S-Parameters math confusion
1st ..... What is | S11 | ^ 2


\$\left|S_{11}\right|^2\$ is a power ratio. -47 dB means a power ratio of roughly 2/100,000 or 0.00002 (because -50 dB would be 1/100,000 and +3 dB is about 2x)

You could also get this result by simply changing your network analyzer to display a linear scale.


  Is this accurate? g = 20 log_10(-47.661) dB


Since you haven't defined \$g\$, it's not possible to answer this.

But it would be very unusual to take the log of a decibel value. 
You get decibels by taking the log of a power ratio. Taking the log again doesn't generally give a meaningful result.

If you want to convert a decibel value \$X\$ back to a power ratio, you'd use
$$\frac{P_2}{P_1} = 10^{X/10}.$$

If you want to convert a decibel value \$X\$ back to a voltage ratio (assuming equal reference impedances for both signals), you'd use
$$\frac{V_2}{V_1} = 10^{X/20}.$$