ECE201Lect-22

# ECE201Lect-22 - L as short circuit find I L(0 – and/or V...

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ECE201 Lect-22 1 Second-Order Circuits Cont’d

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ECE201 Lect-22 2 Important Concepts The differential equation for the circuit Forced (particular) and natural (complementary) solutions Transient and steady-state responses 1st order circuits: the time constant ( τ ) • 2nd order circuits: natural frequency 0 ) and the damping ratio (ζ)
ECE201 Lect-22 3 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition to be developed: Particular and complementary solutions Effects of initial conditions Roots of the characteristic equation

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ECE201 Lect-22 4 The second-order ODE has a form of To find the natural solution, we solve the characteristic equation : • Which has two roots: s 1 and s 2 . Second-Order Natural Solution 0 2 2 0 0 2 = + + ϖ ζϖ s s 0 ) ( ) ( 2 ) ( 2 0 0 2 2 = + + t x dt t dx dt t x d
ECE201 Lect-22 5 Step-by-Step Approach 1. Assume solution (only dc sources allowed): i. x ( t ) = K 1 + K 2 e -t/ τ ii. x ( t ) = K 1 + K 2 e s 1 t + K 3 e s 2 t i. At t =0 , draw circuit with C as open circuit and

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Unformatted text preview: L as short circuit; find I L (0 – ) and/or V C (0 – ) ii. At t =0 + , redraw circuit and replace C and/or L with appropriate source of value obtained in step #2, and find x (0)=K 1 +K 2 (+K 3 ) iii. At t = ∞ , repeat step #2 to find x ( ∞ )=K 1 ECE201 Lect-22 6 Step-by-Step Approach 5. Find time constant ( τ ), or characteristic roots ( s ) i. Looking across the terminals of the C or L element, form Thevenin equivalent circuit; τ =R Th C or τ =L/R Th ii. Write ODE at t >0; find s from characteristic equation 5. Finish up 1. Simply put the answer together. 2. Typically have to use d x ( t )/d t │ t =0 to generate another algebraic equation to solve for K 2 & K 3 (try repeating the circuit analysis of step #5 at t =0 + , which basically means using the values obtained in step #3) ECE201 Lect-22 7 Class Examples • Learning Extension E7.10 • Learning Extension E7.11...
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ECE201Lect-22 - L as short circuit find I L(0 – and/or V...

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