{ "title": "sri answer 1 - Dual loop control AC-DC converter (Score 100)", "taskId": "bf9df235-dfd3-4476-b834-d46f805c51d8", "type": "SOLUTION", "timestamp": 1792163000000, "doc": { "type": "doc", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Given data :" } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " The three phase AC-DC converter operates with the following specifications:" } ] }, { "type": "bulletList", "content": [ { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Supply voltage " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "400V" } ] }, { "type": "text", "text": ", 3 -phase " } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "supply frequency " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "50Hz" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Load power " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "10kW" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Filter inductance " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "L=10.2mH" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "series resistance " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "R_s=10mOmega" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Switching frequency " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "f_(sw)=10kHz" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Sampling time " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_1=10mus" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "DC output voltage " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "V_o=800V" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "feedback gain " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_1=0.0326" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "controller voltage " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "V_c=1V" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "damping rati " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "zeta=0.707" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Required : determine the intergarl gain " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_(i" } ] }, { "type": "text", "text": " of the inner current PI controller , rounded to three decimal places ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "marks": [ { "type": "bold" } ], "text": "Step 1 :Determine the total delay time " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The total delay includes the sampling time and the effective switching delay . for the assumed converter model , the switching delay is half the switching period ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_d=T_(sw)/2=1/(2f_(sw" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "substituting " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "f_(sw)=10kHz" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_d = 1/(2(10*10^3)" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "= 50*10^-6s" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Hence, the total delay is " } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_Sigma = T_1+T_2" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "= 10mus+50mus" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "= 60*10^-6s" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " should consider both sampling and switching delays ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "marks": [ { "type": "bold" } ], "text": "Step 2 : " }, { "type": "text", "text": "Calculate the proportional gain " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_p" } ] }, { "type": "text", "text": " using the gain-design equation " } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_(p = L/(2(V_o/2)K_1T_Sigma" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "= L/(V_oK_1T_Sigma" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "substitute given values " } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_p = (10.2*10^-3)/(800*0.0326*60*10^-6" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "= 0.0102/(0.0015648" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_p = 6.518" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The proportional gain is determined using the total delay and the converter's output voltage and feedback gain ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "marks": [ { "type": "bold" } ], "text": "Step 3:" }, { "type": "text", "text": "Calculate the electric time constant " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The electric time constant of the filter's RL circuit is " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "tau_L=L/R_s" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "tau_L = (10.2*10^-3)/(10*10^-3" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "= 1.02s" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The electrical time constant represents the ratio of inductance to series resistance . The problem specifies that the PI controller's zero cancels this electrical time constant , so the controller integral time is set equal to " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "tau_L" } ] }, { "type": "text", "text": "." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Step 4: Determine the intergal gain " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The transfer function pf a conventional PI controller is " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "G_c(s)=K_p(1+1/(sT_i))" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "where " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_i" } ] }, { "type": "text", "text": " is the integral time constant ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Expanding " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "G_c(s)=K_(p)+K_p/(sT_i" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "comparing withe the standard PI form ," } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "G_c(s)=K_p+K_i/s" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "we obtain " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_i=K_p/T_i" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Since the controller zero cancels the electrical time constant ," } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_i=T_L=1.02s" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Thus " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_i=K_p/tau_L" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Substitute the calculated values " } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_i = 6.518/1.02" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "center" }, "content": [ { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "= 6.390196" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Rounding to three decimal places " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_i=6.390" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Final answer " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_i=6.390" } ] } ] } ] }, "raw_blocks": [ { "type": "TEXT", "block": { "editorContentState": { "type": "doc", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Given data :" } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " The three phase AC-DC converter operates with the following specifications:" } ] }, { "type": "bulletList", "content": [ { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Supply voltage " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "400V" } ] }, { "type": "text", "text": ", 3 -phase " } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "supply frequency " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "50Hz" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Load power " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "10kW" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Filter inductance " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "L=10.2mH" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "series resistance " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "R_s=10mOmega" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Switching frequency " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "f_(sw)=10kHz" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Sampling time " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_1=10mus" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "DC output voltage " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "V_o=800V" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "feedback gain " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_1=0.0326" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "controller voltage " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "V_c=1V" } ] } ] } ] }, { "type": "listItem", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "damping rati " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "zeta=0.707" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Required : determine the intergarl gain " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_(i" } ] }, { "type": "text", "text": " of the inner current PI controller , rounded to three decimal places ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "marks": [ { "type": "bold" } ], "text": "Step 1 :Determine the total delay time " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The total delay includes the sampling time and the effective switching delay . for the assumed converter model , the switching delay is half the switching period ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_d=T_(sw)/2=1/(2f_(sw" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "substituting " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "f_(sw)=10kHz" } ] } ] } ] }, "validationError": null, "version": "2.0.0" }, "id": "NZhE7HsmGZ-v4NtsxUYRy" }, { "type": "EQUATION_RENDERER", "block": { "lines": [ { "left": "T_d", "right": "1/(2(10*10^3)", "operator": "=" }, { "left": "", "right": "50*10^-6s", "operator": "=" } ], "isShowingEquationNumbers": false, "version": "1.2.0" }, "id": "T9oH5oWIZQKfqQ5QiRbJ8" }, { "type": "TEXT", "block": { "editorContentState": { "type": "doc", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Hence, the total delay is " } ] } ] }, "validationError": null, "version": "2.0.0" }, "id": "31I1ytvc8fdvWDVI6vT85" }, { "type": "EQUATION_RENDERER", "block": { "lines": [ { "left": "T_Sigma", "right": "T_1+T_2", "operator": "=" }, { "left": "", "right": "10mus+50mus", "operator": "=" }, { "left": "", "right": "60*10^-6s", "operator": "=" } ], "isShowingEquationNumbers": false, "version": "1.2.0" }, "id": "na4L34Rkd7Q0t-Z5vZpki" }, { "type": "TEXT", "block": { "editorContentState": { "type": "doc", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " should consider both sampling and switching delays ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "marks": [ { "type": "bold" } ], "text": "Step 2 : " }, { "type": "text", "text": "Calculate the proportional gain " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_p" } ] }, { "type": "text", "text": " using the gain-design equation " } ] } ] }, "validationError": null, "version": "2.0.0" }, "id": "9cT5TnzBXp3GSpJiaPNuu" }, { "type": "EQUATION_RENDERER", "block": { "lines": [ { "left": "K_(p", "right": "L/(2(V_o/2)K_1T_Sigma", "operator": "=" }, { "left": "", "right": "L/(V_oK_1T_Sigma", "operator": "=" } ], "isShowingEquationNumbers": false, "version": "1.2.0" }, "id": "LWaxOOu8W1WdqALBKRQFA" }, { "type": "TEXT", "block": { "editorContentState": { "type": "doc", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "substitute given values " } ] } ] }, "validationError": null, "version": "2.0.0" }, "id": "H7iiDPagDr8C-h03m7xZz" }, { "type": "EQUATION_RENDERER", "block": { "lines": [ { "left": "K_p", "right": "(10.2*10^-3)/(800*0.0326*60*10^-6", "operator": "=" }, { "left": "", "right": "0.0102/(0.0015648", "operator": "=" }, { "left": "K_p", "right": "6.518", "operator": "=" } ], "isShowingEquationNumbers": false, "version": "1.2.0" }, "id": "pJHT0yZKF8ZZhjSbHGgAc" }, { "type": "TEXT", "block": { "editorContentState": { "type": "doc", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The proportional gain is determined using the total delay and the converter's output voltage and feedback gain ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "marks": [ { "type": "bold" } ], "text": "Step 3:" }, { "type": "text", "text": "Calculate the electric time constant " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The electric time constant of the filter's RL circuit is " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "tau_L=L/R_s" } ] } ] } ] }, "validationError": null, "version": "2.0.0" }, "id": "9DbmYIDzDEz_OM3V71yQ4" }, { "type": "EQUATION_RENDERER", "block": { "lines": [ { "left": "tau_L", "right": "(10.2*10^-3)/(10*10^-3", "operator": "=" }, { "left": "", "right": "1.02s", "operator": "=" } ], "isShowingEquationNumbers": false, "version": "1.2.0" }, "id": "EtqDoEzoFl8kNH_ayP278" }, { "type": "TEXT", "block": { "editorContentState": { "type": "doc", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The electrical time constant represents the ratio of inductance to series resistance . The problem specifies that the PI controller's zero cancels this electrical time constant , so the controller integral time is set equal to " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "tau_L" } ] }, { "type": "text", "text": "." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Step 4: Determine the intergal gain " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "The transfer function pf a conventional PI controller is " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "G_c(s)=K_p(1+1/(sT_i))" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "where " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_i" } ] }, { "type": "text", "text": " is the integral time constant ." } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Expanding " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "G_c(s)=K_(p)+K_p/(sT_i" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "comparing withe the standard PI form ," } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "G_c(s)=K_p+K_i/s" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "we obtain " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_i=K_p/T_i" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Since the controller zero cancels the electrical time constant ," } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "T_i=T_L=1.02s" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Thus " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_i=K_p/tau_L" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Substitute the calculated values " } ] } ] }, "validationError": null, "version": "2.0.0" }, "id": "d6QmnJtYRmeEF8MgwzA7M" }, { "type": "EQUATION_RENDERER", "block": { "lines": [ { "left": "K_i", "right": "6.518/1.02", "operator": "=" }, { "left": "", "right": "6.390196", "operator": "=" } ], "isShowingEquationNumbers": false, "version": "1.2.0" }, "id": "amFBgFzlL1nKm8lZVyBmn" }, { "type": "TEXT", "block": { "editorContentState": { "type": "doc", "content": [ { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Rounding to three decimal places " } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": " " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_i=6.390" } ] } ] }, { "type": "paragraph", "attrs": { "textAlign": "left" } }, { "type": "paragraph", "attrs": { "textAlign": "left" }, "content": [ { "type": "text", "text": "Final answer " }, { "type": "inlineMath", "attrs": { "mathType": "asciimath" }, "content": [ { "type": "text", "text": "K_i=6.390" } ] } ] } ] }, "validationError": null, "version": "2.0.0" }, "id": "XBcVnpGCi2zCIOKwnccc1" } ] }