ECE201_Lecture24

ECE201_Lecture24 - 0.1 v 1 i c 2 0 0.1 v 1 C 2 dv 2 dt 0...

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1 ECE201 Linear Circuit Analysis I Lecture 24 Topic: Second-order circuits: RLC source-free (with complex characteristic roots) 1. The problem d 2 x dt 2 2 dx dt n 2 x 0 where : x ( t ) v c ( t ) or i L ( t ) n 1 LC R 2 L (series RLC) or 1 2 RC (parallel RLC) characteristic equation : s 2 2 s n 2 s s 1 s s 2 0 solutions : s 1 , s 2   2 n 2 today’s topic: < n (under-damped response) 2. From last lecture (1) s 1 ≠ s 2 , the roots are distinct but complex. let d n 2 2 ( d is real, “damped oscillation frequency”) then s 1,2 = ± j d Because s 1 and s 2 are distinct, x ( t ) K 1 e s 1 t K 2 e s 2 t (like the over-damped case) Because s 1 and s 2 are complex, )] sin( ) cos( [ ) ( t B t A e t x d d t A, B are real constants, determined by initial conditions. R L C R OR L C
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2 3. Example Given that v 1 (0) = 10 V , v 2 (0) = 0 V , find v 1 (t) for t > 0. Step 1 Derive circuit equation for v 1 (t). KCL (left side): i R i c 1 0.1 v 2 v 1 R C 1 dv 1 dt 0.1 v 2 (1) KCL (right side):
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Unformatted text preview: 0.1 v 1 i c 2 0 0.1 v 1 C 2 dv 2 dt 0 (2) To eliminate v 2 from (1) &amp; (2): differentiate both sides of (1): 1 R dv 1 dt C 1 d 2 v 1 dt 2 0.1 dv 2 dt from (2): dv 2 dt 0.1 C 2 v 1 thus 1 R dv 1 dt C 1 d 2 v 1 dt 2 0.01 C 2 v 1 Rearrange terms: d 2 v 1 dt 2 1 RC 1 dv 1 dt 0.01 C 1 C 2 v 1 0 R 1 k , C 1 C 2 1 F . d 2 v 1 dt 2 10 3 dv 1 dt 10 10 v 1 0 3 Step 2 Determine form of v 1 (t) (i.e., over-, under-, or critically-damped) Step 3 Determine initial conditions v 1 (0) and v 1 (0) . Step 4 Determine A &amp; B from initial conditions v 1 (0) = 10 V and v 1 (0) 10 4 V Finally...
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ECE201_Lecture24 - 0.1 v 1 i c 2 0 0.1 v 1 C 2 dv 2 dt 0...

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