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Inc.

A. Inc. G. [. ,. ,. ,. ,. ,. ]. ,. ,. i(t). If Time Marches on why is Time Constant ?. V. V. V. V. V. V. 0. 2. 1. 0. R. C. (a). (b). (a). (b). +. R. C. 1. 1. +. R. v. - v (t). [. ,. ,. ,. ,. ,. ]. 0. C. ,. ,. +. v(t). L. C. X. R. X. R.

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  1. A Inc. G [ , , , , , ] , , i(t) If Time Marches on why is Time Constant? V V V V V V 0 2 1 0 R C (a) (b) (a) (b) + R C 1 1 + R v - v (t) [ , , , , , ] 0 C , , + v(t) L C X R X R L I t t I 1 0 C 1 L 1 1 1 V V V V V V V 3 1 1 0 2 0 0

  2. A Inc. G If Time Marches on why is Time Constant? [ , , , , , ] , , i(t) From the capacitor’s perspective; - v (t) v V 0 C 3 M M 1 2 HS (a) (b) 5 0 10 6 0 10 R C 1 1 + V V V V 1 2 0 1 + V 2 C R X t I 0 1 C 1 V V 0 1

  3. -1 A Inc. G If Time Marches on why is Time Constant? (2 f )(C) [ ] Model to Memorize [ , , , , , ] , , i(t) dv I (t) = C C Current vs Time dt V 3 1 M = 2 C M HS mA (6ma)(0.368) = 2.2ma 1 4 - (t / ) 5 (1.8, 2.2) (a) (b) 0 e 10 i (t) = I c 0 6 0 10 2 R C - v (t) v 1 1 + 0 C V V V V 1 0 2 1 + v (t) + 1.8ms C Time V i (t) dt t t v v 2 2 1 2 C 1 R X C I I t 0 0 C 1 1 V V 0 1

  4. -1 A Inc. G If Time Marches on why is Time Constant? (2 f )(C) [ ] Model to Memorize M HS M dv 1 2 I (t) = [ , , , , , ] C , , i(t) C dt (a) (b) - (t / ) e i (t) = c (6ma)(0.368) = 2.2ma + 1 = C 4 + ( , i(t)) mA (1.8,2.2) 2 V V V 5 2 1 0 0 + 1.8ms 10 milliseconds 6 0 10 R C 1 1 - v (t) v v (t) 0 C C i (t) dt t v t v 2 1 2 C 1 R C X I t I 0 0 1 C 1 I V V V V V 0 3 1 0 2 1

  5. -1 A Inc. G If Time Marches on why is Time Constant? (2 f )(C) [ ] Model to Memorize M HS M dv 1 2 I (t) = [ , , , , , ] C , , i(t) C dt (a) (b) - (t / ) e i (t) = c (6ma)(0.368) = 2.2ma + 1 = C 4 + ( , i(t)) mA (1.8,2.2) 2 V V V - (t / ) 5 0 2 1 e i (t) = 0 + 1.8ms 10 2 4 6 c milliseconds 6 0 10 - (t / ) Discharging capacitor Energy balance C V R V V R e = 1 2 2 1 1 1 (-1) [ ] [ ] i(t) [ ] [ ] = = (from resistor’s perspective) in out in out - v (t) v (t) v C 0 C 1 i (t) dt i (t) dt t t v t t t t v 2 2 2 1 2 C C 1 1 1 C 1 R X C I t I 1 0 0 C 1 1 I I I 0 V V V V V C 0 0 3 1 1 0 2

  6. -1 A Inc. G If Time Marches on why is Time Constant? (2 f )(C) [ ] Model to Memorize M HS M dv 1 2 I (t) = [ , , , , , ] C , , i(t) C dt (a) (b) - (t / ) e i (t) = c (6ma)(0.368) = 2.2ma + 1 = C 4 + ( , i(t)) mA (1.8,2.2) 2 V V V - (t / ) 5 2 1 0 e i (t) = 0 + 1.8ms 10 2 4 6 c milliseconds 6 0 10 - (t / ) Discharging capacitor Energy balance R R C V R V V e = 2 1 2 1 1 1 1 (-1) [ ] [ ] i(t) [ ] [ ] = = (from resistor’s perspective) in out in out This is a first order differential equation v (t) - v (t) v 0 C C i(t) [ ] + i(t) = 0 + 0 = d d If the time constant, R, and C are known, the Euler numerical method will produce i(t) vs. t plot above. - (t / ) - (t / ) dt dt 1 1 e e i (t) dt i (t) dt t t v t t t t v 2 2 2 1 2 C C 1 1 1 1 C C 1 1 R X C I I t 1 R C 0 0 C 1 1 1 1 I I I I I 0 V V V V V C 0 0 0 0 3 2 1 0 1

  7. -1 A Inc. G If Time Marches on why is Time Constant? (2 f )(C) [ ] Model to Memorize M HS M dv 1 2 I (t) = [ , , , , , ] C , , i(t) C dt (a) (b) - (t / ) e i (t) = c (6ma)(0.368) = 2.2ma + 1 = C 4 + ( , i(t)) mA (1.8,2.2) 2 V V V - (t / ) 5 1 0 2 e i (t) = 0 + 1.8ms 10 2 4 6 c milliseconds 6 0 10 - (t / ) Discharging capacitor Energy balance R R C V V R V R R e = 2 1 2 1 1 1 1 1 1 (-1) [ ] [ ] i(t) [ ] [ ] = = (from resistor’s perspective) in out in out This is a first order differential equation - v (t) v v (t) C 0 C i(t) [ ] + i(t) = 0 + 0 = d d If the time constant, R, and C are known, the Euler numerical method will produce i(t) vs. t plot above. - (t / ) - (t / ) - (t / ) dt dt 1 1 1 e e e i (t) dt i (t) dt t t v t v t t t 2 2 2 1 2 C C 1 1 1 1 C C C 1 1 1 This is always zero when [] term below is zero. - (t / ) R X C I I t e (-1) 1 R C + 0 = 0 0 1 C 1 1 1 I I I I I I I I [ ] 0 + V V V V V e C 0 1 R C = = 0 0 0 0 0 0 0 3 1 1 0 2 R C -(t / ) 1 1 1 C

  8. +1 A Inc. G If Time Marches on why is Time Constant? (2 f )(L) [ ] Model to Memorize [ , , , , , ] , , v(t) L 1 From the Inductor’s perspective; di M V (t) = L 1 M L dt 2 0 -10 +10 (b) (a) 6 0 + 10 V V L R R V 0 L R X t I 1 L 1 V V 0 0

  9. +1 A Inc. G If Time Marches on why is Time Constant? (2 f )(L) [ ] Model to Memorize [ , , , , , ] , , v(t) L 1 From the Inductor’s perspective; di M V (t) = L 1 M L dt 2 0 -10 +10 (b) (a) 6 0 + 10 V V L R R V 0 + L X R I t 1 1 L V V 0 0

  10. +1 A Inc. G If Time Marches on why is Time Constant? (2 f )(L) [ ] Model to Memorize [ , , , , , ] , , v(t) L 1 From the Inductor’s perspective; di M V (t) = L 1 M L dt 2 0 -10 +10 (b) (a) 6 0 + 10 V V L R R V 0 - L X R I t 1 1 L V V 0 0

  11. +1 A A Inc. Inc. G G If Time Marches on why is Time Constant? (2 f )(L) [ ] Model to Memorize di V (t) = L L Voltage vs Time dt = L mv 4 (b) (a) - (t / ) (1.8, 2.2) e V (t)= V L 0 + 2 V V L R R V 0 v (t) + 1.8ms [ , , , , , ] L , , v(t) Time L i (t) dt v t v t 2 2 1 C 1 L X R V t I 1 0 L 1 1 V V 0 0

  12. A A Inc. Inc. G G If Time Marches on why is Time Constant? +1 [ ] (2 f ) (L) Model to Memorize Natural di Response (a) (b) v (t) = L L dt + R - (t / ) 9 e (10mv)(0.368) = 3.7mv V (t) = L i 2 1 V (t) dt = L ( , i(t)) L (3.7, 0.382) i L mv 1 3 1 e - (t / ) = (-1) V L e i (t) = + 1.8ms .4 .8 1.2 L L milliseconds - (t / ) Discharging inductor Energy balance V V L V V R di L R 1 1 L R [ ] [ ] i(t) [ ] [ ] = = (from resistor’s perspective) in out in out dt i (t) [ , , , , , ] L , , v(t) t t t t 2 2 1 1 V R X L t I 1 0 1 1 L I V I 0 V 0 0 0

  13. A Inc. G If Time Marches on why is Time Constant? +1 [ ] (2 f ) (L) Model to Memorize Natural di Response (a) (b) v (t) = L L dt + R - (t / ) 9 e (10mv)(0.368) = 3.7mv V (t) = L i 2 1 V (t) dt = L ( , i(t)) L (3.7, 0.382) i L mv 1 3 1 e - (t / ) = (-1) V L e i (t) = + 1.8ms .4 .8 1.2 L L milliseconds - (t / ) Discharging inductor Energy balance R V V V R L L V di R R L 1 1 1 L R 1 [ ] [ ] i(t) [ ] [ ] = 1 = (from resistor’s perspective) in out in out L dt 1 This is a first order differential equation i (t) [ , , , , , ] L i(t) [ ] , , + i(t) v(t) = 0 + 0 = d d If the time constant, R, and L are known, the Euler numerical method will produce i(t) vs. t plot above. - (t / ) - (t / ) dt dt e e t t t t 2 2 1 1 V X R L t I 1 0 L 1 1 I V I I I 0 V 0 0 0 0 0

  14. A Inc. G If Time Marches on why is Time Constant? +1 [ ] (2 f ) (L) Model to Memorize Natural di Response (a) (b) v (t) = L L dt + R - (t / ) 9 e (10mv)(0.368) = 3.7mv V (t) = L i 2 1 V (t) dt = L ( , i(t)) L (3.7, 0.382) i L mv 1 3 1 e - (t / ) = (-1) V L e i (t) = + 1.8ms .4 .8 1.2 L L milliseconds - (t / ) Discharging inductor Energy balance R R V V V V L L L L R R di R R L 1 1 1 1 1 1 1 R L 1 [ ] [ ] i(t) [ ] [ ] = 1 = (from resistor’s perspective) in out in out L dt 1 This is a first order differential equation i (t) [ , , , , , ] L i(t) [ ] + , , i(t) v(t) = 0 + 0 = d d If the time constant, R, and L are known, the Euler numerical method will produce i(t) vs. t plot above. - (t / ) - (t / ) - (t / ) dt dt -1 -1 e e e - (t / ) t t t t e (-1) 2 2 + 0 = 1 1 This is always zero when [] term below is zero. V R X L t I 1 0 L 1 1 I L V I I I I I I [ ] 0 + V 1 e 0 = = 0 0 0 0 0 0 0 0 -(t / ) RL R 1

  15. Summary - Time Constant A Inc. G +1 [ ] (2 f ) (L) Natural Response (a) (b) [ , , , , , ] , , i(t) + R 9 - (t / ) e i (t) = (10mv)(0.368) = 3.7mv - (t / ) Model to Memorize c e V (t) = L 1 (3.7, 0.382) i = 2 1 mv V (t) dt = C L ( , i(t)) ( , i(t)) L i L 3 1 di V (t) = L L dt + 1.8ms .4 .8 1.2 V V V milliseconds 2 1 0 = L R V V C L R 1 1 (6ma)(0.368) = 2.2ma Model to Memorize 4 (a) (b) mA (1.8,2.2) v I (t) i (t) v (t) v (t) - v (t) [ , , , , , ] L 0 L C C , , C v(t) + 2 + dv V (t) dt i (t) dt i (t) dt v t t v t t i t t v t v i t I (t) = C 1 2 2 2 2 2 1 1 2 2 C C L 1 1 1 1 C dt + 1.8ms 2 4 6 V R L C X R X I t I I t milliseconds 1 0 0 0 C L 1 1 1 1 V I V V V V V V = C 0 0 3 0 1 2 0 1

  16. A Inc. G [ , , , , , ] , , i(t) End of Presentation If Time Marches on why is Time Constant? V V V V V V 0 C R 0 2 1 C R 1 1 v - v (t) (a) (b) (a) (b) [ , , , , , ] 0 C , , v(t) + + R + L R X X R C I I t t 1 0 1 C 1 1 1 L L V V V V V V V 3 1 2 0 0 1 0

  17. = 20 Amperes A Inc. G Natural Model to Memorize Response (a) (b) di v (t) = L L dt + 10 ohm - (t / ) e i (t) = L 15 2 Henry A 9 3 .4 .8 1.2 seconds V V R L I I 0 0 I 0

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