1. Chapter 7 – First-Order Circuits
In Chapters 1-5, when we apply KVL or KCL: v = Ri
In Chapter 7,
 Differential eqns.
First-Order Circuit:
One energy storage element (C or L)  1st order diff eq
Energy dissipation element/s (R)
Excitation (source of energy)
Energy initially stored in L or C  source-free
Vs or Is

2. Source-Free RC Circuit
Given: R & C, vC(0) = Vo
Find: v(t) and i(t)

3. Natural Response: behavior w/o external sources  exponential decay
Smaller time constant  faster decay

4. Example 1.
Given: vC(0) = 15 V
Find: vC, vx and ix

5. Example 2.
Given: Switch closed for a long time, and opened at t=0.
Find: vC, for t>0.

6. Source-Free RL Circuit
Given: R & L, iL(0) = Io
Find: i(t)

7. Example 3.
Given: i(0) = 10 A
Find: i(t), and ix(t)

8. Example 4.
Given: Switch open for a long time, and I is closed at t=0.
Find: i(t), io(t) and vo(t) for t>0.

9. Singularity Functions
either discontinuous, or have discontinuous derivatives.
useful in switching
Unit Step Function:

10. Example:

11. Unit Impulse Function:
An applied or resulting shock
A very short duration pulse.

12. Unit Ramp Function:

13. Step Response of an RC Circuit
-excitation is a step function
Given: v(0) = Vo;
Switch closed at t=0.

14. Vo – Initial voltage – from I.C. – “no jump” condition for vc
Complete Response = Natural Response
+ Forced Response
Complete Response = Steady-state Response
+ Transient Response
3 Parameters:
Vo – Initial voltage – from I.C. – “no jump” condition for vc
V∞ - steady-state response – at steady state, C becomes an open circuit
Time constant

15. Example 5.
Given: Switch has been at position A for a long time, and moves to B at t=0.
Find: v(t) for t>0.

16. Step Response of RL Circuits
Given: i(0) = Io;
Switch closed at t=0.

17. Example 6.
Given: Switch has been closed for a long time, and opens at t=0.
Find: vL(t) for t>0.

19. Ex. 7. Find i(t) for t >0.

20. First-Order Circuits with Op Amps
Method 1: Set up and solve ODE.
Method 2: Find RTH as seen by C or L – replace with test V or I.
Example: Given: R1 = 20kΩ, Rf = 80kΩ, C=5µF, and v(0)=3 V
Find: vo(t)
Method 1: See Lab #4 –
Op Amp differentiator

21. Method 2:
Find vo(0+):
Find RTH: