RC filter

On the analog side, tbe filter is often the conventional resistor-capacitor or RC filter. However, other possibihties exist. For example, one type of A/D converter is called an integrating A/D because the converter basically integrates the input signal over a fixed intei val of time. By making the intei val I/60th second, this approach provides excellent rejection of any 60-Hz electrical noise. 

As our first example we shall define what is known as a gaussian Markov process. This process, as we shall see later, is a good model for thermal noise or vacuum-tube-generated noise that has been passed through an RC filter with time constant a 1. We begin by defining two functions / and Q as follows... 

As an example of the use of Eq. (3-321), we shall calculate the output power density spectrum of an RC filter whose input consists of white noise. The RC filter in question is shown in Fig. 3-14. It is a simple matter to verify that the impulse response of this filter is given by... 

Fig. 3-14. An RC Filter Whose Input is a Current with Reference Direction Indicated by the Arrow, and Whose Output is a Voltage with Reference Polarity as Shown.

This result can now be used to verify our earlier statement that the gaussian Markov process defined by Eq. (3-218) is a good model for RC filtered vacuum tube noise. We have already seen that vacuum tube noise is essentially gaussian (as long as n is large) and that its spectrum is essentially white .70 A reasonable model for RC filtered... 

When simple electrical RC filters are treated, the truncated exponential e, x,H(x) is indispensable. Its transform is given by (2n) 1/2(1 — jco)/( 1 + co2). If the truncated exponential is reflected about the origin, eliminating H(x) and leaving e x, the imaginary part of the transform disappears. We obtain the transform (2/7c)1/2/(l + co2). This is the resonance contour, Cauchy distribution, or Lorentzian shape encountered previously in Section III.B. 

A brief treatment of the major contributors to optical spreading is certainly in order because it will clarify some of the considerations in designing and adjusting the spectrometer. In some cases it will also allow us to compute a useful approximation to the optical spread function. In the same vein, we later describe the behavior of the most common, useful, and easily described form of electrical noise smoothing the single-stage RC filter. 

The design of optimum electrical filters is a highly developed discipline that we shall not address here. It is worthwhile, however, to discuss briefly the simple single-stage RC filter that is widely used. When all factors are... 

As discussed in Chapter 1, any linear analog filter network may be used. Its performance may always be described by convolution with a filter function, here called sE. Many modern digital filters may be thus described. The instrument employed in the present work used a simple single-stage RC filter of the type analyzed in Section II.D of Chapter 2. 

Returning to the mainstream discussion of data preparation, we note that, for a 6-dB-per-octave-rolloff RC filter network in a lock-in amplifier, the continuous scan rate amounts to approximately one time constant per data point or 10 time constants per resolution element (Blass, 1976a). Some time is saved if only six data points are taken per resolution element. We have tried acquiring in this fashion, with no visible negative effects. 

Several parameters in Equation 29.19 can be adjusted to optimize the sensitivity of the spectrometer. For example, the incident power can be increased provided that the absorption is not saturated, and the extent of RC filtering can be increased (i.e., the Af decreased). For small samples it is sometimes feasible to use a higher frequency (higher field) spectrometer and thus increase the population difference between the two energy levels (cf. Eq. 29.11). But it can be shown that if the same geometry is maintained, the minimum detectable concentration is theoretically proportional to Vq1/2, so for electrochemical experiments the sensitivity actually decreases at higher frequencies, all other factors being equal. 

As to the rejection of charging and interfacial current contributions, Rosamilia and Miller [71, 72] were able to extend the scan rate this required also to increase the modulation frequency but ensuring always p< 1. Also the determination of / involves one band passfilter, one RC filter, and a full wave rectifier which leads to a lag in the 17 /E curve relative to that of the corresponding I/E curve. 

Sometimes, more effective control IC decoupling may be required — in which case we can use a small resistor (typically 10 to 22 C) from the input (supply) rail, going to a (separate) ceramic capacitor placed directly across the input and ground pins of the IC. This constitutes a small RC filter for the IC supply. 

Figure 7-4 Analyzing the First-order Low-pass RC Filter in the Frequency Domain...

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