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EGR 261 – Inverse Laplace Transforms using MATLAB Application of Laplace Transforms: Circuit Analysis MATLAB is a powerful tool for analyzing circuits using Laplace transforms. One approach might be: • Determine the s-domain circuit (find initial conditions first) • Use MATLAB to find complex impedances (XC = 1/(sC), XL = sL) • Write any required circuit equations (KVL, KCL, etc) • Use solve( ) solve the circuit equations. The result will be functions of s. • Use ilaplace( ) to find the corresponding time-domain expressions. In class the s-domain relationships for each type of circuit element are developed. They are summarized on the following slide. 1 EGR 261 – Inverse Laplace Transforms using MATLAB 2 s-domain circuit models time-domain s-domain R C L R 1/(sC) v(0)/s sL Li(0) 10V 10/s 2mA 0.002/s Lecture #22 EGR 267 – Engineering Analysis Tools 3 Procedure: Circuit Analysis using the Laplace-transformed Circuit 1) Draw the circuit at t = 0-. • Assume that the circuit is in steady state. • Draw inductors as short circuits and capacitors as open circuits. • Find vC(0-) and iL(0-) – these are needed for step 2. 2) Draw the s-domain circuit for t > 0. 3) Analyze the circuit as you might analyze a DC circuit (using any circuit analysis method). Recall that the s-domain impedances sL and 1/(sC) act essentially like resistors. Determine the desired result in the s-domain (V(s), I(s), etc). 4) Convert the result back to the time domain. In other words, use inverse Laplace transforms to find v(t) or i(t) from V(s) or I(s). Note: If the circuit has zero initial conditions then the voltage sources in the capacitor and inductor models will be zero. EGR 261 – Inverse Laplace Transforms using MATLAB Example 1: Use Laplace transforms and MATLAB to determine i(t) and vC(t) in the circuit shown below (for t > 0). Assume that all initial conditions are zero. i(t) 28 ohms 160 V + - 4H 0.025 F + VC(t) _ 4 EGR 261 – Inverse Laplace Transforms using MATLAB Example 1 (continued) 5 EGR 261 – Inverse Laplace Transforms using MATLAB Example 2: Use Laplace transforms and MATLAB to determine ia(t) and ib(t) in the circuit shown below (for t > 0). Assume that all initial conditions are zero. 5 15 ia(t) 10V ib(t) 4H 20V 6 EGR 261 – Inverse Laplace Transforms using MATLAB Example 2 (continued) 7 EGR 261 – Inverse Laplace Transforms using MATLAB Example 3: (class example) Use Laplace transforms and MATLAB to determine i(t) and v(t) in the circuit shown below (for t > 0). Assume that vC(0) = 50V and i(0) = 100 mA. 8 mH i(t) 336 V + - 10 uF + v(t) 4k 2k 8