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Problem Sheet 5 Question 4

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1 Problem Sheet 5 Question 4
Problem: For the circuit shown in the figure below, the switch is in open position for a long time before ๐‘ก=0, when it is closed instantaneously.

2 Problem Sheet 5 Question 4
a) Let us examine what happens for ๐’•<๐ŸŽ. We know that ๐‘– ๐ถ ๐‘ก =๐ถ ๐‘‘ ๐‘ฃ ๐ถ (๐‘ก) ๐‘‘๐‘ก (or ๐‘ฃ ๐ถ ๐‘ก = 1 ๐ถ โˆ’โˆž ๐‘ก ๐‘– ๐ถ (๐œ) ๐‘‘๐œ) and ๐‘ฃ ๐ฟ ๐‘ก =๐ฟ ๐‘‘ ๐‘– ๐ฟ (๐‘ก) ๐‘‘๐‘ก . If the system is in steady state, the current ๐‘ฆ 2 ๐‘ก across the capacitor is 0 since the voltage across the capacitor is constant. Therefore, ๐‘ฆ โˆ’ =0 ๐ด. Furthermore, the current ๐‘ฆ 1 ๐‘ก which flows through the left loop (Loop 1) is constant and therefore, the voltage across the inductor is 0. Based on the above two points, the voltage across the capacitor ๐‘ฃ 0 ๐‘ก is the same as the voltage across the 4 ฮฉ resistor, i.e., ๐‘ฆ 1 ๐‘ก โˆ™4 ๐‘‰. In that case for the left loop (Loop 1) we have: ๐‘ฆ 1 ๐‘ก โˆ™ =40๐‘‰โ‡’ ๐‘ฆ 1 ๐‘ก =4 ๐ด, ๐‘ก<0 ๐‘ฃ 0 ๐‘ก = ๐‘ฆ 1 ๐‘ก โˆ™4 ๐‘‰=16๐‘‰, ๐‘ก<0 Initial conditions: ๐‘ฆ โˆ’ =4 ๐ด, ๐‘ฆ โˆ’ =0 ๐ด, ๐‘ฃ โˆ’ =16 ๐‘‰

3 Problem Sheet 5 Question 4 cont.
Write loop equations in time domain for ๐’•โ‰ฅ๐ŸŽ. Loop 1: ๐ฟ ๐‘‘ ๐‘– ๐ฟ (๐‘ก) ๐‘‘๐‘ก +4โˆ™ ๐‘– ๐ฟ ๐‘ก +1 โˆ™๐‘ฆ 1 ๐‘ก =40 ๐‘ข(๐‘ก) ๐‘– ๐ฟ ๐‘ก = ๐‘ฆ 1 ๐‘ก โˆ’ ๐‘ฆ 2 ๐‘ก ๐ฟ( ๐‘‘ ๐‘ฆ 1 ๐‘ก ๐‘‘๐‘ก โˆ’ ๐‘‘ ๐‘ฆ 2 ๐‘ก ๐‘‘๐‘ก )+4( ๐‘ฆ 1 ๐‘ก โˆ’๐‘ฆ 2 ๐‘ก )+1 โˆ™๐‘ฆ 1 ๐‘ก =40 ๐‘ข(๐‘ก) 2 ๐‘‘ ๐‘ฆ 1 ๐‘ก ๐‘‘๐‘ก โˆ’2 ๐‘‘ ๐‘ฆ 2 ๐‘ก ๐‘‘๐‘ก +5 ๐‘ฆ 1 ๐‘ก โˆ’4๐‘ฆ 2 ๐‘ก =40๐‘ข(๐‘ก) Loop 2: ๐ฟ ๐‘‘ ๐‘ฆ 1 ๐‘ก ๐‘‘๐‘ก โˆ’ ๐‘‘ ๐‘ฆ 2 ๐‘ก ๐‘‘๐‘ก +4 ๐‘ฆ 1 ๐‘ก โˆ’๐‘ฆ 2 ๐‘ก = 1 ๐ถ โˆ’โˆž ๐‘ก ๐‘– ๐ถ ๐œ ๐‘‘๐œ= 1 ๐ถ โˆ’โˆž ๐‘ก ๐‘ฆ 2 ๐œ ๐‘‘๐œ

4 Problem Sheet 5 Question 4 cont.
b) Loop 1 equation in time and Laplace domain for ๐’•โ‰ฅ๐ŸŽ. Loop 1: 2 ๐‘‘ ๐‘ฆ 1 ๐‘ก ๐‘‘๐‘ก โˆ’2 ๐‘‘ ๐‘ฆ 2 ๐‘ก ๐‘‘๐‘ก +5 ๐‘ฆ 1 ๐‘ก โˆ’4๐‘ฆ 2 ๐‘ก =40 ๐‘ข(๐‘ก) ๐‘ฆ โˆ’ =4 ๐ด ๐‘ฆ โˆ’ =0 ๐ด 2 [๐‘ ๐‘Œ 1 ๐‘  โˆ’ ๐‘ฆ โˆ’ ]โˆ’2 [๐‘ ๐‘Œ 2 ๐‘  โˆ’ ๐‘ฆ โˆ’ ]+5 ๐‘Œ 1 ๐‘  โˆ’4 ๐‘Œ 2 ๐‘  =40/๐‘ โ‡’ 2๐‘ +5 ๐‘Œ 1 ๐‘  โˆ’(2s+4) ๐‘Œ 2 ๐‘  =8+40/s

5 Problem Sheet 5 Question 4 cont.
b) Loop 2 equation in time and Laplace domain for ๐’•โ‰ฅ๐ŸŽ. Loop 2: ๐ฟ ๐‘‘ ๐‘ฆ 1 ๐‘ก ๐‘‘๐‘ก โˆ’ ๐‘‘ ๐‘ฆ 2 ๐‘ก ๐‘‘๐‘ก +4 ๐‘ฆ 1 ๐‘ก โˆ’๐‘ฆ 2 ๐‘ก = โˆ’โˆž ๐‘ก ๐‘ฆ 2 ๐œ ๐‘‘๐œ ๐‘ฆ โˆ’ =4 ๐ด ๐‘ฆ โˆ’ =0 ๐ด ๐‘ฃ โˆ’ =16 ๐‘‰ โˆ’โˆž ๐‘ก ๐‘ฅ ๐œ ๐‘‘๐œโ‡” ๐‘‹ ๐‘  ๐‘  + 1 ๐‘  โˆ’โˆž 0 โˆ’ ๐‘ฅ ๐‘ก ๐‘‘๐‘ก โ„’ โˆ’โˆž ๐‘ก ๐‘ฆ 2 ๐œ ๐‘‘๐œ = ๐‘Œ 2 ๐‘  ๐‘  + 1 ๐‘  1 ๐ถ โˆ’โˆž 0 โˆ’ ๐‘ฆ 2 ๐‘ก ๐‘‘๐‘ก= ๐‘Œ 2 ๐‘  ๐‘  + 1 ๐‘  ๐‘ฃ โˆ’ 2 [๐‘ ๐‘Œ 1 ๐‘  โˆ’ ๐‘ฆ โˆ’ ]โˆ’2 [๐‘ ๐‘Œ 2 ๐‘  โˆ’ ๐‘ฆ โˆ’ ]+ 4 ๐‘Œ 1 ๐‘  โˆ’4 ๐‘Œ 2 ๐‘  = ๐‘Œ 2 ๐‘  ๐‘  + 16 ๐‘  โˆ’ 2๐‘ +4 ๐‘Œ 1 ๐‘  +(2s+4+ 1 ๐‘  ) ๐‘Œ 2 ๐‘  =โˆ’8โˆ’ 16 s

6 Problem Sheet 5 Question 4 cont.
Merge equations for Loops 1 and 2 in a matrix form ๐’•โ‰ฅ๐ŸŽ 2๐‘ +5 โˆ’(2๐‘ +4) โˆ’(2๐‘ +4) 2s+4+ 1 ๐‘  ๐‘Œ 1 ๐‘  ๐‘Œ 2 ๐‘  = s โˆ’8โˆ’ 16 s By solving the above system we obtain: ๐‘Œ 1 ๐‘  = 4(6 ๐‘  2 +13๐‘ +5) ๐‘ ( ๐‘  2 +3๐‘ +2.5) = 8 ๐‘  + 16๐‘ +28 (๐‘  2 +3๐‘ +2.5) ๐‘Œ 2 ๐‘  = 20(๐‘ +2) ( ๐‘  2 +3๐‘ +2.5)

7 Problem Sheet 5 Question 4 cont.
Find ๐’š ๐Ÿ ๐’• , ๐’•โ‰ฅ๐ŸŽ ๐‘Œ 1 ๐‘  = 4(6 ๐‘  2 +13๐‘ +5) ๐‘ ( ๐‘  2 +3๐‘ +2.5) = 8 ๐‘  + 16๐‘ +28 ๐‘  2 +3๐‘ +2.5 We use Property 10c from Laplace Properties tables. ๐’“ ๐’† โˆ’๐’‚๐’• ๐œ๐จ๐ฌ ๐’ƒ๐’•+๐œฝ ๐’–(๐’•)โ‡” ๐‘จ๐’”+๐‘ฉ ๐’” ๐Ÿ +๐Ÿ๐’‚๐’”+๐’„ ๐‘Ÿ= ๐ด 2 ๐‘+ ๐ต 2 โˆ’2๐ด๐ต๐‘Ž ๐‘โˆ’ ๐‘Ž 2 ๐œƒ= tan โˆ’1 ๐ด๐‘Žโˆ’๐ต ๐ด ๐‘โˆ’ ๐‘Ž 2 ๐‘= ๐‘โˆ’ ๐‘Ž 2 ๐ด=16, ๐ต=28, ๐‘Ž=1.5, ๐‘=2.5 ๐‘ฆ 1 ๐‘ก =[ ๐‘’ โˆ’1.5๐‘ก cos 0.5๐‘กโˆ’ ฮฟ ] ๐‘ข(๐‘ก)

8 Problem Sheet 5 Question 4 cont.
b) Find ๐’š ๐Ÿ ๐’• , ๐’•โ‰ฅ๐ŸŽ ๐‘Œ 2 ๐‘  = 20(๐‘ +2) ( ๐‘  2 +3๐‘ +2.5) = 20๐‘ +40 ( ๐‘  2 +3๐‘ +2.5) Property 10c ๐‘Ÿ ๐‘’ โˆ’๐‘Ž๐‘ก cos ๐‘๐‘ก+๐œƒ ๐‘ข(๐‘ก)โ‡” ๐ด๐‘ +๐ต ๐‘  2 +2๐‘Ž๐‘ +๐‘ ๐‘Ÿ= ๐ด 2 ๐‘+ ๐ต 2 โˆ’2๐ด๐ต๐‘Ž ๐‘โˆ’ ๐‘Ž 2 ๐œƒ= ๐‘ก๐‘Ž๐‘› โˆ’1 ๐ด๐‘Žโˆ’๐ต ๐ด ๐‘โˆ’ ๐‘Ž 2 ๐‘= ๐‘โˆ’ ๐‘Ž 2 ๐ด=20, ๐ต=40, ๐‘Ž=1.5, ๐‘=2.5 ๐‘ฆ 2 ๐‘ก = ๐‘’ โˆ’1.5๐‘ก cos 0.5๐‘กโˆ’ 45 ฮฟ ๐‘ข(๐‘ก)


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