Chapter 6 Second-Order Circuit

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

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

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

§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)

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

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

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

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

and

S(t=0) Uo i i Uc(0)=U0 IL(0)=0 t O

非振荡放电 (过阻尼Overdamped Case) S(t=0) Uo i i Uc(0)=U0 IL(0)=0 t O 非振荡放电 (过阻尼Overdamped Case)

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

Then t O Uo i

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：

δ ω ω0  δ和ω决定衰减快慢

∵

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

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

等幅振荡（无阻尼） Undamped Case ωt L C + -

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

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)

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.

§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:

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

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

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。

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：