Pole-zero cancellation

This produces the (pole-zero cancellation) root locus plot shown in Figure 5.18. When run, exampSlO.m allows the user to select the value of K that corresponds to ( = 0.7, and then uses this selected value to plot the step response. The text that appears in the command window is... 

A couple of quick observations First, Gc is the reciprocal of Gp. The poles of Gp are related to the zeros of Gc and vice versa—this is the basis of the so-called pole-zero cancellation.1 Second, the choice of C/R is not entirely arbitrary it must satisfy the closed-loop characteristic equation ... 

We used the term pole-zero cancellation at the beginning of this section. We should say a few more words to better appreciate the idea behind direct synthesis. Pole-zero cancellation is also referred to as cancellation compensation or dominant pole design. Of course, it is unlikely to... 

The idea is that we may cancel the (undesirable open-loop) poles of our process and replace them with a desirable closed-loop pole. Recall in Eq. (6-20) that Gc is sort of the reciprocal of Gp. The zeros of Gc are by choice the poles of Gp. The product of GcGp cancels everything out—hence the term pole-zero cancellation. To be redundant, we can rewrite the general design equation as... 

Since the system characteristic equation is 1 + GcGp = 0, our closed-loop poles are only dependent on our design parameter xc. A closed-loop system designed on the basis of pole-zero cancellation has drastically different behavior than a system without such cancellation. 

Too small a value of xc means too large a Kc and therefore saturation. System response is subject to imperfect pole-zero cancellation. 

Note 2 As we reduce the integral time constant from Xi = 3 min to exactly 2 min, we have the situation of pole-zero cancellation. The terms in the closed-loop characteristic equation cancel... 

The ensuing consequence of these pole-zero cancellations onto the corresponding amplitudes of Froissart resonances can be seen at once from the explicit formulae for df in terms of all the recovered poles and zeros ... 

For these prostate spectra, theoretically it would be possible to retrieve all 27 input resonances at Np = 54, since there are 54 unknowns (27 complex frequencies and 27 complex amplitudes) with 54 linear equations and 54 signal points needed. In fact, at Np = 54, precisely 27 resonances were retrieved. However, 15 of fhese were spurious for the normal glandular prostate, and 16 were spurious for the normal stromal prostate and malignant prostate. The pole-zero cancellations with zero-valued amplitudes were used to identify these resonances as spurious. 

The materials and structures associated with primary sensors contain dissipative, storage and inertial elements. These translate into the time derivatives appearing in the differential equation that models the sensor system. Hence another major defect is represented by the time (or frequency) response. The means to neutralise this imperfection involves filtering, which may be thought of in terms of pole-zero cancelation. If the device has a frequency response H s) then a cascaded filter of response G s) = 1/H s) will compensate for the non-ideal time response. The realisation of such a filter in analogue form presents a major obstacle that is greatly diminished in the digital case. 

Figure 4.20 Pole-zero cancellation in practice, showing the schematic circuit and the effect of cancellation on the tail of the pulse...

Some high specification amplifiers have built-in facilities to help with pole-zero cancellation. For example, the ORTEC 672 Spectroscopy Amplifier provides automatic cancellation at the press of a button. Some amplifiers provide LED over/under indicators to assist with the adjustment. Nevertheless, because of the importance of this setting I would advocate that every gamma spectrometry laboratory should have ready access to an oscilloscope. With experience, a quick look at the pulses coming from the preamplifier or the amplifier can quickly reassure one that everything is normal, or lead one to a solution if it isn t. 

Points to note are the pole-zero cancellation circuit at the start of the chain, so that the amplifier is matched to the... 

There are, however, difficulties. Accurate correction for pulse loss depends upon the pulser pulses accurately mimicking the detector pulses. The rise time and the fall times of the pulser pulses should be identical to those of detector pulses. Leaving aside the fact that preamplifier output pulses have a variable rise time, none of the readily available pulsers allow detailed control of the fall time. Bear in mind that the fall time of the preamphfier pulses depends upon the time constant of the feedback circuit in the preamplifier, and that pole-zero cancellation within the amplifier matches the shaping circuits to the input pulse fall time. The consequence of this is that it may not be possible to pole-zero correct the pulser pulses and the detector pulses together. At anything more than a low count rate, many detector pulses may be incorrectly measured by the ADC if they occur close in time to a pulser pulse. 

For best resolution, it is crucial that the pole-zero cancellation should be adjusted correctly. If high throughput is required, consider a trade-off between resolution and throughput. 

It is worth reminding the reader that, before getting too excited about poor charge collection, under-compensated pole-zero cancellation will also cause a tail on the low-energy side of peaks. Whatever the circumstances, the FIRST diagnostic check to make should be pole-zero cancellation. 

As with the preamplifier, the scintillation amphfier need not be of such a demanding low noise specification as would be needed for semiconductor systems. In the manufacturers catalogues, a distinction is commonly made between amplifier , suitable for low-resolution spectrometry, and spectroscopy amplifier intended for high-resolution spectrometry using semiconductor detectors. Typical simple amplifier modules provide pole-zero cancellation and automatic base line restoration. The pulse shaping time options provided are often limited on such instruments and may need to be selected internally. Because of the faster rise time of scintillation pulses, the time constants provided are usually within the range 0.2 to 2 or 3 (its. 

An oscilloscope for looking at pulses. Unless your amplifier has automatic correction built in to it, this item is essential for pole-zero cancellation. The specification does not need to be high, but must have a 50 MHz band-width, the ability to measure rise times down to a few nanoseconds and have a voltage sensitivity of 5mV per division. An oscilloscope with a storage option is a bonus, allowing a more considered examination of pulses. 

If the resolution of a detector has degraded with no apparent reason and all the usual checks and adjnstments, such as pole-zero cancellation, fail to cure the problem it may be that a thermal cycle is necessary. ... 

Determine the best time constant for the measurements intended, taking into account the required throughput and acceptable loss of resolution. For semi-Gaussian shaping at high count rates, this would not normally be greater than 2p,s. For RF preamplifier systems, pole zero cancellation would need to be checked. 

---