Filter, high frequency module, and communication device

Abstract

To improve the frequency characteristics of filters.SOLUTION: A filter having a pass band and a stop band includes a first resonator having a first inductor and a first capacitor, and a second resonator having a second inductor and a second capacitor. The resonant frequency of the first resonator is located in the stop band on the low frequency side of the pass band, and the resonant frequency of the second resonator is located in the stop band on the high frequency side of the pass band. The propagation characteristic coefficient γ of the filter is different from 1. γ can be calculated by different calculation methods depending on the configuration and connection relationship of the first resonator and the second resonator.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present disclosure relates to filters. [Background technology]

[0002] The following Patent Document 1 discloses an example of a filter configuration having a plurality of resonant circuits. [Prior art documents] [Patent documents]

[0003] [Patent Document 1] JP 2005-328262 A Summary of the Invention [Problem to be solved by the invention]

[0004] It is desirable to improve the frequency response of filters. [Means for solving the problem]

[0005] A filter according to an embodiment of the present disclosure is a filter having a pass band and a stop band, the filter comprising: a first resonator having a first inductor and a first capacitor; and a second resonator having a second inductor and a second capacitor, wherein a resonant frequency of the first resonator is located within the stop band on a lower frequency side of the pass band, and a resonant frequency of the second resonator is located within the stop band on a higher frequency side of the pass band, and γ defined as a propagation characteristic coefficient of the filter is γ≠1; In a first pattern, the filter includes a first series arm and a second series arm adjacent to the first series arm, the first resonator is located in the first series arm of the first pattern and is a Type 1 first parallel resonator in which the first inductor and the first capacitor are connected in parallel, the second resonator is located in the second series arm of the first pattern and is a Type 1 second parallel resonator in which the second inductor and the second capacitor are connected in parallel, and γ in the first pattern is determined by any one of the following first to fifth calculation methods, [First calculation method] The inductance value of the first inductor is represented as L1 [H], The inductance value of the second inductor is represented as L2 [H], The capacitance value of the first capacitor is represented as C1 [F], When expressed as a capacitance value C2 [F] of the second capacitor, β1=L1 / C1 β2=L2 / C2 β1 and β2 are defined as γ = (β2 / β1) 1 / 2 γ is defined as: [Second calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ = (f2 / f1)·(L2 / L1) γ is defined as: [Third calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], The capacitance value of the first capacitor is represented as C1 [F]; When the capacitance value of the second capacitor is expressed as C2 [F], γ = (f1 / f2)·(C1 / C2) γ is defined as: [4th calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], The reactance value of the first inductor at f1 is represented as X1 [Ω], When the reactance value of the second inductor at f2 is expressed as X2 [Ω], γ=X2 / X1 γ is defined as: [5th ​​calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], Denote the susceptance value of the first capacitor at f1 as B1 [1 / Ω]; When the susceptance value of the second capacitor at f2 is expressed as B2 [1 / Ω], γ=B1 / B2 γ is defined as: In a second pattern, the filter includes a first parallel arm and a second parallel arm adjacent to the first parallel arm, the first resonator is located in the first parallel arm in the second pattern and is a Type 1 first series resonator in which the first inductor and the first capacitor are connected in series, the second resonator is located in the second parallel arm in the second pattern and is a Type 1 second series resonator in which the second inductor and the second capacitor are connected in series, and γ in the second pattern is determined by any of the following sixth to tenth calculation methods, [6th calculation method] The inductance value of the first inductor is represented as L1 [H], The inductance value of the second inductor is represented as L2 [H], The capacitance value of the first capacitor is represented as C1 [F], When expressed as a capacitance value C2 [F] of the second capacitor, β1=L1 / C1 β2=L2 / C2 β1 and β2 are defined as γ = (β2 / β1) 1 / 2 γ is defined as: [7th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ = (f2 / f1)·(L2 / L1) γ is defined as: [8th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], The capacitance value of the first capacitor is represented as C1 [F]; When the capacitance value of the second capacitor is expressed as C2 [F], γ = (f1 / f2)·(C1 / C2) γ is defined as: [9th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], The reactance value of the first inductor at f1 is represented as X1 [Ω], When the reactance value of the second inductor at f2 is expressed as X2 [Ω], γ=X2 / X1 γ is defined as: [10th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], Denote the susceptance value of the first capacitor at f1 as B1 [1 / Ω]; When the susceptance value of the second capacitor at f2 is expressed as B2 [1 / Ω], γ=B1 / B2 γ is defined as: In a third pattern, the filter includes a series arm and a parallel arm adjacent to the series arm, the first resonator is located in the series arm of the third pattern and is a Type 2 first parallel resonator in which the first inductor and the first capacitor are connected in parallel, the second resonator is located in the parallel arm of the third pattern and is a Type 2 second series resonator in which the second inductor and the second capacitor are connected in series, In the third pattern, The resonant frequency of the type 2 first parallel resonator is represented as f1 [Hz], When the resonant frequency of the type 2 second series resonator is expressed as f2 [Hz], β0=2500·(f2 / f1) β0 in the third pattern is defined as In the third pattern, γ is determined by any one of the following 11th to 16th calculation methods, [11th calculation method] The inductance value of the first inductor is represented as L1 [H], When the capacitance value of the second capacitor is expressed as C2 [F], γ=(L1 / C2) / β0 γ is defined as: [12th calculation method] The resonant angular frequency of the type 2 second series resonator is represented as ω2 [rad / s], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ = (ω2 2 ·L1·L2) / β0 γ is defined as: [13th calculation method] The resonant angular frequency of the type 2 first parallel resonator is represented as ω1 [rad / s], The capacitance value of the first capacitor is represented as C1 [F]; When the capacitance value of the second capacitor is expressed as C2 [F], γ={1 / (ω1 2 ·C1·C2)} / β0 γ is defined as: [14th calculation method] The reactance value of the first inductor at a predetermined frequency within the passband is represented as X1 [Ω], When the susceptance value of the second capacitor at the predetermined frequency is expressed as B2 [1 / Ω], γ=(X1 / B2) / β0 γ is defined as: [15th calculation method] The reactance value of the first inductor at f2 is represented as X1 [Ω], When the reactance value of the second inductor at f2 is expressed as X2 [Ω], γ=(X1·X2) / β0 γ is defined as: [16th calculation method] Denote the susceptance value of the first capacitor at f1 as B1 [1 / Ω]; When the susceptance value of the second capacitor at f1 is expressed as B2 [1 / Ω], γ={1 / (B1·B2)} / β0 γ is defined as: In a fourth pattern, the filter includes a series arm and a parallel arm adjacent to the series arm, the first resonator is located in the parallel arm of the fourth pattern and is a Type 2 first series resonator in which the first inductor and the first capacitor are connected in series, and the second resonator is located in the series arm of the fourth pattern and is a Type 2 second parallel resonator in which the second inductor and the second capacitor are connected in parallel, In the fourth pattern, The resonant frequency of the type 2 first series resonator is represented as f1 [Hz], When the resonant frequency of the type 2 second parallel resonator is expressed as f2 [Hz], β0=2500·(f1 / f2) β0 in the fourth pattern is defined as In the fourth pattern, γ is determined by any one of the following 17th to 22nd calculation methods, [17th calculation method] The inductance value of the second inductor is represented as L2 [H], When the capacitance value of the first capacitor is expressed as C1 [F], γ=(L2 / C1) / β0 γ is defined as: [18th calculation method] The resonant angular frequency of the type 2 first series resonator is represented as ω1 [rad / s], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ = (ω1 2 ·L1·L2) / β0 γ is defined as: [19th calculation method] The resonant angular frequency of the type 2 second parallel resonator is represented as ω2 [rad / s], The capacitance value of the first capacitor is represented as C1 [F]; When the capacitance value of the second capacitor is expressed as C2 [F], γ={1 / (ω2 2 ·C1·C2)} / β0 γ is defined as: [20th calculation method] The reactance value of the second inductor at any frequency within the pass band is represented as X2 [Ω], When the susceptance value of the first capacitor at the predetermined frequency is expressed as B1 [1 / Ω], γ=(X2 / B1) / β0 γ is defined as: [21st calculation method] The reactance value of the first inductor at f1 is represented as X1 [Ω], When the reactance value of the second inductor at f1 is expressed as X2 [Ω], γ=(X1·X2) / β0 γ is defined as: [22nd calculation method] Denote the susceptance value of the first capacitor at f2 as B1 [1 / Ω]; When the susceptance value of the second capacitor at f2 is expressed as B2 [1 / Ω], γ={1 / (B1·B2)} / β0 γ is defined as: Effect of the Invention

[0006] According to one aspect of the present disclosure, the frequency characteristics of a filter can be improved. [Brief description of the drawings]

[0007] [Figure 1] 3 shows an example of the configuration of a filter in the first embodiment. [Diagram 2] 3 shows an example of a filter configuration as a comparative example in the first embodiment. [Diagram 3] 4 shows examples of frequency characteristics of the filter in the first embodiment and the filter in the comparative example. [Figure 4] 4 shows an example of frequency characteristics of a filter in a comparative example of the first embodiment. [Diagram 5] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 6] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 7] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 8] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 9] 2 shows an example of a two-port circuit in the first embodiment. [Figure 10] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 11] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 12] An example of αS1 and αS2 is shown diagrammatically. [Figure 13] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 14] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 15] 4 shows another example of the configuration of the filter in the first embodiment. [Figure 16] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 17] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 18] 4 shows an example of frequency characteristics of a filter in the first embodiment. [Figure 19] 4 shows yet another configuration example of the filter in the first embodiment. [Figure 20] 4 shows yet another configuration example of the filter in the first embodiment. [Figure 21] 21 shows an example of the frequency characteristic of the composite filter of FIG. 20. [Figure 22] 13 shows an example of the configuration of a filter in the second embodiment. [Diagram 23] 13 shows an example of a filter configuration as a comparative example in the second embodiment. [Figure 24]6 shows examples of frequency characteristics of a filter in the second embodiment and a filter in a comparative example. [Diagram 25] 13 shows an example of frequency characteristics of a filter in a comparative example of the second embodiment. [Figure 26] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Figure 27] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Figure 28] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Figure 29] 13 shows an example of a two-port circuit in the second embodiment. [Diagram 30] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Diagram 31] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Diagram 32] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Diagram 33] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Diagram 34] 13 shows another example of the configuration of the filter in the second embodiment. [Diagram 35] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Diagram 36] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Figure 37] 6 shows an example of frequency characteristics of a filter in the second embodiment. [Figure 38] 13 shows yet another configuration example of the filter in the second embodiment. [Figure 39] 13 shows yet another configuration example of the filter in the second embodiment. [Diagram 40] 13 shows an example of the configuration of a filter in the third embodiment. [Diagram 41] 13 shows an example of frequency characteristics of a filter in the third embodiment. [Diagram 42] 13 shows an example of frequency characteristics of a filter in the third embodiment. [Diagram 43]13 shows another example of the configuration of the filter in the third embodiment. [Diagram 44] 13 shows an example of frequency characteristics of a filter in the third embodiment. [Diagram 45] 13 shows an example of frequency characteristics of a filter in the third embodiment. [Figure 46] 13 shows an example of frequency characteristics of a filter in the third embodiment. [Figure 47] 13 shows an example of frequency characteristics of a filter in the third embodiment. [Figure 48] 13 shows an example of frequency characteristics of a filter in the third embodiment. [Figure 49] 13 shows an example of frequency characteristics of a filter in the third embodiment. [Figure 50] 13 shows yet another configuration example of the filter in the third embodiment. [Figure 51] 13 shows yet another configuration example of the filter in the third embodiment. [Figure 52] 13 shows an example of the configuration of a filter in the fourth embodiment. [Diagram 53] 13 shows an example of frequency characteristics of a filter in the fourth embodiment. [Figure 54] 13 shows an example of frequency characteristics of a filter in the fourth embodiment. [Figure 55] 13 shows another example of the configuration of the filter in the fourth embodiment. [Figure 56] 13 shows an example of frequency characteristics of a filter in the fourth embodiment. [Figure 57] 13 shows an example of frequency characteristics of a filter in the fourth embodiment. [Figure 58] 13 shows an example of frequency characteristics of a filter in the fourth embodiment. [Figure 59] 13 shows an example of frequency characteristics of a filter in the fourth embodiment. [Figure 60] 13 shows an example of frequency characteristics of a filter in the fourth embodiment. [Figure 61] 13 shows an example of frequency characteristics of a filter in the fourth embodiment. [Figure 62] 13 shows yet another configuration example of the filter in the fourth embodiment. [Figure 63] 13 shows yet another configuration example of the filter in the fourth embodiment. [Figure 64] 13 illustrates a schematic configuration of a communication device according to a fifth embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0008] [Embodiment 1] The first embodiment will be described below. For convenience of explanation, components having the same functions as those described in the first embodiment will be given the same reference numerals in the following embodiments, and the explanations thereof will not be repeated. For the sake of brevity, the explanations of well-known technical matters will also be omitted as appropriate.

[0009] Each component and each numerical value described in this specification is merely an example unless otherwise stated. Therefore, for example, unless otherwise stated, the positional relationship and connection relationship of each component are not limited to the examples in each figure. Also, each figure is not necessarily drawn to scale. In this specification, unless otherwise stated, the notation "A~B" for two numbers A and B means "greater than or equal to A and less than or equal to B."

[0010] In this specification, the term "connected" means "electrically connected" in the context of an electric circuit. Therefore, the term "connected" does not necessarily mean a physical connection. In this specification, the inductance value of a certain inductor L is also written as L. The capacitance value of a certain capacitor C is also written as C.

[0011] In this specification, unless otherwise specified, inductance values ​​are in [H] units, capacitance values ​​are in [F] units, frequency values ​​are in [Hz] units, and angular frequency values ​​are in [rad / s] units. Impedance values ​​and reactance values ​​are in [Ω] units, and admittance values ​​and susceptance values ​​are in B1 [1 / Ω] units. In this specification, the unit [1 / Ω] may also be abbreviated as [S].

[0012] (One configuration example of Filter 1) FIG. 1 shows one configuration example of Filter 1 in Embodiment 1. Filter 1 may be configured to have a passband and a stopband. Filter 1 has a first resonator 11 and a second resonator 12. Also, Filter 1 has a series arm SL.

[0013] The first resonator according to one aspect of the present disclosure may have a first inductor and a first capacitor. The first resonator 11 in the example of FIG. 1 has an inductor L1 and a capacitor C1. The inductor L1 and the capacitor C1 are examples of the first inductor and the first capacitor in Embodiment 1, respectively.

[0014] The second resonator according to one aspect of the present disclosure may have a second inductor and a second capacitor. The second resonator 12 in the example of FIG. 1 has an inductor L2 and a capacitor C2. The inductor L2 and the capacitor C2 are examples of the second inductor and the second capacitor in Embodiment 1, respectively.

[0015] In this specification, the resonance frequency of the first resonator 11 is denoted as f1, and the resonance frequency of the second resonator 12 is denoted as f2. In this specification, it is assumed that f1 is located within the stopband on the low-frequency side with respect to the passband of Filter 1, and f2 is located within the stopband on the high-frequency side with respect to the passband of Filter 1 (see, for example, the example of FIG. 3 shown later). Thus, in this specification, the case where f1 < f2 will be described. As an example, f2 may be 2 to 10 times f1.

[0016] And, in this specification, the low-frequency end frequency and the high-frequency end frequency of the passband of Filter 1 are denoted as fp1 and fp2, respectively. As an example, fp2 may be 1.12 to 4 times fp1. Also, f2 may be 2 times or more fp1 and 3 times or less fp2.

[0017] In this specification, a pattern in which the first resonator and the second resonator are connected in a series arm is referred to as a first pattern. In the first embodiment, the first pattern will be described. As shown in Fig. 1, in the first pattern, the filter 1 may have, as the series arm SL, a first series arm SL1 and a second series arm SL2 adjacent to the first series arm SL1.

[0018] 1, the first resonator 11 is located in the first series arm SL1. In the first resonator 11, an inductor L1 and a capacitor C1 are connected in parallel. The first resonator 11 is also referred to as a type 1 first parallel resonator.

[0019] The second resonator 12 is located in the second series arm SL2. In the second resonator 12, an inductor L2 and a capacitor C2 are connected in parallel. The second resonator 12 is also called a Type 1 second parallel resonator. As described above, FIG. 1 illustrates a filter 1 having two LC parallel resonators.

[0020] Fig. 2 shows a configuration example of a filter as a comparative example in the first embodiment. The filter 1X in Fig. 2 has only an inductor L1 and a capacitor C1 connected in parallel. That is, the filter 1X has only a first resonator 11. The filter 1Y in Fig. 2 has only an inductor L2 and a capacitor C2 connected in parallel. That is, the filter 1Y has only a second resonator 12.

[0021] In this specification, the value obtained by dividing a certain inductance value L by a certain capacitance value C is called the LC ratio. In this specification, the LC ratio is represented as β. Therefore, β=L / C β is defined as follows.

[0022] In this specification, in order to evaluate the performance of a filter according to one aspect of the present disclosure, attention is focused on the propagation characteristic coefficient γ of the filter. As described in each embodiment, γ may be determined by various calculation methods. In the first embodiment, a calculation method of γ in the first pattern will be described. γ in the first pattern may be determined by any one of the following first to fifth calculation methods.

[0023] [First calculation method] In the first calculation method, β1=L1 / C1 β2=L2 / C2 Focus on β1 and β2, which are defined as follows: β1 and β2 are also referred to as the first LC ratio and the second LC ratio, respectively.

[0024] In the first calculation method, γ is determined as the square root of β2 divided by β1 (in other words, 1 / 2 power). That is, γ = (β2 / β1) 1 / 2 …(G-1) γ is defined as:

[0025] [Second calculation method] In the second calculation method, attention is paid to f2 / f1, which is a value obtained by dividing f2 by f1, and L2 / L1, which is a value obtained by dividing L2 by L1. In the second calculation method, γ is determined as the product of f2 / f1 and L2 / L1. That is, γ=(f2 / f1)·(L2 / L1) …(G-2) γ is defined as: [Third calculation method] The third calculation method is a method that is paired with the second calculation method. In the third calculation method, attention is paid to f1 / f2, which is a value obtained by dividing f1 by f2, and C1 / C1, which is a value obtained by dividing C1 by C2. In the third calculation method, γ is determined as the product of f1 / f2 and C1 / L2. That is, γ = (f1 / f2) · (C1 / C2) … (G-3) γ is defined as: [4th calculation method] In the fourth calculation method, attention is paid to the reactance value X1 of the first inductor at the resonant frequency f1 and the reactance value X2 of the second inductor at the resonant frequency f2. In the fourth calculation method, γ is determined as a value obtained by dividing X2 by X1. That is, γ=X2 / X1 …(G-4) γ is defined as:

[0026] [5th ​​calculation method] The fifth calculation method is a method paired with the fourth calculation method. In the fifth calculation method, attention is paid to a susceptance value B1 of a capacitor at a resonance frequency f1 and a susceptance value B2 of a second capacitor at a resonance frequency f2. In the fifth calculation method, γ is determined as a value obtained by dividing B1 by B2. That is, γ=B1 / B2 …(G-5) γ is defined as:

[0027] (Additional information on calculation methods 1 to 5) The resonant angular frequency ω0 of an LC parallel filter formed by connecting an inductor L and a capacitor C in parallel is given by: ω0=(L C) -1 / 2 Therefore, the resonant angular frequency ω1 of the first resonator 11 and the resonant angular frequency ω2 of the second resonator 22 are expressed as follows: ω1=(L1 C1) -1 / 2 ω2=(L2·C2) -1 / 2 As is clear from this, the γ's expressed by the above formulas (G-1) to (G-3) are all mathematically equivalent.

[0028] In addition, X2 / X1 in the fourth calculation method is, X2 / X1=(ω2·L2) / (ω1·L1)=(ω2 / ω1)·(L2 / L1) =(f2 / f1)·(L2 / L1) Therefore, γ in the fourth calculation method is equivalent to γ ​​in the second calculation method.

[0029] In addition, B1 / B2 in the fifth calculation method is B1 / B2=(ω1·C1) / (ω2·C2)=(ω1 / ω2)·(C1 / C2) =(f1 / f2)·(C1 / C2) Therefore, γ in the fifth calculation method is equivalent to γ ​​in the third calculation method.

[0030] From the above, the γ values ​​expressed by the above formulas (G-1) to (G-5) are all mathematically equivalent.

[0031] (Example of frequency characteristics of each filter) Reference numerals 310 to 340 in Fig. 3 show examples of the frequency characteristics of filter 1, filter 1X, and filter 1Y. Graphs of reference numerals 320 to 340 are enlarged portions of the graph of reference numeral 310. The horizontal axis in the graph of Fig. 3 represents frequency, and the vertical axis represents transmission.

[0032] In the graph of Fig. 3, the legend "Double" corresponds to filter 1, "Single Low" corresponds to filter 1X, and "Single High" corresponds to filter 1Y. In the example of Fig. 3, the settings are f1=3.3GHz, f2=12GHz, L1=1.0nH, and L2=0.63nH. Therefore, in the example of Fig. 3, β1=0.427, and β2=2.236.

[0033] For example, as shown by the arrows 320 to 340, filter 1 achieves lower loss in the passband compared to filters 1X and 1Y. In this way, by using two LC parallel filters having different LC ratios, it is possible to achieve better frequency characteristics than when one LC parallel filter is used.

[0034] Reference numerals 410 to 440 in Fig. 4 show other examples of frequency characteristics of a filter (e.g., filter 1X or filter 1Y) according to a comparative example. In the example of Fig. 4, α is determined as a value obtained by dividing a certain frequency f by a predetermined frequency f0. That is, α is determined as α=f / f0. In the example of Fig. 4, f0 is the resonant frequency of the filter.

[0035] The horizontal axis of the graph 410 represents logα, and the vertical axis represents the transmission coefficient. In this specification, unless otherwise specified, the base of log is 10. The graph 420 is an enlarged version of the graph 410 in the vicinity of logα=0. As shown in the graphs 410 to 420, the transmission coefficient at a certain α may vary depending on β. Δlogα in the graph 420 represents the absolute value of the difference between two values ​​of logα at which the transmission coefficient is −10 dB.

[0036] In the graph of reference numeral 430, the horizontal axis represents β, and the vertical axis represents the −10 dB cutoff frequency fractional bandwidth. Δlogα It is defined as -1. As shown in the graph, the -10 dB cutoff frequency fractional bandwidth can vary depending on β. Specifically, it was confirmed that the -10 dB cutoff frequency fractional bandwidth tends to increase with an increase in β.

[0037] The horizontal axis of the graph of reference numeral 440 represents β, and the vertical axis represents the −0.2 dB cutoff frequency ratio. The −0.2 dB cutoff frequency ratio is determined as the value of α at which the transmission coefficient is −0.2 dB. The legend “1LC(Low)” in the graph of reference numeral 440 corresponds to filter 1X, and “1LC(High)” corresponds to filter 1Y.

[0038] As shown in the graph, the -0.2 dB cutoff frequency ratio may vary depending on β. Specifically, in filter 1X, it was confirmed that the -0.2 dB cutoff frequency ratio tends to increase as β increases. On the other hand, in filter 1Y, it was confirmed that the -0.2 dB cutoff frequency ratio tends to decrease as β increases.

[0039] Reference numerals 510 to 540 in Fig. 5 show another example of the frequency characteristics of the filter 1. Fig. 5 is a diagram paired with Fig. 4. In the example of Fig. 5, β1=β2 is set, and γ=1. As shown in the graph of reference numeral 510, in the filter 1, two different attenuation poles are obtained due to the presence of two LC filters.

[0040] In the filter 1, f0 is defined as the center frequency of the first resonator and the second resonator. In the example of the first embodiment, the center frequency of the filter 1 is defined as the geometric mean of f1 and f2. Therefore, in the example of the first embodiment, f0 = (f1 f2) 1 / 2 Thus, f0 in filter 1 is defined as:

[0041] The legend "2LC" in 530 corresponds to filter 1. In the graphs 530 and 540, the data for "1LC" described above is plotted for comparison with filter 1.

[0042] Figures 6 and 7 correspond to Figure 5. In the example of Figure 6, β1=β2 / 1.5 2 In the example of FIG. 7, β1=β2 / 2 2 where γ=2. As shown in FIGS. 5 to 7, the frequency characteristics of the filter 1 can differ depending on γ.

[0043] Reference numerals 810 to 840 in Fig. 8 show yet another example of the frequency characteristics of the filter 1. In the example of Fig. 8, α in the first resonator is represented as α1, and α in the second resonator is represented as α2. α1 and α2 are expressed as follows: α1=f1 / f0 α2=f2 / f0 In this specification, the value obtained by dividing α2 by α1 is represented as δ. That is, δ=α2 / α1 δ is defined as: δ is also called the resonant frequency ratio.

[0044] The horizontal axis of the graph 810 represents logα, and the vertical axis represents the transmission coefficient. The graph 820 is an enlarged version of the graph 810 in the vicinity of logα = 0. As shown in the graphs 810 to 820, the frequency characteristics of the filter 1 can differ depending on δ.

[0045] The horizontal axis of the graph of reference numeral 830 represents δ, and the vertical axis represents the frequency ratio based on α1. The legend "Center" in the graph represents the value obtained by dividing the center frequency f0 of filter 1 by α1. The legend "-0.2 dB Cutoff" represents the value obtained by dividing the -0.2 dB cutoff frequency of filter 1 by α1.

[0046] As shown in the graph of reference numeral 830, both the center frequency and the -0.2 dB cutoff frequency of filter 1 increase with an increase in δ. However, the increase in the -0.2 dB cutoff frequency is more gradual than the increase in the center frequency. From this, it is considered that the increase in the -0.2 dB cutoff frequency is not so remarkable even when the passband of filter 1 is expanded.

[0047] In the graph of reference numeral 840, the horizontal axis represents δ, and the vertical axis represents the -0.2 dB cutoff frequency ratio. In the graph, the -0.2 dB cutoff frequency ratio is defined as the -0.2 dB cutoff frequency divided by f0. As shown in the graph, the -0.2 dB cutoff frequency ratio decreases with an increase in δ. From this, it can be considered that the low loss band of filter 1 expands as the pass band of filter 1 expands.

[0048] (Preliminary study on frequency characteristics of filter 1) To mathematically consider the frequency characteristics of the filter 1, we first refer to the two-port circuit shown at 910 in Figure 9. The circuit portion shown at 910 has an admittance y. The admittance matrix [Y] in the example shown at 910 is given by

number

[0049] The S-parameter matrix [S] corresponding to this admittance matrix [Y] is

number

[0050] In the first embodiment, as an example of the circuit portion indicated by reference numeral 910, an LC parallel circuit indicated by reference numeral 920 in Fig. 9 is considered. Then, attention is paid to S21, which is an S parameter corresponding to the output from port 1, which is an input port, to port 2, which is an output port. S21 is expressed as follows:

number

[0051] From this, the absolute value of S21, |S21|, is

number

[0052] (Study on frequency characteristics of filter 1) The inventors of the present application (hereinafter simply referred to as "the inventors") have mathematically studied the frequency characteristics of the filter 1 of Fig. 1 based on the preliminary study in the example of Fig. 9. An example of the study results will be described below.

[0053] First, S21 in filter 1 is

number

[0054] Therefore, |S21| in filter 1 is

number

[0055] According to equation (6), in filter 1,

number

[0056] In this specification, α that gives the maximum value of |S21| is denoted as αS21max. αS21max in Filter 1 is expressed as follows:

number

[0057] Therefore, αS21max for filter 1 is

number

[0058] An example of αS21max derived by the inventors based on formula (9) is shown in Fig. 10. Specifically, a contour diagram showing the distribution of αS21max is shown in Fig. 10. The horizontal axis in Fig. 10 represents log γ, the vertical axis represents α1, and the height axis represents αS21max.

[0059] In Fig. 10, as an example, the case where f1 = 3.3 GHz and f2 = 12 GHz is focused on. In this case, f0 is approximately 6.29 GHz. Therefore, α1 = 0.524 and α2 = 1.907.

[0060] In the example of FIG. 10, α1p and αp2 are αp1=fp1 / f0 αp2=fp2 / f0 10, fp1=5 GHz, and fp2=7.5 GHz. Therefore, when f0 is the above value, αp1=0.795, and αp2=1.192.

[0061] In the example of Fig. 10, when α1 = 0.524, the relationship αp1 ≦ αS21max ≦ αp2 holds for log γ in the range of -0.274 to 0.364. In other words, when α1 = 0.524, the relationship αp1 ≦ αS21max ≦ αp2 holds for γ in the range of 0.532 to 2.312. Fig. 10 shows that the maximum value of |S21| can be obtained within the passband of filter 1.

[0062] By the way, from the above formula (6),

number

[0063] By rearranging equation (10),

number

[0064] Next, by transforming equation (11),

number

[0065] By raising both sides of equation (12) to the power of 1 / 2,

number

[0066] Then, by expressing the right-hand side of equation (13) without using absolute value symbols,

number

[0067] Next, the left side of equation (14) is expressed in a form that does not use absolute value symbols, and the left side is expanded to obtain

number

[0068] FIG. 11 shows an example of the frequency characteristic of filter 1 derived by the inventors. Specifically, FIG. 11 shows a contour diagram showing the distribution of index values ​​corresponding to the -0.2 dB bandwidth (hereinafter, referred to as BW) of filter 1. BW is an example of a low-loss passband width. In the graph of reference numeral 1110 in FIG. 11, the horizontal axis represents β1, the vertical axis represents γ, and the height axis represents the index value. In the graph of reference numeral 1120 in FIG. 11, the horizontal axis represents α1, the vertical axis represents γ, and the height axis represents the index value.

[0069] The index value in FIG. 11 is given by dividing BW by the above-mentioned δ. That is, the index value is Index value=BW / δ=BW / (α2 / α1)=BW·α1 2 In the example at 1110, a contour plot is shown for each of three different α1 and δ. In the example at 1120, a contour plot is shown for each of two different β1.

[0070] 11, for example, when β=1100, an index value of 0.4 or more is obtained when δ is in the range of approximately 3 to 10. Therefore, when β=1100, it is considered that a low-loss passband width of 40% or more can be obtained in the above range.

[0071] By the way, by substituting αS1 and αS2 for α in the above (12),

number

number

[0072] As an example, αS1 and αS2 may be α corresponding to fp1 and fp2, respectively. Therefore, αS1 may be set to a value equal to αp1, and αS2 may be set to a value equal to αp2. αS2 / αS1 in this case is also referred to as a low-loss band frequency ratio. FIG. 12 shows a schematic example of αS1 and αS2 in this case.

[0073] In this specification, the value obtained by dividing the low-loss band frequency ratio by δ is expressed as δ0. That is, δ0=(αS2 / αS1) / δ=(αS2 / αS1) / (α2 / α1) δ0 is given as δ0, which is also called the passband ratio.

[0074] Here, since the relationship α2=1 / α1 and αS2=1 / αS1 holds, δ0 is δ0=α1 2 / αS1 2 It can also be expressed as:

[0075] In addition, αS2 can also be expressed as αS2 = δ·α1. Therefore, equation (16) becomes:

number

[0076] And from equation (18),

number

[0077] Next, by multiplying both sides of equation (19) by 1 / 2,

number

[0078] Solving equation (20) for γ gives

number

number

[0079] On the other hand,

number

number

[0080] By the way, from the above formula (18),

number

[0081] By solving equation (25) for β1,

number

[0082] Moreover, the above formula (21) is

number

[0083] If the frequency at which the transmission coefficient of filter 1 is maximized between f1 and f2 is represented as fs0, then αS0 is expressed as follows: αs0=fs0 / f0 It can also be expressed as:

[0084] And equation (22) becomes:

number

[0085] Moreover, equation (23) is

number

[0086] And equation (24) becomes:

number

[0087] From this, β1 in equation (26) is

number

[0088] As can be seen from Figs. 13 and 14 described below, the loss of the filter 1 can be reduced with a decrease in β. For example,

number

[0089] Moreover, equation (32) is

number

[0090] Fig. 13 shows an example of a graph showing the relationship between γ and β1 in filter 1. The horizontal axis of the graph represents γ, and the vertical axis represents β1. The downward arrows in the graph in Fig. 13 correspond to the inequality signs in the above-mentioned equation (32) or equation (33).

[0091] In Fig. 13, reference numeral 1310 denotes a graph in the case where δ0 is constant, and reference numeral 1320 denotes a graph in the case where the passband width of the filter 1 is constant. As shown in Fig. 13, the relationship between γ and β1 can vary depending on α1. In the example of Fig. 13 and Fig. 14 described below, |S21|=10 -0.01 This value of |S21| corresponds to -0.2 dB. In the example of the reference numeral 1310, δ0=0.4. In the example of the reference numeral 1420, 1 / αS1 2 = 1.5. Therefore, in the example of the reference numeral 1420, αS1 = 0.8165. This αS1 corresponds to a passband width of 50%.

[0092] FIG. 14 shows various example graphs of γ in filter 1. Reference numeral 1410 in FIG. 14 shows a graph showing an example of the relationship between γ and β1. As shown in the graph, the relationship between γ and β1 may vary depending on α1 and δ0. In the example graph, α1 and δ0 are selected such that β1=2000 when γ=1. In the example graph, the smallest β1 is obtained when α1=0.707 and δ0=0.559.

[0093] 14, reference numeral 1420 indicates a graph showing the relationship between γ and β1 when α1=0.707. As shown in the graph, the relationship between γ and β1 can differ depending on δ0 when α1=0.707.

[0094] Next, based on the results of the graph of reference numeral 1410, the inventors derived, by interpolation, a relational expression between α1 and δ0 that satisfies β1=2000 when γ=1. Reference numeral 1430 in FIG. 14 shows an example of an interpolation expression (approximation expression) derived by the inventors. In the example of FIG. 14, δ0 = 0.7062α1 2 + 0.0392α1 + 0.1778 …(H - 1) That is. By selecting α1 and δ0 that satisfy this interpolation formula (H - 1), it is considered that the loss of filter 1 can be reduced.

[0095] (Another configuration example of the filter in Embodiment 1) FIG. 15 shows another configuration example of the filter in Embodiment 1. The mutual inductance M between inductor L1 and inductor L2 is M = k·(L1·L2) 1 / 2 expressed as. k is the coupling coefficient. For the sake of clarity of explanation, in FIG. 1 described above, the case where inductor L1 and inductor L2 do not have mutual inductance is illustrated. That is, in the example of FIG. 1, it is assumed that k = 0.

[0096] On the other hand, in filter 1A in FIG. 15, inductor L1 and inductor L2 have positive mutual inductance. That is, in filter 1A, 0 < k < 1 and M > 0. Therefore, inductor L1 and inductor L2 in filter 1A increase each other's self - inductance. Thus, inductor L1 and inductor L2 may be in series - aiding coupling.

[0097] Also, in filter 1B, inductor L1 and inductor L2 have negative mutual inductance. That is, in filter 1B, - 1 < k < 0 and M < 0. Therefore, inductor L1 and inductor L2 in filter 1B decrease each other's self - inductance. Thus, inductor L1 and inductor L2 may be in differential coupling.

[0098] FIG. 16 shows an example of the frequency characteristics of filters 1, 1A, and 1B. In the example of FIG. 16, k=0 corresponds to filter 1, k=0.14 corresponds to filter 1A, and k=-0.14 corresponds to filter 1B. In the example of FIG. 16, f1=3.3GHz, f2=12GHz, L1=1.0nH, and L2=0.63nH are set. Therefore, in the example of FIG. 16, γ=2.289.

[0099] In Fig. 16, the graph 1610 shows the relationship between k and S21 in the -10 dB cutoff frequency band. The graphs 1620, 1640, and 1650 show the relationship between frequency and transmission coefficient, respectively. The graph 1630 shows the relationship between frequency and reflection coefficient. The frequency range in the graph 1630 is the same as the frequency range in the graph 1620. As shown in Fig. 16, the frequency characteristics of the filter may vary depending on the coupling coefficient.

[0100] Figures 17 and 18 correspond to Figure 16. In the example of Figure 17, L1 is set to 1.5 nH and γ is set to 1.528. In the example of Figure 18, L1 is set to 2.0 nH and γ is set to 1.148.

[0101] (Still another configuration example of the filter in embodiment 1) 19 and 20 each show another example of the configuration of the filter in the first embodiment. In Fig. 19 and Fig. 20, a composite filter 100 including a filter 1 is illustrated. The composite filter 100 also falls within the category of a filter according to one aspect of the present disclosure. The composite filter 100 has a series arm and a parallel arm.

[0102] 19, the composite filter 100 includes an acoustic wave filter 90A having an acoustic wave resonator Reso. The acoustic wave resonator Reso in the acoustic wave filter 90A is located in a series arm of the composite filter 100. In this manner, the acoustic wave filter 90A may have the acoustic wave resonator Reso as a series resonator. In the example of FIG. 19, the anti-resonance frequency of the acoustic wave resonator Reso may be located between fp1 or fp2 and the pass band of the composite filter 100.

[0103] 20, the composite filter 100 includes an acoustic wave filter 90B. An acoustic wave resonator Reso in the acoustic wave filter 90B is located in a parallel arm of the composite filter 100. In this manner, the acoustic wave filter 90B may have the acoustic wave resonator Reso as a parallel resonator. In the example of FIG. 20, the resonant frequency of the acoustic wave resonator Reso may be between fp1 or fp2 and the passband of the composite filter 100.

[0104] Fig. 21 shows an example of the frequency characteristic of the composite filter 100. Fig. 21 illustrates the frequency characteristic of the composite filter 100 of Fig. 20. In the example of Fig. 21, f1 = 2.355 GHz, f2 = 12.125 GHz. Therefore, f0 = 5.344 GHz, α1 = 0.4407, and α2 = 2.269.

[0105] In the example of Fig. 21, fp1 is set to 5.15 GHz and fp2 is set to 7.125 GHz. Therefore, αp1 is set to 0.963 and αp2 is set to 1.333. In addition, in the example of Fig. 21, C1 is set to 2.85 pF and C2 is set to 0.32 pF. Therefore, γ is set to 1.7297.

[0106] Calculating S21max based on the above numerical values ​​gives S21max=0.8333, which corresponds to 4.453GHz. Calculating β1 based on the above numerical values ​​gives β1=562.3.

[0107] (Additional information about composite filters) In this specification, αp0=fp0 / f0 In this case, the attenuation pole of the acoustic wave resonator Reso may be located within the stopband of the composite filter when the frequency is normalized by αp0.

[0108] As an example, in the first embodiment and the fourth embodiment described later, an attenuation pole of the acoustic wave resonator Reso may be located within the stop band of the composite filter on the side where 1 / γ, the inverse of γ, is located with respect to f0 as the reference value, and γ is located with respect to 1 as the reference value, in the second and third embodiments described later, an attenuation pole of the acoustic wave resonator Reso may be located within the stop band of the composite filter on the side where γ is located with respect to f0 as the reference value, and

[0109] (Supplementary Note on the First Embodiment) As described above, in the first embodiment, γ can be calculated by any one of the first to fifth arithmetic expressions described above. In the first embodiment, the filter may be designed so that γ≠1. When γ≠1, the frequency characteristics of the filter can be improved compared to when γ=1 (see also the second to fourth embodiments described below). As shown in the above-mentioned Figs. 16 to 18, in the first embodiment, when γ>1, the frequency characteristics of the filter can be improved compared to when γ≦1.

[0110] [Embodiment 2] (An example of a configuration for filter 2) 22 shows a configuration example of a filter 2 in the second embodiment. The filter 2 has a parallel arm PL in addition to a series arm SL. The first resonator and the second resonator in the filter 2 are denoted as a first resonator 21 and a second resonator 22, respectively. In this specification, a pattern in which the first resonator and the second resonator are connected to the parallel arm is referred to as a second pattern. In the second embodiment, the second pattern will be described.

[0111] As shown in FIG. 22, in the second pattern, the filter 1 may have, as the parallel arm PL, a first parallel arm PL1 and a second parallel arm PL2 adjacent to the first parallel arm PL1.

[0112] 22, the first resonator 21 is located in the first parallel arm PL1. In the second resonator 22, an inductor L1 and a capacitor C1 are connected in series. The first resonator 21 is also referred to as a Type 1 first series resonator.

[0113] Moreover, the second resonator 22 is located in the second parallel arm PL2. In the second resonator 22, an inductor L2 and a capacitor C2 are connected in series. The second resonator 22 is also called a Type 1 second series resonator. As described above, FIG. 22 illustrates a filter 2 having two LC series resonators.

[0114] Fig. 23 shows a configuration example of a filter as a comparative example to the filter 2. The filter 2X in Fig. 23 has only an inductor L1 and a capacitor C1 connected in series. That is, the filter 2X has only a first resonator 21. The filter 2Y in Fig. 23 has only an inductor L2 and a capacitor C2 connected in parallel. That is, the filter 2Y has only a second resonator 22.

[0115] In the second embodiment, a method for calculating γ in the second pattern will be described. γ in the second pattern may be determined by any one of the following sixth to tenth calculation methods.

[0116] [6th calculation method] The sixth calculation method corresponds to the first calculation method described above. That is, in the sixth calculation method, β1=L1 / C1 β2=L2 / C2 β1 and β2 are defined as γ = (β2 / β1) 1 / 2 …(G-6) γ is defined as:

[0117] [7th calculation method] The seventh calculation method corresponds to the above-mentioned second calculation method. That is, in the seventh calculation method, γ = (f2 / f1) · (L2 / L1) … (G-7) γ is defined as:

[0118] [8th calculation method] The eighth calculation method corresponds to the above-mentioned third calculation method. That is, in the eighth calculation method, γ=(f1 / f2)·(C1 / C2) …(G-8) γ is defined as:

[0119] [9th calculation method] The ninth calculation method corresponds to the fourth calculation method described above. That is, in the ninth calculation method, γ=X2 / X1 …(G-9) γ is defined as:

[0120] [10th calculation method] The tenth calculation method corresponds to the fifth calculation method described above. That is, in the tenth calculation method, γ=B1 / B2 …(G-10) γ is defined as:

[0121] (Additional information on calculation methods 6 to 10) The resonant angular frequency ω0 of an LC series filter formed by connecting an inductor L and a capacitor C in series is also given by ω0=(L C) -1 / 2 Therefore, the resonant angular frequency ω1 of the first resonator 21 and the resonant angular frequency ω2 of the second resonator 22 are also expressed as follows: ω1=(L1 C1) -1 / 2 ω2=(L2·C2) -1 / 2 Therefore, as can be understood from the explanation in the first embodiment, γ expressed by the above formulas (G-6) to (G-10) are all mathematically equivalent.

[0122] (Example of frequency characteristics of each filter) FIG. 24 shows an example of the frequency characteristics of each of filter 2, filter 2X, and filter 2Y. FIG. 24 corresponds to FIG. 3. The legend "Double" in the graph of FIG. 24 corresponds to filter 2, "Single Low" corresponds to filter 2X, and "Single High" corresponds to filter 2Y. In the example of FIG. 24, f1=3.3GHz, f2=12GHz, L1=5.0nH, and L2=1.0nH are set. Therefore, in the example of FIG. 24, β1=10.81, and β2=5.236.

[0123] For example, as shown by the arrows 2420 to 2440 in Fig. 24, filter 2 achieves lower loss in the pass band compared to filters 2X and 2Y. In this way, by using two LC series filters having different LC ratios, it is possible to achieve better frequency characteristics than when one LC series filter is used.

[0124] Fig. 25 shows another example of frequency characteristics of a filter (e.g., filter 1X or filter 1Y) according to a comparative example of embodiment 2. Fig. 25 corresponds to Fig. 4. The legend "1LC(Low)" in the graph of reference numeral 2540 in Fig. 25 corresponds to filter 2X, and "1LC(High)" corresponds to filter 2Y.

[0125] 26 to 28 show examples of frequency characteristics of the filter 1. FIGS. 26 to 28 correspond to FIGS. 5 to 7. In the example of FIG. 26, β1 is set to β2, and γ is set to 1. In the example of FIG. 27, β1 is set to β2 / 0.8. 2 In the example of FIG. 28, β1=β2 / 0.5 is set as γ=0.8. 2 and γ=0.5.

[0126] (Preliminary study on frequency characteristics of filter 2) In order to mathematically consider the frequency characteristics of filter 2, first refer to the two-port circuit shown in FIG. 29 as reference numeral 2910. The circuit portion shown in reference numeral 2910 has an impedance z. The impedance matrix [Z] in the example of reference numeral 2910 is given by

number

[0127] The S-parameter matrix [S] corresponding to this impedance matrix [Z] is

number

[0128] In the second embodiment, a case will be considered in which the circuit portion indicated by reference numeral 2910 in Fig. 29 is an LC series circuit indicated by reference numeral 2920. S21 in the example of Fig. 29 is as follows:

number

[0129] From this, |S21| in the example of FIG. 29 is

number

[0130] (Study on frequency characteristics of filter 2) The inventors have mathematically studied the frequency characteristics of the filter 2 in Fig. 22 based on the preliminary study in the example of Fig. 29. An example of the study results will be described below.

[0131] First, S21 in filter 2 is

number

[0132] Therefore, |S21| in filter 2 is

number

[0133] According to equation (39), for filter 2,

number

[0134] Therefore, αS21max in filter 2 is

number

[0135] Therefore, αS21max for filter 2 is

number

[0136] FIG. 30 shows an example of αS21max derived by the inventors based on equation (42). FIG. 30 corresponds to FIG. 10. In the example of FIG. 30, when α1=0.524, the relationship αp1≦αS21max≦αp2 holds for log γ in the range of −0.364 to 0.274. In other words, the relationship αp1≦αS21max≦αp2 holds for γ in the range of 0.433 to 1.879. FIG. 30 shows that the maximum value of |S21| can be obtained within the passband of filter 2.

[0137] By the way, from the above equation (39),

number

[0138] By rearranging equation (43),

number

[0139] Next, by rearranging equation (44),

number

[0140] By multiplying both sides of equation (45) by 1 / 2,

number

[0141] Then, by expressing the right-hand side of equation (46) without using absolute value symbols,

number

[0142] Next, the left side of equation (47) is expressed without using absolute value symbols, and the left side is expanded to obtain

number

[0143] Fig. 31 shows an example of the frequency characteristics of filter 2 derived by the inventors. Fig. 31 corresponds to Fig. 11. As shown in Fig. 11, for example, when β=8000, an index value of 0.4 or more is obtained when δ is in the range of approximately 3 to 10. Therefore, when β=8000, it is considered that a low-loss passband width of 40% or more can be obtained in the above range.

[0144] By the way, by substituting αS1 and αS2 for α in the above (45),

number

number

[0145] Equation (49) is

number

[0146] And from equation (51),

number

[0147] Next, by multiplying both sides of equation (52) by 1 / 2,

number

[0148] Solving equation (53) for γ gives

number

number

[0149] On the other hand,

number

number

[0150] By the way, from the above equation (51),

number

[0151] By solving equation (58) for β1,

number

[0152] Moreover, equation (54) is

number

[0153] And equation (55) becomes:

number

[0154] Moreover, equation (56) is

number

[0155] And equation (57) becomes:

number

[0156] From this, β1 in equation (59) is

number

[0157] As can be seen from Figs. 32 to 33 described below, the loss of the filter 2 can be reduced with an increase in β. For this reason, for example,

number

[0158] Moreover, equation (65) is

number

[0159] An example of a graph showing the relationship between γ and β1 in filter 1 is shown in Fig. 32. Fig. 32 corresponds to Fig. 13. The upward arrow in the graph in Fig. 32 corresponds to the inequality sign in the above-mentioned equation (65) or equation (66).

[0160] FIG. 33 shows examples of various graphs for γ in the filter 2. FIG. 33 corresponds to FIG. 14. Reference numeral 3310 in FIG. 33 indicates a graph representing an example of the relationship between γ and β1. As shown in the graph, α1 and δ0 are selected such that β1 = 3200 at γ = 1. In the example of the graph, the maximum β1 is obtained when α1 = 0.707 and δ0 = 0.56.

[0161] Based on the results of the graph of reference numeral 3310, the inventors derived an expression for the relationship between α1 and δ0 that satisfies β1 = 3200 at γ = 1 by interpolation. Reference numeral 3330 in FIG. 33 shows an example of the interpolation formula (approximate formula) derived by the inventors. In the example of FIG. 33, δ0 = 0.7128α1 2 + 0.0285α1 + 0.1829 …(H-2) is obtained. By selecting α1 and δ0 that satisfy this interpolation formula (H-2), it is considered that the loss of the filter 2 can be reduced.

[0162] (Another configuration example of the filter in Embodiment 2) FIG. 34 shows another configuration example of the filter in Embodiment 2. FIG. 34 corresponds to FIG. 15. In the filter 2 in FIG. 22 described above, k = 0. On the other hand, in the filter 2A in FIG. 34, 0 < k < 1, and the inductor L1 and the inductor L2 have a positive mutual inductance. In the filter 2B, -1 < k < 0, and the inductor L1 and the inductor L2 have a negative mutual inductance.

[0163] FIGS. 35 to 37 correspond to FIGS. 16 to 18. In the example of FIG. 35, f1 = 3.3 GHz, f2 = 12 GHz, L1 = 5.0 nH, and L2 = 1.0 nH are set. Therefore, in the example of FIG. 35, γ = 0.724. In the example of FIG. 36, L1 = 4.0 nH is set, and γ = 0.905. In the example of FIG. 37, L1 = 3.0 nH is set, and γ = 1.207.

[0164] (Still another configuration example of the filter in the second embodiment) 38 to 39 each show another example of the filter configuration according to the second embodiment. In Fig. 38 to Fig. 39, a composite filter 200 including a filter 2 is illustrated. Fig. 38 to Fig. 39 correspond to Fig. 19 to Fig. 20, respectively. The composite filter 200 in the example of Fig. 38 includes an acoustic wave filter 90A. The composite filter 200 in the example of Fig. 39 includes an acoustic wave filter 90B.

[0165] (Supplementary Note on the Second Embodiment) As described above, in the second embodiment, γ can be calculated by any one of the above-mentioned 6th to 10th arithmetic expressions. As shown in the above-mentioned Figs. 35 to 37, in the second embodiment, when γ<1, the frequency characteristics of the filter can be improved compared to when γ≧1.

[0166] [Embodiment 3] Fig. 40 shows a configuration example of the filter 3 in the third embodiment. A first resonator 31 and a second resonator 32 in Fig. 40 are the first resonator and the second resonator in the filter 3, respectively. In this specification, a pattern in which the first resonator is located in the series arm SL and the second resonator is located in the parallel arm PL adjacent to the series arm SL is referred to as a third pattern. In the third embodiment, the third pattern will be described.

[0167] In the first resonator 31 in the example of Fig. 40, an inductor L1 and a capacitor C1 are connected in parallel. The first resonator 31 is also called a type 2 first parallel resonator. On the other hand, in the second resonator 32 in the example of Fig. 40, an inductor L2 and a capacitor C2 are connected in series. The second resonator 32 is also called a type 2 second series resonator.

[0168] 40 illustrates a filter 3 having one LC parallel resonator and one LC series resonator. In the third embodiment, the resonant frequency f1 of the first resonator 31, which is an LC parallel resonator, is lower than the resonant frequency f2 of the second resonator 32, which is an LC series resonator.

[0169] In the third embodiment, for convenience of explanation, L1 and C1 may be written as Lp and Cp, respectively, and L2 and C2 may be written as Ls and Cs, respectively. Therefore, in the third embodiment, f1 and f2 may be written as fp and fs, respectively, and ω1 and ω2 may be written as ωp and ωs, respectively.

[0170] In the third embodiment, β0=2500·(f2 / f1) β0 is determined as follows. This β0 is also referred to as a reference LC ratio in the third pattern. γ in the third pattern may be determined by any one of the following 11th to 16th calculation methods. In the third pattern, unlike the first and second patterns, γ is determined using β0.

[0171] [11th calculation method] In the eleventh calculation method, attention is paid to L1 / C2, which is a value obtained by dividing L1 by C2. In the eleventh calculation method, γ is determined as a value obtained by dividing this value by β0. That is, γ=(L1 / C2) / β0 …(G-11) γ is defined as:

[0172] [12th calculation method] In the 12th calculation method, ω2 is multiplied by the square of ω2 and L1 and L2. 2 Focus on ·L1·L2. In the 12th calculation method, γ is determined as the value obtained by dividing the product by β0. That is, γ = (ω2 2 ·L1·L2) / β0 …(G-12) γ is defined as:

[0173] [13th calculation method] The 13th calculation method is a method that is paired with the 12th calculation method. In the 13th calculation method, the square of ω1 and the reciprocal of the product of C1 and C2, 1 / (ω1 2 In the 12th calculation method, γ is determined as the value obtained by dividing the reciprocal by β0. That is, γ={1 / (ω1 2 ·C1·C2)} / β0 …(G-13) γ is defined as:

[0174] [14th calculation method] The fourteenth calculation method focuses on X1 / B2, which is the value obtained by dividing the reactance value X1 of the first inductor at an arbitrary frequency within the passband of the filter by the susceptance value B2 of the second capacitor at the same frequency. In the fourteenth calculation method, γ is determined as the value obtained by dividing this value by β0. That is, γ=(X1 / B2) / β0 …(G-14) γ is defined as:

[0175] [15th calculation method] In the fifteenth calculation method, attention is focused on X1·X2, which is the product of the reactance value X1 of the first inductor and the reactance value X2 of the second inductor at f2. In the fifteenth calculation method, γ is determined as the value obtained by dividing this product by β0. That is, γ=(X1·X2) / β0 …(G-15) γ is defined as:

[0176] [16th calculation method] The 16th calculation method is a method that is paired with the 15th calculation method. The 16th calculation method focuses on 1 / (B1·B2), which is the reciprocal of the product of the susceptance value B1 of the first capacitor and the susceptance value B2 of the second capacitor at f1. In the 16th calculation method, γ is determined as the value obtained by dividing the reciprocal by β0. That is, γ={1 / (B1·B2)} / β0 …(G-16) γ is defined as:

[0177] (Additional information on calculation methods 11 to 16) ω2 in the 12th calculation method 2 ·L1·L2 are ω2 2 ·L1·L2={1 / (L2·C2)}·(L1·L2)=L1 / C2 Therefore, the 12th calculation method is equivalent to the 11th calculation method.

[0178] 1 / (ω1 2 ·C1·C2) is 1 / (ω1 2 ·C1·C2)=(L1·C1) / (C1·C2)=L1 / C2 Therefore, the 13th calculation method is equivalent to the 11th calculation method.

[0179] If the angular frequency corresponding to any frequency in the 14th calculation method is expressed as ω, X1 / B2 in the 14th calculation method is X1 / B2=(ω·L1) / (ω·C2)=L1 / C2 Therefore, the 14th calculation method is equivalent to the 11th calculation method.

[0180] X1·X2 in the 15th calculation method is X1·X2=(ω2·L1)·(ω2·L2)=ω2 2 (L1 L2) ={1 / (L2·C2)}·(L1·L2)=L1 / C2 Therefore, the 15th calculation method is equivalent to the 11th calculation method.

[0181] In the 16th calculation method, 1 / (B1 B2) is 1 / (B1·B2)=1 / {(ω1·C1)·(ω1·C2)} =1 / {(ω1 2 ·(C1 ·C2)} =(L1 C1) / (C1 C2)=L1 / C2 Therefore, the 16th calculation method is equivalent to the 11th calculation method.

[0182] From the above, the γ values ​​expressed by the above equations (G-11) to (G-16) are all mathematically equivalent.

[0183] (Study on frequency characteristics of filter 3) S21 in filter 3 is

number

[0184] From this, |S21| is

number

[0185] Bcs is the susceptance value of Cs at f0. Therefore, Bcs in the third embodiment is the susceptance value of the second capacitor. On the other hand, Bcs in the fourth embodiment is the susceptance value of the first capacitor.

[0186] βps is, βps=Lp / Cs=Lp / {Ls / βs}=Lp·Ls / ωs 2 =βp Cp / Cs=1 / {Cp Cs ωp 2} It can be rewritten as: γ=βps / {(Z0 2 fs) / fp}=α12 ·βps / Z0 2 It is.

[0187] According to equation (68), αS21max in filter 3 is

number

[0188] Therefore, αS21max for filter 3 is

number

[0189] Moreover, by modifying the above equation (68),

number

[0190] By rearranging equation (71),

number

[0191] By multiplying both sides of equation (72) by 1 / 2,

number

[0192] Then, by expressing the right-hand side of equation (73) without using absolute value symbols,

number

[0193] Next, the left side of equation (74) is expressed without using absolute value symbols, and the left side is expanded to obtain

number

[0194] By the way, by substituting αS1 and αS2 for α in the above equation (72),

number

[0195] Equation (76) is

number

[0196] And from equation (77),

number

[0197] By multiplying both sides of equation (78) by 1 / 2,

number

[0198] Solving equation (79) for γ gives

number

number

[0199] On the other hand,

number

number

[0200] By the way, from the above equation (77),

number

[0201] By solving equation (84) for Bcs,

number

[0202] Moreover, the above equation (80) is

number

[0203] And equation (81) becomes:

number

[0204] Moreover, equation (82) is

number

[0205] And equation (83) becomes:

number

[0206] From this, Bcs in equation (85) is

number

[0207] As can be seen from Figs. 41 and 42 described below, the loss of the filter 3 can be reduced with a decrease in Bcs. For this reason, for example,

number

[0208] Moreover, equation (91) is

number

[0209] Fig. 41 shows an example of a graph showing the relationship between γ and Bcs in filter 3. Fig. 41 corresponds to Fig. 13. The downward arrow in the graph in Fig. 41 corresponds to the inequality sign in the above-mentioned equation (91) or equation (92).

[0210] FIG. 42 shows examples of various graphs for γ in the filter 3. FIG. 42 corresponds to FIG. 14. Reference numeral 4210 in FIG. 42 indicates a graph representing an example of the relationship between γ and Bcs. As shown in the graph, α1 and δ0 are selected such that Bcs = 0.0125 at γ = 1. In the example of the graph, the minimum Bcs is obtained when α1 = 0.707 and δ0 = 0.559.

[0211] Based on the results of the graph of reference numeral 4210, the inventors derived the relational expression between α1 and δ0 that satisfies Bcs = 0.0125 at γ = 1 by interpolation. Reference numeral 4230 in FIG. 42 shows an example of the interpolation formula (approximate formula) derived by the inventors. In the example of FIG. 42, δ0 = 0.3321α1 2 + 0.6732α1 - 0.083 …(H - 3) is obtained. By selecting α1 and δ0 that satisfy this interpolation formula (H - 3), it is considered that the loss of the filter 3 can be reduced.

[0212] (Another configuration example of the filter in Embodiment 3) FIG. 43 shows another configuration example of the filter in Embodiment 3. FIG. 43 corresponds to FIG. 15. In the filter 3 in FIG. 40 described above, k = 0. On the other hand, in the filter 3A in FIG. 43, 0 < k < 1, and the inductor L1 and the inductor L2 have a positive mutual inductance. In the filter 3B, -1 < k < 0, and the inductor L1 and the inductor L2 have a negative mutual inductance.

[0213] FIGS. 44 to 49 correspond to FIGS. 16 to 18. In the example of FIG. 44, f1 = 3.3 GHz, f2 = 12 GHz, L1 = 0.15 nH, and L2 = 4.0 nH are set. Therefore, in the example of FIG. 44, γ = 0.375.

[0214] In the examples of Figures 45 and 46, only L1 is different from the example of Figure 44. In the example of Figure 45, L1 is set to 0.2 nH and γ is set to 0.500. In the example of Figure 46, L1 is set to 0.3 nH and γ is set to 0.750.

[0215] On the other hand, in the examples of Figures 47 to 49, L1 and L2 are different from the example of Figure 44. In the examples of Figures 47 to 49, L2 is set to 6.0 nH. In the example of Figure 47, L1 is set to 1.0 nH, and γ is 3.752. In the example of Figure 48, L1 is set to 1.5 nH, and γ is 5.628. In the example of Figure 49, L1 is set to 2.0 nH, and γ is 7.504.

[0216] (Still another configuration example of the filter in the third embodiment) 50 to 51 each show another example of the filter configuration according to the third embodiment. In Fig. 50 to Fig. 51, a composite filter 300 including a filter 3 is illustrated. Fig. 50 to Fig. 51 correspond to Fig. 19 to Fig. 20, respectively. The composite filter 300 in the example of Fig. 50 includes an acoustic wave filter 90A. The composite filter 300 in the example of Fig. 51 includes an acoustic wave filter 90B.

[0217] (Supplementary Note on the Third Embodiment) As described above, in the third embodiment, γ can be calculated by any one of the above-mentioned 11th to 16th arithmetic expressions. As shown in the above-mentioned Figs. 44 to 49, in the third embodiment, when γ<1, the frequency characteristics of the filter can be improved compared to when γ≧1.

[0218] [Embodiment 4] Fig. 52 shows a configuration example of the filter 4 in the fourth embodiment. A first resonator 41 and a second resonator 42 in Fig. 52 are the first resonator and the second resonator in the filter 4, respectively. In this specification, a pattern in which the first resonator is located in the parallel arm PL and the second resonator is located in the series arm SL adjacent to the parallel arm PL is referred to as a fourth pattern. In the fourth embodiment, the fourth pattern will be described. The fourth pattern is a pattern that forms a pair with the third pattern.

[0219] In the first resonator 41 in the example of Fig. 52, an inductor L1 and a capacitor C1 are connected in series. The first resonator 41 is also called a type 2 first series resonator. On the other hand, in the second resonator 42 in the example of Fig. 52, an inductor L2 and a capacitor C2 are connected in parallel. The second resonator 42 is also called a type 2 second parallel resonator.

[0220] In the fourth embodiment, the resonant frequency f1 of the first resonator 41, which is an LC series resonator, is lower than the resonant frequency f2 of the second resonator 42, which is an LC parallel resonator. In the fourth embodiment, for convenience of explanation, L1 and C1 may be written as Ls and Cs, respectively, and L2 and C2 may be written as Lp and Cp, respectively. Therefore, in the fourth embodiment, f1 and f2 may be written as fs and fp, respectively, and ω1 and ω2 may be written as ωs and ωp, respectively.

[0221] In embodiment 4, β0=2500·(f1 / f2) β0 is determined as follows. This β0 is also referred to as the reference LC ratio in the fourth pattern. Note that β0 in the fourth pattern is different from β0 in the third pattern. γ in the fourth pattern may be determined by any one of the following 17th to 22nd calculation methods.

[0222] [17th calculation method] The 17th calculation method is a method that is paired with the 11th calculation method. The 17th calculation method focuses on L2 / C1, which is a value obtained by dividing L2 by C1. In the 17th calculation method, γ is determined as a value obtained by dividing this value by β0. That is, γ=(L2 / C1) / β0 …(G-17) γ is defined as:

[0223] [18th calculation method] The 18th calculation method is a method that is paired with the 12th calculation method. In the 18th calculation method, ω1, which is the square of ω1 and the product of L1 and L2, is used. 2 Focus on ·L1·L2. In the 12th calculation method, γ is determined as the value obtained by dividing the product by β0. That is, γ = (ω1 2 ·L1·L2) / β0 …(G-18) γ is defined as:

[0224] [19th calculation method] The 19th calculation method is a method that is paired with the 13th calculation method. In the 19th calculation method, the square of ω2 and the reciprocal of the product of C1 and C2, 1 / (ω2 2 In the 19th calculation method, γ is determined as the value obtained by dividing the reciprocal by β0. That is, γ={1 / (ω2 2 ·C1·C2)} / β0 …(G-19) γ is defined as:

[0225] [20th calculation method] The 20th calculation method is a method that is paired with the 14th calculation method. The 20th calculation method focuses on X2 / B1, which is the value obtained by dividing the reactance value X2 of the second inductor at an arbitrary frequency within the pass band of the filter by the susceptance value B1 of the first capacitor at the same frequency. In the 20th calculation method, γ is determined as the value obtained by dividing this value by β0. That is, γ=(X2 / B1) / β0 …(G-20) γ is defined as:

[0226] [21st calculation method] The 21st calculation method is a method that is paired with the 15th calculation method. The 21st calculation method focuses on X1·X2, which is the product of the reactance value X1 of the first inductor and the reactance value X2 of the second inductor at f1. The 15th calculation method determines γ as the value obtained by dividing this product by β0. That is, γ=(X1·X2) / β0 …(G-21) γ is defined as:

[0227] [22nd calculation method] The 22nd calculation method is a method that is paired with the 16th calculation method. The 22nd calculation method focuses on 1 / (B1·B2), which is the reciprocal of the product of the susceptance value B1 of the first capacitor and the susceptance value B2 of the second capacitor at f2. In the 22nd calculation method, γ is determined as the value obtained by dividing the reciprocal by β0. That is, γ={1 / (B1·B2)} / β0 …(G-22) γ is defined as:

[0228] (Additional information on calculation methods 17 to 22) ω1 in the 18th calculation method 2 ·L1·L2 are ω1 2 ·L1·L2={1 / (L1·C1)}·(L1·L2)=L2 / C1 Therefore, the 18th calculation method is equivalent to the 17th calculation method.

[0229] 1 / (ω2 in the 19th calculation method 2 ·C1·C2) is 1 / (ω2 2 ·C1·C2)=(L2·C2) / (C1·C2)=L2 / C1 Therefore, the 19th calculation method is equivalent to the 11th calculation method.

[0230] If the angular frequency corresponding to any frequency in the 20th calculation method is expressed as ω, X2 / B1 in the 20th calculation method is X2 / B1=(ω·L2) / (ω·C1)=L2 / C1 Therefore, the 20th calculation method is equivalent to the 17th calculation method.

[0231] X1·X2 in the 21st calculation method is X1·X2=(ω1·L1)·(ω1·L2)=ω1 2 (L1 L2) ={1 / (L1·C1)}·(L1·L2)=L2 / C1 Therefore, the 21st calculation method is equivalent to the 17th calculation method.

[0232] In the 22nd calculation method, 1 / (B1 B2) is 1 / (B1·B2)=1 / {(ω2·C1)·(ω2·C2)} =1 / {(ω2 2 ·(C1 ·C2)} =(L2 C2) / (C1 C2)=L2 / C1 Therefore, the 22nd calculation method is equivalent to the 17th calculation method.

[0233] From the above, the γ values ​​expressed by the above equations (G-17) to (G-22) are all mathematically equivalent.

[0234] (Study on frequency characteristics of filter 4) S21 in filter 4 is

number

[0235] From this, |S21| is

number

[0236] According to equation (94), αS21max in filter 4 is

number

[0237] Therefore, αS21max for filter 4 is

number

[0238] Moreover, by modifying the above equation (94),

number

[0239] By rearranging equation (97),

number

[0240] By multiplying both sides of equation (98) by 1 / 2,

number

[0241] Then, by expressing the right hand side of equation (99) without using absolute value symbols,

number

[0242] Next, the left side of equation (100) is expressed in a form that does not use absolute value symbols, and the left side is expanded to obtain

number

[0243] By the way, by substituting αS1 and αS2 for α in the above equation (98),

number

[0244] And from equation (102),

number

[0245] Next, from equation (103),

number

[0246] By multiplying both sides of equation (104) by 1 / 2,

number

[0247] Solving equation (105) for γ gives

number

number

[0248] On the other hand,

number

number

[0249] By the way, from the above equation (103),

number

[0250] By solving equation (110) for Bcs,

number

[0251] Moreover, the above equation (106) is

number

[0252] And, γ in equation (107) is

number

[0253] Moreover, the above equation (108) is

number

[0254] And γ corresponding to equation (109) is

number

[0255] From this, Bcs in equation (111) is

number

[0256] As can be seen from Figs. 53 and 54 described below, the loss of the filter 4 can be reduced with a decrease in Bcs. For this reason, for example,

number

[0257] Moreover, equation (117) becomes

number

[0258] Fig. 53 shows an example of a graph showing the relationship between γ and Bcs in filter 4. Fig. 53 corresponds to Fig. 13. The downward arrow in the graph in Fig. 53 corresponds to the inequality sign in the above-mentioned equation (117) or equation (118).

[0259] FIG. 54 shows examples of various graphs for γ in the filter 4. FIG. 54 corresponds to FIG. 14. Reference numeral 5410 in FIG. 54 indicates a graph showing the relationship between γ and Bcs in the passband tolerance value. As shown in the graph, α1 and δ0 are selected such that Bcs = 0.0125 at γ = 1. In the example of the graph, the minimum Bcs is obtained when α1 = 0.707 and δ0 = 0.5595.

[0260] Based on the results of the graph of reference numeral 5410, the inventors derived the relational expression between α1 and δ0 that satisfies Bcs = 0.0125 at γ = 1 by interpolation. Reference numeral 5430 in FIG. 54 shows an example of the interpolation formula (approximate formula) derived by the inventors. In the example of FIG. 54, δ0 = 0.3304α1 2 + 0.676α1 - 0.0837 …(H-4) That is. By selecting α1 and δ0 that satisfy this interpolation formula (H-4), it is considered that the loss of the filter 4 can be reduced.

[0261] (Another configuration example of the filter in Embodiment 4) FIG. 54 shows another configuration example of the filter in Embodiment 4. FIG. 54 corresponds to FIG. 15. In the filter 4 in FIG. 52 described above, k = 0. On the other hand, in the filter 4A in FIG. 54, 0 < k < 1, and the inductor L1 and the inductor L2 have positive mutual inductance. Also, in the filter 4B, -1 < k < 0, and the inductor L1 and the inductor L2 have negative mutual inductance.

[0262] FIGS. 56 to 61 correspond to FIGS. 16 to 18. In the example of FIG. 56, f1 = 3.3 GHz, f2 = 12 GHz, L1 = 8 nH, and L2 = 0.4 nH are set. Therefore, in the example of FIG. 56, γ = 2.001.

[0263] In the examples of Figures 57 and 58, only L2 is different from the example of Figure 56. In the example of Figure 57, L2 is set to 0.3 nH and γ is set to 1.500. In the example of Figure 58, L2 is set to 0.2 nH and γ is set to 1.000.

[0264] On the other hand, in the examples of Figures 59 to 61, L1 and L2 are different from the example of Figure 56. In the examples of Figures 59 to 61, L1 is set to 5nH. In the example of Figure 59, L2 is set to 0.2nH, and γ is 0.625. In the example of Figure 60, L2 is set to 0.15nH, and γ is 0.469. In the example of Figure 61, L2 is set to 0.1nH, and γ is 0.312.

[0265] (Still another configuration example of the filter in the fourth embodiment) Fig. 62 to Fig. 63 each show another example of the filter configuration according to the fourth embodiment. Fig. 62 to Fig. 63 show a composite filter 400 including a filter 4. Fig. 62 to Fig. 63 correspond to Fig. 19 to Fig. 20, respectively. The composite filter 400 in the example of Fig. 62 includes an acoustic wave filter 90A. The composite filter 400 in the example of Fig. 63 includes an acoustic wave filter 90B.

[0266] (Supplementary Note on the Fourth Embodiment) As described above, in the fourth embodiment, γ can be calculated by any one of the above-mentioned 17th to 22nd arithmetic expressions. As shown in the above-mentioned Figs. 56 to 61, in the fourth embodiment, when γ>1, the frequency characteristics of the filter can be improved compared to when γ≦1.

[0267] [Embodiment 5] 64 illustrates a schematic configuration of a communication device 900 according to the fifth embodiment. The communication device 900 is an application example of an acoustic wave filter according to an aspect of the present disclosure, and performs wireless communication using radio waves. The communication device 900 may include a filter (e.g., filter 1) according to an aspect of the present disclosure. For example, the communication device 900 may include one filter 1 as a transmission filter 109 and another filter 1 as a reception filter 111.

[0268] Moreover, a high-frequency module according to an embodiment of the present disclosure may include a filter according to an embodiment of the present disclosure, and a communication device 900 may include the high-frequency module.

[0269] In the communication device 900, a transmission information signal TIS including information to be transmitted may be modulated and frequency-raised (converted into a high-frequency signal having a carrier frequency) by an RF-IC (Radio Frequency-Integrated Circuit) 153, and converted into a transmission signal TS. A band-pass filter 155 may remove unnecessary components from the TS other than the transmission passband. Next, the TS after removing the unnecessary components may be amplified by an amplifier 157 and input to the transmission filter 109.

[0270] The transmission filter 109 may remove unnecessary components outside the transmission passband from the input transmission signal TS. The transmission filter 109 may output the TS after removing the unnecessary components to the antenna 159 via the antenna terminal. The antenna 159 may convert the TS, which is an electrical signal input to itself, into radio waves as a wireless signal, and transmit the radio waves to the outside of the communication device 900.

[0271] Furthermore, the antenna 159 may convert the received radio wave from the outside into a reception signal RS, which is an electric signal, and input the RS to the reception filter 111 via the antenna terminal. The reception filter 111 may remove unnecessary components outside the reception passband from the input RS. The reception filter 111 may output the reception signal RS after the unnecessary components have been removed to the amplifier 161. The output RS may be amplified by the amplifier 161. The bandpass filter 163 may remove unnecessary components outside the reception passband from the amplified RS. The RS after the unnecessary components have been removed may be frequency-downgraded and demodulated by the RF-IC 153, and converted into a reception information signal RIS.

[0272] The TIS and RIS may be low-frequency signals (baseband signals) containing appropriate information. For example, the TIS and RIS may be analog voice signals or digitized voice signals. The passband of the wireless signal may be set appropriately and may comply with various known standards.

[0273] 〔summary〕 A filter according to a first aspect of the present disclosure is a filter having a pass band and a stop band, comprising: a first resonator having a first inductor and a first capacitor; and a second resonator having a second inductor and a second capacitor, wherein a resonant frequency of the first resonator is located within the stop band on a lower frequency side of the pass band, and a resonant frequency of the second resonator is located within the stop band on a higher frequency side of the pass band, and γ defined as a propagation characteristic coefficient of the filter is γ≠1; In a first pattern, the filter includes a first series arm and a second series arm adjacent to the first series arm, the first resonator is located in the first series arm of the first pattern and is a Type 1 first parallel resonator in which the first inductor and the first capacitor are connected in parallel, the second resonator is located in the second series arm of the first pattern and is a Type 1 second parallel resonator in which the second inductor and the second capacitor are connected in parallel, and γ in the first pattern is determined by any one of the following first to fifth calculation methods, [First calculation method] The inductance value of the first inductor is represented as L1 [H], The inductance value of the second inductor is represented as L2 [H], The capacitance value of the first capacitor is represented as C1 [F], When expressed as a capacitance value C2 [F] of the second capacitor, β1=L1 / C1 β2=L2 / C2 β1 and β2 are defined as γ = (β2 / β1) 1 / 2 γ is defined as: [Second calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ = (f2 / f1)·(L2 / L1) γ is defined as: [Third calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], The capacitance value of the first capacitor is represented as C1 [F]; When the capacitance value of the second capacitor is expressed as C2 [F], γ = (f1 / f2)·(C1 / C2) γ is defined as: [4th calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], The reactance value of the first inductor at f1 is represented as X1 [Ω], When the reactance value of the second inductor at f2 is expressed as X2 [Ω], γ=X2 / X1 γ is defined as: [5th ​​calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], Denote the susceptance value of the first capacitor at f1 as B1 [1 / Ω]; When the susceptance value of the second capacitor at f2 is expressed as B2 [1 / Ω], γ=B1 / B2 γ is defined as: In a second pattern, the filter includes a first parallel arm and a second parallel arm adjacent to the first parallel arm, the first resonator is located in the first parallel arm in the second pattern and is a Type 1 first series resonator in which the first inductor and the first capacitor are connected in series, the second resonator is located in the second parallel arm in the second pattern and is a Type 1 second series resonator in which the second inductor and the second capacitor are connected in series, and γ in the second pattern is determined by any of the following sixth to tenth calculation methods, [6th calculation method] The inductance value of the first inductor is represented as L1 [H], The inductance value of the second inductor is represented as L2 [H], The capacitance value of the first capacitor is represented as C1 [F], When expressed as a capacitance value C2 [F] of the second capacitor, β1=L1 / C1 β2=L2 / C2 β1 and β2 are defined as γ = (β2 / β1) 1 / 2 γ is defined as: [7th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ = (f2 / f1)·(L2 / L1) γ is defined as: [8th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], The capacitance value of the first capacitor is represented as C1 [F]; When the capacitance value of the second capacitor is expressed as C2 [F], γ = (f1 / f2)·(C1 / C2) γ is defined as: [9th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], The reactance value of the first inductor at f1 is represented as X1 [Ω], When the reactance value of the second inductor at f2 is expressed as X2 [Ω], γ=X2 / X1 γ is defined as: [10th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], Denote the susceptance value of the first capacitor at f1 as B1 [1 / Ω]; When the susceptance value of the second capacitor at f2 is expressed as B2 [1 / Ω], γ=B1 / B2 γ is defined as: In a third pattern, the filter includes a series arm and a parallel arm adjacent to the series arm, the first resonator is located in the series arm of the third pattern and is a Type 2 first parallel resonator in which the first inductor and the first capacitor are connected in parallel, the second resonator is located in the parallel arm of the third pattern and is a Type 2 second series resonator in which the second inductor and the second capacitor are connected in series, In the third pattern, The resonant frequency of the type 2 first parallel resonator is represented as f1 [Hz], When the resonant frequency of the type 2 second series resonator is expressed as f2 [Hz], β0=2500·(f2 / f1) β0 in the third pattern is defined as In the third pattern, γ is determined by any one of the following 11th to 16th calculation methods, [11th calculation method] The inductance value of the first inductor is represented as L1 [H], When the capacitance value of the second capacitor is expressed as C2 [F], γ=(L1 / C2) / β0 γ is defined as: [12th calculation method] The resonant angular frequency of the type 2 second series resonator is represented as ω2 [rad / s], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ = (ω2 2 ·L1·L2) / β0 γ is defined as: [13th calculation method] The resonant angular frequency of the type 2 first parallel resonator is represented as ω1 [rad / s], The capacitance value of the first capacitor is represented as C1 [F]; When the capacitance value of the second capacitor is expressed as C2 [F], γ={1 / (ω1 2 ·C1·C2)} / β0 γ is defined as: [14th calculation method] The reactance value of the first inductor at a predetermined frequency within the passband is represented as X1 [Ω], When the susceptance value of the second capacitor at the predetermined frequency is expressed as B2 [1 / Ω], γ=(X1 / B2) / β0 γ is defined as: [15th calculation method] The reactance value of the first inductor at f2 is represented as X1 [Ω], When the reactance value of the second inductor at f2 is expressed as X2 [Ω], γ=(X1·X2) / β0 γ is defined as: [16th calculation method] Denote the susceptance value of the first capacitor at f1 as B1 [1 / Ω]; When the susceptance value of the second capacitor at f1 is expressed as B2 [1 / Ω], γ={1 / (B1·B2)} / β0 γ is defined as: In a fourth pattern, the filter includes a series arm and a parallel arm adjacent to the series arm, the first resonator is located in the parallel arm of the fourth pattern and is a Type 2 first series resonator in which the first inductor and the first capacitor are connected in series, and the second resonator is located in the series arm of the fourth pattern and is a Type 2 second parallel resonator in which the second inductor and the second capacitor are connected in parallel, In the fourth pattern, The resonant frequency of the type 2 first series resonator is represented as f1 [Hz], When the resonant frequency of the type 2 second parallel resonator is expressed as f2 [Hz], β0=2500·(f1 / f2) β0 in the fourth pattern is defined as In the fourth pattern, γ is determined by any one of the following 17th to 22nd calculation methods, [17th calculation method] The inductance value of the second inductor is represented as L2 [H], When the capacitance value of the first capacitor is expressed as C1 [F], γ=(L2 / C1) / β0 γ is defined as: [18th calculation method] The resonant angular frequency of the type 2 first series resonator is represented as ω1 [rad / s], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ = (ω1 2 ·L1·L2) / β0 γ is defined as: [19th calculation method] The resonant angular frequency of the type 2 second parallel resonator is represented as ω2 [rad / s], The capacitance value of the first capacitor is represented as C1 [F]; When the capacitance value of the second capacitor is expressed as C2 [F], γ={1 / (ω2 2 ·C1·C2)} / β0 γ is defined as: [20th calculation method] The reactance value of the second inductor at any frequency within the pass band is represented as X2 [Ω], When the susceptance value of the first capacitor at the predetermined frequency is expressed as B1 [1 / Ω], γ=(X2 / B1) / β0 γ is defined as: [21st calculation method] The reactance value of the first inductor at f1 is represented as X1 [Ω], When the reactance value of the second inductor at f1 is expressed as X2 [Ω], γ=(X1·X2) / β0 γ is defined as: [22nd calculation method] The susceptance value of the first capacitor at f2 is expressed as B1 [1 / Ω]; When the susceptance value of the second capacitor at f2 is expressed as B2 [1 / Ω], γ={1 / (B1·B2)} / β0 γ is defined as:

[0274] In the filter according to the second aspect of the present disclosure, in the first aspect, γ>1 may be satisfied in at least one of the first pattern and the fourth pattern.

[0275] In the filter according to the third aspect of the present disclosure, in the first or second aspect, γ may be smaller than 1 in at least one of the second pattern and the third pattern.

[0276] In the filter according to Aspect 4 of the present disclosure, in any one of Aspects 1 to 3, a resonant frequency of the second resonator may be two or more times and ten or less times the resonant frequency of the first resonator.

[0277] In the filter according to a fifth aspect of the present disclosure, in any one of the first to fourth aspects, the high end frequency of the pass band may be 1.12 times or more and 4 times or less the low end frequency of the pass band.

[0278] In a filter according to a sixth aspect of the present disclosure, when the resonant frequencies of the first resonator and the second resonator in the first pattern or the fourth pattern are represented as f1 and f2, respectively, in any one of the first to fifth aspects, f0 = (f1 f2) 1 / 2 f0 [Hz] may be defined as α1=f1 / f0 α2=f2 / f0 α1 and α2 may be defined as αS21max={(γ·α1+α2) / (α1+γ·α2)} 1 / 2 α may be defined as When the lower end frequency and the upper end frequency of the passband are expressed as fp1 [Hz] and fp2 [Hz], respectively, αp1=fp1 / f0 αp2=fp2 / f0 αp1 and αp2 may be defined as αp1≦αS21max≦αp2 It may be.

[0279] In the filter according to the seventh aspect of the present disclosure, when the resonant frequencies of the first resonator and the second resonator in the second pattern or the third pattern are represented as f1 and f2, respectively, in any one of the first to sixth aspects, f0 = (f1 f2) 1 / 2 f0 [Hz] may be defined as α1=f1 / f0 α2=f2 / f0 α1 and α2 may be defined as αS21max={(α1+γ·α2) / (γ·α1+α2)} 1 / 2 α may be defined as When the lower end frequency and the upper end frequency of the passband are expressed as fp1 [Hz] and fp2 [Hz], respectively, αp1=fp1 / f0 αp2=fp2 / f0 αp1 and αp2 may be defined as αp1≦αS21max≦αp2 It may be.

[0280] In a filter according to an eighth aspect of the present disclosure, in any one of the first to seventh aspects, when the resonant frequencies of the first resonator and the second resonator in the third pattern are represented as f1 [Hz] and f2 [Hz], respectively, f0 = (f1 f2) 1 / 2 f0 [Hz] may be defined as If the frequency at which the transmission coefficient of the filter is maximized is expressed as fs0 [Hz] between f1 and f2, then α1=f1 / f0 αs0=fs0 / f0 α1 and αs0 may be defined as The susceptance value BCs of the second capacitor at f0 [Hz] may satisfy the above-mentioned formula (92) with respect to γ ​​calculated in the third pattern, where |S21|=10 -0.01 Z0=50[Ω] δ0=0.3321α1 2 +0.6732α1-0.083 It is.

[0281] In a filter according to a ninth aspect of the present disclosure, in any one of the first to eighth aspects, when the resonant frequencies of the first resonator and the second resonator in the fourth pattern are represented as f1 [Hz] and f2 [Hz], respectively, f0 = (f1 f2) 1 / 2 f0 [Hz] may be defined as If the frequency at which the transmission coefficient of the filter is maximized is expressed as fs0 [Hz] between f1 and f2, then α1=f1 / f0 αs0=fs0 / f0 α1 and αs0 may be defined as The susceptance value BCs of the first capacitor at f0 [Hz] may satisfy the above-mentioned formula (118) with respect to γ ​​calculated in the fourth pattern, where |S21|=10 -0.01 Z0=50[Ω] δ0=0.3304α1 2 +0.676α1-0.0837 It is.

[0282] In a filter according to a tenth aspect of the present disclosure, in any one of the first to ninth aspects, when the resonant frequencies of the first resonator and the second resonator in the first pattern are expressed as f1 [Hz] and f2 [Hz], respectively, f0 = (f1 f2) 1 / 2 f0 [Hz] may be defined as If the frequency at which the transmission coefficient of the filter is maximized is expressed as fs0 [Hz] between f1 and f2, then α1=f1 / f0 αs0=fs0 / f0 α1 and αs0 may be defined as The β1 in the first pattern may satisfy the above formula (33) with respect to the γ calculated in the first pattern, where |S21|=10 -0.01 Y0=0.02[S] δ0=0.7062α1 2 +0.0392α1+0.1778 It is.

[0283] In a filter according to an eleventh aspect of the present disclosure, in any one of the first to tenth aspects, when the resonant frequencies of the first resonator and the second resonator in the second pattern are represented as f1 [Hz] and f2 [Hz], respectively, f0 = (f1 f2) 1 / 2 f0 [Hz] may be defined as If the frequency at which the transmission coefficient of the filter is maximized is expressed as fs0 [Hz] between f1 and f2, then α1=f1 / f0 αs0=fs0 / f0 α1 and αs0 may be defined as The β1 in the second pattern may satisfy the above formula (66) with respect to the γ calculated in the second pattern, where |S21|=10 -0.01 Z0=50[Ω] δ0=0.7128α1 2 +0.0285α1+0.1829 It is.

[0284] In a filter according to aspect 12 of the present disclosure, in any one of aspects 1 to 11, the resonant frequency of the second resonator may be more than twice the low end frequency of the pass band and may be less than three times the high end frequency of the pass band.

[0285] A thirteenth aspect of the present disclosure relates to a filter in any one of the first to twelfth aspects, wherein the first inductor and the second inductor have a positive mutual inductance.

[0286] In the filter according to a fourteenth aspect of the present disclosure, in any one of the first to twelfth aspects, the first inductor and the second inductor may have a negative mutual inductance.

[0287] The filter of aspect 15 of the present disclosure, in any one of aspects 1 to 14, may further include a series arm and an elastic wave resonator located in the series arm, and the anti-resonance frequency of the elastic wave resonator may be located between the resonant frequency of the first resonator or the resonant frequency of the second resonator and the pass band.

[0288] The filter according to aspect 16 of the present disclosure, in any one of aspects 1 to 14, may further include a parallel arm and an elastic wave resonator located in the parallel arm, and the resonant frequency of the elastic wave resonator may be located between the resonant frequency of the first resonator or the resonant frequency of the second resonator and the pass band.

[0289] In the filter according to aspect 17 of the present disclosure, when the resonant frequencies of the first resonator and the second resonator in aspect 15 or 16 are represented as f1 and f2, respectively, f0 = (f1 f2) 1 / 2 f0 [Hz] may be defined as When the lower end frequency and the upper end frequency of the passband are expressed as fp1 [Hz] and fp2 [Hz], respectively, fp0 = (fp1 fp2) 1 / 2 fp0 [Hz] may be defined as αp0=fp0 / f0 αp0 may be defined as An attenuation pole of the elastic wave resonator may be located within the stop band when the frequency is normalized by αp0.

[0290] In a filter according to an eighteenth aspect of the present disclosure, in any one of the fifteenth to seventeenth aspects, when the resonant frequencies of the first resonator and the second resonator in the first pattern or the fourth pattern are represented as f1 and f2, respectively, f0 = (f1 f2) 1 / 2 f0 [Hz] may be defined as An attenuation pole of the elastic wave resonator may be located within the stopband on a side where the inverse number 1 / γ of γ is located with respect to 1 with respect to f0.

[0291] In a filter according to a nineteenth aspect of the present disclosure, in any one of the fifteenth to eighteenth aspects, when the resonant frequencies of the first resonator and the second resonator in the second pattern or the third pattern are represented as f1 and f2, respectively, f0 = (f1 f2) 1 / 2 f0 [Hz] may be defined as An attenuation pole of the acoustic wave resonator may be located within the stopband on the side where γ is located relative to 1 with respect to f0.

[0292] A high-frequency module according to a twentieth aspect of the present disclosure may include the filter according to any one of the first to nineteenth aspects.

[0293] A communication device according to a twenty-first aspect of the present disclosure may include a filter according to any one of the first to twenty aspects.

[0294] [Additional Notes] The invention according to the present disclosure has been described above based on the drawings and examples. However, the invention according to the present disclosure is not limited to the above-mentioned embodiments. In other words, the invention according to the present disclosure can be modified in various ways within the scope of the present disclosure, and the embodiments obtained by appropriately combining the technical means disclosed in the different embodiments are also included in the technical scope of the invention according to the present disclosure. In other words, it should be noted that a person skilled in the art can easily make various modifications or corrections based on the present disclosure. It should also be noted that these modifications or corrections are included in the scope of the present disclosure. [Explanation of symbols]

[0295] 1~1B Filter (first pattern filter) 11 First resonator (Type 1 first parallel resonator) 12 Second resonator (type 1 second parallel resonator) 2~2B Filter (filter in the second pattern) 21 First resonator (Type 1 first series resonator) 22 Second resonator (type 1 second series resonator) 3~3B Filter (filter in the third pattern) 31 First resonator (Type 2 first parallel resonator) 32 Second resonator (Type 2 first series resonator) 4~4B Filter (filter in the 4th pattern) 41 First resonator (Type 2 first series resonator) 42 Second resonator (Type 2 first parallel resonator) 100~400 Composite filter (filter) 900 Communication Equipment L1 Inductor (first inductor) L2 inductor (second inductor) C1 Capacitor (First Capacitor) C2 Capacitor (Second Capacitor) SL Series arm SL1 First series arm SL2 Second series arm PL parallel arm PL1 First parallel arm PL2 Second parallel arm Reso Elastic wave resonator

Claims

1. A filter having a passband and a stopband, a first resonator having a first inductor and a first capacitor; a second resonator having a second inductor and a second capacitor, a resonant frequency of the first resonator is located within the stop band on the lower frequency side of the pass band, a resonant frequency of the second resonator is located within the stop band on the high frequency side of the pass band, γ defined as a propagation characteristic coefficient of the filter is γ ≠ 1; In the first pattern, the filter includes a first series arm and a second series arm adjacent to the first series arm, the first resonator is located in the first series arm of the first pattern and is a Type 1 first parallel resonator in which the first inductor and the first capacitor are connected in parallel, the second resonator is located in the second series arm of the first pattern and is a Type 1 second parallel resonator in which the second inductor and the second capacitor are connected in parallel, The γ in the first pattern is determined by any one of the following first to fifth calculation methods, [First calculation method] The inductance value of the first inductor is represented as L1 [H], The inductance value of the second inductor is represented as L2 [H], The capacitance value of the first capacitor is represented as C1 [F], When expressed as a capacitance value C2 [F] of the second capacitor, β1=L1 / C1 β2=L2 / C2 β1 and β2 are defined as γ=(β2 / β1) １／２ γ is defined as: [Second calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ=(f2 / f1)・(L2 / L1) γ is defined as: [Third calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], The capacitance value of the first capacitor is represented as C1 [F], When the capacitance value of the second capacitor is expressed as C2 [F], γ=(f1 / f2)・(C1 / C2) γ is defined as: [Fourth calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], The reactance value of the first inductor at f1 is expressed as X1 [Ω], When the reactance value of the second inductor at f2 is expressed as X2 [Ω], γ=X2 / X1 γ is defined as: [Fifth calculation method] The resonant frequency of the Type 1 first parallel resonator is represented as f1 [Hz], The resonant frequency of the type 1 second parallel resonator is represented as f2 [Hz], Denote the susceptance value of the first capacitor at f as B [1 / Ω]; If the susceptance value of the second capacitor at f2 is expressed as B2 [1 / Ω], then γ=B1 / B2 γ is defined as: In the second pattern, the filter includes a first parallel arm and a second parallel arm adjacent to the first parallel arm, the first resonator is located in the first parallel arm of the second pattern and is a Type 1 first series resonator in which the first inductor and the first capacitor are connected in series, the second resonator is located in the second parallel arm of the second pattern and is a Type 1 second series resonator in which the second inductor and the second capacitor are connected in series, The γ in the second pattern is determined by any one of the following sixth to tenth calculation methods, [Sixth calculation method] The inductance value of the first inductor is represented as L1 [H], The inductance value of the second inductor is represented as L2 [H], The capacitance value of the first capacitor is represented as C1 [F], When expressed as a capacitance value C2 [F] of the second capacitor, β1=L1 / C1 β2=L2 / C2 β1 and β2 are defined as γ=(β2 / β1) １／２ γ is defined as: [7th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ=(f2 / f1)・(L2 / L1) γ is defined as: [8th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], The capacitance value of the first capacitor is represented as C1 [F], When the capacitance value of the second capacitor is expressed as C2 [F], γ=(f1 / f2)・(C1 / C2) γ is defined as: [9th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], The reactance value of the first inductor at f1 is expressed as X1 [Ω], When the reactance value of the second inductor at f2 is expressed as X2 [Ω], γ=X2 / X1 γ is defined as: [10th calculation method] The resonant frequency of the Type 1 first series resonator is represented as f1 [Hz], The resonant frequency of the Type 1 second series resonator is represented as f2 [Hz], Denote the susceptance value of the first capacitor at f as B [1 / Ω]; If the susceptance value of the second capacitor at f2 is expressed as B2 [1 / Ω], then γ=B1 / B2 γ is defined as: In the third pattern, the filter includes a series arm and a parallel arm adjacent to the series arm, the first resonator is located in the series arm of the third pattern and is a type-2 first parallel resonator in which the first inductor and the first capacitor are connected in parallel, the second resonator is located in the parallel arm of the third pattern and is a Type 2 second series resonator in which the second inductor and the second capacitor are connected in series, In the third pattern, The resonant frequency of the type-2 first parallel resonator is represented as f1 [Hz], When the resonant frequency of the type 2 second series resonator is expressed as f2 [Hz], β0=2500・(f2 / f1) β0 in the third pattern is defined as The γ in the third pattern is determined by any one of the following 11th to 16th calculation methods, [11th calculation method] The inductance value of the first inductor is represented as L1 [H], When the capacitance value of the second capacitor is expressed as C2 [F], γ=(L1 / C2) / β0 γ is defined as: [12th calculation method] The resonant angular frequency of the type 2 second series resonator is represented as ω2 [rad / s], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ=(ω2) ２ ・L1・L2) / β0 γ is defined as: [13th calculation method] The resonant angular frequency of the type-2 first parallel resonator is represented as ω1 [rad / s], The capacitance value of the first capacitor is represented as C1 [F], When the capacitance value of the second capacitor is expressed as C2 [F], γ={1 / (ω1) ２ ・C1・C2)} / β0 γ is defined as: [14th calculation method] The reactance value of the first inductor at a predetermined frequency within the pass band is represented as X1 [Ω], When the susceptance value of the second capacitor at the predetermined frequency is expressed as B2 [1 / Ω], γ=(X1 / B2) / β0 γ is defined as: [15th calculation method] The reactance value of the first inductor at f2 is represented as X1 [Ω], When the reactance value of the second inductor at f2 is expressed as X2 [Ω], γ=(X1・X2) / β0 γ is defined as: [16th calculation method] Denote the susceptance value of the first capacitor at f as B [1 / Ω]; When the susceptance value of the second capacitor at f1 is expressed as B2 [1 / Ω], γ={1 / (B1・B2)} / β0 γ is defined as: In the fourth pattern, the filter includes a series arm and a parallel arm adjacent to the series arm, the first resonator is located in the parallel arm of the fourth pattern and is a type-2 first series resonator in which the first inductor and the first capacitor are connected in series, the second resonator is located in the series arm of the fourth pattern and is a Type 2 second parallel resonator in which the second inductor and the second capacitor are connected in parallel, In the fourth pattern, The resonant frequency of the type 2 first series resonator is represented as f1 [Hz], When the resonant frequency of the type-2 second parallel resonator is expressed as f2 [Hz], β0=2500・(f1 / f2) β0 in the fourth pattern is defined as In the fourth pattern, γ is determined by any one of the following 17th to 22nd calculation methods, [17th calculation method] The inductance value of the second inductor is represented as L2 [H], When the capacitance value of the first capacitor is expressed as C1 [F], γ=(L2 / C1) / β0 γ is defined as: [18th calculation method] The resonant angular frequency of the type 2 first series resonator is represented as ω1 [rad / s], The inductance value of the first inductor is represented as L1 [H], When the inductance value of the second inductor is represented as L2 [H], γ=(ω1 ２ ・L1・L2) / β0 γ is defined as: [19th calculation method] The resonant angular frequency of the type-2 second parallel resonator is represented as ω2 [rad / s], The capacitance value of the first capacitor is represented as C1 [F], When the capacitance value of the second capacitor is expressed as C2 [F], γ={1 / (ω2) ２ ・C1・C2)} / β0 γ is defined as: [20th calculation method] The reactance value of the second inductor at an arbitrary frequency within the pass band is represented as X2 [Ω], When the susceptance value of the first capacitor at the predetermined frequency is expressed as B1 [1 / Ω], γ=(X2 / B1) / β0 γ is defined as: [21st calculation method] The reactance value of the first inductor at f1 is represented as X1 [Ω], When the reactance value of the second inductor at f1 is expressed as X2 [Ω], γ=(X1・X2) / β0 γ is defined as: [22nd calculation method] Denote the susceptance value of the first capacitor at f2 as B1 [1 / Ω]; If the susceptance value of the second capacitor at f2 is expressed as B2 [1 / Ω], then γ={1 / (B1・B2)} / β0 γ is defined as: filter.

2. The filter of claim 1 , wherein in at least one of the first and fourth patterns, γ>1.

3. The filter of claim 1 , wherein in at least one of the second and third patterns, γ<1.

4. The filter according to claim 1 , wherein the resonant frequency of the second resonator is equal to or greater than two times and equal to or less than ten times the resonant frequency of the first resonator.

5. 2. The filter according to claim 1, wherein an upper end frequency of the pass band is equal to or greater than 1.12 times and equal to or less than 4 times a lower end frequency of the pass band.

6. When the resonant frequencies of the first resonator and the second resonator in the first pattern and the fourth pattern are represented as f1 and f2, respectively, <h2 style=";text-align:left;direction:ltr">I'm sorry<h2 style=";text-align:left;direction:ltr"> １／２ f0 [Hz] is defined as α1=f1 / f0 α2=f2 / f0 α1 and α2 are defined as αS21max={(γ・α1+α2) / (α1+γ・α2)} １／２ αS21max is defined as When the lower end frequency and the upper end frequency of the pass band are expressed as fp1 [Hz] and fp2 [Hz], respectively, αp1=fp1 / f0 αp2=fp2 / f0 αp1 and αp2 are defined as αp1≦αS21max≦αp2 2. The filter of claim 1 , wherein:

7. When the resonant frequencies of the first resonator and the second resonator in the second pattern and the third pattern are represented as f1 and f2, respectively, <h2 style=";text-align:left;direction:ltr">I'm sorry<h2 style=";text-align:left;direction:ltr"> １／２ f0 [Hz] is defined as α1=f1 / f0 α2=f2 / f0 α1 and α2 are defined as αS21max={(α1+γ・α2) / (γ・α1+α2)} １／２ αS21max is defined as When the lower end frequency and the upper end frequency of the pass band are expressed as fp1 [Hz] and fp2 [Hz], respectively, αp1=fp1 / f0 αp2=fp2 / f0 αp1 and αp2 are defined as αp1≦αS21max≦αp2 2. The filter of claim 1 , wherein:

8. When the resonant frequencies of the first resonator and the second resonator in the third pattern are represented as f1 [Hz] and f2 [Hz], respectively, <h2 style=";text-align:left;direction:ltr">I'm sorry<h2 style=";text-align:left;direction:ltr"> １／２ f0 [Hz] is defined as If the frequency at which the transmission coefficient of the filter is maximized between f1 and f2 is expressed as fs0 [Hz], α1=f1 / f0 αs0=fs0 / f0 α1 and αs0 are defined as The susceptance value BCs of the second capacitor at f0 [Hz] satisfies the following conditional expression regarding γ calculated in the third pattern: [0010] meets the following criteria: Where: |S21|=10 －０．０１ Z0 = 50 [Ω] δ0=0.3321a1 ２ +0.6732a1-0.083 2. The filter of claim 1 , wherein:

9. When the resonant frequencies of the first resonator and the second resonator in the fourth pattern are represented as f1 [Hz] and f2 [Hz], respectively, <h2 style=";text-align:left;direction:ltr">I'm sorry<h2 style=";text-align:left;direction:ltr"> １／２ f0 [Hz] is defined as If the frequency at which the transmission coefficient of the filter is maximized between f1 and f2 is expressed as fs0 [Hz], α1=f1 / f0 αs0=fs0 / f0 α1 and αs0 are defined as The susceptance value BCs of the first capacitor at f0 [Hz] satisfies the following conditional expression regarding γ calculated in the fourth pattern: [0025] meets the following criteria: Where: |S21|=10 －０．０１ Z0 = 50 [Ω] δ0=0.3304a1 ２ +0.676a1-0.0837 2. The filter of claim 1 , wherein:

10. When the resonant frequencies of the first resonator and the second resonator in the first pattern are represented as f1 [Hz] and f2 [Hz], respectively, <h2 style=";text-align:left;direction:ltr">I'm sorry<h2 style=";text-align:left;direction:ltr"> １／２ f0 [Hz] is defined as If the frequency at which the transmission coefficient of the filter is maximized between f1 and f2 is expressed as fs0 [Hz], α1=f1 / f0 αs0=fs0 / f0 α1 and αs0 are defined as β1 in the first pattern satisfies the following conditional expression regarding γ calculated in the first pattern: [0030] meets the following criteria: Where: |S21|=10 －０．０１ Y0=0.02 [S] δ0=0.7062a1 ２ +0.0392a1+0.1778 2. The filter of claim 1 , wherein:

11. When the resonant frequencies of the first resonator and the second resonator in the second pattern are represented as f1 [Hz] and f2 [Hz], respectively, <h2 style=";text-align:left;direction:ltr">I'm sorry<h2 style=";text-align:left;direction:ltr"> １／２ f0 [Hz] is defined as If the frequency at which the transmission coefficient of the filter is maximized between f1 and f2 is expressed as fs0 [Hz], α1=f1 / f0 αs0=fs0 / f0 α1 and αs0 are defined as β1 in the second pattern satisfies the following conditional expression regarding γ calculated in the second pattern: [0045] meets the following criteria: Where: |S21|=10 －０．０１ Z0 = 50 [Ω] δ0=0.7128a1 ２ +0.0285a1+0.1829 2. The filter of claim 1 , wherein:

12. 2. The filter according to claim 1, wherein the resonant frequency of the second resonator is equal to or greater than twice the lower end frequency of the pass band and equal to or less than three times the upper end frequency of the pass band.

13. The filter of claim 1 , wherein the first inductor and the second inductor have a positive mutual inductance.

14. The filter of claim 1 , wherein the first inductor and the second inductor have a negative mutual inductance.

15. The piezoelectric transducer further includes a series arm and an acoustic wave resonator located in the series arm, 2. The filter according to claim 1, wherein an anti-resonance frequency of the acoustic wave resonator is located between a resonant frequency of the first resonator or a resonant frequency of the second resonator and the pass band.

16. The piezoelectric transducer further includes a parallel arm and an elastic wave resonator located in the parallel arm, 2. The filter according to claim 1, wherein a resonant frequency of the acoustic wave resonator is located between a resonant frequency of the first resonator or a resonant frequency of the second resonator and the pass band.

17. When the resonant frequencies of the first resonator and the second resonator are represented as f1 and f2, respectively, <h2 style=";text-align:left;direction:ltr">I'm sorry<h2 style=";text-align:left;direction:ltr"> １／２ f0 [Hz] is defined as When the lower end frequency and the upper end frequency of the pass band are expressed as fp1 [Hz] and fp2 [Hz], respectively, fp0=(fp1・fp2) １／２ fp0 [Hz] is defined as αp0=fp0 / f0 αp0 is defined as 17. The filter according to claim 15, wherein an attenuation pole of the elastic wave resonator is located within the stop band when the frequency is normalized by αp0.

18. When the resonant frequencies of the first resonator and the second resonator in the first pattern and the fourth pattern are represented as f1 and f2, respectively, <h2 style=";text-align:left;direction:ltr">I'm sorry<h2 style=";text-align:left;direction:ltr"> １／２ f0 [Hz] is defined as 17. The filter according to claim 15, wherein an attenuation pole of the elastic wave resonator is located within the stop band on a side where the inverse number 1 / γ of γ is located with respect to f0, relative to 1 as a reference.

19. When the resonant frequencies of the first resonator and the second resonator in the second pattern and the third pattern are represented as f1 and f2, respectively, <h2 style=";text-align:left;direction:ltr">I'm sorry<h2 style=";text-align:left;direction:ltr"> １／２ f0 [Hz] is defined as 17. The filter according to claim 15, wherein an attenuation pole of the elastic wave resonator is located within the stop band on a side where γ is located relative to 1 with respect to f0.

20. A high frequency module comprising the filter according to claim 1.

21. A communication device comprising the filter of claim 1.

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