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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.


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