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
\upsilon_{X}(t)
and
\upsilon_{Y}(t)
can be modeled by
\upsilon_{0} = K[\upsilon_{X}\upsilon_{Y} -T_{A}\dot{\upsilon}_{X}\upsilon_{Y} - T_{B}\upsilon_{X}\dot{\upsilon}_{Y}]
. The three model parameters
K, T_{A}
, and
T_{B}
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
\upsilon_{0}= K \upsilon_{X} \upsilon_{Y}
.