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number, such that C 2 ! 4bC 1 =a
2 , and the resistors are then found by
R 1;2 ¼
aC 2 Ç
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðaC 2 Þ
2 À 4bC 1 C 2
q
4pf c C 1 C 2
ð6:70Þ
The filter cutoff frequency is set by
f c ¼
1
2p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
R 1 R 2 C 1 C 2
p
ð6:71Þ
The unit-gain Sallen–Key high-pass filter is shown in Figure 6.36b. The design switches the
resistors and capacitors from the low-pass filter. For high-pass filters, we can simplify the design and
set C 1 ¼ C 2 ¼ C. The high-pass filter transfer function is
GðsÞ ¼
1
1 þ a=s þ b=s 2
ð6:72Þ
where a ¼ 2=v c CR 1 and b ¼ 1=v
2
c C
2
R 1 R 2 . Values for a and b to meet Butterworth or Bessel filter
characteristics are given in Table 6.3 with the resistor values set by R 1 ¼ 1=pf c aC and
R 2 ¼ a=4pf c bC.
At some frequency, the reactive elements in a filter circuit resonate, leading to a peak in the
response behavior. This is behavior predicted by its Q-factor, an inverse damping ratio-like
parameter. For Q ¼ 1=2 response is critically damped, for Q < 1=2 response is overdamped,
and for Q > 1=2 response is underdamped. A higher Q-factor generally allows for sharper roll-off
and is used in place of using a higher number of stages in a design. Like its mechanical system
analog (Q ¼
ffiffiffiffiffiffi ffi
mk
p =z), an underdamped filter shows ringing and has a resonance behavior. The Qfactors for Butterworth and Bessel filters are shown in Table 6.3, where
Q ¼
ffiffi ffi
b
p =a
ð6:73Þ
6.9 GROUNDS, SHIELDING, AND CONNECTING WIRES
The type of connecting wires used between electrical devices can have a significant impact on the
noise level of the signal. Low-level signals of <100 mVare particularly susceptible to errors induced
by noise. Some simple rules help to keep noise levels low: (1) keep the connecting wires as short as
possible, (2) keep signal wires away from noise sources, (3) use a wire shield and proper ground, and
(4) twist wire pairs along their lengths.
Ground and Ground Loops
The voltage at the end of a wire that is connected to a rod driven far into the soil would likely be at
the same voltage level as the earth—a reference datum called zero or earth ground. A ground is
simply a return path to earth. Now suppose that wire is connected from the rod through an electrical
box and then through various building routings to the ground plug of an outlet. Would the ground
potential at the outlet still be at zero? The answer is probably not. The network of wires that form the
return path to earth would likely act as antennae and pick up some voltage potential relative to earth
ground. Any instrument grounded at the outlet would be referenced back to this voltage potential,
not to earth ground. The point is that an electrical ground does not represent an absolute value.
Ground values vary between ground points because the ground returns pass through different
250 Chapter 6 Analog Electrical Devices and Measurements
11:55:7 Page 250
number, such that C 2 ! 4bC 1 =a
2 , and the resistors are then found by
R 1;2 ¼
aC 2 Ç
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðaC 2 Þ
2 À 4bC 1 C 2
q
4pf c C 1 C 2
ð6:70Þ
The filter cutoff frequency is set by
f c ¼
1
2p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
R 1 R 2 C 1 C 2
p
ð6:71Þ
The unit-gain Sallen–Key high-pass filter is shown in Figure 6.36b. The design switches the
resistors and capacitors from the low-pass filter. For high-pass filters, we can simplify the design and
set C 1 ¼ C 2 ¼ C. The high-pass filter transfer function is
GðsÞ ¼
1
1 þ a=s þ b=s 2
ð6:72Þ
where a ¼ 2=v c CR 1 and b ¼ 1=v
2
c C
2
R 1 R 2 . Values for a and b to meet Butterworth or Bessel filter
characteristics are given in Table 6.3 with the resistor values set by R 1 ¼ 1=pf c aC and
R 2 ¼ a=4pf c bC.
At some frequency, the reactive elements in a filter circuit resonate, leading to a peak in the
response behavior. This is behavior predicted by its Q-factor, an inverse damping ratio-like
parameter. For Q ¼ 1=2 response is critically damped, for Q < 1=2 response is overdamped,
and for Q > 1=2 response is underdamped. A higher Q-factor generally allows for sharper roll-off
and is used in place of using a higher number of stages in a design. Like its mechanical system
analog (Q ¼
ffiffiffiffiffiffi ffi
mk
p =z), an underdamped filter shows ringing and has a resonance behavior. The Qfactors for Butterworth and Bessel filters are shown in Table 6.3, where
Q ¼
ffiffi ffi
b
p =a
ð6:73Þ
6.9 GROUNDS, SHIELDING, AND CONNECTING WIRES
The type of connecting wires used between electrical devices can have a significant impact on the
noise level of the signal. Low-level signals of <100 mVare particularly susceptible to errors induced
by noise. Some simple rules help to keep noise levels low: (1) keep the connecting wires as short as
possible, (2) keep signal wires away from noise sources, (3) use a wire shield and proper ground, and
(4) twist wire pairs along their lengths.
Ground and Ground Loops
The voltage at the end of a wire that is connected to a rod driven far into the soil would likely be at
the same voltage level as the earth—a reference datum called zero or earth ground. A ground is
simply a return path to earth. Now suppose that wire is connected from the rod through an electrical
box and then through various building routings to the ground plug of an outlet. Would the ground
potential at the outlet still be at zero? The answer is probably not. The network of wires that form the
return path to earth would likely act as antennae and pick up some voltage potential relative to earth
ground. Any instrument grounded at the outlet would be referenced back to this voltage potential,
not to earth ground. The point is that an electrical ground does not represent an absolute value.
Ground values vary between ground points because the ground returns pass through different
250 Chapter 6 Analog Electrical Devices and Measurements