Coupled Resonators & Bandpass Coupling
When two resonant LC circuits are brought near each other they exchange energy through mutual inductance or capacitive coupling. This splits the single resonant frequency into two modes and produces a characteristic double-peaked bandpass response. Understanding coupled resonators is essential for designing IF transformers, bandpass filters, and resonator-based sensors.
Coupling Mechanisms
- Magnetic (inductive): mutual inductance M between the two coils. k = M/L for identical resonators (M < L, so 0 < k < 1). Dominant in IF transformers and wound coil pairs.
- Electric (capacitive): a gap capacitor Cm between the two tanks. Used in high-Q cavity and ceramic resonator filters.
- Mixed: both mechanisms simultaneously; common in printed resonators.
Mode Splitting
For two identical resonators (same L, C, resonant frequency f₀ = 1/(2π√LC)), coupling splits the resonant frequency into an even mode (lower, f₋) and an odd mode (upper, f₊):
The mode splitting Δf = f₊ − f₋ grows with coupling coefficient k. For small k: Δf ≈ k·f₀.
Passband Bandwidth
For high-Q resonators the −3 dB passband bandwidth of the coupled pair equals:
A larger coupling coefficient k gives a wider, flatter passband; a smaller k gives a narrower passband with higher loaded Q.
Critical Coupling
The transfer response shape depends on the ratio of coupling to resonator loss. Define the critical coupling coefficient:
- Under-coupled (k < k_crit): single peaked response, insertion loss at f₀
- Critically coupled (k = k_crit): maximally flat single peak, maximum power transfer at f₀
- Over-coupled (k > k_crit): double-peaked response, dip at f₀
Bandpass Filter Design
A two-resonator bandpass filter is designed by choosing k to achieve the desired bandwidth and response shape. For a maximally flat (Butterworth) 2-pole bandpass filter, critical coupling gives the flattest passband. Chebyshev response requires slight over-coupling. Multi-resonator filters chain additional coupled pairs, with each coupling coefficient k_{i,i+1} set from the prototype g-values.
IF Transformers
The coupled resonator principle is the basis of the IF transformer found in superheterodyne receivers. The primary and secondary coils are wound on a shared ferrite core and are slightly over-coupled to achieve a flat-topped passband around the IF frequency (commonly 455 kHz or 10.7 MHz). Adjusting the core position changes the coupling coefficient and therefore the bandwidth.