Bilateral Design
Understanding Bilateral Design
Bilateral behavior means a network transmits signals equally in both directions. The Lorentz reciprocity theorem guarantees this for any linear, time-invariant, isotropic network. Bilateral design takes advantage of this symmetry: input and output matching can be designed independently, stability is inherently guaranteed, and power conservation (|S21|² + |S11|² = 1 for lossless networks) simplifies filter synthesis.
Active devices break bilaterality through their gain mechanism (S21 >> S12), introducing feedback from output to input that couples the matching networks. The unilateral figure of merit U indicates whether this coupling can be safely ignored.
Stability and Matching Equations
K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12S21|)
Unconditional: K > 1 AND |Δ| < 1
Unilateral Figure of Merit:
U = |S12S21S11S22| / |(1−|S11|²)(1−|S22|²)|
U < 0.1: unilateral valid (<0.5 dB error)
Input Impedance (Bilateral Device):
Γin = S11 + S12S21ΓL / (1 − S22ΓL)
Bilateral vs. Unilateral Comparison
| Property | Bilateral (S12=S21) | Unilateral (S12=0) | Non-Bilateral (Active) |
|---|---|---|---|
| Examples | Filters, couplers | Ideal amplifier | Real transistor |
| Stability | K → ∞ | K → ∞ | K may be < 1 |
| Matching | Independent ports | Independent ports | Coupled (iterative) |
| Power | |S21|²+|S11|²=1 | N/A (gain) | N/A (gain) |
Frequently Asked Questions
Bilateral vs. unilateral?
Bilateral: S12 = S21, all passive networks. Unilateral: S12 = 0, ideal only. Real transistors: 15–40 dB isolation. U < 0.1 means safe to approximate as unilateral (<0.5 dB gain error).
Stability impact?
Non-zero S12 feeds output to input. Rollett K factor decreases with increasing |S12|. Bilateral passive: K → ∞. Active: check K > 1 AND |Δ| < 1 at all frequencies before matching.
Matching simplification?
Bilateral: ΓS = S11*, ΓL = S22* independently. Non-bilateral: simultaneous conjugate match required (iterative). Power conservation |S21|²+|S11|²=1 enables filter synthesis from reflection alone.