COUPLED INDUCTORS – A BASIC FILTER BUILDING BLOCK
Robert Balog
Philip T. Krein
David C. Hamill
Dept. of Electrical and Computer
Engineering
University of Illinois
Urbana, Illinois, USA
balog@ece.uiuc.edu
Dept. of Electrical and Computer
Engineering
University of Illinois
Urbana, Illinois, USA
krein@ece.uiuc.edu
Surrey Space Centre
University of Surrey
Guildford, UK
d.hamill@surrey.ac.uk
Abstract: Coupled magnetics and coupled filters can
provide smoothing in power converter applications.
This paper describes a general-purpose coupled
inductor filter building block. The coupled inductor
circuit model is presented, starting with the basic
topology with ideal circuit elements. Real circuit
elements are then substituted for the ideal elements
and the effects of ESR discussed. Experimental results
are provided for a typical application. Lastly, the
effect of a turns ratio between the "ac" and "dc"
windings is explored.
I.
Introduction
Coupled-inductor and other integrated-magnetic techniques have existed for many years, but most power
electronics engineers are uncomfortable with them. This
may be because of limited experience with coupled
magnetics. Or it could be explained by the level of
complexity in many treatments of coupled magnetic
devices. Unlike most networks, coupled-inductor
techniques involve simultaneous parallel energy-transfer
pathways: electrical and magnetic. Despite the difficulty,
the circuits are useful and deserve to be better
appreciated.
The technique described in this paper replaces a series
smoothing choke with a “smoothing transformer” (a pair
of coupled inductors) and a dc blocking capacitor.
Together, they form a basic filter building block shown in
Figure 1.
Ldc
+
Vn
+
Lac
C
-
+
VC
-
Vq
-
Figure 1: Coupled inductor building block
Because these components form a linear two-port filter,
the building block can be implemented in any dc circuit to
reduce the ripple current wherever a choke is currently
used. Thus it may be applied to the dc input of a
converter, its dc output, or an internal dc link (in
applications such as motor drives or HVDC transmission).
The concept described here is not a new one. It has
surfaced in many forms over the years, but it nonetheless
appears to be little known, and is not well understood.
The purpose of this paper is fourfold:
•
•
•
•
To bring the technique to the attention of a wider
audience;
To give a simple explanation of its operation,
avoiding magnetic theory;
To show how it can be usefully applied in practice;
To give an indication of broader possibilities for
coupled magnetic devices.
II.
History and Reinvention
Coupled magnetics were more common early in the
history of rectification. The circuit of Figure 1 was
known in 1930 [1], [2]. Since that time, it has been
described in various texts [3], [5], yet authors have
claimed to “invent” it on multiple occasions since then
(refer to PESC paper [6].) Even as recently as 1998 [7],
the circuit was introduced as new. The recent power
electronics work seems to treat the smoothing circuit of
Figure 1 as an integral part of a converter, missing the
point that it is a linear two-port which can be analyzed
independently and added to a wide variety of circuits.
This points to a need for analysis and modeling, to allow
the routine application of such circuits.
III.
Principle of Operation
The coupled inductor is schematically represented in
Figure 2. The addition of a dc blocking capacitor forms a
two port linear filter. The goal is to produce a low ac
current (ideally zero) at the quiet port. It can be
misleading to think of the smoothing transformer’s
windings as a primary and secondary. Hence we label
them the “dc” and “ac” windings to indicate their purpose
in the circuit. The dc winding carries a heavy direct
current (like a smoothing choke), while the ac winding
carries only a small ac ripple current.
The port labeled Vn in Figure 1 is connected to the noisy,
unfiltered, source represented in as a dc source with ac
noise. In the case of a power supply, this could be the
output of the switching circuit. Superposition allows us to
write Vn as the sum of a dc component (VDC) and an ac
component (vac). The port labeled Vq in Figure 1 is the
filtered "quiet" port.
vac
Ldc
+ vac -
Lac
+ vac C= ∞
Vdc
+
+ Vdc
Vdc
- -
Figure 2: Coupled inductor voltage diagram
If C is large then the ac voltage (vac) across it due to ripple
current becomes zero.
1
iC dt = 0
lim VC =
(1)
C ®¥
C
Applying KVL, the ripple voltage, vac appears across the
inductor Lac. Assuming perfect coupling and a one to one
turns ratio, the ac ripple voltage in the Lac winding will
transfer to the Ldc winding. Applying KVL on the quiet
port confirms that Vq is only the dc component of the
noisy port voltage Thus, the coupled inductor can be
thought of as serving two functions:
ò
1)
2)
An energy storage element for the switching power
supply.
A "signal transformer" for ripple current steering.
Our goal is to take advantage of this current steering
ability.
IV.
Analysis and Simulation
We begin the examination of the performance of the
coupled inductor with the basic model and add elements
one by one until we have the full realization, including
imperfections such as ESR effects.
A. Finite blocking capacitance
The standard "T" equivalent for magnetic circuits used to
aid in analyzing the coupled inductor is shown in Figure
3.
Ldc-M
M
Lac-M
C
Figure 3: Equivalent "T" model plus the capacitor
M is the mutual inductance of the two coils and is related
to the coupling coefficient.
M = k Lac Ldc
(2)
The factor k, the coupling coefficient represents the flux
linkage between the two windings. Permissible values of
k are 0<k<1. A value of 1 means that all the flux linking
coil 1, also links coil 2. In practice, values of k >0.9 are
easily achievable. Since the coupling coefficient depends
on geometry, changes in wire spacing or inconsistent
winding techniques can cause variations from sample to
sample. As we will see, the operating regime of the
coupled inductor is dependent on k.
The transfer function of the equivalent
Figure 1 is:
æ
1 + s 2 C × Lac × ç1 - k ×
Vq
è
=
2
Vn
1 + s C × Lac
circuit in terms of
Ldc ö
Lac ÷ø
(3)
The structure of the transfer function appears to be similar
to that of the second order filter with a frequency
dependent term s2CLac appearing in both the numerator
and denominator. The location of the transfer function
zero depends on the value of the coupling coefficient k.
At low frequencies, the gain is unity and at high
frequencies the gain in dB is given by the expression:
20 log
1
k.
L dc
L ac
(4)
B. Second order low pass mode
An interesting mode of operation occurs at the null
condition. The location of the zero in the transfer
function depends on the coupling coefficient, k. There
exists a k that causes the infinite frequency gain to equal
zero:
Lac
(5)
knull =
Ldc
When k=knull, the s2LacC term disappears from the
numerator. The coupled inductor degenerates into a
second order filter with the characteristic -40 dB/ decade
rolloff.
Vq
1
=
(6)
Vn 1 + s 2 C Lac
It should be noted that the null condition second order low
pass filter is composed of Lac and C rather than Ldc and C
as one might expect in a typical low pass filter. Since Lac
is typically less than Ldc (see section on construction of
coupled inductors), the cutoff frequency for the null
condition will be slightly higher than that of the simple
second order filter Ldc and Cdc.
Further, it turns out that knull is a boundary condition for
the formation of a notch. For values of k < knull, the zero
frequency is greater than the pole frequency and a notch
appears. We can take advantage of the coupled inductor
in this mode as a notch filter. For values of k > knull, there
is no notch and worse high frequency attenuation than a
simple second order filter. This mode of operation is not
useful.
C. Ideal capacitor with finite load resistance
+
M
L DC -M
+
L AC -M
R Load
Vn
-
C AC
+
V AC
-
Vq
-
Figure 5: Frequency response for circuit in figure 4
Figure 4: Finite capacitance with resistive output
In practice, the filter will be connected to a finite load
impedance as in Figure 4. Assuming that the load is
purely resistive, the transfer function is realized as:
Vq
=
Vn
æ
L ö
1 + s 2 Cac Lac ç1 - k dc ÷
Lac ø
è
1+ s
(
Cac Lac Ldc 1 - k
Ldc
+ s 2 Cac Lac + s 2
Rload
Rload
2
)
(7)
Mode of operation
D. Second order notch mode
Consider the numerator of equation (3). Lac and Cac form
a series resonant circuit, reducing the impedance of that
leg to, in principle, zero for the frequency given by
equation (8) obtained by setting the numerator of the
transfer function to zero.
1
æ
L ö
Cac Lac ç 1 - k Ldc ÷
ac ø
è
The coupled inductor in this basic form appears to offer
an alternative to the second order filter. However, the
penalty paid for the deep notch response is a reduced high
frequency response.
Table 1
When the load resistance is large, the limit becomes
equation (3).
w notch =
The value of the coupling coefficient k is affected by the
manufacturing process of the inductor and can vary from
lot to lot. To ensure that the coupled inductor always
operates in the notch mode, the design of the inductor
should place k well below knull.
(8)
The value of k√(Ldc/Lac) must not be too close to unity, or
the notch frequency will be extremely sensitive to
changes in k, Ldc and Lac. On the other hand, k√(Ldc/Lac)
should be close to unity for good attenuation of
frequencies above ωnotch as the asymptotic high frequency
gain is 1 – k√(Ldc/Lac). In a practical design, a compromise
is required. When k = knull, ωnotch moves to infinity, giving
the low-pass mode. If k > knull there is no real value of
ωnotch and no notch appears.
Figure 5 compares the effect of a terminating load
resistance for various values of k using a coupled inductor
with Lac=240 µH, Ldc=250 µH and a capacitor C=100µF.
The null condition occurs when k= 0.98 (from equation
(4).) The graph illustrates the sensitivity of the notch to k.
Second order filter
Notch w/ k= 0.97
Difference
Attenuation at
frequency of notch
-40 dB
-120 dB
80 db
If the notch can be designed at an arbitrary frequency, the
fundamental switching frequency of a power converter for
example, significant attenuation can be achieved.
E. ESR effect of a real capacitor
In a practical circuit, the windings of the inductor will
have copper loss and the capacitors will have equivalent
series resistances (ESR). All these add damping and affect
the ripple attenuation.
We have observed some of the affects of circuit resistance
in Figure 5. In addition to providing another system pole,
the load resistance also helps to dampen the overshoot at
the cutoff frequency of the filter.
In the previous section, the circuit model used an ideal
capacitor. The resonance set up by the series L and C is
the mechanism responsible for the formation of the notch.
From circuit theory, we recall that series resistance in an
LC circuit dampens out the sharp resonance.
+
M
Ldc-M
G. Fourth order mode
At frequencies above the notch, the second order mode
response is governed by the pole formed by Ldc and Rload.
A second pole can be introduced by connecting a
capacitor Cdc across the “quiet port.”
+
Lac-M
Vn
C
Vq
Ldc
-
ESRac
-
+
+
Figure 6: "ac" branch capacitor ESR
Figure 6 shows the "T" model including the ESR of the
capacitor and coil resistance. The transfer function is:
Vq
=
Vn
æ
L ö
1 +sCR + s 2 C Lac ç1 - k dc ÷
Lac ø
è
1 + sR + s 2 C Lac
Lac
Vn
Cdc
Vq
Cac
-
-
Figure 8: Coupled inductor with output capacitor
(9)
Figure 7 examines the damping effect of the ESR in the
capacitor. Notice that for ESR values as little as 0.1Ω, the
notch disappears.
The true benefit of the coupled inductor now becomes
apparent: more than one inductance is integrated into one
magnetic structure, which makes a fourth order filter
possible using only three components.
+
Vn
-
M
LDC-M
LAC-M
CAC
+
Rload
CDC
Vq
+
VAC
-
-
Figure 9: Fourth order coupled inductor "T" model
The value of Cdc should be chosen such that the Cdc, Ldc
pole is placed at a higher frequency than the pole formed
by Cac, Lac.
Figure 7: ESR effect
Without a notch mode, our simple model of the coupled
inductor offers no benefit over the simple second order
filter. Even worse, the high frequency response flattens
out (equation (4)) while the simple second order filter
continues to roll off at 40 dB per decade.
F. Real capacitor and output resistance
Adding output resistance to the circuit model does not
affect the filter performance significantly provided that at
the frequencies of interest, and well above dc, the
impedance of the ac branch is much less than the
impedance of the dc branch such that the ripple current
diverts predominantly into the ac branch and away from
the load. Practically, this translates into a required
minimum load impedance.
The equation for the circuit model with ideal components
is:
æ
L ö
1 + s 2Cac Lac ç1 - k dc ÷
Lac ø
Vq
è
(10)
=
Vn 1 + s 2 (Cac Lac + Cdc Ldc ) + s 4Cac Cdc Lac Ldc 1 - k 2
(
)
With Cac zero, the coupled inductor becomes a simple
second order filter with Ldc and Cdc as the circuit elements
and is shown in Figure 10 for comparison purposes.
Once again we notice that for values of k less than the null
condition, there is the appearance of the frequency notch.
At high frequencies, the filter slope is governed by the
Cdc, Ldc pole and rolls off at –40 db / decade. If the
frequency of the notch can be arbitrarily designed, then
the coupled inductor can yield better results than the
simple second order filter while using the same magnetic
volume. A word of caution: a frequency peak occurs just
below the notch frequency. If the fundamental switching
frequency of the converter varies then the proximity of
the peak may actually worsen the performance of the
output filter.
Figure 12: Full realization of coupled inductor - simulated
Figure 10: Fourth order coupled inductor
Imposing the null condition once again presents an
unusual condition. After the second peak, the filter slope
is now governed by four poles and rolls off at the –80 dB
/ decade slope of a fourth order filter. This operating
condition may be used to provide a fourth order filter
effect using only one magnetic component. However, the
design is extremely sensitive to values of k. In this case
k=0.98 is the null condition. k=0.97 and 0.99 eliminated
the fourth order behavior and in fact, as depicted on the
graph, may even move a peak into the frequency range
that we are interested in attenuating the most.
V.
In Figure 12, at 50 kHz the coupled inductor provides,
theoretically, an additional 5 dB of suppression compared
to the second order filter. Notice that due to the ESR
effects, the notch is not apparent. However, the circuit
still provides a benefit.
Experimental Test and Comparison
We examine the complete model of the coupled inductor
including capacitor ESR and load resistance:
M
LDC-M
+
+
CDC
LAC-M
Vn
CAC
-
RLOAD
+
ESRAC
VAC
Vq
ESRDC
Figure 13: Bode plot comparing second order filter with
coupled inductor - measured
The plot from a network analyzer in Figure 13 shows the
performance of the filter to closely match the calculated
theoretical.
-
Figure 11: Completed non-ideal coupled inductor model
The parameters of the coupled inductor used were
Lac=240µH, Ldc=250µH, and k=0.98. The inductor's
coupling coefficient is below knull=0.9997 indicating notch
mode regime. The capacitors were selected such that
Cdc=Cac= 10µF and provided a simple second order cutoff
frequency of 3kHz. The value of the ESR was determined
based on the coil dc resistance and the ESR of the
capacitor.
VI.
Typical Application: Buck Converter
Although not the only application for the coupled
inductor, the buck converter is an ideal choice to
demonstrate the usefulness of the coupled inductor.
Figure 14 illustrates the power electronics portion of a
buck converter topology. The traditional inductor has
been replaced with a coupled inductor and capacitor used
in the previous section. The gate drive, generated
elsewhere, has a fundamental switching frequency of 50
kHz.
Ldc
+
+
PWM
Gate Drive
Vn
Lac
Cdc
Vq
Cac
-
-
Inductance is proportional to the number of turns squared.
Therefore, in a 1:5 ratio, the inductance of the ac winding
is increased by a factor of 25. It is desired to keep the
same frequency response (i.e., same Lac-Cac resonant
frequency). Therefore, the capacitor must decrease by a
factor of 25. Physical size is related to capacitance, so a
smaller value capacitor means a physically smaller
device.
Figure 14: Buck converter
Figure 15 shows the circuit waveform without Cdc (only
the coupled inductor and capacitor from Figure 1.) As
expected, the inductor current (lower trace) is triangular.
The upper trace, the voltage across Cac, is also triangular
and includes an ESR voltage jump.
Using the parameters for the 1:5 coupled inductor from
Table 3 and equation (8), the value of Cac needed to tune
the coupled inductor filter for a notch a 50 kHz is 3.35 nF.
The theoretical filter response is shown in Figure 16.
VCac
ILac
Figure 15: Capacitor voltage and inductor current
Figure 16: Theoretical 1:5 coupled filter response
Table 2 compares the output voltage ripple for three
circuit configurations.
The second configuration is
recognizable as the typical second order output filter.
From the circuit model using ideal components (no ESR),
the notch is calculated to have 35 dB more attenuation
that the simple second order filler. Including typical ESR
values, the 1:5 filter provides a 15 dB notch with respect
to the simple second order filter.
Table 2: Comparison of output voltage ripple
1
2
3
Circuit
Ldc only
Ldc, Cdc low pass filter
LPF + Cac
∆Vout
1.6 Vpk-pk
0.094 Vpk-pk
0.055 Vpk-pk
It is observed that with the addition of the second
capacitor, Cac, the output filter provides an additional
4.6dB reduction in the output ripple voltage, consistent
with the 5db predicted by the model.
VII.
Coupled inductor with turns ratio
So far we have examined coupled inductors with turns
rations of 1:1, that is, the dc inductor and ac inductor have
the same number of turns and therefore approximately the
same inductance. With the addition of a third component,
the "ac" side capacitor, a reduction in output ripple
voltage was achieved. Recall that the coupled inductor
may be thought of as a signal transformer. It may be
useful to exploit this fact and use a turns ratio to "scale"
the required value of the ac side capacitor.
second order filter
1:5 notch mode
Figure 17: Bode plot of actual 1:5 coupled inductor filter
The Bode plot of the experimental result is shown in
Figure 17 and confirms the theoretical model. The second
notch around 250 kHz is the result of circuit parasitic and
stray capacitances.
VIII. Measuring k
Figure 18 shows the output ripple voltage of the buck
converter from section VI with a simple second order low
pass filter. As expected the 50 kHz switching frequency
is the dominant component.
For successful implementation, it is vital to know the
coupling coefficient k. Here we give two methods for
measuring k experimentally. All measurements must be
taken at a low enough frequency that parasitic capacitance
is negligible. The methods do not need calibrated
inductance measurements, as they depend only on ratios,
although it is useful to perform the measurements with a
dc current source in place to set the correct bias level.
A. Open/short-circuit inductance method
The inductance of winding 1 is measured with winding 2
open-circuited (L1) and short-circuited (L1,sc). The
coupling coefficient is calculated from:
L1, sc
(11)
k = 1−
L1
Figure 18: Simple filter output voltage
Using the 1:5 coupled filter, the 50 kHz fundamental
frequency is severely attenuated. The output waveform in
Figure 19 now looks significantly different. With the
absence of the 50 kHz components, the higher order
harmonics are more pronounced.
A second value of k can be obtained by measuring
winding 2, and the two averaged. This method is
particularly suitable when k is close to unity, as it then
gives an accurate result despite possible difficulty in
measuring L1,sc. However, it is unsuitable when the
winding resistances are substantial.
B. Series aiding/opposing inductance method
In this method, four inductance readings are taken. First,
with the other winding open-circuited, the individual
winding self-inductances are measured (L1 and L2). Then
the two windings are connected in series and their
combined inductance is measured, in series-aiding
connection (Laid) and series-opposing (Lopp). (Note that
Laid > Lopp always, and that Laid – Lopp = 4M). The
coupling coefficient is calculated from:
Laid − Lopp
k=
(12)
4 L1 L2
An advantage is low sensitivity to winding resistance.
This method is unsuitable when k is small, as in that case
it depends on the difference between two similar
quantities.
IX.
Construction
The coupled inductors used in this paper were constructed
on a 2616PA250-3B9 pot core. This core has an effective
permeability of 78µo.
Figure 19: 1:5 coupled inductor output voltage
The dc port inductor was wound onto the core first.
Dielectric tape was wrapped around the dc winding,
followed by the ac port winding. Since the ac coil sees
only the ripple current it can be wound using small
diameter wire. As a result, the coupled inductor can
typically use the same core as a single inductor depending
on the fill factor and window size.
Table 3: Sample coupled inductors
Parameter
DC Inductor
Turns
Inductance
Wire Gauge
Resistance
AC Inductor
Turns
Inductance
Wire Gauge
Resistance
Coupling Coefficient k
X.
1:1 inductor
1:5 inductor
25
151.9 µH
26
165.7 µH
22 AWG
0.077 Ω
0.074 Ω
25
151.8 µH
0.252 Ω
.9802
130
4.095 mH
28
1.578 Ω
.985
Conclusions
By integrating two inductors into one magnetic structure,
it is possible to achieve fourth order filtering using only
three components. The interaction between these two
inductors has interesting properties that vary with the
operating regime. By taking advantage of a turns ratio,
the coupled inductor becomes a tuned notch filter with
minimal component count.
XI Acknowledgements
R. Balog is supported through the Grainger Center for
Electric Machinery and Electromechanics at the
University of Illinois. P. Krein was supported in part as a
Fulbright Scholar through the Fulbright Commission in
the United Kingdom.
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[2]
[3]
[4]
[5]
[6]
[7]
O.K. Marti and H. Winograd, Mercury Arc Rectifiers,
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G.B. Crouse, Electrical Filter, US Patent 1,920,948,
August, 1933.
R. Lee, Electronic Transformers and Circuits. New York:
John Wiley, 1947, p. 97.
R.W. Landee, D.C. Davis and A.P. Albrecht, Electronic
Designers Handbook, McGraw-Hill, 1957, p. 15-21, Fig.
15.22.
R.P. Severns and G.E. Bloom, Modern DC-to-DC
Switchmode Power Converter Circuits, New York: Van
Nostrand Reinhold, 1985, Figs. 8.5A, 12.13, 12.14 and
12.16.
D. C. Hamill and P. T. Krein, “A ‘Zero’ ripple technique
applicable to any converter” in Rec., IEEE Power
Electronics Specialists Conf., 1999, pp. 1165-1171.
D.K.W. Cheng, X.C. Liu and Y.S. Lee, “A new improved
boost converter with ripple free input current using
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pp. 592–599.
Robert Balog received his BSEE degree at the College of
Engineering, Rutgers University in 1996. He began his
professional career as a design engineer with Lutron
Electronics. He is currently pursuing a MSEE at the
University of Illinois at Urbana-Champaign. His research
interests are in the areas of power electronics and analog
circuits.
Philip T. Krein received the B.S. degree in electrical
engineering and the A.B. degree in economics and
business from Lafayette College, Easton, Pennsylvania,
and the M.S. and Ph.D. degrees in electrical engineering
from the University of Illinois, Urbana. He was a design
engineer at Tektronix in Beaverton, Oregon. In 1987 he
returned to the University of Illinois, where he is now
Professor of Electrical and Computer Engineering and
Director of the Grainger Center for Electric Machinery
and Electromechanics. Dr. Krein is President of the IEEE
Power Electronics Society and is a Fellow of the IEEE.
His research interests are in power electronics and
electromechanics. In 1997-98, he was a Fulbright Scholar
at the University of Surrey, Guildford, UK.
David C. Hamill was born in London, England. He
received his bachelor's and master's degrees from the
University of Southampton (Southampton, UK) and his
PhD from the University of Surrey (Guildford, UK).
After practicing as a design engineer and a consultant, he
was technical director of PAG Ltd (London, UK) for
seven years. In 1986 he joined the University of Surrey,
where he is currently a Senior Lecturer in the Surrey
Space Centre. His research interests include dc-dc
conversion, space power systems and nonlinear dynamics.
Dr Hamill was an Associate Editor of the IEEE
Transactions on Power Electronics, 1994-96. He is a
Senior Member of the IEEE, a member of the IEE, and a
Chartered Engineer.