Two-Port Networks and Interconnected Systems

Introduction

Many electrical systems are best studied through their terminal behavior rather than their internal components. An amplifier may contain dozens of devices, yet its interaction with a source and load can be described through two input terminals and two output terminals. Filters, transmission-line sections, transformers, matching networks and feedback subsystems can be treated similarly.

A two-port network groups four terminals into an input port and an output port. Each port satisfies the port condition: the current entering one terminal equals the current leaving the other. Two-port models reduce a circuit to relationships among four variables—port voltages V₁,V₂ and port currents I₁,I₂. Different parameter sets are advantageous for different measurements and interconnections.

Learning Objectives

After completing this chapter, students should be able to:

• establish consistent voltage and current reference directions for a two-port network;
• derive impedance, admittance, hybrid and transmission parameters;
• obtain parameters from open-circuit and short-circuit tests;
• convert between parameter representations;
• test networks for reciprocity and symmetry;
• calculate loaded input and output impedances, voltage gain and current gain;
• analyze cascaded networks using transmission matrices;
• apply two-port models to amplifiers, filters and transmission subsystems; and
• recognize model limitations, measurement concerns and high-frequency alternatives.

1. Two-Port Variables and Sign Conventions

The standard convention used in this chapter defines both I₁ and I₂ as entering the network. Port voltages are measured from the upper terminal to the lower reference terminal.

Figure: Reference directions for two-port variables; both port currents enter the network.

For a passive load Z_L connected to port 2, the load current is I_L=-I₂. Therefore,

This sign convention must be maintained during parameter conversion. A silent change in the direction of I₂ is a common source of incorrect signs.

The models developed below normally assume a linear network at a fixed frequency. In sinusoidal steady state, voltages, currents and parameters may be complex phasors. Resistances and impedances have units of ohms (Ω), while conductances and admittances have units of siemens (S).

2. Impedance Parameters

The impedance, or z-parameter, model expresses port voltages as functions of currents:

Thus,

All four parameters have units of ohms. They are obtained through open-circuit tests:

The z model is convenient for series-connected two-ports because, under compatible port references, their z matrices add. It may be unsuitable where a true open circuit cannot be produced safely or where open-circuit operation causes instability.

3. Admittance Parameters

The admittance, or y-parameter, model expresses currents as functions of voltages:

All y parameters have units of siemens. They are measured using short-circuit conditions:

The y representation is useful for parallel-connected two-ports, whose y matrices add when the corresponding port voltages are common. Physical short circuits may nevertheless be impractical at high power or high frequency.

If the z matrix is nonsingular, then \mathbf{Y}=\mathbf{Z}^-1. Defining

the conversion is

This conversion requires \Delta_z\ne0.

4. Hybrid Parameters

Hybrid, or h parameters, use an input current and output voltage as independent variables:

Hence,

The parameters have mixed dimensions:

• h₁₁: input impedance in Ω with the output short-circuited;
• h₁₂: reverse voltage ratio, dimensionless, with the input open-circuited;
• h₂₁: forward current ratio, dimensionless, with the output short-circuited;
• h₂₂: output admittance in S with the input open-circuited.

Hybrid parameters are useful for small-signal transistor and amplifier modeling because input impedance and forward current gain appear explicitly.

From z parameters, provided z₂₂\ne0,

Worked Example 1: Conversion of a z Matrix

A reciprocal two-port has

Its determinant is \Delta_z=(10)(8)-(4)(4)=64\ \Omega². Therefore,

The h parameters are

Mixed units are shown explicitly; an h matrix does not possess one common unit.

5. Transmission Parameters

Transmission parameters—also called chain or ABCD parameters—are especially effective for cascaded systems. The convention adopted here is

Because -I₂ is the current delivered from the two-port into the next stage, this form allows direct matrix multiplication. Parameters A and D are dimensionless, B is in Ω, and C is in S.

From z parameters, provided z₂₁\ne0,

Conversely, if C\ne0,

A conversion formula requiring division by zero does not imply that the physical network is invalid; it means that the selected algebraic representation is singular for that case.

Elementary Transmission Matrices

For the transmission line, \gamma is the propagation constant in m⁻¹, l is length in metres, and Z₀ is characteristic impedance in Ω. For a lossless line, \gamma=j\beta, so hyperbolic functions reduce to trigonometric forms.

Worked Example 2: Cascaded Series–Shunt–Series Network

A network consists, from input to output, of a 4 Ω series resistor, a 0.5 S shunt conductance and a 2 Ω series resistor. Its transmission matrix is

Thus A=3, B=10\ \Omega, C=0.5\ \text{S} and D=2. The determinant is AD-BC=6-5=1.

With Z_L=10\ \Omega,

The loaded voltage ratio is

6. Reciprocity, Symmetry and Passivity

These properties must not be confused.

A linear bilateral two-port is reciprocal if excitation and response can be interchanged appropriately. Equivalent tests include

The ABCD test applies to the sign convention used in this chapter. Dependent sources, active devices and nonreciprocal magnetic components may violate reciprocity.

A two-port is symmetric if interchanging its ports leaves its terminal behavior unchanged. Common tests are

Symmetry does not automatically establish reciprocity.

Passivity means that a network cannot deliver more net energy than it has previously stored or received. A reciprocal matrix is not necessarily passive: an active reciprocal circuit can be constructed using controlled sources. At a single sinusoidal frequency, passivity imposes positive-real or power inequalities that cannot generally be inferred from one scalar parameter alone.

7. Loading and Gain

7.1 Loaded Input Impedance

For a load Z_L at port 2, the z equations and I₂=-V₂/Z_L give

The expression requires z₂₂+Z_L\ne0. In ABCD form,

provided the denominator is nonzero.

7.2 Loaded Output Impedance

To determine output impedance, an independent voltage source at the input is set to zero and replaced by its internal impedance Z_S. Dependent sources remain active. The result is

This formula assumes the source termination is the only external connection at port 1 and z₁₁+Z_S\ne0.

7.3 Voltage and Current Ratios

For an ABCD model terminated by Z_L,

These are loaded two-port ratios, not necessarily overall source-to-load gains. If a Thévenin source V_S with impedance Z_S drives the network, then

Source loading must therefore be included when system-level gain or power is required.

Worked Example 3: Loaded Amplifier Using h Parameters

An amplifier has h₁₁=1\ \text{k}\Omega, h₁₂=0, h₂₁=50 and h₂₂=20\ \mu\text{S}. It drives Z_L=2\ \text{k}\Omega.

Using I₂=-V₂/Z_L in the second h equation,

Because h₁₂=0, V₁=h₁₁I₁. Hence

The negative sign indicates phase inversion under the chosen references. Nonzero reverse feedback h₁₂ would alter both the loaded input impedance and gain.

8. Cascaded Networks

If two sections have transmission matrices \mathbf{T}₁ and \mathbf{T}₂ in physical order from source to load, then

For N stages,

Matrix multiplication is not commutative. Reversing two dissimilar stages can change loading and overall behavior even when each individual stage is reciprocal.

Figure: Transmission matrices multiply in the physical order of cascaded sections.

9. Filter and Transmission Applications

For a low-pass L-section comprising a series inductor followed by a shunt capacitor,

where L is inductance in henries, C is capacitance in farads and \omega is angular frequency in rad/s. The section matrix is

The symbol Z_ind is deliberately distinguished from load impedance Z_L. Frequency response follows by substituting the frequency-dependent matrix into the loaded voltage-ratio formula. Component losses can be incorporated using complex impedances rather than ideal values.

For a lossless line of characteristic impedance Z₀ and electrical length \theta=\beta l,

This representation includes phase delay and impedance transformation. Lumped models become inaccurate when physical dimensions are no longer small relative to wavelength.

10. Illustrative Case: Instrumentation Transmission Subsystem

Background

This fictional case concerns a 50 Ω laboratory sensor interface. A sinusoidal source with open-circuit RMS voltage 1 V and source resistance 50 Ω drives a 50 Ω load through a lossless 30° transmission-line section and a matched 6.02 dB attenuator. The attenuator has voltage ratio N=2 when both ports are matched.

Problem

The engineer must verify input matching and determine load voltage and power without reconstructing the internal attenuator circuit.

Analysis and Approach

For a reciprocal attenuator matched to Z₀=50\ \Omega,

Its determinant is (1.25)²-(37.5)(0.015)=1, confirming reciprocity. The 30° lossless line has

Both sections are matched to 50 Ω, so their cascade also presents 50 Ω when terminated in 50 Ω. The source therefore divides equally between its 50 Ω resistance and the subsystem:

The lossless line changes phase but not matched-load magnitude. The attenuator divides voltage by two, giving

The load power is

Outcome

The subsystem remains matched, delivers 0.25 V RMS and dissipates 1.25 mW in the load. The source’s available power is 5 mW, so the 6.02 dB attenuator reduces power by a factor of four, as expected.

Learning Points

• Compatible ABCD matrices permit modular system analysis.
• Matching prevents reflections but does not imply zero insertion loss.
• Voltage gain must be distinguished from source-to-load gain.
• Phase and magnitude effects can be allocated to different subsystem blocks.

11. Measurement, Limitations and Contemporary Practice

Parameter values depend on frequency, bias, temperature and operating amplitude. Linear small-signal parameters are valid only near the bias point at which they were obtained. Parasitic capacitance, lead inductance and fixture effects can dominate high-frequency measurements.

At microwave frequencies, open- and short-circuit tests become difficult and may create strong reflections or instability. Scattering parameters use incident and reflected traveling waves and are measured with vector network analyzers. For equal real reference impedances Z₀, the matrix relation is

provided the inverse exists and wave definitions use the same reference impedance. Modern software can convert parameter sets, de-embed fixtures and optimize matching networks, but automated results remain vulnerable to wrong port orientation, reference impedance or calibration data.

Professional best practice includes documenting reference directions, frequency, bias, temperature, calibration plane and units. Power and voltage limits must be respected during open- and short-circuit tests. In regulated communication, medical and aerospace systems, models should support traceable verification, electromagnetic-compatibility requirements and safe failure behavior. AI-assisted fitting may identify compact two-port models from measurements, but validation data should be independent and physically impossible predictions—such as unexplained active gain from a passive device—must be investigated.

Chapter Summary

A two-port model describes a linear network through terminal voltage and current relationships. Impedance parameters suit open-circuit characterization and series interconnections; admittance parameters suit short-circuit characterization and parallel interconnections; hybrid parameters are useful for amplifier models; and ABCD parameters simplify cascades. Correct sign conventions are essential, particularly because the chain model uses -I₂.

Parameter matrices can be converted when the required determinants or divisors are nonzero. Reciprocity, symmetry and passivity are distinct properties. Loading modifies input impedance, output impedance and gain, so source and load terminations must be included in system calculations. Two-port techniques support modular analysis of amplifiers, filters, attenuators and transmission lines, while scattering parameters are generally preferred at microwave frequencies.

Key Terms

Two-port network: A four-terminal network organized as an input port and an output port.

Port condition: Requirement that current entering one terminal of a port leaves through the other.

Impedance parameters: Parameters relating port voltages to port currents.

Admittance parameters: Parameters relating port currents to port voltages.

Hybrid parameters: Mixed-dimension parameters using input current and output voltage as independent variables.

Transmission parameters: Parameters relating input variables to output voltage and delivered output current.

ABCD matrix: The matrix containing transmission parameters A, B, C and D.

Loading: Change in circuit behavior caused by connection of a finite source or load impedance.

Reciprocity: Property permitting interchange of excitation and response under specified conditions.

Symmetry: Invariance of two-port behavior when its ports are interchanged.

Passivity: Inability of a network to generate net energy.

Cascade: Ordered connection in which the output of one stage drives the input of another.

Characteristic impedance: Voltage-to-current ratio of a single traveling wave on a uniform line.

Propagation constant: Complex quantity describing attenuation and phase change per unit length.

Insertion loss: Reduction in delivered signal power caused by inserting a network.

Small-signal model: Linearized model valid near a specified operating point.

Scattering parameters: Traveling-wave parameters used especially at radio and microwave frequencies.

De-embedding: Removal of fixture and interconnection effects from measured data.

Review Questions

Short Answer Questions

1. State the current reference convention used for both ports in this chapter.
2. How is each z parameter measured experimentally?
3. Why does the ABCD definition use -I₂ rather than I₂?
4. Distinguish reciprocity from symmetry.
5. Why are scattering parameters preferred at microwave frequencies?

Long Answer Questions

1. Derive the conversion from z parameters to ABCD parameters and state all required conditions.
2. Explain the physical meaning, dimensions and measurement conditions of the four h parameters.
3. Derive the loaded input impedance in both z-parameter and ABCD forms.
4. Discuss how frequency, bias, temperature and fixture parasitics limit a measured two-port model.
5. Compare z, y, h, ABCD and scattering representations for laboratory and design applications.

Analytical / Application Questions

1. A two-port has \mathbf{Z}=\begin{bmatrix}12&3\\3&6\end{bmatrix}\ \Omega. Convert it to y and ABCD forms, then test reciprocity and symmetry.
2. Cascade a 5 Ω series impedance, a 0.1 S shunt admittance and a 10 Ω load. Determine the ABCD matrix, input impedance and loaded voltage ratio.
3. Design an analysis procedure for a multistage sensor interface containing an amplifier, filter and transmission cable. Specify suitable parameter sets, conversions, measurements and validation checks.

Multiple Choice Questions with Answers

Question 1

Which parameter set is most directly suited to a cascade of two-port sections?

A. Impedance parameters

B. Admittance parameters

C. Transmission parameters

D. Open-circuit parameters only

Correct Answer: C. Transmission parameters

Question 2

Under the current convention of this chapter, a load connected at port 2 satisfies which relation?

A. V₂=Z_L I₂

B. V₂=-Z_L I₂

C. V₁=-Z_L I₂

D. I₁=-V₂/Z_L

Correct Answer: B. V₂=-Z_L I₂

Question 3

For a reciprocal two-port represented by ABCD parameters under the stated convention, which condition applies?

A. A+D=1

B. A=D

C. B=C

D. AD-BC=1

Correct Answer: D. AD-BC=1

Question 4

A two-port has z₁₁=10\ \Omega, z₁₂=2\ \Omega, z₂₁=2\ \Omega and z₂₂=8\ \Omega. With Z_L=12\ \Omega, what is its input impedance?

A. 9.8 Ω

B. 10.2 Ω

C. 8.0 Ω

D. 12.0 Ω

Correct Answer: A. 9.8 Ω

Question 5

Which hybrid parameter has units of siemens?

A. h₁₁

B. h₁₂

C. h₂₁

D. h₂₂

Correct Answer: D. h₂₂

Question 6

The ABCD matrix of a 3 Ω series impedance is which of the following?

A. \begin{bmatrix}1&3\\0&1\end{bmatrix}

B. \begin{bmatrix}1&0\\3&1\end{bmatrix}

C. \begin{bmatrix}3&0\\0&1\end{bmatrix}

D. \begin{bmatrix}1&0\\1/3&1\end{bmatrix}

Correct Answer: A. \begin{bmatrix}1&3\\0&1\end{bmatrix}

Question 7

A network has A=2, B=6\ \Omega, C=0.25\ \text{S} and D=1.25. With Z_L=8\ \Omega, what is Z_in?

A. 4.0 Ω

B. 6.0 Ω

C. 8.0 Ω

D. 22.0 Ω

Correct Answer: C. 8.0 Ω

Question 8

Which condition indicates symmetry in an ABCD representation?

A. A=D

B. B=C

C. A+B=C+D

D. AD=BC

Correct Answer: A. A=D

Question 9

For an ideal inductor used as a series filter element, which impedance notation and value are appropriate?

A. Z_L=j\omega C

B. Z_ind=1/(j\omega L)

C. Z_ind=j\omega L

D. Y_cap=j\omega L

Correct Answer: C. Z_ind=j\omega L

Question 10

Two cascaded sections have matrices \mathbf{T}₁ and \mathbf{T}₂ in source-to-load order. What is the total matrix?

A. \mathbf{T}₁+\mathbf{T}₂

B. \mathbf{T}₂\mathbf{T}₁

C. \mathbf{T}₁^-1\mathbf{T}₂

D. \mathbf{T}₁\mathbf{T}₂

Correct Answer: D. \mathbf{T}₁\mathbf{T}₂