Passive One-Port Network Synthesis

Introduction

Network synthesis is the inverse of network analysis. In analysis, a known circuit is used to determine its impedance, admittance, transfer function, or response. In synthesis, a prescribed mathematical function is converted into a physical network of resistors, inductors, capacitors, and, where necessary, ideal transformers or coupled elements.

A passive one-port contains no independent energy source and is accessed through one pair of terminals. Its driving-point impedance is

where s=σ+jω is the complex frequency. The corresponding driving-point admittance is Y(s)=1/Z(s). For finite lumped networks with constant positive element values, these functions are real-rational and positive real.

Passive synthesis is central to filters, matching networks, equalizers, oscillation suppression, power-system equivalents, sensor interfaces, electromagnetic compatibility, and macromodel implementation. The essential problem is not merely to obtain a circuit with the right algebraic expression, but to guarantee passivity, physical realizability, acceptable sensitivity, and practical component values.

Learning Objectives

After studying this chapter, the reader should be able to:

• explain the positive-real condition and its relationship to passivity;
• test rational driving-point functions for necessary realizability properties;
• synthesize LC, RC, RL, and suitable general RLC functions using Foster forms;
• apply continued-fraction expansion to obtain Cauer ladders;
• recognize when direct Foster or Cauer extraction is insufficient;
• describe Brune and Bott–Duffin synthesis for general positive-real functions;
• perform impedance and frequency scaling;
• verify a synthesized network analytically and computationally;
• evaluate practical limitations associated with tolerances, parasitics, losses, and passive-model fitting.

1. Passive One-Ports and Positive-Real Functions

1.1 Energy interpretation

For a passive one-port initially at rest, the energy delivered to the port cannot be negative:

This time-domain inequality leads, under linear time-invariant lumped-network assumptions, to the positive-real property of the driving-point immittance.

A real-rational function F(s) is positive real, abbreviated PR, if:

1. F(s) is real for real positive s;
2. it is analytic for Re(s)>0;
3. Re\{F(s)\}\ge 0 for every Re(s)>0.

For real-rational functions, realness is equivalent to having real coefficients after cancellation of common factors.

A finite passive RLC driving-point impedance or admittance is PR. Conversely, every real-rational PR function has a finite passive realization, although a simple Foster or Cauer ladder may not exist. Brune synthesis provides a realization using ideal transformers or coupled inductors; the Bott–Duffin theorem establishes transformerless RLC realizability, generally at the cost of a more complicated topology.

1.2 Boundary properties

Useful consequences of positive realness include:

• no pole lies in the open right half-plane;
• a pole on the imaginary axis, including the origin, must be simple;
• the residue at an imaginary-axis pole is real and nonnegative;
• complex poles and zeros occur in conjugate pairs;
• for a nonzero PR function, its reciprocal is also PR;
• a pole at infinity, if present in a rational lumped RLC immittance, is at most simple.

Thus properness is not required. For example,

is the impedance of an inductor and is improper by one degree. Here L>0 is measured in henries.

Similarly, Z(s)=1/(sC) has a simple pole at the origin and represents a capacitor of capacitance C>0.

1.3 Element immittances

Here R is in ohms, L in henries, C in farads, and s in s⁻¹. Positive values make each element passive.

1.4 Practical PR testing

Let

be reduced to coprime real polynomials. A reliable test proceeds as follows:

1. Confirm that N(s) and D(s) have real coefficients.
2. Confirm that D(s) has no zero in Re(s)>0.
3. Inspect imaginary-axis poles and their residues.
4. Evaluate the real part on s=jω wherever the function is finite:

1. Verify nonnegativity for all real ω, including limiting behavior at zero and infinity.

Boundary testing alone must be combined with pole and residue checks. In numerical work, passivity should be tested over a continuum by polynomial, state-space, or Hamiltonian methods rather than inferred solely from a coarse frequency sweep.

2. Reactance and Susceptance Functions

A lossless LC one-port has an impedance that is purely imaginary on the frequency axis except at poles. Such an impedance is called a reactance function. It is a real-rational odd function:

The poles and zeros of an LC reactance function are simple, lie on the imaginary axis, and strictly interlace. Its slope with respect to frequency is positive between singularities. Dual statements apply to LC susceptance functions.

These strong properties make LC functions especially suitable for Foster and Cauer canonical realizations. Resistive loss moves poles into the left half-plane and removes the simple pole-zero interlacing structure.

3. Foster Synthesis

Foster synthesis expresses an immittance as a sum of elementary positive-real terms. An impedance expansion leads to a series connection; an admittance expansion leads to a parallel connection.

3.1 Foster form I for LC impedances

A general LC reactance impedance can be written, subject to the poles actually present, as

where L_∞\ge0, C₀>0, A_k>0, and ω_k>0. Terms with zero coefficients are omitted.

The resonant term

is the impedance of a parallel LC branch. Its element values are

Because s and ω_k have units s⁻¹, A_k has units Ω/s, equivalently F⁻¹. Consequently, 1/A_k is in farads and A_k/ω_k² is in henries.

All branches are connected in series, together with L_∞ and C₀. This is Foster form I.

3.2 Foster form II for LC networks

Instead of expanding impedance, expand admittance:

Each resonant term is the admittance of a series LC branch with

The branches are connected in parallel. Foster forms I and II realize the same immittance class but usually have different topologies and sensitivities.

3.3 Standard RC and RL Foster expansions

An RC driving-point impedance having simple negative-real poles can take the Foster form

where R_∞\ge0, K₀\ge0, K_k>0, and p_k>0. The term K₀/s is a capacitor with C₀=1/K₀. Each term K_k/(s+p_k) is a parallel RC branch with

Here K_k has units Ω/s.

A common RL impedance form is

where R_k>0 and p_k>0. Each rational term is a parallel RL branch having

Dual forms are obtained by expanding admittance.

3.4 Foster treatment of general RLC functions

For a general RLC rational impedance, real partial-fraction expansion may contain constant, polynomial, real-pole, and conjugate-pole terms. A particularly useful admissible decomposition is

with all displayed parameters positive. The last term is the impedance of a parallel RLC branch because

Therefore, for

the element values are

where A_r is in Ω/s, a_r in s⁻¹, and ω_r in s⁻¹.

This decomposition gives a direct series Foster realization when every individual term has the required positive coefficient. However, an arbitrary PR RLC function need not split into independently PR partial-fraction terms. A general quadratic numerator αs+β may not correspond to one uncoupled parallel RLC branch, and residues associated with left-half-plane complex poles can be complex. In such cases, ordinary term-by-term Foster realization is not universal; Brune, Bott–Duffin, Darlington, or other structured synthesis is required.

Worked Example 1: General RLC Foster realization

Consider

All numerical values are expressed in consistent SI units. The terms give:

• 2s: a series inductor L=2\ H;
• 3: a series resistor R=3\ Ω;
• 4/(s+2): a parallel RC branch with

• 5s/(s²+2s+25): a parallel RLC branch with

The four sections are connected in series. Independent verification follows from substitution of the element values into their impedance formulas.

Passivity can also be checked term by term. For the damped resonant term,

when A>0 and a>0. The real-pole RC term likewise has a nonnegative real part. Because sums of PR functions are PR, the complete impedance is PR.

4. Cauer Synthesis by Continued Fractions

Cauer synthesis repeatedly extracts the impedance or admittance of an element from a rational function. It naturally produces ladder networks and is based on the Euclidean algorithm.

4.1 Extraction principles

For impedance expansion about s=∞, the leading behavior may identify a series inductor or resistor:

The appropriate limit depends on the degree and network class. After extracting a series element, the remainder is inverted to expose a shunt element. Alternating subtraction and inversion generates a continued fraction.

Expansion about s=0 instead extracts low-frequency elements first. These alternatives are often called Cauer forms I and II, although naming conventions differ among texts; the expansion point should therefore always be stated.

Worked Example 2: RC Cauer ladder

Let

At high frequency,

Subtracting this resistor gives

Invert the remainder:

Thus the first shunt element is C₁=0.5\ F. Inverting the residual admittance gives

Hence the final series arm contains R₂=4\ Ω and a capacitor satisfying 1/(C₂s)=2/s, so C₂=0.5\ F. The continued fraction is

Figure: Cauer RC ladder obtained by high-frequency continued-fraction expansion.

Direct recombination of the ladder expression reproduces the original rational impedance, completing the algebraic verification.

5. General Positive-Real Synthesis

5.1 Why simple extraction can fail

A function may be PR even though its first polynomial-division coefficient, residue, or continued-fraction coefficient is unsuitable for a positive uncoupled element. This does not disprove realizability; it only shows that the chosen canonical extraction is inappropriate.

Common causes include:

• complex left-half-plane poles;
• partial-fraction terms that are not individually PR;
• negative coefficients produced by an unsuitable expansion;
• a topology constraint inconsistent with the prescribed function;
• near-cancellations that make numerical extraction unreliable.

5.2 Brune synthesis

Brune synthesis applies to arbitrary real-rational PR impedances. In outline:

1. Determine the minimum resistance

including relevant limiting values.

1. Extract R_min as a series resistor.
2. The remainder has zero real part at at least one frequency.
3. Extract a reactive section, generally involving tightly coupled inductors or an ideal transformer.
4. Iterate on a lower-degree PR remainder.

The detailed Brune section must preserve positive realness and cannot generally be replaced by an arbitrary uncoupled resonator.

5.3 Bott–Duffin and Darlington approaches

The Bott–Duffin construction proves that any rational PR impedance can be realized using only positive R, L, and C elements without transformers. Its networks are usually less economical than canonical ladders.

Darlington synthesis embeds a prescribed PR impedance in a lossless two-port terminated by a resistor. It is important in filter and matching-network design because dissipation can be concentrated in the termination while the embedding network remains reactive.

6. Scaling and Practical Realization

Canonical examples often produce inconvenient element values. If a normalized network is transformed using impedance scale R_s and frequency scale ω_s, the physical elements are

where subscript n denotes normalized values. These relations preserve the network shape while moving its impedance and frequency levels.

A practical synthesis workflow is:

1. reduce the rational function and record units;
2. test positive realness without imposing an incorrect properness requirement;
3. classify the function as LC, RC, RL, or general RLC;
4. inspect pole-zero locations and limiting behavior;
5. choose Foster, Cauer, Brune, Darlington, or Bott–Duffin synthesis;
6. extract elements while checking signs and dimensions after every step;
7. recombine the network symbolically;
8. compare frequency responses numerically;
9. scale values and select standard components;
10. evaluate tolerance, parasitic, thermal, voltage, current, and quality-factor constraints.

7. Illustrative Application Case: Passive Macromodel for a Sensor Interface

Background

An instrumentation team obtains a rational impedance model from measured frequency-response data for a piezoelectric sensor and its cable. This is an illustrative case. The model must be inserted into a circuit simulator used to design a low-noise interface.

Problem

An unconstrained curve-fitting algorithm matches the measured samples closely but produces a small frequency band in which Re\{Z(jω)\}<0. When connected to an active amplifier model, the fitted one-port generates nonphysical energy and causes unstable simulation.

Analysis

A good frequency fit does not guarantee passivity. Measurement noise, insufficient model order, delayed behavior, and numerical ill-conditioning can all produce a non-PR rational approximation. A negative real part represents active behavior and invalidates direct passive RLC synthesis.

Approach and solution

The team:

1. removes unreliable out-of-band samples and estimates measurement uncertainty;
2. obtains a stable rational approximation using vector fitting;
3. applies passivity enforcement to perturb residues while preserving the measured response within tolerance;
4. verifies right-half-plane analyticity and nonnegative real part using a state-space passivity test;
5. decomposes simple real-pole terms into RC branches;
6. uses damped RLC resonators for admissible quadratic terms;
7. applies a structured general synthesis method to the remaining coupled terms;
8. performs impedance and frequency scaling to obtain manufacturable values;
9. validates the macromodel against independent measurements and worst-case component tolerances.

Outcome

The corrected model remains passive when connected to any passive termination, eliminates artificial simulation growth, and reproduces the measured terminal response within the specified uncertainty band.

Learning points

Passivity is a system property, not merely a visual feature of a sampled response. Data-driven synthesis should preserve stability, causality, realness, and passivity simultaneously. Model order and topology must also be chosen with regard to implementation cost and parameter sensitivity.

8. Verification, Limitations, and Contemporary Practice

8.1 Verification methods

A synthesized network should be checked by at least two independent methods:

• symbolic recombination of element immittances;
• numerical comparison over a dense frequency grid;
• pole-zero and residue inspection;
• time-domain energy tests;
• state-space passivity analysis;
• Monte Carlo tolerance analysis.

Agreement at a few frequency points is not sufficient because algebraic or topological errors can remain hidden.

8.2 Physical limitations

Ideal synthesis assumes frequency-independent resistors, lossless inductors and capacitors, exact element values, and negligible electromagnetic coupling. Real components have equivalent series resistance, dielectric loss, self-resonance, saturation, temperature coefficients, and finite quality factor. At microwave frequencies, distributed transmission-line synthesis may be more appropriate than lumped RLC synthesis.

Canonical realizations are also not equally robust. A mathematically valid network may require extreme element ratios or exhibit severe sensitivity near a pole-zero cancellation. Network transformations, order reduction, and optimization can improve manufacturability.

8.3 Technology and computational developments

Modern synthesis combines classical PR theory with computer algebra, convex optimization, semidefinite programming, vector fitting, and state-space passivity enforcement. Machine-learning tools may assist topology search or surrogate modeling, but any proposed model still requires deterministic checks of causality, stability, passivity, units, and physical constraints.

8.4 Professional, safety, and governance considerations

Incorrect passive models can underpredict current, voltage stress, heating, or resonance. Engineers should document the frequency range, measurement uncertainty, synthesis method, tolerances, and model limitations. Components must satisfy applicable insulation, electromagnetic-compatibility, thermal, and safety requirements. Deliberately presenting a non-passive fitted model as a physical passive device is both technically misleading and potentially unsafe.

Chapter Summary

Passive one-port synthesis converts a prescribed driving-point impedance or admittance into a realizable network. The fundamental realizability concept is positive realness: a passive rational immittance is real-rational, analytic in the open right half-plane, and has a nonnegative real part there. Properness is not required; an inductive impedance may contain a first-degree polynomial term.

Foster synthesis uses partial-fraction expansions. LC Foster forms produce series or parallel combinations of resonant sections, while RC and RL functions use real negative-axis poles. Suitable general RLC decompositions may include damped parallel RLC sections, but arbitrary PR functions do not necessarily separate into independently passive partial-fraction terms. Cauer synthesis uses continued fractions to generate ladder networks. General PR functions may require Brune, Bott–Duffin, or Darlington methods.

Reliable practice requires dimensional checking, symbolic reconstruction, numerical passivity testing, scaling, tolerance analysis, and consideration of component nonidealities. Contemporary data-driven macromodels must be explicitly constrained or corrected to preserve passivity.

Key Terms

Positive-real function: A real-rational function analytic in the open right half-plane with nonnegative real part there.

Driving-point impedance: The ratio of terminal voltage to current at a one-port.

Driving-point admittance: The reciprocal of driving-point impedance.

Reactance function: A lossless LC driving-point impedance that is purely imaginary on the frequency axis away from poles.

Residue: The coefficient of the singular term associated with a simple pole.

Foster form I: A realization obtained by partial-fraction expansion of impedance and series connection of sections.

Foster form II: A realization obtained by partial-fraction expansion of admittance and parallel connection of sections.

Cauer synthesis: Ladder synthesis based on continued-fraction expansion and successive element extraction.

Interlacing: Alternation of simple poles and zeros along the imaginary axis or negative real axis, as applicable.

Immittance: A collective term for impedance or admittance.

Brune synthesis: A general PR synthesis procedure that may use transformers or coupled inductors.

Bott–Duffin synthesis: A transformerless RLC realization method applicable to every rational PR function.

Darlington synthesis: Realization by embedding an impedance in a lossless two-port terminated by a resistor.

Passivity enforcement: Modification of a fitted model so that it satisfies a mathematical passivity condition.

Frequency scaling: Transformation that moves a normalized network to a desired frequency range.

Impedance scaling: Transformation that changes network impedance levels while preserving normalized behavior.

Parasitic element: An unintended resistance, inductance, or capacitance affecting practical behavior.

Macromodel: A reduced mathematical representation reproducing the terminal behavior of a physical system.

Review Questions

Short Answer Questions

1. Define a positive-real rational function.
2. Why is properness not a necessary condition for passive impedance functions?
3. State the pole and residue conditions at an imaginary-axis pole of a PR function.
4. Distinguish Foster form I from Foster form II.
5. What physical network is represented by As/(s²+as+ω₀²) when all parameters are positive?

Long Answer Questions

1. Derive the element values of the parallel LC branch represented by As/(s²+ω₀²) and verify their units.
2. Explain a rigorous workflow for testing and synthesizing a rational passive one-port impedance.
3. Compare Foster, Cauer, Brune, Bott–Duffin, and Darlington synthesis in terms of applicability and topology.
4. Discuss the pole-zero properties of an LC reactance function and explain their physical significance.
5. Explain how component losses, tolerances, parasitics, and self-resonance affect an ideal synthesized network.

Analytical / Application Questions

1. Synthesize Z(s)=5+6/(s+3)+8s/(s²+4s+16) as a series connection of elementary Foster sections and determine every R, L, and C value.
2. Obtain a high-frequency Cauer expansion of Z(s)=(2s²+5s+1)/(s(s+1)) and verify the result by recombination.
3. A fitted impedance has stable poles but exhibits Re\{Z(jω)\}<0 over a narrow measured band. Assess whether it is passively realizable and propose a technically defensible correction and validation process.

Multiple Choice Questions with Answers

Question 1

Which statement about a rational passive driving-point impedance is correct?

A. It must always be strictly proper.

B. It may be improper by one degree, as in Z(s)=sL.

C. It may have poles anywhere in the right half-plane.

D. Its imaginary-axis pole residues must be negative.

Correct Answer: B. It may be improper by one degree, as in Z(s)=sL.

Question 2

In the LC Foster term As/(s²+ω₀²), what are the units of A?

A. Ω·s

B. Ω/s, equivalently F⁻¹

C. H/s

D. S·s

Correct Answer: B. Ω/s, equivalently F⁻¹

Question 3

The term 4/(s+2) is realized by a parallel RC branch. What are its values?

A. R=2\ Ω, C=0.25\ F

B. R=4\ Ω, C=0.5\ F

C. R=0.5\ Ω, C=2\ F

D. R=8\ Ω, C=4\ F

Correct Answer: A. R=2\ Ω, C=0.25\ F

Question 4

For Z(s)=As/(s²+as+ω₀²), which parallel RLC values are correct?

A. C=A, R=a/A, L=ω₀²/A

B. C=1/A, R=A/a, L=A/ω₀²

C. C=A/ω₀², R=Aa, L=1/A

D. C=1/a, R=ω₀/A, L=A/ω₀

Correct Answer: B. C=1/A, R=A/a, L=A/ω₀²

Question 5

For Z(s)=(s²+3s+1)/(s(s+1)), what is the first element extracted by a high-frequency Cauer expansion?

A. A 1 H series inductor

B. A 0.5 F shunt capacitor

C. A 1 Ω series resistor

D. A 4 Ω series resistor

Correct Answer: C. A 1 Ω series resistor

Question 6

What condition must hold for a simple pole of a PR function on the imaginary axis?

A. Its residue must be real and nonnegative.

B. Its residue must be purely imaginary.

C. It must be accompanied by a double zero.

D. Its residue must have a negative real part.

Correct Answer: A. Its residue must be real and nonnegative.

Question 7

Why can ordinary partial-fraction Foster synthesis fail for a general PR RLC function?

A. Every PR function is necessarily non-rational.

B. Its partial-fraction terms need not be individually positive real.

C. Passive networks cannot contain complex-conjugate poles.

D. Foster synthesis prohibits resistors.

Correct Answer: B. Its partial-fraction terms need not be individually positive real.

Question 8

Which method provides a general realization of rational PR functions and may require transformers or coupled inductors?

A. Fourier synthesis

B. Nodal elimination

C. Brune synthesis

D. Root-locus synthesis

Correct Answer: C. Brune synthesis

Question 9

Under impedance scale R_s and frequency scale ω_s, how does a normalized capacitor C_n transform?

A. C'=R_sω_sC_n

B. C'=R_sC_n/ω_s

C. C'=ω_sC_n/R_s

D. C'=C_n/(R_sω_s)

Correct Answer: D. C'=C_n/(R_sω_s)

Question 10

What topology results from a Foster expansion of admittance?

A. A parallel connection of the extracted admittance branches

B. A series connection of all extracted admittance branches

C. A bridge containing only resistors

D. A transmission line with no lumped elements

Correct Answer: A. A parallel connection of the extracted admittance branches