Lecture 19 •Review: •First order circuit step response •Steady-state response and DC gain •Step response examples •Related educational materials: –Chapter 7.5 First order system step response • Block diagram: y(0) = y0 A×u0(t) System y(t) • Governing differential equation and initial condition: First order system step response • Solution is of the form: • Initial condition: • Final condition: • Note on previous slide that we can determine the solution without ever writing the governing differential equation – Only works for first order circuits; in general we need to write the governing differential equations – We’ll write the governing equations for first order circuits, too – give us valuable practice in our overall dynamic system analysis techniques Notes on final condition • Final condition can be determined from the circuit itself • For step response, all circuit parameters become constant • Capacitors open-circuit • Inductors short circuit • Final conditions can be determined from the governing differential equation • Set and solve for y(t) 2. Checking the final response • These two approaches can be used to double-check our differential equation 1. Short-circuit inductors or open-circuit capacitors and analyze resulting circuit to determine y(t) 2. Set in the governing differential equation and solve for y(t) • The two results must match Steady-state step response and DC gain • The response as t is also called the steady-state response • The final response to a step input is often called the steady-state step response • The steady state step response will always be a constant • The ratio of the steady-state response to the input step amplitude is called the DC gain • Recall: DC is Direct Current; it usually denotes a signal that is constant with time DC gain – graphical interpretation • Input and output signals: • Block Diagram: y(0) = 0 A×u0(t) DC gain = System y(t) Dy Suggested Overall Approach 1. 2. 3. 4. Write governing differential equation Determine initial condition Determine final condition (from circuit or diff. eqn) Check differential equation • Check time constant (circuit vs. differential equation) • Check final condition (circuit vs. differential equation) 5. Solve the differential equation Example 1 • The circuit below is initially relaxed. Find vL(t) and iL(t) , t>0 • Determine initial AND final conditions on previous slide. Example 1 – continued • Circuit for t>0: Example 1 – checking results Example 1 – continued again • Apply initial and final conditions to determine K1 and K2 Governing equation: Form of solution: Example 1 – sketch response Example 1 – Still continued… • Now find vL(t). Example 2 (alternate approach to example 1) • Find vL(t) , t>0 Example 2 – continued Example 3 (still another approach to example 1) • Find iL(t) , t>0 Example 4 • For the circuit shown: determine: 1. The differential equation governing v(t) 2. The initial (t=0+) and final (t) values of v(t) 3. The circuit’s DC gain 4. C so that =0.1 seconds 5. v(t), t>0 for the value of C determined above Example 4 – Part 1 • Determine the differential equation governing v(t) Example 4 – Parts 2 and 3 Determine the initial and final values for v(t) and the circuit’s DC gain Example 4 – Checking differential equation • Governing differential equation (Part 1): • Final Condition (Part 2): Example 4 – Part 4 Determine C so that =0.1 seconds Example 4 – Part 5 Determine v(t), t>0 for the value of C determined in part 3 Governing equation, C = 0.01F: Form of solution: Initial, final conditions: ;