Synthesis of passband filters with asymmetric transmission zeros Giuseppe Macchiarella Polytechnic of Milan, Italy Electronic and Information Department
Passband filters with asymmetric zeros Placing asymmetric zeros respect the centre of the passband (f 0 ) produce a response which is no more geometrically symmetric around f 0 An asymmetric response allows a more flexible assignment of selectivity requirements, allowing at the same time to reduce the overall filter order The synthesis techniques based on the lowpass bandpass classical transformation cannot be directly employed (they implies a geometric symmetry in the bandpass domain)
Extension of the circuit components class In addition to capacitors, inductors, resistors and inverters, a new component is now introduced: The frequency-invariant reactance (FIR) / susceptance (FIB): Z=jX, Y=jB Circuits including this new component present network functions with more general properties. In particular, the response around zero frequency can be asymmetric The synthesis of a lowpass prototype with FIR (FIB) components allows to obtain an asymmetric response (around f 0 ) in the passband domain (after application of the classical lowpass bandpass frequency transformation)
Are FIR significant from a physical point of view? Strictly speaking FIR are not physically realizable, so synthesized networks containing FIR are not meaningul This is especially true around the zero frequency, where a FIR can not be even approximated with real component (concentrated or distributed) In the bandpass domain however, it is possible to obtain a reactance (susceptance) which does not present a relevant variation in a small range of frequencies. So a synthesized bandpass network containing FIR is significant from a practical point of view because it can be approximated with real components
Positive and Positive-real functions Impedances (admittances) of networks with FIR (FIB) components are positive function in the complex frequency variable s (not positive-real as in case of R,L,C networks). A rational function f(s) in s is a positive function if Re{f(s)} 0 for Re{s} 0. (It is a positive-real function if f(s) is real for s real) The main differences between the two class of functions are: The coefficient of polynomials at numerator and denominator of a positive function are complex (real for positive-real functions) The roots of such polynomials occur in complex conjugate pairs for positive-real function (not for positive function)
Characteristic polynomials for positive networks Assuming to be in the normalized domain s, the characteristic polynomials define the scattering parameters of a lossless 2- port network: Fs () R Ps () F2 () s R S11 s, S21, S22 s Es () Es () Es () All polynomials are assumed monic. The coefficients and R are real number which are related each other. Conditions to be verified for positivy: -Coefficients of E and F in general complex (same degree n) -Roots of E have negative real part (Hurwitz polynomial) -Degree of P(s) n
Unitary of S matrix (Lossless condition) SS * U * * S s S s S s S s 11 11 21 21 * * 11 12 21 22 1 S22 s S22 s S12 s S12 s 1 * * S s S s S s S s 0 Paraconjugation (s=j): Q(s)*=Q*(s*)=Q*(-s) Qs q qsqs qs 2 ( ) 0 1 2... n Qs ( ) Q( s) q qsqs... qs ( neven) * * * * * 2 * 0 1 2 n n n qs * n n ( nodd) The roots zq of Q(s)* are those of Q(s) with the real part of opposite sign: zq=-zq*. * n Qs () 1 szq * k n k 1
Characteristic polynomials of lossless networks Roots of P: -Imaginary -Complex pairs with opposite real part Polynomial P (degree n z ) must be multiplied by j when (n-n z ) is even Roots of F 2 : - Equal to the negative conjugate of the roots of F: * ( ) ( 1) n ( ) * zf2 zf F2 s F s Feldtkeller equation: PsPs () () FsFs () () * * 2 2 r EsEs () () E(s) is defined once P(s) and F(s) are known *
Relationship between and r When n z <n: P( j) F( j) S21( j) 0, S11( j) 1 R =1 E( j) E( j) When n z =n (fully canonical condition) * * P( j) P( j) F( j) F( j) 1 1 1 1 E( j) E( j) E( j) E( j) 2 * 2 * 2 2 r r r 2 1 R Let remember that is determined by RL=-10log( S 11 (±j) )
A The approximation problem: the characteristic function Cn and polynomials P, F 1 C 2 2 2 n 2 S21 j 1 E j E j P j P j F j F j F P j P j P 2 2 2 1 Cn 2 Applying the analytic continuation (js): C n s F s P s Given C n () (order n and trasmission zeros imposed), it is possible to compute the characteristic polynomials 2 2
C The generalized Chebycheff characteristic function n ( ) 1 1, nz 1 1 zk, cos nnz cos ( ) Re cos 1 k zk, 1 1, nz 1 1 zk, cosh n nz cosh ( ) Re cosh 1 k zk, z,k are the assigned transmission zeros: zp k are the roots of P(s), which must be imaginary or complex pairs with opposite real part. C n () can be expressed in terms of the roots of P(s) and F(s) C n k 1 nz n k 1 zf zp k k j j zp k j z, k
Evaluation of polynomials P(s), F(s) given C n () Assign the order n and the transmission zeros zp k Evaluate C n () (with z,k =zp k /j) for i = 1, N, in the interval -1< i <1 (N>2n) Generate the vector F C zp j i n i i k k 1 Find the coefficient of polynomial F () by fitting F ( i ) with a polynomial of order n Find the roots F,k of F (): F,k =zf k /j zf k =j F,k Generate the polynomials F(s) and P(s) from their roots (zp k, zf k ). Remember that P(s) must be multiplied by j if n-n z is even n z
Evaluation of E(s) using lossless condition From the imposed RL the constant is computed: RL 10 2 10 RL 10 1 10 If n z =n r is evaluated with the expression previously shown. Otherwise, r =1. The polynomial E 2 (s) is then evaluated: 2 * PsPs () () FsFs () () E() s EsEs () () 2 2 * * The roots of E 2 are computed and those with negative real part define the roots of E(s) Finally E(s) is obtained from its roots (it is monic) r
More efficient method (for imaginary zf k ) Let consider the following factorization of E 2 : * * 2 * Ps () Fs () Ps () Fs () E () s E() s E() s Ea Eb r r The equality holds if: * * Ps () Fs () PsFs () () * * 0 P() s F() s P() s F() s r The last equation is verified if the roots of F are imaginary or symmetric with respect the imaginary axis. The roots of E(s) are obtained from the roots of E a (or E b ), by assigning all the real part negative (i.e. by changing the sign of those which are positive)
Example: n=6, RL=26dB, zp={1.12i, 1.31i} 0-10 -20-30 -40-50 -60-70 -80-90 -100-2 -1.5-1 -0.5 0 0.5 1 1.5 2 P={1.4224, -3.4564i, -2.0869} =0.7031 F={1, -1.0794i, 0.81597, -1.0023i, -0.0079246, -0.096402i} E={1, 2.64-1.079i, 4.3-3.16i, 3.624-5.58i, 0.5864-5.316i, -1.1655-1.7338i}
Group Delay Evaluation Group Delay in domain: S21 PE nz, P jzp E jze k k1 k1 The phase of P() is independent of : in fact the roots zp k are on imaginary axis (0 contribute) or in pair with opposite real part (contributes of opposite sign). Then: n Im n 1 k S21 Etan k 1 RezEk E n ze Re zek ze Im ze 2 2 k 1 Re k k k
Complex zero for phase equalizing Group Introducing Delay of two the complex previous zeros example: (±0.25-1.05i): Max value in in passband: 149 sec 30 25 Sec. 25 20 20 15 15 10 10 5 0-2 -1.5-1 -0.5 0 0.5 1 1.5 Normalized Frequency
Attenuation worsens! 0 Attenuation/ReturnLoss -5-10 -15-20 db -25-30 -35-40 -45-50 -3-2 -1 0 1 2 3 Normalized Frequency
Synthesis of the filter in the normalized domain () Filtering networks presenting transmission zeros can be classified in two very general categories: Crossed-coupled networks: the transmission zeros are generated by means of multiple paths which allow the output signal to vanish at some frequencies Extracted pole networks: each transmission zero (pure imaginary) is realized by means of a suitable impedance (admittance) which blocks the transmission between input at output at a specific frequency The networks synthesized in are called prototypes. To obtain the network in the final bandpass domain it is necessary to perform a de-normalization process. Prototypes with a number of transmission zeros equal to the number of poles are called fully canonical
Cross-coupled prototype networks General topology: conventional representation 2 7 Source (g 0 =1) 1 3 4 6 8 9 Load (g 0 =1) i Node C i =1 jb i i q J i,q Line Given a set of polynomials F,P,E defining the generalized Chebyceff characteristic, it is always possible to synthesize a cross-coupled network with the above components. Such network is called normalized prototype
Minimum path rule Given a cross-coupled topology, the maximum number of transmission zeros which can be accommodated is determined by the minimum path rule : The maximum number of transmission zeros is equal to the prototype order (n) minus the number of nodes touched for going from the source to the load (np) n z =n-np Source 1 2 3 4 7 6 8 9 n z =9-4=5 Load n=9, np=4
1.. Canonical Prototypes There are particular prototype networks with a specific topology which can be always synthesized from the given characteristic polynomials. These prototypes are called canonical Most important canonical prototypes: S L n S 2 L 2 n-1.. 1.. 3 2 4 3 1 L n-2 S n n/2 n/2+1 Folded Transversal Wheel n n-1
Notes on Canonical Prototypes Not all the couplings are different from zero (this is true only for transversal prototype) In folded and wheel prototypes there are two kinds of couplings: the direct (those connecting sequential nodes) and the cross (those between not-consecutive nodes). The number of not-zero cross couplings depends on the number of transmission zeros (according to the minimum path rule) In the folded prototype the cross couplings involving source and load are necessary when n z >n-2 For fully canonical prototypes (n z =n) a coupling between load and source is requested
Examples L 6 5 4 n=6, np=3 nz=3 S 2 1 3 2 3 4 1 L 5 n=7, np=4 nz=3 S 6 7
Canonical prototypes with symmetric response Symmetric response in the normalized domain is obtained with transmission zeros symmetrically placed around real axis The diagonal elements of M are null The corresponding prototype does not include FIR (FIS) elements (positive-defined network) The canonical prototypes networks have specific properties: Folded: oblique cross couplings are zero Wheel: cross couplings terminating on the load vanish alternately Transversal: couplings M 1,k and M k,n+2 have the same magnitude
Example N=10, transmission zeros: [±1.23i, ±0.3±0.1i], RL=25-10 -20-30 -40-50 -2.5-2 -1.5-1 -0.5 0 0.5 1 1.5 2 2.5 Normalized Frequency 6 7 8 9 5 L L 10 1 2 3 4 5 6 7 8 9 10 4 S S 3 2 1
Circuit analysis of cross-coupled prototypes: formation of the (n+2) x (n+2) admittance matrix Y J 01 1 J 1n J 2n J13 3 J3n n J 12 2 J Jn,n+1 23 J 3n 1 1 1 jb 11 jb 2 jb 3 1 jb n 1 0 1 2 3 n (n+2)x(n+2) n+1 1 Admittance Matrix 1 Y Y i, j i, j 1,1 n2, n2 ii, i0 Y jj Y s jb i 0
Normalized Coupling Matrix M 0 jj01 0... 0 jj01 s jb1 jj12... jj 1n YjJ02 jj12 s jb2... jj 2n sun jm............... 0 0... jjnn, 1 0 M 0 J01 0... 0 J b J... J J J b... J............... 0 0... Jnn, 1 0 01 1 12 1n 02 12 2 2n Normalized Coupling Matrix
Evaluation of the scattering parameters from M The Y matrix is computed at frequencies s=j k : Y j U jm k The Z matrix is obtained by inverting Y: 1 Z Y k k n k The matrix Z (2x2) is extracted from Z k by cancelling all rows and columns except the first and last: Z0,0 Z0, n1 Zk Z0, n1 Z n1, n1 The scattering matrix of the prototype is computed from Z : S Z U Z U 1 k k k
De-normalization of prototype networks De-normalization consists in the network transformation from the normalized domain to the bandpass domain f, using the classical frequency transformation: =(f 0 /B)(f/f 0 -f 0 /f) At circuit level, this transformation is obtained by replacing the unit capacitance with a shunt resonator. If also the external loads are scaled from 1 to G 0 the correspondence between normalized and de-normalized components are is the following: 1 jb J jb c,f 0 J G 2 B 0 c, b bg0, J JG0
De-normalized bandpass network The de-normalized network is constituted by coupled resonators with the following coupling coefficients: J i, j B B ki, j Ji, j Mi, j c f f 0 0 0 The resonant frequency of i-th shunt admittance results: 2 2 B b i B b i B Mii, B Mii, fris, i 1 1 f0 2 f0 2 f0 2 f0 2 External Q produced by the q-th resonator coupled to source (load): f0 Bc 1 1 QEq, 2 2 2 J G B f J B f M 0, q 0 0 0, q 0 0, q
Dependence of filter response on the coupling parameters Once the parameters k i,j, f ris,i and Q E,q are defined, also the filter response is uniquely determined. This means that there are infinite networks, differing for the circuit component values but with same coupling parameters, which present the same response (identical scattering parameters) The circuit component values have however an influence on the voltages and currents along the filter; moreover, there could be some combinations of component values which result in a more easy implementation while other values may even not allow the physical realization of the filter
De-normalized coupling matrix M The matrix M is defined: M B f 0 M The off main diagonal elements M q,k represent the coupling coefficient k q,k The diagonal elements M q,q determine the resonance frequencies of q-th node: 2 f M 2 M 2 1 risq, qq, qq, The external Q if q-th resonator coupled to source (load) is given by: QEq, B f 0 M 0, q 2
Evaluation of the de-normalized response from M including losses The Y matrix vs. frequency (for G 0 =1) can be written as: f0 1 f f 0 Y f j B U n Un M Q0 f0 f f f f M 0 1 ris, i j diag B Un Q 0 fris, i f where Q 0 is the unloaded Q of the resonators. The approximated expression assumes that the frequency invariant elements are represented by de-tuned resonators (from f 0 to f ris,i, see next slide); M is then obtained from M by putting the element of the main diagonal to zero.
Approximated de-normalized resonators jb q c,f 0 c,f ris,q 2 f M 2 M 2 1 risq, qq, qq, The resonators in a cross-coupled bandpass filter are in general no synchronous. The approximation is typically acceptable for (B/f 0 )<<1. Note that the b q do not influence the coupling coefficients, which must be evaluated at f 0
Synthesis of canonical prototypes: the circuit approach Starting point: evaluation of Chain Matrix from characteristic polynomials: 1 As () Bs () ABCD jps Cs () Ds () 2 Es ( ) As ( ) Bs ( ) Cs ( ) Ds ( ), 2 F As ( ) Bs ( ) Cs ( ) Ds ( ) As ( ) Ee( s) Fe( s), Bs ( ) Eo( s) Fo( s) Cs ( ) E () s F (), s D() s E () s F () s o o e e Where the subscript e,o define the even and odd part of a polynomial: Qs ( ) Q( s) Q( s) e Qs () Qs () Qs () Qs () Qe() s, Qo() s 2 2 o * *
Synthesis by extraction The synthesis is performed by subsequent extractions from the [ABCD] matrix of the prototype: Extracted element Remainder network [ABCD] [ABCD] matrix Suitable rules are available for the synthesis of specific canonical prototypes (see the works of Cameron on the folded prototype)
Example of synthesis (folded prototype) S L C L C1 J c,b S L C2 J c,b S c,b J L C3 J c,b S c,b J J L C4 J c,b S c,b J J J c,b L C5 J c,b S c,b J J J c,b L C6 J c,b S c,b J J J c,b J c,b J
Scaling of resonator nodes The circuit synthesis does not produce in general a normalized prototype (i.e. the capacitances are not all equal to 1) Using the conservation of the coupling coefficient k i,j, it is easy to evaluate the elements of the coupling matrix M i,j resulting from synthesized components J i,j, c i, c j : Ji, j M i, j ci cj The elements M i,i resulting from the synthesized frequency-invariant b i are given by: bi M ii, c i
Direct Synthesis of the Coupling Matrix This technique consists in the direct evaluation of the coupling matrix M without resort to an explicit circuit synthesis The method has been proposed by Cameron and allows the evaluation of the transversal canonical prototype Details of the method, which rely on the relationship between the Coupling matrix and the short circuited Admittance Matrix of the transversal prototype, can be found in the literature The computation procedure, can be easily automated in a computer program (input data: the characteristic polynomials)
Coupling Matrix reconfiguration Once a canonical prototype is available, it is possible to derive other topologies by performing suitable transformations of the synthesized coupling matrix The transformation must conserve the response of the network A class of topological transformations with such a property is represented by the Similarity Transform (Given s Rotation) Starting from the Transversal Prototype, specific transformations are available for obtaining the other canonical prototypes (folded, wheel)
SYNPROT: A software for the prototype synthesis