Simulation and measurement of image charge detection with printed-circuit-board detector and differential amplifier

Image charge detection (ICD) is a technique for non-destructively observing a charged particle in motion and can be used to infer the particle’s charge, velocity, mass, and other characteristics. In its simplest form, ICD functions as follows: As a charged particle passes near a conducting material, a charge is induced on the surface of that material. This induced charge varies in magnitude as the particle continues its trajectory, generating a small electrical current. Such a signal can then be manipulated and analyzed using current amplifying electronics.

ICD has been used in a variety of applications including ion-traps,^1,2 ion implantation,³ and various forms of mass spectrometry.^4–7 The device presented here is intended for use in charge detection mass spectrometry (CDMS). CDMS is a technique for the non-destructive analysis of charged particles such as dust, polymers,^8–10 biomolecules,^11,12 and aerosols.¹³ CDMS is particularly useful in the detection of large particles because its mass detection limit is much higher than that of traditional mass spectrometry.^11,14,15 In CDMS applications, the charge of the particle is first determined directly from the signal it induces. Then, the particle’s mass can be measured by accelerating the particle with an electric field and using the time of arrival of the resulting signal peaks to determine the particle’s acceleration. Acceleration, combined with the particle’s charge and the applied electric field, can then be used to deduce mass.¹⁶ This technique is known as time-of-flight (TOF) CDMS.

ICD is the primary mechanism for CDMS. ICD for the analysis of microparticles was first reported in 1960.¹⁷ This result featured pairs of conducting plates inside a shielded cylinder. In 1995, Fuerstenau and Benner presented the first biological application of this technique.¹¹ Their design consisted of a conductive sensing cylinder co-linearly flanked by two grounded cylinders of the same size. Since then, many ICD and virtually every CDMS system has utilized cylindrical electrodes, often with multiple sensing and grounded stages to lower the charge detection limit through repeated measurements.^2,3,7,18 Improved noise performance through repeated measurements is also achieved through the use of ion traps.^14,19,20 In 2013, the ability to use copper electrodes on printed circuit boards (PCBs) for detecting image charges in CDMS was reported in Ref. 18. This development is significant because it greatly simplifies electrode alignment and manufacturing. It also allows the charge-sensitive electronics to be integrated directly with the detector, eliminating the need for excessive wires or cables and reducing complexity and parasitic capacitance.

Despite the widespread use of image charge detection, little work has been done by way of modeling interactions between charged particles, sensing electrodes, and amplifying electronics. This has led to an incomplete understanding of how the signal output from the sensing electronics relates back to the charged particle. For example, the peak amplitude and the area under the transient output signal curve are both affected by the charge of the input ion; however, it is unclear which of those two measures are most relevant in determining ion charge. While there has been some speculation about the effect that the spacing between adjacent electrodes has on the rise time of the current signal,¹⁸ questions about the relationship between electrode geometry, particle trajectory, and the resulting induced signal still remain unanswered. A detailed model would illuminate some of these unanswered questions. A simple model was proposed in Refs. 21 and 22 to predict the electrical current produced in an electrode by a moving charge. Their method is extremely powerful and is often used in conjunction with software simulators because closed-form mathematical solutions to their proposed equations only exist for certain simple geometries. Recently, modeling methods relying on this theorem have been proposed for ICD using simulators such as SIMION.^3,23,24 However, such methods have typically only been reported using cylindrical electrode geometries.

Another fundamental challenge of ICD is reduction of the detection limit, with the ideal detector having the capacity to detect a single elementary charge. One approach for reducing this limit is by cascading multiple sensing electrodes in series. This allows for noise from the electronics to be averaged out across the multiple signals acquired, lowering the limit of detection by a factor of $N$ (N being the number of sensing electrodes).^16,25,26 Similarly, the noise limit can also be lowered by repeated passes across the sensing electrodes through the use of electrostatic mirror electrodes.^4,27,28

While the above technique is extremely useful and can easily be employed by the design presented here, the fundamental noise limit is set by the electronics themselves. A common approach for improving the performance of the electronics is to cool the device or the input transistor.²⁹ Since most ICD systems use common singled-ended amplifier topologies, cooling is often the only option explored,^9,18,26 resulting in expensive and energy hungry systems. However, the signals induced onto the sensing and grounded electrodes are near differential (as will be shown later) and lend the detector to use with a differential amplifier.^2,25 Figure 1 shows a diagram of a typical PCB detector. In a single-ended ICD system, V_i,neg would be connected to ground and V_i,pos to the input of the amplifier. For a differential amplifier, both sets of electrodes are connected to amplifier inputs. We show that by utilizing a differential topology, the noise floor of the ICD system can be significantly reduced. For the current work, we used the differential amplifier reported by Ref. 30 and report direct experimental comparisons between the singled-ended and differential topologies in order to highlight the improved noise performance of the latter. The noise values reported here are not given using averaging of repeated stages. This is to highlight the performance of the electronics themselves.

We also present a novel simulation method for modeling the interaction between the charged particle and sensing electrodes. This method not only gives insight into basic questions about ICD, but it also fills the gap between the ion–electrode interaction and charge-sensing electronics. The output signal from the simulation can easily be imported into circuit simulation software, providing a comprehensive view of how charge-sensitive electronics will behave when charged particles pass through the detector. The insight provided by this simulation technique facilitates design improvements not only to the electronics but also to the sensing electrodes themselves. The result of such understanding will be highly efficient, low-noise ICD systems.

The first goal in developing an accurate simulation model is to better understand the signal that is induced onto the electrodes by the charged particle and how that signal is processed by the electronics. Almost all ICD systems consist of a charge-sensitive preamplifier followed by subsequent shaping and filtering electronics.^{5,9,16,18,26,31} The shaping stages are typically used to convert the preamplifier output to a series of peaks, making it easier to extract time-of-flight information from the signal. However, in order to definitively answer whether the relevant information from the output waveform is the peak amplitude or area under the curve, we focus directly on the output of the preamplifier before the subsequent shaping stage. The second primary objective of this model is to gain insight into how electrode geometry and particle trajectory affect the induced signal. For example, Fig. 1 shows detecting electrodes identically printed on two PCBs. The PCBs are spaced apart from each other to create a channel through which charged particles can travel. Figure 1 also depicts a particle aligned to travel along the center axis of the detector. We seek to understand what the induced signal will look like for particle trajectories that do not follow the center axis. If the particle’s path is shifted along the x-axis, for example, how will that affect the induced charge on the electrodes? A thorough simulation technique provides insight into the design of electrodes in such a way so as to minimize the uncertainty in how much charge from the particle is induced onto the sensing electrodes. This technique also allows us to design the electrodes such that the resulting signal is compatible with the charge sensitive electronics. For example, the simulation can predict how the spacing between electrodes affects the rise time of the induced signal. The bandwidth of the sensing amplifiers determines what range of rise times the device can tolerate. Thus, given the expected velocity of the incoming particle, we can determine what electrode spacing we need to ensure that the resulting signal falls within the bandwidth of our amplifier. The flexibility in design afforded by the PCB electrodes enables us to easily make such adjustments. Therefore, the simulation model presented here, coupled with the PCB sensing electrodes, allows us to easily explore new ideas and verify their functionality.

The chosen electrostatic simulator was ANSYS Maxwell 3D, which uses a finite element method (FEM) for calculating electric fields.³² We verified the accuracy of the software by simulating the interaction between a charged particle and a single, square copper plate measuring 1 × 1 cm². Although this geometry is not realistic for ICD, it was chosen to match that of Ref. 33, in which Nelson et al. compared the accuracy of the image charge method with a FEM for calculating the surface charge density induced on a finite conducting plate by a charged particle. We then plotted the induced charge distribution on the plate and compared the results. The simulated plots are shown in Fig. 2. The white shading denotes areas of high charge density and the black denotes those of low charge density. The particle is also shown in each plot. For clarity, a charge density vs position plot of a cross section taken through the middle of the plate is shown in Figs. 2(c) and 2(d). As expected, when the particle is close to the plate, the peak of the charge density is found directly below the particle. However, as the particle moves further away from the plate, charge density peaks accumulate on the edges of the plate, with the highest peaks found in the corners. These results are consistent with those found in Ref. 33.

In order to produce signals generated by a moving particle, we created multiple Maxwell files, each with the charged particle located at an incremental position along the particle trajectory. The accuracy of the method is maintained because the particles in these applications travel far slower (typically on the order of 10 m/s) than the speed of light. Because of the computational cost of running so many simulations (over 1000 simulations per particle trajectory), the analysis is performed using the Fulton supercomputer at Brigham Young University. The total charge induced onto the plate is then extracted from each of these files by integrating the charge density over the surface of the plate, producing a total induced charge vs position plot. Further processing of this plot along with an assumed particle velocity allows for differentiation of the charge waveform, producing a current vs time plot. A flowchart summarizing this process is shown in Fig. 3. To compare this numerical simulation with a closed form solution taken from image charge theory, we used this process to generate charge vs position and current vs time plots for a charged particle passing over a single conducting plate 1 cm by 1 cm in size. The image charge theory solution is given as follows:³³ 

where Q is the particle charge; x_pi is the initial x position of the particle; x_o and y_o are the x and y positions of the particle, respectively; h is the height of the particle above the plate; and v is the particle velocity. The calculated charge density was integrated over the size of the plate to obtain charge vs particle position. The resulting plots are shown in Fig. 4 with the black lines representing the image theory plots and the gray lines showing those derived from simulation (dotted line represents the location of the plate). A particle speed of 50 m/s was assumed in order to produce the current vs time result. The plots were normalized by the charge on the particle to show the percentage of the original charge induced on the plate. When compared to a FEM, image theory should underestimate the total charge induced by the particle and the discrepancy between the two methods should become more pronounced as the particle moves further away from the plate;³² this is confirmed by our simulation and shown in Fig. 4(a). This also causes image theory to predict a much faster rise time, which reveals higher current peaks once the charge signal is differentiated, as shown in Fig. 4(b).

Not only is the simulation method more accurate than image theory, but also it can easily be applied to actual detector geometries that are too complicated to analyze using image theory, for instance, designs with multiple electrodes. The following signals were derived using the electrode design of Fig. 1. As shown in Fig. 1, the first and last electrodes are connected to the negative input terminal of the amplifier, while the middle electrode is connected to the positive input. Both the top and bottom PCBs have the same connections such that when a particle passes through the space between PCBs, the induced signals on the top and bottom electrodes are routed to the same amplifier input. The dimensions of the detector are labeled in Fig. 1. Using the same process described in Fig. 3, the black curves in Fig. 5 show the signals induced on V_i,pos from a particle traveling through the center axis of the detector, as shown in Fig. 1 (trajectory 1), with Fig. 5(a) showing the normalized charge vs position and Fig. 5(b) showing the current vs time. The current vs time plot is the most relevant for our application as this is the signal that will be input into the preamplifier. The gray plot of Fig. 5 will be explained later.

It is worth noting that the computational cost of Maxwell is so high because it employs a FEM in solving Gauss’s law in the problem region. Because SIMION utilizes the Shockley–Ramo theorem, it can compute induced image currents in a fraction of the time. We ran the charged-particle simulation using the geometry of Fig. 1 in SIMION to compare its results with Maxwell. The resulting current vs position plots are shown in Fig. 6(a). While there are some differences between the two signals, they both produce current peaks of the same general shape. These signals were integrated, resulting in the induced charge plots of Fig. 6(b). Again, there are subtle differences between the two plots. SIMION estimates less than the complete charge induced onto the detector. However, the purpose of the current work is not to determine which method is more accurate but to establish a link between the detector geometry, input current signal, and charge sensitive electronics. For the remainder of this work, we continue to use Maxwell because it also has the capability to simulate detector capacitance, which is crucial for minimizing the noise floor of the ICD system.

The amount of charge that the particle induces on the electrode is wholly determined by the geometry of the conducting electrodes and the trajectory of the charged particle. If we reduce the sensing electrodes to two simple copper squares spaced a very large distance apart and place a charged particle with charge Q exactly in between the squares, there would be almost no charge induced on the electrodes. However, as the electrodes move in toward each other, maintaining the charged particle in the middle, then more and more charge from the particle will be induced on the plates until each plate holds a total charge of Q/2. If we were to then move the particle toward the edge of the electrodes, while maintaining it on a plane parallel to and evenly spaced between the plates, eventually less than the total particle charge would be induced onto the plates. Thus, there is a region on this plane through which if the particle passes, all of its charge will be induced onto the electrodes. Maintaining a constant electrode size, this region grows as the plates come closer together. Likewise, the region grows if the distance between plates is held constant, but the size of the individual plates increases. If the charged particle travels within this region of complete induction, then it can vary in distance from one plate to the other and its full charge will still be induced on the plates collectively, enabling an accurate charge reading. For the geometry of Fig. 1, this region is 28 mm wide and is represented by the light gray, cross-hatched rectangle shown overlaying the bottom electrodes. The gray plot in Fig. 5 resulted from simulating a particle that passed through the plates outside of this region (trajectory 2). Trajectory 2 (depicted in Fig. 1) is simply trajectory 1 shifted in the x direction to the edge of the electrodes. As shown in Fig. 5, this particle induces a little over half of its charge onto the plates, causing the resulting current peak to be smaller and subsequently producing an inaccurate charge reading. This is an outlying case as we can expect most particles to pass through the region of complete induction. We can therefore infer that an output signal that contains peaks of varying heights from a single particle is potentially due to non-ideal particle trajectories, and to a certain extent, we can estimate that trajectory based on the peak heights.

Finally, in order to model the response of our amplifier to the detector signals, we simulated the induced current using the PCB geometry of Fig. 1. The result is shown in Fig. 7. The gray plot represents the current induced on the negative electrodes, while the black represents that of the positive. Simulation confirms that the current peaks correspond to the edge of the electrode and that a smaller spacing between electrodes causes a shorter rise time, which leads to higher amplitude current peaks. This can be seen by noting that the first gray peak in the current vs time plot is slightly lower in amplitude than the first black peak.

A differential amplifier is well suited for our ICD system because the signals produced by the repeated electrode pattern of the detector are nearly differential (Fig. 7). As the charged particle leaves the vicinity of a plate, the induced charge on that plate begins to recede, causing a negative current peak to occur. Charge is simultaneously building on the subsequent plate, producing a positive current peak. These two peaks are identical in shape and are separated by a small time interval caused by the spacing between the electrodes. Since this time interval is small, the signal is effectively doubled when the positive input is subtracted from the negative, enabling all the advantages that come with using a differential amplifier such as robustness against common-mode input noise, supply noise, and coupling.

The single-ended and differential configurations of the amplifier can be modeled as shown in Figs. 8(a) and 8(b), respectively. The detector, wire, and PCB parasitic capacitances are lumped together and represented as C_p1 in the single-ended amplifier model. The differential amplifier model lumps the parasitic shunt capacitance as C_p2 and the parasitic coupling capacitance between the two inputs as C_x. The parasitic capacitance is strongly determined by the electrode geometry as the detector is typically the largest contributor of capacitance. To study the effects of the parasitic input capacitance on the charge–voltage gain (A_Q-V) and input-referred noise (IRN) of the amplifier, we sweep the input capacitance (C_p1 for the single-ended amplifier, C_p2 and C_x are scaled by the same factor and correspond to the differential amplifier) from 1 pF to 100 pF and observe the change in A_Q-V and IRN using Cadence simulations³⁴ (Fig. 9). The solid line represents the differential amplifier, and the dashed line represents the single-ended amplifier. In theory, the gain of a differential configuration should be double that of a single-ended configuration, and the IRN of a differential configuration should outperform the single-ended configuration by a factor of $2$⁠. This is confirmed by our simulation. Both amplifiers show reduced A_Q-V and IRN as the input capacitance increases. Because the detector capacitance in the differential configuration adds to the capacitance between input nodes, rather than from one input node to ground, the differential amplifier’s response to higher detector capacitance is less severe than that of the single-ended amplifier. This is a major advantage of using a differential amplifier.

The current signals shown in Fig. 7 were imported into Cadence to simulate the response of our charge amplifier to a charged particle passing through the PCB electrodes. Input signals were derived from particles with assumed speeds of 20 m/s, 50 m/s, and 70 m/s. For all the simulations in this work, the amplifier feedback capacitance was selected at 10 fF. The results are shown in Fig. 10. The simulation in Cadence was performed with C_x and C_p2 values set to 2 pF each, which is realistic for this type of device. At this input capacitance, A_Q-V is expected to be 16.28 µV/e⁻. The output of the amplifier in response to a charged particle aligns with this gain. With a charge of 1000 e⁻ on the particle, the output ramp has a peak-to-peak amplitude of roughly 16.3 mV. The decrease in voltage that occurs after a peak is due to a feedback resistor (128 GΩ, realized with a pseudo-resistor³⁵) in parallel with the feedback capacitors.

The charged particle simulation paired with Cadence definitively reveals that the peak amplitude of the output waveform, not the area under the curve, is proportional to the charge on the input particle. Although the input current peak is a function of particle velocity, accurate particle charge readings can still be obtained because the amplifier’s output voltage amplitude is velocity independent, as demonstrated in Fig. 10. This same simulation technique can be easily employed to predict the response of other amplifier topologies. The detailed knowledge of the output waveform of the first amplifying stage can also be hugely beneficial in designing subsequent shaping electronics and predicting their performance.

Finally, to physically verify the advantages of using a differential amplifier, along with the accuracy of the discussed simulation method, we designed an experimental setup, as shown in Fig. 11. The primary challenge associated with designing such a system is separating charged particles before they pass through the detector as the presence of multiple particles leads to a signal overlap and greatly reduces the likelihood for accurate charge measurement. To create charged particles, we place glass spheres ranging in diameter from 100 μm to 500 µm on a conductive plate connected to a 2-kV source. The particles then enter the inlet tube, carried by the flow of air produced by the Venturi vacuum generator. The purpose of the inlet tube is to increase the probability of detecting a single charged particle. This is done through two mechanisms: 1. the tube selects only a small group of particles at a time and 2. as particles of various surface areas and masses are exposed to roughly the same amount of force (provided by the airflow) for an extended period of time, they naturally separate by mass. The speed of the particle can be adjusted by tuning the output flow from the vacuum generator or by varying the pressure of the compressed air source.

We focus on two comparisons in our tests. First, the widely used Amptek A250^{9,18,26,31,36} vs the presented custom amplifier in the differential configuration. The Amptek features a feedback capacitance and resistance of 1 pf and 1 GΩ, respectively, and the device was used with an uncooled input junction gate field-effect transistor (JFET) that had a capacitance of roughly 8 pF.³⁷ Both devices function as preamplifiers and thus largely set the noise floor of the charge-sensing system. Second, we compare the custom amplifier in a single-ended configuration vs a differential configuration in order to show the superiority of the latter in this particular application. Figure 12 shows the diagram of the PCB used to take these measurements. As shown in the diagram, the single-ended detector consists of two sensing electrodes separated by grounding electrodes. The differential detector features two negative and two positive input electrodes. By cascading the single-ended and differential sensing electrodes, a comparison can be made between the two amplifiers as they respond to the same particle. The data were acquired using an oscilloscope. The result of the Amptek vs custom design test is shown in Fig. 13. These plots were low-pass filtered digitally using a 5 kHz cutoff frequency before they were analyzed. The gain of Amptek was measured to be 0.082 µV/e⁻, while the gain of the differential amplifier is 7.96 µV/e⁻. Using those gains to refer the noise to the input, we find that the measured IRN of the custom design outperforms that of Amptek by a factor of 13, with the custom amplifier exhibiting an IRN of 1700 e⁻ and the Amptek exhibiting 23 000 e⁻. The sensitivity of the custom charge amplifier is so much greater than that of the Amptek that even charges practically invisible to the Amptek appear distinctively in the custom amplifier output.

We validate our simulation method by comparing the simulation and measurement results. We generated simulated input current signals using the PCB in Fig. 12. Those signals were then input into the custom amplifier in differential and single-ended modes. The particle’s charge and velocity used in the simulation were chosen to match the experimental result in Fig. 14(a). The output of the Cadence simulation is shown in Fig. 14(b), with the experimental results shown in Fig. 14(a). The experimental signals were filtered digitally using a low-pass filter with a cutoff frequency of 5 kHz. Overall, the waveform shape and noise characteristics of the experimental data are in excellent agreement with simulation. For the differential signal, however, there is a slight discrepancy in shape between simulation and experiment. This can be explained by the differences in the amplifier feedback resistance, resulting in a slightly different leakage rate. The exact resistance of the feedback pseudo-resistor is a strong function of process variations; as large as a couple of orders of magnitude resistance variations are not uncommon.³⁵ The feedback capacitance and resistance in the simulations are 10 fF and 2 TΩ, respectively. The experimental signal also contains some low-frequency noise that is unique to the test bench and not easily simulated. Such noise also has a minor effect on waveform shape.

In terms of gain, the differential geometry outperforms the single-ended amplifier by a factor of roughly 2 and the Amptek amplifier by a factor of 97. This is expected as the custom amplifier utilizes a much lower feedback capacitance (10 fF) than that of the Amptek (1 pF). For the Amptek vs differential test, we estimate a charge of about 115 000 electrons on the particle that induced the signal. For the differential vs single-ended, a charge of about 110 000 electrons is estimated. Both of these values are based on the peak voltage of the output waveform and the confirmed gain of the custom amplifier³⁰. The observed noise floor of the single-ended and differential amplifiers at the given input capacitance agrees with theory and is 1705 and 1030 e⁻, respectively. This was calculated by measuring the rms voltage of the flat part of the signal after the particle leaves the detector and dividing that number by the charge-voltage gain. The noise performance of the amplifiers can be improved by removing the extraneous grounding electrodes that are used solely for the purposes of the comparison setup. We have simulated, for example, that with an input capacitance of 1 pF, the noise floor is reduced below 140 e⁻.

We have presented a study on the interaction between a charged particle and an ICD detector using both electrostatic and circuit simulations. This study includes modeling the interaction between the particle and the electrodes and the induced signal as it is processed by sensing electronics. We have also demonstrated the natural compatibility between ICD sensing electrodes and a differential amplifier topology. We have shown the superiority of the differential topology in comparison to two different single-ended amplifiers, which was demonstrated in simulation and verified in testing with charged particles with diameters ranging from 100 µm to 500 µm. The differential custom amplifier outperformed the commonly used Amptek by a factor of about 13 in terms of input-referred noise. The custom amplifier in the differential configuration also outperformed the single ended configuration of the same device by a factor of 2 in terms of gain.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors are grateful for financial support of this project from NASA (Grant No. 80NSSC17K0101).