E1C03 09/14/2010
15:24:54 Page 106
3.5 PHASE LINEARITY
We can see from Figures 3.16 and 3.17 that systems having a damping ratio near 0.7 possess the broadest
frequency range over which M(v) will remain at or near unity and that over this same frequency range the
phase shift essentially varies in a linear manner with frequency. Although it is not possible to design a
measurement system without accepting some amount of phase shift, it is desirable to design a system such
that the phase shift varies linearly with frequency. This is because a nonlinear phase shift is accompanied by
a significant distortion in the waveform of the output signal. Distortion refers to a notable change in the
shape of the waveform from the original, as opposed to simply an amplitude alteration or relative phase
shift. To minimize distortion, many measurement systems are designed with 0.6 z 0.8.
Signal distortion can be illustrated by considering a particular complex waveform represented
by a general function, u(t):
u t
ð Þ ¼
X 1
n¼1
sin nvt ¼ sin vt þ sin 2vt þ Á Á Á
ð3:31Þ
Suppose during a measurement a phase shift of this signal were to occur such that the phase shift
remained linearly proportional to the frequency; that is, the measured signal, v(t), could be
represented by
v t
ð Þ ¼ sin vt À F
ð
Þþsin 2vt À 2F
ð
ÞþÁÁÁ
ð 3:32Þ
Or, by setting
u ¼ vt À F
ð
Þ
ð3:33Þ
we write
v t
ð Þ ¼ sin u þ sin 2u þ Á Á Á
ð3:34Þ
We see that v(t) in Equation 3.22 is equivalent to the original signal, u(t). If the phase shift were not
linearly related to the frequency, this would not be so. This is demonstrated in Example 3.11.
Example 3.11
Consider the effect of the variations in phase shift with frequency on a measured signal by
examination of the signal defined by the function
u t
ð Þ ¼ sin t þ sin 5t
Suppose this signal is measured in such a way that a phase shift that is linearly proportional to the
frequency occurs in the form
v t
ð Þ ¼ sin t À 0:35
ð
Þþsin 5t À 5 0:35
ð
Þ
½
Both u(t) and v(t) are plotted in Figure 3.21. We can see that the two waveforms are identical except
that v(t) lags u(t) by some time increment.
Now suppose this signal is measured in such a way that the relation between phase shift and
frequency was nonlinear, such as in the signal output form
w t
ð Þ ¼ sin t À 0:35
ð
Þþsin 5t À 5
ð
Þ
The w(t) signal is also plotted in Figure 3.21. It behaves differently from u(t), and this difference is
the signal distortion. In comparison of u(t), v(t), and w(t), it is apparent that distortion is caused by a
nonlinear relation between phase shift and frequency.
106 Chapter 3 Measurement System Behavior
15:24:54 Page 106
3.5 PHASE LINEARITY
We can see from Figures 3.16 and 3.17 that systems having a damping ratio near 0.7 possess the broadest
frequency range over which M(v) will remain at or near unity and that over this same frequency range the
phase shift essentially varies in a linear manner with frequency. Although it is not possible to design a
measurement system without accepting some amount of phase shift, it is desirable to design a system such
that the phase shift varies linearly with frequency. This is because a nonlinear phase shift is accompanied by
a significant distortion in the waveform of the output signal. Distortion refers to a notable change in the
shape of the waveform from the original, as opposed to simply an amplitude alteration or relative phase
shift. To minimize distortion, many measurement systems are designed with 0.6 z 0.8.
Signal distortion can be illustrated by considering a particular complex waveform represented
by a general function, u(t):
u t
ð Þ ¼
X 1
n¼1
sin nvt ¼ sin vt þ sin 2vt þ Á Á Á
ð3:31Þ
Suppose during a measurement a phase shift of this signal were to occur such that the phase shift
remained linearly proportional to the frequency; that is, the measured signal, v(t), could be
represented by
v t
ð Þ ¼ sin vt À F
ð
Þþsin 2vt À 2F
ð
ÞþÁÁÁ
ð 3:32Þ
Or, by setting
u ¼ vt À F
ð
Þ
ð3:33Þ
we write
v t
ð Þ ¼ sin u þ sin 2u þ Á Á Á
ð3:34Þ
We see that v(t) in Equation 3.22 is equivalent to the original signal, u(t). If the phase shift were not
linearly related to the frequency, this would not be so. This is demonstrated in Example 3.11.
Example 3.11
Consider the effect of the variations in phase shift with frequency on a measured signal by
examination of the signal defined by the function
u t
ð Þ ¼ sin t þ sin 5t
Suppose this signal is measured in such a way that a phase shift that is linearly proportional to the
frequency occurs in the form
v t
ð Þ ¼ sin t À 0:35
ð
Þþsin 5t À 5 0:35
ð
Þ
½
Both u(t) and v(t) are plotted in Figure 3.21. We can see that the two waveforms are identical except
that v(t) lags u(t) by some time increment.
Now suppose this signal is measured in such a way that the relation between phase shift and
frequency was nonlinear, such as in the signal output form
w t
ð Þ ¼ sin t À 0:35
ð
Þþsin 5t À 5
ð
Þ
The w(t) signal is also plotted in Figure 3.21. It behaves differently from u(t), and this difference is
the signal distortion. In comparison of u(t), v(t), and w(t), it is apparent that distortion is caused by a
nonlinear relation between phase shift and frequency.
106 Chapter 3 Measurement System Behavior