### 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.

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.