Advances in Power Electronics

Research Article

The Influence of Controller Parameters on the Quality of the Train Converter Current

Table 1

State equations of a 4Q converter.
State variable 𝑖 2 State variable 𝑣 𝑑 Matrix format
First configuration S1, S4 ON; S2, S3 OFF 𝐿 ( 𝑑 𝑖 2 / 𝑑 𝑡 ) = − 𝑅 𝑠 𝑖 2 − 𝑣 𝑑 + 𝐸 2 s i n ( 𝜔 𝑡 ) 𝐶 ( 𝑑 𝑣 𝑑 / 𝑑 𝑡 ) = 𝑖 2 − 𝐼 T R 𝐴 ⋅ ̇ 𝑥 = 𝐵 ⋅ 𝑥 + 𝐾
Second configuration S1, S4 OFF; S2, S3 ON 𝐿 ( 𝑑 𝑖 2 / 𝑑 𝑡 ) = − 𝑅 𝑠 𝑖 2 + 𝑣 𝑑 + 𝐸 2 s i n ( 𝜔 𝑡 ) 𝐶 ( 𝑑 𝑣 𝑑 / 𝑑 𝑡 ) = − 𝑖 2 − 𝐼 T R 𝐴 ⋅ ̇ 𝑥 = 𝐵 ⋅ 𝑥 + 𝐾
Third configuration S1, S2 ON; S3, S4 OFF 𝐿 ( 𝑑 𝑖 2 / 𝑑 𝑡 ) = − 𝑅 𝑠 𝑖 2 + 𝐸 2 s i n ( 𝜔 𝑡 ) 𝐶 ( 𝑑 𝑣 𝑑 / 𝑑 𝑡 ) = − 𝐼 T R 𝐴 ⋅ ̇ 𝑥 = 𝐵 ⋅ 𝑥 + 𝐾
Fourth configuration S1, S2 OFF; S3, S4 ON 𝐿 ( 𝑑 𝑖 2 / 𝑑 𝑡 ) = − 𝑅 𝑠 𝑖 2 + 𝐸 2 s i n ( 𝜔 𝑡 ) 𝐶 ( 𝑑 𝑣 𝑑 / 𝑑 𝑡 ) = − 𝐼 T R 𝐴 ⋅ ̇ 𝑥 = 𝐵 ⋅ 𝑥 + 𝐾
𝜔 is the supply voltage angular frequency, 𝑥 = [ 𝑖 2 𝑣 𝑑 ] 𝑇 is the state vector, ̇ 𝑥 is its time derivative, 𝐴 contains the conservative elements of the converter, 𝐵 , 𝐵 , and 𝐵 are the matrices of the state variables coefficients that represent the specific working conditions, and 𝐾 is the forcing term vector. The switching operation of the 4Q converters realizes three different state conditions (because the third and fourth configurations bring to the same circuit topology) characterized by different matrices 𝐵 , 𝐵 , and 𝐵 .