Chapter 10 Frequency Response Techniques Frequency Response
Techniques
Slide 2
Sinusoidal frequency response: a. system; b. transfer function;
c. input and output waveforms The steady-state output sinusoid is
The system function is given by
Slide 3
Analytical Expression for Frequency response
Slide 4
Frequency response plots for G(s) = 1/(s + 2): separate
magnitude and phase Problem: Find the analytical expression for the
magnitude and phase frequency response for a system G(s)=1/(s+2).
Plot magnitude, phase and polar diagrams. Solution: substituting
s=jw, we get The magnitude frequency response is The phase
frequency response is The magnitude diagram is And the phase
diagram is
Slide 5
Frequency response plots for G(s)= 1/(s + 2): polar plot The
polar plot is a plot of For different values of w
Slide 6
Asymptotic Approximations: Bode plots Consider the transfer
function The magnitude frequency response is Converting the
magnitude response into dB, we obtain We could make an
approximation of each term that would consists only of straight
lines, then combine these approximations to yield the total
response in dB
Slide 7
a. magnitude plot; b. phase plot. Asymptotic Approximations:
Bode plots The straight-line approximations are called asymptotes.
We have low frequency asymptote and high frequency asymptote a is
called the break frequency Bode plots for G(s) = (s + a): At low
frequency when approaches zero The magnitude response in dB is
20log M=20log a where and is constant and plotted in Figure (a)
from =0.01a to a. At high frequencies where a The magnitude
response in dB is
Slide 8
Table 10.1 Asymptotic and actual normalized and scaled
frequency response data for (s + a)
Slide 9
Figure 10.7 Asymptotic and actual normalized and scaled
magnitude response of (s + a)
Slide 10
Figure 10.8 Asymptotic and actual normalized and scaled phase
response of (s + a)
Slide 11
Normalized and scaled Bode plots for a. G(s) = s; b. G(s) =
1/s;
Slide 12
Normalized and scaled Bode plots for c. G(s) = (s+a); d. G(s) =
1/(s+a);
Slide 13
Closed-loop unity feedback system Example Problem : Draw the
Bode plot for the system shown in figure where Solution: The Bode
plot is the sum of the Bode plot for each first order term. To make
it easeier we rewrite G(s) as
Slide 14
Bode log-magnitude plot for Example: a. components; b.
composite
Slide 15
Table 10.2 Bode magnitude plot: slop contribution from each
pole and zero in Example
Slide 16
Bode phase plot for Example: a. components; b. composite
Slide 17
Table 10.3 Bode phase plot: slop contribution from each pole
and zero in Example 10.2
Slide 18
Bode asymptotes for normalized and scaled G(s) = At low
frequencies G(s) becomes The magnitude M, in dB is therefore, At
high frequencies And The log-magnitude is The log-magnitude
equation is a straight line with slope 40dB/decade The low freq.
asymptote and the high freq. asymptote are equal when Thus is
called the break frequency for second order polynomial.
Slide 19
Bode asymptotes for normalized and scaled G(s) = a. magnitude;
b. phase
Slide 20
Table 10.4 Data for normalized and scaled log-magnitude and
phase plots for (s 2 + 2 n s + n 2 ). Mag = 20 log(M/ n 2 ) (table
continues) Freq. n Mag (dB) = 0.1 Phase (deg) = 0.1 Mag (dB) = 0.2
Phase (deg) = 0.2 Mag (dB) = 0.3 Phase (deg) = 0.3
Slide 21
Freq. n Mag (dB) = 0.5 Phase (deg) = 0.5 Mag (dB) = 0.7 Phase
(deg) = 0.7 Mag (dB) = 1 Phase (deg) = 1 Table 10.4 (continued)
Data for normalized and scaled log-magnitude and phase plots for (s
2 + 2 n s + n 2 ). Mag = 20 log(M/ n 2 )
Slide 22
Normalized and scaled log-magnitude response for
Slide 23
Scaled phase response for
Slide 24
Bode Plots for G(s) = 1/ Bode plots for can be derived
similarly to those for We find the magnitude curve breaks at the
natural frequency and decreases at a rate of -40dB/decade. The
phase plot is 0 at low frequencies. At 0.1 begins a decrease of -90
o /decade and continues until, where it levels off at -180 o.
Slide 25
Table 10.5 Data for normalized and scaled log-magnitude and
phase plots for 1/(s 2 + 2 n s + n 2 ). Mag = 20 log(M/ n 2 )
(table continues) Freq. n Mag (dB) = 0.1 Phase (deg) = 0.1 Mag (dB)
= 0.2 Phase (deg) = 0.2 Mag (dB) = 0.3 Phase (deg) = 0.3
Slide 26
Freq. n Mag (dB) = 0.5 Phase (deg) = 0.5 Mag (dB) = 0.7 Phase
(deg) = 0.7 Mag (dB) = 1 Phase (deg) = 1 Table 10.5 (continued)
Data for normalized and scaled log-magnitude and phase plots for
1/(s 2 + 2 n s + n 2 ). Mag = 20 log(M/ n 2 )
Slide 27
Normalized and scaled log magnitude response for
Slide 28
Scaled phase response for
Slide 29
Closed-loop unity feedback system Example Problem : Draw the
Bode plot for the system shown in figure where Solution: The Bode
plot is the sum of the Bode plot for each first order term, and the
2 nd order term. Convert G(s) to show the normalized components
that have unity low-frequency gain. The 2 nd order term is
normalized by factoring out forming Thus,
Slide 30
Bode magnitude plot for G(s) =(s + 3)/[(s + 2)(s 2 + 2s + 25)]:
a. components; b. composite The low frequency value for G(s), found
by letting s = 0, is 3/50 or -24.44dB The Bode magnitude plot
starts out at this value and continues until the first break freq
at 2 rad/s.
Slide 31
Table 10.6 Magnitude diagram slopes for Example
Slide 32
Bode phase plot for G(s) = (s + 3)/[(s +2)(s 2 + 2s + 25)]: a.
components; b. composite
Slide 33
Table 10.7 Phase diagram slopes for Example
Slide 34
Introduction to the Nyquist criterion The Nyquist criterion
relates the stability of a closed-loop system to the open-loop
frequency response and open-loop pole location. We conclude that
The poles of 1+G(s)H(s) are the same as the poles of G(s)H(s), the
open-loop system The zeroes of 1+G(s)H(s) are the same as the poles
of T(s), the closed-loop system
Slide 35
Mapping Contours concept If we take a complex number on the
s-plane and substitute it into a function, F(s), another complex
number results. This process is called Mapping. Consider the
collection of points, called a contour (shown in Figure as Contour
A). Contour A can be mapped through F(s) Into Contour B by
substituting each point in contour A into Function F(s) and
plotting the resulting complex numbers.
Slide 36
Examples of Contour Mapping If contour A maps clockwise then
contour B maps clockwise if: F(s) has just Zeros or Poles that are
not encircled by the contour The contour B maps counterclockwise
if: F(s) has just Poles that are encircled by contour A
Slide 37
Examples of Contour Mapping If the Pole or Zero of F(s) is
enclosed by contour A, then the mapping encircles the origin
Slide 38
Vector Representation of Mapping
Slide 39
Nyquist Contour is a contour which contains the imaginary axis
and encloses the right half- place. The Nyquist contour is
clockwise. A Clockwise Curve Starts at the origin. Travels along
imaginary axis till r = . At r = , loops around clockwise. Returns
to the origin along imaginary axis. The Nyquist Contour
Slide 40
a. Turbine and generator; b. block diagram of speed control
system for Example 10.4 Sketching a Nyquist diagram
Slide 41
Vector evaluation of the Nyquist diagram for Example 10.4 a.
vectors on contour at low frequency; b. vectors on contour around
infinity; c. Nyquist diagram The Nyquist Diagram is plotted by
substituting the points of the contour shown in Figure (a) into
G(s) = 500/[(s+1)(s+3)(s+10)] Its performing complex arithmetic
using vectors of G(s) drawn to points of the contour as shown in
(a) and (b) The resultant vector, R, found at any point is the
product of zero vectors divided by product of pole vectors as in
(c)
Slide 42
Vector evaluation of the Nyquist diagram for Example 10.4 a.
vectors on contour at low frequency; b. vectors on contour around
infinity; c. Nyquist diagram Evaluating G(s) at specified points we
get So from A to C the magnitude is finite goes from 50/3 to 0 and
the angle goes from 0 to -270 o From C to D the magnitude is 0
while there is an angle change of 540 o so goes from -270 o to 270
o
Slide 43
Vector evaluation of the Nyquist diagram for Example 10.4 The
mapping can be explained analytically by Substituting s= j and
evaluating G(j) at specified points Multiplying numerator and
denominator by the complex conjugate of denominator, we obtain At 0
freq G(j)=500/30= 50/3. Thus Nyquist diagram starts at 50/3 at an
angle 0. As increases, the real part remains positive while
imaginary part remains negative. At = 30/14, the real part becomes
negative. At =43 the Nyquist diagram crosses the negative real
axis, the value at the crossing is -0.874. Continuing toward = ,
the real part is negative and imaginary is positive. At = ,
G(j)500j/ 3 or approximately 0 at 90 o
Slide 44
Slide 45
Detouring around open-loop poles: a. poles on contour; b.
detour right; c. detour left
Slide 46
a. Contour for Example 10.5; b. Nyquist diagram for Example
10.5 Nyquist Diagram for open-loop function with poles on contour
Problem: Sketch the Nyquist diagram of unity feedback system where
G(s)= (s+2)/ S 2 Solution: The system two poles at origin are on
the contour and must be by-passed as in (a). Mapping DE is mirror
of AB
Slide 47
Applying the Nyquist Criterion to determine stability a.
contour does not enclose closed-loop poles; b. contour does enclose
closed-loop poles If a contour A, that encircles the entire right
half-plane is mapped through G(s)H(s), then the number of closed
loop poles, Z, in the right half- plane equals the number of
open-loop poles, P, that are in the right half-plane minus the
number of counterclockwise revolutions, N, around -1 of the
mapping; that is Z=P-N.
Slide 48
Demonstrating Nyquist stability: a. system; b. contour; c.
Nyquist diagram We could set K=1 and position the critical point at
-1/K rather than -1. Thus the critical point appears to move closer
to the origin as K increases. For this example P=2, and the
critical point must be encircled by the Nyquist diagram to yield
N=2 and a stable system. The system is marginally stable if the
Nyquist diagram intersect the real axis at -1 ( frequency at this
point is the frequency at which the root locus crosses the jw
axis.)
Slide 49
a. Contour for Example 10.6; b. Nyquist diagram Range of gain
for stability via Nyquist criterion Problem: for the unity feedback
system with G(s) = K/[s(s+3)(s+5)], find the range for stability,
instability and the value of gain for marginal stability. Solution:
first set K=1 and sketch Nyquist diagram for the system. Using the
contour shown in (a), for all points on the imaginary axis, Next
find the point where the diagram intersects the negative real axis
by setting the imaginary part equal to zero. We find. and real part
= -0.0083. Finally From the contour, P=0; for stability N must
equal zero. So the system is stable if the critical point lies
outside the contour. K can be increased by 1/0.0083 =120.5 before
the diagram encircles -1. Hence for stability K