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

We present the “Virtual VNA 3.0” technique for estimating the scattering matrix of a nonreciprocal, linear, passive, and time-invariant device under test (DUT) with N monomodal ports using a single measurement setup involving a vector network analyzer (VNA) with only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text{A}}\lt N$ </tex-math></inline-formula> ports—thus eliminating the need for any reconnections. We partition the DUT ports into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text{A}}$ </tex-math></inline-formula> “accessible” and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text{S}}$ </tex-math></inline-formula> “not-directly-accessible” (NDA) ports. We connect the accessible ports to the VNA and 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 closed-form and gradient-descent methods to estimate the complete scattering matrix of the nonreciprocal DUT from measurements conducted with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text{A}}$ </tex-math></inline-formula>-port VNA under various NDA-port terminations. We validate both methods experimentally for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text{A}}=N_{\text{S}}=4$ </tex-math></inline-formula>, 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 nonreciprocal DUT with a few-port VNA (only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text{A}}\gt 1$ </tex-math></inline-formula> 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.