The concept of reactance one-port function stems from electrical network theory: In arbitrary passive (that is RLC-) networks we find the following relations between the complex voltage Uν(s**) and the complex current** Iν (s):

![0_image_0.png](0_image_0.png)

![0_image_2.png](0_image_2.png)

$${\mathrm{an~inductance~}}L_{l}$$
Ur(s) = Rr · Ir(s**) for a resistor** Rr

![0_image_1.png](0_image_1.png)

$$\mathbf{\Phi}(s)\,=s\cdot L_{l}\,\cdot\,$$

Ul(s) = s · Ll· Il(s**) for an inductance** Ll

$U_k(s)=\frac{1}{s\cdot C_k}\cdot I_k(s)$  for a capacity $C_k$ . 

An impedance (complex resistor) or admittance (complex conductance) composed of network elements R, L and C only is called a (RLC-) one-port **function,**
an impedance or admittance composed of network elements L and C only is called a reactance one-port function. Conversely, for every one-port function Z(s**) there exists at least one one-port, whose impedance or admittance is equal**
to Z(s):
Hence, theorem 1 reduces stability checking to the considerable easier task to synthesize a one-port solely using inductors (L) and capacitors **(C), that is to** synthesize a reactance one-port. To this end there exist easy procedures like for example Routh's method to construct a chain one-port [Unb93].

It turns out that one-port functions Z(s) are exactly the real positive rational functions. For a real function we have that for real s also Z(s**) is real**7, and a positive function means that ℜ{s} > 0 implies ℜ{Z(s)} > **0. A reactance** one-port function is a one-port function, that is in addition odd. The property of being positive is closely connected to Hurwitz polynomials:

## Theorem 2. **([Unb93])**

Let f(x) be a real polynomial with degree n ≥ **1. Furthermore let all coefficients**
of f(x) be greater than 0. Let fe(x) and fo(x**) denote the even part resp. the** odd part of f(x). Assume thatfe(x) and fo(x**) have no common roots and that**
Z(x) = fe(x)/fo(x**) is positive. Then**
(i) ℜ{Z(x)} ≥ 0 for all x with ℜ{x} = 0
(ii) fe(x) + fo(x**) is a Hurwitz polynomial.**
7 This condition implies that the coefficients of Z(s) are real. In network theory, however, this definition is used.