Bandpass mode analysis

The proposed MMR and its even- and odd-mode equivalent circuit are shown in Fig. 1. Y _e and Y _o are the input admittance of the even mode and odd mode, respectively. Even-odd mode method can be used to analyze this structure since the structure is symmetrical along its middle plane. The resonant frequencies of the MMR can be obtained by.

(1)$${\rm Im} (Y_{11}) = 0,$$

(2)$$Y = \left( {\matrix{ {Y_{11}} & {Y_{12}} \cr {Y_{21}} & {Y_{22}} \cr}} \right) = \left( {\matrix{ {\displaystyle{{Y_e + Y_o} \over 2}} & {\displaystyle{{Y_e-Y_o} \over 2}} \cr {\displaystyle{{Y_e-Y_o} \over 2}} & {\displaystyle{{Y_e + Y_o} \over 2}} \cr}} \right).$$

Fig. 1.

The simplified structure of the proposed multiple-mode resonator and its even- and odd-mode equivalent structure.

According to equations (1) and (2), the total input admittance can be derived from its even- and odd-mode input admittance. It should be noticed that the characteristic impedance of the microstrip line in this design is a constant value Z, so this parameter is not shown in Fig. 1. The derived even- and odd-mode input admittance equations are as follows:

(3)$$Y_e = Y \cdot j\tan \theta _1 + Y_1 + Y_2 + Y_{3e},$$

(4)$$Y_o = -Y \cdot j\cot \theta _1 + Y_1 + Y_2 + Y_{3o},$$

where

(5)$$Y_1 = Y \cdot \displaystyle{{\,j\omega C_1 + jY\tan \theta _5} \over {Y-\omega C_1\tan \theta _5}},$$

(6)$$Y_2 = Y \cdot \displaystyle{{\,j\omega C_3 + jY\tan \theta _2} \over {Y-\omega C_3\tan \theta _2}},$$

(7)$$Y_{3e} = Y \cdot \displaystyle{{Y_L + j\omega Y\tan \theta _3} \over {Y + j\omega Y_L\tan \theta _3}},$$

(8)$$Y_{3o} = Y \cdot \displaystyle{{2j\omega C_4 + Y\cot \theta _4 + jY\tan \theta _3} \over {Y-(2\omega C_4-jY\cot \theta _4)\tan \theta _3}},$$

(9)$$Y_L = j\omega (C_4 + \displaystyle{{C_5} \over 2}) + Y \cdot j\displaystyle{{\omega \displaystyle{{C_2} \over 2} + Y\tan \theta _4} \over {Y-\omega \displaystyle{{C_2} \over 2}an\theta _4}}.$$

A rough dimension θ ₁ = θ, θ ₂ = 1.3θ, θ ₃ = 1.2θ, θ ₄ = 3.4θ, θ ₅ = 1.5θ is given thus we can offer a basic estimation on the multiple modes resonant character by calculating the input admittances. A numerical study based on the above equations can be done to demonstrate the resonant frequency of each mode versus the capacitance value, which can provide a clear guideline for us to understand the MMR's tuning principle. The initial capacitance values are set as C ₁ = 1 pF, C ₂ = 30 pF, C ₃ = 0.1 pF, C ₄ = 0.1 pF and C ₅ = 1 pF. The characteristic impedance is set to be 120 Ω.

Bandpass resonant modes

As shown in Fig. 2(a), four resonant modes (defined as: f _mode1 < f _mode2 < f _mode3 < f _mode4) can be observed, which form the entire passband for the wide bandwidth bandpass mode. Two transmission zeros are located at both the upper and lower side of the passband to improve the selectivity of the filter. As shown in Fig. 2(b), a transmission zero locates between the lower two resonant modes and upper two transmission modes, which is different from Fig. 2(a). This state is proposed for narrow bandwidth bandpass mode, only the upper two resonant modes will form the narrow passband while the lower two resonant modes can be suppressed by impedance mismatching, therefore a narrow band response can be achieved. The switching between 2 resonant modes and 4 resonant modes are realized by tuning C ₁ and C ₂. Our goal is to tune these resonant frequencies independently, therefore, the tuning principle should be studied by sweeping the capacitance values and observing the trend of the resonant frequencies.

Fig. 2.

(a) Four resonant modes state (b) two resonant modes state.

From Fig. 3(a), it can be observed that although all four modes are affected by C ₁, the impact on the lower two modes is more significant than the upper two modes. Actually, C ₁ is mainly used to tune the external quality factor in this design, which will be illustrated in the section “Tunable External Quality Factor”. Fig. 3(b) shows that C ₂ can only change the mode1's frequency and keep other modes stable, which means that this capacitor is a proper choice to tune the lower cut off frequency. Fig. 3(c) shows a similar character to Fig. 3(b) that C ₃ can be used to tune the upper cut off frequency. C ₄ and C ₅ are not shown in this figure because the two capacitors are used in embedded-stopband tuning, which will be demonstrated in the section “Bandpass-with-Embedded-Stopband Mode Analysis”.

Fig. 3.

Simulated resonant frequencies versus different capacitance values.

Tunable external quality factor

Q _e is the external quality factor of an end resonator when it is coupled to port termination and it plays a critical role for tunable filter design. According to traditional microwave theory, the passband performance is related to the quality factor as shown in equation (10).

(10)$$Q_e = \displaystyle{{\omega _0} \over {\Delta \omega _{3dB}}},$$

where ω ₀ is the angular frequency of the passband and Δω _3dB is the 3 dB bandwidth of the passband. Actually, the essence of this theory is similar to impedance matching theory, if the external quality factor is not suitable for current filtering response, it can also be regarded as impedance mismatching between the feedline and the resonator. We always call this mismatching as over coupling or under coupling when discussing external quality factor. Fig. 4 shows two typical filtering responses with improper external quality factor. As can be observed, under coupling state suffers severely from insertion loss between resonant modes while the attenuation out of band of the over-coupling is terrible.

Fig. 4.

Filtering response with improper external quality factor.

To achieve an ultra-wide bandwidth tuning range at a fixed center frequency, the external quality factor must change as the bandwidth changes or the passband performance will badly deteriorate.

To tune the external quality factor, a tunable K inverter should be adopted as shown in equation (11). From Fig. 5 we can observe that the tunable K inverter is achieved by a tunable capacitor C ₁, which means that C ₁ is used in both resonant frequency tuning and external quality factor tuning. Therefore we can conclude that the variable capacitor C ₁ is used to control the external coupling value, external quality factor or impedance matching, the three nouns are essentially the same.

(11)$$Q_e = K^2\displaystyle{1 \over {Z_0}}b_s,$$

where $b_s = \displaystyle{\omega \over 2} \cdot \displaystyle{{\Delta B_s(\omega )} \over {\Delta \omega}} \vert {_{\omega = \omega_0}} $ and $K = \sqrt {\displaystyle{1 \over {{(\omega _0C_1)}^2}} + Z_0^2}, \,\omega _0$ is the angular frequency of the passband, B _s(ω) is the susceptance of MMR and Z ₀ is the characteristic impedance.

Fig. 5.

The tunable K inverter.

Based on the above analysis shown in the section “Bandpass Resonant Modes and Tunable External Quality Factor”, we can give a brief summary on the tuning mechanism for different tuning states under bandpass mode. There are four resonant modes with wide bandwidth and two resonant modes with narrow bandwidth under bandpass mode, as shown in Fig. 3. C ₂ and C ₃ can be used to tune the lower and upper cutoff frequency of the passband, which can be observed from Fig. 3. C ₁ is used to control the external quality factor, from equation (11) it is obvious that the increase of capacitance will lead to the increase of external quality factor.

To tune the center frequency of the passband, C ₂ and C ₃ should change in the same direction. In this way, the lower cutoff frequency (controlled by C ₂) and upper cutoff frequency (controlled by C ₃) will move to the different center frequency direction simultaneously with a stable bandwidth. C ₁ also need to be slightly tuned to fit different external quality factors under different center frequency.

To tune the bandwidth of the passband, C ₂ and C ₃ should change in the opposite direction. The lower cutoff frequency(controlled by C ₂) and upper cutoff frequency(controlled by C ₃) will move to the opposite direction accordingly. In this way, the bandwidth can be tuned under a fixed center frequency. To achieve a wide bandwidth tuning range, C ₁ needs to be adjusted as bandwidth changes. Otherwise, the passband performance will badly deteriorate for the improper external quality factor.

Bandpass-with-embedded-stopband mode analysis

The bandpass-with-embedded-stopband mode can be obtained by controlling C ₄ and C ₅, which will introduce another two resonant modes (defined as mode5 and mode6) and another two transmission zeros (defined as TZ1 and TZ2). The six resonant modes form a constant passband and two transmission zeros form the embedded-stopband. The two transmission zeros are induced by the T capacitor junction consisting of C ₄ and C ₅. The reason of their occurrence is that there must be a 180° phase difference between the main signal transmission path and the T capacitor junction signal transmission path at some certain frequencies and it is obvious that a 180° phase difference will cause a transmission. Once the frequencies with 180° phase differences can be controlled by C ₄ and C ₅, the center frequency and bandwidth of the embedded-stopband can, therefore, be controlled. The T capacitor junction is also symmetrical along the middle plane, thus the odd- and even-mode method is still useful in this analysis. Based on the even-odd mode analysis, C ₄ can control two transmission zeros induced by both even mode and odd mode from the capacitor junction, C ₅ can only control one transmission zero induced by only even mode from the capacitor junction. In this way, the center frequency and bandwidth of the embedded-stopband can be tuned independently.

As shown in Fig. 6, when C ₄ and C ₅ are relative low, mode5 and mode6 locate far away from mode1 to mode4. This is the bandpass mode as black line shows. As C ₄ and C ₅ become larger, mode5 and mode6 will move closer to mode4 and two TZs will appear between mode4 and mode5,6(f _mode1 < f _mode2 < f _mode3 < f _mode4 < f _TZ1 < f _TZ2 < f _mode5 < f _mode6), as shown in Fig. 6(a). Under this state, the lower and upper cutoff frequency of the passband is mode1 and mode6, respectively, which is different from bandpass mode. As shown in Fig. 6(b), keeping C ₄ at a fixed value while tuning C ₅, only one transmission zero moves. This situation accords with our previous expectation in the first paragraph. The center frequency and bandwidth of the stopband can be tuned independently because of the two controllable TZs.

Fig. 6.

(a) Stopband center frequency tuning (b) stopband bandwidth tuning.

In this section, we also give a brief summary on the tuning mechanism for different tuning states under bandpass-with-embedded-stopband mode. The passband is formed by totally six modes while two transmission zeros are embedded into the passband as “embedded-stopband.”

The center frequency of the embedded-stopband can be tuned by simply controlling C ₄ because two transmission zeros all affected by C ₄. C ₅ can be used to tune the bandwidth of the embedded-stopband because only one transmission zero is affected by this capacitor. During this tuning process, the passband maintains constant.

Design technique and tuning logic

Although this design is mainly accomplished by iterative optimization, it is still necessary to provide design steps for readers who are interested in redo this work at a particular working frequency. We need to build a rough structure first and observe the location of resonant modes and transmission zeros, and then optimize the structure's dimension towards our desired frequency based on the varactors' capacitance tuning range. Several design techniques are listed as follows for different tuning states:

1. (1) For the determination of wide passband center frequency: The center frequency of the passband in this state is determined by all four resonant modes. If the length of microstrip line L ₁ and L ₅ is increased (as shown in Fig. 7) or the line width W is decreased, all four modes will move to the lower side. In this way the center frequency of the wide passband can be adjusted to a particular value.

   Fig. 7.

   Two-stage cascaded multiple-mode resonator for higher order filter design.

   

2. (2) For the determination of narrow passband center frequency tuning range: The center frequency of the passband in this state is determined by the upper two resonant modes. If L ₂ is increased then the center frequency tuning range of the narrow passband will move to the lower side.

3. (3) For the determination of bandwidth tuning range: The attainable widest bandwidth is determined by the gap between the lower two modes (can be adjusted by L ₆) and upper two modes, but it is also limited by the external quality factor value.

4. (4) For the determination of embedded-stopband center frequency and bandwidth: the two transmission zeros, which forms the stopband, always located near the four resonant mode when C ₄ and C ₅ is relative low. The only impact factor on the tuning range of embedded-stopband is the varactors' capacitance tuning range and the T varactor junction's location.

A brief tuning logic of the bandpass-to-bandpass-with-embedded-stopband filter is provided, as shown in Table 1.

Table 1.

Tuning logic of the proposed filter

Higher order filter design

This design can be expanded to higher-order filters by cascading several replicas of the proposed multiple-mode resonator shown in Fig. 1 with small modifications.

As an example, Fig. 7 shows a two-stage cascaded multiple-mode resonator for higher order filter design. Two multiple mode resonators are cascaded in series as “two stage” which can provide a higher order passband response. The previous T capacitor junction is also modified by adding a new capacitor-loaded stub, which can provide a higher order stopband response. Fig. 8 shows the simulated fully matched bandpass and bandpass-with-embedded-stopband response of this proposed higher-order design. From Figs 8(a) and (b), a wider passband can be observed from the passband response, which indicates that more resonant modes are included in this response. From Figs8(c) and (d), a three-stage stopband can be observed from the bandpass-with-embedded-stopband response, which suits the expectation of modified capacitor junction.

Fig. 8.

Simulated response of higher order multiple-mode resonator for bandpass response and bandpass-with-embedded-stopband response.