1. Lecture 20 (parts A & B) First order circuit step response
Nonzero initial conditions and multiple sources
Steady-state response and DC gain
Bias points and nominal operating conditions
Introduction to second order systems
Related educational materials:
Chapters 7.5, 8.1

2. First order system step response
Block diagram:
So far, we have considered only circuits which are initially relaxed  y(0) = 0
We now consider circuits with non-zero initial conditions

3. Example 1
The switch moves from A to B at time t=0
Find v(t), t>0

4. Sketch input function on previous slide

5. Example 1 – initial condition

6. Example 1 – Differential equation for t>0

7. Example 1 – Check , steady-state response

8. Example 1 – circuit response
Differential equation:
Initial, final conditions: ,
Form of solution:

9. Example 1 – sketch input, output

10. Alternate representation of example 1
The circuit of example 1 can be written as:
Now determine the response using superposition

11. Annotate previous slide to show input function

12. Example 1 – superposition approach Response to (constant) 2V source

13. Example 1 – superposition approach (cont’d) Response to 3V step input
Input-output equation:

14. Example 1 – superposition approach (cont’d) Response to 3V step input
Governing equation:
Form of solution:
Initial condition:
Final condition:

15. Example 1 – superposition approach (cont’d) Overall response

16. Note on overall approach
Both the input and output can be decomposed into a constant value and a time-varying value
It is sometimes convenient to analyze these components independently
For example, the DC gain of the system applies to both the constant input and the time varying input

17. Graphical interpretation
The system DC gain =

18. Why is this approach useful?
Decomposing the input and output into constant and time-varying components can simplify analysis and interpretation of results
The constant part of the input and output is the bias point or nominal operating point
The system dynamic response is often characterized by the time-varying part of the input-output relationship
A nonlinear system, for example, can be approximated as a linear system with a bias point

19. Introduction to second order systems
Second order systems are governed by second order differential equations
Input-output relation contains a second order derivative term, but no derivatives higher than second order
The physical system has two independent energy storage elements
The natural response of a second order system can oscillate with time (but doesn’t necessarily have to)
The response can overshoot its final value

21. Introduction to second order systems – continued
The oscillations in the natural response are due to energy being traded between the energy storage elements
Increasing energy dissipation reduces the amplitude of the oscillations (the system is said to be more highly damped)
If energy dissipation is above a critical value, the response will no longer oscillate
In general, increasing the energy dissipation will also cause the system to respond to changes more “slowly”

23. On previous slide, talk about damping and energy dissipation
Example: suspension system in car

24. Example: Series RLC circuit
Write the differential equation governing iL(t)

25. Series RLC circuit – continued

26. Example: Parallel RLC circuit
Write the differential equation governing vC(t)

27. Parallel RLC circuit – continued