2 lec Eng
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Frequency-Response Analysis
Introduction
Frequency-response methods were developed in 1930s and 1940s by Nyquist, Bode, Nichols, and many others. The frequency-response methods are very powerful in control theory.
g(t)
Input signal
System
y(t)
Output signal
By the term frequency response, we mean the steady-state response of a system to a sinusoidal input.
In frequency-response methods, we vary the frequency of the input signal over a certain range and study the resulting response.
Introduction
Let a system is described by the Stable aperiodic link:
g t Ain sin t |
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Ts 1 |
Transient part
Steady-state part
Advantages
•One advantage of the frequency-response approach is that we can use the data obtained from measurements on the physical system without deriving its mathematical model.
•frequency-response tests are, in general, simple and can be made accurately by use of readily available sinusoidal signal generators and precise measurement equipment.
•Often the transfer functions of complicated components can be determined experimentally by frequencyresponse tests.
•Stability and Performance analysis.
Frequency Characteristics
g(t)
Input signal
System
y(t)
Output signal
If the input g(t) is a sinusoidal signal, the steady-state output will also be a sinusoidal signal of the same frequency, but with possibly different magnitude and phase
angle: |
g t Ain sin t |
Ain |
Aout |
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yss t Aout sin t
Frequency Characteristics
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g(t) |
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y(t) |
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System |
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Input signal |
Output signal |
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g t Ain sin t |
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yss t Aout sin t |
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Aout Aout |
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Aout |
- Magnitude frequency characteristic (amplitude |
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H Ain |
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ratio of the output sinusoid to the input sinusoid) |
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- Phase frequency characteristic (phase shift of the |
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output sinusoid with respect to the input sinusoid) |
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A positive phase angle is called phase lead, and a negative phase angle is called phase lag.
Mathematical Interpretation of
the Frequency Characteristics
g t Ain sin t |
W s |
yss t Aout sin t |
After waiting until steady-state conditions are reached, the frequency response can be calculated by replacing s in the transfer function by jω.
s j |
W j u jv H e j |
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where H |
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arctan |
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u2 v2 |
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The Transfer function in the frequency domain is a complex quantity, and can be represented by the magnitude and phase angle with frequency as a parameter.
Mathematical Interpretation of
the Frequency Characteristics
g t Ain sin t W s yss t Aout sin t
s j
W j H e j
yss t Ain H sin t
Nyquist diagram (polar plot)
W j u jv H e j
The polar plot (Nyquist
Re W j diagram) is the locus of vectors W(jω) as ω is varied from zero to infinity.
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Each point on the polar plot |
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of W(jω) represents the terminal |
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Im W j |
point of a vector at a particular |
W j |
value of ω. In the polar plot, it is |
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important to show the frequency |
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graduation of the locus. |
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The projections of W(jω) on |
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the real and imaginary axes are |
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its real and imaginary |
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components. |
Note that in polar plots a positive (negative) phase angle is measured counterclockwise (clockwise) from the positive real axis.
Bode Diagrams (Logarithmic Plots)
A Bode diagram consists of two graphs:
• a plot of the logarithm of the magnitude of a sinusoidal transfer function;
• a plot of the phase angle.
Both are plotted against the frequency on a logarithmic
scale.