SParameters and the Reflection Coefficient
Key Takeaways

Sparameters are crucial for characterizing electrical networks and calculating parameters like Gain, Return loss, and VSWR.

S_{11} parameters are pivotal in understanding reflection behavior, particularly in antenna design.

Return loss, reflection coefficients, and Sparameters are interconnected measurements that describe wave reflection in networks.
Frequency plot of absolute values S_{11} scattering parameters equal to the return loss. Notice 0 dB at approximately 2 GHz.
Sparameters are a valuable tool for calculating reflection coefficients and transmission gains for a twoport network's input and output sides. This fundamental concept determines Sparameters for multiport networks, which calculate parameters such as return loss, VSWR, and insertion loss. In this context, the term Sparameters or "scattering" parameters refer to how traveling currents or voltages are influenced when they encounter a disruption in a transmission line. Read on as we explore sparameters and their relation to the reflection coefficient.
Return Loss, Sparameters, and reflection coefficient relationship summary
Return Loss Definition in terms of reflected and incident (forward) power and voltages 

Return Loss Definition in terms of the reflection coefficient 

S_{11} parameter in terms of the reflection coefficient and return loss 
SParameters Short Summary
The Sparameter matrix for a 2port network is one of the most widely used applications. It is the foundational element for constructing higherorder matrices for more extensive networks. The relationship between the outgoing ("reflected") and incident waves and the Sparameter matrix can be expressed in the following formula and multiplied to get individual equations for b1 and b2. The values a and b are the incident and reflected power waves, respectively.
SParameters on a decibel (dB) scale
Scattering parameters are often expressed on a decibel scale. What does S_{11 }at 10dB signify? S_{11 }represents the return loss of a device, indicating how much of the input power supplied to the device reflects back to the input port. Ideally, there should be no reflected power, and 100% of the power should be delivered to the device.
When S_{11} is at 10dB, it means that at least 90% of the input power is effectively delivered to the device, with less than 10% being reflected. In practical applications, especially in the context of antennas, S_{11} is a crucial parameter. If S_{11} is 0 dB, it indicates that all the power is reflected from the antenna, and none is radiated. Conversely, when S_{11} is 10 dB, it implies that if 3 dB of power is fed into the antenna, only 7 dB is reflected, with the remainder being either radiated or absorbed as losses within the antenna. Antennas are typically designed to minimize losses, ensuring that most of the power delivered to the antenna is radiated.
For example, consider the S_{11 }plot in the figure above. This plot is typically obtained using a Vector Network Analyzer (VNA), which can measure and display S_{11}. The figure indicates that the antenna's optimal radiation occurs at approximately 2 GHz, where S_{11 }reaches 12 dB. Conversely, at 1.2 GHz, the antenna hardly radiates any power as S_{11} approaches 0 dB, signifying that almost all power is reflected. Furthermore, the antenna's bandwidth can be determined from this plot. If the bandwidth is defined as the frequency range where S_{11} is less than 6 dB, then the bandwidth spans approximately 1 GHz, with 2.5 GHz as the upper limit and 1.5 GHz as the lower limit of the frequency band.
SParameters and Reflection Coefficient, Relationship
S_{11} parameters play a crucial role in characterizing the reflection behavior of a network, defined as the reflection coefficient observed when looking from the source end to the load end, considering the port impedance and the network's input impedance. This definition emphasizes the importance of understanding how waves propagate at the input port of a transmission line. In essence, S_{11} is proportional to the reflection coefficient that accounts for the behavior of propagating waves at the transmission line's input.
Sparameters often used interchangeably with return loss, insertion loss, and reflection coefficient, can sometimes be a source of confusion without proper context. All of these concepts serve as measurements related to the reflection of a propagating wave of a load, whether it's a terminated transmission line or a circuit network.

Return loss is directly proportional to the reflection coefficient. Return loss yields a positive dB value since the reflection coefficient (Γ) is always less than 1.

The negative sign is frequently omitted in graphical representations of the return loss formula, leading to interchangeability with the S_{11} parameter. However, formally, S_{11} is defined as the negative return loss and thus carries a negative dB value.
For long transmission lines, S_{11} can be set equal to the reflection coefficient between the source/load and the transmission line characteristic impedance. However, this holds true only for specific situations. In general, the line's input impedance should be considered, which may coincide with the load impedance in certain circuit networks, such as short transmission lines. However, it's crucial to note that circuits with propagating waves will eventually demonstrate S_{11 }values that converge to the reflection coefficient, emphasizing the interconnectedness of these parameters.
Relation to VSWR
The voltage standing wave ratio (VSWR) measures how well the port is matched, akin to return loss. However, VSWR is distinct in that it is a scalar linear parameter, representing the ratio between the maximum and minimum voltages of the standing wave. Consequently, VSWR is directly connected to the magnitude of the voltage reflection coefficient, which, in turn, correlates with either S_{11} for the input port or S_{22} for the output port.
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