Electronics and Communications in Japan, Part 2, Vol. 90, No. 9, 2007
Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J89-C, No. 12, December 2006, pp. 962–968
Coupling Coefficient of Resonators—An Intuitive Way of Its
Understanding
Ikuo Awai and Yangjun Zhang
Department of Electronics & Informatics, Ryukoku University, Otsu, 520-2194 Japan
paper, the concepts that have been considered “common
sense” by microwave researchers without clear reasoning
are spotlighted. Foundations are provided for those that are
correct and correction for those that are inadequate.
As the theoretical process, a simple method is chosen
based on the coupled mode theory and the electromagnetic
fields of the coupled resonator system are given in terms of
a linear combination of the electromagnetic fields of the
individual noncoupled resonators [3, 4]. As a result, two
new representations are obtained for the coupling coefficient [3]. One is expressed in terms of the energy exchange
period between two resonators in the time domain. The
other is expressed in terms of the overlap integral of the
electromagnetic fields in the uncoupled condition. The
former emphasizes that the coupling is stronger if the
energy exchange between two resonators is faster. This
expression provides a physical interpretation that is clearer
than the conventional method based on the frequencies split
by coupling. Hence, the authors consider that this is indeed
appropriate for the definition of the coupling coefficient
between resonators [5]. On the other hand, the latter is a
concept often used when coupling between resonators is
discussed qualitatively. For instance, one physical interpretation is “when the parts of resonators with strong electric
fields approach each other, electric field coupling is dominant over magnetic field coupling, so that the overall coupling becomes capacitive.”
In the present paper we revisit the “physical interpretation” related to various examples based on the overlap
integral. According to the present method, the electric component and the magnetic component of the coupling coefficient can be calculated separately to provide a basis of the
SUMMARY
A new representation of the coupling coefficient of
resonators has been obtained by the authors using the
coupled mode theory. In this expression, the coupling coefficient is given by an overlap integral of the electric field
and the magnetic field. This expression is extremely useful
for explanation of the physical meaning of coupling. In the
present paper, various microstrip resonators are discussed
and their coupling mechanisms are explained by the above
theory resulting in a clear interpretation that includes phenomena hitherto not adequately explained. © 2007 Wiley
Periodicals, Inc. Electron Comm Jpn Pt 2, 90(9): 11–18,
2007; Published online in Wiley InterScience (www.
interscience.wiley.com). DOI 10.1002/ecjb.20342
Key words: resonator; coupling coefficient; overlap integral; electric coupling; magnetic coupling.
1. Introduction
The concept of coupling has a broad meaning and is
used for transmission lines in addition to resonators. In the
field of mechanics, coupling pendulums and coupling
springs are well known. Further, in quantum mechanics,
electrons in the molecule are considered to have various
forms of coupling. As a result, various types of coupling
exist in chemistry. A keyword common to all of these
couplings is energy. From this point of view, the authors
have reported various observations [1–3]. In the present
© 2007 Wiley Periodicals, Inc.
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where
above “physical interpretation.” The separation of the coupling coefficient is described in Refs. 5 and 6 in a slightly
different form, which is derived empirically and not from
the basic equation, so that the basis is weak. In this sense,
the present report may claim to provide the first interpretation of coupling based on Maxwell’s equations.
(2)
Here E1 and E2 are the electric field distributions of the two
resonators prior to coupling, ε1 is the dielectric profile of
resonator 1 prior to coupling, and ε is the dielectric profile
of the coupled system. For the value of E and the integration
range, see Fig. 1. This expression can be rewritten in
approximate form as follows [4]:
2. Overlap Integral
In coupled mode theory, in order to express the electromagnetic field of two coupled resonators, an appropriate
sum of the electromagnetic fields of the eigenmode of each
resonator is used. The authors’ discussion uses the simplest
form, with one eigenmode each. In coupled mode theory,
the cause of the mode coupling lies in the spatial overlap of
two mode functions. Therefore, the method can be used
only for resonators with electromagnetic fields that leak to
the exterior to form evanescent fields, or open-type resonators. Initially, the authors used normalized real number
eigen-electromagnetic fields [3]. However, such an approach is inconvenient for actual calculations. Therefore,
subsequently the theory was reconfigured by using the
nonnormalized complex electromagnetic field representations [4, 7, 8].
In each method, the coupling can be classified into
two cases.
(3)
Therefore, it is possible to attach the physically reasonable
meaning that the coupling coefficient is proportional to the
polarization of the dielectric resonators.
(2) Conducting resonators
Let us next study structures in which the presence of
the conductors plays the fundamental role in the formation
of the resonators, in such cases as microstrip lines and
coplanar waveguides. The coupling coefficient is given by
the following [7, 8]:
(1) Dielectric resonators
The coupling coefficient k is given as follows if the
two resonators are identical:
(4)
(1)
Aside from the normalization constant in the denominator,
this equation states that the coupling coefficient is given by
the difference between the overlapping integral of the magnetic fields and that of the electric fields.
In the present paper, the coupling coefficient of a
microstrip resonator is discussed by using Eq. (4). First, in
order to test the reliability of Eq. (4), the coupling coefficient of coupled λ/2 open-ended microstrip resonators as
shown in Fig. 2 is calculated and compared with that
obtained by the conventional split frequency method. The
simulation software is HFSS (Ansoft) and an independently
developed program is used for calculation of the integrals.
Figure 2 indicates that the two methods provide excellent agreement for all resonator spacing values. Next, it
is shown that the signs of the electric field coupling and of
the magnetic field coupling are identical and the latter is
always larger. Since the overall coupling coefficient is given
by the difference between the two, it is smaller than the
Fig. 1. How to calculate overlap integral.
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3. General Rule of Overlap Integral
Method
From Eq. (4), the terms generally valid are summarized.
(1) The difference between the magnetic coupling
and the electric coupling provides the
overall coupling coefficient
In Ref. 5, the coefficient of the electric field coupling
in Eq. (4) is made positive:
(5)
However, since no derivation of equations is given, even in
Ref. 6 by the same author, it is conjectured that this expression is derived empirically as an extension of
Fig. 2. Coupling of microstrip open-ended
half-wavelength resonators with broader
sides facing each other.
(6)
derived in the LC circuit.
Equation (4) derived by the present authors defines
the overall coupling coefficient as the difference between
the magnetic and electric couplings, whereas Eq. (5) of
Hong and Lancaster uses the sum as the overall coupling
coefficient. As shown in the previous section, correct values
cannot be obtained unless the authors’ equation is used.
However, there exists another example in which the overall
coupling coefficient is expressed by the difference, although this is for an LC circuit [9]. This representation is
considered as the low-frequency limit or the lumped element representation of the authors’ expression.
The physical explanation of why the overall coupling
coefficient is given by the difference rather than the sum of
the magnetic and electric couplings is unknown at this time.
Lentz’s law in magnetics may be related, but no further
ideas are available.
Further, if Eq. (4) cannot be applied to dielectric
resonators as explained in the previous section, there is a
problem. So long as the coupled mode theory is used, Eq.
(3) is valid for the dielectric resonators. It is not very
convincing that the equation is different for different resonators. If another theory is used, Eq. (4) may be valid for
dielectric resonators.
individual values. The reason why the electric field coupling is smaller than the magnetic field coupling is considered to be that large electric field components occur upward
and downward from both ends of the two resonators, and
that these components do not overlap substantially (see Fig.
3).
Fig. 3. Electric lines of force do not overlap much.
Magnetic lines of force overlap.
13
coefficient is always used in the design of bandpass filters.
The coupling coefficient is needed for determination of the
relative bandwidth. The sign presents no problem. If k is
negative, the absolute value is taken as
(2) The individual signs of the magnetic coupling
and the electric coupling are of no concern
In the integrals in Eq. (4), the subscripts 1 and 2 refer
to the resonators and the electromagnetic fields are those
prior to coupling. This is natural in view of the fact that the
coupled mode theory is basically a perturbation technique.
The electromagnetic fields prior to coupling are nothing but
the eigenmode electromagnetic fields. For instance, there
are degrees of freedom in the signs of the eigenmode
electric field E1 of resonator 1 and E2 of resonator 2.
Therefore, it is permissible to attach ± arbitrarily to E1 or
E2. Of course, once E1 or E2 is fixed, then H1 and H2 are
determined automatically and cannot be altered arbitrarily.
As a result of the above discussion, it is unknown if
the sign of each term on the right-hand side of Eq. (4) is
positive or negative due to the way the first eigenmode is
determined. However, if the first term is positive and the
second term is negative, then the second term is positive
once the sign of the eigenmode is changed, so that the first
term is negative.
(8)
Here i denotes the i-th stage and gi is the element value of
the i-th stage of the low-pass prototype filter. The sign is of
interest only when an elliptical BPF is designed. Thus, only
the difference in the sign is of importance, not the signs of
the coupling coefficients of each stage.
4. Role Change of Magnetic/Electric
Coupling
When open-ended half-wave resonators are shifted in
parallel as shown in Fig. 4(a), the ratio of magnetic and
electric coupling varies according to the change of the
overlapped segment and the total coupling coefficient
changes in a complex manner as shown in Fig. 5. When the
changes of individual magnetic and electric couplings are
observed, the following is found. When the overlapped
segment s is small, the edges of the two resonators are close,
so that the contribution of the electric coupling is strong. At
s = 10 mm, where the resonators are perfectly aligned, the
total coupling coefficient should be 0 in the simple model
shown in Fig. 6(b). However, this is not the case: the
magnetic coupling is stronger, as illustrated in Fig. 3.
Let us next consider quarter-wave resonators in the
interdigital arrangement shown in Fig. 4(b). The electromagnetic field distribution of the structure is identical to
that of half-wave resonators in Fig. 4(a). Hence, the overlap
segments are almost identical but the internal energy is
(3) Magnetic coupling and electric coupling do not
always cancel each other
Since the coupling coefficient cannot be separated
into two in the conventional theory, various inappropriate
statements have been made on the basis of empirical estimations with regard to the magnitude and sign of the
magnetic/electric component. For instance, it has been asserted that “the comb line coupling is weak because the
magnitudes of the magnetic and electric coupling are almost equal and their signs are opposite.” Although this
statement is correct, it is not possible to explain why interdigital coupling is strong unless the signs of the magnetic
and electric couplings can be either positive or negative. As
described later, the magnetic and electric couplings do not
cancel each other, but are additive in interdigital coupling.
Also, in microstrip line resonators, “it is considered common knowledge that the narrow-side coupling is weak and
the broad-side coupling is strong.” Since there are examples
in which the broad-side coupling is not very strong, some
comments are provided later.
(4) The total coupling coefficient can be negative
As a necessary consequence of item (2) above, the
coupling coefficient defined by Eq. (4) can be negative. In
contrast, the ordinary coupling coefficient is treated as a
positive quantity because it is calculated by
(7)
Fig. 4. Open-ended half-wavelength resonator and
corresponding quarter-wavelength resonator
(substrate thickness 0.7 mm, εr = 3.27).
Here ωh and ωl are the higher and lower of the resonant
angular frequencies separated by coupling. The coupling
14
resonators. The simulation results in Fig. 5 support this
claim.
5. Comb Line and Interdigital Alignments
Figure 7 shows the quarter wave resonators with
comb line and interdigital alignment used in the calculations. The results for the comb line are shown in Fig. 8 and
those of the interdigital alignment are given in Fig. 9. In
order to intuitively explain these results, simplified models
similar to the previous ones are presented in Fig. 10. In the
case of the comb line, the absolute values of the magnetic
coupling and the electric coupling are larger than those for
the interdigital case, because the locations of the maximum
amplitude of the electric field and the magnetic field of the
resonators are close to each other. However, since the signs
are opposite, the results cancel each other and a large
coupling coefficient cannot be obtained even if the resonators are placed close together. Figure 8 exhibits such a
situation.
On the other hand, in the interdigital case, the locations of the maximum values for the electric/magnetic fields
are interchanged and hence the absolute values of the
magnetic and electric couplings are rather small. However,
since they make additive contributions, a large overall value
can be achieved. Figure 9 illustrates large coupling coefficients as the resonators approach each other.
Fig. 5. Coupling coefficients as functions of resonator
shift.
smaller by half. Thus, if Eq. (4) is used, the coupling
coefficient is expected to be twice that of the half-wave
6. Narrow-Side/Broad-Side Couplings
Figure 11 shows the results obtained by changing the
resonator spacing with broad-side coupling of the quarterwave resonators in a comb line alignment such as those in
Fig. 7 or 8. The magnetic/electric couplings are extremely
strong and are much larger than the narrow-side coupling
in Fig. 8. But the overall coupling coefficient is not much
stronger than that in narrow-side coupling since both cancel
(a) Shifted case
(b) Overlapped case
Fig. 6. Configurations of two resonators and
electromagnetic field distribution. (If ± potential is
provided as shown, currents flow as indicated by arrows.
However, since the overlap integrals of the electric field
are contributions with different signs, the electric field
distribution of the lower resonator is correspondingly
shown to be inverted, and similarly in the following
cases.)
(a) Comb line coupling
(b) Interdigital coupling
Fig. 7. Comb line and interdigital alignment (substrate
thickness = 0.7 mm).
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Fig. 8. Coupling coefficient for comb line structure.
Fig. 11. Coupling coefficient of broadside-coupled
quarter-wavelength resonators. (The resonator is placed
in a vacuum.)
each other. Hence, it is now understood that the signs of the
electric and magnetic couplings must be considered in
addition to placing the two resonators in close proximity in
order to strongly couple two resonators.
7. Rotation of Open Ring Resonators
Fig. 9. Coupling coefficient for interdigital structure.
Much attention has been paid to open ring resonators
in connection with their application to metamaterials [10].
By means of broad-side coupling, an extremely large coupling coefficient can be obtained [11]. Therefore, it has been
proposed to make use of this structure for an ultrawide-band
filter [12]. The present paper shows significant variations
of the coupling coefficient by mutually rotating broadsidecoupled resonators.
Figure 12 presents the coupling coefficient of broadside-coupled circular open ring resonators as a function of
the angle between the gap locations. It is seen that the
overall coupling coefficient varies severalfold as a result of
variations of the magnetic and electric couplings, especially
large variations of the electric coupling. As in the previous
section, variations of the magnetic field and the electric
field along the resonator are shown in Fig. 13 graphically.
This resonator is essentially a half-wave resonator with zero
currents at both ends, and the direction of the current does
not change along the resonator. Therefore, the magnitude
of the magnetic coupling does not change much as a result
Fig. 10. Electromagnetic field distribution for comb
line and interdigital alignment. (In correspondence with
the negative sign of ke, the electric field of the lower
resonator has an inverted sign.)
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netic field are interchanged. However, the variations of the
overall coupling coefficient suggest the same tendency
discussed above while only the signs are interchanged.
8. Conclusions
The physical meanings of the coupling coefficient of
resonators have been interpreted using the recently derived
expressions by the present authors. Although such interpretations have been made in the past, theoretical foundations
have not been provided as far as the authors know. In this
sense, the present paper provides a consistent explanation.
Even though the paper is based on the coupled mode theory,
simple and instructive observations are provided by using
overlap integrals. It is hoped that the paper will be useful in
an educational environment.
Fig. 12. Coupling coefficient of rotated open ring
resonators. (Resonator length = 28.2 mm, width = 1 mm,
gap = 0.1 mm, resonators are in vacuum.)
REFERENCES
of mutual rotation. In contrast, the electric field has opposite
signs at the two ends so that the sign changes at the center.
Hence, the electric coupling changes substantially in both
magnitude and sign as a function of the angle of rotation.
Hence, if open ring resonators with both ends
grounded are formed, the magnetic coupling changes significantly while the electric coupling is almost constant,
because the distributions of the electric field and the mag-
0°
90°
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17
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AUTHORS (from left to right)
Ikuo Awai (member: fellow) graduated from the Department of Electronic Engineering, Kyoto University, in 1963,
completed the doctoral program in 1968, and became a research associate there. He was engaged in research on magnetic waves
in magnetic materials and optical integrated circuits. In 1984, he became an engineering director at Uniden, where he was
responsible for the development of various wireless devices. In 1990, he was appointed a professor on the Faculty of Engineering
at Yamaguchi University. He has been engaged in research on magnetostatic wave devices, dielectric filters, planar filters,
superconducting filters, and artificial materials. In 2004, he became a professor at Ryukoku University. He received a Best Paper
Award from IEICE in 2002. He was the chairman of the IEEE Microwave Group, IEEE MTT-S Japan chapter chairman, and
IEEE Hiroshima Section chairman. He is a member of IEEE.
Yangjun Zhang (member) completed the M.S. program at Shanghai Institute of Technology, China, in 1992, completed
the doctoral program in electronic science at Shizuoka University, and became a research associate there. In 2003, he became
a research associate at Ryukoku University, and has been engaged in research on microwave moisture measurement and
microwave resonators. He holds a D.Eng. degree, and is a member of IEEE.
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