Third-Order Filters with Four Transmission Zeros

Fig. 3.6 (a) plots the layout of the microstrip trisection filter based on the lowpass prototype of Fig. 3.2(a) with the admittance matrix in (3.2.4). The in-band ripple is 0.1 dB and ∆ = 6%. All experimental circuits are built on a substrate with ε_r

= 2.2 and thickness = 0.508 mm. The frequency-dependent microstrip J-inverter is the same as that simulated in section 3.3. The gap D12 is determined by the coupling coefficient K₁₂ = 0.0573 from (3.2.18). The tap point, indicated by L_f, is chosen to match the Qsi of a stepped-impedance resonator [34] to Qext = 17.72 from (3.2.19).

Fullwave software package IE3D is used to simulate the circuits before fabrication.

The theoretic, simulated, and measured responses are shown in Fig. 3.6(c) and (d). It can be seen that four zeros occur in the stopband. Zeros f_z1 and f_z4 are caused by tapped-line while fz2 and fz3 are by the trisection. When the gap width D13 is too large to provide significant E coupling, it is interesting to note that the J₁₃-inverter eventually becomes frequency-independent with only M coupling, which can generate only one zero in lower stopband for this trisection configuration. In measurements, the passband insertion loss is about 1.75 dB, return loss is better than 15dB, and the lobe level in stopband is about -29.5dB. The measured and simulated group delays (τ) are also given. The broadband response shows that the measured 20dB-rejection can be up to 5.75 GHz or 2.3 f₀. It is found that the peak at 6.57 GHz is caused by the first spurious mode of resonator 2, which impedance ratio is slightly lower that of the other resonators and results in lower spurious frequency.

A second realization of the filter is based on (3.2.9), in which the negative J-inverter is implemented by electric coupling. The responses and photo are given in

Fig. 3.7. When the gap width D13 is too large to provide significant E coupling, the J₁₃-inverter has only M coupling, and hence the circuit has only a zero in the upper stopband.

L' 1

W₁

D₁₂

Resonator 2 W₃ 2L ₃ Port 1

L f

Port 2

1 L

Resonator 1

D'¹³ D₁₃

W₄ L ₄

W₂ L ₂ Resonator 3

(a) (b)

|S | ,|S |(dB)

Frequency (GHz) 2.1

1.7 1.9 -70

-50 -60

1121 -40

f z1

3.3 3.1 2.7

2.5 2.9

2.3

f z2 f _z3

f _z4 0

-10

-30 -20

Measurement Simulation Calculation from (11)

(c)

Measurement

1121|S | ,|S |(dB)

2.7 2.5 -60 2.3

Frequency (GHz) 2

-701

4

3 5 6 7

-30

-50 -40

(ns)

τ 2

0 4 6 -20

-10 Simulation

0

(d)

Fig. 3.6. A third-order filter with four zeros. (a) Layout. Dimensions in mm: D₁₂ = 0.47, D₁₃ = 0.27, D’₁₃ = 0.79, L₁ = 7.79, L’₁= 7.19, L₂ = 7.59, L₃ = 8.58, L₄ = 7.42, L_f

= 1.95, W₁ = W₃ = 0.4, W₂ = 2, W₄ = 1.74. (b) Photo. (c) |S₂₁| and |S₁₁| responses. (d) Group delay and broadband responses.

-60

Frequency (GHz)

1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5

|S | ,|S | (dB)21

-50 -40

11

-30 -20

Measurement Simulation

-10 0

(a)

(b)

Fig. 3.7. The alternative third-order filter with four zeros. (a) |S21| and |S11| responses.

(b) Photo.

3.4.2 Third-order Filters with a Pair of Real Zeros and Two Other Zeros Created by Tapped-Line

This circuit demonstrates the design to create a pair of zeros on the real axis.

The zeros are located at s = ± 1.6 and the in-band ripple is 0.1dB. The admittance matrix can be found in (3.2.6). When the fractional bandwidth ∆ = 5%, it can be

obtained the coupling coefficients K₁₂ = –K₂₃ = 4.03% and external factors (Q_ext)_i = (Qext)o = 20.38 from (3.2.18) and (3.2.19), respectively. Fig. 3.8(a) shows that two dips occur at 1.9 and 3.5 GHz in |S₂₁| response, which are caused by tapped input/output. As shown in Fig. 3.8(b), the group-delay response is equalized since a pair of zeros exists on the real axis. From 2.45 to 2.55 GHz, it can be seen that the group delay is within 4.8 and 5.2 ns. Fig. 3.8(c) shows that the broadband results. It is found that the 20-dB rejection level in the upper band can reach to about 8GHz (3.2f₀).

Equivalent Circuit

|S | ,|S | (dB)

-20

Frequency (GHz) -40

-501.5 1.7 1.9 2.1 2.3 2.5 -30

2111

2.7 2.9 3.1 3.3 3.5 0

-10 Simulation Measurement

(a)

2.2

(ns)

3

1 0 2 τ

5 4 6

2.6 Frequency (GHz)

2.5 2.4

2.3 2.7 2.8

Equivalent Circuit Simulation Measurement

(b)

Measurement

-10 -20

11

Simulation

0

|S | ,|S | (dB)

Frequency (GHz)

-50_{1 2 3 4 5 6} 7

-30 -40

21

10 9

8 11 12

(c)

(d)

Fig. 3.8. A third-order filter with two zeros at real frequencies and two others at imaginary frequencies. (a) |S₂₁| and |S₁₁| responses. (b) Group delay (c) Broadband response. (d) Photo.

3.4.3 A Fourth-order Filter with Three Transmission Zeros

This section demonstrates a fourth-order filter based on the prototypes in Fig.

3.2(b) with the admittance matrix in (3.2.13) and uniform-impedance resonators.

The J₂₄-inverter can be designed by the method developed in section 3.3 except one of the tapped-line in the test circuit is replaced by a coupled feed-line. Equation (3.3.6) is used in calculating Kj,j+2 since b11 is not equivalent to b22. The in-band ripple is 0.1 dB and ∆ = 5%. The layout and filter responses are drawn in Fig. 3.9. It is noted that one arm of resonator 2 is used to implement J₂₄-inverter while the other

arm is used to couple with resonators 1 and 3 by L₁₂ and L₂₃, respectively. The frequency of fz1 is 0.4 GHz lower than the prediction because the actual values of microstrip J₂₄-inverter at low frequencies are lower than the specified value. In measurements, the passband insertion loss is about 1.96 dB. Due to the first spurious harmonic at 2f₀, the stopband of a rejection level of 30 dB is only to about 4.62 GHz (1.85 f₀). It can be seen that the zero f_z3 is determined by the tap point of port 1.

There is an alternative tap-point, labeled by T’, which has an identical Qsi value and can produce a zero in lower stopband. Although this circuit has only three zeros, it can control zero at the frequency fz3 by adjusting the length of Lf1 with a proper width of resonator 1 for matching the port impedance.

Dm

D L

Resonator 3

34

L

D23 34

L 23

12 T'D12

Resonator 1 Port 2

J -inverter Resonator 4

24 e

f2

L

Lm

D_x

f1

Port 1 L

Resonator 2 L

De

W

(a)

(b)

f z2

f z1

|S | ,|S |(dB) -50

2.3

Frequency (GHz) 1.7

-60

-80

-70 (ns) 4

1.9 0 τ

2.1 8

2.9 2.5 2.7

f z3

3.1 3.3 -20

-40 -30

1121

0 -10

Calculation from (13) Simulation Measurement

(c)

Fig. 3.9. A fourth-order filter with three transmission zeros. (a) Dimensions in mm:

D12 = 0.64, D23 = 0.46, D34 = 0.3, De = 0.3, Dm = 0.38, Dx = 8.94, L12 = 13.16, L23 = 4.61, L₃₄ = 12.6, L_e = 5.38, L_f1 = 3.35, L_f2 = 17.12, L_m = 3.03, W= 1.55. (b) Photo. (c) Group delay, |S21| and |S11| responses.

3.4.4 A Fourth-order Filter with Five Transmission Zeros

Fig. 3.10 shows the layout and responses of a fourth-order filter based on the prototype in Fig. 3.2(c) with the admittance matrix in (3.2.17) and stepped-impedance resonators. Its in-band ripple and ∆ are the same as those in the second example. Note that there are two identical frequency-dependent J-inverters, i.e., J₁₃ and J₂₄ inverters, in this circuit. For establishing the coupling between resonators 2 and 3, the two trisections have a back-to-back arrangement with a shift by a distance S. This shift can result in the split of the upper zero due to tapped-line, i.e. the zeros at f_z4 and f_z5. The upper rejection band is broadened and a total of five zeros are generated accordingly. In measurement, the insertion loss is 2.34 dB at f0, 0.38 dB higher than that of the second circuit due to the use of high-impedance line in stepped-impedance resonators. It is found that rejection level of better than 40 dB

can be achieved within the band covering from 2.64 to 4.91 GHz, and the stopband with a 30dB-rejection level is up to 6.03 GHz (2.4f0). The circuit size is 3.8×3.3 cm², about 55% of the area 5.8×3.9 cm² for circuit in section 3.4.3.

Port 2

Resonator 4 J -inverter

Resonator 3

Resonator 2

J -inverter₂₄

W1

Port 1 Resonator 1

S D₁₂

L' ₁ L _f

L 1

1 13

D'₁₃

D ¹³

L ₂ W2

(a) (b)

2.3

Frequency (GHz) -70

-60

-801.7

z1 τ f 1.9

0 2.1

8

(ns)

4 f _z4

2.9 2.5 2.7

f z5

3.1 3.3 -20

-30

-50

|S | ,|S |(dB)1121 -40

f z2

0 -10

Calculation from (15)

f z3

Simulation Measurement

(c)

Circuit 2 -70

Frequency (GHz) -801

3 2

Measurement 6 5

4 7

-30

|S | ,|S |(dB)

21 -50

-60

11

-40 -20 -10 0

Simulation Measurement Circuit 3

(d)

Fig. 3.10. A fourth-order filter with five zeros. (a) Layout. Dimensions in mm: D₁₂ = 0.4, D13 = 0.19, D’13 = 0.79, L1 = 7.79, L’1 = 7.19, L2 = 7.64, Lf = 1.95, S = 3.0, W1 = 0.4, W2 = 2. (c) Group delay, |S21| and |S11| responses. (d) Broadband response.

CHAPTER 4

Compact Inline Stepped-Impedance Resonator Filters with a Quasi-Elliptic Function Response

In this chapter, we explore a simple filter structure using the stepped-impedance resonators as building blocks. Fig. 4.1 plots two fourth-order circuits with both symmetric and skew-symmetric feeds. The circuit exhibits several attractive properties, such as compact size, wide upper stopband, elliptic function passband response, and plural transmission zeros. In Fig. 4.1, all resonators form an in-line array so that the circuit occupies a compact area. When order is increased, the circuit size grows only in the direction of the width, which is usually much smaller than the length of the resonator. Although the analysis becomes more complicated, creation of certain transmission zeros in the filter response indeed relies on the nonadjacent coupling. Besides, it is found that the nonadjacent coupling is frequency-dependent. Thus, this filter can be designed by the theory developed in chapter 3.

B'

2

B

4 3

B' B

A

1

A

Fig. 4.1. Two in-line fourth-order filters with two tapped input/output schemes:

symmetric (A–B) and skew-symmetric (A–B’) feeds.

Z W2

2

D1

W P'

1

Z1

θ1

L ( ) θ2

L ( )2 1

P L ( )_f θ_f

D₂

Fig. 4.2. Generic coupling structure of the in-line bandpass filter.

Here, we limit ourselves to exploring in-line stepped- impedance resonator filters only of orders four, six and eight. Some results for circuits of lower orders can be referred to [46]. The circuit in [46] also possesses a quasi-elliptic function response. Its analysis by the theory of multiple coupled microstrips, however, lacks for design concept for filter synthesis. In the following, Section 4.1 briefly describes the passband synthesis procedure and discusses the coupling properties among the in-line resonators. Section 4.2 explores the existence of the zeros in terms of Y-parameter matrix by taking the adjacent and nonadjacent coupling into account.

Section 4.3 addresses the creation of zero by the tapped input/output structure, Section 4.4 presents measured results for three experimental circuits.

4.1 Passband Synthesis

Fig. 4.2 shows the generic basic coupling structure for the filters in Fig. 4.1.

Each resonator has one high-Z and two low-Z sections. The former has physical (electric) lengths 2L1 (2θ₁) and each of the later has L2 (θ₂), and their respective widths are W₁ and W₂ with corresponding characteristic impedances Z₁ and Z₂. In addition to D2, the gap size D1 can be tuned to establish necessary coupling for synthesizing the passband. Choice of the geometrical dimensions for the resonator

has been extensively studied in [39, 40]. The impedance ratio R = Z₁/Z₂ and length ratio u = θ_1/(θ_{1 +} θ₂) are the key parameters to determine its resonant spectrum. If R and u are properly chosen, the first spurious resonance can be pushed far beyond twice the fundamental frequency or 2f_o [40]. For example, if the first higher order resonance occurring at 3fo is desired, u = 0.5 and R = 2.5 can be used. When fo = 2.5 GHz, geometry parameters can be L₁ = L₂ = 7.6 mm, W₁ = 0.4 mm and W₂ = 2.0 mm for a substrate with ε_r = 2.2 and thickness = 0.508 mm.

Next step is to determine spacing between each pair of adjacent resonators. The coupling coefficient between the jth and (j + 1)th resonators, K_j,j+1, is given by [1]:

1 1

,

+ +

= ∆

j j j

j g g

K (4.1.1)

3.0 D = D +2(W - W )

1.5

Distance D (mm) 0 0.5 1.0

1

2.0 2.5

D = 0.2, 0.4, 0.6, 0.8, 1.2₂

2 1

3.5 4.0 D = D_{1 2}

2 1

D _z

K100

-10 -5 0 5 15

10

Fig. 4.3. Coupling coefficients of two stepped-impedance resonators against D1 for various D2. L1 = L2 = 7.6, W1 = 0.4, W2 = 2.0, all in mm. Substrate: ε_{r = 2.2,} thickness = 0.508 mm.

where gj is the jth element value of the low-pass filter prototype and ∆ the fractional bandwidth. To realize this coefficient for coupled resonators in Fig. 4.2, the test method in [10] can be invoked. Through weak gap feeds to the coupled resonators, the simulated transmission response will present two peaks. If the peaks are at f_a and fb, the coefficient can be calculated as

2 2

2 2 1 ,

a b

a b j

j f f

f K f

+

−

+ = (4.1.2)

For the fourth-order circuits in Fig. 4.1, the coupling matrix is symmetric about its two main diagonals, i.e., K12 = K21 = K34 = K43, K13 = K31 = K24 = K42, K23 = K32

and K14 = K41. Thus, only K12, K13 and K23 need specifying since all diagonal entries Kjj are zero and K₁₄ is negligible owing to the relatively large space between resonators 1 and 4.

It can be anticipated that change of D₁ will not significantly alter the first resonance of the resonator, but will change magnitude, and even polarity of the coupling coefficient of two coupled resonators. This property is useful for adjusting the resonator geometry when more than one coupling coefficients have to be simultaneously considered in filter synthesis. An example will be given in section 4.3 for such demonstration. For D2 (mm) = 0.2 to 1.2, Fig. 4.3 plots coupling coefficients of two resonators against D₁. Except for D₁ = D₂, each curve runs from positive to negative values when D1 is increased up to 4 mm. Generally speaking, the structure consists of both electric and magnetic coupling, called the mixed coupling. When D1 is small, magnitude of magnetic coupling due to current on the thin sections is larger than that of electric coupling between the low-Z sections at

both ends. When D₁ is increased to be large enough for small D₂, on the other hand, electric coupling becomes dominant. The coefficient calculated by (2) is the net coupling which can be electric (K < 0) or magnetic (K > 0).

The use of curves in Fig. 4.3 can be demonstrated as follows. Suppose we are designing a fourth-order Chebyshev filter with a 0.1-dB ripple and ∆ = 8%. From (4.1.1), the three interstage coupling coefficients are K₁₂ = K₃₄ = 0.06648 and K₂₃ = 0.05261. If D2 = 0.2 mm is chosen, we have –0.089 ≤ K ≤ 0.15, and the zero-crossing point is at D1 = 0.856 mm. It is obvious that both electric and magnetic coupling can be used to realize each K value. Thus, there are at least two possible designs: one uses K₁₂ = K₃₄ < 0 and K₂₃ > 0, and the other uses K₁₂ = K₃₄ >

0 and K23 < 0. The former and the latter are respectively referred as the M- and E-type filters herein.

For the input and output coupling, the tap positions, i.e., Lf in Fig. 4.2, should be determined by matching the singly loaded Q (Q_si) of the tapped resonator with the passband specification. The singly loaded Q (Q_si) is defined as

d o

R dB Q_{si L}

ω ω

ω 2

= 0 (4.1.3)

where RL is the impedance seen by the resonator looking toward the source, ω_{o is the} operation frequency, and B is the input susceptance of the resonator seen at the tap point. The derivation of (4.1.3) for a stepped-impedance resonator can be referred to [40]. Both M- and E-type circuits can be designed with the symmetric (A–B) or the skew-symmetric (A–B’) feeds [41] with identical Qsi values and hence identical passband responses. In the rejection bands, nevertheless, they exhibit quite different

characteristics. Figs. 4.4(a) and 4.4(b) show the simulated |S21| responses of the fourth-order M- and E-type filters, respectively.

f z2

Frequency (GHz) -801

2 3 4 5 6 7

M-type

|S | (dB)

-70 -60

21

-50 -40

f z1

-20 -30 0 -10

B B'

A-B' A-B A

(a)

|S | (dB)

f f A-B' -70 f

Frequency (GHz)

z1

-801

zL

3

2 5

z2

4 6

21 -40

-50 -60

f zH

-20 -30

A-B

B' B

7 0

-10 E-type A

(b)

Fig. 4.4. Simulation responses of the two fourth-order filters. (a) M-type: D₁₂ = D₃₄ = 0.28, D₂₃ = 1.0, L_f = 3.2. (b) E-type: D₁₂ = D₃₄ = 0.82, D₂₃ = 0.37, L_f = 3.2, all in mm.

The four passbands, say before |S21| ≥ –30 dB, show very good agreement. In Fig. 4.4(a), both the M-type filters have a transmission zero (fz2) in the upper stopband. The circuit with the symmetric (A–B) feed, however, has one more zero (f_z1) in the lower stopband. In Fig. 4.4(b), the two E-type filters exhibit sharp transition bands, like those of an elliptic function response, since two zeros, fzL and f_zH, are created on both sides of the passband. In addition, there is an extra transmission zero fz1 in the lower stopband and fz2 in the upper stopband for the (A–B) and the (A–B’) feeds, respectively. Obviously, the E-type filters possess better frequency selectivity in the stopband than the M-type ones. Thus, the E-type filters are investigated in detail as follows.

4.2 Transmission Zeros Due to Frequency-Dependent Cross Coupling

The occurrence of the zeros in the quadruplet relies on the coupling between the first and the last resonator, which causes two split signals to be out of phase at the output port. In the E-type filters, however, the elliptic function-like response is clearly resulted from a different scheme, since the K₁₄ in Fig. 4.1 can be negligible while the nonadjacent coupling coefficients K13 and K24 should be taken into account.

In addition, for predicting the zeros of the particular filter configuration, based on Y-parameter of equivalent circuit of the filter, an analysis method is developed as

follows.

The equivalent lumped-circuit model of a filter with two coupled-resonators is shown in Fig. 4.5. Each resonator is modeled with a parallel LC network, and there are magnetic and electric coupling between the inductors and capacitors,

respectively. From the circuit theory, the two-port Y-parameters can be derived as follows:

( )ω λ Lr

j C Y

Y = =

22

11 (4.2.1)

( )ω β Lr

j C Y

Y = =

21

12 (4.2.2) where

( )

ω ω ω ω ω

λ ⁰

0

−

= (4.2.3)

( ) _{m e}

0 0

ω ω ω

ω ω

β = − (4.2.4)

C L_r

1

0 =

ω (4.2.5)

L

m = M (4.2.6)

C

e = E (4.2.7)

(

_m

)

_L

L

L_r = 1− ² ≈ (4.2.8)

E Resonator 1

I I L1

V1 C ^C1 L L I1

M

I I

Resonator 2

L2 C2

C V ² I2

Fig. 4.5. The equivalent circuit of two coupled resonators with coupling.

The coefficients m in (4.2.6) and e in (4.2.7) respectively represent magnetic and electric coupling between the two resonators. They are assumed constants over a certain frequency range centered at the design frequency. The natural frequencies f_a and f_b in (4.1.2) of the coupled system can be determined by two conditions: λ₍ω_{) =}

±β₍ω), which are obtained by enforcing the determinant of the Y-matrix to zero. It can be validated that

e

The approximation is valid since em << 1. The result in (4.2.9) means that the net coupling K calculated by (4.1.2) should be m – e. Note that all Y-parameters in (4.2.1) and (4.2.2) are purely imaginary since the circuit is assumed lossless. In (4.2.2), Y₂₁ has a zero-crossing point at ω _{= m/e}ω_o. Thus, its sign over ω _{< m/e}ω_{o is} opposite to that over ω _{> m/e}ω_o. This property is unusual since in conventional coupling matrix non-diagonal elements are usually assumed independent of frequency. For investigating the possible occurrence of transmission zeros, define the relative phase between Y11 and Y21 as

2.8 Y - Y 21

2.1 0

2

Frequency (GHz) 2.2 2.3 2.4 2.5 2.6 2.7

11

90 180

D2

2.9 3

(a)

D2

Y - Y 21

2.9 0

2 2.1

Frequency (GHz) 2.3

2.2 2.4 2.5 2.6 2.7 2.8 3

90

11

180

(b)

Y - Y 21

2.4

Frequency (GHz) 2

0

2.2

2.1 2.3 2.5 2.6 2.7 2.8 2.9 3

11

90 180

D2

(c)

Fig. 4.6. Responses of ∠Y21 – ∠Y11 for investigating occurrence of the transmission zeros of the E-type filter in Fig. 4(b). (a) Resonators 1 and 2, D2 = 0.82 mm. (b) Resonators 2 and 3, D2 = 0.37 mm. (c) Resonators 1 and 3, D2 = 3.19 mm.

Fig. 4.6 plots the simulated θ responses for the three basic coupled structures of the E-type filter in Fig. 4.4(b). The Y-parameters are obtained by the software package IE3D. Each θ response shows a jump at f = fo due to the phase change of the denominator of (4.2.1). Based on Fig. 4.6(a), one can assure that the coupling between resonators 1 and 2 is magnetic- dominant and K₁₂ ≈ m >> e for 2 GHz < f <

3 GHz. Similarly, the response in Fig. 4.6(b) guarantees K23 ≈ e >> m. In Fig. 4.6(c) there are extra phase jumps at 2.08 GHz and 2.75 GHz. Two important properties of this coupled structure should be identified by the later jump. First, the type of K13

coupling is magnetic, like that is indicated in Fig. 4.6(a). The jump at f = 2.75 GHz indicates that Y₂₁ changes sign, by (4.2.2), since λ₍ω) > 0 when ω _> ω_o. Also, by (4.2.2), the m and e values can be extracted since m/e= 2.75/2.5 = 1.1 and K ≈ m – e is known by (4.1.2) from simulation data. The jump at 2.08 GHz, however, can not

be explained by (4.2.1). It could be due to that the equivalent circuit in Fig. 4.5 has a lower frequency limit for modeling the distributed coupled resonators with a relative large distance in Fig. 4.6(c).

Identifying the type of coupling and value of K13 is further investigated for D2 = 0.6 mm and 1.5 mm by Figs. 4.7(a) and 4.7(b), respectively. From Fig. 4.7(a), K13 is electric coupling and m/e= 2.13/2.5. Similarly, Fig. 4.6(b) indicates that K13 is of the magnetic type and m/e= 2.58/2.5. It is important to identify m and e from the θ responses of coupled resonators in Figs. 4.6(c) and 4.7, since analysis of the transmission zeros relies on it. By analyzing the phase relation of two split signals in the main- and cross-coupled paths, a zero in the upper stopband can occur at f > 2.75 GHz. This zero can also be validated by the Y-matrix method given below. Let the filter bandwidth ∆ = 8% and ripple = 0.1 dB, then the external Q (Q_e) = 13.86. The Y-matrix for the circuit, normalized with respect toj C/L_r , can be expressed as:

( )

Y - Y 21

Frequency (GHz) 0

2.2

2 2.1 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 180

90

11

(a)

Y - Y 21

2.4

Frequency (GHz) 2

0

2.2

2.1 2.3 2.5 2.6 2.7 2.8 2.9 3

11

90 180

(b)

Fig. 4.7. ∠Y21 – ∠Y11 responses for identification type of coupling between resonators 1 and 3. (a) D2 = 0.6 mm. (b) D2 = 1.5 mm.

where K12 = K34 = 0.06648 and K23 = 0.05261. These values are derived from (1).

The entries in the first off-diagonal use the following approximation. For example, Y₁₂ ≈ K₁₂ω_o/ω _{since K12} ≈ m >> e and the last term in (4.2.4) is neglected. Values of m and e can be obtained by prescribed zeros at (1 ± ξ_{) × fo}. If the two zeros are symmetric about f_o, m/e ≈ 1 is required. Note that the sign of the elements Y₁₃ (Y₂₄) is determined by β₍ω_{). When} ξ = 0.2 and 0.152, m = e = 0.012 and 0.018 can be obtained from the |S₂₁| responses based on (4.2.11), respectively. Fig. 4.8 shows |S₂₁| responses from the matrices with and without the nonadjacent coupling Y13 (= Y24).

Note that when m = e = 0 the response will have no transmission zero. It can be seen from this example that values of m and e can be varied for controlling these two

transmission zeros. One possible way to adjust m and e is to slide or deform the high-Z section of one of the coupled resonators, as shown in Fig. 4.1.

The Y-matrix in (4.2.11) can be easily extended to circuits of order N = 6 and 8 with the quasi-elliptic response. The Y-matrix can be established and the frequencies of the zeros can be predicted. For example, the coupling matrices for N = 6 are

( )

Fig. 4.8. |S21| responses based on coupling matrices in (4.2.11).

3.1 N = 6

-1201.5 2.3

Frequency (GHz) 1.9

1.7 2.1 2.5 2.7 2.9

21|S | (dB)

-60

-100 -80 -40 -20

N = 8

3.5 3.3 0

Fig. 4.9. Responses of higher-order in-line filters with m = e = 0.011.

4.3 Transmission Zeros due to Tapped Input/Output

Fig. 4.10 plots simulation |S21| responses of the E-type filters with skew-symmetric feed for L_f = 3.1, 6 and 9 mm. Impedance transformers are added to keep the Q_si value of each tapped resonator unchanged for the three tap positions. It can be seen that frequencies of fzL and fzH as well as the passband do not vary significantly with the changes of L_f. However, the zero f_z2 moves to higher frequency when tap point is moved away from the center to the edge of the resonator. It reflects the fact that determination of fz2 can be dominated by L_f. In the parallel- coupled stepped-impedance resonator filters in [34], a zero can be created at a frequency where the electric length of the arm between the open end of the tapped resonator and the tap point is one quarter wavelength long. The arm used for coupling with adjacent resonator, however, does not create a zero. For the structure in Fig. 4.1, both open ends of the input and output resonators are coupled with their adjacent resonators. Thus, creation of the zero f_z2 in Fig. 4.10 needs further investigation.

The four-port network in Fig. 4.11(a) is employed for the prediction. Two more ports are added to the circuit in Fig. 4.1 since in analysis the whole circuit can be reduced by half due to the symmetry (dash line). Let I_n be total current flowing into

The four-port network in Fig. 4.11(a) is employed for the prediction. Two more ports are added to the circuit in Fig. 4.1 since in analysis the whole circuit can be reduced by half due to the symmetry (dash line). Let I_n be total current flowing into

在文檔中 具頻率相關耦合之步階阻抗諧振器的準橢圓函數響應濾波器之合成與實現 (頁 60-0)