engineering recuitment GATE EE 2023-24 Test Series Control Systems Basics of Control Systems Mathematical Modeling and Representation of Systems
The mathematical model of an analogous electrical system for the following mechanical system using the force-current analogy is (i – current, v – voltage, L – Inductance, C – Capacitance)
1
\(\frac{1}{{{C_{K1}}}}\smallint \left( {{i_y} - {i_x}} \right)dt = {L_{M2}}\frac{{d{i_x}}}{{dt}} + \frac{1}{{{C_{K2}}}}\smallint {i_x}dt,\;{v_F}\left( t \right) = {L_{M1}}\frac{{d{i_y}}}{{dt}} + \frac{1}{{{C_{K3}}}}\smallint {i_y}dt + \frac{1}{{{C_{K1}}}}\smallint \left( {{i_y} - {i_x}} \right)dt\)
2
\(\frac{1}{{{L_{K1}}}}\smallint \left( {{i_y} - {i_x}} \right)dt = {L_{M2}}\frac{{d{i_x}}}{{dt}} + \frac{1}{{{L_{K2}}}}\smallint {i_x}dt,\;{v_F}\left( t \right) = {L_{M1}}\frac{{d{i_y}}}{{dt}} + \frac{1}{{{L_{K3}}}}\smallint {i_y}dt + \frac{1}{{{L_{K1}}}}\smallint \left( {{i_y} - {i_x}} \right)dt\)
3
\(\frac{1}{{{C_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt = {L_{M2}}\frac{{d{v_x}}}{{dt}} + \frac{1}{{{C_{K2}}}}\smallint {v_x}dt,\;{i_F}\left( t \right) = {L_{M1}}\frac{{d{v_y}}}{{dt}} + \frac{1}{{{C_{K3}}}}\smallint {v_y}dt + \frac{1}{{{C_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt\)
4
\(\frac{1}{{{L_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt = {C_{M2}}\frac{{d{v_x}}}{{dt}} + \frac{1}{{{L_{K2}}}}\smallint {v_x}dt,\;{i_F}\left( t \right) = {C_{M1}}\frac{{d{v_y}}}{{dt}} + \frac{1}{{{L_{K3}}}}\smallint {v_y}dt + \frac{1}{{{L_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt\)