Time constant noise filtering

In general, the longer the sweep time the better the sensitivity since the filter time constant parameter can be set longer with consequent improvement in signal-to-noise ratio. In practice, however, sweep times are usually set in accordance with the expected lifetime of the radical species, the stability of the instrument, and the patience of the operator. Decay of the radical or drift of the spectrometer during a scan is clearly undesirable. The sweep time is most commonly set in the range 4-10 min. 

Filtering. For control variables, when filtering is needed, use a first-order filter to reduce the effects of high-frequency noise. Do not excessively filter measurements unless absolutely necessary. The filter time constant, x, should be much less than feedback dynamics. 

Figure 15.7.1 A low-current transducer for insertion between the working electrode and the current follower (CF) of a potentiostat. Depending on which feedback resistor is chosen in the first stage, the ampUfication factor in this system is 10, 10, or 10. The capacitors in the feedback loops provide some filtering (time constant, 100 fjLs). An inductor-capacitor network was inserted in each power supply connection to minimize noise coupling. [Reprinted with permission from H.-J. Huang, P. He, and L. R. Faulkner, Anal. Chem., 58, 2889 (1986). Copyright 1986, American Chemical Society.]...

After the signal emerges from the lock-in amplifier it still contains a considerable amount of noise. Most of the noise contributions to the signal can be eliminated by passing the signal through a low-pass filter. The filter time constant is a measure of the cutoff frequency of the filter. If accurate linewidth and g-factor... 

We need to consider any noise that may be present in the DV. In our case this was previously only an indication and so any noise was not passed through to a control valve. This will no longer be the case and so filtering may be necessary. Ideally the filter should be put in place before steptesting but if this has been overlooked then we can compensate for its addition by increasing Tl by the filter time constant (t ). 

When noise is present in the measured process output, any derivative action should be accompanied by a filter. However, the problem is often that, as the time constant of the derivative filter is increased to cope with the measurement noise, the closed-loop performance degrades, requiring re-tuning of the controller parameters. This problem is illustrated using the IMC-PID rules appUed to Process A with fd = 0.67. Figure 7.7 shows the closed-loop responses to a negative unit step load disturbance with a derivative filter time constant equal to O.lrp. Figure 7.8 shows the closed-loop responses... 

The filter time constant jp in (17-4) should be much smaller than the dominant time constant of the process Tdom to avoid introducing a significant dynamic lag in the feedback control loop. For example, choosing Tp < 0.1 Tdom generally satisfies this requirement. On the other hand, if the noise amplitude is high, then a larger value of may be required to smooth the noisy measurements. The frequency range of the noise is another important consideration. Suppose that the lowest noise frequency expected is denoted by (Ojv Then Tp should be selected so that (O/r < where (x)p = 1/t. For example, suppose we specify (Op = which corresponds to Tp = 10/(O/. Then... 

Representative results for high frequency sinusoidal noise are shown in Fig. 17.6. The square-wave with additive noise, the signal to be filtered, is shown in Fig. 17.6a, and the performance of two analog exponential filters is shown in Fig. 17.66. Choosing a relatively large filter time constant (t/t = 0.4 min) results in a filtered signal that contains less noise but is more sluggish, compared to the response for ip = 0-1... 

The patch clamp electronic circuit consists of a current to voltage converter provided with a feedback resistor of high value which sets the amplification factor and with a low leakage and low noise input current in order to allow a resolution of a few picoamperes. A high order active filter and a capacitance compensating circuit are usually added in order to improve the noise and time constant performances of the current to voltage converter. 

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... 

By the use of frequency filters (RG circuits) which cut off the high frequency noise from a low frequency signal. Care must be taken to avoid distortion of the signal. An RC circuit as shown below is a low pass filter of time constant t = RG. This gives a rough value of the cut-off frequency. 

Phosphorus was monitored at 214.9nm (second order). The analog signal was taken out through the profile mode of a computer program. In order to filter noise, a 50k 2 resistor and 20 gF condenser were installed before the recording, giving an approximate time constant of Is. 

One final comment regarding the time constant in chromatographic detection limits concerns the noise level observed on the baseline. Large time constants serve as efficient filters for (high frequency) noise. Therefore, if a great reduction of the time constant is not... 

The detector time constant (or digital filter for modern instruments) is used to remove high-frequency noise. If the detector time constant is too slow, the observed peaks will be broadened. The 1/3 rule may be applied here as well As a rule of thumb, the maximum detector time constant (seconds) tolerable is about 1/3 the standard deviation of the peak in seconds. Peaks 1-2 seconds in width require a time constant no larger than 0.1 second. Figure 17-19 illustrates these points. Although noise-free chromatograms are desirable, resolution and sensitivity can be adversely affected by excessively large time constants due to peak distortion. 

The Derivative mode is sometimes referred to as rate because it applies control action proportional to the rate of change of its input. Most controllers use the process measurement, rather than the error, for this input in order to prevent an exaggerated response to step changes in the setpoint. Also, noise in the process measurement is attenuated by an inherent filter on the Derivative term, which has a time constant 1/8 to 1/10 of the Derivative time. Even with these considerations, process noise is a major deterrent to the use of Derivative mode. 

The pulse-shaping methods used today are based on combinations of RC circuits and delay lines. For example, the use of a CR-RC circuit combination produces the pulse shown in Fig. 10.15. The exact shape and size of the output pulse depends on the relative magnitudes of the time constants Cjf and C2f 2-The use of the CR-RC circuit combination provides, in addition to pulse shaping, a better signal-to-noise ratio by acting as high-pass and low-pass filter for undesired frequencies. 

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