Classical Analog Signal Processing ------------------------------------ Revision: 20-Sept-2023 For many detector types, the Physics signal of interest is the total charge that the detector produces in response to an event, e.g. the total charge that comes from a PMT. If the charge comes out of the detector over a period of a usec or a few usec then an integrator stage can be used to generate an analog signal, the step size of which is proportional to the total charge collected. To minimize the pickup of electrical noise and to minimize the parasitic capacitance at the input to such an integrator, it is often installed immediately adjacent to the detector and is sometimes called the Preamplifier for the detector. - The rise-time of the output signal from the integrator is determined by how long the charge (current) flows out of the detector for a given event as seen by the detector. - The "gain" of the integrator is controlled by the value of the capacitor in its feedback network with the simple relation V = C/Q. In many situations a detector will produce only a small amount of charge per event (perhaps even only a few femto-Coulombs per event). In these cases a low value of feedback capacitor is used - perhaps only a few pFd. In other case where lower gain is appropriate a larger value of feedback capacitor is used - in the hundred or few hundred pFd range. . The fall-time of the output signal from the integrator is determined by the value of a resistor that is placed across the capacitor in the integrator's feedback network. This is typically a high value resistor in the Meg Ohm or tens of Meg Ohm range. The exponential fall of the integrator's output means that it has essentially returned to zero in 5 or so time constants. The fall-time of the integrator (aka Preamplifier) is set to be 10 or more times longer than how long it takes to collect all of the charge from the detector for a given event. In this way the peak of the integrator's output closely approximates the total amount of charge collected from the detector. If the average time between events is significantly longer than the integrator's fall-time then most of the signal will be isolated in time from one event to the next. For higher event rates, or when two events randomly happen to occur very close to each other in time, the signal from the 2nd event will pile on top of the falling "tail" of the signal from the first event. - To reduce this pile-up effect, - to remove any DC offset from the integrator, and - to limit the low frequency response of the overall system which often may include a large 1/f noise component, the integrator is followed by a differentiator. This differentiator can be a simple CR circuit followed by a buffer. The rc time-constant of this CR differentiator is set so that the differentiator passes most of the fast rising edge of the integrator output and rejects most of the slowly falling tail of the output signal from the integrator. However, the output of this differentiator will not be a nice symmetric pulse because a small amount of the slowly falling output from the integrator will pass through the differentiator. The falling output from the integrator will push the differentiator's output below zero for as long as the integrator's falling tail lasts. To first order you can make the differentiator's output into a symmetric signal by using pole-zero cancellation. The differentiator responds to the slope of the integrator's falling tail. This falling tail has an exponential shape so its slope is proportional to its amplitude. Thus by just putting an appropriate value resistor across the differentiation capacitor you can remove the below zero section of the differentiator's output. This is called pole-zero cancellation. We now have a more or less symmetric signal and it's the beginning of what is often called a pseudo-Gaussian pulse. Much fancier filtering can be use which is often implemented with active filters. - For example it is often useful to cut-off the high frequency response, which has not yet been controlled in the signal processing steps so far. There is no point in allowing through any frequency components that are higher than what the detector itself can produce. Your can roll off the high frequency response with a RC stage. - One may also want to include a much longer time-constant CR circuit to remove any residual DC off-set. Recall that the pole-zero compensated differentiator will allow through some attenuated DC from the integrator. One of the advantages of the pseudo-Gaussian pulse shape is that it is convenient to digitize - it starts at zero and goes up to a smooth peak that is proportional to the Physics quantity of interest. I believe that this pseudo-Gaussian shape also represents an optimum filtering of the detector's signal in frequency space, i.e. it optimizes the Signal to Noise Ratio of the signal from the detector. This pulse shape is also convenient for use with a discriminator, threshold detector, or window detector. However, this smooth "uni-polar" pseudo-Gaussian shape is not good for making a timing measurement indicating when the detector's output signal happened. Such a timing signal can be made by doing a 2nd differentiation. This 2nd CR differentiator circuit should have an rc time-constant that is similar to half of the length of the pseudo-Gaussian pulse. The "bi-polar" output of the 2nd differentiator will have a nice falling edge zero crossing that can be used to make the timing measurement. In slang this 2nd differentiation is sometime called a "hard" differentiation. Other Points: - Many Integrators (sometimes called Preamplifiers) use a standard exponential fall time for their trailing edge. 50 usec is the common standard. - The Integrator (Preamplifier) is often built as part of the detector itself (e.g. built into the PMT Base). For testing or calibrating all of the electronics after this first stage one needs a pulse generator that makes a pulse shape that looks like the Integrator output pulse with its long falling tail. People call this type of pulser a "Tail Pulser". - This type of analog signal processing was used ahead of a Multi-Channel Analyzer for doing things like Gama ray Spectrometery. An amplifier that includes this string of analog signal processing stages was called a "Spectrometery Amplifier" or a "Research Amplifier". You seeing these amplifiers in the labs of folks who do a lot of work like you are doing now, i.e. figuring out how to best process the signal from a given detector for a given type of event input. - Ceramic Capacitors are not ideal. Besides dielectric absorption they can also have a significant voltage coefficient. For example an X7R ceramic capacitor, which is fine for power supply bypass work, may have a Dissipation Factor of 5 or 10% ---> not good for an Integrator. An X7R ceramic capacitor looses about 30% of its capacitance by the time it is at 1/2 of its rated voltage ---> not good for an Integrator or for an accurate RC or CR circuit. Ceramic Caps with other types of dielectric are fine for these applications. Using a ceramic cap with an operating voltage 10x that of your circuit voltage can help to minimize these problems. Some ceramic capacitor dielectrics are piezo electric and can make noise if vibrated or stressed.