Biomedical Engineering Reference
In-Depth Information
and at
t
¼
0,
v
C
0
10
3
10
3
10
3
ðÞ¼
2
:
5
¼
7
:
718
K
1
2
:
7
K
2
Solving gives
K
1
¼
0
:
04 and
K
2
¼
1
:
04
:
Substituting these values into the solution gives
e
7:71810
3
e
2:710
3
t
t
v
C
ðÞ¼
0
:
04
1
:
04
þ
3V
for
t
0
:
9.11 OPERATIONAL AMPLIFIERS
Section 9.3 introduced controlled voltage and current sources that are dependent on a
voltage or current elsewhere in a circuit. These devices were modeled as a two-terminal
device. In this section, we look at the operational amplifier, also known as an op amp,
which is a multiterminal device. An operational amplifier is an electronic device that con-
sists of large numbers of transistors, resistors, and capacitors. Fully understanding its oper-
ation requires knowledge of diodes and transistors—topics that are not covered in this
topic. However, fully understanding how an operational amplifier operates in a circuit
involves a topic already covered: the controlled voltage source. Circuits involving opera-
tional amplifiers form the cornerstone for any bioinstrumentation, from amplifiers to filters.
Amplifiers used in biomedical applications have very high-input impedance to keep the
current drawn from the system being measured low. Most body signals have very small
magnitudes. For example, an ECG has a magnitude in the millivolts, and the EEG has a
magnitude in the microvolts. Analog filters are often used to remove noise from a signal,
typically through frequency domain analysis to design the filter.
As the name implies, the operation amplifier is an amplifier, but as we will see, when it is
combined with other circuit elements, it integrates, differentiates, sums, and subtracts. One
of the first operational amplifiers appeared as an eight-lead dual-in-line package (DIP),
shown in Figure 9.28.
Differing from previous circuit elements, this device has two input and one output term-
inals. Rather than draw the operational amplifier using Figure 9.28, the operational ampli-
fier is drawn with the symbol
in Figure 9.29. The input
terminals are labeled the
noninverting input (
þ
) and the inverting input (-). The power supply terminals are labeled
V
and V-, which are frequently omitted, since they do not affect the circuit behavior
except in saturation conditions, as will be described. Most people shorten the name of the
operational amplifier to the “op amp.”
Figure 9.30 shows a model of the op amp, focusing on the internal behavior of the input
and output terminals. The input-output relationship is
v
o
¼
Av
p
v
n
þ
Þ
Since the internal resistance is very large, we will replace it with an open circuit to simplify
analysis, leaving us with the op amp model show in Figure 9.31.
With the replacement of the internal resistance with an open circuit, the currents
i
n
¼
i
p
¼
ð
9
:
29
0 A. In addition, current
i
A
, the current flowing out of the op amp, is not zero.
Because
is unknown, seldom is KCL applied at the output junction. In solving op amp
problems, KCL is almost always applied at input terminals.
i
A
