Synthesis of one port and two port networks

Network synthesis is the reverse of analysis: given a desired impedance or transfer function, find a circuit that realises it. Before building a circuit we must check that the function is physically realisable using passive elements. This is done with Hurwitz polynomials and positive real (PR) functions. Then standard canonical forms — Foster (partial fractions) and Cauer (continued fractions) — give LC, RC and RL networks, usually in ladder form. NEC questions focus on properties of Hurwitz and PR functions and on recognising Foster and Cauer forms.

Key concepts

Hurwitz polynomials A polynomial $P(s)$ is Hurwitz if all its roots have negative real parts (strictly Hurwitz) or lie on the jω axis as simple roots (modified Hurwitz). Properties:

• All coefficients are real and positive; no coefficient missing between highest and lowest power, unless the polynomial is entirely even or entirely odd.
• Test: split $P(s) = m(s) + n(s)$ (even + odd parts). The continued-fraction expansion of $m/n$ (or $n/m$) must give all positive quotients, and the number of quotients equals the degree.
• Routh array with no sign changes in the first column also confirms Hurwitz.
• Denominator of a stable network function must be Hurwitz.

Positive real (PR) functions $F(s)$ is PR if (1) $F(s)$ is real when s is real, and (2) $\text{Re}[F(s)] \ge 0$ whenever $\text{Re}(s) \ge 0$. Every driving-point impedance of a passive RLC network is PR. Practical tests:

• Numerator and denominator are Hurwitz polynomials.
• Degrees of numerator and denominator differ by at most 1; lowest powers also differ by at most 1.
• Poles and zeros on the jω axis are simple with real positive residues.
• $\text{Re}[F(j\omega)] \ge 0$ for all ω.
• If F is PR, then $1/F$ is also PR; sum of PR functions is PR.

Properties of LC driving-point functions

• Ratio of even to odd (or odd to even) polynomial.
• Poles and zeros are simple, lie on the jω axis, and alternate.
• There is always a pole or zero at s = 0 and at s = ∞.
• $dX/d\omega > 0$ (reactance always increases with frequency — Foster's reactance theorem).

RC impedance (same rules for RL admittance)

• Poles and zeros on negative real axis, alternating; critical frequency nearest the origin is a pole; $Z_{RC}(0) > Z_{RC}(\infty)$; slope $dZ/d\sigma < 0$. RL impedance (same as RC admittance): critical frequency nearest the origin is a zero; $Z_{RL}(\infty) > Z_{RL}(0)$.

Foster forms (partial fraction expansion)

• Foster I: expand $Z(s)$; each term is a series element or parallel tank, connected in series. For LC: $Z(s) = \frac{k_0}{s} + k_\infty s + \sum \frac{2k_is}{s^2+\omega_i^2}$ → series capacitor, series inductor, and parallel LC tanks.
• Foster II: expand $Y(s)$; branches in parallel (series LC branches).

Cauer forms (continued fraction expansion → ladder networks)

• Cauer I: expand about s = ∞ (divide highest powers first) → series inductors and shunt capacitors for LC (low-pass ladder shape).
• Cauer II: expand about s = 0 (lowest powers first) → series capacitors and shunt inductors (high-pass ladder shape).
• For an LC function, all four canonical forms use the same minimum number of elements, equal to the highest power of s in its numerator or denominator.

Two-port synthesis

• Ladder networks realise transfer functions whose zeros of transmission occur where a series arm becomes open or a shunt arm becomes a short.
• Resistively terminated LC two-ports (single or double terminated) are designed so a lossless ladder sits between source and load resistors — the basis of Butterworth and Chebyshev passive filters.
• Active synthesis uses op-amps with RC elements (e.g. Sallen-Key) to realise any stable transfer function without inductors.

High-yield points

• Hurwitz polynomial: roots in the left half plane (or simple on the jω axis); coefficients positive, none missing unless purely even or odd.
• PR function: numerator and denominator degrees differ by at most 1, and $\text{Re}[F(j\omega)] \ge 0$.
• LC function: simple, alternating poles and zeros on the jω axis, with a pole or zero at both s = 0 and s = ∞.
• RC impedance: critical frequency nearest the origin is a pole; RL impedance: a zero.
• Foster = partial fractions; Cauer = continued fractions (Cauer I about ∞, Cauer II about 0).

Formulas

Foster I LC terms: residue at pole $\pm j\omega_i$ gives tank $L_i = 2k_i/\omega_i^2$, $C_i = 1/(2k_i)$.

Continued fraction (Cauer I): $Z(s) = a_1 s + \cfrac{1}{a_2 s + \cfrac{1}{a_3 s + \cdots}}$ where $a_1$ = series L (H), $a_2$ = shunt C (F), etc.

Worked example

Is $P(s) = s^3 + 2s^2 + 2s + 1$ Hurwitz?

1. Even part $m = 2s^2 + 1$, odd part $n = s^3 + 2s$.
2. $n/m$: $s^3 + 2s = (s/2)(2s^2+1) + 1.5s$ → quotient $s/2$.
3. $(2s^2+1)/(1.5s)$: quotient $(4/3)s$, remainder 1.
4. $1.5s/1$: quotient $1.5s$.
5. Quotients $0.5, 1.333, 1.5$ — three, all positive. Degree is 3. So P(s) is Hurwitz. (Indeed roots are $-1$ and $-0.5\pm j0.866$.)

Exam tips

• Hurwitz polynomial: all coefficients positive and none missing (unless purely even/odd).
• PR function: degree difference of numerator and denominator ≤ 1.
• LC functions: poles/zeros alternate on jω axis; RC impedance has a pole nearest the origin.
• Foster = partial fractions; Cauer = continued fractions (ladder).
• Cauer I expands about infinity; Cauer II about zero.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. The impedance $Z(s) = \dfrac{s^3 + 2s}{s^2 + 1}$ is realised in the Cauer first form (continued fraction about infinity). What is the ladder?

• A. Series 1 H, shunt 2 F, series 0.5 H
• B. Shunt 1 F, series 1 H, shunt 1 F
• C. Series 1 H, shunt 1 F, series 1 H
• D. Series 2 H, shunt 1 F, series 1 H

Answer: C. $Z = s + \dfrac{s}{s^2+1} = s + \dfrac{1}{s + \frac1s}$: series $L = 1$ H, then shunt $Y = s$ ($C = 1$ F), then series $Z = s$ ($L = 1$ H).

2. Which of the following functions is positive real?

• A. $\dfrac{1}{(s+1)^2}$
• B. $\dfrac{s-1}{s+1}$
• C. $\dfrac{s+2}{s+1}$
• D. $\dfrac{s^2}{s+1}$

Answer: C. $(s+2)/(s+1)$ has a LHP pole and zero and $\mathrm{Re}$ part $\ge 0$ on the $j\omega$ axis. The others fail: a RHP zero, a degree difference of 2, and a double zero on the $j\omega$ axis.

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