CHAPTER 8
Coupled Resonator Circuits
Coupled resonator circuits are of importance for design of RF/microwave filters, in
particular the narrow-band bandpass filters that play a significant role in many ap-
plications. There is a general technique for designing coupled resonator filters in the
sense that it can be applied to any type of resonator despite its physical structure. It
has been applied to the design of waveguide filters [1–2], dielectric resonator filters
[3], ceramic combline filters [4], microstrip filters [5–7], superconducting filters
[8], and micromachined filters [9]. This design method is based on coupling coeffi-
cients of intercoupled resonators and the external quality factors of the input and
output resonators. We actually saw some examples in Chapter 5 when we discussed
the design of hairpin-resonator filters and combline filters, and we will discuss
more applications for designing various filters through the remainder of this book.
Since this design technique is so useful and flexible, it would be desirable to have a
deep understanding not only of its approach, but also its theory. For this purpose,
this chapter will present a comprehensive treatment of the relevant subjects.
The general coupling matrix is of importance for representing a wide range of
coupled-resonator filter topologies. Section 8.1 shows how it can be formulated ei-
ther from a set of loop equations or from a set of node equations. This leads to a
very useful formula for analysis and synthesis of coupled-resonator filter circuits in
terms of coupling coefficients and external quality factors. Section 8.2 considers
the general theory of couplings in order to establish the relationship between the
coupling coefficient and the physical structure of synchronously or asynchronously
tuned coupled resonators. Following this, a discussion of a general formulation for
extracting coupling coefficients is given in Section 8.3. Formulations for extracting
the external quality factors from frequency responses of the externally loaded in-
put/output resonators are derived in Section 8.4. The final section of this chapter de-
scribes some numerical examples to demonstrate how the formulations obtained
can be applied to extract coupling coefficients and external quality factors of mi-
crowave coupling structures from EM simulations.
235

··· – j
␻
L
1n
i
n
= e
s
– j
␻
L
21
i
1
+
΂
j
␻
L
2
+
ᎏ
j
␻
1
C
2
ᎏ
΃
i

+
΃
i
n
= 0
in which L
ij
= L
ji
represents the mutual inductance between resonators i and j, and
the all loop currents are supposed to have the same direction as shown in Figure
8.1(a), so that the voltage drops due to the mutual inductance have a negative sign.
This set of equations can be represented in matrix form
1
ᎏ
j
␻
C
n
1
ᎏ
j
␻
C
1
236
COUPLED RESONATOR CIRCUITS
L
2
i

n
R
n
()
a
()
b
Two-port n-coupled
resonator filter
V
1
V
2
I
1
I
2
a
1
a
2
b
1
b
2
FIGURE 8.1 (a) Equivalent circuit of n-coupled resonators for loop-equation formulation. (b) Its net-
work representation.
8.1 GENERAL COUPLING MATRIX FOR COUPLED-RESONATOR FILTERS
237
R

ᎏ
j
␻
1
C
2
ᎏ
··· –j
␻
L
2n
΄΅
΄΅
=
΄΅
(8.2)
ӇӇӇӇ
–j
␻
L
n1
–j
␻
L
n2
··· R
n
+ j
␻
L

1
= C
2
= ··· C
n
. The imped-
ance matrix in (8.2) may be expressed by
[Z] =
␻
0
L·FBW·[Z
ෆ
] (8.3)
where FBW = ⌬
␻
/
␻
0
is the fractional bandwidth of filter, and [Z
ෆ
] is the normalized
impedance matrix, which in the case of synchronously tuned filter is given by
ᎏ
␻
0
L
R
·F
1
BW

ᎏ
FB
1
W
ᎏ
–j
ᎏ
␻
␻
0
ᎏ
ᎏ
L
L
21
ᎏ
·
ᎏ
FB
1
W
ᎏ
p ··· –j
ᎏ
␻
␻
0
ᎏ
ᎏ
L

W
ᎏ
–j
ᎏ
␻
␻
0
ᎏ
ᎏ
L
L
n2
ᎏ
·
ᎏ
FB
1
W
ᎏ
···
ᎏ
␻
0
L
R
·F
n
BW
ᎏ
+ p

0
Ӈ
0
i
1
i
2
Ӈ
i
n
Q
e1
and Q
en
are the external quality factors of the input and output resonators, re-
spectively. Defining the coupling coefficient as
M
ij
= (8.6)
and assuming
␻
/
␻
0
Ϸ 1 for a narrow-band approximation, we can simplify (8.4) as
ᎏ
q
1
e1
ᎏ

q
ei
= Q
ei
· FBW for i = 1, n (8.8)
and m
ij
denotes the so-called normalized coupling coefficient
m
ij
= (8.9)
A network representation of the circuit of Figure 8.1(a) is shown in Figure 8.1(b),
where V
1
, V
2
and I
1
, I
2
are the voltage and current variables at the filter ports, and
the wave variables are denoted by a
1
, a
2
, b
1
, and b
2
. By inspecting the circuit of Fig-

ᎏ
b
1
=
(8.10)
a
2
= 0 b
2
= i
n
͙
R
ෆ
n
ෆ
and hence
S
21
=
Έ
a
2
=0
=
(8.11)
S
11
=
Έ

b
2
ᎏ
a
1
e
s
– 2i
1
R
1
ᎏ
2͙R
ෆ
1
ෆ
M
ij
ᎏ
FBW
L
ij
ᎏ
L
238
COUPLED RESONATOR CIRCUITS
Solving (8.2) for i
1
and i
n

ᎏ
[Z
ෆ
]
n1
–1
where [Z
ෆ
]
ij
–1
denotes the ith row and jth column element of [Z
ෆ
]
–1
. Substituting (8.12)
into (8.11) yields
S
21
=
ᎏ
␻
2
0
͙
L·
R
ෆ
F
1

–1
Recalling the external quality factors defined in (8.5) and (8.8), we have
S
21
= 2
ᎏ
͙
q
ෆ
e
1
1
ෆ
·q
ෆ
en
ෆ
ᎏ
[Z
ෆ
]
n1
–1
(8.13)
S
11
= 1 –
ᎏ
q
2

q
1
e1
ᎏ
+ p – jm
11
–jm
12
··· –jm
1n
–jm
21
p – jm
22
··· –jm
2n
[Z
ෆ
] =
΄΅
(8.15)
ӇӇӇӇ
–jm
n1
–jm
n2
···
ᎏ
q
1

and i
s
is the source current. According to the current law, which is the other one of
Kirchhoff’s two circuit laws and states that the algebraic sum of the currents leaving
a node in a network is zero, with a driving or external current of i
s
the node equa-
tions for the circuit of Figure 8.2(a) are
΂
G
1
+ j
␻
C
1
+
΃
v
1
– j
␻
C
12
v
2
··· – j
␻
C
1n
v

v
n
= 0
(8.16)
Ӈ
– j
␻
C
n1
v
1
– j
␻
C
n2
v
2
··· +
΂
G
n
+ j
␻
C
n
+
ᎏ
j
␻
1

n-1
C
n-1
L
1
v
1
C
1
G
1
G
1
i
s
i
s
L
n
v
n
C
n
G
n
G
n
()a
()b
Two-port n-coupled

␻
1
L
1
ᎏ
–j
␻
C
12
··· –j
␻
C
1n
–j
␻
C
21
j
␻
C
2
+
ᎏ
j
␻
1
L
2
ᎏ
··· –j

ᎏ
or
[Y]·[v] = [i]
in which [Y] is an n × n admittance matrix.
Similarly, the admittance matrix in (8.17) may be expressed by
[Y] =
␻
0
C·FBW·[Y
ෆ
] (8.18)
where
␻
0
= 1/
͙
L
ෆ
C
ෆ
is the midband frequency of filter, FBW = ⌬
␻
/
␻
0
is the frac-
tional bandwidth, and [Y
ෆ
] is the normalized admittance matrix. In the case of syn-
chronously tuned filter, [Y

␻
␻
0
ᎏ
ᎏ
C
C
1n
ᎏ
·
ᎏ
FB
1
W
ᎏ
–j
ᎏ
␻
␻
0
ᎏ
ᎏ
C
C
21
ᎏ
·
ᎏ
FB
1

ᎏ
ᎏ
C
C
n1
ᎏ
·
ᎏ
FB
1
W
ᎏ
–j
ᎏ
␻
␻
0
ᎏ
ᎏ
C
C
n2
ᎏ
·
ᎏ
FB
1
W
ᎏ
···

0
C
i
s
0
Ӈ
0
v
1
v
2
Ӈ
v
n
8.1 GENERAL COUPLING MATRIX FOR COUPLED-RESONATOR FILTERS
241
and assume
␻
/
␻
0
Ϸ 1 for the narrow-band approximation. A simpler expression of
(8.19) is obtained:
ᎏ
q
1
e1
ᎏ
+ p –jm
12

asynchronously tuned, (8.21) and (8.22) become
M
ij
= for i  j (8.23)
ᎏ
q
1
e1
ᎏ
+ p – jm
11
–jm
12
··· –jm
1n
–jm
21
p – jm
22
··· –jm
2n
[Y
ෆ
] =
΄΅
(8.24)
ӇӇӇӇ
–jm
n1
–jm

1
=
ᎏ
2
͙
i
s
G
ෆ
1
ෆ
ᎏ
b
1
=
(8.25)
a
2
=0 b
2
= v
n
͙
G
ෆ
n
ෆ
S
21
=

s
G
1
ᎏ
– 1
2͙G
ෆ
1
G
ෆ
n
ෆ
v
n
ᎏᎏ
i
s
2v
1
G
1
– i
s
ᎏᎏ
2͙G
ෆ
1
ෆ
C
ij

]
11
–1
(8.27)
v
n
=
ᎏ
␻
0
C·
i
F
s
BW
ᎏ
[Y
ෆ
]
n1
–1
where [Y
ෆ
]
ij
–1
denotes the iith row and jth column element of [Y
ෆ
]
–1

=
ᎏ
␻
0
C
2
·
G
F
1
BW
ᎏ
[Y
ෆ
]
11
–1
– 1
which can be simplified as
S
21
= 2
ᎏ
͙
q
ෆ
e
1
1
ෆ

[Y
ෆ
]. This is very important because it implies that we could have a unified formula-
tion for a n-coupled resonator filter regardless of whether the couplings are magnet-
ic or electric or even the combination of both. Accordingly, the equations of (8.13)
and (8.29) may be incorporated into a general one:
S
21
= 2
ᎏ
͙
q
ෆ
e
1
1
ෆ
·q
ෆ
en
ෆ
ᎏ
[A]
n1
–1
(8.30)
S
11
= ±
΂

ii
for an asynchronously tuned filter.
For a given filtering characteristic of S
21
(p) and S
11
(p), the coupling matrix and
the external quality factors may be obtained using the synthesis procedure devel-
oped in [10–11]. However, the elements of the coupling matrix [m] that emerge
from the synthesis procedure will, in general, all have nonzero values. The nonzero
values will only occur in the diagonal elements of the coupling matrix for an asyn-
chronously tuned filter. But, a nonzero entry everywhere else means that in the net-
work that [m] represents, couplings exist between every resonator and every other
resonator. As this is clearly impractical, it is usually necessary to perform a se-
quence of similar transformations until a more convenient form for implementation
is obtained. A more practical synthesis approach based on optimization will be pre-
sented in the next chapter.
8.2 GENERAL THEORY OF COUPLINGS
After determining the required coupling matrix for the desired filtering characteris-
tic, the next important step for the filter design is to establish the relationship be-
tween the value of every required coupling coefficient and the physical structure of
coupled resonators so as to find the physical dimensions of the filter for fabrication.
In general, the coupling coefficient of coupled RF/microwave resonators, which
can be different in structure and can have different self-resonant frequencies (see
Figure 8.3), may be defined on the basis of the ratio of coupled energy to stored en-
ergy [12], i.e.,
k = + (8.31)
where E
and H represent the electric and magnetic field vectors, respectively, and
we now use the more traditional notation k instead of M for the coupling coefficient.

ෆ
͐
ෆ
͐
ෆ
␮
ෆ
|H
ෆ
2
|
ෆ
2
ෆ
d
ෆ
v
ෆ
͐͐͐
␧
E
1
·E
2
dv
ᎏᎏᎏᎏ
͙͐
ෆ
͐
ෆ

2
ෆ
d
ෆ
v
ෆ
244
COUPLED RESONATOR CIRCUITS
Resonator 1
Resonator 2
Coupling
E
1
E
2
H
1
H
2
E
1
E
2
H
1
H
2
FIGURE 8.3 General coupled RF/microwave resonators where resonators 1 and 2 can be different in
structure and have different resonant frequencies.
Note that all fields are determined at resonance, and the volume integrals are over

–1/2
equals the angular resonant frequency of uncoupled
resonators, and C
m
represents the mutual capacitance. As mentioned earlier, if the
coupled structure is a distributed element, the lumped-element circuit equivalence
is valid on a narrow-band basis, namely, near its resonance. The same comment is
applicable for the other coupled structures discussed later. Now, if we look into ref-
erence planes T
1
–TЈ
1
and T
2
–TЈ
2
, we can see a two-port network that may be de-
scribed by the following set of equations:
I
1
= j
␻
CV
1
– j
␻
C
m
V
2

12
= Y
21
= –j
␻
C
m
can easily be found from definitions.
According to the network theory [13] an alternative form of the equivalent cir-
cuit in Figure 8.4(a) can be obtained and is shown in Figure 8.4(b). This form yields
the same two-port parameters as those of the circuit of Figure 8.4(a), but it is more
246
COUPLED RESONATOR CIRCUITS
(a)
(b)
FIGURE 8.4 (a) Synchronously tuned coupled resonator circuit with electric coupling. (b) An alterna-
tive form of the equivalent circuit with an admittance inverter J =
␻
C
m
to represent the coupling.
convenient for our discussions. Actually, it can be shown that the electric coupling
between the two resonant loops is represented by an admittance inverter J =
␻
C
m
. If
the symmetry plane T–TЈ in Figure 8.4(b) is replaced by an electric wall (or a short
circuit), the resultant circuit has a resonant frequency
f

–TЈ
1
and T
2
–TЈ
2
are
V
1
= j
␻
LI
1
+ j
␻
L
m
I
2
(8.37)
V
2
= j
␻
LI
2
+ j
␻
L
m

ෆ
(C
ෆ
–
ෆ
C
ෆ
m
ෆ
)
ෆ
1
ᎏᎏ
2
␲
͙
L
ෆ
(C
ෆ
+
ෆ
C
ෆ
m
ෆ
)
ෆ
8.2 GENERAL THEORY OF COUPLINGS
247

(b)
FIGURE 8.5 (a) Synchronously tuned coupled resonator circuit with magnetic coupling. (b) An alter-
native form of the equivalent circuit with an impedance inverter K =
␻
L
m
to represent the coupling.
f
e
= (8.39)
It can be shown that the increase in resonant frequency is due to the coupling effect
reducing the stored flux in the single resonator circuit when the electric wall is in-
serted in the symmetric plane. If a magnetic wall (or an open circuit) replaces the
symmetry plane in Figure 8.5(b), the resultant single resonant circuit has a resonant
frequency
f
m
= (8.40)
In this case, it turns out that the coupling effect increases the stored flux, so that the
resonant frequency is shifted down.
Similarly, (8.39) and (8.40) can be used to find the magnetic coupling coefficient
k
M
k
M
= = (8.41)
It should be emphasized that the magnetic coupling coefficient defined by (8.41)
corresponds to the definition of the ratio of the coupled magnetic energy to the
stored energy of an uncoupled single resonator. It is also consistent with the defini-
tion given in (8.6) for coupled-resonator filters.

C
(8.42)
Y
12
= Y
21
= j
␻
C
Ј
m
Z
11
= Z
22
= j
␻
L
(8.43)
Z
12
= Z
21
= j
␻
L
Ј
m
where C, L, C
Ј

+
ෆ
L
ෆ
m
ෆ
)Cෆ
1
ᎏᎏ
2
␲
͙(L
ෆ
–
ෆ
L
ෆ
m
ෆ
)Cෆ
8.2 GENERAL THEORY OF COUPLINGS
249