Words and Expressionsintegrator n. 积分器 amplitude n. 幅值slope n 斜率 denominator n. 分母impedance n 阻抗 inductor n. 电感capacitor n 电容 cascade n. 串联passband n 通带 ringing n. 振铃damping n. 阻尼，衰减 conjugate adj. 共轭的stage v. 成为low-pass filters 低通滤波器building block 模块linear ramp 线性斜坡log/log coordinates 对数/对数坐标Bode plot 伯德图transfer function 传递函数complex-frequency variable 复变量complex frequency plane 复平面real component 实部frequency response 频率响应complex function 复变函数Laplace transform 拉普拉斯变换real part 实部imaginary part 虚部angular frequency 角频率frequency response 频率响应transient response 瞬态响应decaying-exponential response 衰减指数响应step function input 阶跃（函数）输入time constant 时间常数first-order filters 一阶滤波器second-order low-pass filters 二阶低通滤波器passive circuit 无源电路active circuit 有源电路characteristic frequency 特征频率quality factor n. 品质因子，品质因数circular path 圆弧路径complex conjugate pairs 共轭复数对switched-capacitor 开关电容negative-real half of the complex plane 复平面负半平面Unit 4 Low-pass FiltersFirst-Order FiltersAn integrator (Figure 2. la) is the simplest filter mathematically, and it forms the building block for most modern integrated filters. Consider what we know intuitively about an integrator. If you apply a DC signal at the input (i.e., zero frequency), the output will describe a linear ramp that grows in amplitude until limited by the power supplies. Ignoring that limitation, the response of an integrator at zero frequency is infinite, which means that it has a pole at zero frequency. (A pole exists at any frequency for which the transfer function's value becomes infinite.)（为什么为极点，为什么低通？）Figure 2.la A simple RC integratorWe also know that the integrator's gain diminishes with increasing frequency and that at high frequencies the output voltage becomes virtually zero. Gain is inversely proportional to frequency, so it has a slope of -1 when plotted on log/log coordinates (i.e., -20dB/decade on a Bode plot, Figure 2. 1b).Figure 2.1 b A Bode plot of a simple integratorYou can easily derive the transfer function as Where s is the complex-frequency variable and is 1/RC. If we think of s as frequency, this formula confirms the intuitive feeling that gain is inversely proportional to frequency.The next most complex filter is the simple low-pass RC type (Figure 2. 2a). Its characteristic (transfer function) isWhen, the function reduces to , i.e., 1. When s tends to infinity, the function tends to zero, so this is a low-pass filter. When, the denominator is zero and the function's value is infinite, indicating a pole in the complex frequency plane. The magnitude of the transfer function is plotted against s in Figure 2. 2b, where the real component of s () is toward us and the positive imaginary part () is toward the right. The pole at - is evident. Amplitude is shown logarithmically to emphasize the function's form. For both the integrator and the RC low-pass filter, frequency response tends to zero at infinite frequency; that is, there is a zero at. This single zero surrounds the complex plane.But how does the complex function in s relate to the circuit's response to actual frequencies? When analyzing the response of a circuit to AC signals, we use the expression for impedance of an inductor and for that of a capacitor. When analyzing transient response using Laplace transforms, we use sL and 1/sC for the impedance of these elements. The similarity is apparent immediately. The in AC analysis is in fact the imaginary part of s, which, as mentioned earlier, is composed of a real part and an imaginary part.If we replace s by in any equation so far, we have the circuit's response to an angular frequency. In the complex plot in Figure 2.2b, and hence along the positive j axis. Thus, the function's value along this axis is the frequency response of the filter. We have sliced the function along the axis and emphasized the RC low-pass filter's frequency-response curve by adding a heavy line for function values along the positive j axis. The more familiar Bode plot (Figure 2.2c) looks different in form only because the frequency is expressed logarithmically.（根据图翻译这两句话）Figure 2.2a A simple RC low-pass filterWhile the complex frequency's imaginary part () helps describe a response to AC signals, the real part () helps describe a circuit's transient response. Looking at Figure 2.2b, we can therefore say something about the RC low-pass filter's response as compared to that of the integrator. The low-pass filter's transient response is more stable, because its pole is in the negative-real half of the complex plane. That is, the low-pass filter makes a decaying-exponential response to a step-function input; the integrator makes an infinite response. For the low-pass filter, pole positions further down the axis mean a higher, a shorter time constant, and therefore a quicker transient response. Conversely, a pole closer to the j axis causes a longer transient response.So far, we have related the mathematical transfer functions of some simple circuits to their associated poles and zeroes in the complex-frequency plane. From these functions, we have derived the circuit’s frequency response (and hence its Bode plot) and also its transient response. Because both the integrator and the RC filter have only one s in the denominator of their transfer functions, they each have only one pole. That is, they are first-order filters.Figure 2.2b The complex function of an RC low-pass filterFigure 2.2c A Bode plot of a low-pass filterHowever, as we can see from Figure 2.1b, the first-order filter does not provide a very selective frequency response. To tailor a filter more closely to our needs, we must move on to higher orders. From now on, we will describe the transfer function using f(s) rather than the cumbersome.Second-Order Low-Pass FiltersA second-order filter has in the denominator and two poles in the complex plane. You can obtain such a response by using inductance and capacitance in a passive circuit or by creating an active circuit of resistors, capacitors, and amplifiers. Consider the passive LC filter in Figure 2.3a, for instance. We can show that its transfer function has the formand if we define and ,then where is the filter's characteristic frequency and Q is the quality factor (lower R means higher Q).Figure 2.3a An RLC low-pass filterThe poles occur at s values for which the denominator becomes zero; that is, when. We can solve this equation by remembering that the roots of are given byIn this case, a = 1, , and .The term () equals, so if Q is less than 0.5 then both roots are real and lie on the negative-real axis. The circuit's behavior is much like that of two first order RC filters in cascade. This case isn't very interesting, so we'll consider only the case where Q > 0.5, which means is negative and the roots are complex.Figure 2.3b A pole-zero diagram of an RLC low-pass filterThe real part is therefore, which is, and common to both roots. The roots' imaginary parts will be equal and opposite in signs. Calculating the position of the roots in the complex plane, we find that they lie at a distance of from the origin, as shown in Figure 2.3b.Varying , changes the poles' distance from the origin. Decreasing the Q moves the poles toward each other, whereas increasing the Q moves the poles in a semicircle away from each other and toward theaxis. When Q = 0.5, the poles meet at on the negative-real axis. In this case, the corresponding circuit is equivalent to two cascaded first-order filters.Now let's examine the second-order function's frequency response and see how it varies with Q. As before, Figure 2.4a shows the function as a curved surface, depicted in the three-dimensional space formed by the complex plane and a vertical magnitude vector. Q =0.707, and you can see immediately that the response is a low-pass filter. The effect of increasing the Q is to move the poles in a circular path toward the axis. Figure 2.4b shows the case where Q = 2. Because the poles are closer to the axis, they have a greater effect on the frequency response, causing a peak at the high end of the passband.Figure 2.4a The complex function of a second-order low-pass filter (Q = 0.707)Figure 2.4b The complex function of a second-order low-pass filter (Q = 2)There is also an effect on the filter's transient response. Because the poles' negative-real part is smaller, an input step function will cause ringing at the filter output. Lower values of Q result in less ringing, because the damping is greater. On the other hand, if Q becomes infinite, the poles reach the axis, causing an infinite frequency response (instability and continuous oscillation) at. In the LCR circuit in Figure 2.3a, this condition would be impossible unless R=0. For filters that contain amplifiers, however, the condition is possible and must be considered in the design process.A second-order filter provides the variablesand Q, which allow us to place poles wherever we want in the complex plane. These poles must, however, occur as complex conjugate pairs, in which the real parts are equal and the imaginary parts have opposite signs. This flexibility in pole placement is a powerful tool and one that makes the second-order stage a useful component in many switched-capacitor filters. As in the first-order case, the second-order low-pass transfer function tends to zero as frequency tends to infinity. The second-order function decreases twice as fast, however, because of the factor in the denominator. The result is a double zero（零点） at infinity.低通滤波器一阶滤波器 从数学公式上讲，积分器（见图2.1a）是最简单的滤波器；它是构成大多数现代滤波器的基本模块。
我们怎么从直观上理解积分器呢？假设在输入端加上一个直流信号（频率为0），那么在输出端将会出现一个线性斜坡信号—其幅度一直增至电源电压如果不考虑电源电压对输出信号的限制，积分器在零频率上的响应将是无穷大，这意味着它在零频率点上存在一个极点（在任何使传递函数为无穷大值的频率点上都存在一个极点） 我们也知道，积分器的增益随频率的增加而减小；在很高频率上的输出电压事实上就是0增益和频率成反比，因此它在对数坐标系中是一条斜率为-1的直线（见图2.1b） 传输函数很容易推导出来（公式略）；其中，s是复频率变量，等于1/RC假如把s当作频率的话，该公式印证了我们有关“增益和频率成反比”的直观推断 简单低通RC滤波器（图2.2a）是一个稍复杂些的滤波器其传输函数如下：（公式略）当s等于0时，函数简化为1；当s趋近于无穷大时，函数简化为0；因此，这是一个低通滤波器当s等于-时，分母为0且函数值为无穷大，这意味着在复频率平面有一个极点图2.2b画出了传输函数的幅度和复频率s的关系，s的实部指向读者，而s的正虚部朝右处的极点非常明显为了突出函数的形状，幅度是用对数值显示的在无穷大频率处，积分器和RC低通滤波器的频率响应都趋近于0，也就是说在s等于无穷大处有一个零点。
这个零点环绕着复平面 但是，s变量复函数和电路的实际频率响应是怎么联系起来的呢？在分析电路对交变信号的响应时，我们用来表示电感的阻抗，用来表示电容的阻抗在使用拉普拉斯变换分析瞬态响应时，我们用sL和1/sC分别表示电感的阻抗和电容的阻抗这种相似性是显而易见的实际上，交流分析中的就是s的虚部；正如上面提到的那样，s是由一个实部和一个虚部组成的 假如将上面任何一个公式中的s用替代，就可以得到电路的角频率响应在图2.2b中，等于0，正j坐标轴上s等于因此，沿着该坐标轴的函数值就是滤波器的频率响应我们沿着轴将函数切割，并且在沿正坐标轴的函数值处加以粗线，从而突出了RC低通滤波器的频率响应曲线因为频率是对数表示的缘故，所以它和人们更熟悉的伯得图在形式上看上去不一样 复频率的虚部有助于描述电路对交流信号的响应，而其实部有助于描述电路的瞬态响应从图2.2b中可以看出，RC低通滤波器和积分器之间的一些区别低通滤波器的瞬态响应更加稳定，因为其极点位于复平面的左半部即对于阶跃函数输入，滤波器的响应是衰减指数形式的；积分器的响应是无穷大的对于低通滤波器而言，极点沿坐标轴离原点越远，意味着越大，时间常数越短，瞬态响应越快。
相反的情况是：极点离坐标轴越近，瞬态响应越慢 到目前为止，我们已经阐述了一些简单电路的数学传递函数与其复频率平面上的极、零点之间的关系从这些函数中，我们可以推导出电路的频率响应（从而可以得到伯得图）及其瞬态响应因为积分器和RC滤波器在其传递函数的分母中都仅有一个s，所以他们仅有一个极点也就是说，它们都是一阶滤波器 然而，从图2.2b中也可以看出：一阶滤波器的频率选择性不是很好为了得到一个更接近需求的滤波器，我们必须增加阶数从现在开始，我们将使用表示传输函数，而不再使用繁琐的二阶低通滤波器 二阶滤波器的分母中出现了，所以在复平面上就有两个极点使用电感和电容的无源电路或者包含电阻、电容和放大器的有源电路都具有这样的响应例如，考虑图2.3a中的无源LC滤波器其传输函数具有如下形式：（公式略），当定义和之后，该式变为：（公式略），此处的是滤波器的特征频率，而Q是品质因数（R越低，Q越高） 极点出现在使分母为零的s值处，即当的时候只要记住的根为（公式略），我们可以解出这个方程 在这个方程中，a等于1，b等于，而c等于这一项等于，如果Q小于0.5，那么两个根都是实数且位于复实轴上。
而电路的性能极似两个一阶滤波器的级联这种情况不是很有意义，所以我们只考虑的情况这意味着小于零，而方程的根都是复数实部为，等于，这部分对于两个根都是一样的两根的虚部互为相反数计算根在复平面的位置，我们就会发现它们位于距零点处（图2.3b） 改变会改变极点距离原点的位置降低Q值会将两个极点靠近，而增加Q值会将极点沿着半圆弧拉开并趋近轴当Q=0.5时，两极点在负实轴上的处汇合在这种情况下，对应的电路久等效于两个一阶滤波器的级联 现在，让我们考察一下二阶函数的频率响应及其随Q变化的情况和以前一样，图2.4a展示了在由复平面和垂直幅度向量构成的三维空间中绘制的函数曲线面图中Q=0.707，马上就可以看出这是一个低通滤波器的响应 增大Q值能将极点沿着圆弧路径移向轴图2.4B 展示的是Q=2的情形由于极点靠近轴，其对频率响应的影响就越大，这导致在通带高频段出现一个极值 这对滤波器的瞬态响应也有影响因为极点的负实部越小，输入阶跃函数就会引起滤波器输出振铃Q值越小，振铃越少，因为阻尼更大另一方面，假如Q为无限，极点抵达轴就会引起在处的频率响应为无限大（不稳定及持续振荡）在图2.3a的LCR电路中，这种情况是不可能出现的，除非R=0。
然而，对于包含放大器的滤波器而言，这种情况是有可能出现的，在设计过程中必须考虑这一点 二阶滤波器提供了和Q这两个变量，这就允许我们将极点放置在复平面上任何需要的地方 然而，这两个极点必须是一对共轭队，二者是不相同而虚部符号相反极点放置的这种灵活性是一种强大的手段，它使二阶滤波器成为众多开关电容滤波器中一个有用部件与一阶低通滤波器传输函数一样，随着频率也趋近于无穷，二阶低通滤波器的传输函数趋于零然而，二阶低通滤波器传输函数的下降速度是一阶低通滤波器传输函数下降速度的两倍；这是分母中出现因子的缘故这样，在无穷大处就存在两个零点。