当前位置:首页 >> 信息与通信 >>

Freq Resp


Frequency Response and Bode Plots
The frequency response of a system is obtained by applying a sinusoidal input and observing the steady-state sinusoidal output. The magnitude of the system gain and the phase angle with respect to the input are recorded for different input frequencies. For a closed loop system with G(s) in the forward path and H(s) in the feedback, the open loop frequency response is the main one of interest. For Laplace transfer functions, the frequency response can be obtained by letting s = j? where ? is frequency in rads / s. Hence, the open loop frequency response is

G( j? )H ( j? ) ? G( j? )H ( j? )

G( j? )H ( j? )

Bode Diagrams are graphs of the magnitude in dB, 20log10 G( j?)H ( j?) dB, and the phase angle ? in degrees plotted against the frequence ?. Plots are usually made on log-linear graphs. (A frequency ratio of 10:1 is a decade, 2:1 is an octave)? Example
? 1 2 2? ? K? s ? s ? 1? ?? 2 ? ?n n ? ? G ( s) H ( s) ? u v s (T1s ? 1) (T2 s ? 1) t
w

w,u, t, v, are 0, –ve or +ve integers. Let s = j?:
w

? 1 ? 2? K? ( j? ) 2 ? j? ? 1? ?? 2 ? ?n n ? G ( j ? ) H ( j? ) ? ? u v ( j? ) ( j?T1 ? 1) ( j?T2 ? 1) t
? 20 log10 G ( j? ) H ( j? ) ? 20 log10 K ? w20 log10 1
2 ?n

( j? ) 2 ?

2?

?n

j? ? 1 ? u 20 log10 ?

?

? v 20 log20 ( j?T1 ? 1)1/ 2 ? t 20 log20 ( j?T2 ? 1)1/ 2

The complete magnitude plot is obtained by adding plots of each factor.

? ? w tan?1

2?? / ? n 1 ? (? / ? n )
2

? u90o ? v tan?1 ?T1 ? t tan?1 ?T2

Overall phase shift is the sum of the individual phase shift terms. 4 basic terms in a transfer function for frequency response.

1. Constant gain, K magnitude is 20 log10 | K | dB. Zero phase shift, ? = 0? 2. s ? 1 or (j?) ? 1 . Mag. is ? 20 log10 ? dB, which is a straight line of slope ? 20 dB / decade passing through 0 dB at ? = 1. Constant phase shift ? = ? 90o .?

Bode Diagrams
40 20

s

Phase (deg); Magnitude (dB)

0

1/s
-20 -40

80 35 -10 -55

s

1/s
-100 -1 10 10
0

10

1

10

2

Frequency (rad/sec)

3. (Ts+1) ? 1 or (j????) ? 1 . . Mag. is ? 20 log10 [1+(?T)2 ]1/2 dB. straight line asymptotes : For ?T << 1, Mag. ? ? 20 log10 1 = 0 dB

Phase angle ? = ? tan-1 ?? .?

This exponential lag (-1) or lead (+1) transfer function has Bode plots which can be represented by

For ?T >> 1, Mag. ? ? 20 log10 ?T which has a slope of ? 20 dB / decade. These mag. asymptotes meet at the point where 20log10 ?T = 0, i.e. ? = 1/T, which is called the corner or break frequency. The max. error with this approximation is 3 dB and it occurs at the break frequency ? = 1/T .

The phase angle plot can be made directly or by a three straight line approximation: For ? < 1/10T For ? > 10/T ?=0 ? = ? 90o
o

For 1/10T < ? < 10/T, a straight line, slope ? 45o per decade passing through ? = 45 break frequency ? = 1/T rad/s.

at the

This linear approximation, which passes through the correct phase at the break frequency, is within 6o of the actual phase curve for all frequencies.?

Bode Plot of 1/(s+1) 10 0 -10 -20 -30 -40 -50 -2 10 10
-1

Magnitude (dB)

10

0

10

1

10

2

0

Phase (deg)

-20 -40 -60 -80 -100 -2 10 10
-1

10 Frequency (rad/sec)

0

10

1

10

2

Bode Plot of (s+1) 50 40 30 20 10 0 -10 -2 10 100 80 10
-1

Magnitude (dB)

10

0

10

1

10

2

Phase (deg)

60 40 20 0 10
-2

10

-1

10 Frequency (rad/sec)

0

10

1

10

2

4. [ (s2/??n ) + (2? ??n) +1 ]

?1

or [ (j?/?n)2 + j2????n) +1 ]

?1

. See plots for

(-1)

. Normalised

second order responses with normalised frequency ?/?n . Mag. has 0 dB for low ?, slopes off at ? 40 dB / decade at high frequency and has a peak value at approx. ?/?n =1. Phase is 0 for low ?, ? 180o for high ? and passes through ? 90o at ?/?n = 1.?

Gain and Phase Margins The Nyquist plot is a polar plot of gain | GH(s) | and phase ? . In a basic Nyquist plot, the Nyquist stability theorem states that if the plot includes the point (–1,0) the system will be unstable in closed loop. The point (–1,0) is the point where the gain | GH(s) | = 1 or 0 dB and ? = –180?. Hence, this point can be used to measure stability margins on a Bode plot. The gain margin GM occurs when ? = –180? and is the amount by which the gain can be increased before the system becomes closed loop unstable. The phase margin PM occurs when the magnitude is 0 dB and is the amount of additional phase lag that can be tolerated before closed loop instability. ?

Gain and Phase Margins Gm=10.458 dB (at 1 rad/sec), Pm=70.772 deg. (at 0.31432 rad/sec)
50

GM
0

Phase (deg); Magnitude (dB)

-50

-100 -50 -100 -150 -200 -250 -300 -2 10
-1 0 1

-180

PM

10

10

10

Frequency (rad/sec)

Frequency Response Design Relations Usually assume closed loop transfer function can be represented as 2nd order. ?after a design performance should be verified by simulation. 1. For ? ? 0.7, PM ? 100 ? degrees. 2. Closed loop bandwidth, ?B. Given ?n, estimate from 2nd order normalised curves at – 3 dB, i.e. ?B / ?n = ??? ??B = ???. ?B can also be (cautiously) estimated from open loop frequency response when mag. is ? – 6 dB.



相关文章:
04 附录MATLAB工具箱函数
freqint freqint2 freqresp freqs freqspace freqz freqz2 fsamp2 隶属度计算 取样响应函数 绘制快速伯德图 利用模糊 C 平均聚集方法找出簇 将两个系统反馈连接 ...
matlab 常用函数汇总
Freqint Bode 图的自动定范围算法 Freqint2 Nyquist 图的自动定范围算法 Freqresp 低级频率响应函数 Givens 旋转 Housh 构造 Householder 变换 Imargin 利用内插...
MATLAB工具箱函数
freqint freqint2 freqresp freqs freqspace freqz freqz2 fsamp2 隶属度计算 取样响应函数 绘制快速伯德图 利用模糊 C 平均聚集方法找出簇 将两个系统反馈连接 ...
重大信号与系统作业
如要投诉违规内容,请到百度文库投诉中心;如要提出功能问题或意见建议,请点击此处进行反馈。 重大信号与系统作业 信号与系统信号与系统隐藏>> Signal and System ...
第三章 系统频率特性
三、频率响应计算函数 MATLAB 控制工具箱中,函数 FREQRESP 用于计算 LTI 系 统的频率响应,它既适用于连续时间系统,也适用于离散时间系 统;既适用于 SISO 系统,...
Si47xx编程指南(部分翻译)
READFREQH[7:0] READFREQL[7:0] RSSI[7:0] SNR[7:0] READANTCAPH[7:0] READANTCAPL[7:0] RESP 1 1 1 1 BIT 7 6:2 1 0 Name BLTF ...
LabView 2011 examples(所有例程文件名)
freqresp.llb apps\tankmntr.llb apps\tempsys.llb apps\testseq.llb comm\access.llb comm\actxpp.llb comm\axevent.llb comm\Bluetooth.llb comm\Calling ...
HyperMesh载荷集解释-Nastran
(vibration or buckling) analysis with the Lanczos method FREQ Defines a set of frequencies to be used in the solution of frequency response problems FREQ...
MATLAB命令大全
Dexresp Dfrqint Dfrqint2 Dmulresp Distsl Dric Dsigma2 Dtimvec Exresp Freqint Freqint2 Freqresp Givens Housh Imargin Lab2ser 喷气式飞机偏航阻尼的...
频响分析
则用 DLOAD 组合 6.loadstep type=freq.resp(direct) SPC——>SPC DLOAD——>RLOAD2 FREQ——>OMEGA 7.定义 set type=SET_GRID 6.control cards ...
更多相关标签: