Design of Narrow Band Filters Part 2

E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Design of Narrow Band Filters Part 2 Thomas Buch Institute of Communications Engineering University of Rostock Th. Buch, Institute of Communications Engineering, University of Rostock, Germany

E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Design of Compact Filters Filter Design, PZ-Map Loss Transformation Modified Transfer Function Determination of the normalized component values of the Continued Fractions Arrangement Inductive and capacitive coupling Denormalization Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 2

Idea of the Procedure E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Idea of the Procedure Design method looked at till now permits only the realization by reactive four-terminal networks. (no losses!) Therefore loss transformation by moving of the jω - axis to the left. s-data s -data Realization of a reactive four-terminal network for s -data. The circuit then carries out the actual data of the s-plane with losses. jω' jω u, u' Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 6

Idea of the Procedure E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Loss transformation Formulation: All inner resonant circuits have the same quality factor and with that the same attenuation. δ 0 The st circle has the attenuation δ > δ 0 Consideration of the internal resistance of the source. The n-th circle has the attenuation δ n > δ 0 Consideration of the input resistance of the following amplifier stage. Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 7

Idea of the Procedure E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Loss transformation 2 s + δ 0 s Input impedance of a re- jω' jω Z e (s) Z e (s ) active four-terminal net- δ 0 work X e (s ) δ 0 must be lower then the smallest absolute value of the real parts of the poles! u, u' Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 8

Continued Fractions Arrangement E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Continued Fractions Arrangement With that becomes the continued fraction decomposition of the input impedance for the coupled bandpass filters: Z e (s ) R /δ = (δ δ 0 ) + s + s + s +... x,2 2 x2,3 2 x 2 n,n (δ n δ 0 ) + s Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 9

Continued Fractions Arrangement E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Simplification Simplification: Z e (s ) = R δ (δ δ 0 ) + s R δ + s δ + R x,2 2 s R x 2,2 δ x 2 2,3 + s δ x 2 2,3 R x 2,2 x2 3,4 +... Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 0

Continued Fractions Arrangement E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Resulting Circuit R ( δ δ 0 ) δ R δ R x δ 2 2 2 x23 n odd δ R 2 x2 δ x R x x 2 23 2 2 2 34 n even Th. Buch, Institute of Communications Engineering, University of Rostock, Germany

Continued Fractions Arrangement E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Remark The input impedance results from the loss transformation for a completed equivalent LP-Reactant from the input impedance of the doubly-terminated lossy bandpass circuit. The normalized components can be won directly by continued fraction decomposition of the fractional rational impedance function Z e (s). It is important to say that only transfer functions without zeros can be realized by coupled resonant circuits. Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 2

Further Design Method E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Further Design Method Transformation of the given BP-DTS into the normalized LP-DTS and determination of the PZ-data. 2 Loss transformation with the aim of generating the PZ -data of the s -plan. 3 Building the transmission function from the PZ -data and outline of the reactive four-terminal network. 4 Continued fraction stripping down of the input impedance. 5 Denormalizing and determination of the components of the coupled bandpass filter. Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 7

An Example shoes the further Design Method Example: Three section bandpass filter with Butterworth approximation. Data: f m = 200 khz B = 4 khz a d = 3 db j Ω' jω Ω PZ-Data of the normalized LP: u 0 u u, u' u k Ω k 0 0,5 0,8660254 δ 0 Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 9

Loss Transformation Loss transformation: Condition: δ 0 < min{ u k } = 0,5 Selected: δ 0 = 0,3 δ 0 = Q 0 Conclusions: Q 0 = δ 0 = Q 0 = 66,6 f m δ 0 B Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 20

PZ-Data If the quality factor Q 0 (e.g. the filter coil) is predefined, then the bandwidth B cannot be chosen freely. It is valid: B f m δ 0 Q 0 PZ data of the loss transformed LP: u k Ω k 0,7 0 0,2 0,8660254 Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 2

Transmission function H B (s K ) = (s + 0,7)(s 2 + 0,4s + 0,2 2 + 0,8660254 2 ) K = s 3 +,s 2 +,07s + 0,553 H B (s ) = K N(s ) K has to be chosen so, that HB (s ) s =jω = HB (jω ) H B (jω') < Ω' Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 22

Design of the reactive four-terminal network H B (s ) H B ( s ) = H E (s ) H E ( s ) H E (s ) H E ( s ) = N(s ) N( s ) K 2 N(s ) N( s ) Determination of the zeros of H E (s ) H E ( s ), since the poles are known. H E (s ) H E ( s ) = s 6 0,93s 4 + 0,077s 2 + 0,305809 K 2 s 6 0,93s 4 + 0,077s 2 + 0,305809 Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 23

Zeros of H E (s ) H E ( s ) Approach: K 2 = 0, Zeros of H E (s ) H E ( s ): = numerical solution! u E Ω E +0,7077 +0,829324 +0,7077-0,829324-0,7077 +0,829324-0,7077-0,829324 +0,646687 0-0,646687 0 H E (p') j Ω' H E (-p') u' Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 24

Remark If K is set too greatly, so that H E (jω ) >,the zeros of H E (s ) and H E ( s ) do not let themselves divide. The zeros then lie on the imaginary axis. Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 25

Continued Fractions Arrangement of Z e (s) It gives up: H E (s ) = s 3 + 0,88084s 2 + 0,8529096s + 0,453644 s 3 +,s 2 +,07s + 0,553 From this the input resistor Z e (s ) is determined. Z e (s ) = r + H E(s ) H E (s ) = 2 s 3 +,98084ss 2 +,9229096s +,006644 0,2959s 2 + 0,270904s + 0,0993586 r Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 26

Continued fraction stripping down of Z e (s )/r Z e (s ) = +9,257945s + r 0,256687s + 0,227433s + 0,3396 From this the normalized LP gives up: R σ σ 0 σ ( ) R σ 2 x 2 2 x 23 R σ R σ 2 x 2 R σ x 2 2 2 x 23 ( σ σ ) 3 0 Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 27

System of Equations The system of equations arises: R δ (δ δ 0 ) = r () R δ = 9,257945 r (2) δ R x 2,2 R x 2,2 δ x 2 2,3 = 0,256687/r (3) = 0,227433 r (4) R x,2 2 δ x2,3 2 (δ 3 δ 0 ) = 0,3396 r (5) One gets from (??): R δ r = 9,257945 This expression is contained in all other equations. = Recursive solution of the system of equations. Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 28

Results Result: δ = 0,4095795 > δ 0! x,2 2 = 0,508098 x,2 = 0,7280559 x2,3 2 = 0,45336369 x 2,3 = 0,673322485 δ 3 =,29060986 > δ 0! Use of standard coils. Specification of an identical inductance for all circles. Selected: L = 0, mh Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 29

Measure of the coupling It is valid: x,2 2 = ωm 2 M,2 2 δ δ 0 = M 2 ωm 2,2 R R 0 B 2 L 2 M,2 = L x,2 General: M k,k+ = L x k,k+ A bandpass filter with an inductive coupling results. Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 30

Application of the capacitive coupling M R, 2 R 2 C R 2 R 2 C L L C L C C 2 L ω 2 m = L C C = C + C 2 C 2 C 2 + C 2 As C 2 << C 2 Approximation: C C + C 2 For the inner circles: Approximation: C C k,k + C k + C k,k+ Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 3

Calculation of the coupling capacities (Philippow: Taschenbuch der Elektrotechnik, Bd. II, S. 580, Bild 7.45) Conversion: It is valid: C k,k+ = C ω m M k,k+ C (ω 2 m M k,k+ C) 2 With that: ω 2 m M k,k+ C = ω 2 m x k,k+ L C = x k,k+ C k,k+ = C x k,k+ ( x k,k+ ) 2 Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 32

Denormalizing Numerical values for the example: C = = 6,332574 nf ωm L 2 C = C C,2 = 6,2422682 nf C 2 = C C,2 C 2,3 = 6,56984 nf C 3 = C C 2,3 = 6,2472898 nf C,2 = 90,3057 pf C 2,3 = 85,284 pf Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 33

Circuit Diagram with Source and Load (C-coupling) I 0 C C 2 3 R L L L R a i C Q = 66,67 Q = 66,67 Q = 66,67 C 2 C 3 U a All circles get the same swinging Q factor. The greater attenuations for the circles and 3 are realized by the input resistor of the source and the load resistor (input resistor of the following amplifier). Required quality factor of the st resonant circuit: Q = / δ The inner resonant circuits have a quality factor: Q 0 > Q Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 34

Calculation of source and load resistance It is valid: R 0p = Q 0 ω m L R p = Q ω m L } R p = R 0p R i R i = R 0p R p R 0p R p R i = Q 0 ω m L Q ω m L Q 0 Q = ω m L ω m L (Q 0 Q ) Q 0 Q = ω m L δ 0 δ It is valid correspondingly: R a = δ 0 = δ ω m L = 6,34395 kω (δ 3 δ 0 ) ω m L = 57,3393 kω (δ δ 0 ) Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 35

Testing the designed Filter Testing the ready filter circuit with the help of a network analyzer software, e. g. PSpice Design-Center (PSpice for Windows) Circuit Diagram for PSpice: C 4 2 3 4 5 C R4 L R C L2 R2 L C R C 2 3 3 3 R 4 G Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 36

Electrical devices Resonant impedances of the oscillating circuits: R p = ω 0 L Q = 2π 2 0 5 0 4 66,67 = 4π 0 66,67 R p = 20,944 kω L = L 2 = L 3 = 0, mh R = R 2 = R 3 = 20,94 kω R 4 = 57,34 kω R 5 = 6,344 kω C = 6,242 nf C 2 = 6,57 nf C 3 = 6,24 nf C 4 = 90,2 pf C 5 = 85,28 pf Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 37

PSpice Plot Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 38

Application Application of a compact filter in the IF amplifier of the HiFi tuner ReVox A76. It becomes a 8-stage Gaussian filter with a linear phase response (constant group delay!) used. Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 39

HiFi tuner ReVox A76 Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 40

Summery E.U.I.T. Telecomunicación 200, Madrid, Spain, 27.09 30.09.200 Summery Consideration of the losses by left shift of the jω-axis. With the new PZ data the transmission function of a reactive four-terminal network is calculated. From the continued fraction expansion of the input impedance the normalized network elements can be dimensioned. From the elements the couple factors and resonant circuit losses can determined. Calculation of the normalized devices. At use of customary coils the capacitive coupling of the circles is more favorable. Th. Buch, Institute of Communications Engineering, University of Rostock, Germany 46