TI高精度实验室-带宽 1.pdf.pdf

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Review: Gain. Linear vs, Decibel 回顾:增益,线性vs.分贝 Linear(vM to Decibels(dB) Decibels (dB)to Linear (VM) G dB=20 xlog(Gv/v) Vv=1020 Example: Convert the closed Example: Convert the open-loop loop gain(Gci of an op amp gain (AoL range of the OPA188 circuit from 100v to decibels to ww Solution: Solution 130dB A@z=1020 3162277 cueB)=20×o0100=40B OdB AOL@2MHz=10 20 HU TEXAS INSTRUMENTS ●于电子领域,我们经常需要表达数值,如 op amp gaIn(运算放大器的增益), signal-to- noise ratio(信号与噪声比), common-mode- rejection ratio(共模抑制比)不 power supply rejection ratio(电源抑制比),其值有非常大的跨度。 ●因此,有一个表示法,可以使用小的数字代表一个大的范围内的值是很重要的。这种 表示式被称为“分贝”或简称“dB”。需要注意的是分贝没有单位。 ●此贞显小了线性增益值与分贝的相互转换公式。 例如,让我们转换运算放大器电路的闭环回路增益为100V/V至分贝。 套入100V/V线性增益于方程式,可得到40分贝 ●同样地,给定以分贝为单位的增益,我们可以使用该方程式转换以线性来表示 ●虽然前面的例子似乎并没显著的改善,表示大的数值,计我们来看看OPA188的open loop gain(开环回路增益),或Aol。在1Hz,开环回路增益为130分贝,这相当于 3,162.,277Ⅴ/Ⅴ线性增益,在2MHz的开回路增益为0dB,这相当」⑩V/Ⅴ的线性 增益。 最终,我们发现使用分贝更容易来表示大范围,而不是以线性的每伏电压值来表 When working with electronics we often need to express quantities such as op amp gain, signal-to-noise ratio, common-mode-rejection ratio, and power supply rejection ratio whose values have very large spans e Therefore it is important to have a mechanism upon which we can represent a large range of values while using small numbers. This mechanism is called the ' decibel or 'db for short. note that decibels have no units This slide shows how to convert linear gain values to dB and vice versa o This equation shows how to convert from a linear gain in volts per volt to decibels For example, let's convert the closed loop gain of an op amp circuit from 100V/ to decibels o Substituting 100V/ for the linear gain in the given equation yields 40dB Similarly, given a gain in decibels we can convert it to a linear representation using this equation o While the previous example may not seem like a significant improvement in representing large numbers, let's look at the open loop gain, or Aol, of the OPA188 At 1Hz, the open loop gain is 130dB, which equates to a linear gain of 3, 162, 277V/ At 2MHz the open loop gain is OdB, which equates to a linear gain of 1V/ o Ultimately we find it's much easier to represent such a large range of values using decibels instead of volts per volt Poles& Zeros standard Form 极点&零点,标准型式 Zeros. Oz10z2 s +1 Roots in the numerator H(S=G 十 Poles o 10p2 Where Roots in the denominate 5=jxw=2xn f Pole- Magnitude in dB Zero-Magnitude in dB 20dB/dec 20dB/dec Freq( hz) Freq(Hz) HU TEXAS INSTRUMENTS 4 现在让我们来看看极点和零点'。 H(s)表示具有两个极点和两个零点的转移函数。 ●'s代表jwj- omega)而w(角频率)等于2pi*f ●在分子」中使该项为苓的角频率值被称为苓点。 ●在分母中使该项为零的角频率值被称为极点。 每个极点和零点被分解为(sMw+1)。这就是所谓的 standard form(标准形式),因为它 可以让你很容易地观察到极点和岺点。此外,请注意,直流增益被独立分解岀来。这 使得标准形式很容易确定低频增益。 ●极点显示于增益与频率的对应在左下角。请注意,该増益可以dB为单位。需要注意 的是增益经过极点以-20dB/ decade(十倍频程)的速率在下降。这是有道理的,因为更 高频率的分母变大造成了 magnitude(幅值)减小。下一张投影片,我们将看到更多这 方面的细节 ●岺点显小于增益与频率的对应在右下角。零点使得增益以+20dB/十倍频程的频率速 度上升。这也是有道理的,因为更高频率的分子变大引起的幅值增加。 Now let's take a look at poles' and zeros o H(s represents a transfer function with two poles and two zeros e The 's' represents jW ( -omega)where omega is 2*pi*f o The frequencies at which each term in the numerator equal zero are called zeros The frequencies at which each term in the denominator equal zero are called poles Each pole and zero is factored to be(s/omega +1). This is called the standard form because it allows you to easily determine the pole and zero by inspection Furthermore, notice that the dc gain is factored out of the transfer function this makes it easy to determine the low frequency gain A plot of gain VS frequency for a pole is shown in the lower left. Notice that the gain is given as a magnitude in dB. Note that the gain of a pole decreases, or rolls off, at a rate of -20dB/decade in frequency this makes sense because for higher frequencies the denominator will become large causing the magnitude to decrease In the next slide we will see more detail on this o A plot of gain Vs frequency for a zero is shown in the lower right. A zero causes the gain to increase at a rate of +20d B/decade in frequency. This also makes sense because for higher frequencies the numerator will become large causing the magnitude to increase Bode plots- Pole 100 0.707*GW=3dB Straight-Line Approximation Magnitude Function 20 dB/Decade G-204 Venture incB Pole Location= fy(Cutoff Freg) 101001k10k100k1MCM Frequency Hz) Magnitude(f <fp)=Gac(.g 100dB) Magnitude(t = fp)=-3dB P Magnitude (f > fp)=-20dB/Decade 101001k10k100k1M1oM Phase(f=fp)=-45° 50 245° Decade e Phase(0.Ifp<f< 10fp)=-45"/Decade 843@fx10 Phase (f>10fp)=-90 45@fs Phase(≤0.1f)=0 HU TEXAS INSTRUMENTS ●此页显示极点的方程式及其相关联的响应。后续,我们将提供一个真实世界的电路为 例。综观方稈式,你可以看到第一个公式的极点为复数。复数有实部和虚部。为实际 电路的复杂函数转换为幅值及 phase(相位)。第二个方程式表示幅值,第三个方程式 表小相位。正如前面讨论过的对线性增益取对数乘20得到10分贝的增益。 ●该图表以dB为单位表示幅值,以角度为单位表示相位。这种类型的图称为Bode ot(波特图)。注意到这两个水平轴和垂轴是对数的。计我们来看看在波特图的一些 关键点。首先,邗P为极点的频率。对于低于邗P频率的增益是定值,表示为GDC,换 句话说,在直流或岺频率处的增益将是GDC。在这个例子中GDC=100分贝。还要注 意,在們P增益衰减了3dB,或0.707倍的直流增益。最后,当频率大于fP,幅值以 20dB/10倍频程的速率下降 ●现在让我们考虑 phase shift(相移)随频率变化的图形。在极点频率的相移为-45度。相 位」小10倍的极点频率开始改变,并于大10倍的极点频率后停止改变。在此区域内 的斜率是每十倍频程下降45度。考虑频率低于小10倍的极点频率,其相移是0度。 对于频率高于大10倍的极点频率且相移为-90度。 ●请注意,波特图和相位图用近似直线来绘制。在现实中,函数会偏离这种近似。例如, 如果你考虑的是极点大J及小」倍频程的点‘。直线近似表示在这些点的值是分别为 0度和-90度。然而,实际的函数会稍微偏离。实际上,仿真软件可以用来获得精确 的值。 This slide illustrates the equations for a pole and its associated response. Later we will provide a real world circuit example for a pole. Looking at the equations, you can see that the first equation represents a pole as a complex number. Complex numbers have a real and imaginary part. For practical circuits the complex function is converted to a magnitude and phase. The second equation shows the magnitude and the third equation shows the phase. As discussed earlier 20 "log base 10 of the linear gain yields the gain in decibels The graphs show the magnitude in dB as well as the phase in degrees. This type of plot is called a Bode plot. Notice that both the horizontal axis and vertical axis are logarithmic. Let's look at some key points on the bode plot. First, the pole frequency as denoted by fP. For frequencies below fp the gain is constant and is denoted GDC. In other words, the gain at dc or zero frequency would be gdc. In this example gDC= 100dB. Also notice that the gain at fP is attenuated by 3db, or is 0.707 times the dc gain finally for frequencies greater than fp the magnitude plot rolls off at a rate of -20dB/decade Now let's consider the graph of phase shift Vs frequency. The phase shift at the pole frequency is -45 degrees. Phase begins to change one decade before the pole and stops changing one decade after the pole. In this region the slope is-45 degrees per decade. Considering frequencies that are lower than one decade below the pole, the phase shift is 0 degrees. For frequencies greater than one decade beyond the pole the phase shift is-90 degrees Notice that bode plots and phase plots are drawn using straight line approximations In reality the function will deviate from this approximation. For example, if you consider the points exactly one decade below and above the pole on the phase curve. The straight line approximation shows the value at these points to be 0 degrees and-90 degrees respectively. However, the actual function deviates slightly from the straight line approximation. In practice, simulation software can be used to obtain the actual values Bode plots- zero 1M前r Straight-Line Approximation +20 dB/Decad G=20!G Manitude ndB Actual Function +3dB 0 10k 100k 1M 10M Zero Location =fz +90 Magnitude(f <f,=odB +45@ 94.3,x10 Magnitude (f =f2=+3dB 7@ P Magnitude(f >fz=+20dB/Decade 45° Decade > Phase(f=fz)=+45° 101001k10k100k1M1M Phase(0. 1f<f 10f2)=+45/Decade 「 requency(Hz) Phase(f> 10fz > Phase(f0.1fz)=0° HU TEXAS INSTRUMENTS 6 ●此页显示零点的方程式及其相关联的响应。看着这些方式式中,可以看出,第一个方 程代表·个零点为复数。复数有实部和虚部。为实际电路的复杂函数转换为值和相 位。第二个方程表示的幅值和第三个方程表示的相位。以20L0g10的值函数以dB 为单位表示幅值。 ●该方程式用于生成波特图和相位图。让我们来看看在波特图的一些关健点。首先,零 点频率被衣示亿时。对于低于亿吋频率的增益是恒定的,衣示为GDC。在这个例了 中GDC=0分贝。还要注意,在亿的增益是在3分贝,或者是1.414倍的直流增益。 最后,对于频率高于亿时,幅值曲线以+20dB/十倍频程的速率增加。 ●现在让我们考虑相移随频率变化的曲线图。在零点频率的相移为45度。相位于小10 倍的极点频率开始改变,并于大10倍的极点频率后停止改变。在此区域内的斜率是 每十倍频程上升45度。考虑蜘率低于小10倍的极点频率,其相移是0度。对于频率 高于大10倍的极点频率且相移为+90度。 至此,我们已经看过频率响应的数学表示后。现在我们将数学连结于电路。 This slide illustrates the equations for a zero and its associated response. Looking at the equations, you can see that the first equation represents a zero as a complex number. Complex numbers have a real and imaginary part. For practical circuits the complex function is converted to a magnitude and phase. The second equation shows the magnitude and the third equation shows the phase. Taking 20 Log 10 of the magnitude function gives the magnitude in dB e The equations were used to generate the bode plot and phase plot. Let's look at some key points on the bode plot. First, the zero frequency is denoted fz. For frequencies below fz the gain is constant and is denoted GDC. In this example GDC =OdB. Also notice that the gain at fZ is at +3db, or is 1.4 14 times the dc gain Finally, for frequencies greater than fz, the magnitude plot increases at a rate of +20d B/decade o Now let's consider the graph of phase shift vs frequency. The phase shift at the zero frequency is +45 degrees. Phase begins to change one decade before the zero and stops changing one decade after the zero. In this region the slope is +45 degrees per decade. Considering frequencies that are lower than one decade below the zero, the phase shift is o degrees. For frequencies greater than one decade beyond the zero the phase shift is +90 degrees o Up to this point we have looked at the mathematics behind frequency response Now we will look at connecting the mathematics to electrical circuits

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