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.