2.12 The frequency response diagram and the Bode Plot
2.12.1 Plotting the frequency response of the open-loop transfer function
Any transfer function can be divided into a number of basic first- or second-order factors which form the numerators or denominators of the element. The magnitude and phase re- sponse of the basic factors are simple to derive and recall. By combining the responses of the factors, the overall frequency response of the transfer function is generated.
G s
s jf G s
G j fH j f G s H s G j fH j f
r t f
f log
20logG j fH j f
jf = G j fH j f
52 Control systems techniques Ch. 2 Any transfer function (with ) contains factors, , of the following types in its numerator or denominator.
= =
(i) from , a scalar gain
(ii) from
(iii) from
(iv) from
Note the forms of the type (iii) and (iv) factors, i.e. the polynomial form , rather than pole-zero form .
Example 9
Find log-magnitude and phase of the open-loop transfer function:
. The log-magnitude and phase responses are:
; . ---
This example demonstrates that multiplication or division of the four types of factors be- come addition or subtraction of their log-magnitudes and of their phase contributions. This simplifies analysis because it involves simple addition or subtraction of the component terms; such operations are the basis for plotting manually the frequency response of transfer functions.
Let us consider the log-magnitude and phase plots of each of the factors as a func- tion of .
2.12.1.1 (i) A factor in the transfer function is a constant gain K.
The transfer function is , and the associated log-magnitude and phase are:
and if , (or if ) respectively. The
magnitude response is shown in Figure 2.12.
s = jf F j f
F j f F s
K K
jf
n sn
1+jfT
m 1+sTm
1 f
n ---
2
– j 2f
n ---
+
p 1 2 s
n --- s2
n2 ---
+ +
p
1+sa1+
s a+
G j H j K jf1+jfT ---
=
LM = 20logGH = 20logK–logjf –log1+jfT Arg GH = Arg K –Arg j f–Arg1+jfT
F j f
f log
F j f = K
LM = 20logK jf = 0 K0 jf = 180 K0
Sec. 2.12 Frequency response diagrams and Bode Plots 53
Figure 2.12 Transfer function of a constant gain K
Increasing or decreasing the gain K results in the plot of moving vertically up or down.
2.12.1.2 (ii) A factor in the transfer function contains pure differentiation or integration of multiplicity n
The transfer function is of the form: , . Differentiation is as- sociated with +n and integration with -n. The log magnitude of the transfer function is:
dB. (2.37)
This asymptote, when plotted against , is a straight-line having a slope . According to (2.37) it intersects the frequency axis when LM = 0 dB at , i.e. at rad/s. The phase response is , i.e. it is a constant for a given n over the entire frequency range. The log-magnitude and phase re- sponses are plotted in Figure 2.13.
Figure 2.13 (a) Magnitude and (b) phase responses of when n=1 2.12.1.3 (iii) A factor in the transfer function contains a real pole / zero of multiplicity n:
The transfer function is of the form: , . Its log magni-
tude is:
20logK LM (dB)
1 10 100 rad/s
0.1
(Horizontal line)
Frequency
20logK
F j f = jfn n = 1 2
LM = 20 log jfn = 20nlogf
f log 20n db/decade
f
log = 0 f = 1 Arg j fn = n90
-n = -1
f
0.1 1 10
+90
-90 LM Arg
+20
-20
(deg) 0
+n = +1
10 1
0.1
+n = +1
-n = -1
f
(a) (b) integration
integration differentiation
differentiation (dB)
j
n
F j f = 1+jfTn n = 1 2
54 Control systems techniques Ch. 2
dB. (2.38)
Consider the asymptotes for (a) a low frequency case when , (b) a high-frequency case when .
(a) , dB. This low-frequency asymptote is a horizontal
line when plotted against .
(b) , dB.
This log-magnitude asymptote, when plotted against , is a straight-line. The slope of the line is for each decade of frequency (i.e. for each unit). The low- frequency and high-frequency straight-line asymptotes of the log-magnitude plots are shown in Figure 2.14 for zeros (+n) or poles (-n). The actual plot is also drawn for a transfer func- tion over a frequency range about the so-called corner frequency where the high- and low-frequency asymptotes intersect; rad/s. The differences be- tween the actual plot and the straight-line asymptotes are easily remembered. In the case of multiple poles the actual log-magnitude plot is 3n dB down at the corner frequency and n dB down at an octave above and below the corner.
Figure 2.14 Magnitude response for a real pole or zero of order n.
Consideration of (2.38) reveals that for multiple zeros the actual log-magnitude plot and the straight-line asymptotes are the mirror image about the frequency axis of those for multiple poles as shown in Figure 2.14.
Consider now the plot of the phase shift for the transfer function . LM = 20logF = 20nlog1+2T21 2/
fT«1
fT»1
fT«1 LM = 20nlog1 = 0
f log
fT»1 LM = 20nlogfT = 20n logf+logT
f log 20n dB
log10 = 1
1 1 +jfTn c
c = 1T
c2
-n dB -3n dB
c1T
Poles:
2c log
LM
-n dB Zeros:
actual plot
slope 20n dB/dec
slope -20n dB/dec 0
(dB)
1+jfT
n
jf 1+jfTn
Sec. 2.12 Frequency response diagrams and Bode Plots 55
(a) for : , hence ;
(b) for : , hence ;
(c) for (corner): , hence .
The phase response for real poles of order n is shown in Figure 2.15. A straight-line approx- imation of the phase response is employed from a decade below the corner, phase , to a decade above the corner at . The straight-line approximation to the response, which passes through the corner frequency at , differs at most from the actual by over a decade in frequency on either side of the corner. The phase re- sponse for multiple zeros is the mirror image about the frequency axis of those for the mul- tiple poles shown in Figure 2.15.
Figure 2.15 Phase response for real poles of order n.
2.12.1.4 (iv) A factor in the transfer function contains a complex pair of poles or zeros of mul- tiplicity n:
The transfer function is of the form: .
The frequency responses of this factor for a single pair of complex poles ( ) are shown in Figure 2.16 for damping ratios . The responses are given for a normal- ised frequency , where is the undamped natural frequency. The straight-line ap- proximations which can be employed are crude and therefore the more accurate plots shown in the figure are used as templates when sketching the frequency responses of com- plex poles or zeros. Note that for a single pair of complex zeros the magnitude and phase plots are those shown in Figure 2.16 rotated a half-turn about their respective frequency ax- es; note that for the associated phase varies between zero and .
fT«1 = natanfT0 0
fT»1 = natanfT n90
fT = 1 = natan1 = n45
0 n
– 90
n – 45 5 to 6 (for n = 1
c2
c = 1T
2c log
Phase
actual plot 0
-n45 -n90
10c 0.1c
n5
n5.7
n5.7
n5
Straight line approximation
F j 1 f
n ---
2
– j 2f
n ---
+
n
=
n = –1 0.1 1
fn n
n = 1 180
56 Control systems techniques Ch. 2
Figure 2.16 Magnitude and phase responses as a function of the normalized frequency for a pair of complex poles.
Example 10. Lead compensation
Lead compensation is often employed as a more practical form of derivative compensation over a range of frequencies. In its application in power system dynamics and control it is used to provide phase lead over a desired range of frequencies; the design of cascade phase- lead compensation for conventional closed-loop control systems is covered in [1].
The form of the lead transfer function is , where is an adjusta- ble gain. The range of values for is typically when the transfer function is im- plemented using analog devices. For the phase lead is . For values of
ξ = 0.10 0.15 0.20
ξ = 0.25 0.30 0.40
ξ = 0.50 0.71 1.0
10−1 100 101
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20 0
Frequency (rad/s)
Phase Response (deg)
10−1 100 101
−30
−25
−20
−15
−10
−5 0 5 10 15 20
Magnitude Response (dB)
fn
A GLD s A 1+sT 1+sT ---
= A
0.1 1
= 0.1 56 0.1
Sec. 2.12 Frequency response diagrams and Bode Plots 57 the additional phase lead provided is small. For example, cascading two identical lead net- works with produces the same maximum phase shift as a single lead network with .
Figure 2.17 Frequency response for a lead transfer function.
The form of the frequency response of the lead compensator is shown in Figure 2.17; the important feature of this response is the phase lead introduced by the compensator. The maximum phase lead, , occurs at the geometric mean of its corners, , where . The maximum phase lead can be shown to be . Note that at the log-magnitude is .
--- 2.12.1.5 Exercise.
Show that the magnitude and phase plots of the lag block, ,
are those shown in Figure 2.17 reflected in their respective frequency axes. It is a practical form of integral compensation over a range of frequencies; identify that range.
--- Example 11. Plotting the frequency response
It is often useful to visualize or sketch the frequency response of a given transfer function based on the straight-line approximations to the frequency responses of the component fac- tors.
Draw the straight-line approximations to the frequency response of the following open-loop transfer function of a unity-feedback control system. Show both the straight-line asymptotes and the actual plot.
= 0.25
= 0.025
1
T--- 1
T---
f 0
Phase + 90
0
LM (dB) 20log1
m
f
(deg)
10 1
---
log
m
m m
m 1
T--- 1 T---
1 2 1 ---T
= =
m 1–
1+ ---
asin
= m LM 10 1
---
log
=
A GLG s A 1+sT 1+sT
--- , 1
=
58 Control systems techniques Ch. 2
. (2.39) Note the linear factors are in the form rather than form required for plot- ting the asymptotes. By dividing the denominator factors by 10 and 100 the form of (2.39)
is changed to ; note the gain is 10, or 20 dB. Set .
The corner frequencies of the transfer function are , i.e and rad/s for the two factors and , respectively. The straight-line representations for the four factors are shown in Figure 2.18. These are com- bined, making allowance for the deviations of the actual responses from the straight-line ap- proximations as shown in Figure 2.14 and Figure 2.15, to form the response of the transfer function.
Figure 2.18 Frequency response plots of the transfer function using straight-line approximations.
B s 104 s s +10s+100 ---
=
s a+
1+sT
B s 10
s1+s0.11+s0.01 ---
= s = jf
c = 1T 1 0.1 = 10
1 0.01 = 100 1+s0.1–1 1+s0.01–1
10−1 100 101 102 103
−100
−50 0 50
Magnitude Response (dB)
Gain x10 (20dB) 1/s 1/(1+0.1s) 1/(1+0.01s)
Transfer function plot Combined straight−line approximation
10−1 100 101 102 103
−250
−200
−150
−100
−50 0
Frequency (rad/s)
Phase Response (deg)
−180 deg
B s
Sec. 2.12 Frequency response diagrams and Bode Plots 59 2.12.2 Stability Analysis of the closed-loop system from the Bode Plot
The transfer function of the closed-loop system shown Figure 2.9 was derived in Section 2.7; the transfer function is .
As mentioned earlier, the important feature of the Bode Plot is that the stability of the closed-loop system can be derived from the plot of the open-loop transfer function
. The theoretical basis for this result requires that no poles or zeros of
lie in the right-half of the complex s-plane, i.e. it is open-loop stable and ‘minimum phase’
[1]. (If zeros of lie in the right-half of the s-plane the latter transfer function is called ‘non-minimum phase’.) The criterion for the stability of closed-loop systems based on the Bode Plot of open-loop stable transfer functions follows from the more generally appli- cable Nyquist criterion that covers both open-loop unstable and non-minimum phase sys- tems [1].
Assume the Bode Plot in Figure 2.19 is drawn in the vicinity of the gain cross-over frequen- cy, , for the open-loop transfer function . (This analysis may be carried out for the transfer function in Figure 2.18; satisfies the condition that it is open- loop stable and minimum phase.)
Figure 2.19 Gain and Phase Margins defined on the Bode Plot of the . It can then be shown that when the phase shift , the corresponding value of the LM must be negative for stability.The amount by which the gain can be increased before instability results, is called the ‘Gain Margin’.
W s
W s G s 1+G s H s ---
=
G s H s G s H s
G s H s
co G j fH j f
B s B s
-180 0 0
-180
|GH|
Gain Margin
Phase Margin
co is the “gain cross-over frequency” (rad/s)
co Log Magnitude Plot (dB)
Frequency Phase Plot (deg)
Arg GH
f
f
G s H s
f = –180
60 Control systems techniques Ch. 2
The term, ‘Phase Margin’ is defined as the difference between the line and the phase plot when the Log Magnitude Plot crosses the zero dB axis, i.e. when . The Phase Margin for a minimum-phase system must be positive for stability. The Phase Margin can also be interpreted as the amount of phase lag that can be introduced at unity loop-gain be- fore instability of the closed-loop system results.
Example 12. Derive the information on stability and dynamic performance of the closed-loop system from the Bode Plot
Assume that the transfer function in the Example 11 represents the open-loop transfer function of a unity gain feedback system, i.e. . The Gain and Phase Margins for this system are illustrated in Figure 2.20 for the Bode Plot for . The following information can be extracted from the Plot.
• Because the Gain and Phase Margins are positive the closed-loop system is stable.
• If the gain in the forward-loop is increased by 21 dB the closed-loop system becomes marginally stable and, ideally, it would oscillate with a constant amplitude at a fre- quency of 32 rad/s. Note that the gain-crossover frequency shifts to the right - increasing in frequency from 8 to 32 rad/s as the magnitude plot shifts vertically upwards.
• Further indicative data on the dynamic performance of the closed-loop system is revealed by the Phase Margin (PM ). A rule of thumb is, if the closed-loop system has a pair of dominant complex poles, the damping ratio of the closed-loop poles is approximately PM /100 [1].
For a good servo-system transient response, the Phase Margin should be about for a closed-loop system that has a dominant pair of complex poles. For such a Phase Margin it would be necessary to reduce the gain by 9.3 dB in the case of Figure 2.20. Note the gain crossover frequency is reduced to 3.3 rad/s; this implies that the frequency of the damped sinusoidal response to a step change at the input of the closed-loop system would also be reduced, possibly to 4 - 5 rad/s.
The significance of this example is that it illustrates how a variety of useful information for the design of the performance of the closed-loop system can be derived from the Bode Plot of the open-loop system.
---
180
– LM = 0
B s
B s = G s H s = G s
G s H s
70
Sec. 2.13 The Q-filter, a passband filter 61
Figure 2.20 Bode Plot of the open-loop transfer function showing the gains and phase margins.
As stated, in applications to power system analysis one should be aware of open-loop sys- tems that are non-minimum phase when using the Bode Plot for stability analysis. An exam- ple of such a case is the model of a Francis turbine in a hydro-electric plant. This model contains a right-half plane zero (which causes the turbine power output to rise initially as the wicket gates are closed).