Methodology for Generation of Brune’s Equivalent Circuit Models for Linear Passive Reciprocal Multi-ports F. Mukhtar∗
J. A. Russer∗
Y. Kuznetsov†
P. Russer∗
Abstract — In this paper Brune’s multi-port synthesis is treated systematically. The method allows to synthesize linear passive reciprocal multi-port circuits using lumped element RLC networks containing only positive circuit elements.
where Aar is an arbitrary constant matrix with complex entries. Right-side of Eq. 1 is a degree determinant of a determinant which would develop into a rational function in complex frequency s.
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Poles and their ranks
Introduction
The application of network methods to electromagnetic field problems can contribute significantly to problem formulation and to an efficient solution methodology [1, 2]. Connection networks in numerical hybridization of different techniques and lumped element models for radiating modes are examples for the application of network methods. A systematic procedure for Brune’s one-port circuit synthesis is discussed in [3]. In this paper we present a methodology for generation of Brune’s equivalent circuit models for linear passive reciprocal multiports. This allows to realize lumped element equivalent circuits exhibiting only positive R-. L- and C- elements and ideal transformers. In the following section, the special class of positive real symmetric matrices which is used to represent impedance or admittance matrices is discussed. It is followed by a discussion of a circuit realization most commonly used in Brune’s synthesis. Finally, different cases of Brune’s algorithms are described. A common symbol W is used to represent either impedance or admittance matrices. 2
The pole of any element of a matrix W(s) is also a pole of that matrix. We shall denote the set of all poles by π(W). The rank of a pole of the matrix is the rank of the matrix of residues of that pole [6, §2]. Zeros and their ranks
When the determinant of a matrix W(s) is zero at a certain complex frequency s, then we say that the network has a zero at that complex frequency. All zeros of a matrix are poles of its inverse [6, §2]. We shall denote the set of all poles of W(s) by ζ(W). The rank of a zero of a matrix is the rank of the matrix of residues of the corresponding poles in the inverse of that matrix. Realizability
This is a set of necessary and sufficient conditions which a matrix has to satisfy to be synthesizable into a passive network (see [6, §2] and [7, Ch. 1]). 1. No pole or zero has to be on the right half plane of complex frequency s. 2. {W(s = jω)} = A(ω) has to be positive definite, where denotes the real part. For A(ω) to be positive definite, Aii (ω) ≥ 0 for all i = 1, 2, · · · , n and all principal minors and the determinant of A(ω) should be positive real. It should be noted that off-diagonal entries can be negative.
Positive Real Symmetric Matrices
Impedance and admittance matrices of linear reciprocal passive n-port circuits belong to the class of positive real symmetric matrices (PRSM) [4], which are characterized by following parameters and conditions: Order or Degree
3. If there are poles on the imaginary axis then for each such pole ωp the matrix of residues Kωp must have real elements and should be positive definite.
The order or degree of a given matrix W(s), denoted as deg(W), is defined as [5, 6] deg(W) = deg (|W(s) + Aar |)
(1)
∗ Institute for Nanoelectronics, Technische Universit¨ at M¨ unchen, Germany, e-mail:
[email protected],
[email protected],
[email protected] † Theoretical Radio Engineering Department, Moscow Aviation Institute, Moscow, Russia email:
[email protected]
Function (ω)
This is a special function used in Burne’s process. We shall denote it by the second letter of the Hebrew alphabet (pronounced ’beth’). If A(ω) = [W]|s=jω then
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Figure 1: Equivalent circuit of rank r impedance Figure 2: Equivalent circuit of rank r admittance matrix with its symbol matrix with its symbol 4 Cases of PRSM and their Extraction Brune’s Algorithm is an iterative algorithm. In |A(ω)| (ω) = , (2) each iteration a part of the given matrix W(s) is M11 (ω) extracted and synthesized, leaving behind a matrix where M11 (ω) is the minor of first element in A(ω). W (s) which is of lesser order. The number of itThe minimum of this function will be used to ex- erations depends on the order of W(s). There are seven cases (Cases 1 to 7) for W(s), tract the resistance from the first port in the algoincluding a stop criterion (Case 0). For each case rithm. the extraction and synthesis procedure is different. 3 Synthesis of Circuit of Type In each iteration, the case of W(s) is checked and W(s) = CW r w(s) corresponding extraction procedure is applied. The synthesis of an n × n matrix W(s) which can Case 0 (Stop Criterion) be represented as a multiplication of the frequency Order δ(W) = 0 which indicates the circuit is a dependent function w(s) and a real matrix CW r is pure resistive / conductive network, and can be considered here. The real matrix CW is symmetr written (as described above) ric (due to the reciprocity of the circuit) and is of r rank r where 1 ≤ r ≤ n. This matrix would turn CW W = 1,i A . out to be the connecting network consisting of ideal i=1 transformers. It connects the circuit realization of w(s) with the n-ports of the circuit. Due to the Figs. 3(a)-(b) show the possible networks for spectral theorem, any real symmetric matrix is di- impedance and admittance respectively. agonalizable by an orthogonal matrix. So we can Case 1 write Wij (s) → ∞ as s → ∞ for any i, j = 1, 2, ..., n, W T where n is the number of ports. For each element of W(s) = Cr w(s) = QΛQ w(s) the matrix having this pole, division of numerator r r T W with denominator will extract the pole. = QΛi Q w(s) = C1,i w(s) . i=1
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W(s) = CW r Ds + W (s) .
where Q is an orthogonal matrix consisting of eigenvectors of CW r and Λ is a diagonal matrix consisting of eigenvalues of CW r . Λ is decomposed into a sum of Λi , where each Λi contains only one nonzero eigenvalue. Fig. 1 and Fig. 2 show the implementation for W = Z and W = Y respectively.
Figs. 3(c)-(d) show the circuits for impedance and admittance respectively. Case 2
|W(s)| → 0 as s → ∞. This shows a pole in the inverse of the matrix. Taking inverse of the matrix
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the remaining matrix is W . Figs. 3(i)-(j) show the possible circuits. Next steps depend on the value of ωmin at which the minimum occurs. There are three possible sitFigs. 3(d)-(c) show the circuits for impedance and uations: Situation 1: ωmin → ∞. In this situation, a zero admittance respectively. has already been created by removing the real value Case 3 A from port 1 and case 2 would follow. Wij (s) → ∞ as s → 0 for any i, j = 1, 2, ..., n, Situation 2: ωmin = 0. In this situation, a zero where n is number of ports. There is a pole at s = has already been created by removing the real value 0. This pole can be extracted from the individual A from port 1 and case 4 would follow. element. Situation 3: ωmin = σ. In this situation, a true zero has not been created yet as the imaginary part WD W(s) = Cr + W (s) . s of the determinant is not zero at ωmin . Let and extracting poles as in case 1, will give −1 −1 W(s) = CW (s) . r Ds + W
1 i.e. The elements C and L will have the value D the reciprocal of the residue of the pole. Figs. 3(e)(f) show the circuits for impedance and admittance respectively.
W (s = jσ) = A (s = jσ) + jI (s = jσ) .
The matrix A (ω = ωmin ) is a rank deficient matrix because of the removal of the real part from port 1 in the previous step, and has some non-zero null Case 4 % vector β. |W(s)| → 0 as s → 0. A × β% = 0 , −1
T where β% = [β1 , β2 , · · · , βn ] . This vector will be −1 WD . + W (s) W(s) = Cr used in extracting the imaginary part to create a s true zero. A rank one symmetric matrix can be Figs. 3(f)-(e) show the circuits for impedance and written as admittance respectively. H = α%hT × %h , T Case 5 where α ∈ {+1, −1} and %h = [h1 , h2 , · · · , hn ] and Wij (s) → ∞ as s → ±jωp for any i, j = 1, 2, ..., n, where values of hi are in such a way that where n is number of ports. Instead of a single pole I − H × β% = 0 at s = 0 or s = ∞, a pole pair occurs at the finite frequency ωp . Extracting this pole pair does not (3) I × β% = α%hT × %h × β% effect the positive real character of the remaining 2 matrix. % %h β% T × I × β% = α β. (4) 2ks W W(s) = Cr 2 + W (s) . From Eq. 4 value of α is determined. This yields s + ωp2 two cases in Situation 3 : Case I: α = −1 Figs. 3(g)-(h) show the circuits for impedance and In this type, a series inductive circuit would be admittance respectively. extracted; followed by extraction of the parallel resCase 6 onance circuit at ωmin and then again series induc|W(s)| → 0 as s → ±jω. tive circuit. [6] has shown that this circuit is equiv −1
alent to the circuit given in Fig. 3 (k). −1 2ks + W (s) . W(s) = CW Case II: α = +1 r s2 + ωp2 This type is opposite to type I, where each inducFigs. 3(h)-(g) show the circuits for impedance and tor is replaced by a capacitor and vice versa. The equivalent circuit is shown in Fig. 3 (l). admittance respectively. A numerical example illustrating the use of this Case 7 procedure is presented in [8] All above cases are related to poles or zeros lying on the imaginary axis of the complex frequency s 5 Conclusion plane. Brune’s process starts by creating a zero on A systematic methodology to synthesize Brune’s the imaginary axis while preserving the P.R. char- equivalent multiport circuits for linear reciprocal passive lossy multiports is presented. The equivaacter of the remaining matrix. Here the function is used. The minimum value lent multiport circuits contain only resistors, capacof this function will be extracted from port 1, and itors and inductors with real positive element values
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Figure 3: Possible Circuits of Brune’s Algorithm tributed one-port and symmetric two-port miand ideal transformers. Starting from numerical crowave circuits,” in GeMIC, Mar. 2011. data the positive real symmetric matrices are obtained by system identification methods and then [4] V. Belevitch, “On the Brune’s process for nare systematically reduced. By this way canoniports,” IRE Trans. Circ. Th., vol. 7, no. 3, pp. cal equivalent lumped element circuit modules are 280–296, Sept. 1960. extracted step by step from the multiport circuit yielding finally the lumped element equivalent mul- [5] R. J. Duffin and D. Hazony, “The degree of a tiport circuit. rational matrix function,” J. So. Indust. Appl. Math., vol. 11, no. 3, pp. 645–658, Sept 1963. Acknowledgement This work has been supported by the Deutsche Forschungsgemeinschaft.
[6] B. D. H. Tellegen, “Synthesis of the 2n-poles by networks containing the minimum number of elements,” J. Math. & Phys., vol. 32, no. 1, pp. 1–18, April 1953.
References [1] P. Russer, Electromagnetics, Microwave Circuit and Antenna Design for Communications En- [7] E. A. Guillemin, Synthesis of Passive Networks. John Wiley & Sons, Inc., 1957. gineering, 2nd ed. Artech House, 2006.
[2] L. B. Felsen, M. Mongiardo, and P. Russer, [8] A. Baev, A. Gorbunova, Y. Kuznetsov, F. Mukhtar, J. Russer, P. Russer, D. Bajon, Electromagnetic Field Computation by Network and S. Wane, “Equivalent circuit model for Methods. Berlin: Springer, Mar. 2009. coupled monolithic integrated millimeter-wave folded antennas,” in ICEAA, 2012, September [3] F. Mukhtar, Y. Kuznetsov, C. Hoffmann, and 2012. P. Russer, “Brune’s synthesis of linear lossy dis-
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