排版有点乱。。。。
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Operational Amplifiers (OAs) are highly stable, high gain dc difference amplifiers. Since
there is no capacitive coupling between their various amplifying stages, they can handle signals
from zero frequency (dc signals) up to a few hundred kHz. Their name is derived by the fact that
they are used for performing mathematical operations on their input signal(s) .
Fig. 2.1 shows the symbol for an OA. There are two inputs, the inverting input (-) and the
non-inverting input (＋) .These symbols have nothing to do with the polarity of the applied input
signals. Connections to power supplies are also shown..
v+ and v- are the signals applied to the non-inverting and to the inverting input, respectively.
Α represents the open loop gain of the OA. A is infinite for the ideal amplifier, whereas for the
various types of real OAs, it is usually within the range of 104 to 106.
OAs require two power supplies to operate, supplying a positive voltage (＋V) and a
negative voltage (－V) with respect to circuit common. This bipolar power supply allows OAs to
generate output signals (results) of either polarity. The output signal (vo) range is not limited. The
voltages of the power supplies determine its actual range. Thus, a typical OA fed with －15 V
and ＋15 V may yield a vo within the (approximately) －13 V to ＋13 V range, which is called
operational range. Any result expected to be outside this range is clipped to the respective limit,
and OA is in a saturation stage.
The connections to the power supplies and to the circuit common symbols, shown in
Fig. 2.1, hereafter will be implied, and they will be not shown in the rest of the circuits for
simplicity.
Because of their very high open loop gain, OAs are almost exclusively used with some
additional circuitry (mostly with resistors and capacitors) required to ensure a negative feedback
loop. Through this loop a tiny fraction of the output signal is fed back to the inverting input. The
negative feedback stabilizes the output within the operational range and provides a much smaller
but precisely controlled gain, the so-called closed loop gain. [1]
Circuits of OAs have been used in the past as analog computers, and they are still in use for
mathematical operations and modification of the input signals in real time. A large variety of
OAs is commercially available in the form of low cost integrated circuits.
There is a plethora of circuits with OAs performing various mathematical operations. Each
circuit is characterized by its own transfer function, i.e. the mathematical equation describing
output signal (vo) as a function of the input signal (vi) or signals (v1, v2, …, vn). Generally,
transfer functions can be derived by applying Kirchhoff’s rules and the following two simplifying
assumptions:
(1) The output signal (vo) acquires a value that (through the feedback circuits) practically
equates the voltages applied to both inputs, i.e. v＋≈v-.
(2) The input resistance of both OA inputs is extremely high (usually within the range
106~1012 MΩ, for the ideal OA it is infinite) , thus no current flows into them.
The basic circuit of the inverting amplifier is shown in Fig. 2.2.
The transfer function is derived as follows: Considering the arbitrary current directions we
have:
1 i s i =(v−v)/Ri and 2 s o f i =(v−v)/R (2-2)
The non-inverting input is connected directly to the circuit common (i.e. v+＝0 V) ,
(considering simplifying assumption 1) vs＝v-＝0 V, therefore:
1 i i i = v / R and 2 o f i = −v /R (2-3)
Since there is no current flow to any input (simplifying assumption 2) , it is:
i1＝i2 (2-4)
Therefore, the transfer function of the inverting amplifier is:
o f i i v = −(R /R)v (2-5)
Thus, the closed loop gain of the inverting amplifier is equal to the ratio of Rf (feedback
resistor) over Ri (input resistor). This transfer function describes accurately the output signal as
long as the closed loop gain is much smaller than the open loop gain A of the OA used (e.g. it
must not exceed 1000) , and the expected values of vo are within the operational range of the OA.
The summing amplifier is a logical extension of the previously described circuit, with two
or more inputs. Its circuit is shown in Fig. 2.3.
The transfer function of the summing amplifier (similarly derived) is:
o 1 1 2 2 f ( / / / ) n n v v R v R = − + + +v R R … (2-6)
Thus if all input resistors are equal, the output is a scaled sum of all inputs, whereas, if they
are different, the output is a weighted linear sum of all inputs.
The summing amplifier is used for combining several signals. The most common use of a
summing amplifier with two inputs is the amplification of a signal combined with a subtraction
of a constant amount from it (dc offset).
Difference amplifier precisely amplifies the difference of two input signals. Its typical
circuit is shown in Fig. 2.4.
If Ri＝
i R′ and Rf＝
f R′ , then the transfer function of the difference amplifier is:
o 2 1 f i v =(v −v)R/R (2-7)
The difference amplifier is useful for handling signals referring not to the circuit common,
but to other signals, known as floating signal sources. Its capability to reject a common signal
makes it particularly valuable for amplifying small voltage differences contaminated with the
same amount of noise (common signal). [3]
In order for the difference amplifier to be able to reject a large common signal and to
generate at the same time an output precisely proportional to the two signals difference, the two
ratios p＝Rf/Ri and q＝
f R′ / i R′ must be precisely equal, otherwise the signal output will be:
o 2 1 v=[q(p+1) /(q+1)]v−pv (2-8)
The differentiator generates an output signal proportional to the first derivative of the input
with respect to time. Its typical circuit is shown in Fig. 2.5.
The transfer function of this circuit is
o i (d v = −RC v / dt) (2-9)
Obviously, a constant input (regardless of its magnitude) generates a zero output signal. A
typical usage of the differentiator in the field of chemical instrumentation is obtaining the first
derivative of a potentiometric titration curve for the easier location of the titration final points
(points of maximum slope). [4]
The integrator generates an output signal proportional to the time integral of the input signal.
Its typical circuit is shown in Fig. 2.6.
In Fig. 2.6, the output remains zero as far as switch S remains closed. The integration starts
(t＝0) when S opens. The output is proportional to the charge accumulated in capacitor C, which
serves as the integrating device. A typical application of the (analog) integrator in chemical
instrumentation is the integration of chromatographic peaks, since its output will be proportional
to the peak area.
If the input signal is stable, then the output from the integrator will be given by the equation:
o i v = −(v /RC)t (2-11)
i.e. the output signal will be a voltage ramp. Voltage ramps are commonly used for generating the
linear potential sweep required in polarography and many other voltammetric techniques.
运算放大器(简称OA)是具有高稳定性、高增益的直流差动放大器。由于运放的不同级
之间没有耦合电容，因而可以对数百赫兹的直流信号进行放大。运放用于对输入的信号进
行数学运算，故此而得名。
图2.1 为运放的符号。其输入端有两个，即反相输入端(-)和同相输入端(+)。这些符号
与加在输入端的信号极性无关。电源的接线在图中也给了出来。
输出信号(电压)vo 由vo A(v v) + − = − 确定。v+和 v-是分别加在同相输入端和反相输入端
的信号。A 表示运放的开环增益，对于理想的运放为无穷大，但对各种实际的运放，其范
围通常在104到106之间。
运放需要两个工作电源，对于电路的公共端分别提供正电压和负电压。这种双极性电
源使得运放可以输出任意极性的信号。输出信号的范围没有限制，实际范围取决于电源电
压。因此，工作于±15V 的典型运放，其输出范围大约在-13V 到+13V，称为工作范围。任
何要超出这一范围的输出都被钳位在相应的极限值上，此时运放处于饱和状态。
图2.1 中所示的电源和电路公共端的接线此后将隐含起来。为了表达简洁，在其余的
电路中就不再画出来。
由于开环增益很高，运放几乎总是与一些用来构成负反馈电路的附加电路(主要是电阻
和电容)一起使用。通过这个回路，输出的一小部分被反馈到反相输入端。负反馈使输出稳
定在其工作范围内，并提供一个较小但控制准确的增益，即所谓闭环增益。
运放电路在以前用于模拟计算机，并用于实时的算术运算和输入信号的调节。在市场
上可以买到大量低成本的运放集成电路。
采用运放来完成各种算术运算的电路数不胜数。每一个电路由其自身的传递函数，即
描述输出信号与输入信号函数关系的数学方程来描述。一般可以通过应用克希霍夫定律和
以下两个简化的假设来得到传递函数：
(1) 输出信号的数值大小(通过反馈回路)实际上等于加在两个输入端上的电压，即v＋
≈v-。
(2) 运放的两个输入端的输入阻抗很高，一般在106～1012 MΩ的范围内，对于理想的
运放则是无穷大。因此输入端没有输入电流。
反相放大器的基本电路如图2.2 所示。考虑到电流方向的任意性，其传递函数可按以
下推导而得到：
1 i s i =(v−v)/Ri 且2 s o f i =(v−v)/R (2-1)
同相输入端直接连接到电路的公共端(即v+＝0 V)，因此有
1 i i i = v / R and 2 o f i = −v /R (2-2)
由于输入端没有电流(简化假设2)，即i1＝i2，因此，反相放大器的传递函数为
o f i i v = −(R /R)v 。
这样，反相放大器的闭环增益等于反馈电阻Rf与输入电阻Ri之比。只要所用运放的闭
环增益比开环增益A 小得多(即不能超过1000)，这个传递函数就能准确地表示输出信号，
且vo的预期值在运放的工作范围之内。
加法放大器有两个或更多输入，是对上述电路的逻辑扩展，如图2.3 所示。
加法放大器的传递函数(用类似的推导方法)为：
o 1 1 2 2 f ( / / / ) n n v = − v R +v R + +v R R …
这样，如果输入电阻都相等，输出就是所有输入的标量和。反之，如果输入电阻不相
等，输出就是各个输入信号的加权线性和。
加法放大器用于对多个信号进行组合。两输入加法放大器最常见的应用是放大一个减
去固定常数(直流偏差)的信号。
差动放大器准确地放大两个输入信号的偏差。其典型电路如图2.4所示。
如果Ri＝ i R′且Rf＝ f R′，则差动放大器的传递函数为o 2 1 f i v =(v −v)R/R 。
差动放大器对于处理不是以电路公共地为参考端，而是以通常所说的浮动信号源等其
他信号为参考端的信号非常有用。差动放大器抑制共模信号的能力使它特别适于放大受同
样大小噪声(共模信号)干扰的小压差信号。
为使差动放大器能够抑制较大的共模信号并同时产生一个正好正比于两个信号偏差的
输出，两个比值p＝Rf/Ri和q＝ f R′ / i R′必须完全相等。否则信号输出就是：
o 2 1 v=[q(p+1) /(q+1)]v−pv
微分器产生一个正比于输入对时间的第一次微分的输出信号。其典型电路如图2.5所示。
该电路的传递函数为o i v = −RC(dv / dt)。
显然，一个恒定的输入(不管幅度大小)产生的输出信号为0。微分器在化工仪表领域
中的典型应用是获取为了便于确定最后滴定点(最大斜率点)在电压滴定曲线的位置的第一
个微分值。
积分器产生与输入信号对时间积分成比例的输出信号。其典型电路如图2.6所示。
在图2.6 中，只要开关S 闭合，输出就保持为0。当开关S断开时，积分过程开始。输
出与作为积分器件的电容C中所充的电荷成比例。在化工仪表中，模拟积分器的典型应用
是色谱分析的峰值，因为其输出与峰值区成比例。
如果输入信号不变，积分器的输出将由如下方程确定：
vo = −(vi/RC)t
即输出信号将是一个斜坡电压。斜坡电压一般用于产生极谱法和许多其他伏安法技术中所
需要的线性电压扫描信号。