Multi-state calibration of a dual VI sensor for microsecond-level impedance measurement in plasma systems

Abstract

This paper presents a multi-state calibration for a dual Voltage-Current (VI) sensor system designed for real-time impedance measurement in plasma applications. The proposed approach addresses key limitations of conventional VI sensors, including susceptibility to parasitic effects, and impractical calibration routines. The proposed system leverages a multi-state calibration methodology to extract and compensate for key error sources (e.g., stray capacitance, propagation constant, and characteristic impedance) through measurements under varying standing-wave ratios, eliminating the need for risk-prone short-open-load calibration. The sensor design incorporates a capacitive voltage divider and a Rogowski coil for accurate radio frequency (RF) voltage and current measurement. Digital demodulation algorithms implemented on a field-programmable gate array (FPGA) enable rapid impedance computation via direct RF sampling, achieving response times on the order of sub-microsecond through the use of non-IQ sampling and coordinate rotation digital computer (CORDIC)-based processing. The calibration process involves solving an overdetermined system of equations to minimize residuals, ensuring high accuracy at the target frequency of 13.56 MHz. Experimental results at 13.56 MHz demonstrate the system's efficacy under impedance mismatch conditions (voltage standing wave ratio, VSWR, up to 3.0), with voltage and current measurements showing less than 0.3% error compared to theoretical values.