CN117977141A - Band-stop filter with quasi-elliptic response and design method thereof - Google Patents
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Abstract
The invention discloses a band-stop filter with quasi-elliptic response, comprising: the invention also discloses another band-stop filter with quasi-elliptic response, which comprises: the invention further discloses a design method of the band-stop filter with quasi-elliptic response, which can design expected stop band inhibition and electric length under stop band cut-off frequency on the premise of meeting the quasi-elliptic response, thereby obtaining expected stop band bandwidth. The invention has the characteristics of generating enough transmission zero and reflection zero, controlling stop band bandwidth and stop band inhibition and having more compact circuit size.
Description
Technical Field
The invention relates to the technical field of band-stop filters, in particular to a band-stop filter with quasi-elliptic response and a design method thereof.
Background
In the current design of the band-stop filter, the equiripple response is usually obtained by repeatedly debugging impedance parameters, and a structure with a large number of zero points is difficult to process in the mode. In comparison, it is very convenient to design a band reject filter with a specific response using an existing quasi-elliptic or elliptic function. In the prior art, researchers have directly established a relationship between a quasi-elliptic function and a distributed circuit parameter, so that extra optimization and approximation links are avoided, and the impedance parameter of the distributed quasi-elliptic band-stop filter can be accurately calculated, however, in the proposed design method, the band-stop filter can only be designed through the electric length (theta c S11) at the given passband cutoff frequency, passband Return Loss (RL) and transmission zero point (theta TZ) of the stopband, which results in that the designed band-stop filter can only design the expected theta c S11, RL and theta TZ, but cannot design the expected Stopband Rejection (SR) and the electric length (theta c S21) of the stopband cutoff frequency. In fact, in stop band filter design, the latter two design parameters (SR, θ c S21) are more important than the first three design parameters (θ c S11、RL、θTZ).
The existing design method of the band-stop filter can not directly and accurately design the expected stop band inhibition and stop band bandwidth, in addition, most of the existing band-stop filters do not have enough transmission zero points and reflection zero points, and no comprehensive design theory about the increase of the number of the reflection zero points exists, so that the frequency selectivity of the band-stop filter is poor, the structure of most of circuits is not compact, the circuit size is large, and the band-stop filter is not beneficial to being applied to a communication system.
Disclosure of Invention
The invention aims to design and develop a band-stop filter with quasi-elliptic response, which can generate a plurality of transmission zeros and reflection zeros and realize wide stop band performance and good frequency selectivity.
The invention also designs and develops a design method of the band-stop filter with quasi-elliptic response, so that the expected stop band inhibition and the electric length under the stop frequency of the stop band can be designed on the premise of meeting the quasi-elliptic response.
The invention aims to design and develop another band-stop filter with quasi-elliptic response, which can generate a plurality of transmission zeros and reflection zeros, realize wide stop band performance and good frequency selectivity and reduce the size of a circuit.
The invention also designs and develops another design method of the band-stop filter with quasi-elliptic response, so that the expected stop band inhibition and the electric length under the stop frequency of the stop band can be designed on the premise of meeting the quasi-elliptic response.
The technical scheme provided by the invention is as follows:
a band reject filter having a quasi-elliptical response, comprising:
The first port and the second port can be an input port or an output port;
Two open branch node lines connected in parallel to both ends of the first port and the second port;
Two transmission lines connected in series between the first port and the second port;
The ladder impedance branch line is connected in parallel to the centers of the two transmission lines, comprises a first characteristic impedance and a second characteristic impedance, and is close to the centers of the two transmission lines.
A design method of a band-stop filter with quasi-elliptic response uses the band-stop filter with quasi-elliptic response, which comprises the following steps:
Step one, determining any three values of the electric length theta c S11 under the cut-off frequency of a passband, the passband return loss RL, the electric length theta c S21 under the cut-off frequency of the stopband, the expected stopband rejection SR and the electric length theta TZ at the transmission zero point according to a band-stop filter with quasi-elliptic response;
step two, obtaining a circuit transfer function after calculating the odd mode input impedance and the even mode input impedance of the band-stop filter:
in the method, in the process of the invention, As a circuit transfer function of the band reject filter, Z 1 is a characteristic impedance of an open branch line, Z 2 is a characteristic impedance of a transmission line, Z 3 is a first characteristic impedance, Z 4 is a second characteristic impedance, S 11 is a ratio of a reflected voltage wave amplitude of the first port to an incident voltage wave amplitude of the first port, and S 21 is a ratio of a reflected voltage wave amplitude of the second port to an incident voltage wave amplitude of the first port;
And thirdly, restraining the circuit transfer function by using a theoretical quasi-elliptic response transfer function of the band-stop filter to obtain the characteristic impedance of two open-circuit branch lines, the characteristic impedance of two transmission lines and the characteristic impedance of the stepped impedance branch line of the band-stop filter.
A band reject filter having a quasi-elliptical response, comprising:
The first port and the second port can be an input port or an output port;
Two open branch node lines connected in parallel to both ends of the first port and the second port;
two transmission lines connected in series between the first port and the second port;
the two ladder impedance branch node lines are respectively connected in parallel with the centers of the two transmission lines, each ladder impedance branch node line comprises a first characteristic impedance and a second characteristic impedance, the two first characteristic impedances are close to the centers of the two transmission lines, and the two first characteristic impedances and the two second characteristic impedances are respectively coupled with the two transmission lines and the two open branch node lines in a one-to-one correspondence manner.
A design method of a band-stop filter with quasi-elliptic response uses the band-stop filter with quasi-elliptic response, which comprises the following steps:
Step one, determining any two values of the electric length theta c S11 under the cut-off frequency of a passband, the passband return loss RL, the electric length theta c S21 under the cut-off frequency of the stopband, the expected stopband rejection SR and the electric length theta TZ at the transmission zero point according to a band-stop filter with quasi-elliptic response;
step two, obtaining a circuit transfer function after calculating the odd mode input impedance and the even mode input impedance of the band-stop filter:
in the method, in the process of the invention, As a circuit transfer function of the band elimination filter, Z e1 is a coupled line even mode characteristic impedance formed by coupling a second characteristic impedance with an open circuit branch line, Z o1 is a coupled line odd mode characteristic impedance formed by coupling a second characteristic impedance with an open circuit branch line, Z e2 is a coupled line even mode characteristic impedance formed by coupling a first characteristic impedance with a transmission line, Z o2 is a coupled line odd mode characteristic impedance formed by coupling a first characteristic impedance with a transmission line, S 11 is a ratio of a reflected voltage wave amplitude of a first port to an incident voltage wave amplitude of the first port, and S 21 is a ratio of a reflected voltage wave amplitude of the second port to an incident voltage wave amplitude of the first port;
And thirdly, restraining the circuit transfer function by using a theoretical quasi-elliptic response transfer function of the band-stop filter to obtain the characteristic impedance of the coupling line formed by coupling the two ladder impedance branch lines, the two transmission lines and the two open-circuit branch lines of the band-stop filter.
Preferably, the two open-circuited branch lines, the two transmission lines, the two stepped impedance branch lines and the two pairs of coupling structures formed by the two open-circuited branch lines and the two transmission lines each have an electrical length of 90 ° at the center frequency.
Preferably, the ratio of the reflected voltage wave amplitude of the first port to the incident voltage wave amplitude of the first port and the ratio of the reflected voltage wave amplitude of the second port to the incident voltage wave amplitude of the first port satisfy:
in the method, in the process of the invention, Z 0 is the load of the first port or the second port, which is the odd-mode input impedance of the band-stop filter,/>Is the even mode input impedance of the band reject filter and i= I, II, when i=i, the band reject filter according to claim 1 is used and when i=ii, the band reject filter according to claim 3 is used.
Preferably, the odd mode input impedance of the band-stop filter and the even mode input impedance of the band-stop filter satisfy:
or (b)
Where j is an imaginary unit and θ is the electrical length of the open branch node line at the center frequency.
Preferably, the constraint condition in the third step is:
or (b)
In the method, in the process of the invention,Is the theoretical quasi-elliptic response transfer function of the band reject filter, epsilon is the first intermediate parameter.
Preferably, the theoretical quasi-elliptic response transfer function of the band reject filter satisfies:
Where Φ is a second intermediate parameter, ζ is a third intermediate parameter, γ is a fourth intermediate parameter, m i is a fifth intermediate parameter, and when i=i, m I =1, and when i=ii, m II =2.
Preferably, the theoretical quasi-elliptic response transfer function of the band reject filter satisfies:
Where θ D is the electrical length in the stop band at |s 21 |=sr.
The beneficial effects of the invention are as follows:
(1) The band-stop filter with quasi-elliptic response can generate a plurality of transmission zeros and reflection zeros, and achieves wide stop band performance and good frequency selectivity.
(2) The design method of the band-stop filter with the quasi-elliptic response, which is designed and developed by the invention, enables the expected stop band parameters (SR, theta c S21) to be designed on the premise of meeting the quasi-elliptic response.
(3) The other band-stop filter with quasi-elliptic response is designed and developed by the invention, a compact coupling structure is introduced, the design principle of adding an additional reflection zero point is disclosed, and meanwhile, the band-stop filter has very compact circuit size, and the miniaturization design requirement of a circuit is met.
(4) According to the design method of the band-stop filter with the quasi-elliptic response, which is developed by the design method, on the premise of meeting the quasi-elliptic response, expected stop band parameters (SR, theta c S21) can be designed.
Drawings
Fig. 1 is a schematic circuit topology of a band reject filter i with quasi-elliptic response according to the present invention.
Fig. 2 is a schematic circuit topology of a band reject filter ii with quasi-elliptic response according to the present invention.
Fig. 3 is a schematic diagram of general S parameters of the band-stop filter i and the band-stop filter ii according to the present invention.
Fig. 4 is a schematic diagram of an even mode equivalent circuit corresponding to the band reject filter i according to the present invention.
Fig. 5 is a schematic diagram of an odd mode equivalent circuit corresponding to the band reject filter i according to the present invention.
Fig. 6 is a schematic diagram of an even mode equivalent circuit corresponding to the band reject filter ii according to the present invention.
Fig. 7 is a schematic diagram of an odd mode equivalent circuit corresponding to the band reject filter ii according to the present invention.
Fig. 8 is a schematic diagram of S-parameters of a band reject filter i according to an embodiment of the invention.
Fig. 9 is a schematic diagram of S-parameters of a band reject filter ii according to an embodiment of the invention.
Detailed Description
The present invention is described in further detail below to enable those skilled in the art to practice the invention by reference to the specification.
As shown in fig. 1, the band-stop filter i with quasi-elliptic response provided by the present invention includes:
The first port #1, the second port #2, two open circuit branch lines, two transmission lines and a stepped impedance branch line, wherein the first port #1 and the second port #2 can be used as an input port or an output port; the two open branch node lines are connected in parallel at two ends of the first port #1 and the second port #2, and the characteristic impedance is Z 1; the two transmission lines are connected in series between the first port #1 and the second port #2, and the characteristic impedance is Z 2; the stepped impedance branch line is connected in parallel at the center of the two transmission lines, i.e. at the center of the entire circuit structure, and comprises a first characteristic impedance Z 3 and a second characteristic impedance Z 4, the first characteristic impedance Z 3 being close to the center of the two transmission lines.
As shown in fig. 2, the present invention also provides another band-stop filter ii having a quasi-elliptic response, including:
The structure of the first port #1, the second port #2, the two open-circuit branch lines, the two transmission lines and the two ladder impedance branch lines is the same as that of the band-stop filter I, the ladder impedance branch lines in the band-stop filter I are equally divided into two parallel structures and are respectively coupled with the two open-circuit branch lines and the two transmission lines in one-to-one correspondence, namely, the band-stop filter II comprises two pairs of coupling structures, the two first characteristic impedances are close to the centers of the two transmission lines, the two first characteristic impedances and the two second characteristic impedances are respectively coupled with the two transmission lines and the two open-circuit branch lines in one-to-one correspondence, the characteristic impedances are respectively marked as Z e1 (the coupling line even mode characteristic impedance formed by coupling of the second characteristic impedance and the open-circuit branch lines), Z o1 (the coupling line even mode characteristic impedance formed by coupling of the second characteristic impedance and the open-circuit branch lines) and Z e2 (the coupling characteristic impedance formed by coupling of the first characteristic impedance and the coupling mode impedance o2 of the first characteristic impedance and the coupling mode impedance of the first characteristic impedance and the first coupling mode impedance o2.
In this embodiment, the loads of the first port #1 and the second port #2 of the band-stop filter i and the band-stop filter ii are Z 0, and the electrical lengths θ of all the open branch lines, the transmission lines, and the coupling structures at the center frequency are 90 °.
The two band-stop filters with quasi-elliptic response designed and developed by the invention can generate a plurality of transmission zero points and reflection zero points, realize wide stop band performance and good frequency selectivity, particularly the band-stop filter II, introduce a compact coupling structure, disclose the design principle of adding additional reflection zero points, have very compact circuit size, meet the miniaturization design requirement of a circuit and are suitable for realizing the bandwidth exceeding 100 percent.
The invention also provides a design method of the corresponding band-stop filter I and the band-stop filter II, which specifically comprises the following steps:
first, as shown in fig. 3, general S parameters of the band-stop filter i and the band-stop filter ii are as follows: s 11 is the ratio of the reflected voltage wave amplitude of the first port to the incident voltage wave amplitude of the first port, S 21 is the ratio of the reflected voltage wave amplitude of the second port to the incident voltage wave amplitude of the first port, RL is the passband return loss, SR is the desired stopband rejection, θ c S11 is the electrical length at the cutoff frequency under the passband, θ c S21 is the electrical length at the cutoff frequency under the stopband, θ TZ is the electrical length at the transmission zero point, θ D is the electrical length at |s 21 |=sr in the stopband, FBW S21 is the bandwidth of S 21, and satisfies:
FBWS21＝(90°-θc S21)/90°×200％ (1)
Next, as shown in FIGS. 4-7, And/>The input impedances of the odd mode and the even mode of the band-stop filter I and the band-stop filter II respectively can be calculated by the following formulas (2 a) to (2 d):
where j is an imaginary unit.
The ratio S 11 of the reflected voltage wave amplitude of the first port to the incident voltage wave amplitude of the first port, the ratio S 21 of the reflected voltage wave amplitude of the second port to the incident voltage wave amplitude of the first port, the circuit transfer functions of the band-stop filter i and the band-stop filter ii satisfy:
in the method, in the process of the invention, Is the circuit transfer function of the band-stop filter,/>Is the odd mode input impedance of the band reject filter,/>Is the even mode input impedance of the band-stop filter, and i= I, II, when i=i, the band-stop filter I,/>, is usedFor/>When i=ii, a band reject filter II,/>, is usedIs that
Further, the circuit transfer functions of the band-stop filter i and the band-stop filter ii can be generalized as:
Where p a and g b are the a (a= 0,1,2,3,5) order coefficients and b b =0, 1,2,3,5, 7) order coefficients corresponding to sin θ, respectively, and p a and g b satisfy:
The theoretical quasi-elliptic response transfer function can be summarized as equation (6):
Where Φ is a second intermediate parameter, ζ is a third intermediate parameter, γ is a fourth intermediate parameter, m i is a fifth intermediate parameter, and when i=i, m I =1, and when i=ii, m II =2.
Then, several variables are defined as follows:
β＝αsinθTZ (7a)
Wherein α is a sixth intermediate parameter, β is a seventh intermediate parameter, x is a seventh intermediate parameter, y is an eighth intermediate parameter, and z is a ninth intermediate parameter.
The cosine term in equation (6) can be expanded as:
cos(2φ+ξ+2mγ)＝(T2(x)T1(y)-V2(x)V1(y))T2m(z)-(V2(x)T1(y)+T2(x)V1(y))V2m(z)(8)
Wherein T n(x)、Tn(y)、Tn (z) is a first Chebyshev polynomial, V n(x)、Vn(y)、Vn (z) is related to a second Chebyshev polynomial U n(x)、Un(y)、Un (z), and
The relationships among θ c S11、θc S21、θTZ、θD, SR, and RL can be easily summarized from the formulas (10 a) to (10 d), and SR and θ c S21 can be calculated from the formulas (10 a) to (10 d):
where ε is a first intermediate parameter.
Finally, when m in the band-stop filters I and II is 1 and 2 respectively, a theoretical quasi-elliptic response transfer function matching the circuit transfer function can be obtained, and equations (7 a) - (7 d) are substituted into equation (8) to obtain
Wherein, P c and G d are the c (c= 0,1,2,3,5) sub-term coefficient and d (d=0, 1,2, 3) sub-term coefficient corresponding to sin θ, respectively, and the detailed equations of Pc and G d are:
Based on the above discussion, the general design conditions of the quasi-elliptic response band-stop filter I and the band-stop filter II are summarized as:
For the proposed band reject filter I, when 3 out of 5 design indices (θ c S11、θc S21、θTZ, SR, RL) are given, 4 simultaneous equations and 4 variable parameters (Z 1、Z2、Z3 and Z 4) can be obtained. Thus, from (13 a) a set of unique solutions for Z 1、Z2、Z3 and Z 4 can be calculated.
For the proposed band reject filter II, 5 simultaneous equations and 4 variable parameters (Z e1、Zo1、Ze2 and Z o2) are available from (13 b), and since the number of simultaneous equations is greater than the number of variable parameters, 1 out of the 5 design indices (θ c S11、θc S21、θTZ, SR, RL) needs to be selected as an additional variable parameter, i.e. when 2 out of the 5 design indices (θ c S11、θc S21、θTZ, SR, RL) are given, a set of unique solutions for Z e1、Zo1、Ze2 and Z o2 can be calculated from (13 b).
Examples
To verify the correctness of the design method, for the proposed band-stop filter I, θ c S21, SR, RL are selected from 5 design indexes (θ c S11、θc S21、θTZ, SR, RL) and specific values are given, and a unique solution can be calculated, as shown in fig. 8, for three design examples, specific design parameters are shown in table one:
Design parameters of a band reject filter I
| Z1(Ω) | Z2(Ω) | Z3(Ω) | Z4(Ω) | θc S21(°) | SR(dB) | RL(dB) | |
| Example I | 30.29 | 105.88 | 82.15 | 48.29 | 36.0 | 20 | 20 |
| Example II | 35.79 | 114.42 | 57.35 | 48.16 | 40.5 | 25 | 25 |
| Example III | 39.74 | 121.37 | 43.23 | 51.23 | 45.0 | 30 | 30 |
For the proposed band-reject filter II, the unique solution can be calculated by selecting RL from 5 design indexes (θ c S11、θc S21、θTZ, SR, RL) as additional variable parameters and selecting θ c S21, SR to give specific values, as shown in fig. 9, for three examples, the specific design parameters are shown in table two:
design parameters of a Table II band reject filter
| Ze1(Ω) | Zo1(Ω) | Ze2(Ω) | Zo2(Ω) | θc S21(°) | SR(dB) | RL(dB) | |
| Example IV | 48.27 | 39.53 | 166.58 | 65.83 | 36.0 | 20 | 28.0 |
| Example V | 57.74 | 51.76 | 153.07 | 61.24 | 40.5 | 20 | 30.4 |
| Example VI | 68.58 | 68.21 | 142.68 | 57.52 | 45.0 | 20 | 33.6 |
As shown in table one and fig. 8, it can be seen that the desired stop band parameters (θ c S21, SR, RL) can be obtained on the premise of satisfying the quasi-ellipse, as required, and as shown in table two and fig. 9, the desired stop band parameters (θ c S21, SR) can be obtained on the premise of satisfying the quasi-ellipse, as required.
The design method of the band-stop filter with the quasi-elliptic response is designed and developed, so that expected stop band parameters (SR, theta c S21) can be designed on the premise of meeting the quasi-elliptic response, and in addition, a comprehensive design theory about the increase of the number of reflection zeros is provided.
Although embodiments of the present invention have been disclosed above, it is not limited to the details and embodiments shown, it is well suited to various fields of use for which the invention is suited, and further modifications may be readily made by one skilled in the art, and the invention is therefore not to be limited to the particular details and examples shown and described herein, without departing from the general concepts defined by the claims and the equivalents thereof.
Claims (10)
1. A band reject filter having a quasi-elliptical response, comprising:
The first port and the second port can be an input port or an output port;
Two open branch node lines connected in parallel to both ends of the first port and the second port;
Two transmission lines connected in series between the first port and the second port;
The ladder impedance branch line is connected in parallel to the centers of the two transmission lines, comprises a first characteristic impedance and a second characteristic impedance, and is close to the centers of the two transmission lines.
2. A method of designing a band stop filter having a quasi-elliptical response using the band stop filter having a quasi-elliptical response of claim 1, comprising the steps of:
Step one, determining the electrical length at the cut-off frequency under the passband according to a band stop filter with quasi-elliptical response Pass band return loss RL, electrical length at stop band cut-off frequency/>Any three values of the desired stop band rejection SR and the electrical length at transmission zero theta TZ;
step two, obtaining a circuit transfer function after calculating the odd mode input impedance and the even mode input impedance of the band-stop filter:
in the method, in the process of the invention, As a circuit transfer function of the band reject filter, Z 1 is a characteristic impedance of an open branch line, Z 2 is a characteristic impedance of a transmission line, Z 3 is a first characteristic impedance, Z 4 is a second characteristic impedance, S 11 is a ratio of a reflected voltage wave amplitude of the first port to an incident voltage wave amplitude of the first port, and S 21 is a ratio of a reflected voltage wave amplitude of the second port to an incident voltage wave amplitude of the first port;
And thirdly, restraining the circuit transfer function by using a theoretical quasi-elliptic response transfer function of the band-stop filter to obtain the characteristic impedance of two open-circuit branch lines, the characteristic impedance of two transmission lines and the characteristic impedance of the stepped impedance branch line of the band-stop filter.
3. A band reject filter having a quasi-elliptical response, comprising:
The first port and the second port can be an input port or an output port;
Two open branch node lines connected in parallel to both ends of the first port and the second port;
two transmission lines connected in series between the first port and the second port;
the two ladder impedance branch node lines are respectively connected in parallel with the centers of the two transmission lines, each ladder impedance branch node line comprises a first characteristic impedance and a second characteristic impedance, the two first characteristic impedances are close to the centers of the two transmission lines, and the two first characteristic impedances and the two second characteristic impedances are respectively coupled with the two transmission lines and the two open branch node lines in a one-to-one correspondence manner.
4. A method of designing a band stop filter having a quasi-elliptical response using the band stop filter having a quasi-elliptical response of claim 3, comprising the steps of:
Step one, determining the electrical length at the cut-off frequency under the passband according to a band stop filter with quasi-elliptical response Pass band return loss RL, electrical length at stop band cut-off frequency/>Any two values of the desired stop band rejection SR and the electrical length at transmission zero θ TZ;
step two, obtaining a circuit transfer function after calculating the odd mode input impedance and the even mode input impedance of the band-stop filter:
in the method, in the process of the invention, As a circuit transfer function of the band elimination filter, Z e1 is a coupled line even mode characteristic impedance formed by coupling a second characteristic impedance with an open circuit branch line, Z o1 is a coupled line odd mode characteristic impedance formed by coupling a second characteristic impedance with an open circuit branch line, Z e2 is a coupled line even mode characteristic impedance formed by coupling a first characteristic impedance with a transmission line, Z o2 is a coupled line odd mode characteristic impedance formed by coupling a first characteristic impedance with a transmission line, S 11 is a ratio of a reflected voltage wave amplitude of a first port to an incident voltage wave amplitude of the first port, and S 21 is a ratio of a reflected voltage wave amplitude of the second port to an incident voltage wave amplitude of the first port;
And thirdly, restraining the circuit transfer function by using a theoretical quasi-elliptic response transfer function of the band-stop filter to obtain the characteristic impedance of the coupling line formed by coupling the two ladder impedance branch lines, the two transmission lines and the two open-circuit branch lines of the band-stop filter.
5. The method of designing a band reject filter having a quasi-elliptical response of claim 2 or 4, wherein the two open-circuited branch lines, the two transmission lines, the two stepped impedance branch lines, and the two pairs of coupling structures formed by the two open-circuited branch lines and the two transmission lines each have an electrical length of 90 ° at the center frequency.
6. The method of designing a band reject filter with quasi-elliptical response of claim 5, wherein the ratio of the reflected voltage wave amplitude at the first port to the incident voltage wave amplitude at the first port and the ratio of the reflected voltage wave amplitude at the second port to the incident voltage wave amplitude at the first port satisfy:
in the method, in the process of the invention, Z 0 is the load of the first port or the second port, which is the odd-mode input impedance of the band-stop filter,/>Is the even mode input impedance of the band reject filter and i= I, II, when i=i, the band reject filter according to claim 1 is used and when i=ii, the band reject filter according to claim 3 is used.
7. The method of designing a band reject filter with quasi-elliptical response of claim 6, wherein the odd mode input impedance of the band reject filter and the even mode input impedance of the band reject filter satisfy:
Where j is an imaginary unit and θ is the electrical length of the open branch node line at the center frequency.
8. The method of designing a band reject filter with quasi-elliptical response of claim 7, wherein the constraints in step three are:
in the method, in the process of the invention, Is the theoretical quasi-elliptic response transfer function of the band reject filter, epsilon is the first intermediate parameter.
9. The method of designing a band reject filter having a quasi-elliptical response of claim 6, wherein the theoretical quasi-elliptical response transfer function of the band reject filter satisfies:
Where Φ is a second intermediate parameter, ζ is a third intermediate parameter, γ is a fourth intermediate parameter, m i is a fifth intermediate parameter, and when i=i, m I =1, and when i=ii, m II =2.
10. The method of designing a band reject filter having a quasi-elliptical response of claim 9, wherein the theoretical quasi-elliptical response transfer function of the band reject filter satisfies:
Where θ D is the electrical length in the stop band at |s 21 |=sr.
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