Network Functions, Poles, Zeros and Stability

Introduction and Context

Network functions provide the mathematical bridge between a physical electrical network and its frequency- or time-domain behavior. They describe how a linear network transforms an excitation into a response, while poles and zeros reveal the mechanisms that shape magnitude, phase, transient response, resonance, bandwidth, and stability. In network synthesis, these concepts are also used in the reverse direction: a desired function is specified and a realizable network is constructed.

At postgraduate level, it is important to distinguish three related but different ideas. A network function is a ratio relating a response to an excitation under specified initial conditions. Poles and zeros are properties of that function, but they also reflect the natural modes and transmission properties of the physical network. Stability is a system-level property that depends on pole locations and on the class of response being considered. A passive network, for example, has strong restrictions on its poles and zeros, whereas an active network may exhibit right-half-plane poles and instability.

The discussion assumes lumped, linear, time-invariant (LTI) networks unless stated otherwise. For sinusoidal steady-state analysis, phasor methods are used. For transient and stability analysis, the Laplace transform is the principal tool.

Learning Objectives

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

1. Define driving-point and transfer network functions precisely.
2. Derive network functions using impedances, admittances, nodal analysis, and state-space representations.
3. Identify poles, zeros, pole-zero cancellations, and their physical significance.
4. Relate pole locations to natural response, transient behavior, resonance, and stability.
5. Apply the Routh–Hurwitz criterion and frequency-domain reasoning to assess stability.
6. Explain restrictions imposed by passivity and positive-realness.
7. Interpret network-function data for design, synthesis, measurement, and fault diagnosis.
8. Recognize numerical and practical limitations in pole-zero analysis.

1. Network Functions: Definitions and Classification

Consider an LTI network with an input excitation X(s) and an output response Y(s). With all independent initial conditions set to zero, the network function is

where s = σ + jω is the complex frequency variable. The units of H(s) depend on the selected input and output. A voltage-to-voltage transfer function is dimensionless; a voltage-to-current function has units of ohms; a current-to-voltage function has units of siemens.

The zero-initial-condition condition is essential. If stored energy is present initially, the total response includes a zero-input response that cannot generally be represented by the same input-output ratio.

1.1 Driving-point functions

A driving-point function relates a variable at a port to another variable at the same port:

• Driving-point impedance: Z(s) = V(s)/I(s)
• Driving-point admittance: Y(s) = I(s)/V(s)

For a one-port network, Z(s) and Y(s) are reciprocal where both are finite and nonzero. A driving-point impedance is not a transfer function between two distinct ports, although both are network functions.

1.2 Transfer functions

A transfer function relates a response at one port or branch to an excitation at another location. Common examples include:

The selected variables and reference directions must be stated because changing a current reference direction can change the sign of a transfer function.

1.3 Rational form

For a finite lumped network containing ideal resistors, inductors, capacitors, dependent sources, and independent sources, a network function can usually be expressed as

where K is a constant gain factor, z_k are zeros, p_m are poles, and repeated roots are included according to multiplicity. An equivalent polynomial form is

where N(s) and D(s) are polynomials in s. Before interpreting roots, common factors should be examined carefully. A common factor may represent an exact mathematical cancellation, but it does not always indicate that the corresponding physical mode has disappeared internally.

2. Deriving Network Functions

2.1 Element relations in the s-domain

Under zero initial conditions, the Laplace-domain impedances of the basic passive elements are:

Here R is in ohms, L in henries, C in farads, and s in radians per second. The expressions assume zero initial energy. For nonzero initial inductor current or capacitor voltage, additional source terms appear in the s-domain model.

2.2 Example: a first-order voltage divider

Consider a series resistor R and capacitor C, with the output taken across the capacitor. The divider relation gives

Substitution of Z_C = 1/(sC) yields

Thus the pole is

There is no finite zero. The pole is in the left half of the s-plane for R > 0 and C > 0, so the natural response decays exponentially. The time constant is τ = RC, measured in seconds, and the pole may also be written p = −1/τ.

For R = 1 kΩ and C = 1 μF,

and p = −1000 rad/s. The value of the transfer function at s = jω determines the sinusoidal steady-state response.

2.3 Nodal and mesh formulations

For larger networks, direct series-parallel reduction is generally insufficient. In nodal analysis, the s-domain equations can be written as

where Y(s) is the nodal admittance matrix, V(s) is the vector of unknown node voltages, and I(s) is the vector of injected currents. Since resistor, inductor, and capacitor admittances are respectively 1/R, 1/(sL), and sC, the entries of Y(s) are rational functions of s.

A transfer function is obtained by solving for the required output variable and dividing by the specified input. The determinant of the system matrix commonly contributes to the denominator. However, algebraic cancellations may occur, so the determinant itself is not always the reduced denominator of an externally observed transfer function.

In state-space form,

where x is the state vector, u the input, y the output, and A, B, C, D are system matrices. With zero initial conditions, the transfer function is

provided sI − A is nonsingular. The eigenvalues of A are the natural-mode poles of a minimal realization. A nonminimal realization may contain unobservable or uncontrollable modes that cancel from an input-output transfer function.

3. Poles and Zeros

3.1 Definitions

A finite zero is a value s = z at which the numerator of the reduced network function becomes zero, while the denominator is nonzero. A finite pole is a value s = p at which the denominator becomes zero, while the numerator is nonzero. Poles and zeros at infinity are associated with relative degree and high-frequency behavior.

If

then z_1 and z_2 are zeros and p_1 and p_2 are poles. Repeated roots have multiplicity greater than one and produce stronger effects on magnitude and time response.

3.2 Physical interpretation of poles

Poles are associated with natural modes. If a system is excited and then disconnected from its source, its zero-input response is formed from terms such as

For a simple real pole p_k. For repeated poles, polynomial factors in t occur. For a complex-conjugate pair p = σ ± jω_d, the real response contains damped sinusoidal terms of the form

where σ is the exponential growth or decay rate and ω_d is the damped angular frequency.

3.3 Physical interpretation of zeros

Zeros identify frequencies or complex-frequency conditions at which a specified response vanishes despite a nonzero excitation. They may result from cancellation between multiple signal paths, conservation laws, bridge balance, or output measurement location. A zero on the jω-axis produces zero response at that sinusoidal frequency, provided no pole is simultaneously present there.

Zeros influence magnitude and phase but do not, by themselves, determine internal stability. A right-half-plane zero can produce nonminimum-phase behavior, including an initial response opposite to the eventual steady-state direction in some systems.

3.4 Pole-zero plots

A pole-zero plot places zeros and poles in the complex s-plane. The horizontal axis is σ in radians per second and the vertical axis is jω in radians per second. For real-coefficient networks, nonreal poles and zeros occur in conjugate pairs. The plot provides qualitative information:

• Poles near the jω-axis create sharp frequency-selective behavior and long transients.
• Poles far into the left half-plane decay rapidly.
• Right-half-plane poles produce exponentially growing natural responses.
• A zero near a pole can reduce the externally observed effect over a limited condition.
• Symmetry about the real axis is required for real-coefficient rational functions.

4. Frequency Response and Pole-Zero Geometry

The frequency response is obtained by evaluating the network function on the imaginary axis:

provided the evaluation is meaningful and the system is stable in the relevant sense. If the factored form is used,

Thus the magnitude is proportional to the product of distances from the point jω to the zeros, divided by the product of distances to the poles. The phase is

with phase interpreted continuously or modulo 360°, according to the application.

For a first-order factor 1/(1 + sτ), the break frequency is ω_b = 1/τ rad/s. Its magnitude is

and its phase is −tan^(−1)(ωτ). At ω = ω_b, the magnitude is 1/√2 of the low-frequency value, corresponding to −3.01 dB.

A pole contributes approximately −20 dB/decade to the asymptotic magnitude slope above its break frequency, while a zero contributes approximately +20 dB/decade. These asymptotic rules are local approximations and do not replace exact evaluation near break frequencies.

5. Stability

5.1 BIBO and internal stability

For a continuous-time rational transfer function, bounded-input bounded-output (BIBO) stability requires that every pole of the reduced transfer function lie strictly in the open left half-plane, assuming a proper transfer function and no poles on the jω-axis. A pole at the origin or on the imaginary axis generally leads to a nondecaying response and violates asymptotic stability; repeated imaginary-axis poles are more severely unstable.

Internal or state stability concerns every natural mode of a realization. A pole canceled in a transfer function may remain as an unobservable or uncontrollable internal mode. Therefore, transfer-function pole inspection is sufficient for BIBO stability of a minimal realization, but not necessarily for internal stability of a nonminimal realization.

For a continuous-time state matrix A, asymptotic state stability requires

For a discrete-time system, the corresponding condition is |λ_i(A)| < 1, because modes evolve as λ_i^n. This chapter focuses primarily on continuous-time networks.

5.2 Stability classification from pole locations

Practical circuits may contain losses, parasitic elements, saturation, noise, and nonlinearities. Consequently, an ideal marginally stable model may not accurately predict long-term hardware behavior.

5.3 Routh–Hurwitz criterion

Suppose the characteristic polynomial is

with real coefficients and a_n > 0. The Routh–Hurwitz criterion determines the number of roots in the right half-plane from a tabulation of coefficients. A necessary condition for strict stability is that all coefficients be positive, although coefficient positivity alone is not sufficient for order three or higher.

For a cubic polynomial

with a_3 > 0, strict stability requires

These conditions assume no degeneracy such as an entire row of zeros. Such cases require special Routh-array procedures and may indicate symmetric pole pairs on the imaginary axis.

#### Worked example: cubic stability test

Consider

Here a_3 = 1, a_2 = 4, a_1 = 5, and a_0 = K. The cubic conditions give K > 0 and

Therefore, the strict-stability range is

At K = 20, a pair of roots reaches the imaginary axis, marking a stability boundary under the ideal polynomial model. For K ≤ 0, at least one condition fails and the polynomial is not strictly stable.

5.4 Nyquist and feedback interpretation

For a feedback network with loop transfer function L(s), closed-loop poles satisfy

The Nyquist criterion assesses closed-loop stability by relating encirclements of the critical point −1 + j0 to open-loop right-half-plane poles and closed-loop right-half-plane poles. This is especially useful when the open-loop transfer function has delays, high order, or frequency-response data rather than a convenient polynomial representation.

The criterion requires consistent contour orientation and careful treatment of poles on the imaginary axis. It also distinguishes absolute stability from relative stability: a system may technically be stable while having poles so close to the imaginary axis that its transient response is unacceptable.

6. Passivity, Positive-Realness and Synthesis Constraints

A passive one-port cannot deliver net energy to an external load. Its driving-point impedance Z(s) is positive real if it satisfies, in the standard rational-network setting:

1. Z(s) is real for real s.
2. Z(s) has no poles in the open right half-plane.
3. Re{Z(s)} ≥ 0 for Re{s} > 0, wherever Z(s) is finite.

Additional pole-residue restrictions apply to poles on the jω-axis. Positive-realness is a necessary condition for the impedance of a passive network composed of R, L, and C elements, and it underlies classical synthesis procedures such as Foster and Cauer forms.

For a passive network, poles and zeros exhibit interlacing properties in several important one-port classes. For example, a lossless LC driving-point reactance has poles and zeros on the jω-axis that alternate, subject to endpoint and multiplicity conditions. Resistive losses move poles into the left half-plane and alter the ideal interlacing picture.

These restrictions are central to synthesis. Not every rational function can be realized using only passive RLC elements. A proposed function must satisfy appropriate reality, positivity, degree, pole, zero, and residue conditions. Active elements enlarge the realizable function class but introduce power-supply dependence, noise, finite gain, bandwidth limits, and possible instability.

7. Pole-Zero Cancellation and Model Reduction

Suppose

Algebraically, the reduced transfer function is

The input-output response does not display a pole at −2 in exact arithmetic. However, exact cancellation requires exact parameter matching. In a physical circuit, component tolerances, parasitic coupling, sensor limitations, and nonlinear effects prevent perfect cancellation. A nearly canceled unstable pole is particularly dangerous: the measured transfer function may appear benign while a weakly observable internal state grows.

Model reduction should therefore preserve the dominant poles, relevant zeros, low-frequency or high-frequency gains, and stability margins. Balanced truncation and singular-perturbation methods are more reliable than deleting factors merely because they appear near one another. In control and network modeling, minimal realization should be established before interpreting transfer-function poles as the complete internal modal set.

8. Practical Computation and Measurement

Network functions are obtained in practice through symbolic derivation, numerical circuit simulation, impedance analyzers, vector network analyzers, and system identification. Each method has limitations.

Symbolic algebra provides exact structural insight but can suffer from expression growth and ill-conditioned coefficient forms. Numerical eigenvalue computation is sensitive to scaling, especially when poles differ by many orders of magnitude. Frequency-response fitting can produce spurious pole-zero pairs if the frequency range is too narrow, measurements are noisy, or the fitting order is excessive.

Best practices include:

• Normalize variables and scale matrices before numerical computation.
• Verify poles by substituting them into the characteristic polynomial.
• Check conjugate symmetry for real-coefficient models.
• Compare pole-zero predictions with time-domain simulation and measured frequency response.
• Use physically meaningful units and document reference directions.
• Test sensitivity to component tolerances and parasitic elements.
• Distinguish a mathematical pole from a robustly observable physical mode.

9. Illustrative Case: Stability of an Active Two-Stage Filter

Background

Consider an illustrative active two-stage filter whose small-signal closed-loop characteristic polynomial is

The gain-setting parameter K is adjustable through a resistor ratio. The design objective is to obtain a stable filter with adequate transient decay while avoiding operation close to the oscillation boundary.

Problem

Determine the allowable range of K for strict stability and assess two candidate settings: K = 8 and K = 24.

Analysis

Applying the cubic Routh–Hurwitz conditions gives

and

Hence 0 < K < 20. The setting K = 8 lies inside the stable range. The setting K = 24 lies outside it and must produce at least one right-half-plane pole.

For K = 8, the polynomial is s^3 + 4s^2 + 5s + 8. Its Routh first column is positive:

There are no sign changes, so there are no right-half-plane roots. The model is strictly stable.

For K = 24, the s¹ first-column entry is (20 − 24)/4 = −1. The first column is 1, 4, −1, 24, which contains two sign changes. Thus two roots lie in the right half-plane, and the filter is unstable under the model.

Approach and solution

A prudent engineering choice is not merely any K between 0 and 20. Operation too near K = 20 gives low damping and high sensitivity to component tolerances. The designer should select a value with a stability margin, simulate parasitics and amplifier limitations, and verify the result using measured frequency response and step response. K = 8 is preferable to K = 19 if other performance specifications permit.

Outcome

The Routh test quickly identifies the admissible design region. It does not by itself specify bandwidth, overshoot, noise, or robustness. Those properties require pole locations, zeros, gain and phase margins, and hardware verification.

Learning points

1. A positive-coefficient polynomial is not automatically stable for order three or higher.
2. Stability boundaries are design constraints, not desirable operating points.
3. Component tolerances can move poles across a theoretical boundary.
4. Internal amplifier poles and parasitic modes must be included in a realistic model.

10. Design, Professional and Governance Implications

Network-function analysis supports filter design, impedance matching, power-electronic control, sensor interfaces, communication networks, biomedical instrumentation, and fault diagnosis. In safety-critical systems, stability claims must be traceable to documented models, test conditions, tolerances, and verification procedures.

Engineers should not report a stable result without specifying whether the claim concerns BIBO stability, internal state stability, small-signal stability, or nonlinear large-signal behavior. A small-signal pole calculation around one operating point does not guarantee stability under saturation, switching, thermal variation, or component aging.

Computer-aided design and AI-assisted model extraction can accelerate pole-zero identification, but automated results require expert validation. An algorithm may fit noise with spurious poles, miss lightly damped modes, or conceal unstable dynamics through an apparently accurate but nonminimal model. Responsible practice requires reproducibility, uncertainty reporting, version-controlled models, and independent checks.

Challenges and Limitations

The ideal LTI model omits nonlinearities, time variation, distributed effects, electromagnetic coupling, quantization, delays, and uncertain parameters. At high frequencies, lumped-element assumptions may fail and poles or zeros associated with transmission-line behavior may appear. At low frequencies, leakage and parasitic resistances can dominate ideal capacitor or inductor predictions.

Pole locations are also parameter-dependent. A nominally stable system can become unstable under tolerance extremes. Robust stability analysis therefore considers uncertain coefficients, interval polynomials, structured uncertainties, or parameter-dependent state matrices. Frequency-domain margins and worst-case analysis are often more informative than a single nominal pole plot.

Chapter Summary

A network function is a zero-state input-output relationship in the Laplace domain. Driving-point functions describe behavior at one port, whereas transfer functions relate variables at distinct locations. Rational network functions are characterized by poles, zeros, gain, and possible poles or zeros at infinity.

Poles represent natural modes and determine exponential decay, growth, and oscillation. Zeros describe conditions under which a specified response vanishes and shape magnitude and phase. Frequency response is obtained by evaluating the function on the jω-axis, and pole-zero geometry provides a direct interpretation of gain and phase.

For continuous-time systems, strict BIBO stability requires poles of the reduced transfer function in the open left half-plane. Internal stability requires all modes of a suitable realization to decay, so hidden canceled modes must not be ignored. The Routh–Hurwitz criterion provides an algebraic stability test, while Nyquist analysis is valuable for feedback systems and measured frequency data.

Passivity restricts driving-point functions through positive-realness and associated pole-zero properties. Practical analysis must account for cancellation sensitivity, numerical conditioning, parasitics, uncertainty, and model validity. In professional design, nominal stability is only one requirement; robustness, observability, reproducibility, and experimental verification are equally important.

Key Terms

Network function: A zero-initial-condition ratio of a selected response to a selected excitation.

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

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

Transfer function: A ratio relating an output variable to an input variable.

Pole: A root of the reduced denominator of a rational network function.

Zero: A root of the reduced numerator of a rational network function.

Natural response: The response produced by stored energy or system modes without an applied input.

Frequency response: The value of a network function at s = jω.

BIBO stability: Stability condition in which every bounded input produces a bounded output.

Internal stability: Stability of every state or natural mode of a realization.

Minimal realization: A realization that is both controllable and observable, with no removable internal modes.

Routh–Hurwitz criterion: An algebraic method for counting right-half-plane roots.

Nyquist criterion: A frequency-domain method for assessing feedback stability.

Positive-real function: A function satisfying reality and nonnegative real-part conditions required for passive driving-point behavior.

Passivity: The property that a network cannot generate net energy for external delivery.

Pole-zero cancellation: Algebraic removal of a common numerator and denominator factor.

Damping: The rate at which oscillatory or transient energy decays.

Stability margin: A measure of distance from an instability boundary, often expressed through gain, phase, or pole-location measures.

Nonminimum-phase zero: Typically a right-half-plane zero in a continuous-time system, associated with restrictive transient and phase behavior.

Review Questions

Short Answer Questions

1. What is meant by a network function under zero initial conditions?
2. Distinguish a driving-point impedance from a transfer impedance.
3. What physical behavior is associated with a pole in the left half-plane?
4. State the continuous-time pole condition for strict BIBO stability.
5. Why can pole-zero cancellation be misleading in a physical circuit?

Long Answer Questions

1. Derive the transfer function of a series RC circuit when the output is taken across the capacitor, and interpret its pole and frequency response.
2. Explain the relationship between state-space eigenvalues and poles of a network transfer function. Discuss controllability and observability.
3. Describe the Routh–Hurwitz criterion and apply it to a general cubic characteristic polynomial.
4. Explain positive-realness and its role in passive network synthesis.
5. Compare BIBO stability, internal stability, marginal stability, and robust stability.

Analytical / Application Questions

1. A network has characteristic polynomial D(s) = s³ + 6s² + 11s + K. Determine the range of K for strict stability using the cubic Routh–Hurwitz conditions.
2. For H(s) = (s + 4)/[(s + 1)(s + 10), identify the pole and zero locations and explain the expected low-frequency and high-frequency behavior.
3. A measured transfer function appears stable because an unstable pole is canceled by a numerator zero. Explain why this conclusion may be unsafe and propose at least three verification steps.

Multiple Choice Questions with Answers

Question 1

Which condition defines a network function in the Laplace domain?

A. It includes arbitrary nonzero initial conditions

B. It is a zero-initial-condition response-to-excitation ratio

C. It is always dimensionless

D. It applies only to passive networks

Correct Answer: B. It is a zero-initial-condition response-to-excitation ratio

Question 2

For a series RC circuit with output across the capacitor, the transfer function is:

A. H(s) = 1 + sRC

B. H(s) = sRC/(1 + sRC)

C. H(s) = 1/(1 + sRC)

D. H(s) = R/(sC)

Correct Answer: C. H(s) = 1/(1 + sRC)

Question 3

What is the principal physical significance of a pole?

A. It identifies a natural mode of the network

B. It always identifies a frequency of zero output

C. It specifies the input resistance only

D. It guarantees passivity

Correct Answer: A. It identifies a natural mode of the network

Question 4

For strict continuous-time stability, the poles of a minimal transfer function must lie:

A. In the right half-plane

B. On the imaginary axis

C. In the open left half-plane

D. At the origin

Correct Answer: C. In the open left half-plane

Question 5

For D(s) = s³ + 4s² + 5s + K, the strict-stability range is:

A. K < 0

B. 0 < K < 20

C. K > 20

D. K = 20

Correct Answer: B. 0 < K < 20

Question 6

A right-half-plane zero most directly indicates:

A. A guaranteed unstable natural mode

B. Nonminimum-phase behavior

C. A passive resistor

D. A pole at infinity only

Correct Answer: B. Nonminimum-phase behavior

Question 7

Which expression gives the state-space transfer function under zero initial conditions?

A. H(s) = A(sI − C)^(−1)B + D

B. H(s) = C(sI − A)^(−1)B + D

C. H(s) = B(sI − D)^(−1)C + A

D. H(s) = (sI − A)CB + D

Correct Answer: B. H(s) = C(sI − A)^(−1)B + D

Question 8

Why is an exact pole-zero cancellation potentially unsafe in hardware?

A. It makes all poles move to the origin

B. Component tolerances can reveal the canceled internal mode

C. It always violates Kirchhoff’s laws

D. It eliminates every zero in the network

Correct Answer: B. Component tolerances can reveal the canceled internal mode

Question 9

For a first-order low-pass factor 1/(1 + sτ), the break angular frequency is:

A. τ

B. 1/τ

C. 2πτ

D. τ²

Correct Answer: B. 1/τ

Question 10

A positive-real driving-point impedance is associated primarily with:

A. Passive network realizability

B. Guaranteed right-half-plane poles

C. Nonlinear saturation

D. Discrete-time sampling

Correct Answer: A. Passive network realizability