Lossless Network Definition
A lossless (non-dissipative) network conserves power: all incident power is either reflected or transmitted with no absorption as heat. This constrains the S-matrix to be unitary:
[S]†[S] = [I] (unitary condition) [S]† = complex conjugate transpose of [S] Physical meaning: columns of [S] are orthonormal vectors Consequence: Σⱼ |Sᵢⱼ|² = 1 for all i (row sum of squares = 1) Example (2-port): |S₁₁|² + |S₂₁|² = 1 (all power either reflected or transmitted from port 1) |S₁₂|² + |S₂₂|² = 1 (same from port 2)
Lossless Examples and Power Budget
| Device | S11 | S21 | |S11|²+|S21|² |
|---|---|---|---|
| Ideal transmission line | 0 (matched) | 1 (no loss) | 0+1=1 ✓ |
| Ideal bandpass filter (passband) | 0 (matched) | 1 (passes all) | 1 ✓ |
| Ideal bandpass filter (stopband) | 1 (total reflection) | 0 (blocks all) | 1+0=1 ✓ |
| Real SAW filter (passband) | 0.2 (−14 dB) | 0.8 (−1.9 dB) | 0.04+0.64=0.68 <1 (lossy!) |
Real Filters Are Not Lossless
No real filter is truly lossless because resistive losses in the resonator elements dissipate power. The degree of lossiness determines insertion loss. In a lossless bandpass filter:
In passband: S11→0, S21→1 (all power transmitted) In stopband: S21→0, S11→1 (all power reflected, none lost) Real SAW: stopband S21 ≈ −40 dB, S11 ≈ −0.5 dB → NOT lossless (1% absorbed) This "leakage" causes the filter to heat up under high TX power.
RF View: Check losslessness with RF View: overlay |S11|² + |S21|² vs frequency. For a good low-loss filter, this should be close to 1 in the passband (≥0.95) and close to 1 in the stopband. Deviation shows dissipation. Free on Android.