The question asks for the percentage of power dissipated in a lossy passive two-port network at the frequency where the magnitudes of the S-parameters, $|S_{11}|$ and $|S_{21}|$, intersect at -4 dB.

S-Parameter Magnitude to Power Ratio Conversion

The magnitude of an S-parameter given in decibels (dB) can be converted to a power ratio using the formula:

$P_{ratio} = |S_{ij}|^2 = 10^{\frac{S_{dB}}{10}}$

where $S_{dB}$ is the S-parameter magnitude in dB.

Calculating Power Ratios at Intersection

At the intersection frequency for the lossy network, we have $|S_{11}|_{dB} = -4 \text{ dB}$ and $|S_{21}|_{dB} = -4 \text{ dB}$.

The corresponding power ratios are:

• Power ratio for reflection: $|S_{11}|^2 = 10^{\frac{-4}{10}} = 10^{-0.4}$
• Power ratio for transmission: $|S_{21}|^2 = 10^{\frac{-4}{10}} = 10^{-0.4}$

Calculating the value:

$10^{-0.4} \approx 0.3981$

Therefore, at the intersection frequency:

• $|S_{11}|^2 \approx 0.3981$
• $|S_{21}|^2 \approx 0.3981$

Determining Power Dissipation

For a passive network, the sum of the power ratios ($|S_{11}|^2 + |S_{21}|^2$) represents the fraction of incident power that is either reflected or transmitted. The remaining power is dissipated.

The total power ratio accounted for by reflection and transmission is:

$|S_{11}|^2 + |S_{21}|^2 \approx 0.3981 + 0.3981 = 0.7962$

The fraction of power dissipated ($P_{diss\_ratio}$) is:

$P_{diss\_ratio} = 1 - (|S_{11}|^2 + |S_{21}|^2)$

$P_{diss\_ratio} \approx 1 - 0.7962 = 0.2038$

Final Percentage Calculation

To find the percentage of power dissipated, we multiply the dissipated fraction by 100%:

$ \text{Percentage Dissipated} = P_{diss\_ratio} \times 100\% $

$ \text{Percentage Dissipated} \approx 0.2038 \times 100\% = 20.38\% $

Rounded to two decimal places, the percentage of power dissipated is 20.38%.