Realizability and Positive-Real Functions

Introduction and context

Network synthesis asks a fundamental inverse question: given a prescribed impedance, admittance, transfer function, or driving-point response, can a physically realizable electrical network be constructed using specified elements? Network analysis determines behavior from a circuit; network synthesis determines a circuit from behavior.

A mathematical function may satisfy algebraic requirements yet fail to represent a passive physical network. For example, a passive one-port cannot deliver unlimited net energy to an external load. Consequently, its driving-point impedance must obey restrictions associated with passivity, causality, stability, and real-valued physical components. These restrictions are expressed most completely through the theory of positive-real functions.

The subject is central to passive filter design, impedance matching, power electronics, microwave networks, electrochemical models, control-oriented circuit models, and macromodeling of interconnects. At postgraduate level, realizability should be viewed not merely as a test, but as a bridge between complex analysis, energy principles, and constructive synthesis procedures.

Learning objectives

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

1. distinguish network functions, driving-point functions, and transfer functions;
2. explain realizability in terms of passivity, causality, and stability;
3. define and test positive-real functions;
4. derive important necessary conditions for positive-realness;
5. apply pole-zero and frequency-domain tests;
6. synthesize simple RLC one-port networks using Foster and Cauer forms;
7. interpret limitations caused by losslessness, nonminimum-phase behavior, and nonpassive data;
8. evaluate a practical impedance model for physical realizability.

1. Network functions and physical realizability

For a linear time-invariant network, a network function is commonly written as

where s = σ + jω is the complex frequency, N(s) and D(s) are polynomials, and j = √(-1). The variable σ has units of inverse seconds and ω is angular frequency in radians per second.

A driving-point impedance is

and a driving-point admittance is

where voltage and current are measured at the same port. A transfer function, such as V₂(s)/V₁(s), relates variables at different ports and is not generally required to be positive real. Positive-real theory applies most directly to driving-point immittance functions: impedance and admittance.

A function is real-rational if it is a ratio of polynomials with real coefficients. Physical networks containing R, L, and C elements produce real-rational functions. The coefficients are real because real excitation signals produce real voltage and current signals.

Realizability depends on the allowed element set:

The same mathematical function may be realizable with active components but unrealizable as a passive RLC network. Thus, realizability is always relative to a network class.

2. Passivity, causality, and stability

2.1 Passivity

A one-port is passive if it cannot supply more energy than it has initially stored. With port voltage v(t) and current i(t), the absorbed power is p(t) = v(t)i(t). The net absorbed energy over an interval is

A passive network satisfies an energy inequality of the form

where W_s(t₀) represents initially stored energy under the selected sign convention. A resistor dissipates energy, whereas an inductor and capacitor store and return energy. An ideal passive network cannot create net energy.

2.2 Causality

Causality requires that the response at time t depend only on present and past inputs, not future inputs. For rational lumped networks, causality is reflected in proper transfer behavior and physically meaningful initial conditions. A noncausal frequency response cannot be implemented as an ordinary real-time passive network.

2.3 Stability

For a continuous-time rational function, poles in the open right half-plane generally indicate exponential growth and instability. Passive networks do not possess such poles. Poles on the imaginary axis may occur in lossless networks, but they must be appropriately simple; repeated imaginary-axis poles are incompatible with ordinary passive RLC driving-point behavior.

Passivity is stronger than stability. A stable active network may still generate energy and therefore fail the positive-real test.

3. Positive-real functions

3.1 Definition

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

1. F(s) is real for real positive s;
2. F(s) is analytic in the open right half-plane Re(s) > 0;
3. its real part is nonnegative there:

For rational functions, the first condition is ensured by real coefficients. The second excludes right-half-plane poles. The third expresses nonnegative average power under sinusoidal or complex exponential excitation.

If Z(s) is an impedance, then passivity requires Z(s) to be PR. If Y(s) is an admittance, then Y(s) must be PR. The reciprocal of a nonzero PR impedance is also PR, so impedance and admittance descriptions are equivalent for a passive one-port.

For sinusoidal operation, set s = jω. The average power absorbed by an impedance is proportional to

where I is the peak phasor current. Thus, a passive impedance must satisfy Re Z(jω) ≥ 0 wherever finite.

3.2 Characterization theorem for rational functions

A real-rational function is positive real if and only if:

• it has no poles in Re(s) > 0;
• it has real coefficients;
• Re F(jω) ≥ 0 at every finite frequency where the function is defined;
• any pole on the imaginary axis is simple and has a nonnegative real residue.

The last condition is essential. Merely checking the real part away from poles is insufficient because poles on the imaginary axis correspond to ideal energy storage.

For an imaginary-axis pole at s = jω₀, the residue is

and PR behavior requires K to be real and nonnegative. At the origin, the same rule applies using K = lim_s→0 sF(s).

4. Necessary structural conditions

4.1 Pole and zero locations

For a nonconstant PR rational function:

• no pole lies in the open right half-plane;
• poles on the imaginary axis are simple;
• poles and zeros on the imaginary axis interlace, subject to cancellation conventions;
• residues at imaginary-axis poles are nonnegative;
• numerator and denominator degrees differ by no more than one for many standard immittance forms, although the precise degree condition depends on the chosen impedance or admittance normalization.

A right-half-plane zero is not automatically forbidden for every transfer function, but driving-point PR functions have much stronger interlacing restrictions. Confusing transfer-function minimum phase with driving-point positive realness is a frequent error.

4.2 Frequency-domain test

Write

where R(ω) and X(ω) are real-valued. A necessary boundary condition is

for all real ω at which F(jω) is finite. This is not by itself sufficient unless pole locations and imaginary-axis residues are also checked.

4.3 Symmetry

For real-coefficient functions,

where the asterisk denotes complex conjugation. Consequently, the real part is even and the imaginary part is odd:

Failure of these symmetry properties in measured or fitted data indicates noise, inconsistent reference directions, non-real coefficients, or an invalid model.

5. Important classes of PR functions

5.1 Reactance functions

A reactance function is the impedance of a lossless one-port and has purely imaginary values on the imaginary axis:

Its poles and zeros are simple, lie on the imaginary axis, and interlace. The reactance is odd in frequency. Pure inductors and capacitors are examples:

where L is inductance in henries and C is capacitance in farads.

5.2 Brune and Bott–Duffin realizability

Every positive-real rational function can be realized using a finite passive RLC network. Brune’s synthesis establishes this constructively by extracting resistive and reactive sections until a realizable remainder is obtained. Bott–Duffin synthesis broadens the procedure and avoids some restrictions associated with series-parallel forms, although it may introduce transformers or more elaborate interconnections.

This theorem is important conceptually: positive realness is not merely necessary; for rational one-port functions it is also sufficient for passive RLC realizability, provided ideal transformers or the permitted synthesis elements are admitted where required.

6. Foster and Cauer synthesis

6.1 Foster forms

Foster synthesis uses partial-fraction expansions. For an impedance, a typical first Foster form is

where terms must be interpreted according to the actual pole structure and all extracted element values must be nonnegative. The term R_∞ is a series resistance, C₀ is a capacitance, ω_k are resonant frequencies, and K_k are nonnegative constants related to inductance-capacitance branches.

A more direct and commonly used form is obtained by recognizing elementary impedances. For example,

is a series RLC network. Parallel Foster forms are often more convenient when the admittance is expanded instead.

6.2 Cauer forms

Cauer synthesis uses continued fractions. A ladder impedance can be represented as

The exact sequence depends on whether series or shunt elements are extracted first and on whether impedance or admittance is used. Each quotient corresponds to a physical element. Positive-realness guarantees nonnegative extracted values under a valid extraction sequence.

Cauer form is especially useful for ladder realization, sensitivity analysis, and implementation with a small number of components. Foster form often gives a clearer modal interpretation because each term corresponds to a resonant contribution.

7. Worked example: testing and realizing a simple impedance

Consider

The constants have units: resistance is 2 Ω, inductance is 3 H, and capacitance is 4 F because 1/(Cs) has units of ohms.

This function is already the impedance of a series RLC network:

• R = 2 Ω;
• L = 3 H;
• C = 4 F.

For s = σ + jω with σ > 0,

Every term is nonnegative, so Re Z(s) > 0 in the open right half-plane. On the imaginary axis away from the origin,

and therefore Re Z(jω) = 2 ≥ 0. At the origin there is a simple pole with residue

Hence the function is positive real and is directly realizable as a passive series RLC network.

8. Worked example: a non-PR function

Let

The pole at s = -2 is stable, but stability alone is insufficient. On the imaginary axis,

Multiplication by the conjugate denominator gives

Therefore,

For |ω| < √2, this real part is negative. The function would absorb negative average power over that frequency range, meaning that it behaves as an active one-port. It is stable but not positive real and cannot be realized as a passive RLC driving-point impedance.

9. A practical identification case

Background

Consider an illustrative engineering case involving a battery-equivalent impedance model. Frequency-response measurements are made over a broad range to support charger control and transient prediction. The measured data are fitted by a rational impedance model.

Problem

The first fitted model is

The negative coefficient in the last term produces a negative-residue mode. Although the model may fit noisy data well over a limited frequency interval, it can predict energy generation near a resonance and become unsuitable for a passive simulation or hardware-in-the-loop test.

Analysis

The constant 0.08 Ω represents high-frequency ohmic resistance. The term 0.12/(s+4) is a decaying polarization contribution. The term -0.05/(s+20) has the wrong sign for a simple passive relaxation mode. A rigorous test should evaluate pole locations, imaginary-axis behavior, and Re Z(jω) over the entire intended band. Sampling only a few frequencies is not a proof of positive-realness.

Approach and solution

A constrained vector-fitting procedure is used with:

1. stable poles restricted to the left half-plane;
2. nonnegative residues for terms representing passive relaxation modes;
3. a nonnegative instantaneous resistance;
4. enforcement of Re Z(jω) ≥ 0 on a dense frequency grid;
5. a final analytical PR test rather than reliance on sampled data alone.

The corrected model is represented as

This correction may slightly increase fitting error, but it produces a physically credible passive model. If the fit must reproduce genuinely active behavior—such as an actively controlled converter—the correct solution is not to force a PR model; instead, the active subsystem must be represented explicitly.

Outcome and learning points

The constrained model is safer for time-domain simulation, preserves passivity under interconnection, and gives interpretable energy-storage modes. The case demonstrates that model accuracy must be judged together with physical admissibility. A low numerical fitting error does not guarantee realizability.

10. Limitations and advanced considerations

10.1 Finite-band data

Positive-realness is a global property, whereas measurements are available only over a finite bandwidth. Extrapolation outside the measured band can introduce nonphysical poles or negative resistance. Engineers should report the verified frequency range and the assumptions used for extrapolation.

10.2 Approximate versus exact synthesis

Exact synthesis may require very large or very small component values, ideal transformers, or a high network order. Approximation reduces complexity but may violate passivity. Passivity-preserving model reduction is therefore preferable to unconstrained least-squares reduction.

10.3 Distributed and nonrational systems

Transmission lines and electromagnetic structures can have transcendental, delay-dependent, or distributed-parameter descriptions. Rational PR approximations are useful, but approximation error must not create artificial energy generation.

10.4 Numerical conditioning

Near pole-zero cancellations, coefficient-based tests can be ill-conditioned. State-space passivity tests, Hamiltonian or dissipativity-based methods, and frequency sweeps with certified margins are often more reliable for high-order models. Floating-point roundoff can make a theoretically zero real part appear slightly negative; tolerances must be selected relative to scaling and measurement uncertainty.

10.5 Active networks

Negative impedance converters, controlled sources, and power-electronic converters may intentionally violate PR conditions. In such systems, passivity may hold only after including a controller, a source impedance, or a restricted operating region. The analyst must define the port, operating point, and perturbation model before applying a PR test.

11. Best practices in engineering synthesis

• Specify the port variables and sign convention before forming an immittance.
• Distinguish a transfer function from a driving-point function.
• Check real coefficients, pole locations, boundary real-part conditions, and imaginary-axis residues.
• Use symbolic algebra for exact low-order tests and numerically scaled methods for high-order models.
• Preserve passivity during vector fitting, interpolation, and model reduction.
• Validate both frequency-domain power behavior and time-domain energy behavior.
• Document bandwidth, tolerances, component restrictions, and whether ideal transformers are permitted.
• Treat a small negative real-part margin as a modeling issue requiring investigation, not automatic proof of activity.

Chapter Summary

Realizability concerns whether a prescribed network function can be produced by a specified physical network. For passive RLC one-ports, the central mathematical condition is positive realness. A real-rational function must have no right-half-plane poles, must have nonnegative real part in the right half-plane, and must satisfy special residue conditions at simple imaginary-axis poles. Frequency-domain nonnegative resistance is necessary but must be combined with pole and residue checks.

Positive-real rational functions can be synthesized using Foster, Cauer, Brune, or Bott–Duffin procedures. Foster forms emphasize partial-fraction and modal structure, while Cauer forms produce ladder networks through continued fractions. Stability alone does not imply passivity, and accurate finite-band fitting does not guarantee global realizability. Modern identification and model-reduction workflows therefore impose passivity constraints and verify the final model analytically.

Key Terms

Positive-real function: A real-rational function with no right-half-plane poles and nonnegative real part in the right half-plane.

Realizability: The possibility of constructing a physical network having a specified response.

Driving-point impedance: The ratio of port voltage to port current.

Driving-point admittance: The ratio of port current to port voltage.

Passivity: The inability of a network to generate net energy.

Causality: Dependence of response only on present and past excitation.

Lossless network: A network that stores but does not dissipate energy.

Reactance function: A lossless driving-point impedance with purely imaginary frequency response.

Imaginary-axis pole: A pole satisfying Re(s)=0.

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

Foster form: A synthesis based on partial-fraction expansion.

Cauer form: A ladder synthesis based on continued fractions.

Brune synthesis: A constructive passive synthesis procedure for PR functions.

Bott–Duffin synthesis: A general passive synthesis method for PR rational functions.

Immittance: A collective term for impedance and admittance.

Positive-realness margin: The minimum real-part margin of a model over a specified domain.

Vector fitting: A rational approximation method for frequency-response data.

Passivity-preserving reduction: Model reduction that retains dissipativity or PR behavior.

Review Questions

Short Answer Questions

1. Define a positive-real function.
2. Why is stability alone insufficient for passive realizability?
3. State the residue condition at a simple imaginary-axis pole.
4. Distinguish a driving-point function from a transfer function.
5. What is the principal difference between Foster and Cauer synthesis?

Long Answer Questions

1. Explain the relationship among passivity, causality, stability, and positive realness.
2. Derive the frequency-domain condition for nonnegative average power absorbed by an impedance.
3. State and explain the rational positive-real characterization theorem.
4. Compare Foster, Cauer, Brune, and Bott–Duffin synthesis methods.
5. Discuss the difficulties of proving realizability from finite-band measured data.

Analytical / Application Questions

1. Test Z(s) = 1 + s + 2/s for positive realness and identify its passive realization.
2. Determine whether Z(s) = (s+1)/(s+4) can be a passive driving-point impedance by evaluating Re Z(jω) and checking poles.
3. Design a verification workflow for a high-order rational battery impedance model obtained by vector fitting, including analytical and numerical passivity checks.

Multiple Choice Questions with Answers

Question 1

Which condition is fundamental for a rational driving-point impedance to be positive real?

A. Its real part is negative in the right half-plane.

B. It has at least one right-half-plane pole.

C. Its real part is nonnegative in the right half-plane and it has no right-half-plane poles.

D. All of its zeros must lie at the origin.

Correct Answer: C. Its real part is nonnegative in the right half-plane and it has no right-half-plane poles.

Question 2

For a passive impedance, the average sinusoidal power is proportional to which quantity?

A. Im[Z(jω)]

B. Re[Z(jω)]

C. The pole count only

D. The phase angle only

Correct Answer: B. Re[Z(jω)]

Question 3

What is required of a positive-real function at an imaginary-axis pole?

A. The pole must be repeated and have negative residue.

B. The pole must be simple with a nonnegative real residue.

C. The pole must be in the right half-plane.

D. The residue must be purely imaginary.

Correct Answer: B. The pole must be simple with a nonnegative real residue.

Question 4

Which synthesis method is primarily based on partial-fraction expansion?

A. Foster synthesis

B. Cauer synthesis

C. Fourier synthesis

D. Newton synthesis

Correct Answer: A. Foster synthesis

Question 5

Consider Z(s)=2+3s+1/(4s). What is its series resistance?

A. 0.25 Ω

B. 2 Ω

C. 3 Ω

D. 4 Ω

Correct Answer: B. 2 Ω

Question 6

For Z(s)=(s-1)/(s+2), at which condition is the real part negative?

A. For |ω|<√2

B. For every frequency

C. Only at ω=0 after removing the pole

D. Never

Correct Answer: A. For |ω|<√2

Question 7

Which statement about stability and passivity is correct?

A. Every stable network is passive.

B. Passivity and stability are identical properties.

C. A stable active network may fail the positive-real test.

D. An unstable network is always positive real.

Correct Answer: C. A stable active network may fail the positive-real test.

Question 8

What does a Cauer representation normally produce?

A. A ladder network from a continued fraction.

B. Only a parallel resistor network.

C. A noncausal filter.

D. A frequency-independent transfer gain.

Correct Answer: A. A ladder network from a continued fraction.

Question 9

Why can unconstrained rational fitting be unsafe for passive modeling?

A. It always produces too few poles.

B. It may fit data accurately while creating negative real parts or nonphysical residues.

C. It cannot represent any resistance.

D. It necessarily produces a lossless model.

Correct Answer: B. It may fit data accurately while creating negative real parts or nonphysical residues.

Question 10

For a real-coefficient network function, which symmetry holds on the imaginary axis?

A. F(-jω)=F(jω) always

B. F(-jω)=-F(jω) always

C. F(-jω)=F*(jω)

D. F(-jω)=0

Correct Answer: C. F(-jω)=F*(jω)