Network Parameters of a Two-Port Filter

Contents Network Parameters of a Two-Port Filter S Parameters in the s-domain Relation between ε and ε R 2 2 Derivation of S 22 3 2 Network Parameters in the jω-domain 3 2 S Parameters in the jω-domain 3 22 ABCD Parameters in the jω-domain 4 23 Y Parameters in the jω-domain 4 24 Z Parameters in the jω-domain 4 2 Lowpass Prototype Filters 5 2 Basic Components of a Lowpass Prototype Filter 5 22 Electric and Magnetic Couplings 5 23 ε and ε R Values for the Prototype Filters 7 24 Alternating Pole Method for Determination of E (jω) 7 25 The N Coupling Matrix 25 Analysis of the General N Coupling Matrix 25 Series Type LPP 252 Shunt Type LPP 252 Synthesis of the General N Coupling Matrix 2 252 Series Type LPP 2 2522 Shunt Type LPP 5 26 The N + 2 Coupling Matrix 5 26 Analysis of the General N + 2 Coupling Matrix 5 26 Series Type LPP 5 262 Shunt Type LPP 7 262 Synthesis of the General N + 2 Coupling Matrix 7 262 Series Type LPP 7 2622 Shunt Type LPP 8 263 Synthesis of the N + 2 Transversal Matrix 8

Chapter Network Parameters of a Two-Port Filter Figure : S parameter representation of a linear, loss-less and reciprocal (LLR) device S Parameters in the s-domain For a LLR device, if the scattering parameter S is given in s (σ + jω) domain as S (s) ε R F (s) E (s), () then, from the unitary property of scattering matrices of LLR devices, If S 2 (s) can be written as ε S 2 (s) S 2 (s) S (s) S (s) ε R 2 P (s) E(s), then P (s) 2 F (s) E (s) ε R 2 E (s) 2 F (s) 2 ε R 2 E (s) 2 (2) ε 2 ε R 2 E (s) 2 F (s) 2 ε R 2 (3) All polynomials we are dealing in this section are assumed to be normalized (by ε and ε R ) such that their highest degree coefficients are equal to 2

Also, re-arranging (2) and (3) gives where F (s) P (s) S 2 (s) 2 + ε 2 F (s) ε R P (s) 2, (4) is referred to as characteristic function Before proceeding further, some important properties of the polynomials E (s), F (s) and P (s) are given 2 here E (s) is a Hurwitz polynomial of degree N All its roots lie in the left half-plane of s 2 The roots of F (s) lie on the imaginary axis where the degree is N These roots are known as reflection zeros 3 Zeros of the polynomial P (s) are known as transmission zeros (TX zeros) All TX zeros lie on the imaginary axis or appear as pairs of zeros located symmetrically with respect to the imaginary axis 3 As a result, the polynomials E (s), F (s) and P (s) have the forms E (s) s N + e N s N + e N 2 s N 2 + e N 3 s N 3 + + e, F (s) s N + jf N s N + f N 2 s N 2 + jf N 3 s N 3 + + f and P (s) s n fz + jp nfz s n fz + p nfz 2s n fz 2 + jp nfz 3s n fz 3 + + p (5) In the above equations, all the coefficients e i are complex Except f and p, all other parameters f i and p i are real Since the coefficients of F (s) and P (s) alternate between real and imaginary numbers, f and p are real if N and n fz are even (and imaginary if their orders are odd) Also, the following statements can be proved without much difficulty { S f ε at s : R e S 2 p (6) ε e S ε R at s ±j : S 2, if n fz < N (7) S 2, if n ε fz N Relation between ε and ε R As mentioned before, ε and ε R are just some complex numbers for normalizing all the polynomials The relation between these two parameters can be derived from the equation S (s) S (s) + S 2 (s) S 2 (s) ε R 2 F (s) F (s) E (s) E (s) + ε 2 P (s) P (s) E (s) E (s) (8) 2 For the time being, these properties are stated without any proof 3 If n fz, the degree of the polynomial P (s) is zero, then all transmission zeros are located at s ±j (eg, conventional Butterworth, Chebyshev, etc) 2

At s, (8) becomes Similarly, at s ±j, S (s) S (s) + S 2 (s) S 2 (s) 2 f ε R 2 e + 2 p ε 2 e (9) { εr, if n fz < N +, if n ε R 2 ε 2 fz N () So, from (9) and () { εr, if n fz < N : ε 2 p 2 () e 2 f 2 ε 2 f 2 p 2, f if n fz N : 2 e 2 ε R 2 f 2 p (2) 2 e 2 p 2 It is important to understand that only ε R and ε are related to each other, but not ε and ε R The phases can be changed arbitrarly by shifting the reference planes of the two ports 2 Derivation of S 22 Since S, S 2 and S 2 are already known, one can derive S 22 from the following S-parameter unitary property: S (s) S 2 (s) + S 2 (s) S 22 (s) ( ) ε F (s) P (s) S 22 E (s) P (s) (3) Further it can be showed that ( S 22 ( S 22 ε εε R ) ε p f εε R e p εε R, at s ) ( ) N+nfz+, as s ±j (4) 2 Network Parameters in the jω-domain 2 S Parameters in the jω-domain Till now, all the scattering parameters have been dealt in the s domain Such an analysis provides information regarding the physical realizability 4 of the filter Once the filtering functions are 4 from the view point of placement of the roots of polynomials E (s), F (s) and P (s) 3

made sure to be physically realizable, one needs to worry about the jω domain only So, S parameters can be written in jω domain as F (jω) P (jω) S (jω) S 2 (jω) ε R E(jω) ε E(jω) S 2 (jω) S 22 (jω) P (jω) (5) ε E(jω) ( ) +n fz ε εε R F (jω) E(jω) 22 ABCD Parameters in the jω-domain ABCD parameters of the device can be obtained from (5) and are as given below: R S A (jω) B (jω) ε R L (EF + + EF + ) RS R L (EF + EF + ) C (jω) D (jω) 2P (jω) R RS R L (EF EF ) L R S (EF + EF ) where EF + E (jω) + F (jω), ε R ( ) EF + ( ) n fz ε E (jω) + F (jω) ε ε R EF E (jω) F (jω) and ε R ( ) EF ( ) n fz ε E (jω) F (jω) ε ε R,, (6) 23 Y Parameters in the jω-domain Once ABCD parameters are known, Y parameters (for a reciprocal network) can be easily obtained as shown below: Y (jω) Y 2 (jω) D (jω) Y 2 (jω) Y 22 (jω) B (jω) A (jω) R S (EF + EF ) 2P RS R L ε (EF + EF + ) 2P (7) RS R L ε R L (EF + + EF + ) 24 Z Parameters in the jω-domain Similarly, Z parameters also can be obtained from the ABCD parameters as shown below: Z (jω) Z 2 (jω) A (jω) Z 2 (jω) Z 22 (jω) C (jω) D (jω) 2P RS (EF + + EF + ) RS R L ε 2P (EF EF ) RS R L R ε L (EF + EF ) (8) 4

Chapter 2 Lowpass Prototype Filters 2 Basic Components of a Lowpass Prototype Filter It is customary in the filter design to synthesize the lowpass prototype (LPP) first From the designed LPP, components of the actual filter can be obtained by using frequency transformations Two general LPP configurations are shown in Fig 2 and 22 In these figures, the colored components are assumed to be frequency invariant (ie, do not change with frequency transformations) All the other components change according to the actual filter response required (such as bandpass, bandstop, etc) and their transformed values are shown in Fig 24 Also, characteristics of the immittance inverters (both & ) used in the LPPs are shown in Fig 25 For impedance and admittance inverters, Z in 2 Z L and Y in 2 Y L, respectively 22 Electric and Magnetic Couplings The concept of immittance inverters has been mentioned briefly in the previous section One can physically realize immittance inverters in several ways Out of all the possible ways, two methods namely electric and magnetic coupling methods are very important in filter designing These two coupling phenomenas are described in Fig 26 and 27 VL and CL equations related to both electric as well as magnetic coupling are as given below: ( Magnetic coupling : jωl + ) Electric coupling : jωc ( jωc + jωl i + ji 2, where ωl m ) v + jv 2, where ωc m (2) In Fig 26( and 27, if each ) resonator is isolated from the other, then their resonant frequencies are equal f 2π However, when these two resonators are brought closer to each other, LC coupling between them yields two distinct resonant frequencies, usually known as f even and f odd (see Table 2) All the theory given in this section is related to synchronous coupling (ie, the two isolated resonators resonate at the same frequency) For mixed and asynchronous couplings, see S Hong 5

Figure 2: A series type LPP Figure 22: A shunt type LPP Figure 23: Normalized, equivalent circuit of Fig 2 6

23 ε and ε R Values for the Prototype Filters In chapter, the relationship between the magnitudes of ε and ε R was given () In addition, it was said that for a general two-port device, no such relation can be derived for the phases of ε and ε R However, if the device under consideration is restricted to be one of the LPP configurations considered in this chapter (eg, Fig 2 and 22), then a more relaxed relation exists between these two parameters For example, consider the LPP configuration shown in Fig 2 As s ±, it can be showed that both S and S 22 tends to the value Therefore, from (7) and (4), { εr, and ( ) (N+n fz+) (22) From the above equation, it can be seen that { ε ±ε re, when (N + n fz + ) is even ε ε ε ±jε re, when (N + n fz + ) is odd where ε re is some real number (when (N + n fz + ) is odd, (N n fz ) is even) Similarly, for the LPP configuration shown in Fig 22, ε and ε R values are given as ε R ε ±ε re, when (N + n fz + ) is even (24) ε ±jε re, when (N + n fz + ) is odd Similar results can be obtained for fully canonical filter configurations 24 Alternating Pole Method for Determination of E (jω) Now that ε, ε R, F (jω) and P (jω) are known 2, one needs to evaluate E (jω) Re-writing (8) in the jω domain gives ε 2 ε R 2 E (jω) E (jω) ε R P (jω) 2 + εf (jω) 2 ε R P (jω) + εf (jω) ε RP (jω) + ε F (jω) ε ε R εr (23) ε P (jω) F (jω) + ε R ε F (jω) P (jω) (25) For the LPP configurations shown in Fig 2 and 22, ε R is a real number and ε ( ) (N+n fz+) ε So, the second term on the right hand side of the above equation becomes ε ε R εr P (jω) F (jω) + ε ε R ε F (jω) P (jω) ε ε RP (jω) F (jω) + ( ) (N+n fz+) F (jω) P (jω) F (jω) P (jω) ε ε RP (jω) F (jω) + ( ) (N+n fz+) ( ) (N+n fz) 2 F (jω) and P (jω) are given from the desired filter responce (26) 7

Figure 24: Frequency transformation from LPP to bandpass, bandstop, etc 8

Figure 25: Immittance inverters: (a) Impedance inverter and (b) Admittance inverter Figure 26: Magnetic coupling; (a), (b) and (c) are equivalent Figure 27: Electric coupling; (a), (b) and (c) are equivalent Electric Coupling f even 2π C(L+L m) f odd 2π C(L L m) Magnetic Coupling 2π C(C C m) 2π C(C+C m) L ml or Cm C f 2 odd f 2 even f 2 odd +f 2 even f 2 even f 2 odd f 2 even +f 2 odd Table 2: Equations related to electric and magnetic couplings 9

So, ε 2 ε R 2 E (jω) E (jω) ε R P (jω) + εf (jω) ε RP (jω) + ε F (jω) (27) On the imaginary axis (ie, s jω), (27) is equivalent to ε 2 ε R 2 E (s) E (s) ε R P (s) + εf (s) ε RP (s) + ε F (s) (28) Rooting (in s domain) one of the two terms on the RHS of (28) results in a pattern of singularities alternating between left-half and right-half planes Also, rooting the other term will give the complementary set of singularities So, it is sufficient to find roots of only one term and then reflect the right-half plane zeros to the left side 25 The N Coupling Matrix 25 Analysis of the General N Coupling Matrix 25 Series Type LPP VL equations corresponding to Fig 2 are given in matrix form as v j v N ωl + X,2,N,2 ωl 2 + X 2 2,N },N 2,N {{ ωl N + X N } Z i i 2 i N (29) After multiplying the first row by L, the above matrix representation becomes v L j,n L ω L + X L,2 L,2 ωl 2 + jx 2 2,N i i 2 v N,N 2,N ωl N + X N i N Now, multiplying the first column by L gives v L v N j,n L ω + X,2 L L,2 L ωl 2 + X 2 2,N,N L 2,N ωl N + X N i L i 2 i N

Thus all elements in the above matrix are normalized with respect to L After several similar steps, the final normalized matrix representation is given as v L v N LN ω + X j j L,2 L L 2,2 L L 2 ω + X 2 L 2,N L L N 2,N L2 L N,N 2,N L L N L2 L N ω + X N L N ω + X,2,N,2 ω + X 2 2,N },N 2,N {{ ω + X N } Z norm jm+ωi i L i 2 L2 i N LN i L i 2 L2 i N LN, (2) where X,,2, etc are the normalized values From (29) and (2), it is evident that Fig 2 and 23 are equivalent Re-writing (2) gives i L i 2 L2 i N LN i L i N LN i i N Znorm v L v N LN Z norm Z norm N Z norm N Z norm L Znorm N L L N Z norm NN Znorm N L L N Z norm NN L N v L v N LN v v N (2) 252 Shunt Type LPP CL equations corresponding to Fig 22 are given in matrix form as i j ωc + B,2,N,2 ωc 2 + B 2 2,N v v 2 (22) i N },N 2,N {{ ωc N + B N } Y v N

Normalizing the above matrix gives i C i N CN ω + B j j C,2 C C 2,2 C C 2 ω + B 2 C 2,N C C N 2,N C2 C N,N 2,N C C N C2 C N ω + B N C N ω + B,2,N,2 ω + B 2 2,N },N 2,N {{ ω + B N } Y norm jm+ωi where B,,2, etc are the normalized values Re-writing (23) gives v C v 2 C2 Ynorm v N CN v C v N CN Y norm N Y norm v C v N i C i N CN Y norm Y norm N Y norm Y norm N C C N Y norm N C C N Y norm NN C N NN v C v 2 C2 v N CN i C v C v 2 C2 v N CN i N CN i i N, (23) (24) 252 Synthesis of the General N Coupling Matrix 252 Series Type LPP From (2), Z norm L y M + ω I jl y (25) Since, M is a real and reciprocal matrix, all of its eigenvalues are real So, using the eigenvalue decomposition, the above equation can be written as T Λ T t + ω I jl y, (26) where Λ diag λ, λ 2,, λ N, λ i are the eigenvalues of M and T is an orthogonal matrix (ie, T T t I) In addition, the columns of T are eigenvectors of M The general solution 2

for (i, j) th element of the left-hand side matrix of (26) is given as T Λ T t + ω I T Λ T t + I ω I ij ij T Λ T t + T T t ω T T t Therefore from (27) and (7), T Λ + T t ω T T t ij T Λ + ω I T t ij T Λ + ω I T t ij T ik T jk, i, j, 2,, N (27) ij T 2 k jl y T 2 k jl R S (EF + EF ) (EF + EF + ) (28) In the above equation, it can be observed that the numerator is always one degree less than the denominator (ie, there wont be any constant term when (28) is expanded as partial fractions) Similarly, from (2), Znorm N L L N y 2 M + ω I N j L L N y 2 T Λ T + ω I N j L L N y 2 T k T Nk j L L N R S R L 2P ε (EF + EF + ) (29) From (28) and (29) it can be said that λ k values are zeros of (EF + EF + ), and T k &T Nk values are related to the residues at those zeros Determination of R S L and R L L N So far, the known parameters are ε, ε R, F (jω), P (jω) and E (jω) In addition, from the properties of orthogonal matrices, { N T 2 k N T 2 Nk (22) 3

If it is assumed that G 2 k j (EF + EF ) (EF + EF + ), then from (28), T k G k L R S (22) Combining (22) and (22) gives 3 L R S G 2 k R S L G 2 k (222) Similarly, it can be showed that LN T Nk G Nk, where R L R L G 2 L Nk (223) N So, for a given filter response, one can determine the matrix Λ and st and N th rows of the matrix T If the matrix T is a 3 rd order matrix, then the remaining row (ie, the second row) can be easily determined No such simple unique solution exists if the order of the filter is greater than 3 So, usually those remaining rows are found by using the Gram-Schmidt orthonormalization process with starting independent vectors 4 as (T, T 2,, T N ), (T N, T N2,, T NN ), (,,,, ), and (,,,, ), All these vectors are independent as long as T and T N2 Otherwise, a different set of vectors should be chosen 3 It is the ration between R S and L that is important, not their actual values So, without loss of generality, many authors simply assume that L H 4 Except for the first and last, all the other independent vectors are chosen here (kind of) randomly One can choose any other combination of vectors if he/she wants! 4

2522 Shunt Type LPP From (24) and (8), Y norm C z M + ω I jc z T Λ T t + ω I jc z Tk 2 jc z T 2 k jr S C (EF + + EF + ) (EF EF ) (224) Once again, Λ diag λ, λ 2,, λ N, λ i are the eigenvalues of M, and T is an orthogonal matrix Similarly, from (24) and (8), Y norm N C C N z 2 M + ω I N j C C N z 2 T Λ T + ω I N j C C N z 2 26 The N + 2 Coupling Matrix T k T Nk j R S R L C C N 2P ε (EF EF ) (225) Till now, it is assumed that coupling is an intra resonator phenomena In addition, if coupling between source/load to inner resonators is allowed, then fully canonical filter responses (ie, n fz N) too can be achieved A LPP configuration with source/load to inner resonator couplings is shown in Fig 28 26 Analysis of the General N + 2 Coupling Matrix 26 Series Type LPP VL equations corresponding to Fig 28 are given in matrix form as v S S, S,2 S,N S,L S, ωl + X,2,N,L S,2,2 ωl 2 + X 2 2,N 2,L j S,N,N 2,N ωl N + X N N,L v L } S,L,L 2,L {{ N,L } Z i S i i 2 i N i L (226) 5

Figure 28: Series type LPP with source/load to inner resonator couplings Normalizing the above matrix gives v S v L j S, S,2 L L2 S, L ω + X L L 2 L,2 S,2 L2,2 L L 2 ω + X 2 L 2 S,N LN S,L,N,L L L N L 2,N 2,L L2 L N L2 S,N,N L L N 2,N L2 L N ω + X N N,L LN L N N,L LN,L S,L 2,L L L2 }{{} Z norm jm+ωi i S i L i 2 L2 i N LN i L (227) Re-writing (227) gives i S i L i 2 L2 i N LN i L is is i L i L Z norm v S v L Z norm, Z norm,n+2 Z norm N+2, Z norm N+2,N+2 Z norm, Z norm,n+2 Z norm N+2, Z norm N+2,N+2 vs v L v L vs (228) 6

Figure 29: Shunt type LPP with source/load to inner resonator couplings 262 Shunt Type LPP From the duality principle and (228), vs v L vs v L Y norm, Y norm,n+2 Y norm N+2, Y norm N+2,N+2 Y norm, Y norm,n+2 Y norm N+2, Y norm N+2,N+2 is is i L i L, (229) where Y norm j S, S,2 C C2 S, C ω + B C C 2 C,2 S,2 C2,2 C C 2 ω + B 2 C 2 S,N CN S,L,N,L C C N C 2,N 2,L C2 C N C2 S,N,N C C N 2,N C2 C N ω + B N S,L,L C 2,L C2 N,L CN C N N,L CN (23) 262 Synthesis of the General N + 2 Coupling Matrix 262 Series Type LPP From (228), Z norm y M + ω I jy (23) 7

Following the theory presented in Sec 252, N+2 where, M T Λ T t Similarly, 2622 Shunt Type LPP From (229) and (8), T 2 k jy N+2 Tk 2 j (EF + EF ) R S (EF + EF + ), (232) Z norm,n+2 y 2 N+2 T k T N+2,k Y norm z N+2 where, M T Λ T t Similarly, T 2 k jz j 2P RS R L ε (EF + EF + ) (233) N+2 Tk 2 (EF + + EF + ) jr S (EF EF ) (234) N+2 Y norm,n+2 z 2 T k T N+2,k j R S R L 2P ε (EF EF ) (235) 263 Synthesis of the N + 2 Transversal Matrix Till now, all the synthesis techniques needed the Gram-Schmidt orthonormalization step This step can be avoided if one starts with a simpler transversal LPP configuration shown in Fig 2 In this configuration, no coupling exists between the resonators Only coupling that exists for each resonator is the corresponding interaction with source/load ABCD parameter matrix corresponding to the highlighted two port section is given as A B C D N j S,N j S,N ( ) N,L jωc N +jb N S,N S,N N,L jωc N + jb N S,N N,L j N,L j N,L 8

Figure 2: Canonical transversal LPP configuration Converting ABCD parameters to Y parameters gives y y 2 y 2 y 22 N d b a jωc N + jb N 2 S,N S,N N,L S,N N,L 2 N,L Since all two port sections are connected parallely, overall Y parameter matrix is given as y y 2 y 2 y 22 total From (236) and (7) j S,L j S,L + (EF + EF ) R S (EF + EF + ) 2P RS R L ε (EF + EF + ) jωc k + jb k j S,L + 2 S,k S,k k,l 2 S,k jωc k + jb k and S,k k,l 2 k,l (236) S,k k,l jωc k + jb k (237) So, one can obtain first and last rows (and columns), and all the diagonal elements of (23) by equating poles and residues on both sides of (237) In addition, all other elements are zero for transveresal prototype 9