Inverse & Image Parameters

Definition

Inverse parameters (also known as g-parameters) represent a set of hybrid parameters that describe a two-port network by relating the input current and output voltage to the input voltage and output current. Image parameters are a set of constants used to characterize a symmetrical or asymmetrical two-port network based on the impedance that provides a matched condition at both ports, ensuring maximum power transfer and minimal reflections.

Main Content

1. Inverse Hybrid Parameters (g-parameters)

The g-parameters are defined by the equations: $I_1 = g_{11}V_1 + g_{12}I_2$ and $V_2 = g_{21}V_1 + g_{22}I_2$.
These parameters are particularly useful for networks that are more easily analyzed with an input voltage and output current as independent variables.

2. Image Impedances ($Z_{I1}$ and $Z_{I2}$)

Image impedance is defined as the impedance seen looking into a port when the other port is terminated by its own corresponding image impedance.
For a two-port network: text Input Port (1) Network Output Port (2) <--------------> <----------> <--------------> | | | | | | V1 | | | V2 | | Z_I1 | | | | Z_I2 | | | | | | |

3. Image Transfer Constant ($\gamma$)

The image transfer constant describes the change in magnitude and phase of the signal as it passes through the network when the network is terminated in its image impedances.
It is expressed as $\gamma = \alpha + j\beta$, where $\alpha$ is the attenuation constant and $\beta$ is the phase shift constant.

Working / Process

1. Determining Inverse (g) Parameters

To find $g_{11}$ and $g_{21}$, set the output port to open circuit ($I_2 = 0$). Then, $g_{11} = I_1/V_1$ and $g_{21} = V_2/V_1$.
To find $g_{12}$ and $g_{22}$, set the input port to short circuit ($V_1 = 0$). Then, $g_{12} = I_1/I_2$ and $g_{22} = V_2/I_2$.

2. Calculating Image Impedances

Measure the open-circuit impedance ($Z_{oc}$) and short-circuit impedance ($Z_{sc}$) at each port.
Calculate $Z_{I1} = \sqrt{Z_{oc1} \cdot Z_{sc1}}$ and $Z_{I2} = \sqrt{Z_{oc2} \cdot Z_{sc2}}$.

3. Deriving the Transfer Constant

Once image impedances are determined, apply a matched load to the output.
Calculate the ratio of the voltages or currents at the input and output ports: $\gamma = \ln\sqrt{V_1I_1 / V_2I_2}$.

Advantages / Applications

Inverse parameters simplify the analysis of circuits containing transistors, especially in common-base configurations.
Image parameters are essential in the design of electric wave filters and transmission lines where impedance matching is critical to avoid signal reflections.
They allow engineers to cascade multiple networks efficiently by matching the image impedances of adjacent stages.

Summary

Inverse parameters characterize two-port networks using hybrid input/output variables, while image parameters define the specific impedances required at ports to achieve a matched, reflection-free state. Together, these tools are vital for designing high-performance communication circuits and filtering systems. Important terms to remember include open-circuit impedance, short-circuit impedance, attenuation constant, and phase shift.