Q-Bandwidth Relationship
For a matching network: Approximate −10 dB S11 bandwidth: BW ≈ f₀ / Q_design L-network: Q_design = Q_min = √(Z_high/Z_low − 1) [cannot be reduced] Pi/T-network: Q_design ≥ Q_min [can be set higher, but not lower than Q_min] Higher Q → narrower bandwidth → more frequency selective Lower Q → wider bandwidth → less frequency selective (limited by Bode-Fano)
Bode-Fano Maximum Bandwidth Limit
For a parallel RC load (capacitive device at RF): ∫₀^∞ ln(1/|Γ(ω)|) dω ≤ π/(R·C) This sets the MAXIMUM achievable BW for a given reflection target. A PA output with Z_out = 5 + j3Ω at 3.5 GHz: C_equiv = Im(Z_out) / (ω_0 × |Z_out|²) ≈ 3.5 pF R_equiv = Re(Z_out) = 5Ω Max bandwidth for −10 dB S11: BW_max ≈ 1/(π × R × C) × ln(1/|Γ_max|) ≈ 1/(π × 5 × 3.5pF) × ln(3.16) ≈ 6.9 GHz → Comfortably covers 5G n78 band (500 MHz)
Matching Network Bandwidth Comparison
| Topology | Elements | BW for 10:1 Z ratio |
|---|---|---|
| L-network | 2 | ~30% FBW (fixed by Q_min=3.0) |
| Pi-network (Q=2) | 3 | ~50% FBW (lower Q by design) |
| 2-section L-network | 4 | ~60% FBW |
| 3-section Chebyshev | 6 | ~80% FBW |
Verification in RF View
After building matching network in Circuit Simulator:
- Send simulation to S-Parameter Plot
- Select S11 display → activate BW Marker at −10 dB threshold
- Read: f_low, f_high, BW = f_high − f_low
- Verify BW covers required system bandwidth (e.g., 500 MHz for 5G n78)
- If BW insufficient: switch from L-network to Pi/T → re-simulate
RF View BW Verification: BW Marker at −10 dB threshold reads matching network bandwidth directly from S11. Compare simulated BW against system requirement. Iterate topology until BW is sufficient. Free on Android.