E1C03 09/14/2010
15:24:53 Page 93
of the input signal. The steady response of any system to which a periodic input of frequency, v, is
applied is known as the frequency response of the system. The frequency affects the magnitude of
amplitude B and also can bring about a time delay. This time delay, b 1 , is seen in the phase shift, FðvÞ, of
the steady response. For a phase shift given in radians, the time delay in units of time is
b 1 ¼
F
v
that is, we can write
sin vt þ F
ð
Þ¼sin v t þ
F
v
!
¼ sin v t þ b 1
ð
Þ
½
The value for b 1 will be negative, indicating that the time shift is a delay between the output and input
signals. Since Equation 3.9 applies to all first-order measuring systems, the magnitude and phase shift
by which the output signal differs from the input signal are predictable.
We define a magnitude ratio, MðvÞ, as the ratio of the output signal amplitude to the input signal
amplitude, MðvÞ ¼ B=KA. For a first-order system subjected to a simple periodic input, the
magnitude ratio is
MðvÞ ¼
B
KA
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ ðvtÞ
2
q
ð3:10Þ
The magnitude ratio for a first-order system is plotted in Figure 3.12, and the corresponding phase
shift is plotted in Figure 3.13. The effects of both system time constant and input signal frequency on
frequency response are apparent in both figures. This behavior can be interpreted in the following
manner. For those values of vt for which the system responds with values of MðvÞ near unity, the
measurement system transfers all or nearly all of the input signal amplitude to the output and with
very little time delay; that is, B will be nearly equal to KA in magnitude and FðvÞ will be near zero
degrees. At large values of vt the measurement system filters out any frequency information of the
2
1
0
Signal
y(t)
F(t)
1
Time, t
A
B
Figure 3.11 Relationship
between a sinusoidal input and
output: amplitude, frequency,
and time delay.
3.3 Special Cases of the General System Model 93
15:24:53 Page 93
of the input signal. The steady response of any system to which a periodic input of frequency, v, is
applied is known as the frequency response of the system. The frequency affects the magnitude of
amplitude B and also can bring about a time delay. This time delay, b 1 , is seen in the phase shift, FðvÞ, of
the steady response. For a phase shift given in radians, the time delay in units of time is
b 1 ¼
F
v
that is, we can write
sin vt þ F
ð
Þ¼sin v t þ
F
v
!
¼ sin v t þ b 1
ð
Þ
½
The value for b 1 will be negative, indicating that the time shift is a delay between the output and input
signals. Since Equation 3.9 applies to all first-order measuring systems, the magnitude and phase shift
by which the output signal differs from the input signal are predictable.
We define a magnitude ratio, MðvÞ, as the ratio of the output signal amplitude to the input signal
amplitude, MðvÞ ¼ B=KA. For a first-order system subjected to a simple periodic input, the
magnitude ratio is
MðvÞ ¼
B
KA
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ ðvtÞ
2
q
ð3:10Þ
The magnitude ratio for a first-order system is plotted in Figure 3.12, and the corresponding phase
shift is plotted in Figure 3.13. The effects of both system time constant and input signal frequency on
frequency response are apparent in both figures. This behavior can be interpreted in the following
manner. For those values of vt for which the system responds with values of MðvÞ near unity, the
measurement system transfers all or nearly all of the input signal amplitude to the output and with
very little time delay; that is, B will be nearly equal to KA in magnitude and FðvÞ will be near zero
degrees. At large values of vt the measurement system filters out any frequency information of the
2
1
0
Signal
y(t)
F(t)
1
Time, t
A
B
Figure 3.11 Relationship
between a sinusoidal input and
output: amplitude, frequency,
and time delay.
3.3 Special Cases of the General System Model 93