If Time Marches on why is Time Constant? + + A G + [ ] , [ ] , - v (t)

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Presentation transcript:

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.

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.

-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? [ ] -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) 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 [ ] [ ] [ ] [ ] 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 [ ] 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 )(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; + [ ] (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; + [ ] (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) 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.

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.

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.

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 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

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.

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.