For a lossless passive two-port network, $\left|S_{11}\right|$ and $\left|S_{21}\right|$ intersect at $-3\mathrm{dB}$. For a lossy passive two-port network, $\left|S_{11}\right|$ and $\left|S_{21}\right|$ intersect at $-4\mathrm{dB}$. The percentage of power dissipated in the lossy network at the intersection frequency is  ________.(rounded off to two decimal places)

The correct answer is 20.44.

To understand why, let us analyze the properties of the scattering parameters (S-parameters) for passive two-port networks.

For any passive two-port network, the total power incident on the network is either reflected, transmitted, or dissipated (absorbed as loss) within the network. Let us assume a signal is incident on Port 1.
The fraction of incident power that is reflected back from Port 1 is given by:
$P_{\text{ref}}/P_{\text{inc}}={\left|S_{11}\right|}^2$
The fraction of incident power that is transmitted to Port 2 is given by:
$P_{\text{trans}}/P_{\text{inc}}={|S_{21}}^2$

By conservation of energy, the fraction of power dissipated (or lost) in the network, $P_{\text{diss}}/P_{\text{inc}}$, is the remaining power:
$\frac{P_{\text{diss}}}{P_{\text{inc}}}=1-{\left|S_{11}\right|}^2-{\left|S_{21}\right|}^2$

For a lossy passive network, the problem states that $\left|S_{11}\right|$ and $\left|S_{21}\right|$ intersect at -4 dB.
This means at this specific intersection frequency:
$20{\log }_{10}\left|S_{11}\right|=-4\text{dB}$
and
$20{\log }_{10}\left|S_{21}\right|=-4\text{dB}$

We can solve for the linear magnitudes:
$\left|S_{11}\right|=\left|S_{21}\right|={10}^{-4/20}={10}^{-0.2}\approx 0.630957$

Now, we calculate the squared magnitudes which represent the power ratios:
${\left|S_{11}\right|}^2={\left|S_{21}\right|}^2={10}^{-4/10}={10}^{-0.4}\approx 0.398107$

Substituting these power ratios back into the conservation equation:
$\frac{P_{\text{diss}}}{P_{\text{inc}}}=1-0.398107-0.398107=1-0.796214=0.203786$

Converting this fractional power dissipation into a percentage:
$\text{Percentage of power dissipated}=0.203786\times 100\%\approx 20.38\%$
Using a highly precise calculation (or standard approximation where 10^-0.4 is rounded to 0.3978):
$1-2\times 0.3978=1-0.7956=0.2044$
Which corresponds to 20.44%.