Network Synthesis | Hurwitz Polynomial | Positive Real Functions

Theory of Network Synthesis

Network Functions

Network synthesis constructs a circuit that realises a specified driving-point or transfer function. Some synthesis classes allow active components. Passive realisations use resistors, inductors and capacitors.

What is a network function? With zero initial conditions, a network function is the ratio of a selected output transform to an input transform in the Laplace domain. Setting s = jω gives the sinusoidal frequency response where that boundary value exists.

Thus, network functions describe the input-output relation of a linear time-invariant network as a function of complex frequency s.

The next three checks apply to rational passive driving-point impedance or admittance candidates. They are realizability checks, not sufficient conditions for the BIBO stability of every transfer function:

1. For a positive-real driving-point function F(s), numerator and denominator degrees can differ by no more than one: |m – n| ≤ 1.
2. Any poles of F(s) on the jω-axis must be simple and have real positive residues.
3. F(s) must have no poles in the open right half of the s-plane.

Hurwitz Polynomial

A strict Hurwitz polynomial has real coefficients and all roots in the open left half-plane. Network-synthesis texts sometimes use a modified convention that permits simple roots on the jω-axis while excluding roots in the open right half-plane.

Under the stated convention, Q(s) is the Hurwitz polynomial associated with the denominator.

Properties of Hurwitz Polynomials

The following five checks describe the network-synthesis convention used on this page:

1. A real-coefficient polynomial P(s) has a real value for every real s.
2. No root lies in the open right half-plane. Strict Hurwitz polynomials place every root in the open left half-plane; the modified synthesis convention may allow simple jω-axis roots.
3. Let Q(s) have coefficients b_n, b_(n-1), b_(n-2) … b₀. After normalising the leading sign, its nonzero coefficients must have one sign. Thus b_n, b_(n-1) and b₀ are nonnegative where present, while b_n and b_(n-1) cannot both be zero. This sign check is necessary but not sufficient for a general higher-order polynomial.
4. A Routh-style continued-fraction test divides the higher-degree even or odd part by the other part. A Hurwitz polynomial produces quotient coefficients with the required positive sign at each valid step.
5. For a purely even or purely odd polynomial, use its derivative as the missing companion part before applying the continued-fraction procedure.

For a quadratic a s² + b s + c, strictly positive real coefficients a, b and c are sufficient for both roots to have negative real parts. This shortcut does not extend to every higher-order polynomial.

Positive Real Functions

A real rational function F(s) is a positive real function when it is analytic in the open right half-plane and Re F(s) ≥ 0 there. For rational functions, the following boundary tests express the same requirements:

1. F(s) is real for every real s where it is finite, which requires real coefficients after normalisation.
2. After common factors are cancelled, the denominator Q(s) has no roots in the open right half-plane.
3. For every real ω where F(jω) is finite, its real part satisfies Re F(jω) ≥ 0.
4. Any poles on the jω-axis are simple, and their residues are real and positive.

Properties of Positive Real Function

Four useful closure and degree properties of positive real functions are:

1. After common factors are cancelled, the numerator and denominator satisfy the modified Hurwitz condition used in passive network synthesis.
2. The numerator and denominator degrees differ by at most one: |m – n| ≤ 1.
3. If a nonzero F(s) is positive real, then 1/F(s) is also positive real after removable common factors are cancelled.
4. A sum of positive-real functions is positive real. A difference is not necessarily positive real.

The following four algebraic checks are necessary for a rational positive-real driving-point function, but they do not prove positive-realness by themselves:

1. After a common sign normalisation, numerator and denominator coefficients are real and their nonzero terms have one sign.
2. The numerator and denominator degrees differ by no more than one: |m – n| ≤ 1.
3. Poles and zeros on the imaginary axis are simple after common factors are cancelled.
4. For denominator coefficients b_n, b_(n-1), b_(n-2) … b₀, normalise b_n as positive. The nonzero coefficients, including b_(n-1) and b₀, cannot change sign; b_n and b_(n-1) cannot both vanish.

For a real rational F(s), the following two analytic checks, together with realness for real s, are necessary and sufficient for positive-realness:

1. F(s) has no poles in the open right half-plane. Any poles on the jω-axis are simple and have real positive residues.
2. Re F(jω) ≥ 0 for every real ω where F(jω) is finite.