
Fig. 2
multiplied by the volume of the core. The magnetic field decreases as
inductor current ramps down, tracing a different path (following the
direction of the arrows in
Fig. 2) for magnetic flux density.
Most of the energy goes to the load, but the difference between stored
energy and delivered energy equals the energy lost. Energy lost in the
core is the area traced out by the B-H loop multiplied by the core's
volume, and the power lost is this energy (E
T) multiplied by the switching frequency.
Hysteresis loss varies as a function of ΔB
n,
where (for most ferrites) “n” lies in the range 2.5 to 3. This
expression applies on the conditions that the core is not driven into
saturation, and the switching frequency lies in the intended operating
range. The shaded area in
Fig. 2, which occupies the first
quadrant of the B-H loop, represents the operating region for positive
flux-density excursions, because typical buck and boost converters
operate with positive inductor currents.
The
second type of core loss is due to eddy currents, which are induced in
the core material by a time-varying flux dΦ/dt. According to Lenz's Law,
a changing flux induces a current that itself induces a flux in
opposition to the initial flux. This eddy current flows in the
conductive core material and produces an I
2R, or V
2/R, power loss.
That effect also can be seen via Faraday's Law. If you imagine the core as a lumped resistive element with resistance R
C, then the voltage v
I(t) induced across R
C according to Faraday's Law is:
where A
C is the cross-sectional area of the core. The power loss in the core due to eddy currents is
This
power loss is proportional to the square of the rate of change of flux
in the core. Since the rate of change of flux is directly proportional
to the applied voltage, the power loss due to eddy currents increases as
the square of the applied inductor voltage and directly with its pulse
width. Thus:
where V
L is the voltage applied to the inductor, t
APPLIED is the on or off time, and T
P
is the switching period. Because the core material has high resistance,
losses due to eddy currents in the core are usually much less than
those due to hysteresis. The data given for core losses usually includes
the effects of both hysteresis and core eddy currents.
Core-loss
measurements are difficult because they require complicated setups for
measuring flux density and because they involve the estimation of
hysteresis-loop areas. Many inductor manufacturers do not supply this
data, but curves are available from ferrite manufacturers to help you
approximate the core loss in an inductor. Such curves indicate power
loss in W/kg or W/cm
3 as a function of peak-to-peak flux density, B(t), and frequency (f).
The magnetics division of Spang and Co. in Pittsburgh supplies ferrite material for inductor manufacturers. From the website
www.mag-inc.com,
you can obtain material data sheets that include curves for core loss
versus flux density at various frequencies. If you know the particular
ferrite material and the volume of the inductor's core, these curves
enable you to make a good estimate of core loss.
Such curves for a given ferrite material (
Fig. 3)
are taken with a sinusoidal applied voltage using bipolar flux swings.
When estimating the core loss for dc-dc converters that operate with
unipolar flux swings and rectangular applied voltages, which consist of
higher-frequency harmonics, you can approximate the loss using the
fundamental frequency and one-half the peak-to-peak flux density:
The core volume can usually be estimated with a rough measurement.
A
few inductor manufacturers do offer core-loss graphs or equations that
enable more accurate estimations of core power loss. For example, Pulse
Engineering in San Diego provides inductor core-loss equations in some
of its inductor data sheets (see
www.pulseeng.com).
See SMT power inductors P1172/P1173 for examples. These data sheets
include an equation using constants (K-factors) that enable the
calculation of core loss as a function of frequency and peak-to-peak
ripple in the inductor current.
On
the other hand, Coiltronics, headquartered in Boynton Beach, Fla.,
presents core loss for many of its inductors in graph form (see FLAT-PAC
3 series power inductors for example at
www.coiltronics.com).
Fig. 4 shows the curve for core power loss versus flux density and frequency from a Coiltronics Flat-Pac 3 data sheet.
Power Loss in Inductor Windings
The
preceding discussion presented losses in the inductor core, but losses
also occur in the inductor windings. Power loss in the windings at dc is
due to the windings' dc resistance and the RMS current through the
inductor (I
RMS2 · R
DC). Resistance (R) is defined as:
where r is the resistivity of the winding material. This material is usually copper, for which ρ=1.724 ·10
-8(1+.0042
· (T°C-20°C))Ωm). Physically smaller inductors typically use smaller
wire, and thus exhibit a higher dc resistance due to the smaller
cross-sectional area of the wire. Larger-value inductors have more turns
of wire, and therefore also have higher resistance due to the longer
length.
Winding losses at dc are due to the dc resistance (R
DC)
of the windings and are given in the inductor data sheet. With
increasing frequency, the winding resistance increases due to a
phenomenon called skin effect, caused by a changing i(t) within the
conductor. The changing current induces a changing flux (dΦ/dt)
perpendicular to the current that induced it.
According
to Lenz's Law, the changing flux induces eddy currents that induce a
flux themselves, in opposition to the initial changing flux. These eddy
currents are of a polarity opposite that of the initial current. The
induced flux is strongest at the conductor's center and weakest at the
surface, causing the current density at the center to decline from its
dc value with increasing frequency. As a result, current gets pushed to
the surface of the conductor, producing a lower current density at the
center and a higher current density at the surface. Resistance increases
because the resistivity of copper remains constant and the conductor's
effective current carrying area decreases.
The
windings' ac resistance is found by determining the depth, known as
penetration depth, to which current exists in the conductor at a
particular frequency. Current density at that point falls to 1/e times
the current density at the surface, or at dc. This depth (D
PEN) can be calculated as:
where r is the resistivity of the conductor (usually copper) and µ is the conductor's permeability (µ = µ
0 · µ
R, where µ
R
= 1 for copper). This calculation is accurate when the conductor is a
flat surface or when the radius of the conductor is much larger than the
penetration depth. Note that ac resistance (R
AC) acts as a
power loss only to the ac current, which for buck and boost converters
is the inductor-current ripple. DC current in the inductor only creates
power loss in R
DC.
You find R
AC
by calculating the effective conducting area of the copper wire at a
given frequency. For conductors that have radii larger then the skin
depth at the given operating frequency, the effective conducting area is
the surface area of a conducting ring with thickness equal to the skin
depth. Because resistivity remains constant, the ratio of R
AC to R
DC is simply the ratio of the two areas:
Furthermore, R
AC/R
DC multiplied by R
DC is the effective resistance at a given frequency for a straight wire in free space.
Eddy
currents in the inductor windings are also induced by other nearby
conductors, a phenomenon known as the proximity effect. For inductors
with many overlapping wire turns and adjacent wires, the increased eddy
currents cause a resistance considerably higher than that from the skin
effect alone. The proximity effect becomes complicated, however, due to
the various configurations and distances with which conductors can be
placed relative to each other. Because such calculations are beyond the
scope of this article, the reader should refer to the references
provided.
A simple circuit illustrates losses in the inductor (
Fig. 4). R
C represents the core losses, and R
AC and R
DC represent the ac- and dc-dependent winding losses. R
C is determined by core loss calculations or estimates, while R
DC is the dc winding resistance and R
AC
is the ac resistance due to skin effect, proximity effect or both. An
example of this loss model can be developed using the MAX5073 switching
power supply. We operate the MAX5073 as a buck converter with V
IN = 12 V, V
OUT = 5 V, f
SW =1 MHz, and I
OUT = 2 A. A 4.7-µH inductor (FP3-4R7 from Coiltronics) produces an inductor current ripple (ΔI(t)) of 621 mA.
A graph of core loss versus flux density and frequency is shown in
Fig. 4.
Peak-to-peak flux density (ΔB) is what matters. It traces out a small
hysteresis loop within the larger hysteresis loop (see the inner loop in
Fig. 2). You can find ΔB using the equation given in the inductor data sheet:
where K is a constant given in the data sheet (K = 105 in our case), and L is the inductance in microhenries. In this example:
As
an alternative, you can estimate ΔB(t) using the inductor volt-second
product divided by the number of turns and the core area within the
turns:
Going to the FP3 data sheet, core loss at 613 gauss and f
SW= 1 MHz is approximately 470 mW. R
C in
Fig. 5
is the equivalent parallel resistance that accounts for power loss in
the inductor core. That resistance is calculated from the RMS voltage
across the inductor and the core power loss:
R
C is then 60.1 V
2/0.470 W=128 Ω, where V
IN × √
D is the RMS value of a rectangular wave with duty cycle D and amplitude V
IN.
R
DC from the data sheet is 40 mΩ, assuming a zero temperature rise for the inductor, which would otherwise increase the value of R
DC. The penetration depth for a 1-MHz switching frequency, using only the fundamental of the triangular current ripple at T
A = +20°C, is 0.065 mm. A rough measurement of the conductor's radius gives 0.165 mm, which results in an R
AC value of:
This resistance only dissipates power due to the RMS ac current. The RMS value of inductor current ripple is:
