By Travis Eichhorn, Applications Engineer,
Maxim Integrated Products, Sunnyvale, Calif.
Inductors dissipate power in the core and in the
windings. Although exact calculations of these
losses can be complex and difficult, they can be
readily estimated using data sheet parameters
available from magnetic component suppliers.
H(t), magnetic flux ⌽(t), magnetic-field density B(t),
permeability ␮, 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 switchmode 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, ␮:
switch-mode power supply incurs loss
in many areas of its circuitry, including the MOSFETs, input and output capacitors, quiescent controller current
and inductor. The power dissipated in
the inductor arises from two separate sources: the losses
associated with the inductor core and those associated with
the inductor windings. Though determining these losses
with precision can require complex measurements, an easier
alternative exists. Inductor losses may be estimated using
readily available data from core and inductor suppliers along
with the relevant power supply application parameters.
A
Inductor Basics
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
Power Electronics Technology April 2005
Magnetic flux, which is measured in webers, equals the
magnetic-field density, B(t), multiplied by the cross-sectional area of the core, AC:
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 mate14
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INDUCTOR POWER LOSS
rial to pass more
magnetic
flux.
Permeability is a
product:
in which ␮ 0 is
the permeability of
free space (␮0 = 4␲
⫻ 10-7 H/m) and ␮R
is the material’s
relative permeability (a dimensionless Fig. 1. This magnetic circuit (a) is represented by the equivalent circuit model (b).
quantity). For example, ␮R for iron is approximately 5000
ment (ℜAIR) in series with a low-reluctance ferrite material
and ␮R for air—the other extreme—is 1. The core of a
(ℜFe), thereby locating the bulk of the magnetomotive force,
power inductor contains an air gap and ferrite material, so
ni(t), at a desired location—that of the air gap. The inducits effective ␮ is somewhere between that of ferrite and air.
tor value is calculated as:
Magnetomotive force, F(t), is approximated in our case
as the magnetic-field strength, H(t), multiplied by the
effective length of the core, lE:
F (t )=H (t )⋅ l E
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 ␮ value allows for a smaller cross-sectional area.
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, ⌽(t), and therefore depends
on the physical construction of the core. Substitution of
the above equations for F(t) and ⌽(t) yields the following
equation for reluctance:
Inductor Operation
The power inductor in a buck or boost converter operates as follows. Turning on the primary switch applies a
source voltage VIN across the inductor, causing the current
to increase as:
di (t ) VIN
=
dt
L
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 (lE). That assumption lets us write Ampere’s Law as:
This changing current, di(t)/dt, induces a changing
magnetic field in the core material according to Ampere’s
Law:
dH (t ) n di (t )
= ⋅
dt
l E dt
H (t )⋅ l E =n ⋅ i (t )
In turn, magnetic flux through the inductor’s core increases 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:
and that increase can be rewritten in terms of magnetic-field density:
dB (t )
n di (t )
=
⋅
dt
A ⋅ ℜ dt
where ⌽(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 elePower Electronics Technology April 2005
The primary switch opens during the off time and removes VIN, causing the magnetic field to decrease. In re16
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INDUCTOR POWER LOSS
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.
Minor hysteresis loop
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 (ET) into
the inductor over one switching period is:
Using Ampere’s Law:
Fig. 2. A plot of magnetic field density B(t) versus magnetic field strength H(t) reveals the
major and minor hysteresis loops associated with an inductor core.
sponse, a decreasing d⌽/dt in the inductor’s core induces
(according to Faraday’s Law) a voltage -n d⌽/dt across the
inductor.
lE ⎞
⎛
⎜⎝ i (t )=H (t )⋅ ⎟⎠
n
and Faraday’s Law:
dB (t )⎞
⎛
⎜⎝ v (t )=n ⋅ A ⋅ dt ⎟⎠
the equation for ET can be rewritten as:
E T =A ⋅ l E ∫ H ⋅ dB
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 (ET) multiplied by the switching frequency.
Hysteresis loss varies as a function of ⌬Bn, 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 d⌽/dt. According to Lenz’s Law, a changing flux in-
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INDUCTOR POWER LOSS
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:
PE ∝
where VL is the voltage applied to the inductor, tAPPLIED
is the on or off time, and TP 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/cm3 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:
Fig. 3. AC core loss for a particular ferrite material is plotted as a
function of flux density at different frequencies. (Data courtesy of
Spang and Co.)
duces 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 I2R, or V2/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 RC, then the voltage vI(t) induced across RC according to Faraday’s Law is:
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
where AC is the cross-sectional area of the core. The
power loss in the core due to eddy currents is
2
PE =
v I (t )
RC
This power loss is proportional to the square of the rate
Power Electronics Technology April 2005
VL2 t APPLIED
⋅
RC
TP
20
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INDUCTOR POWER LOSS
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 (IRMS2 RDC). Resistance (R) is defined as:
z
Fig. 4. AC core loss for a particular inductor is plotted versus flux density at different
frequencies. (Data courtesy of Coiltronics.)
where ␳ is the resistivity of the winding material. This
material is usually copper, for which ␳=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 (RDC)
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 chang-
ing i(t) within the conductor. The changing current induces a changing flux (d⌽/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, pro-
z
z
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cores. Eddy currents, caused by the fringing flux
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lead to excessive heat due to heavy copper losses.
The distributed air gaps inherent in Kool Mµ can
provide a much cooler inductor.
Kool Mµ E cores are available in many industry
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If you are using gapped ferrite E cores for inductor
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1-800-245-3984
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email: magnetics@spang.com
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Power Electronics Technology
April 2005
INDUCTOR POWER LOSS
ducing 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 (DPEN) can be calculated as:
face area of a conducting ring with thickness equal to the
skin depth. Because resistivity remains constant, the ratio
of RAC to RDC is simply the ratio of the two areas:
Furthermore, RAC/RDC multiplied by RDC 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). RC represents the core losses, and RAC and RDC represent
the ac- and dc-dependent winding losses. RC is determined
by core loss calculations or estimates, while RDC is the dc
winding resistance and RAC 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 VIN = 12 V, VOUT = 5 V, fSW =1 MHz, and IOUT = 2
A. A 4.7-␮H inductor (FP3-4R7 from Coiltronics) produces
an inductor current ripple (⌬I(t)) of 621 mA.
A graph of core loss versus flux density and frequency
is shown in Fig. 4. Peak-to-peak flux density (⌬B) 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
⌬B using the equation given in the inductor data sheet:
where ␳ is the resistivity of the conductor (usually copper) and ␮ is the conductor’s permeability (␮ = ␮0 ␮R,
where ␮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 (RAC) 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 RDC.
You find RAC 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 surz
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 ⌬B(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
fSW= 1 MHz is approximately 470 mW. RC 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:
VRMS =VIN ⋅ D=12 V ⋅ 0.417=7.75 VRMS
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INDUCTOR POWER LOSS
Fig. 5. An equivalent loss model for a power inductor includes terms
representing the ac- and dc-dependent winding losses (RAC and RDC)
and the core losses (RC).
RC is then 60.1 V2/0.470 W=128 Ω, where VIN ⫻
D is
the RMS value of a rectangular wave with duty cycle D
and amplitude VIN.
RDC from the data sheet is 40 mΩ, assuming a zero temperature rise for the inductor, which would otherwise increase the value of RDC. The penetration depth for a 1-MHz
switching frequency, using only the fundamental of the
triangular current ripple at TA = +20°C, is 0.065 mm. A
rough measurement of the conductor’s radius gives 0.165
mm, which results in an RAC value of:
This resistance only dissipates power due to the RMS
ac current. The RMS value of inductor current ripple is:
Thus, the total estimated losses are:
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References
1. Erickson, Robert W., and Dragan Maksimovic. Fundamentals of Power Electronics, 2001. Chapters 13 and 14, pp. 491562.
2. Kassakian, John G., Martin F. Schlecht, and George C.
Verghese. Principles of Power Electronics, 1991. Chapter
20, pp. 565-601.
3. Dixon, Lloyd H. Magnetics Design for Switching Power
Supplies. Sections 1-5.
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