Equivalent Circuit Modeling of Foil-Mediated Dissipative Coupling in Microwave Cavities with Enhanced Phase Response

We formulate and validate an equivalent circuit model describing mutual resistive coupling between three microwave cavity resonators interconnected via thin metallic foils. Each cavity is represented as a lumped LCR circuit, while the foils act as a dissipative interface that mediates energy exchange via mutual resistance. This coupling mechanism produces interference effects and a controllable anti-resonance when the input resonators are amplitude- and phase-balanced, a behavior not achievable with standard microwave antenna probes. All three resonators operated in the TM$_{010}$ mode, where two input resonators each excited the third via a thin copper foil. Analytical expressions are derived for the mutual resistance and coupling coefficient of these foils in this geometry. Under balanced conditions, a sharp anti-resonance emerges with a near order-of-magnitude enhanced phase sensitivity at the resonant frequency of the output cavity, consistent with model predictions. The experimentally extracted mutual coupling coefficients, $\Delta_{13}=(5.00\pm0.01)\times10^{-6}$ and $\Delta_{23}=(4.10\pm0.01)\times10^{-6}$, fall within the calculated range $\Delta_{n3}\approx(1\text{--}48)\times10^{-6}$ derived from the foil's electromagnetic properties, where the spread is dominated by the estimated foil thickness uncertainty of $(9\pm1)\,\mu\mathrm{m}$. These results confirm that resistive coupling can occur across a number of skin depths of a metallic interface, providing a new means of engineering controlled interference in multi-resonator systems. The approach offers potential applications in precision microwave experiments, phase-sensitive detection, and tests of fundamental electromagnetic interactions.