Unit 4
Design in frequency
domain
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Bode plot
The Bode plot or the Bode diagram consists of two plots −
• Magnitude plot
• Phase plot
In both the plots, x-axis represents angular frequency (logarithmic scale). Whereas,
yaxis represents the magnitude (linear scale) of open loop transfer function in the
magnitude plot and the phase angle (linear scale) of the open loop transfer
function in the phase plot.
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Procedure for Bode plot
Step 1: Identify the corner frequency.
Step 2: Draw the asymptotic magnitude plot. The slope will change at each corner frequency by
+20 db / dec. for zero and - 20 db / dec for pole. For complex conjugate pole and zero theslope
will change by 40 db/ decade.
Step 3:
(i)
For type zero system draw a line upto first ( lowest) corner frequency having 0 dbdec.
slope.
(ii) For type one system draw a line having slope - 20 db/ dec. upto u = K. Mark first(lowest)
corner frequency.
(iii) For type two system draw the line having slope -40 db/ dec upto e =K and so on.Mark first
corner frequency.
Step 4: Draw a line upto second corner frequency by adding the slope of next pole or zero to
theprevious slope and so on.
Step 5: Calculate phase angle for different values of o from the equation 4.10 and join all points.
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Phase margin and gain margin on Bode
plot
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Polar plot or Nyquist plot
• The polar plot of a sinusoidal transfer function 𝐺(𝑗𝜔) is a plot of the magnitude
of 𝐺(𝑗𝜔) versus the phase angle of 𝐺(𝑗𝜔) on polar coordinates as v is varied
from zero to infinity.
• Thus, the polar plot is the locus of vectors 𝐺 𝑗𝜔 ∠𝐺(𝑗𝜔) as 𝜔 is varied from
zero to infinity.
• In both the plots, x-axis represents angular frequency (logarithmic scale).
Whereas y-axis represents the magnitude (linear scale) of open loop transfer
function in the magnitude plot and the phase angle (linear scale) of the open
loop transfer function in the phase plot.
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Phase margin and gain margin on Polar
plot
Phase Margin
The phase margin is that amount of additional
phase lag at the gain cross-over frequency
required to bring the system to the verge of
instability.
∅𝑚 = 180 + ∅
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Phase margin and gain margin on Polar plot contd.
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Phase margin and gain margin on Polar plot contd.
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Relation between Damping ratio and Phase
Margin
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Relation between Damping ratio and Phase Margin
contd.
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Relation between Damping ratio and Phase Margin
contd.
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Transient Response via Gain Adjustment
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Transient Response via Gain Adjustment
contd.
• In previous section, the relationship between damping ratio (equivalently
percent overshoot) and phase margin was derived for
• Thus, if we can vary the phase margin, we can vary the percent overshoot.
Looking at Figure 11.1, we see that if we desire a phase margin, ΦM,
represented by CD, we would have to raise the magnitude curve by AB.
Thus, a simple gain adjustment can be used to design phase margin and,
hence, percent overshoot.
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Transient Response via Gain Adjustment contd.
Design Procedure
1. Draw the Bode magnitude and phase plots for a convenient value of gain.
2. Determine the required phase margin from the percent overshoot.
3. Find the frequency, ωΦM , on the Bode phase diagram that yields the desired phase
margin, CD, as shown on Figure 11.1.
4. Change the gain by an amount AB to force the magnitude curve to go through 0
dB at ωΦM . The amount of gain adjustment is the additional gain needed to produce
the required phase margin.
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Lag compensators
The function of the lag compensator as seen on Bode diagrams is to
(1) improve the static error constant by increasing only the lowfrequency gain without any resulting instability, and (2) increase the
phase margin of the system to yield the desired transient response.
The transfer function of the lag compensator is
where α > 1
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Lag compensators contd.
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Lag compensators contd.
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Lead compensators
The lead compensator increases the bandwidth by increasing the gain
crossover frequency. At the same time, the phase diagram is raised at
higher frequencies. The result is a larger phase margin and a higher
phase-margin frequency.
where β < 1
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Lead compensators contd.
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Lead Compensator Frequency Response
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Lead Compensator Frequency Response contd.
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Lead Compensator Frequency Response contd.
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Lead compensators contd.
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Lead compensators contd.
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Lag Lead compensators
we designed lag-lead compensation to improve the transient response and steady-state error. One
method is to design the lag compensation to lower the high-frequency gain, stabilize the system, and
improve the steady-state error and then design a lead compensator to meet the phase-margin
requirements.
where γ > 1. The first term in parentheses produces the lead compensation, and the second
term in parentheses produces the lag compensation. The constraint thatwemust follow here
is that the single value γ replaces the quantity α for the lag network in Eq. (11.2) and the
quantity β for the lead network in Eq. (11.6). For our design, α and β must be reciprocals
of each other.
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Lag-Lead compensators contd.
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Lag-Lead compensators contd.
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