CR-RC Pulse Shaping

CR-RC Pulse Shaping

The simplest concept for pulse shaping is the use of a CR high-pass filter followed by an RC low-pass filter. Although this rudimentary filter is rarely used, it encompasses the basic concepts essential for understanding the higher-performance, active filter networks.

In the amplifier, the preamplifier signal first passes through a CR, high-pass filter (Fig. 7). This improves the signal-to-noise ratio by attenuating the low frequencies, which contain a lot of noise and very little signal. The decay time of the pulse is also shortened by this filter. For that reason, it is often referred to as a "CR differentiator." (Note that the differentiation function is not a true mathematical differentiation.)

Just before the pulse reaches the output of the amplifier, it passes through an RC low-pass filter (Fig. 8). This improves the signal-to-noise ratio by attenuating high frequencies, which contain excessive noise. The rise time of the pulse is lengthened by this filter. Although this filter does not perform an exact mathematical integration, it is frequently called an "RC integrator."

Figure 9 demonstrates the effect of combining the high-pass and low-pass filters in an amplifier to produce a unipolar output pulse. Typically, the differentiation time constant t_D = C_DR_D is set equal to the integration time constant t_I = R_IC_I, i.e., t_D = t_I = t. In that case, the output pulse rises slowly and reaches its maximum amplitude at 1.2t. The decay back to baseline is controlled primarily by the time constant of the CR differentiator. In this simple circuit there is no compensation for the long decay time of the preamplifier. Consequently, there is a small amplitude undershoot starting at about 7t. This undershoot decays back to baseline with the long time constant provided by the preamplifier output pulse.

This pulse-shaping technique can be used with scintillation detectors. For that application, the shaping time constant t should be chosen to be at least three times the decay time constant of the scintillator to ensure complete integration of the scintillator signal. The disadvantage in using CR-RC shaping with scintillation detectors is the much longer pulse duration compared with that of single-delay-line shaping.

On silicon and germanium detectors, the electronic noise at the preamplifier input makes a noticeable contribution to the energy resolution of the detector. This noise contribution can be minimized by choosing the appropriate amplifier shaping time constant. Figure 10 shows the effect. At short shaping time constants, the series noise component of the preamplifier is dominant. This noise is typically caused by thermal noise in the channel of the field-effect transistor, which is the first amplifying stage in the preamplifier. At long shaping time constants the parallel noise component at the preamplifier input dominates. This component arises from noise sources that are effectively in parallel with the detector at the preamplifier input (e.g., detector leakage current, gate leakage current in the field-effect transistor, and thermal noise in the preamplifier feedback resistor). The total noise at any shaping time constant is the square root of the sum of the squares of the series and parallel noise contributions. Consequently, the total noise has a minimum value at the shaping time constant where the series noise is equal to the parallel noise. This time constant is called the "noise corner time constant." The time constant for minimum noise will depend on the characteristics of the detector, the preamplifier, and the amplifier pulse shaping network. For silicon charged-particle detectors, the minimum noise usually occurs at time constants in the range from 0.5 to 1 µs. Generally, minimum noise for germanium and Si(Li) detectors is achieved at much longer time constants (in the range from 6 to 20 µs). Such long time constants impose a severe restriction on the counting rate capability. Conse-quently, energy resolution is often compromised by using shorter shaping time constants, in order to accommodate higher counting rates.

Figure 11 demonstrates the bipolar output pulse obtained when a second differentiator is inserted just before the amplifier output. Double differentiation produces a bipolar pulse with equal area in its positive and negative lobes. It is useful in minimizing baseline shift with varying counting rates when the electronic circuits following the amplifier are ac-coupled. It is also convenient for zero-crossover timing applications. The drawbacks of double differentiation relative to single CR differentiation are a longer pulse duration and a worse signal-to-noise ratio.

Figure 7. CR Differentiation.

Figure 8. RC Integration.

Figure 9. CR-RC Pulse Shaping.

Figure 10. The Dependence of the Preamplifier Noise Contribution
on the Amplifier Shaping Time Constant.

Figure 11. Doubly-Differentiated CR-RC-CR Shaping.