s 0.068μ s Rearrange the function into a more convenient form and verify that it is still equal to the original.
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1 Title: TCS Traner Function Author: Eric Warmbier Decription: Thi document derive the variou traner unction or the TCS ytem on the IRTF. The ytem i broken down into block in a Viio document. A traner unction will be developed or thee block (or group o block) to be ued in Simulink. Not all block will be preent in thi document. Simple block may only be preented in the Viio ile. Contant: μ 6 n 9 p 2 k 3 a BLOCK # & #2 π 68 rad Thee block are inverting umming ampliier with ome iltering and ome compenation on one input. Each input can be calculated eparately and then ummed. Generically, the traner unction i: G_comp() Where: = Z_eedback() Z_ource() Z_G_comp_eedback() Z_G_comp_ource() 3k 3k +.33μ 22p 22 p + 56k + 68k 68k +.68μ.68 μ G_comp() Z_G_comp_eedback() Z_G_comp_ource() G_comp() e e e e e e27 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page o 33
2 G_comp_rearranged() Graph to veriy that the unction i imiliar to what i expected. It i a bandpa ilter with ome compenation. 2 TCS Block Compenated Input HA Traner Function 2 log G_comp( j 2 π ) 2 log G_comp_rearranged( j 2 π ) Generically, the other input traner unction i: G_other() Where: = Z_eedback() Z_ource() Z_G_other_eedback() 3k 3k + 22p 22 p Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 2 o 33
3 Z_G_other_ource k G_other() Z_G_other_eedback() Z_G_other_ource() G_other() Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G_other_rearranged() Graph to veriy that the unction i imiliar to what i expected. It i a lowpa ilter. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 3 o 33
4 2 TCS Block Other Input HA Traner Function 2 log G_other( j 2 π ) 2 log G_other_rearranged( j 2 π ) BLOCK #3 Thee block are dierence ampliier with ome low pa iltering. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page o 33
5 Generically, the traner unction i: G3() = Where: Z_eedback() Z_ource() Z_G3_eedback() k k +.22μ.22 μ Z_G3_ource 7k G3() Z_G3_eedback() Z_G3_ource() G3() collect Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 5 o 33
6 G3_rearranged() Graph to veriy that the unction i imiliar to what i expected. It i a low pa with le than unity gain. TCS Block 3 HA Traner Function 5 2 log G3( j 2 π ) 2 log G3_rearranged( j 2 π ) The tachometer alo have a gain (converion rom motion, in radian, to volt). From teting, the reult below were obtained. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 6 o 33
7 TCS Tachometer Voltage v. Speed y =.885x R 2 = Voltage (V) y =.865x -.27 R 2 = Eat Wet Linear (Wet) Linear (Eat) Speed (a/) Aume that a "nominal" tach will have a gain equal to the average o thee two lope. V V a a nom_tach_gain 2 nom_tach_gain =.87 V a The unit above are in V(arcecond/). The input will be in radian/, o thi need to be converted into V/(radian/). nom_tach_gain = V rad BLOCK # Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 7 o 33
8 Thee block are inverting umming ampliier with ome iltering on the irt tage. A traner unction or each input can be determined and then added to together. Generically, the traner unction or each o the irt tage input (Z3) i: G() = Where: Z_eedback() Z_ource() Z_G_eedback() 33.2k 33.2k +.7μ.7 μ Z_G_ource 68.k G() Z_G_eedback() Z_G_ource() G() 3.36e9.36e e9 Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 8 o 33
9 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G_rearranged() Graph to veriy that the unction i imiliar to what i expected - a low pa ilter with gain o ~.5. TCS Block HA Traner Function 2 log G( j 2 π ) 2 log G_rearranged( j 2 π ) BLOCK #7 The block i a group o inverting umming ampliier. There are two tage. The irt tage ha two inverting ampliier without iltering that are ummed into the inal tage that ha iltering. The only tage that will be calculated i the inal tage or only the command input. The mathematic behind thi circuit create a magnitude output limited circuit with a gain o. The irt tage are jut imple addition. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 9 o 33
10 Generically, the traner unction or the econd tage CMD input (Z9) i: G7() = Where: Z_eedback() Z_ource() Z_G7_eedback() 5k 5k +.68μ.68 μ Z_G7_ource 5k G7() Z_G7_eedback() Z_G7_ource() Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page o 33
11 G7() 3 2.9e e8 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G7_rearranged() Graph to veriy that the unction i imiliar to what i expected - a low pa ilter with gain o (db). TCS Block HA Traner Function 2 log G7( j 2 π ) 2 log G7_rearranged( j 2 π ) BLOCK #9 The block i an inverting umming ampliier with 5 input, ome iltering, and output magntiude limiting. The "Joytick" input will be ignored ince it will not be ued in the ervo analyi. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page o 33
12 Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 2 o 33
13 Generically, the traner unction or each input i: G9() = Where: Z_eedback() Z_ource() Z_G9_eedback() 7k 7k +.μ. μ The ource impedance or each input will vary or each input. For "proportional output": Z_G9_prop_ource() 56k 56k k.7μ.7μ + 5.k G9_proportional() Z_G9_eedback() Z_G9_prop_ource() G9_proportional() 3 5.6e + 6.2e5 5.93e e e5 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G9_prop_rearranged() Graph to veriy that the unction i imiliar to what i expected - a low pa ilter with another pole. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 3 o 33
14 TCS Block HA Traner Function 2 2 log G9_proportional( j 2 π ) 2 log G9_prop_rearranged( j 2 π ) 2. 3 For "integrator output": Z_G9_int_ource 56k G9_integrator() Z_G9_eedback() Z_G9_int_ource() G9_integrator() e e8 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G9_int_rearranged() Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page o 33
15 Graph to veriy that the unction i imiliar to what i expected - a low pa ilter. TCS Block 9 HA Traner Function 2 log G9_integrator( j 2 π ) 2 log G9_int_rearranged( j 2 π ) For "EAST and WEST" lew: Z_G9_lew_ource 7.5k G9_lew() Z_G9_eedback() Z_G9_lew_ource() G9_lew() 3.25e e8 Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 5 o 33
16 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G9_lew_rearranged() Graph to veriy that the unction i imiliar to what i expected - a low pa ilter. TCS Block 9 HA Traner Function 2 log G9_integrator( j 2 π ) 2 log G9_int_rearranged( j 2 π ) BLOCK # Thi block i an inverting ampliier with iltering: Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 6 o 33
17 Generically, the HA axi traner unction i: G() Where: = Z_eedback() Z_ource() Z_G_HA_eedback() k k +.μ. μ Z_G_HA_ource 5.k G_HA() Z_G_HA_eedback() Z_G_HA_ource() G_HA() 3 3.3e e9 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G_HA_rearranged() Graph to veriy that the unction i imiliar to what i expected - a low pa ilter with gain. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 7 o 33
18 6 TCS Block HA Traner Function 2 log G_HA( j 2 π ) 2 log G_HA_rearranged( j 2 π ) 2. 3 The DEC axi ha a lightly dierent gain. Generically, the traner unction i: G () = Where: Z_eedback() Z_ource() Z_G_DEC_eedback() k k +.μ. μ Z_G_DEC_ource 7.5k G_DEC() Z_G_DEC_eedback() Z_G_DEC_ource() Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 8 o 33
19 G_DEC() 3 2.3e e9 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G_DEC_rearranged() Graph to veriy that the unction i imiliar to what i expected - a low pa ilter with gain. 3 TCS Block DEC Traner Function 2 2 log G_DEC( j 2 π ) 2 log G_DEC_rearranged( j 2 π ). 3 BLOCK #2 Thi block conit o a umming, inverting integrator that can be cleared (cap horted with 3Ω). Each input will be calculated eparately and thee reult can be ummed together or the complete unction. When the integrator i put in parallel with a reitor, the gain or all the input become very mall and the ilter ormed ha a relatively low cuto requency. Normal operation i with the capacitor NOT in parallel with R7. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 9 o 33
20 ALL calculation below are with no R7 in parallel with the capacitor (normal operation) until noted. Generically, the traner unction or each input i: G () = Where: Z_eedback() Z_ource() Z2_eedback() μ For the error input: Z2_error_ource 62k G2_error() Z2_eedback() Z2_error_ource() Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 2 o 33
21 G2_error() 6.73 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G2_err_rearranged().62 Graph to veriy that the unction i imilar to what i expected - an integrator. TCS Block DEC Traner Function 5 2 log G2_error( j 2 π ) 2 log G2_err_rearranged( j 2 π ) 5. For both EAST and WEST eed orward input: Z2_FF_ource 5k G2_FF() Z2_eedback() Z2_FF_ource() Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 2 o 33
22 G2_FF() Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G2_FF_rearranged().5 Graph to veriy that the unction i imiliar to what i expected: an inverting integrator. 2 TCS Block DEC Traner Function 2 log G2_FF( j 2 π ) 2 log G2_FF_rearranged( j 2 π ) 2. ALL calculation below are with R7 in parallel with the capacitor. Generically, the traner unction or each input i: G () = Where: Z_eedback() Z_ource() Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 22 o 33
23 Z2_eedback_R7() 3 μ 3 + μ For the error input: Z2_error_ource 62 G2_error_R7() Z2_eedback_R7() Z2_error_ource() G2_error_R7().852e2 3.e9 +.e23 Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G2_err_R7_rearranged() Graph to veriy that the unction i imilar to what i expected: inverting ampliier with low gain & ilter. Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 23 o 33
24 TCS Block 2 Error, R7 Traner Function 2 log G2_error_R7( j 2 π ) 2 log G2_err_R7_rearranged( j 2 π ) 5. 3 For both EAST and WEST eed orward input: Z2_FF_ource_R7 5k G2_FF_R7() Z2_eedback_R7() Z2_FF_ource_R7() G2_FF_R7() Rearrange the unction into a more convenient orm and veriy that it i till equal to the original. G2_FF_rearranged_R7() Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 2 o 33
25 Graph to veriy that the unction i imilar to what i expected: inverting ampliier with low gain & ilter. 3 TCS Block 2 Feed Forward, R7 Tran. Function 32 2 log G2_FF_R7( j 2 π ) 2 log G2_FF_rearranged_R7( j 2 π ) Mathcad TCS Traner Function 27_7.xmcdZ:\public_html\Preentation\ Lat Save: /27/27 Page 25 o 33
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