### Section 8: S-Parameters equations

By understanding the fundamentals of S-Parameters, one can better apply them to the design process. Since Circuit Mason and Mason Plot both use an advanced math parser, one can optimize and plot not just for simple S-Parameters but instead potentially more relevant figures of merit. Simple S-Parameter terms such as return loss or simple gain are not defined below.

These are just a few figures of merits commonly found. To make it easier to copy and paste into Circuit Mason or Mason Plot, all these equations will be in text form.

Insertion Loss

Whereas the traditional S-Parameter gain penalizes the device for mismatch, the insertion loss only looks at the loss (or gain) within the device itself.

IL = 10*log10(pow(abs(S_2_1),2) / (1-pow(abs(S_1_1),2)))

VSWR (Voltage Standing Wave Ratio)

Optimizing for VSWR is the same as optimizing for return loss, but one may want to plot VSWR.

VSWR = (1+abs(S_1_1)) / (1-abs(S_1_1))

Relative Phase

To compare phasing, it is nice to look at relative phase shifts, and to compare these plots on top of each other you may need to add some fixed phase delay.

arg(S_3_1/S_2_1)+90 [S_3_1 relative to S_2_1, shifted by 90-degrees]
arg(S_5_1/S_2_1)-90 [S_5_1 relative to S_2_1, shifted by 90-degrees]

Mason Plot does phase unwrapping in degrees (using the Arg plot option). However, mostly as an example, this is an alternative way to unwrap by equations (using the dB plot option).

if(arg(S_4_1/S_2_1)+180<180,arg(S_4_1/S_2_1)+180,arg(S_4_1/S_2_1)-180)
[S_5_1 relative to S_2_1, shifted by +/-180-degrees]

Main Point

Understanding how and when to apply the different figures of merit is an important factor in design.