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闭环系统的过冲和相位裕度关系的分

闭环系统的过冲和相位裕度关系的分
闭环系统的过冲和相位裕度关系的分

Overshoot as a Function of Phase Margin

J. C. Daly

Electrical and Computer Engineering

University of Rhode Island

4/19/03

Figure 1

Figure 2 Amplifier frequency response.

PM w

t /w

eq

Q%OS

55o0.700 0.925 13.3% 60o0.580 0.817 8.7% 65o0.470 0.717 4.7% 70o0.360 0.622 1.4% 75o0.270 0.527 0.008%

When an amplifier with a gain A(s) is put in a

feedback loop as shown in Figure 1, the closed loop gain, V o /V in = A CL

(1)

The system is unstable when the loop gain, ? A(s), equals -1. That is, ? A(s) has a magnitude of one and a phase of -180

degrees. An unstable system oscillates. A system close to being unstable has a large ringing overshoot in response to a step input.

The phase margin is a measure of how close the phase of the loop gain is to -180 degrees, when the magnitude of the loop gain is one. The phase margin is the additional phase required to bring the phase of the loop gain to -180 degrees. Phase Margin = Phase of loop gain - (-180). The loop gain has a dominant pole at

. Higher order poles can be

represented by an equivalent pole at . The amplifier is approximated by a function with two poles as shown in Equation 2.

(2)

Since for frequencies of interest where the loop gain magnitude is close to unity,

(3)

And,

(4)

(5)

Defining ,

Table I

? PM is the phase margin.

? w t is the unity gain frequency (rad/sec).

? w eq is the frequency of the equivalent higher order pole (rad/sec).

? Q is the system Quality factor.

? OS is the Over Shoot.

(6)

For frequencies of interest (frequencies close to the unity gain frequency), the amplifier gain can be written,

(7)

Plugging Equation 7 into Equation 1 results in the following expression for the closed loop gain.

(8)

Equation 8 is the transfer function for a second order system. The general form for the response of a second order system, where system properties are described by its Q and resonant frequence w o, is shown in Equation 10.

(9)

By comparing Equation 8 to Equation 9 we can get an expression for the resonant frequency and Q of the amplifier closed loop gain. (Equate coefficients of like powers of s in the dominators.)

(10)

(11)

The loop gain is the feedback factor ? multiplied by the amplifier gain A(s).

(12)

The phase margin is a function of the phase of the loop gain at the frequency where the magnitude of the loop gain is unity.

(13)

where is the loop gain unity gain frequency. It follows from Equations 12 and 13 that,

(14)

Also, solving for w ta and dividing by w eq,

(15)

It follows from Equations 11 and 15 that,

(16)

The phase of the loop gain (Equation 13) is.

Phase of loop gain(17)

The phase margin is the additional phase required to bring the phase of the loop gain to -180 degrees.

Phase Margin = Phase of loop gain - (-180).

Phase Margin(18)

A well known property of second order systems is that the percent overshoot is a function of the Q and is given by,

(19)

Both phase margin (Equation 18) and Q (Equation 16) are a function of w t / w eq. This allows us to use Equation 19 to create tables and plots of percent overshoot as a function of phase margin. As shown in Figures 3 and 4, and in Table I.

Figure 3 Overshoot as a function

of phase margin. Plot generated

using

MATLAB code.

Figure 4 Q as a function of phase margin. Plot generated using MATLAB code.

闭环系统的过冲和相位裕度关系的分

Overshoot as a Function of Phase Margin J. C. Daly Electrical and Computer Engineering University of Rhode Island 4/19/03 Figure 1 Figure 2 Amplifier frequency response. PM w t /w eq Q%OS 55o0.700 0.925 13.3% 60o0.580 0.817 8.7% 65o0.470 0.717 4.7% 70o0.360 0.622 1.4% 75o0.270 0.527 0.008%

When an amplifier with a gain A(s) is put in a feedback loop as shown in Figure 1, the closed loop gain, V o /V in = A CL (1) The system is unstable when the loop gain, ? A(s), equals -1. That is, ? A(s) has a magnitude of one and a phase of -180 degrees. An unstable system oscillates. A system close to being unstable has a large ringing overshoot in response to a step input. The phase margin is a measure of how close the phase of the loop gain is to -180 degrees, when the magnitude of the loop gain is one. The phase margin is the additional phase required to bring the phase of the loop gain to -180 degrees. Phase Margin = Phase of loop gain - (-180). The loop gain has a dominant pole at . Higher order poles can be represented by an equivalent pole at . The amplifier is approximated by a function with two poles as shown in Equation 2. (2) Since for frequencies of interest where the loop gain magnitude is close to unity, (3) And, (4) (5) Defining , Table I ? PM is the phase margin. ? w t is the unity gain frequency (rad/sec). ? w eq is the frequency of the equivalent higher order pole (rad/sec). ? Q is the system Quality factor. ? OS is the Over Shoot.

幅值裕量和相位裕量

一般来说,)(ωj G 的轨迹越接近与包围-1+j001j +-点,系统响应的震荡性越大。因此,)(ωj G 的轨迹对01j +-点的靠近程度,可以用来度量稳定裕量(对条件稳定系统不适用)。在实际系统中常用相位裕量和增益裕量表示。 Re Positive Phase Margin Negative Gain Margin Negative Stable System Unstable System (ωj G

64 ω Log ω Log ω Log ω Log ?-90? -270?-180Positive Gain Margin Positive Phase Margin Negative Gain Margin Negative Phase Margin Stable System Unstable System dB ? -90? -270?-1800 dB 图1 稳定系统和不稳定系统的相位裕度和幅值裕度 相位裕度、相角裕度(Phase Margin)γ 设系统的截止频率(Gain cross-over frequency)为c ω 1)()()(==c c c j H j G j A ωωω 定义相角裕度为 )()(180c c j H j G ωωγ+?= 相角裕度的含义是,对于闭环稳定系统,如果开环相频特性再滞后γ度,则系统将变为临界稳定。 当0>γ 时,相位裕量相位裕度为正值;当0<γ时,相位裕度为负值。为了使最小相位系统稳定,相位裕度必须为正。在极坐标图上的临界点为0分

贝和-180度。?-180 增益裕度、幅值裕度(Gain Margin)h 设系统的穿越频率(Phase cross-over frequency) πωωω?)12()()()(+== k j H j G x x x ,Λ,1,0±=k 定义幅值裕度为 ) ()(1 x x j H j G h ωω= 幅值裕度h 的含义是,对于闭环稳定系统,如果系统开环幅频特性再增大h 倍,则系统将变为临界稳定状态。 若以分贝表示,则有 )()(log 20)(x x j H j G dB h ωω-= 当增益裕度以分贝表示时,如果1>h ,则0)(>dB h 增益裕度为正值;如果1

相位裕度

闭环极点法是以系统左平面共轭复极点到原点斜率的倒数的绝对值β来判断系统稳定性的方法,β越大,系统就越稳定。在存在减幅振荡的时候,能较好的描述和量化系统的稳定性。 若一个闭环系统阶跃响应出现减幅振荡,系统的闭环传输函数必然会出现左平面共轭复数根s = σ±ωj,阶跃响应会出现一个衰减指数项,形式是K × exp(σt) × sin(ωt) , 可以看作一个衰减的指数项乘以一个正常的sin函数。 现在以图1的常见二级运放为例,说明如何在spectre中运用闭环极点法来分析运放的稳定性,在这里只调整电容的值来改变运放的稳定性, 对运放进行闭环AC和pole-zero分析,再print pole-zero summary就可以看到零极点了。下面先把图表公式全部列出来,然后再进行具体分析。 图1 常见二级运放 实数极点共轭复极点 左平面单调指数减幅(稳定)减幅震荡(可能不稳定,视情况而定) 右平面单调指数增幅(不稳定)增幅震荡(不稳定)

注释: β: 比例因子,σ / ω的绝对值 PM : 相位裕度 σ:闭环极点的实部,可以由spectre仿真得到 ω:闭环极点的虚部,可以由spectre仿真得到 公式1:u(t)= K1+K2 × exp(σt) × sin(ωt) , σ是减幅震荡的衰减因子,ω是减幅震荡的频率公式2 :ωT= 2π,T是减幅震荡的周期 公式3:σ= -1 / τ , τ为时间常数

图2 PM=45度时的阶跃响应 图3 PM=45度时的极点分布 首先来分析PM=45的情况,阶跃响应和闭环极点如图2和图3所示,系统出现了左平面上的共轭复根, 时域上出现了减幅振荡。肉眼能分辨的震荡包括三个上凸,两个下凹,最后一个上凸不很明显,合共2.5个振荡周期T,这可以说明什么呢?其实一旦出现减幅振荡,理论上再过10年,振荡也不会变为0 ,但无论是考虑到噪声也好,波形软件能够到达的精度也好,减幅振荡一旦衰减到一定的程度,例如1% ,就能够认为振荡消失了。可以尝试计算下经过一个振荡周期波形能衰减到多少。这里经过的时间为t=2.5T,由公式1 和公式2 及β=0.36 可得t=2.5T=2.5×0.36 ×2π×τ=5.7×τ , 就是说指数项经过5.7τ的衰减变为原来的exp(σ×5.7×τ)=0.3% ,这说明指数衰减到约0.3%后,减幅震荡就消失了。

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