Synthesis of a Finite Two‐terminal Network whose Driving‐point Impedance is a Prescribed Function of Frequency

Sharp inequalities for positive functions

“…Observe that Brune's result [4] implies that if C is the circular arc z = re iθ , |θ| ≤ µ < π/2, then the curve p(C) lies in the sector |θ| ≤ µ for all positive real functions.…”

“…(1) Brune [4] proved that to every positive real function with a finite number of zeros and poles in C\H + , there corresponds a finite physically realizable electric network. It is important to note that all our extremal functions are precisely of this form.…”

“…The collection of all functions p(z) analytic in H + and satisfying e p(z) > 0 for z ∈ H + is called the class of positive functions and is denoted by P . The class of positive functions which satisfy p(z) = p(z) is called the class of positive real functions, denoted by P R. Apart from their mathematical interest, positive real functions arise as the complex impedance of passive two-terminal networks where z is the complex frequency (see [4] or [1]). In particular, it was shown by O. Brune [4] in 1931 that each such system gives rise to a positive real function and conversely.…”

Synthesis of SEM-Derived Equivalent Circuits for Energy-Collecting Structures.

“…APerhaps a more systematic approach to SEM equivalent circuit synthesis was propsoed by Baum [1976a] (some of this material appeared recently in [Baum, 1978] [Brune, 1931] synthesis method to derive equivalent circuits. As usual in Brune's method, perfect transformers were employed.…”

“…They utilized the BottDuffin (Bott and Duffin, 1949] (Brune, 1931] functions of frequency s and speculated on the PRness of the admittances associated with individual current eigenmode contributions to total current. They also briefly discussed the source synthesis issue, but they did not present any actual active circuits.…”