It is well known that networks containing both lumped and distributed elements can be treated by the theory of mpr's (multi-variable positive real functions). In the theories of cascade synthesis so far developed for those mixed networks, it has been emphasized that the value of network functions depend only on one variable at any transmission zero. So that, the component sections were composed of J-element (Jun-element) pi and its I-element (Inverse-element) pi1. However, a certain class of multi-variable transfer functions does generally have transmission zeros depending on a set of several variables. In the present paper, the separation of the sections which produce transmission zeros depending on two variables, are discussed. In the result, the separation of the four reactance sections are discussed. These four sections correspond to A (or B), Richards, Brune and Hazony-Youla sections for one-variable case.

The present paper deals with the Richards' transformation which converts a multi-variable positive (real) function to another multi-variable positive (real) function. This transformation is based on the fact that the network function reduces to a constant for all zeros of an irreducible polynomial h (p1, p2, , pn) (1degpi h). And its application to the cascade synthesis or the Bott-Duffin type synthesis of a class of multi-variable driving-point impedances, is discussed. Finally, we present the corresponding transformation for the multi-variable positive (real) matrices.

This paper presents two types of two-variable analog filters with maximally flat magnitude-squared attenuation response in the two-dimensional pass region. These are applied in order to obtain five types for the distribution of two-dimensional pass regions with respect to the design of microwave band pass filters consisting of a cascade of commensurate-line filter and lumped LC filter or a cascade of two commensurate-line filters in different propagation times.

We show that the necessary and sufficient condition for the existence of a balanced C4-bowtie decomposition of the complete multi-graph λKn is λ(n - 1) 0 (mod 16) and n 7. Decomposition algorithms are also given.

This paper presents a novel dual-band comb-line filter using a pair of hybrid resonators. The resonator consists of a half-wavelength stripline resonator short-circuited at both ends and a quarter-wavelength resonator of coplanar waveguide that is nested in the half-wavelength resonator. Numerical calculations by an electromagnetic simulator clarify the characteristics of dual-frequency resonance of the hybrid resonator when the structural parameters are changed. The surface current density on the resonator is also investigated at the resonant frequencies. A typical model of the resonator is fabricated and its resonance frequency characteristics are measured.

We show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the complete multigraph λKn is n 5 and λ(n-1) 0 (mod 12). Decomposition algorithms are also given.

We show that the necessary and sufficient condition for the existence of a balanced quatrefoil decomposition of the complete multigraph λKn is n 9 and λ(n - 1) 0 (mod 24). Decomposition algorithms are also given.

The following, which is related to the design of the microwave filters, is mainly presented: (1) certain useful approximation which can be obtained by double-resistive- terminated 2-ports consisting of a cascade of two 1-variable 2-ports in different variables, and (2) an approach for filter design from 2-variable viewpoint. Approximations presented provide useful magnitude responses in 2-D domain. Hence it is discussed that how the provided 2-D responses can be used for the design of the microwave filters. Furthermore, properties of the 2-variable transfer functions resulting in such circuits are given.

First, we show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the complete tripartite multi-graph λ Kn1,n2,n3 is (i) n1=n2=n3 0 (mod 6) for λ 1,5 (mod 6), (ii) n1=n2=n3 0 (mod 3) for λ 2,4 (mod 6), (iii) n1=n2=n3 0 (mod 2) for λ 3 (mod 6), and (iv) n1=n2=n3 2 for λ 0 (mod 6). Next, we show that the necessary and sufficient condition for the existence of a balanced trefoil decomposition of the complete tripartite multi-graph λ Kn1,n2,n3 is (i) n1=n2=n3 0 (mod 9) for λ 1,2,4,5,7,8 (mod 9), (ii) n1=n2=n3 0 (mod 3) for λ 3,6 (mod 9), and (iii) n1=n2=n3 3 for λ 0 (mod 9).

We show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the symmetric complete multi-digraph is n 5 and λ(n-1) 0 (mod 6). Decomposition algorithms are also given.

We show that the necessary and sufficient condition for the existence of a balanced C4-trefoil decomposition of the complete multi-graph λKn is λ(n-1) ≡ 0 (mod 24) and n ≤ 10. Decomposition algorithms are also given.

Let t and n be positive integers. We show that the necessary and sufficient condition for the existence of a balanced t-foil decomposition of the complete graph Kn is n 1 (mod 6t). Decomposition algorithms are also given.