11.2 Z, Y, h, and ABCD parameters

Two-port networks are essential in electrical circuit analysis. They help us understand how signals flow through complex systems. Z, Y, h, and ABCD parameters are different ways to describe these networks, each with unique advantages.

These parameters let us model network behavior using simple equations. By choosing the right parameter set, we can simplify calculations and gain insights into circuit performance. Understanding how to use and convert between these parameters is key to mastering two-port network analysis.

Two-Port Network Parameters

Impedance and Admittance Parameters

• Z-parameters (Impedance parameters) define voltage-current relationships in a two-port network
  • Expressed as ratios of voltage to current
  • Measured with open-circuit conditions at one port
  • Useful for analyzing series-connected networks
• Y-parameters (Admittance parameters) represent the inverse of Z-parameters
  • Expressed as ratios of current to voltage
  • Measured with short-circuit conditions at one port
  • Advantageous for analyzing parallel-connected networks
• Both Z and Y parameters utilize 2x2 matrices to describe network behavior
  • Z-parameter matrix: $\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}$
  • Y-parameter matrix: $\begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}$

Hybrid and Transmission Parameters

• h-parameters (Hybrid parameters) combine voltage and current ratios
  • Utilize both open-circuit and short-circuit conditions for measurement
  • Particularly useful for analyzing transistor circuits
  • h-parameter matrix: $\begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix}$
• ABCD parameters (Transmission parameters) describe overall network transmission characteristics
  • Also known as chain parameters
  • Relate input voltage and current to output voltage and current
  • Facilitate analysis of cascaded networks
  • ABCD parameter matrix: $\begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix}$

Parameter Characteristics

Open and Short Circuit Conditions

• Open-circuit impedance parameters measured with one port open
  • Z11: Input impedance with output open-circuited
  • Z22: Output impedance with input open-circuited
  • Z12 and Z21: Transfer impedances
• Short-circuit admittance parameters measured with one port shorted
  • Y11: Input admittance with output short-circuited
  • Y22: Output admittance with input short-circuited
  • Y12 and Y21: Transfer admittances
• Open and short circuit conditions provide simplified measurement scenarios
  • Allow isolation of specific parameter effects
  • Enable accurate determination of individual parameter values

Network Symmetry and Reciprocity

• Reciprocity in two-port networks indicates bidirectional behavior
  • Applies to passive networks without dependent sources
  • Characterized by Z12 = Z21, Y12 = Y21, h12 = -h21, and AD - BC = 1
• Symmetry in two-port networks implies identical behavior from either port
  • Occurs when network structure is mirrored around its center
  • Indicated by Z11 = Z22, Y11 = Y22, h11 = h22, and A = D
• Both reciprocity and symmetry simplify network analysis
  • Reduce the number of parameters needed to fully describe the network
  • Allow for more straightforward calculations and modeling

Parameter Relationships

Parameter Conversion and Transformations

• Parameter conversion enables switching between different parameter sets
  • Facilitates using the most appropriate parameters for specific analyses
  • Involves matrix algebra and determinant calculations
• Z to Y parameter conversion: $Y = Z^{-1}$
  • Y11 = Z22 / det(Z), Y12 = -Z12 / det(Z)
  • Y21 = -Z21 / det(Z), Y22 = Z11 / det(Z)
• Y to Z parameter conversion: $Z = Y^{-1}$
  • Z11 = Y22 / det(Y), Z12 = -Y12 / det(Y)
  • Z21 = -Y21 / det(Y), Z22 = Y11 / det(Y)
• h-parameter conversions involve combinations of Z and Y parameters
  • h11 = Z11 - (Z12 * Z21 / Z22)
  • h12 = -Z12 / Z22
  • h21 = Y21 - (Y12 * Y11 / Y22)
  • h22 = 1 / Z22
• ABCD parameter conversions relate to both Z and Y parameters
  • A = Z11 / Z21, B = det(Z) / Z21
  • C = 1 / Z21, D = Z22 / Z21
• Parameter transformations allow for flexible network analysis
  • Enable selection of most suitable parameter set for given problem
  • Facilitate comparison of different network representations