Variable Bandwidth Double-Coupled Double-Tuned Filter with Capacitive Couplings

Abstract

In this paper, an adjustable bandwidth high IP3 (third-order intercept point) sub-octave double-coupled double-tuned filter is presented. The proposed filter consists of two capacitively coupled resonators. Its center frequency and bandwidth are tuned by the varactors with two voltages. The filter characteristics are shaped by two transmission zeros located in its lower and upper transition bands. The frequency of the lower transmission zero is tuned simultaneously with the center frequency, and the upper transmission zero frequency is independently controlled. The achieved center frequency tuning range is from 300 MHz to 446 MHz, and the bandwidth swing is from 35 MHz to 75 MHz. In the whole operating band, the IP3 of the filter varies from 22 dBm to 29 dBm, and the insertion loss is from 1.4 dB to 3.3 dB. The filter has been full-wave analyzed and optimized. The electric size of 0.05 × 0.058 ${\lambda }_g^2$ makes the proposed filter the smallest among the filters compared. The obtained measurement results are in very good agreement with the simulations.

1. Introduction

Tunable bandpass filters are most commonly used as input blocks of radio receivers that cover a wide frequency range and are utilized from the HF (high-frequency) band to microwaves [1]. Their role is selective prefiltering signals coming from the receiving antenna that potentially increase the receiver’s immunity to strong out-of-band interferences and suppress possible parasitic channels resulting from frequency conversion [2]. However, in addition to selectivity, the input filter must maintain high linearity [3,4,5,6,7]; otherwise, in the presence of strong signals, it can be a source of interferences appearing in the operating band caused by intermodulation phenomena.

The linearity of continuously tuned filters can be achieved by the electromechanical tuning through the use of variable inductors and/or capacitors and/or distributed resonators [8,9,10]. This can also be obtained at microwave frequencies by the adoption of magnetically tunable YIG (yttrium iron garnet) resonators [11,12]. Such solutions make possible the achievement of a high linearity, which corresponds to a high IP3 value but at the expense of a significant increase in the size and weight of the filter. Another method of tuning involves the application of varactors [4,5,6,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] or MEMS (micro-electromechanical system) capacitors [3,7,31]. Circuits tuned with MEMS capacitors are characterized by large IP3 values, but they could be applied only for microwave bands due to the small capacitances obtained [3,4]. Varactors, because of the wide range of their initial capacitances, can be used for the continuous tuning of filters in nearly all bands. Varactor-tuned filters offer small size, short tuning time, and low cost [13]. However, their application requires the use of filter structures that not only achieve a desired tuning range but at the same time have high IP3s. Further improvements to varactor-tuned filter designs can bring the incorporation of metamaterials. They can lead to innovative filter designs that achieve high performance in a smaller footprint, enhanced tunability, and improved filters’ overall performance metrics [32,33].

Most varactor-tuned bandpass filters do not have a constant absolute bandwidth (CABW), especially when all varactors are tuned with a single voltage; however, it is possible [25,28,29]. The lack of CABW or bandwidth tuning is a disadvantage if the filters are adopted in devices utilizing frequency conversion, where it is a reason of deterioration of the mirror channel attenuation that occurs within the tuning process due to the fixed distance between the useful and mirror channels. Therefore, a desirable feature of tunable receiver input filters, especially in the case of a wide tuning range, is the ability to adjust the bandwidth. This helps to adopt the bandwidth to the signal receiving condition or to maintain CABW of the filter, which optimizes its impedance matching within the operating band and lets the attenuation of the mirror channel to be kept high. The filter bandwidth adjustment operation is performed by modifying the coupling coefficient that occurs between its resonators [4,5,20,21,25,26,28] and/or adding one or more tuned transmission zeros (TZs) in the transition band(s) [6,14,16,18,19,23,27,34]. As in the case of filter tuning circuits, this can be done electromechanically or electronically [8,9], and the application of varactors influences the linearity of the filter [5,6,28] in a similar way. The simultaneous use of varactors to tune the center frequency of the filter and its bandwidth makes obtaining large IP3 values over a wide tuning range a considerable challenge [5,6,28,35,36,37].

In the design of bandpass microstrip filters, a common approach is to transition from a lumped element prototype to a distributed element network by replacing inductors and capacitors with transmission line segments [21]. This approach enables the construction of new filters by modifying and transforming known lumped structures. The lumped element equivalent circuit models are also adopted in the analysis of complex microwave filters to create their simplified mathematical models [38]. Another approach is the use of the coupling matrix method [5,39], which allows for the direct synthesis of microstrip filters with desired properties that can produce networks previously unknown. In this work, the first approach is adopted, together with initial double-tuned filter transformations, preserving the coupling coefficients between the resonators [35,36,37,40]. This approach also leads to new networks whose parameters are in the bandpass similar to those of the initial filter, but in the transition and stop bands, they can be significantly different.

This paper presents a new double-coupled double-tuned bandpass filter. The filter adopts varactors for center frequency and bandwidth adjustments and has the coupling networks providing transmission zeros in the lower and upper transition bands to shape the response. The upper transmission zero is adjustable, which allows for the control of the bandwidth. The principle of filter operation is explained by means of the circuit model. The results of the parametric analysis showing the impact of individual filter elements on their scattering parameters are presented. An exemplary filter was fabricated using sections of transmission lines and grounded vias that are the inductors’ equivalents. The frequency tuning range of the filter prototype is 300 MHz to 446 MHz. Its electrical size is 0.05 × 0.058 ${\lambda }_g^2$, where ${\lambda }_g$ is the guided wavelength at the lowest operating frequency. The IP3 obtained is not less than 22 dBm within the whole frequency and bandwidth tuning range. The measurements show that the proposed bandwidth variable filter can operate with a constant absolute bandwidth from 35 MHz to 75 MHz for the whole tuning range. The parameters obtained predestine the filter for application in the input blocks of high interference immunity wideband radio receivers.

2. Theory

The filter described in this article is a modification of the double-coupled double-tuned filter with inductive coupling, shown in Figure 1, described in the earlier work of the authors [35]. The input and output resonators are colored green and red, respectively. $L_{A0}$ series inductors were added to the network to increase its out-of-band attenuation, as described in [36,37]. The capacitively coupled filter shown in Figure 2 is obtained by a transformation that involves replacement of the inductors within the network of Figure 1 with the capacitors and simultaneously the capacitors with the inductors in such a way that their reactance modules have the same values for the center frequency $f_0$

$$2\pi f_0{L}_A={\displaystyle \frac{1}{2\pi f_0{C}_B}}\mathrm{and}{\displaystyle \frac{1}{2\pi f_0{C}_A}}=2\pi f_0{L}_B$$

(1)

The result filter shown in Figure 3 has in the coupling branches of the resonators two additional included inductors $L_{C3}$ in series with capacitors $C_{C3}$, which were included to modify the character of the coupling networks in the higher frequency range.

The scattering parameters versus the frequency of the discussed filters are compared in Figure 4, for element values: $L_{A0}=66.85$ nH, $L_{A1}=6.68$ nH, $L_{A2}=49$ nH, $L_{A3}=4.46$ nH, $C_{A1}=3.59$ pF, $C_{B0}=C_{C0}=2.2$ pF, $C_{B1}=C_{C1}=22$ pF, $C_{B2}=C_{C2}=3$ pF, $C_{B3}=C_{C3}=33$ pF, $L_{B1}=L_{C1}=41$ nH, and $L_{C3}=0.8$ nH. All the filters have center frequencies $f_0=415$ MHz. It can be seen that the center frequencies of the A, B, and C filters are close, respectively, to $f_{0A}$, $f_{0B}$, and $f_{0C}$:

$$f_{0A}={\displaystyle \frac{\sqrt{L_{A0}+L_{A2}+L_{A3}}}{2\pi \sqrt{C_{A1}(L_{A2}(L_{A0}+L_{A3})+L_{A1}(L_{A0}+L_{A2}+L_{A3})+L_{3A}(2L_{A0}+L_{A3}))}}}$$

(2)

$$f_{0B}={\displaystyle \frac{\sqrt{C_{B3}(C_{B0}{C}_{B3}+C_{B2}(C_{B0}+C_{B3})+C_{B1}(C_{B3}(C_{B0}+C_{B3})+C_{B2}(C_{B0}+2C_{B3})))}}{2\pi \sqrt{L_{B1}\left(C_{B1}{C}_{B3}(C_{B0}{C}_{B3}+C_{B2}(C_{B0}+C_{B3}))\right)}}}$$

(3)

$$f_{0C}\approx f_{0B}$$

(4)

which are the resonant frequencies of the unloaded resonators used. Filters A and B have mirrored frequency characteristics with respect to the frequency $f_0$ that are the result of the applied transformation (1). In particular, both filters have a single transmission zero in their $s_{21}$ characteristics. For filter A, it is located in the upper transition band and for filter B in the lower. The $s_{21}$ characteristic of filter C has two transmission zeros. The first TZ is placed as in filter B, and its frequency is given by the approximate formula (with $L_{C3}$ omitted).

$$f_1\approx {\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{2C_{C1}+C_{C2}+2C_{C3}}{L_{C1}{C}_{C1}(C_{C2}+2C_{C3})}}}$$

(5)

The second TZ is located in the upper transition band that occurs at the frequency

$$f_2={\displaystyle \frac{1}{2\pi \sqrt{L_{C3}{C}_{C3}}}}$$

(6)

of the series resonance of coupling networks built of $C_{C3}$ and $L_{C3}$. It can be seen that the addition of the second transmission zero in the upper transition band narrows the filter bandpass. The lower transition band remains virtually unchanged.

The capacitors $C_{C0}$ and $C_{C2}$ are elements of impedance transformation networks and are responsible for the source and load matching to the resonators. The influence of the $C_{C0}$ capacitor on the filter scattering parameters is depicted in Figure 5, and the influence of the $C_{C2}$ capacitor is depicted in Figure 6. Decreasing the capacitance of $C_{C0}$ or increasing $C_{C2}$ causes a weaker loading of both resonators within the filter, resulting in ripples in the operating band. Increasing the capacitance of $C_{C0}$ or $C_{C2}$ slightly decreases the operating frequency of the filter. The frequency of the upper transmission zero is independent of $C_{C0}$ and $C_{C2}$ variantions. The frequency of the lower transmission zero is independent of $C_{C0}$ but weakly depends on $C_{C2}$ variations. Increasing $C_{C0}$ or decreasing $C_{C2}$ results in deterioration of out-of-band attenuation.

The capacitor $C_{C1}$ and the inductor $L_{C1}$ are the main elements of both resonators, which have a decisive influence on the filter operating frequency. As can be seen in Figure 7 and Figure 8, increasing the capacitance $C_{C1}$ or inductance $L_{C1}$ decreases the filter center frequency, together with the frequency of the lower transmission zero. An increase in capacitance $C_{C1}$ or a decrease in inductance $L_{C1}$ widens the filter bandwidth. Variations in the values of the components $C_{C1}$ and $L_{C1}$ do not affect the upper transmission zero frequency. Tuning the filter center frequency by $C_{C1}$ or $L_{C1}$ has a significant effect on the matching of its ports within the operating band. Port matching becomes worse as the maximum or minimum operating frequencies of the filter are approached.

The capacitors $C_{C3}$ connected in series with the inductors $L_{C3}$ constitute the coupling networks between the filter resonators. The variation in the impedance of the coupling networks influences the operating bandwidth, as shown in Figure 9 and Figure 10. Within the bandpass, the reactance of the coupling networks is capacitive; thus, increasing their reactance widens the bandwidth. For a particular value of coupling elements $L_{C3}$ and $C_{C3}$, increasing the center frequency narrows the operating bandwidth. At series resonant frequency, the coupling networks reach a minimum impedance, which results in the appearance of an upper transmission zero in $s_{21}$ of the filter. Tuning the upper transmission zero affects the upper transition band and the operating bandwidth of the filter. Changing the coupling coefficient affects the filter ports matching within its bandwidth.

Table 1. The influence of filter elements on the main filter parameters.

Element Variation	Center Frequency	Bandwidth	Insertion Loss	Lower TZ Frequency	Upper TZ Frequency	Low Freq. Attenuation	HIGH Freq. Attenuation
$C_{C0}\uparrow$	↓	↓	↑	—	—	$\downarrow \downarrow \downarrow$	$\downarrow \downarrow$
$C_{C2}\uparrow$	$\downarrow \downarrow$	↑	↓	—	—	↓	↑
$L_{C1}\uparrow$	$\downarrow \downarrow \downarrow$	$\downarrow \downarrow$	—	$\downarrow \downarrow$	—	↓	$\downarrow \downarrow$
$C_{C1}\uparrow$	$\downarrow \downarrow \downarrow$	$\uparrow \uparrow$	—	$\downarrow \downarrow \downarrow$	—	$\uparrow \uparrow$	—
$L_{C3}\uparrow$	↓	↓	—	—	$\downarrow \downarrow \downarrow$	—	$\downarrow \downarrow \downarrow$
$C_{C3}\uparrow$	$\downarrow \downarrow$	$\downarrow \downarrow \downarrow$	—	↓	$\downarrow \downarrow \downarrow$	$\uparrow \uparrow$	$\downarrow \downarrow$

(—): no or very little impact, (↑/$\downarrow )$: weak impact, ($\uparrow \uparrow$/$\downarrow \downarrow$): medium impact, and ($\downarrow \downarrow \downarrow$): strong impact.

summarizes the impact of variations in filter element values on the center frequency, the bandwidth, the insertion loss, the frequencies of lower and upper TZs, and the out-of-band attenuation.

3. Tunable Microstrip Filter

The implementation of lumped filter from Figure 3 as a microstrip network with voltage-tunable center frequency and bandwidth is depicted in Figure 11. Tuning of the filter requires replacing the selected capacitors with the varactors. Implementation in microstrip technology demands replacing inductors with the transmission line segments and grounded vias. The center frequency of the filter is tuned by the voltage V₁ by means of two push-pull connected groups of diodes $D_1$, replacing the capacitors $C_{C1}$. The capacitors $C_{C3}$ responsible for the bandwidth tuning are replaced by two groups of diodes $D_3$, which are connected in parallel and tuned by the voltage V₂. The capacitors $C_{C0}$ and $C_{C2}$ are replaced by the capacitors $C_0$ and $C_2$, respectively. The $C_4$ capacitors are the DC blocking capacitors that allow the correct biasing of diodes $D_3$.

Each $L_{C1}$ inductor is replaced with four transmission line sections: two TL1a and two TL1b. The TL1a and TL1b interconnections are grounded by polarizing resistors $R_3$. Between the TL1b sections are included varactor diode groups $D_1$. The relation between the lengths $l_{1A}$ and $l_{1B}$ of the transmission line sections TL1a and TL1b and $L_{C1}$ inductances is given by the following equation

$$\begin{array}{ccc}\hfill \pi f{L}_{C1}& =Z_{0}tan\left({\displaystyle \frac{2\pi (l_{1A}+l_{1B})}{\lambda }}\right)\hfill & \hfill \mathrm{for}{f}_{\max },{\displaystyle \frac{l_{1A}+l_{1B}}{\lambda }}\ll 1\end{array}$$

(7)

describing the input reactance of the shorted transmission line, where $Z_0$ is the characteristic impedance, and $\lambda$ is the wavelength. The $L_{C3}$ inductors are replaced by parallel groups of grounded vias connected in series with $D_3$ diodes leads. The resistors $R_1$ and $R_2$ are used to polarize the diode groups.

It was assumed that the designed filter should be characterized by a high IP3 value and less than 12 V tuning voltages swing. For this purpose, to obtain the octave frequency coverage with the adoption of the couple of filters, a filter center frequency relative tuning range of about 1.4 was assumed, and BBY66-02V varactors were chosen. According to the datasheet, their capacitance swing is from 12 pF to 68 pF for polarization voltage change from 1 V to 4.5 V, their series resistance is 0.25 Ω, and their lead inductance is 0.6 nH. Due to the application of lumped capacitors, the filter operating band was chosen to be in the lower UHF (ultra high frequency) from 300 MHz to 420 MHz and the bandwidth adjustment from 35 MHz to 75 MHz. The layout of the exemplary filter is shown in Figure 12. The filter was manufactured on an RO4003C substrate with a thickness of 0.8 mm, a relative dielectric constant ${\epsilon }_r$ = 3.38, and a loss tangent equal to 0.0021. In order to reduce the size, the microstrip meander lines were applied. The total length of the individual sections of TL1a and TL1b is 19 mm, and their width is 0.25 mm, which gives the characteristic impedance of 123 Ω.

Before prototyping, the filter layout was full-wave analyzed utilizing Simulia CST Studio Suite 2024 software. To set the assumed tuning range and operating bandwidth, the dimensions of the meander lines and the number and diameter of the vias were adjusted. The necessity of verifying and correcting the length and shape of meander lines is due to the fact that their electrical properties depend on their layout proportion. The application of a meander line introduces parasitic capacitances. Improper proportions of the meander line elements can lead to couplings between their sections, significantly modifying the line impedance and phase properties, which can make it difficult to replace the inductors. For a maximum capacitance of $D_1$ varactors of 80 pF and a minimum bandwidth of 35 MHz, the length of the meander lines was selected to achieve a minimum operating frequency of 300 MHz. The number and diameter of the vias were determined to achieve a filter bandwidth swing from 35 MHz to 75 MHz over the entire assumed filter tuning range. The results of parametric study of filter scattering parameters showing their dependence on grounded vias configuration in $L_{C3}$ inductors are depicted in Figure 13 and Figure 14. The figures present analysis of single and two vias variants of $L_{C3}$ in terms of their diameter. Adopting the study, it was decided to apply two vias with diameters of 0.15 mm each that are equivalent of 1.1 nH. The final dimensions of the layout elements are shown in Figure 12. The values of the lumped elements applied are $C_0=4.7$ pF, $C_2=6.6$ pF, $C_4=100$ pF, and $R_1=R_2=R_3=10$ kΩ.

To better illustrate the filter principles, the results of the full-wave simulation of the current density are shown in Figure 15, Figure 16 and Figure 17 for the center frequency 415 MHz and the lower and upper transmission zero frequencies 324 MHz and 816 MHz, respectively. For the center frequency, both resonators were equally excited, and the signal propagated from the input to the output with minimal attenuation. For lower and upper transmission zero, the resonators were excited unevenly. For the frequency of the upper TZ, the signal at the input and output was shorted to ground by two groups of series resonant circuits consisting of $D_3$ varactors and inductances of vias $L_{C3}$. Due to this, the signal did not emerge from the input to the output of the filter, despite the feeble excitation of current in the input resonator. For the frequency of the lower TZ, the input resonator was more excited than the output one, and signal compensation occurred in the output port, resulting in high filter attenuation.

4. Prototype

The designed filter was fabricated using standard two-layer printed circuit board technology. A photo of the prototype with the SMA connectors installed is shown in Figure 18. Filter scattering parameters were measured with the Agilent N5230A vector network analyzer for selected tuning voltages. Figure 19 shows the comparison of the results of full-wave simulations and measurements of the scattering parameters. As can be seen, the tolerance of the manufacturing process did not cause any visible deterioration of the filter characteristics. Good agreement was obtained in attenuation within the bandwidth and in the positions of the transmission zeros. The differences that can be observed are the attenuation values obtained for the TZs, and the $s_{11}$ values obtained in the bandwidth for which the measurements gave slightly worse results than those predicted in the simulations. The noticeable differences occurring out of band of the filter stem from imperfect models of the lumped elements applied in simulations, mainly from underestimating the equivalent resistances of the varactors and capacitors.

Figure 20 shows the $s_{21}$ characteristics of the filter up to 1.5 GHz obtained from measurements and full-wave simulations. As can be seen, the out-of-band attenuation of the filter was better than 30 dB for the second and third harmonics of the input signals. Above 1.4 GHz, there was a fixed spurious passband attenuated at 12 dB. The differences between measurements and simulations above 900 MHz came from the inaccuracy of the diode and capacitor models applied in simulations.

Figure 21 shows the operating frequency tuning characteristics as a function of the voltage V₁ and bandwidth characteristics as a function of voltage V₂. These characteristics confirm that the filter can be used both as a variable or constant absolute bandwidth filter. The frequency tuning characteristic correspond to the capacitance–voltage characteristic of the applied BBY66-02V varicaps. An increase in voltage V₂ shifted the operating frequency up. The slope of the bandwidth tuning characteristic increased with a decrease in voltage V₁.

Figure 22 shows the frequency response of the scattering parameters of the tuned bandwidth filter. During the measurements, the voltage V₂ was changed from 0.45 V to 1.87 V, while the V₁ voltage was selected from the range of 2.06 V to 0.42 V, respectively, to maintain a constant center frequency of 315 MHz. The bandwidth widens as the capacitance of the $D_3$ varactor groups decreases. The achieved bandwidth swing was from 35 MHz to 75 MHz. The observed attenuation variation connected with tuning of the bandwidth was from 3.3 dB to 1.4 dB. The characteristics for the center frequencies of 375 MHz and 420 MHz are shown in Figure 23 and Figure 24, respectively. For the 375 MHz, the V₂ voltage was changed from 1.08 V to 2.07 V and the V₁ from 4.43 V to 3.48 V, and for the 420 MHz, the V₂ was changed from 0.26 V to 2.27 V and the V₁ from 6.97 V to 5.52 V. It can be seen from the characteristics that the bandwidth are related with the position of the upper transmission zero. During bandwidth tuning, the position of the lower transmission zero remained almost the same, but tuning affected the impedance matching of the ports. When the filter bandwidth was increased, an improvement in port matching was observed as a result of changing the coupling coefficient, together with the frequency of the upper transmission zero. Improvement in the port matching was observed up to the bandwidth of about 60 MHz. For wider bandwidths, two minima of the $s_{11}$ parameter in the bandpass, separated by a local maximum occurring near the center frequency, started to appear, which increased $s_{11}$ to about −10 dB for the highest operating frequency.

The proposed filter allows for simultaneous tuning of the center frequency and bandwidth to achieve a CABW within its tuning range. The scattering parameters versus the frequency characteristics of the CABW filter for 50 MHz bandwidth are shown in Figure 25. During the measurements, the center frequency of the filter was increased by decreasing the capacitance of the $D_1$ varactors. As shown in Figure 8, decreasing their capacitance simultaneously resulted in narrowing the bandwidth. Therefore, to keep the bandwidth constant, the capacitance of the $D_3$ varactors was lowered, because decreasing capacitance of $D_3$ widened the bandwidth, as shown in Figure 10. The tuning range from 300 MHz to 446 MHz was achieved for the $D_1$ control voltage swing from 0.26 V to 10.1 V. The shape of all characteristics of the presented filter was almost insensitive to the tuning of their center frequency. The $s_{21}$ was flat within the bandwidth. The attenuation was about 1.7 dB, and good impedance matching of the filter was observed. For the fixed bandwidth, the center frequency of the filter and the frequencies of the lower and upper transmission zero were tuned concurrently. Together with increasing the center frequency of the filter, the values of $s_{21}$ for the frequency of the lower transmission zero increased from −68 dB to −51 dB, whereas for the upper transmission zero, it remained around −50 dB.

The IP3 measurements for the viariable center frequency and bandwidth are shown in Figure 26. The IP3 was measured with two HP 8665B signal generators, a directional combiner, and Agilent MXA N9020A signal analyzer. Measurements were carried out with a two-tone offset of 1 MHz and a signal level of −10 dBm at the combiner output. They were performed for a fixed filter bandwidth of 50 MHz in the frequency range 300 MHz to 446 MHz and for a variable bandwidth for frequencies of 315 MHz, 375 MHz and 420 MHz. It could be observed that IP3 values greater than 22 dBm were achieved throughout the tuning range for any bandwidth.

5. Discussion

Table 2. Comparison between the proposed and the reference filters.

Ref.	f (GHz)	${\mathit{f}}_{\mathbf{max}}/{\mathit{f}}_{\mathbf{min}}$	BW (MHz)	IL (dB)	NTZs	NCV	Size 1 (${\mathit{\lambda }}_{\mathit{g}}^2$)	IP3 (dBm)
[15]	0.58–1.22	2.1	65–180	1.8–4.6	2	3	0.0096	17.6
[16]	1.27–1.6	1.26	65–116	1.5–3.4	2	3	0.0176	14–21
[4]	0.7–1	1.43	60–150	1.5–7	-	3	0.027	8–20
[5]	0.95–1.48	1.55	114–120	3.5–4.4	2	2	-	25.2–34.9
[25]	1.15–2	1.74	115 ± 4	2.4–3.6	2	1	0.015	15.2–25.2
[6]	1.2–1.58	1.31	133 ± 1	<3	2	2	0.13	19.8–32.3
[29]	0.76–1.78	2.34	84 ± 14	2–4.5	2	1	0.029	11–20
[28]	0.43–0.72	1.67	75 ± 4	1.31–2.92	2	1	0.0224	22–27.6
[34]	1.4–2	1.42	126–179	<4	2	2	0.034	12.5–28.4
This Work	0.3–0.446	1.45	35–75	1.4–3.3	2	2	0.0027	22–29

¹ Calculated for the lowest operating frequency (given in the second column).

compares the proposed filter with varactor-tuned CABW microstrip filters [5,6,25,28,29] and with tunable bandwidth microstrip filters [4,15,16,34] described in the literature. It provides data on the tuning range, the relative tuning range ($f_{\max }/f_{\min }$), the bandwidth (BW), the insertion loss (IL), the number of transmission zeros (NTZ), the number of control voltages (NCV), the electrical size, and the IP3 of the filters.

All filters operated in the UHF band, and their center frequencies were placed in the 0.3–2 GHz range. The relative tuning range of the filters was placed between 1.26 and 2.34. The majority of them are sub-octave filters as the presented filter. The insertion loss for the compared filters depends on the bandwidth and center frequency. The minimum value of the IL obtained among the compared filters was 1.31 dB for the CABW and 1.5 dB for the tunable bandwidth filters. The maximum obtained insertion loss was 4.5 dB for the fixed bandpass filter and 7 dB for the variable bandpass filter. The proposed filter insertion loss was about 1.4 dB for BW = 75 MHz and 3.3 dB for BW = 35 MHz, which was the lowest value among the tunable bandpass filters compared. The compared fixed bandwidth filters were tuned with single or two voltages, but filters having adjustable bandwidth were tuned with two or three voltages. The proposed filter was tuned with two voltages; the first of them mainly regulated BW and the other the center frequency. All filters, except the filter [4], had a characteristic shaped by two TZs placed in the upper and lower transition bands. For compared filters, their area related to the wavelength, calculated for the lowest operating frequency, varied in the range from 0.0027 ${\lambda }_g^2$ to 0.13 ${\lambda }_g^2$. The presented filter was the smallest of all compared, and the closest in terms of the area to the proposed one was three times larger. The IP3 measured for the compared filters ranged from 8 dBm to 34.9 dBm, with the largest IP3 variation observed within the operating band of 15.9 dB for the [34] filter. The proposed filter was characterized by a minimum IP3 equal to 22 dBm and an IP3 variation of 7 dB, which gave it the second position.

6. Conclusions

This paper describes the variable-bandwidth double-tuned microstrip bandpass filter with capacitive coupling between resonators. Due to the structure of the coupling networks, two transmission zeros were achieved in the $s_{21}$ filter characteristics. The TZs were located on both sides of the bandpass, which significantly increased the $s_{21}$ slopes in the transition bands of the filter. The lower TZ was tuned simultaneously with the operating frequency. The frequency of the upper TZ could be controlled independently, which was used to adjust the filter bandwidth. The example filter was tuned from 300 MHz to 446 MHz. It had a variable bandwidth from 35 MHz to 75 MHz and low in-band insertion loss ranging from 1.4 dB to 3.3 dB. The IP3 obtained was not less than 22 dBm within the whole frequency and bandwidth tuning range. The filter structure has small electrical dimensions of 0.05 × 0.058 ${\lambda }_g^2$ and is easy to manufacture. The parameters obtained make the proposed filter useful for applications in the input blocks of high interference immunity wideband UHF receivers. Future works will focus on the implementation of a similar filter for higher frequencies by replacing lumped capacitors with printed ones.