CC BY
25Method of synthesis and electric characteristics of two resonators rebuilt quaisipolynomial bandstop filters on lumped
and distributed elements
Ounrou N.E. (onro@ref.nstu.ru), Grigorjev E.V.
NSTU, Novosibirsk
Abstract. In the work the technique of calculation of two resonators quasipolynomials bandstop filters (QBSFs) on the distributed and lumped elements are presented. Electric characteristics of the QBSFs in a range of frequencies and in a range of tuning are considered. The opportunity of construction of two resonators rebuilt bandstop filters (BCFs) on the lumped and distributed elements with satisfactory electric characteristics in a range of rebuilt is shown. Results of computing and natural experiments are submitted.
The statement of the problem. The polynomial BCFs have found wide application in radio electronics, measuring engineering and communication. They are used, for example, in transmitters for suppression of harmonics of output signal or in receivers for suppression of undesirable frequency of input signal. In this situation use of BCFs can appear more effective in comparison with use of bandpass filters [1, 2].
Methods of synthesis of polynomial BSFs on the lumped and distributed elements are well known [1, 3, and 4]. However these filters at their performance on the lumped elements demand for the realization of value of inductor and capacities in the longitudinal and cross branches distinguished approximately in f°/Af time, where fo =y]f_s ' f+ S - the central frequency of the
band of stop (BS); Af = f_s _ f+ s - width BS; f_s and f+ s - top and bottom boundary frequencies of BS. The second essential negative property of polynomials BSFs is complexity of their rebuilt on frequency (because of complexity of realization of rebuilding devices).
The elements of polynomial BSF on the lumped elements are defined through the elements of the LP-prototype under formulas [1, 3, and 4]:
C¡ =-1-, L =-1—-- - for longitudinal branches (parallel contours) and
a-2 nAf-R2 (2 n- f0) • Cj
R2 1
L¡ =-, C¡ =----for cross branches (serials contours), where a - value
a-2-n-Af (2 •n-f0)2-Li
of an element of the LP-prototype, R2 - resistance of loading. From here it is visible, that for maintenance of a constancy absolute of BS Af in a range of tuning it is necessary parallel contours of polynomial BSF to rebuilt inductance, and serial contours - in capacity. For maintenance of a constancy relative of BS, it is necessary to use for rebuilding of contours both capacity and inductance. The listed above requirements to the device of rebuilding of polynomial BSF are similar to requirements which are showed to the device of rebuilding of polynomial bandpass filter [5], and are practically difficultly sold.
In the known scientific and technical literature questions of synthesis of QBSF on the lumped elements are covered only in part [6, 7], and on distributed - are not covered in general. Nevertheless there is a practical requirement for the BSF, especially in rebuilt. The expediency of consideration of ways of synthesis and electric properties of such QBSFs which would have identical resonators from here follows and allowed to realize their frequency reorganization rather simply.
Method of solution. The similar problem was faced already before by the developers of bandpass filters. Its successful solution was connected with the usage of inverters of resistance and conductivities [1, 3, 4]. Using them, it is possible to convert parallel resonators in cross-sectional
branches to series resonators in longitudinal branches and visa versa. Received in outcome bandpass filters are named quasipolynomial.
Unfortunately, this method cannot be used for development of a calculation technique of quasipolynomial BSF, because, as it seen in fig. 1,a, the scheme of polynomial BSFs contains of a series contours (resonators) in cross branches and parallel resonators in longitudinal branches accordingly. We shall be limited in the given work only to consideration of the symmetric QBSFs of 2-nd order with serials resonators in cross branches and we not begin yet consider other possible circuits of construction of the QBSF, for example, with parallel resonators in serials branches as they are badly realized on VHF and UHF because of parasitic inductors and capacities [5].
To start the decision of a task in view we shall write down expression for function of working attenuation [8] of polynomials two contours BSF (fig. 1,a) and for some most simple circuits two resonators QBSFs on the lumped and distributed elements (fig. 1 and 2)
2
L(œ) =
R2 4R1
1 Ri
a11 + a12---+a21 • R1 + a22--
11 12 R2 21 22 R2
(1)
where a> = 2 ■n ■ f; an, aj2, a2i, a22 - elements of a matrix of transfer; R1 and R2 - resistance of a source of a signal and of loading accordingly.
We make system of two (for circuits fig. 1, b-c and fig. 2, e) or three (for circuits fig. 1,d-g, fig. 2,a-d and fig. 2,f-g) the nonlinear equations for a finding of required values of parameters of elements of connection ( L01 = L23, C01 = C23, L12, C12 , r - the not normalized resistance of
connection on a current [8]) and geometrical length of resonators (L1 = L2=L). Thus it is supposed, that trailer capacities of both resonators are identical (C1 = C 2=C). For the circuit fig. 2, a, for example, the mentioned above system of the nonlinear equations can look like
f ~ \
B(f1 )-G
f1,L1, L01 ,L12
B(f2 )-G
B(fs )-G
V f
' j
V f
= 0
f2,L1, L01 ,L12
= 0,
(2)
\
f3,L1,L01, L12
= 0
if three parameters of the circuit (L01 = L23, L1 = L2 L12) are required, or
f ~ >
B(f1 )-G
f1,L1,L12
V f
B(f2)-G f2,L1,L12
= 0
= 0
(3)
if two parameters of the circuit (L1 = L2, L12) are required, and to inductance of connection with external circuits (L01 = L23 ) are set. Values of frequencies f1, f2 and f3 it is necessary to choose
proceeding from conditions f1,f2,f3 G [f+s, f-S] - on these frequencies of value of functions of
working attenuation of polynomial filter B(f ) and of QBSF G(f ) should coincide.
a)
b)
c)
d)
e)
f)
g)
Fig. 1. The circuit of polynomial (a) and quasipolynomial (b - g) two contours BSFs on the lumped elements
At a finding of the solution of systems (2) and (3), as well as at the decision of any system of the nonlinear equations, the important value has a correct choice not only initial approach, but also a method of the decision [9].
As the equations are transcendental for a finding of decisions of these systems not maybe to use analytical methods. Besides generally, values of functions of working attenuation of polynomial BSF and of QBSF can not be equal to two (or three) any points, as their order is various. Therefore it is necessary to replace a task of the decision of system of the nonlinear equations a task of search of a minimum of function of two or three variables.
L01 L12
L23
L01
CI
p. LI
a)
C2
p, L2
CI
Pi LI
b)
L23
C2
p,L2
c)
L01
LI 2
d)
COl
C12
e)
f)
g)
Fig. 2. Circuits of two resonators QBSFs on the distributed elements
Analytical results. Expression for function of working attenuation of two contours polynomial BSF looks like
B(a) =
f, Rl-a2-C1-L2 Rl
1--+ —
R2- t1 - t2 R2
\ 2 f A fa- L2
+
T2-R2
+
a - Cl-Rl Tl
2
(4)
where Tj = 1 -a2 -L1-C1; = 1 - a2 ■ L2 -C2 .
The appropriate expressions for function of working attenuation for circuits of some QBSFs on the lumped and distributed elements G(a) are shown in tables 1 and 2 accordingly where it is
2 1 a ■ L ■ Jsr designated: t = 1 -a ■LC, /u = p^tg 6--— and 6 =-^—, sr - effective dielectric per-
a-C
3-10
meability of environment
. cos 0 J1 = p- sin 0--—, J2 =
Jj + p ■ sin 0
( -r2)
Ja =
J
3
r ■ sin 0
a ■C ~ a-C \pz-r
Jj
J2 aC■ (p2 -r2)
-1
+cos 0, J3 =
r^sin 0
p ■ sin 0
a-C(p2 -r2)
+cos 0,
J2-
a C (p2 - r2)
2
Table 1
The scheme
G(a)
b
V
, a2 C L12 R1■ a2 ■C ■LH R1
1----+ —
t R2 ■ t R2
v
+
( a ■ L12 2 ■ R1 ■ a ■C Rh a3C3^L12^ +-
V
R2
t
t
2
1+-
C
■ + ■
R1 C
• + ■
R1
t ■ C12 R2 ■ C12 ■ t R2
f+(
2 R1 a C
1
a C12 R2
R1 a C t2 C12
2 1
a4 ■N2 ^12^01
1 +
R2
2
a ■ C
— -
. mr 2^RLL01 L12 ■ R11 ,
2 ■ L01 + L12 +-+-1+1 +
R2
R2
R1 R2
+
a5 ■ N2 ■L12^L012 a3 ■C
2
3
R2■ t
2
2 ■ L12 ■ L01 2 ■ L012 R1C^L12)
■ +-+-1 +
R2
R2
( 2 ■ L01 2 ■RhC L12
a A-+-+ -
R2
R2
1 +
R1
,2 f
R2.
2C
i 2C 1+-+ -
C
-+-
C2
C01 t C12 ■ t C12C0L t2
■ + -
C2
-+-
1
C112■ t C01C12■ t2 C12.
|+■ (2+.
+
C
1
R2 a ■C01v
2
2 C
2 +-+
C01 ■ t
C12 ■ t
1 +
R2
,2 (
, 2 a2^L0LC C 1--+ -
a2 ■C2 ^L011
22 a 2 ■L01 + -
a 2 C 2 L01
C12 ■ t
1
C12 ■ t
2
+
a ■ L01 ( - 2C R2 'I t
C12 ■ t C12 ■ t2
1
J
R2 a C12
Rh a ■C ( C ■ +--1 2 + -
C12 ■ t .
g
1 +
~R2
2 f
1 +
2 ■C C01 ■ t
a2 ■ C ■LH a2 ■C2 ■LH1
c0l t
2
+
C2 L12 R2 ■ C012 t2
L12 2 ■ R1 ■C 1
+-+-la -
R2 t J
R1 a 3 C2 L12
2 L12 C R2C0L t
2
R2 C01 a
• +
C
C0L t.
2
c
t
d
2
t
t
2
t
t
2
t
2
e
t
2
t
2
t
2
t
2
t
Table 2
The scheme
G{a)
l +
a-L01 V J
Rl
R2
+1 1 +
2-a-L12 V
2-a-L12 a-L01
R2
R2
2 +
2-a-L12 + a-L01 V
\
Rl
V
i+1-
V
l
~R2
(
2-a-L01 -
a-C12
a-L01
Y
l +
v V J
Rl R2
l -
a-L01
V<a-C12 V V Va-C12 J
Y
2--
1
a-L01- 2 + —- 2-a-L01
_ V a
a-C12
Y
l +
a-L01 V J
f
a-C12
a-L01
l +
v V J
Rl V
2 + -
V
V-a-C12j
l — l
~R2
l
■ + — -
V-a -C01 V
a-L12•
l —
V-a -C01J
a -C01
Rl
R2
l +
a-L12
a-L12-
l —
V-a -C0lJ
a -C0l
-2 +1. V
2
a -C01
-a-L12-
V V-a -C01
1 —
V-a -C01J
L a-L12\ 2 +-
V V J
Rl f^ a-L12^ 2 +-
V
VJ
1 +1-V
1
~R2
a -C12
--1
V-a -C01 J
2
a-C12
■-1
V-a-C01 J
a -C01 1
a-C01
Rl
R2
-2+L. V
1—
2--
V-a -C12 V V V-a -C12 J
2
a-C01 a-C12
-1
V-a-C01 J
Rl V
2 -
V-a-C12J
a
b
c
d
Continuation of table 2
- J
1
r ■ sin 9
J1
a-CJ2 ■ (p2 - r2)
-1
r^sin 9
R1
a
■CJ2 (p2 - r2) R2J2
J3J1 r ■ sin 9
r^sin 9 a-C■ (p2 -r2)
f
R2J
2
r^sin 9--
Ji
2 \
r ■ sin 9
+ Rh
J
J
3
r ■ sin 9
J1
a-CJ2-(p2 - r2)
-1
r ■ sin 9
a2-C2-Jr (p2 - r2)
2
r^sin 9
a-CJ2 ■ (p2 - r2)
J1
a-CJ2-(p2 - r2)
-1
J
1 - a- L01 ■ J4 -—■
1
~R2
-J
1
r sin 9
J1
a-CJ2 ■ (p2 - r2)
-1
r ■ sin 9
r ■ sin 9
R1 R2
J1J3
r ■ sin 9
rJ2-sin 9 2 n2 T i 2 2V 2 a ■C J2( -r )
- + a-L01-J
a-CJ2 ■(p2 - r2)
-a-L0LJ,
4
■a-L01 —
2
J! r ■ sin 9
4
J2~r■ sin 9 J2
a ■ L01 ■
J1J3 + r ■ sin 9
J2 ■r ■sin 9 J2-a-C■ (p2 -r2)
+ R1J
4
r^sin 9
a-CJ2 ■(p2 - r2 )
J1
a-CJ2 ■(p2 - r2)
-1
Jj J4 R1
r ■ sin 9 a C01 R2
J4 J1 ■ J3
r^sin 9
a ■C01 r ■ J2'sin 9 a-CJ2 ■(p2 - r2 )
J_ R2
1
J1
a-CJ2 ■(p2 - r2)
-1
r ■ sin 9
J4
a-C01
JrJq r^sin 9
1 3 - +
J2'r'sin 9 J2 a-C ■(p2 - r2)
a-CJ2 ■(p2 - r2) a -|2
+ R1J4
1
J1 r^sin9 1
—1-+-+-
a ■ C01 J2 ■ r ■ sin 9 J2 a ■ C01
JrJ3 r^sin9
1 3 - + -
J2 'r'sin 9 J2 -a- C ■ (p2 - r2 )
+ R1J4
e
2
g
Results of calculations and experiments.
1. We shall calculate polynomial and quasipolynomials (fig. 1 (a) and (d)) BSFs on the following initial data: f-s = 102 MHz , f+ s = 98 MHz - frequencies BS with a required level of attenuation
ag = 20 dB; f_1 = 107 MHz and f+j = 93 MHz - boundary frequencies of bandpasses,
R1 = R2 = 50 Ohm. We shall choose C1 = C2 = C = 4.43 pF and L01 = 70 nH. As a result of calculations it is received: L1 = 401.9 nH, L2 = 15.76 nH, C1 = 6.305pF, C2 = 160.8pF,
L12 = 89.66 nH, L1 = L2 = L = 572.4 nH . Computational AFC of filters are submitted on fig. 3 and 4. Thus it was accepted, that own good quality of all coils of inductance is equal 200, and good quality of capacity - 100 and is a constant in a range of tuning. 0
-10
-20 -30
-40
-50
80 85 90 95 1 00 105 1 10 1 15 120
f, MHz
Fig. 3. AFC of polynomial and quasipolynomial BSFs on the lumped elements in a near zone
By results of synthesis the breadboard model of the QBSF on the lumped elements was made, the electric basic circuit both experimental AFC and experimental dependence Af on a level damping
as = 20 dB which are given on fig. 5. Dimensions of the filter have made 60*45*25 mm.
2. We shall calculate polynomial BSF and QBSF on the distributed elements on the following initial data: f_g = 1.05 GHz, f+ g = 0.97 GHz - boundary frequencies BS with a required level of
attenuation aS = 20 dB in BS Af = 80 MHz; f_1 = 1.15 GHz , f+1 = 0.87 GHz - boundary frequencies of passbands R1 = R2 = 50 Ohm . Type of AFC - Butterworth. We shall choose, being guided by reasons of convenience of constructive realization, at synthesis of the QBSF (fig. 2,a)
C1 = C2 = C = 2 pF L01 = L23 = 14 nH, sr = 3.89 and p = 50.8 Ohm . Synthesis of the QBSF was
carried out with use of system of the equations (3). As a result of calculations it is received:
- for the polynomial BSF on the lumped elements (fig. 1,a): L1 = 20.1 nH, L2 = 3.094 nH, C1 = 1.24 pF, C2 = 8.04 pF ;
- for the QBSF on the distributed elements under the circuit fig. 2,a: L12 = 11.4 nH and
L = 24 mm .
0
-10
-20
-30
-40
-50
20
X, N -^ N
/
/
1
1
40
60
£ MHz
80
100
120
Fig. 4. Computational AFC of QBSF fig. 1,d in a range of tuning
(C1 = C2 = C = 32pF, 16pF, 8pFand4.43pF)
a)
30 40 50 60 70 80 90 100
fo, MHz
b)
A 1 -V
s £
1 1 n
■ . ■ .
1
■ . ■ .
1 1
J
20 40 60 80 100 120 140 160 180 200
f, MHz
c)
Fig. 5. The electric basic circuit of the rebuilt QBSF on the lumped elements - a), its experimental dependence Af(fo) and AFC - c) at two sets of voltage on varactors: {Utun1=0.5 V, Utun2 = 1.04 V}
and {Utun1=26.8 V, Utun2 = 29.8 V}
Computational AFC of filters fig. 1,a and fig. 2,a in a near zone are shown in a fig. 6. The numerical analysis of AFC is executed in the assumption, that own good quality of all elements of polynomial BSF makes 200, and own good quality of elements of QBSF - 30 (capacities) C1 = C2 and
200 (inductance L1 = L2 and L12). Losses in pieces of lines of transfer here were not taken into account. Computational AFC of the QBSF at tuning and in view of losses in elements are shown on fig. 7.
/V
w
0.8 0.9 1 1.1 1.2
f, GHz
Fig. 6. Theoretical AFC of polynomial BSF- and QBSF
on distributed elements □ in a near zone
f, GHz
Fig. 7. AFC of QBSF on the distributed elements in view of losses at tuning (c 1 = C 2 = C = 2, 4 and 10 pF)
By results of synthesis the breadboard model of the QBSF on the distributed elements, the electric basic circuit was made, and also settlement and experimental which volt - frequency characteristics are given on fig. 8. Dimensions of the filter have made 50*35*15 mm. On fig. 9 it is represented his experimental AFC.
- Utun
Utun., V b)
Fig. 8. The electric basic circuit of the rebuilt QBSF on the distributed elements - a) and his theoretical (a shaped line) and experimental (a continuous line) volt-frequency characteristics - b)
f, MHz
a)
0
-10 -20 3-30 -40 -50 -60
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
f, GHz b)
Fig. 9. Experimental (at Utun = 1 V) - a) and computational (at C1 = C 2=7 pF) - b) AFC of QBSF which circuit is represented on fig. 8,a
Conclusions
1. The method of synthesis of the QBSF which can be used in practice is offered.
2. At synthesis of the QBSF there is some freedom of a choice of value at least one of elements of resonators that facilitates constructive realization of filters.
3. The tuning of the QBSF can be realized by one rebuilt element, one of which conclusions can be earthed, that is the important advantage of rebuilt filters [5].
4. The opportunity of realization on VHF and UHF of two resonators QBSF on the lumped and distributed elements with the simple device of tuning is experimentally shown.
5. The realization of the two contours QBSF at which the law of change of width of BS on a level = 20 dB at tuning in a range of frequencies well enough correspond constant relative is possible.
6. Theoretical and experimental AFC of QBSF on the lumped elements will well enough be coordinated with each other.
7. The AFC of QBSF on the distributed elements within the limits of a strip of damping it is appreciable different from AFC of polynomial BSF, and it distinguishes it from the QBSF on the lumped elements. Appreciable differences between AFC of polynomial BSF and of QBSF place in transitive area and in a distant zone. Thus, for transitive area various steepnesses of slopes of AFC are characteristic, and in a distant zone of QBSF have final attenuation in one of two passbands, that, certainly, limits area of their use a little.
References
[1] G. Matthaei, L. Young and E.M.T. Jones, Microwaves Filters, Impedance-Matching Networks, and Coupling Structures /Translating from English, vol. 2. - Moscow, 1972. - 495 p. (in Russian).
[2] Ounrou N.E. Effectiveness of using of bandpass filters with elliptical amplitude-frequency characteristic for a solution of a problem of EMC of an onboard complex of radio communication. - V international workshop on electromagnetic compatibility and electromagnetic ecology. The collection of the scientific reports. - St.-Petersburg, 2003. pp. 131-134. (In Russian).
[3] L.V. Alecseev, A.E. Znamenski, A.E. Lotcova, Electrical Filters of VHF and UHF. -Moscow, 1976. - 280 p. (in Russian).
[4] G.E. Hansell, Filter Design and Evalution /Translating from English. - Moscow, 1974. - 288 p. (in Russian).
[5] A.E. Znamenski, E.S. Popov, Tuning Electrical Filters. - Moscow, 1979. - 128 p. (in Russian).
[6] E.V. Grigorjev A method of synthesis of two-resonator quasipolynomial bandstop filter. -Proceedings of the regional scientific and technical School-seminar of students, post- graduate students and young scientists "Modern problems of a radio engineering (CQP-2003) " (on November, 26-28 2003, Novosibirsk). - Novosibirsk, 2003. - pp. 101-104. (in Russian).
[7] N.E. Ounrou A method of synthesis and electrical characteristics of two-resonator quasi-polynomial bandstop filters with the lumped elements. - Proceedings of the 4th IEEE - Russia Conference: 2003. Microwave Electronics: Measurements, Identification, Applications. MEMIA 2003. Novosibirsk, December 18 - 21, 2003. - pp. 52-56.
[8] A.L. Feldshtejn, L.R. Javich, Synthesis Two- and Four-ports Microwave Networks. -Moscow, 1982. - 328 p. (in Russian).
[9] Bahvalov N.S., Zhidkov N.P., Kobelkov G.M. Numerical methods - Moscow: Laboratory of Base Knowledge, 2002 - 636 p. (in Russian).
