Page 93 - Mechatronics with Experiments

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6. Repeat until w = w max where w max is the maximum frequency of interest, by incre-
menting w = w +Δw. Δw is the increment of frequency as the experiment sweeps
the frequency range from w to w max .
0
7. Plot B∕A, versus w.
8. Curve fit to B∕A and as function of w and obtain a mathematical expression for the
frequency response as a rational function.

2.9.1 Graphical Representation of Frequency Response

The frequency response of a dynamic system is conveniently represented by a complex
function of frequency. The complex function can be graphically represented in many
different ways. In control systems studies, the three most commonly known representations
are:

1. Bode Plots: plot 20 log |G(jw)| v.s. log (w) and Phase(G(jw)) v.s. log (w).
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2. Nyquist Plots (polar plots): plot Re(G(jw)) v.s. Im(G(jw)) on the complex G(jw) plane
where w-frequency is parameterized along the curve.
3. Log Magnitude versus Phase Plot: plot the 20 log (G(jw)) (y-axis) versus
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Phase(G(jw)(x-axis) and w-frequency is parameterized along the curve.
One can choose to graphically plot the complex frequency response function in many
other ways. The above three representations are the most common ones. With the aid of
CAD-tools, it is a very simple task to plot a given frequency response in any one of the
above forms. However, the ability to plot basic building blocks of transfer functions by
hand sketches still remains a powerful tool in design.
Therefore, we will discuss the manual plotting of various transfer functions next. Let
us consider a general transfer function which has
1. DC gain,
2. zeros and poles at the origin,
3. first-order zeros and poles,
4. second-order zeros and poles.

2
Π(s∕z + 1)Π((s∕w ) + 2 (s∕w ) + 1)
zi
zi
i
zi
G(s) = K 0 (2.45)
2
s ±N Π(s∕p + 1)Π((s∕w ) + 2 (s∕w ) + 1)
pi
pi
pi
i
The rest of the graphical plotting discussions will consider this general form of the transfer
function.
Bode Plots Given a frequency response data either in explicit mathematical form as
G(jw) or as raw experimental data as magnitude and phase information B∕A = |G(jw)| and
(w), one possible graphical represenation is as two plots:
Plot 1, y-axis: 20 log |G(jw)|, x-axis: log w,

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Plot 2, y-axis: Phase(G(jw)), x-axis: log w.

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Such graphical representation is called a Bode plot. The Bode plot is the most commonly
used graphical representation of frequency response information. Let us consider
= G(jw) = |G(jw)|e jψ(w) (2.46)
G(s)| s=jw