# Understanding Scattering Parameters

### Key Takeaways

• Scattering parameters represent power transmission, offering a more practical approach than open or short circuit-based parameters, especially in high signal frequencies.

• Scattering parameters are unique because they use matched loads for characterization, unlike Y, Z, H, T, and ABCD parameters, which rely on open or short circuit conditions.

• S-parameters are crucial in communication systems, antenna design, and amplifier design, offering insights into gain, return loss, VSWR, and amplifier stability.

Illustrative diagram of the reflection coefficient: (x)=V-(x)/V+(x)

Scattering parameters (or S-parameters) allow us to quantify forward- and backward-traveling waves along a transmission line or circuit network, establishing a direct relationship with the flow of power. Within the context of S-parameters, “scattering” refers to how the traveling currents and voltages within a transmission line are influenced when they encounter a disruption introduced by the insertion of a network into the transmission line. This disruption can be likened to the wave encountering an impedance that deviates from the characteristic impedance of the line.

## Scattering Parameters vs. Other Circuit Analysis Parameters

The scattering parameters belong to a group of related parameters that includes Y-parameters, Z-parameters, H-parameters, T-parameters, or ABCD-parameters. What sets S-parameters apart from these others is their unique approach — they do not rely on open or short circuit conditions to describe a linear electrical network. Instead, S-parameters use matched loads for characterization. These matched terminations prove to be significantly more practical at higher signal frequencies compared to the challenges posed by open-circuit and short-circuit terminations. It's important to note that contrary to common misconceptions, these quantities are not measured in terms of power.

### Comparing Scattered Parameters

 Parameter Type Definition Usage Context Relationship with Others Frequency Dependency S-parameters Scattering parameters represent power transmission High-frequency, microwave, RF circuit analysis Can be converted to/from other parameters Highly frequency dependent Y-parameters Admittance parameters, inverse of Z-parameters Low to medium frequency analysis Relatable to Z, H, ABCD parameters Less frequency dependent Z-parameters Impedance parameters, inverse of Y-parameters Low to medium frequency analysis Relatable to Y, H, ABCD parameters Less frequency dependent H-parameters Hybrid parameters, mix of voltage and current ratios Transistor modeling, low-frequency analysis Can be derived from Y, Z parameters Frequency dependent T-parameters Transmission parameters, relate input and output Chain analysis of cascaded networks Can be converted to ABCD parameters Varies with frequency ABCD-parameters Cascade parameters, relate input/output with matrix Power systems, transmission lines Relatable to T, Y, Z parameters Frequency dependent

## Scattering Paramaters: A Quick Summary

When analyzing a network (combination of interconnected electrical components designed to manipulate electrical signals, such as amplifiers, filters, and transmission), employing the two-port network approach allows us to derive relevant information about the network by conducting either short or open circuit measurements at its ports. This method is particularly effective for low-frequency signals. However, when dealing with high-frequency signals, creating a true short or true open at a port becomes challenging due to parasitic inductance and capacitance.

To effectively characterize high-frequency circuits, we turn to scattering parameters, which establish a connection between the traveling voltage waves that occur when an incident wave is reflected and transmitted upon inserting a two-port network into a transmission line.

It's important to note that scattering parameters are frequency-dependent. Therefore S-parameter measurements must specify the frequency at which they are taken, in addition to providing information about the characteristic impedance or system impedance.

## Reflection Coefficient and Scattering Parameters

The exploration of S-parameters commences with an examination of the reflection coefficient, which represents the S-parameter of a one-port network. The reflection coefficient, denoted as Γ (Gamma), for a load can be determined by conducting separate measurements of the forward- and backward-traveling voltages along the transmission line, shown in the image above, with V+ being the forward traveling wave and V- the backward traveling wave.

Left equation: Reflection coefficient at position x along the transmission line is given by the ratio of the backward and forward traveling waves.  Right equation: reflection coefficient for a load impedance ZL  at the end of transmission line with impedance Z0 .

## Two-Port S-Parmaters

The definition of two-port S-parameters involves traveling waves on transmission lines, each connected to the ports of the network with a real characteristic impedance, denoted as Z0.

Scattering parameters in matrix form for a two-port network

### Normalized S-Paramaters

In the context mentioned above, Z0 is termed as the normalization impedance or, equivalently, the reference impedance. To prevent any potential confusion with a transmission line impedance that might differ from the reference impedance, sometimes ZREF is utilized to denote the reference impedance. These S-parameters, as described, are also known as normalized S-parameters, signifying that they have been adjusted or normalized to conform to the same reference impedance at each port. Referring to them as normalized S-parameters also conveys that they are referenced to a single real impedance.

### Simple Two-Port Example with Attenuator

In the example below, we’d like to figure out the scattering parameters associated with a 20 dB attenuator.

Two-port 20dB attenuator diagram

In the context of S-parameters for a 20 dB attenuator with a system impedance denoted as Z0, an ideal attenuator is designed to exhibit no reflection at each of its two ports when integrated into a system with the same impedance. This implies the reflection coefficients at both the input and output, Γin and Γout, are zero. Hence, the S-parameters S11 (input reflection coefficient) and S22 (output reflection coefficient) will be 0, indicating no reflections.

For a 20 dB attenuator, the power supplied to the load impedance Z0 is 20 dB lower than the power accessible from the source. This reduction in power level is represented by S21, the forward transmission coefficient. A 20 dB reduction translates to a value of -20 dB, or approximately 0.1 in linear terms. Since the attenuator is reciprocal, the reverse transmission coefficient S12 equals the forward transmission coefficient S21.

Therefore, the corrected S-parameters for a 20 dB attenuator are as follows:

• S11 = 0 (no input reflection)
• S21 = S12 = -20 dB ≈ 0.1 (forward and reverse transmission coefficients)
• S22 = 0 (no output reflection)

## Scattered Parameters in Design

Scattered parameters offer a versatile means of expressing numerous electrical properties in networks composed of components like inductors, capacitors, and resistors. These properties encompass gain, return loss, voltage standing wave ratio (VSWR), reflection coefficient, and amplifier stability, among others.

• Amplifier Design: The reverse isolation parameter, denoted as S12, plays a critical role in evaluating the amount of feedback from the output of an amplifier to its input, consequently influencing the stability of the amplifier, which refers to its propensity to avoid oscillations. The reverse isolation parameter, in conjunction with the forward gain represented by S21, has a significant impact on the amplifier's stability. In the ideal scenario, an amplifier with input and output ports that are perfectly isolated from each other would exhibit infinite isolation.

• Antenna Design: An antenna serves as a typical one-port network, and when it exhibits small values of S11, it indicates that the antenna is efficient in either radiating electromagnetic energy or dissipating and storing power.

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