Dynamic model for the analog multiplier

Under reasonable assumptions, it is shown that the dynamic behavior of an analog multiplier driven by band-limited signals <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\upsilon_{X}(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\upsilon_{Y}(t)</tex> can be modeled by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\upsilon_{0} = K[\upsilon_{X}\upsilon_{Y} -T_{A}\dot{\upsilon}_{X}\upsilon_{Y} - T_{B}\upsilon_{X}\dot{\upsilon}_{Y}]</tex> . The three model parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K, T_{A}</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T_{B}</tex> can be determined by frequency-domain measurements. This simple equation can in turn be modeled by a circuit containing 2 linear capacitors, 2 linear controlled sources, and an ideal multiplier described by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\upsilon_{0}= K \upsilon_{X} \upsilon_{Y}</tex> .