1. Chapter 6
Second-Order Circuit

2. What is second-order circuit?
A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements.
Typical examples of second-order circuits: a) series RLC circuit, b) parallel RLC circuit, c) RL circuit, d) RC circuit

3. Second-Order Circuit Zero-input Response
Second-Order Circuit Zero-state Response
Second-Order Circuit Complete Response

4. • Application: Filters + - u i E
• Application: Filters
–A lowpass filter with a sharper cutoff than can be obtained with an RC circuit.

5. §6-1 Zero-input Response of the Series RLC Circuit
An understanding of the natural response of the series RLC circuit is a necessary background for future studies in filter design and communications networks.
Assume:
S(t=0) i
Find :uc(t),i(t),uL(t)(t≥0)

6. characteristic equation : 特征方程
In general, a quadratic characteristic equation has two roots:

7. Four distinct possibilities:
Case A: If
two real, unequal negative roots        
Case B: If
two real, equal negative roots                 
Case C: If
two complex conjugate roots
Case D: If
two imaginary conjugate roots

8. Zero-input Response of the Series RLC circuit
Special case: Uc(0)=U0, IL(0)=0
过阻尼
there are two real, unequal negative roots
Overdamped Case

9. According to Initial Conditions :∴
S(t=0) i

10. and

11. S(t=0) Uo i i
Uc(0)=U IL(0)=0 t O

12. 非振荡放电 (过阻尼Overdamped Case)
S(t=0) Uo i i
Uc(0)=U IL(0)=0 t O
非振荡放电
(过阻尼Overdamped Case)

13. Uc(0)=U0, IL(0)=0 （2） 临界阻尼 There are two real, equal roots
Critically damped Case
According to the initial conditions:
Uc(0)=U0, IL(0)=0

14. Then t O Uo i

15. Underdamped Case （3） 欠阻尼 then The roots may be written as：
there are two complex conjugate roots             
when
then
The roots may be written as：

16. δ ω ω0 
δ和ω决定衰减快慢

17. ∵

18. i 欠阻尼 能量转换关系 0 <  t <  uC减小，i 增大 < t < - uC减小，i 减小R L C + -
0 <  t < 
uC减小，i 增大
< t < - R L C + -
uC减小，i 减小 Uo i
- < t <  R L C + -
|uC |增大，i 减小 O ωt
欠阻尼

19. + L C - u t U L C w ) 90
sin( = + \

20. 等幅振荡（无阻尼）
Undamped Case ωt L C + -

21. Note : (1) the behavior of such a network is captured by the idea of damping, which is the gradual loss of the initial stored energy. By adjusting the value of R , the response may be made undamped, overdamped, critically damped, or underdamped.
S(t=0) i

22. uC(t)
(2)Oscillatory response is possible due to the presence of the two types of storage elements. Having both L and C allows the flow of energy back and forth between the two.
overdamped response
critically damped response
underdamped response
(R=0 Undamped Case)

23. uC(t)
critically damped response
overdamped response
underdamped response
(3)The critically damped case is the borderline between the underdamped and overdamped cases and it decays the fastest.

24. §6-2 Second-order Circuit Complete Response
The general second-order linear differential equation with a step function input has the form
The complete response can be found by partitioning y(t) into forced and natural components:

25. yN(t) --- general solution of the homogeneous equation (input set to zero),
∴ yF=A/ao
yF(t) ---a particular solution of the equation
Combining the forced and natural responses
全响应=零输入响应 + 零状态响应

26. Example
Multisim analysis of this circuit
0.5H
t=0
600Ω/6KΩ i + _
24V uc
0.1µF
1KΩ

27. 学习要求
能根据给定的电路列写二阶动态电路的输入输出方程;
根据已知条件确定求解微分方程的初始条件;
能根据电路参数定性地判断R、L、C串联电路的零输入响应的几种放电类型。

28. Summary
1). 一阶电路是单调的响应，时间常数表示过渡过程的时间。
2). 二阶电路用三个参数 ， 和 0来表示动态响应。满足：