Abstract
This paper presents necessary and sufficient conditions for a linear three-phase circuit to have a linear and time-invariant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$dq0$</tex-math></inline-formula> model. More specifically, we show that a circuit with state and input matrices that can be partitioned into three-by-three circulant matrices at the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$abc$</tex-math></inline-formula> frame of reference, can be transformed into a linear and time-invariant model in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$dq0$</tex-math></inline-formula> frame of reference. We also show that under this condition the circuit is symmetrically configured, in the sense that every three-phase balanced input results in a balanced state vector at steady-state. We explain that due to these properties, such circuits may be defined as “symmetrically configured in the narrow sense”.
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