De-embedding a pair of dipoles to calibrate the mutual impedance plasma density diagnostic

For two dipole antennas in a two-port system, an impedance calibration method is presented for situations where the dipole antennas are inaccessible during testing, but environmental changes require recalibration, such as during mutual-impedance measurements of a plasma inside a vacuum chamber. The antenna system of this study is composed of two dipoles connected by rigid coax cables to their respective calibration boxes. The boxes house surface mounted baluns, for balancing the dipoles, and surface mounted standards, for calibrating the cables. The cables connect the system through the chamber walls to the measurement hardware, a Vector Network Analyzer (VNA). The measured S-parameters are simulated in the python package scikit-rf to de-embed the dipoles. This new virtual approach incorporates the balun common mode and eliminates error from the conventional balun characterization where only the differential mode is considered. The presented calibration method has been validated with direct measurements of known cable loads and a COMSOL simulation for a range of 1 MHz–1 GHz, equating to an electron density range of roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mn>1</mml:mn><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> – <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mrow><mml:mn>1</mml:mn><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"><mml:mrow><mml:msup><mml:mrow><mml:mtext>m</mml:mtext></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> . The calibration method de-embeds the two dipoles without using a “through” measurement, allowing for remote calibration and recalibration during testing. Since the eventual goal of this work is to apply the calibration method to an array of dipoles for plasma impedance tomography, the lack of a “through” measurement provides the following key capabilities that removes uncertainty and significantly reduces the setup time: 1) Calibration uncertainty in the cables due to thermal changes and movement are accounted for, which has previously not been possible during plasma testing. 2) The calibration procedure inside the chamber can be fully automated, greatly speeding up the calibration time for multiple antenna pairs. 3) Without the need for a “through” measurement, the number of calibration measurements within the chamber is reduced to the number of antennas <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> rather than the number of unique antenna pairs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> . Extending this method to higher plasma densities could be possible with hardware that performs in the respective GHz range.