1.5 Analysis of One-Port and Two-Port Networks

1.5 Analysis of One-Port and Two-Port Networks

Introduction to Network Analysis

Complex electrical and electronic systems are often constructed by interconnecting simpler functional blocks. The one-port and two-port network models provide a powerful, systematic framework for analyzing these blocks without needing to know their intricate internal details. By characterizing a network solely by the relationships between the voltages and currents at its external terminals (ports), we can predict its overall system behavior, design filters and amplifiers, and efficiently interconnect subsystems. This section explores the foundational concepts of transfer functions and port parameters, which are essential for modern circuit and system design.

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1. Transfer Functions, Poles, and Zeros

1.1 The Transfer Function

1. Definition: For a linear time-invariant (LTI) system, the Transfer Function, $H(s)$, is defined as the ratio of the Laplace transform of the output (response) to the Laplace transform of the input (excitation), assuming all initial conditions are zero. $H(s) = \frac{Y(s)}{X(s)}$ Here, $s = \sigma + j\omega$ is the complex frequency variable from the Laplace transform.

2. Significance:

   • It completely characterizes the system's input-output behavior in the frequency domain.

   • It is independent of the magnitude or nature of the input signal.

   • It allows the prediction of system response (e.g., to steps, sinusoids) and the analysis of stability and frequency response.

1.2 Poles and Zeros

1. General Form: A transfer function of a lumped, linear network is a rational function of $s$: $H(s) = K \frac{(s - z_1)(s - z_2)\dots(s - z_m)}{(s - p_1)(s - p_2)\dots(s - p_n)} \quad \text{where } n \ge m$

2. Zeros:

   • The values of $s$ ($z_1, z_2, \dots, z_m$) that make the numerator (and thus $H(s)$) equal to zero.

   • In the complex s-plane, they are represented by a small circle (◯).

   • Zeros tend to attenuate the system response at frequencies near the zero location.

3. Poles:

   • The values of $s$ ($p_1, p_2, \dots, p_n$) that make the denominator (and thus $H(s)$) equal to infinity.

   • In the complex s-plane, they are represented by a small cross (✕).

   • Poles determine the natural frequencies (modes) of the system and are crucial for stability analysis.

4. Gain Constant (K): Scales the overall magnitude of the response.

1.3 Relationship to System Response

1. Time Domain Response: The poles of $H(s)$ determine the form of the natural (transient) response of the system.

   • Real pole ($s = -\sigma$): Corresponds to an exponential decay, $e^{-\sigma t}$.

   • Complex conjugate poles ($s = -\sigma \pm j\omega_d$): Correspond to a damped sinusoid, $e^{-\sigma t}\cos(\omega_d t + \phi)$.

   • Pole location dictates stability:

     • Stable System: All poles lie in the left half of the s-plane (real part < 0).

     • Marginally Stable: Poles on the imaginary axis (real part = 0), not repeated.

     • Unstable: Any pole in the right half-plane (real part > 0).

2. Frequency Domain Response ($H(j\omega)$):

   • Obtained by evaluating $H(s)$ along the imaginary axis ($s = j\omega$).

   • The magnitude response $|H(j\omega)|$ shows how the system amplifies/attenuates different frequency components.

   • The phase response $\angle H(j\omega)$ shows the phase shift introduced.

   • Poles near the $j\omega$ axis cause peaks in the magnitude response; zeros near the axis cause notches.

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2. One-Port Passive Circuits

2.1 Definition of a One-Port Network

A one-port (or single-port) network is a circuit or device with only two accessible terminals. The external behavior is defined by the relationship between the voltage across these terminals ($V$) and the current flowing into one terminal ($I$).

2.2 Impedance Function, $Z(s)$

1. Definition: The impedance of a one-port is the ratio of the Laplace-transformed terminal voltage to the Laplace-transformed terminal current, with independent sources within the network turned off (zeroed). $Z(s) = \frac{V(s)}{I(s)} \quad \text{(with all internal sources = 0)}$

2. Properties for Passive Networks (containing only R, L, C, M):

   • It is a positive real function.

   • Its poles and zeros lie in the left half of the s-plane or on the $j\omega$ axis, and any on the $j\omega$ axis are simple.

   • The real part of $Z(j\omega)$ is non-negative for all $\omega$: $\operatorname{Re}[Z(j\omega)] \ge 0$.

3. Examples:

   • Series RL: $Z(s) = R + sL$

   • Parallel RC: $Z(s) = \frac{R \cdot (1/sC)}{R + 1/sC} = \frac{R}{1 + sRC}$

2.3 Admittance Function, $Y(s)$

1. Definition: The admittance is the reciprocal of impedance. It is the ratio of current to voltage. $Y(s) = \frac{I(s)}{V(s)} = \frac{1}{Z(s)}$

2. Properties: Shares the same mathematical properties as $Z(s)$ for passive networks (positive real function).

3. Relationship: For a one-port, $Z(s)$ and $Y(s)$ provide equivalent descriptions.

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3. Two-Port Networks and Parameters

3.1 Definition of a Two-Port Network

A two-port network is a circuit or device with two pairs of terminals (four terminals total), where one pair is designated as the input port and the other as the output port. The internal network may be active or passive, linear or nonlinear. For linear networks, the external behavior is completely described by relationships between the four port variables: $V_1, I_1$ (input) and $V_2, I_2$ (output). Convention assumes current enters the positive terminal for each port.

3.2 General Two-Port Parameter Paradigm

Different parameter sets express two of the port variables as linear combinations of the other two. Each set is useful for specific types of interconnections or analyses.

3.3 Short-Circuit Admittance (Y) Parameters

1. Definition: Express the port currents in terms of the port voltages. $\begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}$

2. Parameter Definitions:

   • $y_{11} = \frac{I_1}{V_1} \bigg|_{V_2=0}$: Input Admittance (output short-circuited).

   • $y_{12} = \frac{I_1}{V_2} \bigg|_{V_1=0}$: Reverse Transfer Admittance (input short-circuited).

   • $y_{21} = \frac{I_2}{V_1} \bigg|_{V_2=0}$: Forward Transfer Admittance (output short-circuited).

   • $y_{22} = \frac{I_2}{V_2} \bigg|_{V_1=0}$: Output Admittance (input short-circuited).

3. Conditions: Defined with ports short-circuited ($V=0$). Ideal for analyzing parallel connections of two-ports.

3.4 Open-Circuit Impedance (Z) Parameters

1. Definition: Express the port voltages in terms of the port currents. $\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}$

2. Parameter Definitions:

   • $z_{11} = \frac{V_1}{I_1} \bigg|_{I_2=0}$: Input Impedance (output open-circuited).

   • $z_{12} = \frac{V_1}{I_2} \bigg|_{I_1=0}$: Reverse Transfer Impedance (input open-circuited).

   • $z_{21} = \frac{V_2}{I_1} \bigg|_{I_2=0}$: Forward Transfer Impedance (output open-circuited).

   • $z_{22} = \frac{V_2}{I_2} \bigg|_{I_1=0}$: Output Impedance (input open-circuited).

3. Conditions: Defined with ports open-circuited ($I=0$). Ideal for analyzing series connections of two-ports.

3.5 Transmission (ABCD) Parameters

1. Definition: Express the input variables ($V_1, I_1$) in terms of the output variables ($V_2, I_2$). This is the chain or ABCD matrix. $\begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix}$ Note the negative sign on $I_2$, adhering to the convention that output current exits the positive terminal.

2. Parameter Definitions:

   • $A = \frac{V_1}{V_2} \bigg|_{I_2=0}$: Voltage Ratio (open-circuit reverse).

   • $B = \frac{V_1}{-I_2} \bigg|_{V_2=0}$: Transfer Impedance (short-circuit).

   • $C = \frac{I_1}{V_2} \bigg|_{I_2=0}$: Transfer Admittance (open-circuit).

   • $D = \frac{I_1}{-I_2} \bigg|_{V_2=0}$: Current Ratio (short-circuit reverse).

3. Significance: Extremely useful for analyzing cascaded networks. The overall ABCD matrix is the product of the individual ABCD matrices.

3.6 Hybrid (h) Parameters

1. Definition: Express the input voltage ($V_1$) and output current ($I_2$) in terms of the input current ($I_1$) and output voltage ($V_2$). This is a "mixed" representation. $\begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix}$

2. Parameter Definitions:

   • $h_{11} = \frac{V_1}{I_1} \bigg|_{V_2=0}$: Input Impedance (output short).

   • $h_{12} = \frac{V_1}{V_2} \bigg|_{I_1=0}$: Reverse Voltage Gain (input open).

   • $h_{21} = \frac{I_2}{I_1} \bigg|_{V_2=0}$: Forward Current Gain (output short). (Often denoted $h_{fe}$ for transistors).

   • $h_{22} = \frac{I_2}{V_2} \bigg|_{I_1=0}$: Output Admittance (input open).

3. Application: Naturally suited for modeling bipolar junction transistors (BJTs) in the common-emitter configuration, as these parameters are easy to measure at low frequencies.

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4. Inter-relationships Between Two-Port Parameters

1. Concept: Any complete set of two-port parameters contains the same information about the network. Therefore, it is possible to convert from one parameter set to another.

2. Conversion Tables: Standard conversion formulas exist. For example:

   • From Z to Y: $[Y] = [Z]^{-1}$ (provided the inverse exists).

   • From Y to Z: $[Z] = [Y]^{-1}$.

   • Conversions to/from ABCD, h, etc., involve algebraic manipulation of the defining equations.

3. Condition for Existence: A parameter set exists if the defining conditions can be physically realized (e.g., you cannot measure $y_{11}$ if a short circuit at port 2 causes an internal component to burn out).

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5. Interconnections of Two-Port Networks

Complex networks are built by connecting simpler two-port blocks.

5.1 Series Connection

1. Connection Method: Input ports in series, output ports in series.

2. Applicable Parameters: Z-parameters.

3. Overall Matrix: The overall impedance matrix is the sum of the individual Z-matrices, provided the connection does not alter the individual network's port conditions (a condition known as non-loading or port isolation). $[Z]_{total} = [Z]_{A} + [Z]_{B}$

5.2 Parallel Connection

1. Connection Method: Input ports in parallel, output ports in parallel.

2. Applicable Parameters: Y-parameters.

3. Overall Matrix: The overall admittance matrix is the sum of the individual Y-matrices (subject to port isolation). $[Y]_{total} = [Y]_{A} + [Y]_{B}$

5.3 Cascade Connection

1. Connection Method: Output of the first network feeds directly into the input of the second network. This is the most common interconnection.

2. Applicable Parameters: ABCD (Transmission) Parameters.

3. Overall Matrix: The overall ABCD matrix is the product of the individual ABCD matrices. Order of multiplication matters (matrix multiplication is not commutative). $\begin{bmatrix} A & B \\ C & D \end{bmatrix}_{total} = \begin{bmatrix} A & B \\ C & D \end{bmatrix}_{A} \times \begin{bmatrix} A & B \\ C & D \end{bmatrix}_{B}$

5.4 Other Connections

• Series-Parallel: Uses g-parameters (inverse of h-parameters).

• Parallel-Series: Uses h-parameters.

Conclusion: The one-port and two-port formalism is a cornerstone of linear network theory, enabling a modular, "black-box" approach to system design. By characterizing components and subsystems with impedance, admittance, or ABCD parameters, engineers can predict the behavior of interconnected systems—from cascaded amplifier stages to filter networks and transmission lines—with mathematical rigor and efficiency. The choice of parameter set is dictated by the physical interconnection and the ease of measurement or calculation.