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1、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 p
2、lot 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
3、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 th
4、e 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.,
5、 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
6、 transfer functions 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 integrators 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 sl
7、ope 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 th
8、e 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-pa
9、ss filter. When, the denominator is zero and the functions 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
10、right. The pole at - is evident. Amplitude is shown logarithmically to emphasize the functions 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
11、complex function in s relate to the circuits 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
12、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 circuits response to an angular frequency.
13、In the complex plot in Figure 2.2b, and hence along the positive j axis. Thus, the functions 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 filters frequency-response curve by adding a heavy line for function v
14、alues 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 frequencys imaginary part () helps describe a response to AC signals, the rea
15、l part () helps describe a circuits transient response. Looking at Figure 2.2b, we can therefore say something about the RC low-pass filters response as compared to that of the integrator. The low-pass filters 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;
