Virtual VNA 3.0: Unambiguous Scattering Matrix Estimation for Non-Reciprocal Systems by Leveraging Tunable and Coupled Loads

We present the "Virtual VNA 3.0" technique for estimating the scattering matrix of a \textit{non-reciprocal}, linear, passive, time-invariant device under test (DUT) with $N$ monomodal ports using a single measurement setup involving a vector network analyzer (VNA) with only $N_\mathrm{A}<N$ ports -- thus eliminating the need for any reconnections. We partition the DUT ports into $N_\mathrm{A}$ "accessible" and $N_\mathrm{S}$ "not-directly-accessible" (NDA) ports. We connect the accessible ports to the VNA and the NDA ports to the "virtual VNA ports" of a VNA Extension Kit. This kit enables each NDA port to be terminated with three distinct individual loads or connected to neighboring DUT ports via coupled loads. We derive both a closed-form and a gradient-descent method to estimate the complete scattering matrix of the non-reciprocal DUT from measurements conducted with the $N_\mathrm{A}$-port VNA under various NDA-port terminations. We validate both methods experimentally for $N_\mathrm{A}=N_\mathrm{S}=4$, where our DUT is a complex eight-port transmission-line network comprising circulators. Altogether, the presented "Virtual VNA 3.0" technique constitutes a scalable approach to unambiguously characterize a many-port \textit{non-reciprocal} DUT with a few-port VNA (only $N_\mathrm{A}>1$ is required) -- without any tedious and error-prone manual reconnections susceptible to inaccuracies. The VNA Extension Kit requirements match those for the "Virtual VNA 2.0" technique that was limited to reciprocal DUTs.