ECE201_Exam3_Review_Jung

ECE201_Exam3_Review_Jung - (Undamped) LC Circuit Initial...

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(Undamped) LC Circuit Initial conditions: Goal: KVL and KCL: V-I Relations: Differential equation for v C (t) Differential equation for i L (t) 2 2 C C C L CL di d d di d L L L C LC dt dt dt dt dt           2 2 C L L L LC d d di d i d i i C C C L LC dt dt dt dt dt             L L di t tL dt
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Differential Equations for v C (t) and i L (t) and are both of the general form with solution Check: amplitude phase angular velocity Conclusions: for some K and for some K and undamped oscillation natural (resonant) frequency 2 f 1 f LC Energy stored in sine wave form
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How to determine K and ? for some K and with Use the initial conditions: Use the fact that Take the quotient:       0 0 0 0 sin tan cos C K i i KC 
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Example The switch is closed for a long time before it opens at time t=0 At time 0 For t>0 with
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Another Example The switch is closed for a long time before it flips at time t=0 For t>0 with At time 0
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Serial RLC Circuit are given KCL KVL In terms of v C In terms of i L differentiate v C (t)=?, i L (t)=? 2 2 0 LL L d i di LC RC i dt dt
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General Form of the Differential Equations Equations for v C and i L are of the general form with initial conditions and given characteristic equation Guess : solutions are of the form for some constants K, s 2
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Characteristic Equation Characteristic equation has two roots: Case 1 ( two distinct real roots ): Case 2 ( two identical real roots ): Case 3 ( two distinct complex roots ): (natural frequency) is called the discriminant of the quadratic equation
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Case 1 Example Given the initial conditions with initial conditions: 1. Write the differential equation 2. Solve the characteristic equation
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ECE201_Exam3_Review_Jung - (Undamped) LC Circuit Initial...

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