Class Notes on First Order Circuit - Linear Circuit Analysis I | ECE 20100, Study notes of Electrical Circuit Analysis

Download Class Notes on First Order Circuit - Linear Circuit Analysis I | ECE 20100 and more Study notes Electrical Circuit Analysis in PDF only on Docsity! 1 ECE201 Linear Circuit Analysis I (Prof. H. Z. Tan) Topic: First-order circuits: step response Review (1) general dx(t) dt = λx(t) + f (t) x(to ) = xo ⎧ ⎨ ⎪ ⎩ ⎪ x(t) = eλ (t − to ) xo + to t ∫ f (τ)eλ ( t −τ )dτ (2) zero-input/source-free x(t) = eλ (t − to ) x0 iL(t) = e − R L ( t − to ) iL(to) vc (t) = e − t −t o RC vc(to ) 1. RL circuits with constant input (1) Circuit Equation KVL & Ohm’s Law: vL(t) = Vs − RiL (t) Inductor: vL(t) = L diL(t) dt L diL (t) dt = Vs − RiL(t) diL(t) dt = − R L iL(t) + Vs L Initial condition: iL(to ) ⎧ ⎨ ⎪ ⎩ ⎪ + -Vs R L ↓ iL(t) + vL(t) – 2 (2) Response iL(t) for t ≥ to. iL(t) = e − R L ( t − to ) iL(to) + to t ∫ Vs L ⋅e− RL ( t −τ )dτ = e − R L ( t − to ) iL(to) + Vs L e − R L t ⋅ L R e R L τ t to = e − R L ( t − to ) iL(to) + Vs R 1 − e − R L ( t − to )⎛ ⎝ ⎜ ⎞ ⎠ ⎟ iL(t) = Vs R + iL(to) − Vs R ⎛ ⎝ ⎞ ⎠ e − R L (t − to ) * t-to: elapsed time (3) Interpretation of iL(t). * τ = L R time constant * iL(∞) = Vs R (assuming τ = L R > 0 ) ∴iL (t) = iL (∞) + iL(to ) − iL(∞)[ ]e − t−to τ final initial final e − elapsed time time constant value value value (4) Comments: (i) if iL( to − ) ≠ iL(to + ), use iL( to +) . (ii) iL(∞) can be calculated directly by noting the L looks like a short circuit for a constant source at t = ∞. (iii) time constant can be computed using Rth and Leq. 5 Step 2 Find vc(0+) Step 3 Find vc(t) for t ≥ 0. Step 4 Find ic(t) for t ≥ 0. 6 Example 2 Find iL(t) for t>=0. Relevant time intervals to consider: Step 1: t ∈ Step 2: t ∈ 7 Step 3: t ∈ Step 4: t ∈