非理想运算放大器的发展方程(1)

南京理工大学泰州科技学院毕业设计(论文)外文资料翻译学院 (系)： 电子电气工程学院专 业： 电子信息工程姓 名： 张 国 彬学 号： 0802040149外文出处： Development of the Nonldeal Op Amp Equations附 件：1.外文资料翻译译文；2.外文原文。

注：请将该封面与附件装订成册。

指导教师评语： 签名： 年 月 日 (用外文写)附件1：外文资料翻译译文非理想运算放大器方程的导出1.引言运放中的误差来源有两类，用一般的分类方法可以分为DC误差（直流误差）和AC误差（交流误差）。

DC误差的例子有输入失调电压和输入偏置电流。

DC 误差在运算放大器可使用的频率范围内是保持恒定的。

因此，输入偏置电流在1kHz 下是l0pA，在10KHz下也是l0pA。

由于DC误差具有恒定和可控的性质，所以对它们的讨论我们将会到后面再继续进行。

AC误差是易变的。

我们在这里对AC误差进行分析的方法，是导出一组考虑到AC误差的非理想方程。

AC误差可以在DC状态下出现，但随着工作频率的增加，AC误差会变得越来越糟。

AC误差的一个很好的例子是CMRR(Common Mode Rejection Ratio，共模抑制比)。

大多数远放都有一个确保的CMRR指标，但这个指标只在DC或非常低的频率下才有效。

当我们进一步阅读数据手册时，我们就可以发现CMRR会随着工作频率的增加而下降。

归入AC误差的其他几个指标有输出阻抗.、电源抑制比、峰到峰输出电压、差分增益、差分相位和相位裕度等。

差分增益是最重要的AC 指标，因为其他的AC指标都是由差分增益导出的。

在前面，我们一直把差分增益叫做运放增益或运放开坏增益，我们将继续使用这些术语。

在数据手册中，这个术语叫做差分增益。

正如前面几章中所说的，当频率增加时，运放的增益就降低，误差就增加。

这一章将导出几个方程，用以说明增益的改变所产生的这些影响。

我们将首先回顾基本典范反馈系统的稳定性，因为运放方程的导出使用了相同的技术。

放大器是用像晶体管这样的有源元件构建的。

而晶体管的主要参数，比如晶体管的增益，是易于漂移的，此外，来自许多不同制造商的元件都存在初始的不精确性，因此，用这些元件构建的运放同样易于漂移和不精确。

不过，这种漂移和不精确可以通过使用负反馈而得以减小甚至消除。

运放电路就是采用了这种反馈的结构，使电路的传递方程与放大器本身的参数无关(几乎完全无关)，而只取决于外部无源元件。

外部无源元件可以采购成满足几乎任何漂移或精度的要求，限制这些无源元件使用的只是它们的成本和体积。

运放一旦加有反馈，运放电路就可能变为不稳定。

有些放大器属于一种叫做内部补偿运放的类别，这些运放包含了一些有时被广告里说成可以避免不稳定的内部电容。

虽然工作在指定条件下的内部补偿运放是不应该振荡的，但也有许多这样的电路仍然会有相对的不稳定性，这说明这些电路本身存在着很差的相位响应和易于产生振铃与过冲的问题。

唯一绝对稳定的内部补偿运放只是那种躺在实验台上且未加电源的运放！所有其他的内部补偿运放都会在某些外部电路条件下产生振荡。

非内部补偿或者外部补偿的运放，在没有加上起稳定作用的外部元件时，是不稳定的。

这种情况对许多应用是个缺点，因为这些运放需要增加额外的元件。

然而，没有内部补偿的运放可以使高明的电路设计者完全地发挥出运放的全部性能。

于是，设计者面临两种选择:采用已由IC 制造商做了内部补偿的运放，或者由设计者自己对运放做外部补偿。

除了由运放制造商做好补偿之外，其他的运放都必须在IC 外面做补偿。

非常有意思的是，为了满足要求很高的应用，内部补偿的运放也还需要进行外部补偿。

补偿是通过增加外部元件以修改电路的传递函数来实现的，并以此使电路变成无条件稳定。

对运放进行补偿有几种不同的方法，每种补偿方法都有它的优点和缺点。

我们必须对补偿以后的运放电路进行分析，以确定补偿的效果。

此外，根据补偿对于闭环传递函数的改善效果，往往可以确定出哪种补偿她会有最大的收效。

2.典范方程的回顾输出方程和误差方程重复如下（6-1） （6-2） 由上面的两个方程经过代入法后可以得到下面的（6-3）方程：（6-3） 然后合并同类项之后又可以得到得到（6-4）的方程： IN OUT V A V =+⎪⎪⎭⎫⎝⎛β1 （6-4） OUT IN OUT V V E EAV β-==OUT IN OUT V V A V β-=把上述方程各项经过整理之后，可以得到反馈方程的经典形式（6-5）的方程： （6-5）由上面我们可以看出，当式(6-5) 中的量Aβ变成相对于1 非常大的时候.式(6-5) 即简化为式(6-6）。

式 (6-6) 叫做理想反馈方程.因为它取决于假设Aβ>> 1。

当放大器被认为具有理想品质的时候， 这个方程有着广泛的用途.在Aβ>> 1 的条件下，系统增益将由反愤因子β确定。

我们是用稳定的无源元件来实现这个反馈因子的，因此，理想的闭环增益是可预测的和稳定的，因为β是可预测的和稳定的。

β1=IN OUT V V （6-6） 可以看出Aβ这个量非常重要，因此我们给了它一个特殊的名称:环路增益。

我们来考察图6-2，在把输入电压接地(若电流输入则是断开)和把环路断开之后，所计算出的增益就是环路增益Aβ。

这里需要注意的是，这是一些复数运算，复数有幅度和方向。

当环路增益趋于-1 或者用数学式表示为1角-180度。

的时候，式 (7-5) 就趋于无穷大，因为1/0=>∞。

这时，电路的输出将沿着一条直线方程、以尽可能快的速度奔向无穷大。

如果输出不受能量限制，这个电路将摧毁整个世界，好在电路的能量受到电源的限制，所以，这个世界丝毫米损。

当电路输出趋于电源电压时，电路中的有源器件会表现出非线性，而这种非线性降低了放大器的增益，使环路增益不再等于1 角-180度。

现在，电路可以做两件事:第一，电路可以稳定在电源的一个端电压上，第二，电路可以改变方向(因为存储的电荷会使输出电压继续改变)，奔向电源的另一端。

在第一种情况下，也路稳定在电源的一个端电压上，这叫做锁定。

电路将一直保持在这个锁定状态，直到电源被切断。

在第二种情况下，电路在电源的两个端电压之间来回跳动，这叫做振荡。

我们应该知道，环路增益Ass 是唯一一个确定电路或系统稳定性的因子。

当计算环路增益时，首先要把输入接地或断开，因为输入不会对稳定性有任何影响。

我们将在后面对环路增益这个判别准则进行深入分析。

把式(6-1)和式(6-2)合并和整理之后，可以得到式(6-7) ，这个等式给出了系统βA A V V IN OUT +=1或电路的误差:βA V E IN +=1 (6-7) 首先可以看出，误差是与输入信号成正比的。

这是所期望的结果，因为较大的输入信号产生较大的输出信号，而较大的输出信号需要较大的驱动电压。

其次，环路增益与误差成反比误差会随环路增益的增加而减小.因此，大的环路增益对于减小误差是很有用的. 但大得环路增益同时也降低了稳定性.所以，在误差和稳定性之间总是存在折中关系。

3.同相运算放大器放大器传输过程的方程()B IN OUT V V a V -= (6-8) 同相运放的输出方程可以利用分压器规则导出.在使用分伍德规则的时候，我们实际上假设运放的输出阻扰很低。

0=+=B GF G OUT B forl Z Z Z V V （6-9） 结合上述两个方程我们就可以得到以下的方程 F G OUT G IN OUT Z Z V aZ aV V +-= （6-10） 对式(6- 10) 整理之后，我们又可以得到式 (6-11 )，可以得出了这个同相电路的传递函数：FG G IN OUT Z Z aZ a V V ++=1 （6-11） 我们把式(6-5)重写一遍，编号为式 (6-12)以便子比较两个等式中的对应项: βA A V V IN OUT +=1 （6-12） 通过对式 (6-11)与式（6-12)的比较，可以得到式(6- 13)，这是同相运放的环路增益方程。

这个环路增益方程确定了电路的稳定性。

此外，上面的比较还指出了同相电路的开环增益A 就是运放的开环增益a 。

(6-13) F G GZ Z aZ A +=β把测试电压TEST V 乘以运算放大器开环增益后，获得运放的输出电压TEST aV 。

然后用分压器规则得到式子（6-14）。

然后对（6-14）进行一些数据处理，得到（6-15）。

这是同相运放的环路增益。

而且与（6-13）完全是一样的。

(6-14) (6-15)4.反相功放电路的传递方程A OUT aV V -= (6-16)式 (6-17) 中的节点电压方程是利用叠加定理和分压器规则得到的.式 (6-18) 是利用式(6-16) 和式 (6-17) 得出的。

(6-17) (6-18)式 (6-18) 是反相运放的传递函数。

通过对式（6-18) 与式 (6-12) 的比较，我们又得到了式 (6-13) .而这也是图6-5 中反相运放电路的环路增益方程.通过比较还可以知道，反相电路的开环增益A 是与同相电路的运放开环增益a 不同的。

我们把反馈环路断开后的反相运放回示图6-6 中，这个电路可以用来导出式 (7-19)中的环路增益:(6-19)分析到这里，我们必须对几件事做一说明。

首先，式 (6-11)和式 (6-18) 中的同相和反相方程的传递值数是不同的.对于同样的一组G Z 和F Z 值，无论增益的大小还是极性都是不同的.其次，两个电路的环路增益悬完全一样的，这就是式 (6-13) 和式 (6-19)。

因此，这两个电路在稳定性方面的特性也是完全相同的，虽然两者的传递方程是不同的.这就得出了一个重要的结论:稳定，蚀与电路的输入无关。

最后，图6-1中的增益框A 。

， 对于每一个运放电路都是不同的.通过对式F G G F G F IN OUTB F G G OUT F G F IN A Z Z aZ Z Z aZ V V forl Z Z Z V Z Z Z V V +++-==+++=10G F G TEST RETURNG F G TEST RETURN Z Z aZ A V V Z Z Z aV V +==+=ββA Z Z aZ V V F G G TEST RETUNRN =+=(6-5)、式 (6-11) 和式 (6-18) 的比较.可以看出，同相时a A INV NON =-，反相时()F G F INV Z Z aZ A +=。

附件2：外文原文Development of the Nonldeal Op Amp Equations6.1IntroductionThere are two types of error sources in op amps, and they fall under the general classification of dc and ac errors. Examples of dc errors are input offset voltage and input bias cur- rent. The dc errors stay constant over the usable op amp frequency range; therefore, the input bias current is 10 p A at 1 kHz and it is 10 pA at 10 kHz. Because of their constant and controlled behavior, dc errors are not considered until later chapters.AC errors are flighty, so we address them here by developing a set of nonideal equations that account for ac errors. The ac errors may show up under dc conditions, but they get worse as the operating frequency increases. A good example of an ac error is common– mode rejection ration (CMRR). Most op amps have a guaranteed CMRR specification, but this specification is only valid at dc or very low frequencies. Further inspection of the data sheet reveals that CMRR decreases as operating frequency increases. Several other specifications that fall into the category of ac specifications are output impedance, power-supply rejection-ratio, peak-to-peak output voltage, differential gain, differential phase, and phase margin.Differential gain is the most important ac specification because the other ac specifications are derived from the differential gain. Until now, differential gain has been called op amp gain or op amp open loop gain, and we shall continue with that terminology. Let the data sheet call it differential gain.As shown in prior chapters, when frequency increases, the op amp gain decreases and errors increase. This chapter develops the equations that illustrate the effects of the gain changes. We start with a review of the basic canonical feedback system stability because the op amp equations are developed using the same techniques.Amplifiers are built with active components such as transistors. Pertinent transistor parameters like transistor gain are subject to drift and initial inaccuracies from many sources, so amplifiers being built from these components are subject todrift and inaccuracies. The drift and inaccuracy is minimized or eliminated by using negative feedback. The op amp circuit configuration employs feedback to make the transfer equation of the circuit independent of the amplifier parameters (well almost), and while doing this, the circuit transfer function is made dependent on external passive components. The external pas- sive components can be purchased to meet almost any drift or accuracy specification; only the cost and size of the passive components limit their use.Once feedback is applied to the op amp it is possible for the op amp circuit to become unstable. Certain amplifiers belong to a family called internally compensated op amps; they contain internal capacitors that are sometimes advertised as precluding instabilities. Although internally compensated op amps should not oscillate when operated under specified conditions, many have relative stability problems that manifest themselves as poor phase response, ringing, and overshoot. The only absolutely stable internally compensated op amp is the one lying on the workbench without power applied! All other internally compensated op amps oscillate under some external circuit conditions.Non internally compensated or externally compensated op amps are unstable without the addition of external stabilizing components. This situation is a disadvantage in many cases because they require additional components, but the lack of internal compensation enables the top drawer circuit designer to squeeze the last drop of performance from the op amp. You have two options: op amps internally compensated by the IC manufacturer, or op amps externally compensated by you. Compensation, except that done by the op amp manufacturer, must be done external to the IC. Surprisingly enough, internally compensated op amps require external compensation for demanding applications.Compensation is achieved by adding external components that modify the circuit transfer function so that it becomes unconditionally stable. There are several different methods of compensating an op amp, and as you might suspect, there are pros and cons associated with each method of compensation. After the op amp circuit is compensated,it must be analyzed to determine the effects of compensation. The modifications that compensation have on the closed loop transfer function often determine which compensation scheme is most profitably employed.6.2Review of the Canonical EquationsA block diagram for a generalized feedback system is repeated in Figure 6–1. This simple block diagram is sufficient to determine the stability of any system.The output and error equation development is repeated below. (6-1) (6-2)Combining Equations 6–1 and 6–2 yields Equation 6–3 IN OUT V AV =+⎪⎪⎭⎫ ⎝⎛β1 (6-3) Collecting terms yields Equation 6–4: (6-4)Rearranging terms yields the classic form of the feedback equation.(6-5)Notice that Equation 6–5 reduces to Equation 6–6 when the quantity Ain Equation 6–5 becomes very large with respect to one. Equation 6–6 is called the ideal feedback equation because it depends on the assumption that A>> 1, and it finds extensive use when amplifiers are assumed to have ideal qualities. Under the conditions that A>>1, the system gain is determined by the feedback factor Stable passive circuit components are used to implement the feedback factor, thus the ideal closed loop gain is predictable and stable because is predictable and stable.β1=IN OUT V V (6-6) The quantity A is so important that it has been given a special name, loop gain. Consider Figure 6–2; when the voltage inputs are grounded (current inputs are opened) OUT IN OUT V V E EAV β-==OUT IN OUT V V A V β-=βA A V V IN OUT +=1and the loop is broken, the calculated gain is the loop gain, A Now, keep in mind that this is a mathematics of complex numbers, which have magnitude and direction. When the loop gain approaches minus one, or to express it mathematically 1–180 , Equation 6–5 approaches infinity because 1/0 . The circuit output heads for infinity as fast as it can using the equation of a straight line. If the output were not energy limited the circuit would explode the world, but it is energy limited by the power supplies so the world stays intact.Active devices in electronic circuits exhibit nonlinear behavior when their output ap- proaches a power supply rail, and the nonlinearity reduces the amplifier gain until the loop gain no longer equals 1 –180 . Now the circuit can do two things: first, it could become stable at the power supply limit, or second, it can reverse direction (because stored charge keeps the output voltage changing) and head for the negative power supply rail.The first state where the circuit becomes stable at a power supply limit is named lockup; the circuit will remain in the locked up state until power is removed. The second state where the circuit bounces between power supply limits is named oscillatory. Remember, the loop gain, A , is the sole factor that determines stability for a circuit or system. Inputs are grounded or disconnected when the loop gain is calculated, so they have no effect on stability. The loop gain criteria is analyzed in depth later.Equations 6–1 and 6–2 are combined and rearranged to yield Equation 6–7, which gives an indication of system or circuit error.βA V E IN +=1 (6-7) First, notice that the error is proportional to the input signal. This is the expected result because a bigger input signal results in a bigger output signal, and bigger output signals require more drive voltage. Second, the loop gain is inversely proportional to the error. As the loop gain increases the error decreases, thus large loop gains are attractive for minimizing errors. Large loop gains also decrease stability, thus there is always a trade off between error and stability.6.3 Non inverting Op AmpsA non inverting op amp is shown in Figure 6–3. The dummy variable, VB, is inserted to make the calculations easier and a is the op amp gain.Equation 6–8 is the amplifier transfer equation.()B IN OUT V V a V -= (6-8) The output equation is developed with the aid of the voltage divider ing the voltage divider rule assumes that the op amp impedance is low.0=+=B GF G OUT B forl Z Z Z V V (6-9) Combining Equations 6–8 and 6–9 yields Equation 6–10. F G OUT G IN OUT Z Z V aZ aV V +-= （6-10） Rearranging terms in Equation 6–10 yields Equation 6–11, which describes the transfer function of the circuit.FG GIN OUT Z Z aZ a V V ++=1 （6-11） Equation 6–5 is repeated as Equation 6–12 to make a term by term comparison of the equations easy .βA A V V IN OUT +=1 （6-12） By virtue of the comparison we get Equation 6–13, which is the loop-gain equation for the non inverting op amp. The loop-gain equation determines the stability of the circuit. The comparison also shows that the open loop gain, A, is equal to the op amp open loop gain, a, for the non inverting parison also shows that the open loop gain, A, is equal to the op amp open loop gain, a, for the non inverting circuit.（6-13） The test voltage, V TEST , is multiplied by the op amp open loop gain to obtain the op amp output voltage, a V TEST . The voltage divider rule is used to calculate Equation 6–15, which is identical to Equation 6–14 after some algebraic manipulation.F G G Z Z aZ A +=β（6-14） （6-15） The transfer equation is given in Equation 6–16:A OUT aV V -= （6-16） The node voltage (Equation 6–17) is obtained with the aid of superposition and the volt- age divider rule. Equation 6–18 is obtained by combining Equations 6–16 and 6–17.（6-17） （6-18）Equation 6–16 is the transfer function of the inverting op amp. By virtue of the comparison between Equations 6–18 and 6–14, we get Equation 6–15 again, which is also the loop gain equation for the inverting op amp circuit. The comparison also shows that the open loop gain (A) is different from the op amp open loop gain (a) for the non inverting circuit.The inverting op amp with the feedback loop broken is shown in Figure 6–6, and this circuit is used to calculate the loop-gain given in Equation 6–19.（6-19）Several things must be mentioned at this point in the analysis. First, the transfer functions for the non inverting and inverting Equations, 6–13 and 6–18, are different. For a common set of ZG and ZF values, the magnitude and polarity of the gains are different. Second,The loop gain of both circuits, as given by Equations 6–15 and 6–19, is identical. G F G TEST RETURNG F G TEST RETURN Z Z aZ A V V Z Z Z aV V +==+=βF G G F G F IN OUTB F G G OUT F G F IN A Z Z aZ Z Z aZ V V forl Z Z Z V Z Z Z V V +++-==+++=10βA Z Z aZ V V F G G TEST RETUNRN =+=Thus,the stability performance of both circuits is identical although their transfer equations are different. This makes the important point that stability is not dependent on the circuit in- puts. Third, the A gain block shown in Figure 6–1 is different for each op amp circuit. By comparison of Equations 6–5, 6–11, and 6–18 we see thataA INVNON=-and()FGFINV ZZaZA+=.。