FED Evaluation (April 2001)

The analog behavior of the FED-PMC prototype was evaluated [73] with respect to the digitization of multiplexed APV data. The analog transfer characteristic of a digitizer is an important characteristic with respect to pulse distortion and noise sensitivity. Although clocked with the system frequency of $40\,\rm MHz$, the analog input bandwidth for pulse digitization must be considerably higher to avoid signal loss due to slow transients.

The transfer function of the FED has been obtained by comparing the input from a sine wave generator to the digitized output. According to the sampling theorem by H.NYQUIST^5.2, analog information can only be truly reconstructed when sampled with at least twice the highest frequency contained in the analog data. If this condition is violated, aliasing effectively leads to ``mirroring'' of frequencies above half the sampling rate into the base band. Nevertheless, the amplitude of these signals reveal information about the analog performance.

Figure: Transfer characteristic of the FED-PMC, measured with a sine wave input of $1.5\,\rm V$ amplitude (corresponding to the DC full range of the FED).

Fig.  shows the measured transfer function of the FED-PMC, normalized to the DC gain. A $-3\,\rm dB$ bandwidth of $65\,\rm MHz$ has been extracted from these data. The transfer function has been modelled by a third-order system with two conjugated complex poles and one real pole. In terms of control theory, the model consists of a $\rm PT_2$ element in series with a $\rm PT_1$. Its transfer function in the Laplace plane is given by

$$G(s)=\frac{1}{\left( 1+\frac{s}{\omega_1} \right) \left( 1+\frac{2 D s}{\omega_n}+\frac{s^2}{\omega_n^2} \right) }$$ (5.6)

$${\rm with}\quad\omega_1=\omega_n=2\pi\cdot 80\,{\rm MHz}\quad {\rm and}\quad D=0.6\quad,$$

where $\omega_1$ is the corner frequency of the $\rm PT_1$, $\omega_n$ is the natural oscillation frequency and $D$ is the damping factor of the $\rm PT_2$.

Moreover, the FED response to a rectangular input pulse with the width of one clock cycle was measured. Such a pulse emulates an APV channel with signal information, surrounded by pedestals. The response was measured by progressively delaying the input pulse relative to clock and trigger in steps of $1\,\rm ns$, similar to the ``sequential equivalent-time sampling'' method [74] employed by some digital oscilloscopes. For comparison, the analytical transfer function model has been used to calculate the response to the same rectangular input pulse.

Figure: Normalized FED-PMC large signal response to an input pulse of $25\,\rm ns$ width and $650\,\rm mV$ amplitude.

Fig.  shows both measured and calculated FED responses to an input pulse of $25\,\rm ns$ width and an amplitude corresponding to $3.5\,\rm MIPs$, assuming an APV25 input span of $8\,\rm MIPs$ projected onto the full FED input range. As seen from the matching curves, the model perfectly describes the system. An overshoot of less than $5\%$ is the result of the damping factor being less than one, indicating a pair of complex poles. After the transients have vanished, a relatively flat period of about $5\,\rm ns$ occurs towards the end of the pulse. The actual sampling should be performed in the center of this span, allowing a certain jitter in the ADC clock.

The FED noise was measured by applying stable DC voltages to the FED input. Depending on the input voltage relative to an LSB step, we found a noise figure peaking at $0.5\,\rm ADC$ counts. This includes the digitization noise and the input amplifier to approximately equal parts. Using the nominal calibration of $1500\,{\rm mV}/512\,\rm ADC$, the maximum noise contribution is $1.5\,\rm mV$, referred to the input. With a full range of $8\,\rm MIPs$, the noise figure corresponds to less than $175\,\rm e$ at the APV25 input.

Summarizing these results, the analog input stage of the FED reveals a bandwidth high enough to digitize the APV output without signal loss due to slow transients, while the FED noise contribution can be neglected compared to APV and optical link.