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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 ๐ก=0, when it is closed instantaneously.
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Problem Sheet 5 Question 4
a) Let us examine what happens for ๐<๐. We know that ๐ ๐ถ ๐ก =๐ถ ๐ ๐ฃ ๐ถ (๐ก) ๐๐ก (or ๐ฃ ๐ถ ๐ก = 1 ๐ถ โโ ๐ก ๐ ๐ถ (๐) ๐๐) and ๐ฃ ๐ฟ ๐ก =๐ฟ ๐ ๐ ๐ฟ (๐ก) ๐๐ก . If the system is in steady state, the current ๐ฆ 2 ๐ก across the capacitor is 0 since the voltage across the capacitor is constant. Therefore, ๐ฆ โ =0 ๐ด. Furthermore, the current ๐ฆ 1 ๐ก 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 ๐ฃ 0 ๐ก is the same as the voltage across the 4 ฮฉ resistor, i.e., ๐ฆ 1 ๐ก โ4 ๐. In that case for the left loop (Loop 1) we have: ๐ฆ 1 ๐ก โ =40๐โ ๐ฆ 1 ๐ก =4 ๐ด, ๐ก<0 ๐ฃ 0 ๐ก = ๐ฆ 1 ๐ก โ4 ๐=16๐, ๐ก<0 Initial conditions: ๐ฆ โ =4 ๐ด, ๐ฆ โ =0 ๐ด, ๐ฃ โ =16 ๐
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Problem Sheet 5 Question 4 cont.
Write loop equations in time domain for ๐โฅ๐. Loop 1: ๐ฟ ๐ ๐ ๐ฟ (๐ก) ๐๐ก +4โ ๐ ๐ฟ ๐ก +1 โ๐ฆ 1 ๐ก =40 ๐ข(๐ก) ๐ ๐ฟ ๐ก = ๐ฆ 1 ๐ก โ ๐ฆ 2 ๐ก ๐ฟ( ๐ ๐ฆ 1 ๐ก ๐๐ก โ ๐ ๐ฆ 2 ๐ก ๐๐ก )+4( ๐ฆ 1 ๐ก โ๐ฆ 2 ๐ก )+1 โ๐ฆ 1 ๐ก =40 ๐ข(๐ก) 2 ๐ ๐ฆ 1 ๐ก ๐๐ก โ2 ๐ ๐ฆ 2 ๐ก ๐๐ก +5 ๐ฆ 1 ๐ก โ4๐ฆ 2 ๐ก =40๐ข(๐ก) Loop 2: ๐ฟ ๐ ๐ฆ 1 ๐ก ๐๐ก โ ๐ ๐ฆ 2 ๐ก ๐๐ก +4 ๐ฆ 1 ๐ก โ๐ฆ 2 ๐ก = 1 ๐ถ โโ ๐ก ๐ ๐ถ ๐ ๐๐= 1 ๐ถ โโ ๐ก ๐ฆ 2 ๐ ๐๐
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Problem Sheet 5 Question 4 cont.
b) Loop 1 equation in time and Laplace domain for ๐โฅ๐. Loop 1: 2 ๐ ๐ฆ 1 ๐ก ๐๐ก โ2 ๐ ๐ฆ 2 ๐ก ๐๐ก +5 ๐ฆ 1 ๐ก โ4๐ฆ 2 ๐ก =40 ๐ข(๐ก) ๐ฆ โ =4 ๐ด ๐ฆ โ =0 ๐ด 2 [๐ ๐ 1 ๐ โ ๐ฆ โ ]โ2 [๐ ๐ 2 ๐ โ ๐ฆ โ ]+5 ๐ 1 ๐ โ4 ๐ 2 ๐ =40/๐ โ 2๐ +5 ๐ 1 ๐ โ(2s+4) ๐ 2 ๐ =8+40/s
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Problem Sheet 5 Question 4 cont.
b) Loop 2 equation in time and Laplace domain for ๐โฅ๐. Loop 2: ๐ฟ ๐ ๐ฆ 1 ๐ก ๐๐ก โ ๐ ๐ฆ 2 ๐ก ๐๐ก +4 ๐ฆ 1 ๐ก โ๐ฆ 2 ๐ก = โโ ๐ก ๐ฆ 2 ๐ ๐๐ ๐ฆ โ =4 ๐ด ๐ฆ โ =0 ๐ด ๐ฃ โ =16 ๐ โโ ๐ก ๐ฅ ๐ ๐๐โ ๐ ๐ ๐ + 1 ๐ โโ 0 โ ๐ฅ ๐ก ๐๐ก โ โโ ๐ก ๐ฆ 2 ๐ ๐๐ = ๐ 2 ๐ ๐ + 1 ๐ 1 ๐ถ โโ 0 โ ๐ฆ 2 ๐ก ๐๐ก= ๐ 2 ๐ ๐ + 1 ๐ ๐ฃ โ 2 [๐ ๐ 1 ๐ โ ๐ฆ โ ]โ2 [๐ ๐ 2 ๐ โ ๐ฆ โ ]+ 4 ๐ 1 ๐ โ4 ๐ 2 ๐ = ๐ 2 ๐ ๐ + 16 ๐ โ 2๐ +4 ๐ 1 ๐ +(2s+4+ 1 ๐ ) ๐ 2 ๐ =โ8โ 16 s
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Problem Sheet 5 Question 4 cont.
Merge equations for Loops 1 and 2 in a matrix form ๐โฅ๐ 2๐ +5 โ(2๐ +4) โ(2๐ +4) 2s+4+ 1 ๐ ๐ 1 ๐ ๐ 2 ๐ = s โ8โ 16 s By solving the above system we obtain: ๐ 1 ๐ = 4(6 ๐ 2 +13๐ +5) ๐ ( ๐ 2 +3๐ +2.5) = 8 ๐ + 16๐ +28 (๐ 2 +3๐ +2.5) ๐ 2 ๐ = 20(๐ +2) ( ๐ 2 +3๐ +2.5)
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Problem Sheet 5 Question 4 cont.
Find ๐ ๐ ๐ , ๐โฅ๐ ๐ 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 โ๐๐ ๐๐จ๐ฌ ๐๐+๐ฝ ๐(๐)โ ๐จ๐+๐ฉ ๐ ๐ +๐๐๐+๐ ๐= ๐ด 2 ๐+ ๐ต 2 โ2๐ด๐ต๐ ๐โ ๐ 2 ๐= tan โ1 ๐ด๐โ๐ต ๐ด ๐โ ๐ 2 ๐= ๐โ ๐ 2 ๐ด=16, ๐ต=28, ๐=1.5, ๐=2.5 ๐ฆ 1 ๐ก =[ ๐ โ1.5๐ก cos 0.5๐กโ ฮฟ ] ๐ข(๐ก)
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Problem Sheet 5 Question 4 cont.
b) Find ๐ ๐ ๐ , ๐โฅ๐ ๐ 2 ๐ = 20(๐ +2) ( ๐ 2 +3๐ +2.5) = 20๐ +40 ( ๐ 2 +3๐ +2.5) Property 10c ๐ ๐ โ๐๐ก cos ๐๐ก+๐ ๐ข(๐ก)โ ๐ด๐ +๐ต ๐ 2 +2๐๐ +๐ ๐= ๐ด 2 ๐+ ๐ต 2 โ2๐ด๐ต๐ ๐โ ๐ 2 ๐= ๐ก๐๐ โ1 ๐ด๐โ๐ต ๐ด ๐โ ๐ 2 ๐= ๐โ ๐ 2 ๐ด=20, ๐ต=40, ๐=1.5, ๐=2.5 ๐ฆ 2 ๐ก = ๐ โ1.5๐ก cos 0.5๐กโ 45 ฮฟ ๐ข(๐ก)
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