Contents
Canonical Form of Feedback Network
Sensitivity of some simple transfer functions
Steady State Error and coefficients
Mason's Gain Formula |
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combination of 2 non-touching loops)
combination of 3 non-touching loops)
combination of 4 non-touching loops) …..
the
combination of 2 non-touching loops not touching the forward path)
combination of 3 non-touching loops not touching the forward path) …..
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Sensitivity
of a system with transfer function |
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Sensitivity w.r.t to magnitude and phase |
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Sensitivity overall |
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Unity feedback System: Here closed loop transfer function can be expressed as a function of open loop transfer function.
Sensitivity of
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This shows that a closed system is stable =1, even as open loop gain is infinity. |
Steady State
Error |
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This is direct application of final value theorem. |
Position Error Coefficient |
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Velocity Error Coefficient |
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Acceleration Error Coefficient |
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The type of open loop transfer function is important in determining error due to various signals
Type |
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1 |
2 |
3 |
4 |
Unit Step |
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Ramp |
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Parabolic |
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Type |
0 |
1 |
2 |
3 |
4 |
Unit Step |
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Ramp |
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Parabolic |
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Gain Margin : Reciprocal of gain at phase crossover frequency.
Phase
crossover frequency is the frequency at which phase is |
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Phase Margin : Amount of phase that can be added at gain crossover frequency.
Grain crossover frequency is the frequency at which gain is unity |
Phase Margin =
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Resonant Peak : It is the maximum value of the magnitude of the closed-loop transfer function. |
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Resonant Frequency : It is the frequency at which resonant peak occurs. |
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Cut-off rate : It is the frequency rate at which magnitude ratio changes after cut-off frequency |
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Bandwidth : |
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Delay time : Speed of response |
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1. NP Gives measure of stability of closed loop systems from open-loop transfer function.
2. NP is an extension of Polar plot.
3. Nyquist Plot is the mapping of Nyquist Path into function plane.
1.
In control system
analysis Nyquist Plot is drawn is for the function
2. is the denominator of the closed loop transfer function(CLTF) of
a system.
3. The roots(or zeros) of determines the stability of CLTF.
4. For stability of CLTF, the roots of must not lie in the right half s plane.
If P is the number of poles
of GH(s) inside the Nyquist Path and N is the number of encirclement of point
by the Nyquist Plot then the CLTF is stable if
is
zero where
is
the number of zeros of
that lie inside the Nyquist Path.
Mapping of |
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Plotting of |
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Analyticity of
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If
For e.g. |
If
the angle between the curves
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A Nyquist path
encloses the entire right half of the s-plane. If any pole(s) or zero(s) lie
on the
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Path components 2, 3
and 4 are common to any Nyquist Path but the number of appearances of path
component 1 depends on the number of poles or zeros on the ·
Nyquist Path is
mapped into
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Nyquist Path |
Here Nyquist path is the entire RHS s-plane. A pole lies in the LHS s-plane.
This pole doesn't lie inside Nyquist Path.
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Here
Nyquist path is the entire RHS s-plane with detours around the poles on
A pole and a zero lies in the RHS s-plane.
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Nyquist Plot |
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Nyquist Plot |
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Lead Network |
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Lag Network |
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