Update afe.rst

Author: kcaisley

docs/source/afe.rst

@@ -23,7 +23,11 @@ A typical analog read-out chain - also called analog front-end - for a semicondu
 
 Circuit Implementation
 ======================
-The simplified schematic in the figure below shows the implementation of the signal processing chain. The CSA is built around a low noise op-amp that is fed-back with a small capacitance :math:`C_f` and a large resistance :math:`R_f`. The feedback capacitance :math:`C_f` defines the charge transfer gain and the resistance :math:`R_f` allows for a slow discharge of :math:`C_f` and setting of the DC operating point of the op-amp. The output voltage of the charge sensitive amplifier in response to an input charge *Q* is a step function with an amplitude given by the expression:
+The simplified schematic in the figure below shows the implementation of the signal processing chain on the AFE board.
+
+Charge sensative amplifier
+--------------------------
+The CSA is from a low noise op-amp with a feedback loop comprising a small capacitance :math:`C_f` and a large resistance :math:`R_f`. The feedback capacitance :math:`C_f` defines the charge transfer gain while the resistance :math:`R_f` creates a slow discharge of the parralel :math:`C_f` capacitor and determines the DC operating point of the op-amp. The output voltage of the charge sensitive amplifier in response to an input charge *Q* is a step function with an amplitude given by the expression:
 
 .. math::
 
@@ -161,21 +165,22 @@ The exercises are grouped into three parts. In the first part the basic function
 
 The exercise 0 contains preparatory questions that should be answered before coming to the lab.
 
-.. admonition:: Preparatory questions
+.. admonition:: Pre-lab questions
+
+Please answer the following questions before lab:
 
-  #. The injection circuit generates a charge signal of the size :math:`C_{inj} \cdot V_{inj}`. What is the charge in femto-Coulombs generated by a voltage step of 100 mV with :math:`C_{inj} = 0.1 pF`? What is the charge step size for :math:`V_{inj} = 0.05 mV` (which corresponds to the effective LSB size of the injection voltage DAC)? How could we convert these charge values in Coulombs to units of the elementary charge (electrons)?
-  #. An ideal charge sensitive amplifier generates a step-like output waveform in response to an instantaneous charge signal at the input. What is the **CSA** output step amplitude for an input charge of 1 fC, given the feedback capacitance of 1 pF? If charge sensitivity is defined as the output amplitude per input charge, what is its unit?
+  #. The injection circuit generates a charge signal of the size :math:`C_{inj} \cdot V_{inj}`. What is the charge in femto-Coulombs generated by a voltage step of 100 mV with :math:`C_{inj} = 0.1 pF`? What is the charge step size for :math:`V_{inj} = 0.05 mV`? (This voltage step corresponds to the effective LSB size of the injection voltage DAC.) What is the expression to convert these charge values in Coulombs to units of the elementary charge (electrons)?
+  #. An ideal charge sensitive amplifier generates a step-like voltage output waveform in response to an instantaneous charge signal at the input. What is the **CSA** output step amplitude for an input charge of 1 fC, assuming the feedback capacitance is :math:`C_{f}=1 pF` ? If charge sensitivity is defined as the output amplitude per input charge, what is its unit?
   #. A shaping amplifier responds with a characteristic output pulse to a step-like input waveform. Assuming a step input and a CR-RC (high-pass + low-pass filter) with equal time constants, at what time does the output pulse peak, and what is the amplitude of that peak?
   #. To a first order, what is the charge sensitivity of our analog front-end chain (CSA + SHA)? Put another way, what is the pulse peak amplitude in mV at the shaper output, per fC (or electron) charge at the CSA input? (Note: Use the effective feedback capacitance value :math:`C_{f} = 1.39 pf` for your calculation.)
   #. Following the CSA and SHA, a discriminator with programmable threshold voltage is used to detect the pulse. This discriminator is essentially a comparator with one input voltage set via a DAC with an LSB size of 0.5 mV. What is the equivalent of this LSB size in units of fC or electrons ? Hint: Use the "transfer function" calculated above which relates charge input to voltage output (sensitivity).
   #. The threshold of the comparator should be set in a way that the noise is suppressed and only the signals are detected. What would happen if the threshold was too low, what would happen if it was too high? How could the terms purity and efficiency of the detection process be defined in this context? What happens if baseline and signal fluctuations are getting too close to each other?
   #. The term 'equivalent-noise-charge' (ENC) expresses the voltage noise in a measured output signal, in terms of the equivalent input charge that would produce it. Given the charge sensitivity we calculated above, what would be the ENC for a measured 10mV output noise amplitude?
   #. How are the discriminating comparator's Gaussian distribution and the error-function related? How can one extract the width (sigma) and the mean (lambda) of the underlying Gaussian distribution from a measured error function? How is the noise calculated from the slope of the error function at the 50 % point?
 
-  .. from Q5 commented out, as this is hard
-  .. #. Draw a sketch of an amplitude histogram of an ideal noise-free system. It consist of two delta-like peaks: one for the baseline and one for the signal amplitude produced by a constant input charge. In a real system, however, noise is overlaying the ideal signals, leading to fluctuations of the baseline and signal amplitudes. Modify the amplitude histogram to reflect these fluctuations (assume a Gaussian distribution of the noise).
-.. from Q6: Draw an optimum threshold in your amplitude histogram.
-.. from Q7 #. The term 'equivalent-noise-charge' (ENC) represents the quantity of electrons at the input of an ideal (i.e. noise-free) signal chain that would produce the same  amplitude at the output as the noise alone would in a real system. What is the ENC value for a noise amplitude of 10 mV given the charge sensitivity calculated above?
+#. Draw a sketch of an amplitude histogram of an ideal noise-free system. It consist of two delta-like peaks: one for the baseline and one for the signal amplitude produced by a constant input charge. In a real system, however, noise is overlaying the ideal signals, leading to fluctuations of the baseline and signal amplitudes. Modify the amplitude histogram to reflect these fluctuations (assume a Gaussian distribution of the noise).
+Q6: Draw an optimum threshold in your amplitude histogram.
+Q7 #. The term 'equivalent-noise-charge' (ENC) represents the quantity of electrons at the input of an ideal (i.e. noise-free) signal chain that would produce the same  amplitude at the output as the noise alone would in a real system. What is the ENC value for a noise amplitude of 10 mV given the charge sensitivity calculated above?
   .. Q10 #. Advanced tasks: Calculate and plot the time-over-threshold as a function of the ratio of CR-RC shaper peak amplitude and threshold voltage. You can do that either by inverting the mathematical expression for the shaper pulse waveform (-> Lambert W function) or by implementing a function representing the shaper pulse waveform in Python and numerically evaluating TOT width for a range of amplitude values at a fixed threshold. Note: this function will be useful to fit measured pulse waveforms (see the later exercises). What is the relation between the TOT and the injected charge? What is the effect of the shaping time constant on the TOT? Assume the TOT counter has a resolution of 25 ns and a maximum count of 255. What is the maximum detectable TOT width in this case? Assume the maximum amplitude to threshold ratio is 10. What is the maximum shaping time constant that can be used in this case?
 
 
@@ -185,7 +190,7 @@ The exercise 0 contains preparatory questions that should be answered before com
   #. Implement a scan routine to measure the s-curve of the system. The s-curve is obtained by measuring the hit probability as a function of the injected charge. The charge is varied by changing the injection voltage. The hit probability is calculated by counting the number of hits (using the comparator output pulse) for a given charge step in relation to the total number of injections. Be sure to convert the x-axis of the s-curve from DAC units to charge units (in electrons) **Note:** The effective value of the feedback capacitance is :math:`C_{f}^{eff} = 1.39 pF` due to the parasitic capacitance of the PCB traces and the feedback resistor which add to the nominal value :math:`C_{f} = 1.0 pF`
   #. Acquire s-curves for different shaping time constants. What is the effect of the shaping time on the noise? (Do not yet connect a sensor diode to the CSA, as we just want characterize the AFE circuit in isolation.)
 
-.. .. admonition:: Exercise 1. Waveform measurements
+.. admonition:: Exercise 1. Waveform measurements
 
 ..   This exercise is intended to familiarize you with the analog front-end hardware and the control software. The goal is to observe the different signals of the analog front-end chain (CSA, SHA, COMP) and to understand the effect of the different circuit parameters on the signal shape. To monitor the signal waveform, connect an oscilloscope to the LEMO socket **OUTPUT**. Use the jumper bank in front of the LEMO socket to select the signal to be monitored manually (**CSA, HPF, SHA, COMP**) or use the setting **MUX** to select the signal to be monitored via your program code with the **SPI** interface. Note: As mentioned in the circuit description above, the shaper circuit adds a total gain of 1000 to the CSA output signal. This gain is split in three gain stages with G=10 that are distributed along the signal chain in front of the **CSA**, the **HPF**, and the **SHA** output, respectively. The **CSA** output is amplified by 10, the **HPF** accumulated amplification is 100 and the shaper output **SHA** finally accumulates the total gain of 1000.