Question

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

Hint:

Use power conservation: fraction dissipated = 1 minus |S11|^2 minus |S21|^2, after converting the given dB value to a ratio.

Answer 1

Correct Answer: 20.38

Step 1: Recall power conservation for a two-port network.
For a two-port network fed at port 1 with a matched port 2, the fraction of incident power reflected is \(|S_{11}|^2\) and the fraction transmitted is \(|S_{21}|^2\). The fraction dissipated inside the network is
\[ P_{diss}=1-|S_{11}|^2-|S_{21}|^2 \]

Step 2: Check the lossless case as a sanity check.
For a lossless network, no power is dissipated, so
\[ |S_{11}|^2+|S_{21}|^2=1 \]
At the intersection point \(|S_{11}|=|S_{21}|\), so each term equals \(0.5\), which is \(10\log_{10}(0.5)=-3.01\) dB. This matches the given \(-3\) dB intersection for the lossless case, confirming the power relation used above.

Step 3: Convert the lossy intersection value from dB to a ratio.
For the lossy network, \(|S_{11}|=|S_{21}|\) at \(-4\) dB, so
\[ |S_{11}|^2=|S_{21}|^2=10^{-4/10}=10^{-0.4} \]

Step 4: Evaluate the numeric value.
\[ 10^{-0.4}\approx0.3981 \]

Step 5: Find the total power leaving the network.
\[ |S_{11}|^2+|S_{21}|^2=2\times0.3981=0.7962 \]

Step 6: Find the power dissipated inside the lossy network.
\[ P_{diss}=1-0.7962=0.2038 \]

Step 7: Convert to a percentage.
\[ P_{diss}\times100=20.38\% \]

Step 8: Final answer.
The percentage of power dissipated in the lossy network at the intersection frequency is \[ \boxed{20.38\%} \]