🔦Electrical Circuits and Systems II Unit 11 – Two-Port Networks: Parameters & Analysis

Introduction to Two-Port Networks

• Two-port networks are electrical networks with two pairs of terminals (ports) used to connect to external circuits
• Each port consists of two terminals, an input port and an output port, allowing the network to be analyzed using various parameters
• Enable the analysis and design of complex electrical systems by simplifying them into smaller, more manageable components
• Provide a mathematical framework for understanding the relationship between the input and output voltages and currents
• Commonly used in the analysis of transistor amplifiers, filters, transmission lines, and other electronic circuits
• The behavior of a two-port network can be described using different sets of parameters, such as Z, Y, h, and ABCD parameters
• The choice of parameters depends on the specific application and the available information about the network

Types of Two-Port Parameters

• Two-port parameters are sets of equations that describe the relationship between the input and output voltages and currents of a two-port network
• The four main types of two-port parameters are:
  • Z-parameters (impedance parameters)
  • Y-parameters (admittance parameters)
  • h-parameters (hybrid parameters)
  • ABCD parameters (transmission parameters)
• Each type of parameter has its own unique set of equations and is suitable for different applications and network configurations
• The choice of parameters depends on factors such as the network topology, available measurements, and desired analysis or design objectives
• Two-port parameters allow engineers to analyze and design complex electrical networks by breaking them down into smaller, more manageable components
• The parameters can be determined through various methods, including direct measurement, calculation from network equations, or conversion from other parameter sets

Z-Parameters (Impedance Parameters)

• Z-parameters, also known as impedance parameters, relate the voltages and currents of a two-port network in terms of impedances
• The Z-parameters are defined as:
  • $Z_{11} = \frac{V_1}{I_1}$ (input impedance with output open-circuited)
  • $Z_{12} = \frac{V_1}{I_2}$ (reverse transfer impedance with input open-circuited)
  • $Z_{21} = \frac{V_2}{I_1}$ (forward transfer impedance with output open-circuited)
  • $Z_{22} = \frac{V_2}{I_2}$ (output impedance with input open-circuited)
• The Z-parameter equations can be written in matrix form as: $\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}$
• Z-parameters are particularly useful when the two-port network is composed of series-connected elements or when the input and output impedances are of primary interest
• To measure Z-parameters, open-circuit conditions are applied to the network, and the voltages and currents are measured at each port
• Z-parameters can be converted to other two-port parameter sets, such as Y, h, or ABCD parameters, using matrix transformations

Y-Parameters (Admittance Parameters)

• Y-parameters, also known as admittance parameters, relate the currents and voltages of a two-port network in terms of admittances
• The Y-parameters are defined as:
  • $Y_{11} = \frac{I_1}{V_1}$ (input admittance with output short-circuited)
  • $Y_{12} = \frac{I_1}{V_2}$ (reverse transfer admittance with input short-circuited)
  • $Y_{21} = \frac{I_2}{V_1}$ (forward transfer admittance with output short-circuited)
  • $Y_{22} = \frac{I_2}{V_2}$ (output admittance with input short-circuited)
• The Y-parameter equations can be written in matrix form as: $\begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}$
• Y-parameters are particularly useful when the two-port network is composed of parallel-connected elements or when the input and output admittances are of primary interest
• To measure Y-parameters, short-circuit conditions are applied to the network, and the currents and voltages are measured at each port
• Y-parameters can be converted to other two-port parameter sets, such as Z, h, or ABCD parameters, using matrix transformations

h-Parameters (Hybrid Parameters)

• h-parameters, also known as hybrid parameters, relate the voltages and currents of a two-port network using a mix of impedance and admittance quantities
• The h-parameters are defined as:
  • $h_{11} = \frac{V_1}{I_1}$ (input impedance with output short-circuited)
  • $h_{12} = \frac{V_1}{V_2}$ (reverse voltage gain with input open-circuited)
  • $h_{21} = \frac{I_2}{I_1}$ (forward current gain with output short-circuited)
  • $h_{22} = \frac{I_2}{V_2}$ (output admittance with input open-circuited)
• The h-parameter equations can be written in matrix form as: $\begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix}$
• h-parameters are particularly useful for analyzing transistor circuits, as they directly relate to the input impedance, voltage gain, current gain, and output admittance of the device
• To measure h-parameters, a combination of open-circuit and short-circuit conditions are applied to the network, and the voltages and currents are measured at each port
• h-parameters can be converted to other two-port parameter sets, such as Z, Y, or ABCD parameters, using matrix transformations

ABCD Parameters (Transmission Parameters)

• ABCD parameters, also known as transmission parameters or chain parameters, relate the input and output voltages and currents of a two-port network in a cascaded configuration
• The ABCD parameters are defined as:
  • $A = \frac{V_1}{V_2}$ (voltage ratio with output open-circuited)
  • $B = \frac{V_1}{I_2}$ (transfer impedance with output open-circuited)
  • $C = \frac{I_1}{V_2}$ (transfer admittance with output open-circuited)
  • $D = \frac{I_1}{I_2}$ (current ratio with output open-circuited)
• The ABCD parameter equations can be written in matrix form as: $\begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix}$
• ABCD parameters are particularly useful for analyzing cascaded networks, as the overall ABCD matrix of the cascaded network is the product of the individual ABCD matrices
• The ABCD parameters have a unique property: the determinant of the ABCD matrix is always equal to 1 (AD - BC = 1)
• ABCD parameters can be converted to other two-port parameter sets, such as Z, Y, or h parameters, using matrix transformations

Network Interconnections and Conversions

• Two-port networks can be interconnected in various configurations to create more complex networks
• The three main types of interconnections are:
  • Series connection: The output port of one network is connected to the input port of another network
  • Parallel connection: The input ports and output ports of two networks are connected together
  • Cascade connection: The output port of one network is connected to the input port of another network, forming a chain
• When networks are interconnected, their respective two-port parameters can be combined to determine the overall network parameters
• For series and parallel connections, the Z and Y parameters can be easily combined using matrix addition and inversion
• For cascade connections, the ABCD parameters are particularly useful, as the overall ABCD matrix is the product of the individual ABCD matrices
• Two-port parameter conversions allow engineers to switch between different parameter sets based on the available information and the desired analysis or design objectives
• Conversion formulas exist for transforming between Z, Y, h, and ABCD parameters, enabling flexibility in network analysis and design

Applications and Practical Examples

• Two-port networks and their parameters have numerous applications in electrical engineering, particularly in the analysis and design of electronic circuits
• Transistor amplifiers can be modeled as two-port networks, with h-parameters being particularly useful for analyzing their performance characteristics
  • Example: A common-emitter amplifier can be characterized using h-parameters to determine its input impedance, voltage gain, current gain, and output admittance
• Filters, such as passive LC filters or active filters, can be represented as two-port networks and analyzed using Z or Y parameters
  • Example: A low-pass LC filter can be designed by specifying the desired cutoff frequency and using Z parameters to calculate the required inductor and capacitor values
• Transmission lines, which are used for transmitting high-frequency signals, can be modeled as two-port networks using ABCD parameters
  • Example: A coaxial cable can be characterized by its ABCD parameters, allowing engineers to analyze its propagation characteristics and impedance matching properties
• Impedance matching networks, used to maximize power transfer between a source and a load, can be designed using two-port network theory and parameter conversions
  • Example: A matching network can be designed to transform a complex load impedance to the conjugate of the source impedance, ensuring maximum power transfer
• Two-port network theory is also applied in the analysis and design of power systems, microwave circuits, and communication networks, among other areas