Tài liệu Microstrip bộ lọc cho các ứng dụng lò vi sóng RF (P8) doc

8.1 GENERAL COUPLING MATRIX FOR COUPLED-RESONATOR FILTERS 8.1.1 Loop Equation FormulationShown in Figure 8.1a is an equivalent circuit of n-coupled resonators, where L, C, and R denote t

CHAPTER 8

Coupled Resonator Circuits

Coupled resonator circuits are of importance for design of RF/microwave filters, inparticular 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 thesense that it can be applied to any type of resonator despite its physical structure Ithas 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 andoutput resonators We actually saw some examples in Chapter 5 when we discussedthe design of hairpin-resonator filters and combline filters, and we will discussmore 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 adeep 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 ofcoupled-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 avery useful formula for analysis and synthesis of coupled-resonator filter circuits interms of coupling coefficients and external quality factors Section 8.2 considersthe general theory of couplings in order to establish the relationship between thecoupling coefficient and the physical structure of synchronously or asynchronouslytuned coupled resonators Following this, a discussion of a general formulation forextracting coupling coefficients is given in Section 8.3 Formulations for extractingthe 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 obtainedcan be applied to extract coupling coefficients and external quality factors of mi-crowave coupling structures from EM simulations

235

Microstrip Filters for RF/Microwave Applications Jia-Sheng Hong, M J Lancaster

Copyright © 2001 John Wiley & Sons, Inc ISBNs: 0-471-38877-7 (Hardback); 0-471-22161-9 (Electronic)

8.1 GENERAL COUPLING MATRIX FOR COUPLED-RESONATOR FILTERS 8.1.1 Loop Equation Formulation

Shown in Figure 8.1(a) is an equivalent circuit of n-coupled resonators, where L, C, and R denote the inductance, capacitance, and resistance, respectively; i represents the loop current; and esthe voltage source Using the voltage law, which is one ofKirchhoff ’s two circuit laws and states that the algebraic sum of the voltage dropsaround any closed path in a network is zero, we can write down the loop equations

for the circuit of Figure 8.1(a)

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

8.1 GENERAL COUPLING MATRIX FOR COUPLED-RESONATOR FILTERS 237

[Z]·[i] = [e]

where [Z] is an n × n impedance matrix.

For simplicity, let us first consider a synchronously tuned filter In this case, theall resonators resonate at the same frequency, namely the midband frequency of fil-ter
0= 1/兹L苶C苶, where L = L1= L2= · · · Ln and C = C1= C2= · · · Cn The imped-

ance matrix in (8.2) may be expressed by

i1

i2

⯗

i n

Q e1 and Q enare the external quality factors of the input and output resonators, spectively Defining the coupling coefficient as

the wave variables are denoted by a1, a2, b1, and b2 By inspecting the circuit of

Fig-ure 8.1(a) and the network of FigFig-ure 8.1(b), it can be identified that I1= i1, I2= –in, and V1= es – i1R1 Referring to Chapter 2, we have

a1= 2兹

e s R

苶1苶

(8.10)

a2= 0 b2= i n兹R苶n苶and hence

Solving (8.2) for i1and i n, we obtained

i1= 

0L·

e F

s BW

s BW

苶

W n

The normalized impedance matrix of (8.15) is almost identical to (8.7) except that it

has the extra entries miito account for asynchronous tuning

L ij



兹L苶i苶jL苶8.1 GENERAL COUPLING MATRIX FOR COUPLED-RESONATOR FILTERS 239

8.1.2 Node Equation Formulation

As can be seen, the coupling coefficients introduced in the above section are allbased on mutual inductance, and hence the associated couplings are magnetic cou-plings What formulation of the coupling coefficients would result from a two-port

n-coupled resonator filter with electric couplings? We may find the answer to the dual basis directly However, let us consider the n-coupled resonator circuit shown

in Figure 8.2(a), where v i denotes the node voltage, G represents the conductance, and i sis the source current According to the current law, which is the other one ofKirchhoff ’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 sthe node

equa-tions for the circuit of Figure 8.2(a) are

where Cij = Cji represents the mutual capacitance across resonators i and j Note that

all node voltages are with respect to the reference node (ground), so that the

rents resulting from the mutual capacitance have a negative sign Arrange this set ofequations in matrix form

[Y]·[ v] = [i]

in which [Y] is an n × n admittance matrix.

Similarly, the admittance matrix in (8.17) may be expressed by

where
0= 1/兹L苶C苶 is the midband frequency of filter, FBW = 
/
0is the

frac-tional bandwidth, and [Y苶] is the normalized admittance matrix In the case of

syn-chronously tuned filter, [Y苶] is given by

 + p where p the complex lowpass frequency variable Notice that

and assume
/
0⬇ 1 for the narrow-band approximation A simpler expression of(8.19) is obtained:

Similarly, it can be shown that if the coupled-resonator circuit of Figure 8.2(a) is

asynchronously tuned, (8.21) and (8.22) become

Fig-V2= v n and I1= i s– v1G1 We have

a1= 2兹

i s G

Finding the unknown node voltages v1and v nfrom (8.17)

v1= 

0C·

i F

s BW

s BW

苶

·F

1

B G

苶

W n

1

BW

 [Y苶]11 – 1which can be simplified as

8.1.3 General Coupling Matrix

In the foregoing formulations, the most notable thing is that the formulation of

nor-malized impedance matrix [Z苶] is identical to that of normalized admittance matrix

[Y苶] 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 electrmagnet-ic or even the combination of both Accordingly, the equations of (8.13)and (8.29) may be incorporated into a general one:

trix, which is an n × n reciprocal matrix (i.e., m ij = m ji) and is allowed to have

nonzero diagonal entries m iifor an asynchronously tuned filter

For a given filtering characteristic of S21(p) and S11(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 nonzerovalues 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 quence of similar transformations until a more convenient form for implementation

se-is obtained A more practical synthesse-is approach based on optimization will be sented in the next chapter

After determining the required coupling matrix for the desired filtering 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 ofcoupled resonators so as to find the physical dimensions of the filter for fabrication

characteris-In general, the coupling coefficient of coupled RF/microwave resonators, whichcan be different in structure and can have different self-resonant frequencies (seeFigure 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.

兰兰兰H1·H2d

兹兰苶兰苶兰苶苶|H苶1苶|2苶d苶v苶 ×苶 兰苶兰苶兰苶苶|H苶2苶|2苶d苶v苶

Note that all fields are determined at resonance, and the volume integrals are overall effected regions with permittivity of and permeability of  The first term onthe right-hand side represents the electric coupling and the second term the magnet-

ic coupling It should be remarked that the interaction of the coupled resonators ismathematically described by the dot operation of their space vector fields, which al-lows the coupling to have either positive or negative sign A positive sign would im-ply that the coupling enhances the stored energy of uncoupled resonators, whereas anegative sign would indicate a reduction Therefore, the electric and magnetic cou-plings could either have the same effect if they have the same sign, or have the op-posite effect if their signs are opposite Obviously, the direct evaluation of the cou-pling coefficient from (8.31) requires knowledge of the field distributions andperformance of the space integrals This is not an easy task unless analytical solu-tions of the fields exist

On the other hand, it may be much easier by using full-wave EM simulation orexperiment to find some characteristic frequencies that are associated with the cou-pling of coupled RF/microwave resonators The coupling coefficient can then be de-termined against the physical structure of coupled resonators if the relationship be-tween the coupling coefficient and the characteristic frequencies is established Inwhat follows, we derive the formulation of such relationships Before procedingfurther, it might be worth pointing out that although the following derivations arebased on lumped-element circuit models, the outcomes are also valid for distributedelement coupled structures on a narrow-band basis

8.2.1 Synchronously Tuned Coupled-Resonator Circuits

A Electric Coupling

An equivalent lumped-element circuit model for electrically coupled RF/microwave

resonators is given in Figure 8.4(a), where L and C are the inductance and capacitance, so that (LC)–1/2equals the angular resonant frequency of uncoupled

self-resonators, and Cmrepresents the mutual capacitance As mentioned earlier, if thecoupled structure is a distributed element, the lumped-element circuit equivalence

is valid on a narrow-band basis, namely, near its resonance The same comment isapplicable for the other coupled structures discussed later Now, if we look into ref-

erence planes T1–T1and T2–T2, we can see a two-port network that may be scribed by the following set of equations:

de-I1= j
CV1– j
C m V2

(8.32)

I2= j
CV2– j
C m V1

in which a sinusoidal waveform is assumed It might be well to mention that (8.32)

implies that the self-capacitance C is the capacitance seen in one resonant loop of Figure 8.4(a) when the capacitance in the adjacent loop is shorted out Thus, the

second terms on the R.H.S of (8.32) are the induced currents resulting from the

in-8.2 GENERAL THEORY OF COUPLINGS 245

creasing voltage in resonant loop 2 and loop 1, respectively From (8.32) four Y

pa-rameters

Y11= Y22= j
C

(8.33)

Y12= Y21= –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

(a)

(b)

FIGURE 8.4 (a) Synchronously tuned coupled resonator circuit with electric coupling (b) An tive form of the equivalent circuit with an admittance inverter J =
C mto represent the coupling

alterna-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–T in Figure 8.4(b) is replaced by an electric wall (or a short

circuit), the resultant circuit has a resonant frequency

This resonant frequency is lower than that of an uncoupled single resonator A ical explanation is that the coupling effect enhances the capability to store charge ofthe single resonator when the electric wall is inserted in the symmetrical plane of

phys-the coupled structure Similarly, replacing phys-the symmetry plane in Figure 8.4(b) by a

magnetic wall (or an open circuit) results in a single resonant circuit having a nant frequency

B Magnetic Coupling

Shown in Figure 8.5(a) is an equivalent lumped-element circuit model for cally coupled resonator structures, where L and C are the self-inductance and self- capacitance, and L mrepresents the mutual inductance In this case, the coupling

magneti-equations describing the two-port network at reference planes T1–T1and T2–T2are

V1= j
LI1+ j
L m I2

(8.37)

V2= j
LI2+ j
L m I1

The equations in (8.37) also imply that the self-inductance L is the inductance seen

in one resonant loop of Figure 8.5(a) when the adjacent loop is open-circuited.

Thus, the second terms on the R.H.S of (8.37) are the induced voltages resultingfrom the increasing current in loops 2 and 1, respectively It should be noticed that

the two loop currents in Figure 8.5(a) flow in the opposite directions, so that the

voltage drops due to the mutual inductance have a positive sign From (8.37) we can

find four Z parameters

Z11= Z22= j
L

(8.38)

Z12= Z21= j
L m Shown in Figure 8.5(b) is an alternative form of equivalent circuit having the same network parameters as those of Figure 8.5(a) It can be shown that the magnetic coupling between the two resonant loops is represented by an impedance inverter K

=
L m If the symmetry plane T–T in Figure 8.5(b) is replaced by an electric wall

(or a short circuit), the resultant single resonant circuit has a resonant frequency

(a)

(b)

FIGURE 8.5 (a) Synchronously tuned coupled resonator circuit with magnetic coupling (b) An native form of the equivalent circuit with an impedance inverter K =
L mto represent the coupling.

alter-f e= (8.39)

It can be shown that the increase in resonant frequency is due to the coupling effectreducing 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

C Mixed Coupling

For coupled-resonator structures, with both the electric and magnetic couplings, a

network representation is given in Figure 8.6(a) Notice that the Y parameters are

the parameters of a two-port network located on the left side of reference plane

T1–T1and the right side of reference plane T2–T2, while the Z parameters are the

pa-rameters of the other two-port network located on the right side of reference plane

T1–T1 and the left side of reference plane T2–T2 The Y and Z parameters are

circuit shown in Figure 8.6(b) One can also identify an impedance inverter K =

Lm and an admittance inverter J =
Cm, which represent the magnetic coupling

and the electric coupling, respectively

By inserting an electric wall and a magnetic wall, respectively, into the symmetry

plane of the equivalent circuit in Figure 8.6(b) we obtain

8.6(b) to change sign, we will find that both couplings tend to cancel each other

out

It should be remarked that for numerical computations, depending on the ular EM simulator used, as well as the coupling structure analyzed, it may some-times be difficult to implement the electric wall, the magnetic wall, or even both inthe simulation This difficulty is more obvious for experiments The difficulty can

partic-be removed easily by analyzing or measuring the whole coupling structure instead

of the half, and finding the natural resonant frequencies of two resonant peaks, servable from the resonant frequency response It has been proved that the two nat-

ob-ural resonant frequencies obtained in this way are f e and f m[5] This can also be seen

in the next section when we consider a more general case, namely the nously tuned coupled-resonator circuits

asynchro-8.2.2 Asynchronously Tuned Coupled-Resonator Circuits

Asynchronously tuned narrow-band bandpass filters exhibit some attractive teristics that may better meet the demanding requirements for rapid development of

mobile communications systems (see Chapter 10) In an asynchronously tuned ter, each of the resonators may resonate at different frequencies Hence, in order toachieve an accurate or first-pass filter design, it is essential to characterize cou-plings of coupled resonators whose self-resonant frequencies are different Thiswould seem more important for some filter technologies in which posttuning afterfabrication are not convenient.

fil-In general, two eigenfrequencies associated with the coupling between a pair ofcoupled resonators can be observed, whether or not the coupled resonators aresynchronously or asynchronously tuned In the last section, we derived the formu-

la for extracting coupling coefficients from these two eigenfrequencies for chronously tuned resonators However, if the coupled resonators are asynchro-nously tuned, a wrong result will occur if one attempts to extract the couplingcoefficient by using the same formula derived for the synchronously tuned res-onators Therefore, other appropriate formulas should be sought These will be de-rived below

syn-A Electric Coupling

In the case, when we are only concerned with the electric coupling, an equivalentlumped-element circuit, as shown in Figure 8.7, may be employed to represent thecoupled resonators The two resonators may resonate at different frequencies of
01

= (L1C1)–1/2and
02= (L2C2)–1/2, respectively, and are coupled to each other

electri-cally through mutual capacitance Cm For natural resonance of the circuit of Figure

8.7, the condition is

2Cm 2Cm(C -C )1 m (C -C )2 m

where Z L and Z Rare the input impedances when we look at the left and the right of

reference plane T–T of Figure 8.7 The resonant condition of (8.48) leads to aneigenequation

After some manipulations the equation of (8.49) can be written as

4(L1L2C1C2– L1L2C m2) –
2(L1C1+ L2C2) + 1 = 0 (8.50)

We note that the equation of (8.50) is a biquadratic equation having four solutions

or eigenvalues Among those four, we are only interested in the two positive realones that represent the resonant frequencies that are measurable, namely,

in accordance with the ratio of the coupled electric energy to the average stored

en-ergy, where the positive sign should be chosen if a positive mutual capacitance C misdefined

B Magnetic Coupling

Shown in Figure 8.8 is a lumped-element circuit model of asynchronously tuned

resonators that are coupled magnetically, denoted by mutual inductance Lm The two

resonant frequencies of uncoupled resonators are
= (L C)–1/2 and
=

2

02–
2 01



2

02+
2 01



2

02+
2 01

4



冢

0

0 2

1

 + 

0

0 1