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If Time Marches on why is Time Constant? + + A G + [ ] , [ ] , - v (t)
X L R 1 I i(t) [ ] t , V 1 X C R I (a) (b) (a) (b) + + V 3 - v (t) C v V R V R V V 1 V V + R 1 V C C 1 V 2 V 1 L V 2 A G Inc.
![If Time Marches on why is Time Constant + + A G + [ ] , [ ] , - v (t)](http://slideplayer.com./slide/14825409/90/images/1/If+Time+Marches+on+why+is+Time+Constant+%2B+%2B+A+G+%2B+%5B+%5D+%2C+%5B+%5D+%2C+-+v+%28t%29.jpg "X. L. R. 1. I. i(t) [ ] t. , V. 1. X. C. R. I. (a) (b) (a) (b) + + V v (t) C. v. V. R. V. R. V. V. 1. V. V. + R. 1. V. C. C. 1. V. 2. V. 1. L. V. 2. A. G. Inc.")
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A G From the capacitor’s perspective; + +
If Time Marches on why is Time Constant? i(t) [ ] t , V 1 X C R I From the capacitor’s perspective; M 1 M 2 HS 6 10 5 10 (a) (b) V 3 - v (t) C v + V V 1 V + R 1 C 1 V 1 V 2 V 2 A G Inc.
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-1 (2 f )(C) A G [ ] = = e + If Time Marches on why is Time Constant?
[ ] -1 (2 f )(C) If Time Marches on why is Time Constant? Model to Memorize dv i(t) [ ] t , V 1 X C R I I (t) = C C dt Current vs Time I t 2 = v (t) C 1 i (t) dt v i (t) = I e - (t / ) c (6ma)(0.368) = 2.2ma mA 4 (1.8, 2.2) 5 10 6 + HS (b) (a) C 1 R M 2 V - v (t) v 3 2 + 1.8ms Time A G Inc.
![-1 (2 f )(C) A G [ ] = = e + If Time Marches on why is Time Constant](http://slideplayer.com./slide/14825409/90/images/3/-1+%282+f+%29%28C%29+A+G+%5B+%5D+%3D+%3D+e+%2B+If+Time+Marches+on+why+is+Time+Constant.jpg "[ ] -1. (2 f )(C) If Time Marches on why is Time Constant Model to Memorize. dv. i(t) [ ] t. , V. 1. X. C. R. I. I (t) = C. C. dt. Current vs Time. I. t. 2. = v (t) C. 1. i (t) dt. v. i (t) = I. e. - (t / ) c. (6ma)(0.368) = 2.2ma. mA. 4. (1.8, 2.2) HS. (b) (a) C. 1. R. M. 2. V. - v (t) v ms. Time. A. G. Inc.")
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-1 (2 f )(C) A G [ ] = e + = If Time Marches on why is Time Constant?
[ ] -1 (2 f )(C) If Time Marches on why is Time Constant? Model to Memorize 5 10 6 + HS (b) (a) C 1 V R M 2 - v (t) v 3 dv I (t) = C C i(t) [ ] t , V 1 X C R I dt I - (t / ) i (t) = I e c (6ma)(0.368) = 2.2ma t 2 = v (t) C 1 i (t) dt v 4 mA (1.8,2.2) ( , i(t)) 2 + 1.8ms milliseconds A G Inc.
![-1 (2 f )(C) A G [ ] = e + = If Time Marches on why is Time Constant](http://slideplayer.com./slide/14825409/90/images/4/-1+%282+f+%29%28C%29+A+G+%5B+%5D+%3D+e+%2B+%3D+If+Time+Marches+on+why+is+Time+Constant.jpg "[ ] -1. (2 f )(C) If Time Marches on why is Time Constant Model to Memorize HS. (b) (a) C. 1. V. R. M v (t) v. 3. dv. I (t) = C. C. i(t) [ ] t. , V. 1. X. C. R. I. dt. I. - (t / ) i (t) = I. e. c. (6ma)(0.368) = 2.2ma. t. 2. = v (t) C. 1. i (t) dt. v. 4. mA. (1.8,2.2) ( , i(t)) ms. milliseconds. A. G. Inc.")
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-1 (2 f )(C) e A G [ ] + = e = e [ ] [ [ ] ] [ ]
[ ] -1 (2 f )(C) If Time Marches on why is Time Constant? Model to Memorize 5 10 6 + HS (b) (a) C 1 V R M 2 - v (t) v 3 dv I (t) = C C i(t) [ ] t , V 1 X C R I dt I - (t / ) i (t) = I e c (6ma)(0.368) = 2.2ma t 2 = v (t) C 1 i (t) dt v 4 mA (1.8,2.2) ( , i(t)) 2 i (t) = I e c - (t / ) V 2 = (-1) I C 1 e - (t / ) t + 1.8ms 2 4 6 milliseconds Energy balance Discharging capacitor t 2 C 1 [ V 2 ] = [ [ ] V 1 ] i (t) dt C = [ R 1 i(t) ] (from resistor’s perspective) in out t 1 in out A G Inc.
![-1 (2 f )(C) e A G [ ] + = e = e [ ] [ [ ] ] [ ]](http://slideplayer.com./slide/14825409/90/images/5/-1+%282+f+%29%28C%29+e+A+G+%5B+%5D+%2B+%3D+e+%3D+e+%5B+%5D+%5B+%5B+%5D+%5D+%5B+%5D.jpg "[ ] -1. (2 f )(C) If Time Marches on why is Time Constant Model to Memorize HS. (b) (a) C. 1. V. R. M v (t) v. 3. dv. I (t) = C. C. i(t) [ ] t. , V. 1. X. C. R. I. dt. I. - (t / ) i (t) = I. e. c. (6ma)(0.368) = 2.2ma. t. 2. = v (t) C. 1. i (t) dt. v. 4. mA. (1.8,2.2) ( , i(t)) 2. i (t) = I. e. c. - (t / ) V. 2. = (-1) I. C. 1. e. - (t / ) t ms milliseconds. Energy balance. Discharging capacitor. t. 2. C. 1. [ V. 2. ] = [ [ ] V. 1. ] i (t) dt. C. = [ R. 1. i(t) ] (from resistor’s perspective) in. out. t. 1. in. out. A. G. Inc.")
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-1 (2 f )(C) e A G [ ] + = e = e [ ] [ ] [ ] [ ] e e [ ]
[ ] -1 (2 f )(C) If Time Marches on why is Time Constant? Model to Memorize 5 10 6 + HS (b) (a) C 1 V R M 2 - v (t) v 3 dv I (t) = C C i(t) [ ] t , V 1 X C R I dt I - (t / ) i (t) = I e c (6ma)(0.368) = 2.2ma t 2 = v (t) C 1 i (t) dt v 4 mA (1.8,2.2) ( , i(t)) 2 i (t) = I e c - (t / ) V 2 = (-1) I C 1 e - (t / ) t + 1.8ms 2 4 6 milliseconds Discharging capacitor t 2 Energy balance C 1 [ ] [ ] [ i (t) dt C ] = [ R 1 i(t) ] (from resistor’s perspective) V 2 = V 1 in out t 1 in out This is a first order differential equation d dt C 1 I e - (t / ) I e - (t / ) R 1 i(t) + d dt 1 R C = i(t) [ ] + = If the time constant, R, and C are known, the Euler numerical method will produce i(t) vs. t plot above. A G Inc.
![-1 (2 f )(C) e A G [ ] + = e = e [ ] [ ] [ ] [ ] e e [ ]](http://slideplayer.com./slide/14825409/90/images/6/-1+%282+f+%29%28C%29+e+A+G+%5B+%5D+%2B+%3D+e+%3D+e+%5B+%5D+%5B+%5D+%5B+%5D+%5B+%5D+e+e+%5B+%5D.jpg "[ ] -1. (2 f )(C) If Time Marches on why is Time Constant Model to Memorize HS. (b) (a) C. 1. V. R. M v (t) v. 3. dv. I (t) = C. C. i(t) [ ] t. , V. 1. X. C. R. I. dt. I. - (t / ) i (t) = I. e. c. (6ma)(0.368) = 2.2ma. t. 2. = v (t) C. 1. i (t) dt. v. 4. mA. (1.8,2.2) ( , i(t)) 2. i (t) = I. e. c. - (t / ) V. 2. = (-1) I. C. 1. e. - (t / ) t ms milliseconds. Discharging capacitor. t. 2. Energy balance. C. 1. [ ] [ ] [ i (t) dt. C. ] = [ R. 1. i(t) ] (from resistor’s perspective) V. 2. = V. 1. in. out. t. 1. in. out. This is a first order differential equation. d. dt. C. 1. I. e. - (t / ) I. e. - (t / ) R. 1. i(t) + d. dt. 1. R. C. = i(t) [ ] + = If the time constant, R, and C are known, the Euler numerical method will produce i(t) vs. t plot above. A. G. Inc.")
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-1 (2 f )(C) e A G [ ] + = e = e [ ] [ [ ] [ ] ] e e [ ] e e ] [
[ ] -1 (2 f )(C) If Time Marches on why is Time Constant? Model to Memorize 5 10 6 + HS (b) (a) C 1 V R M 2 - v (t) v 3 dv I (t) = C C i(t) [ ] t , V 1 X C R I dt I - (t / ) i (t) = I e c (6ma)(0.368) = 2.2ma t 2 = v (t) C 1 i (t) dt v 4 mA (1.8,2.2) ( , i(t)) 2 i (t) = I e c - (t / ) V 2 = (-1) I C 1 e - (t / ) t + 1.8ms 2 4 6 milliseconds t 2 Energy balance Discharging capacitor C 1 [ ] [ [ ] [ ] (from resistor’s perspective) V 2 = V 1 ] i (t) dt C = R 1 i(t) in out t 1 in out This is a first order differential equation d dt C 1 I e - (t / ) I e - (t / ) R 1 i(t) + d dt = [ 1 R C ] i(t) + = If the time constant, R, and C are known, the Euler numerical method will produce i(t) vs. t plot above. I e - (t / ) = R 1 + C (-1) This is always zero when [ ] term below is zero. I e = R 1 C -(t / ) + ] [ A G Inc. 1
![-1 (2 f )(C) e A G [ ] + = e = e [ ] [ [ ] [ ] ] e e [ ] e e ] [](http://slideplayer.com./slide/14825409/90/images/7/-1+%282+f+%29%28C%29+e+A+G+%5B+%5D+%2B+%3D+e+%3D+e+%5B+%5D+%5B+%5B+%5D+%5B+%5D+%5D+e+e+%5B+%5D+e+e+%5D+%5B.jpg "[ ] -1. (2 f )(C) If Time Marches on why is Time Constant Model to Memorize HS. (b) (a) C. 1. V. R. M v (t) v. 3. dv. I (t) = C. C. i(t) [ ] t. , V. 1. X. C. R. I. dt. I. - (t / ) i (t) = I. e. c. (6ma)(0.368) = 2.2ma. t. 2. = v (t) C. 1. i (t) dt. v. 4. mA. (1.8,2.2) ( , i(t)) 2. i (t) = I. e. c. - (t / ) V. 2. = (-1) I. C. 1. e. - (t / ) t ms milliseconds. t. 2. Energy balance. Discharging capacitor. C. 1. [ ] [ [ ] [ ] (from resistor’s perspective) V. 2. = V. 1. ] i (t) dt. C. = R. 1. i(t) in. out. t. 1. in. out. This is a first order differential equation. d. dt. C. 1. I. e. - (t / ) I. e. - (t / ) R. 1. i(t) + d. dt. = [ 1. R. C. ] i(t) + = If the time constant, R, and C are known, the Euler numerical method will produce i(t) vs. t plot above. I. e. - (t / ) = R C. (-1) This is always zero when [ ] term below is zero. I. e. = R. 1. C. -(t / ) + ] [ A. G. Inc. 1.")
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+1 (2 f )(L) A G [ ] = From the Inductor’s perspective; +
[ ] (2 f )(L) If Time Marches on why is Time Constant? Model to Memorize dt = V (t) L di [ t , , V , I 1 , v(t) , R 1 , L , X L ] 1 From the Inductor’s perspective; M 1 M 2 6 10 -10 +10 (a) (b) + V R V R V V L L A G Inc.
![+1 (2 f )(L) A G [ ] = From the Inductor’s perspective; +](http://slideplayer.com./slide/14825409/90/images/8/%2B1+%282+f+%29%28L%29+A+G+%5B+%5D+%3D+From+the+Inductor%E2%80%99s+perspective%3B+%2B.jpg "[ ] (2 f )(L) If Time Marches on why is Time Constant Model to Memorize. dt. = V (t) L. di. [ t. , , V. , I. 1. , v(t) , R. 1. , L. , X. L. ] 1. From the Inductor’s perspective; M. 1. M (a) (b) + V. R. V. R. V. V. L. L. A. G. Inc.")
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+1 (2 f )(L) + A G [ ] = From the Inductor’s perspective; +
[ ] (2 f )(L) If Time Marches on why is Time Constant? Model to Memorize dt = V (t) L di v(t) [ ] t , V X L R 1 I From the Inductor’s perspective; M 1 M 2 6 10 -10 +10 (a) (b) + V R V R V + V L L A G Inc.
![+1 (2 f )(L) + A G [ ] = From the Inductor’s perspective; +](http://slideplayer.com./slide/14825409/90/images/9/%2B1+%282+f+%29%28L%29+%2B+A+G+%5B+%5D+%3D+From+the+Inductor%E2%80%99s+perspective%3B+%2B.jpg "[ ] (2 f )(L) If Time Marches on why is Time Constant Model to Memorize. dt. = V (t) L. di. v(t) [ ] t. , V. X. L. R. 1. I. From the Inductor’s perspective; M. 1. M (a) (b) + V. R. V. R. V. + V. L. L. A. G. Inc.")
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- +1 (2 f )(L) A G [ ] = From the Inductor’s perspective; +
[ ] (2 f )(L) If Time Marches on why is Time Constant? Model to Memorize dt = V (t) L di v(t) [ ] t , V X L R 1 I From the Inductor’s perspective; M 1 M 2 6 10 -10 +10 (a) (b) + V R V R V - V L L A G Inc.
![- +1 (2 f )(L) A G [ ] = From the Inductor’s perspective; +](http://slideplayer.com./slide/14825409/90/images/10/-+%2B1+%282+f+%29%28L%29+A+G+%5B+%5D+%3D+From+the+Inductor%E2%80%99s+perspective%3B+%2B.jpg "[ ] (2 f )(L) If Time Marches on why is Time Constant Model to Memorize. dt. = V (t) L. di. v(t) [ ] t. , V. X. L. R. 1. I. From the Inductor’s perspective; M. 1. M (a) (b) + V. R. V. R. V. - V. L. L. A. G. Inc.")
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+1 [ ] (2 f )(L) If Time Marches on why is Time Constant? Model to Memorize di v(t) [ ] t , V X L R 1 I V (t) = L L dt Voltage vs Time V v 2 t 2 v (t) L = L i (t) dt C t 1 V (t)= V e - (t / ) L v 1 mv V (b) L (a) R + 4 (1.8, 2.2) 2 + 1.8ms Time A G Inc. A G Inc.
![+1 [ ] (2 f )(L) If Time Marches on why is Time Constant Model to Memorize.](http://slideplayer.com./slide/14825409/90/images/11/%2B1+%5B+%5D+%282+f+%29%28L%29+If+Time+Marches+on+why+is+Time+Constant+Model+to+Memorize..jpg "di. v(t) [ ] t. , V. X. L. R. 1. I. V (t) = L. L. dt. Voltage vs Time. V. v. 2. t. 2. v (t) L. = L. i (t) dt. C. t. 1. V (t)= V. e. - (t / ) L. v. 1. mv. V. (b) L. (a) R (1.8, 2.2) ms. Time. A. G. Inc. A. G. Inc.")
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If Time Marches on why is Time Constant?
+1 [ ] (2 f ) (L) Model to Memorize + (b) (a) R V L Natural Response di v (t) = L L dt v(t) [ ] t , V X L R 1 I V V (t) = V e - (t / ) L (10mv)(0.368) = 3.7mv 9 i (t) L i 2 1 t 2 1 = V (t) dt L L t 1 (3.7, 0.382) L mv ( , i(t)) 3 i (t) = I e L - (t / ) V L = (-1) I e - (t / ) t 2 1 + 1.8ms .4 .8 1.2 milliseconds Energy balance Discharging inductor di [ ] [ ] [ ] (from resistor’s perspective) V L = [ V R ] = R 1 i(t) L 1 in out dt in out A G Inc. A G Inc.
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e A G + = e = e [ ] [ [ ] [ ] ] e e [ ]
If Time Marches on why is Time Constant? +1 [ ] (2 f ) (L) Model to Memorize + (b) (a) R V L Natural Response di v (t) = L L dt v(t) [ ] t , V X L R 1 I V V (t) = V e - (t / ) L (10mv)(0.368) = 3.7mv 9 i (t) L i 2 1 t 2 1 = V (t) dt L L t 1 (3.7, 0.382) L mv ( , i(t)) 3 i (t) = I e L - (t / ) V L = (-1) I e - (t / ) t 2 1 + 1.8ms .4 .8 1.2 milliseconds Energy balance Discharging inductor di [ ] [ [ ] [ ] (from resistor’s perspective) V L = V R ] = R 1 i(t) L 1 in out dt in out This is a first order differential equation d dt I e - (t / ) I e - (t / ) L 1 i(t) + R 1 d dt = i(t) [ R 1 L ] + = If the time constant, R, and L are known, the Euler numerical method will produce i(t) vs. t plot above. A G Inc.
![e A G + = e = e [ ] [ [ ] [ ] ] e e [ ]](http://slideplayer.com./slide/14825409/90/images/13/e+A+G+%2B+%3D+e+%3D+e+%5B+%5D+%5B+%5B+%5D+%5B+%5D+%5D+e+e+%5B+%5D.jpg "If Time Marches on why is Time Constant +1. [ ] (2 f ) (L) Model to Memorize. + (b) (a) R. V. L. Natural. Response. di. v (t) = L. L. dt. v(t) [ ] t. , V. X. L. R. 1. I. V. V (t) = V. e. - (t / ) L. (10mv)(0.368) = 3.7mv. 9. i (t) L. i t = V (t) dt. L. L. t. 1. (3.7, 0.382) L. mv. ( , i(t)) 3. i (t) = I. e. L. - (t / ) V. L. = (-1) I. e. - (t / ) t ms milliseconds. Energy balance. Discharging inductor. di. [ ] [ [ ] [ ] (from resistor’s perspective) V. L. = V. R. ] = R. 1. i(t) L. 1. in. out. dt. in. out. This is a first order differential equation. d. dt. I. e. - (t / ) I. e. - (t / ) L. 1. i(t) + R. 1. d. dt. = i(t) [ R. 1. L. ] + = If the time constant, R, and L are known, the Euler numerical method will produce i(t) vs. t plot above. A. G. Inc.")
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A G = ] [ e e If Time Marches on why is Time Constant? L 1 V (t) dt
mv milliseconds 9 .4 3 1.2 .8 If Time Marches on why is Time Constant? Energy balance t 2 = L 1 V (t) dt i (t) i (L) (2 f ) [ ] +1 Discharging inductor + 1.8ms ( , i(t)) (3.7, 0.382) (from resistor’s perspective) V R in out ] [ i (t) = I e - (t / ) i(t) d dt + This is a first order differential equation If the time constant, R, and L are known, the Euler numerical method will produce i(t) vs. t plot above. -(t / ) RL (b) (a) Natural Response v (t) di Model to Memorize (-1) -1 v(t) [ ] t , V X L R 1 I V V (t) = V e - (t / ) L (10mv)(0.368) = 3.7mv This is always zero when [ ] term below is zero. A G Inc.
![A G = ] [ e e If Time Marches on why is Time Constant L 1 V (t) dt](http://slideplayer.com./slide/14825409/90/images/14/A+G+%3D+%5D+%5B+e+e+If+Time+Marches+on+why+is+Time+Constant+L+1+V+%28t%29+dt.jpg "mv. milliseconds If Time Marches on why is Time Constant Energy balance. t. 2. = L. 1. V (t) dt. i (t) i. (L) (2 f ) [ ] +1. Discharging inductor ms. ( , i(t)) (3.7, 0.382) (from resistor’s perspective) V. R. in. out. ] [ i (t) = I. e. - (t / ) i(t) d. dt. + This is a first order differential equation. If the time constant, R, and L are known, the Euler numerical method will produce i(t) vs. t plot above. -(t / ) RL. (b) (a) Natural. Response. v (t) di. Model to Memorize. (-1) -1. v(t) [ ] t. , V. X. L. R. 1. I. V. V (t) = V. e. - (t / ) L. (10mv)(0.368) = 3.7mv. This is always zero when [ ] term below is zero. A. G. Inc.")
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A G Summary - Time Constant e + = = = e = + + = = [ ] R V [ ] ,
+1 [ ] (2 f ) (L) + (b) (a) R V L Natural Response V v(t) [ ] t , V X L R 1 I V (t) = V e - (t / ) L 9 (10mv)(0.368) = 3.7mv Model to Memorize i (t) L i 2 1 t 2 (3.7, 0.382) 1 mv = V (t) dt L L t 1 L ( , i(t)) 3 dt = V (t) L di + 1.8ms .4 .8 1.2 v 2 t 2 milliseconds v (t) L = L i (t) dt L I t 1 v 1 - (t / ) i (t) = I e c i(t) [ ] t , V 1 X C R I (6ma)(0.368) = 2.2ma Model to Memorize (a) (b) 4 V 3 t 2 = v (t) C 1 i (t) dt v mA (1.8,2.2) - v (t) C v ( , i(t)) + 2 V 1 V V + dv R 1 I (t) C 1 V 2 V 1 = C C dt + 1.8ms V 2 2 4 6 milliseconds i 2 t 2 I (t) C = C V (t) dt C A G Inc. i 1 t 1
![A G Summary - Time Constant e + = = = e = + + = = [ ] R V [ ] ,](http://slideplayer.com./slide/14825409/90/images/15/A+G+Summary+-+Time+Constant+e+%2B+%3D+%3D+%3D+e+%3D+%2B+%2B+%3D+%3D+%5B+%5D+R+V+%5B+%5D+%2C.jpg "+1. [ ] (2 f ) (L) + (b) (a) R. V. L. Natural. Response. V. v(t) [ ] t. , V. X. L. R. 1. I. V (t) = V. e. - (t / ) L. 9. (10mv)(0.368) = 3.7mv. Model to Memorize. i (t) L. i t. 2. (3.7, 0.382) 1. mv. = V (t) dt. L. L. t. 1. L. ( , i(t)) 3. dt. = V (t) L. di ms v. 2. t. 2. milliseconds. v (t) L. = L. i (t) dt. L. I. t. 1. v (t / ) i (t) = I. e. c. i(t) [ ] t. , V. 1. X. C. R. I. (6ma)(0.368) = 2.2ma. Model to Memorize. (a) (b) 4. V. 3. t. 2. = v (t) C. 1. i (t) dt. v. mA. (1.8,2.2) - v (t) C. v. ( , i(t)) + 2. V. 1. V. V. + dv. R. 1. I (t) C. 1. V. 2. V. 1. = C. C. dt ms. V milliseconds. i. 2. t. 2. I (t) C. = C. V (t) dt. C. A. G. Inc. i. 1. t. 1.")
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End of Presentation + + A G + If Time Marches on why is Time Constant?
v(t) [ ] t , V X L R 1 I i(t) [ ] t , V 1 X C R I (a) (b) (a) (b) + + V 3 - v (t) C v V R V R V 1 V V V + V 1 R 1 C 1 V 2 V C V 2 L A G Inc.
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A G + = e 10 ohm 2 Henry V = 20 Amperes di v (t) dt - (t / ) i (t) =
Natural + (b) (a) V L R Response 10 ohm 2 Henry I = 20 Amperes A seconds 9 .4 3 1.2 .8 - (t / ) i (t) = e L dt = v (t) di Model to Memorize 15 A G Inc.
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