High-resolution timing electronics for fast pixel sensors
PPrepared for submission to JINST
High-resolution timing electronics for fast pixel sensors
Adriano Lai π, and Gian-Matteo Cossu π,π π Istituto Nazionale Fisica Nucleare, Sezione di Cagliari, Cagliari, Italy π Dipartimento di Fisica, UniversitΓ di Cagliari, Cagliari, Italy
E-mail: [email protected]
Abstract: Detectors based on pixels with timing capabilities are gaining increasing importance inthe last years. Next-to-come high-energy physics experiments at colliders requires the use of timeinformation in tracking, due to the increasing levels of track densities in the foreseen experimentalconditions. Various diο¬erent developments are ongoing on solid state sensors to gain high-resolutionperformance at the sensor level, as for example LGAD sensors or 3D sensors. Intrinsic sensor timeresolution around 20 ps have been recently obtained. The increasing performance on the sensorside strongly demands an adequate development on the front-end electronics side, which now risksto become the performance bottle-neck in a tracking or vertex-detecting system. This paper aimsto analyze the ultimate possible performance in timing of a typically-used front-end circuit, theTrans-Impedance Ampliο¬er, considering diο¬erent possible circuit conο¬gurations. Evidence to thepreferable modes of operation in sensor read-out for timing measurement will be given.Keywords: Front-end electronics for detector readout, Timing detectors, Analogue electroniccircuits Corresponding author. ontents π β π >> π‘ π (CS-TIA). 84.1.1 Slope of π ππ’π‘ in CS-TIA 114.1.2 Output Voltage Noise π π in CS-TIA 114.1.3 Front-end Jitter in CS-TIA 134.1.4 Example of CS-TIA circuit optimization 144.2 Condition II: π β π β π‘ π (Fast-TIA). 164.2.1 Slope of π ππ’π‘ in Fast-TIA 174.2.2 Noise in Fast-TIA 184.2.3 Front-end Jitter in Fast-TIA 204.2.4 Example of Fast-TIA circuit 20 An important emerging requirement in experimental high energy physics concerns the need ofintroducing time measurements at the level of the single pixel sensor. As an example, the Ugrade-IIof the LHCb experiment at the CERN LHC, scheduled to take data in about a decade from now,has requirements of concurrent space and time resolutions of the order of 10 π m and at least 50 psrespectively at the single pixel level [1]. Such a trend is foreseen to continue with more severeβ 1 βequirements in the subsequent generation of collider experiments [2], where time resolutionsin the range of 10-20 ps per hit will be necessary. This poses a number of diο¬erent technicalissues both on the sensor and the front-end electronics side. Recent developments in fast siliconsensors [3, 4] demonstrate that time resolutions of 30-20 ps are already reachable on the sensor side.As a consequence, front-end electronics becomes a decisive limiting factor to high time resolution.In order not to degrade the sensor performance, the time resolution requirement for the front-endstage is having an electronic time jitter below 10 ps r.m.s, which is not trivial to obtain.The problem of ps-fast front-end electronics can be attacked from diο¬erent sides and points ofview. A ο¬rst perspective could concern the distinction among diο¬erent circuit solutions and inputstages. A second one concerns the technology choice (for example CMOS versus Si-Ge bipolaror BiCMOS, having superior intrinsic performance in terms of speed, but other limitations, asfor example lower integration capabilities). A third one is about the problem of obtainable timingperformance within limited area and power budget. This last perspective is particularly importantabout the development of pixels in vertex detectors with timing, where besides the input stage also ahigh precision Time-to-Digital-Converter has to be integrated.The design of integrated pixel electronics for high resolution deserves a dedicated treatment andwill be the subject of a separate work. The present paper is dedicated to explore the main requirementson fast input stages from the circuit scheme and characteristics point of view. In particular, therelationship and interaction between the characteristics of sensor and electronics are studied. This isimportant to deο¬ne a clear path between the sensor operation and performance and the front-enddesign. Some simulation results on speciο¬c cases and speciο¬c values of sensor and electronics designparameters are also given, so to gain evidence of the impact of the various solutions explored on thetiming performance of the System. Here and in the following the term System (with capital "S") willbe regularly referred to sensor and front-end electronics, coupled together as a unique device. A solid state pixel sensor or, more commonly, a pixel, can be considered as a capacitive sensor wherethe charge generated by ionization in the sensitive volume is collected at the electrodes by meansof a suitable read-out circuit. The starting process, however, is the generation of current signals byinduction, due to the movement of the charge carriers of both signs, made free by the ionizing tracks.In no-timing applications, where only the amount of collected charge is of interest, it is commonpractise to assimilate such current signals to delta-shaped pulses of inο¬nitesimal duration. This isperfectly justiο¬ed by the common use of relatively slow charge-integrating front-end stages, whichmake the charge collection time generally negligible. On the other hand, when timing issues areconcerned, the ο¬ne structure of signals induced at the electrodes is crucial to understand and decidethe ο¬nal performance of the System. In particular, when design eο¬orts on the sensor side providedevices with charge collection times at the deep sub-ns level, front-end performance should be ableto exploit and not lose such an advantage.When quoting the contributions to uncertainty in the measurement of time, the following mainquantities are normally considered: π t = (cid:2) π + π + π + π + π , (2.1)β 2 βhere π tw ( time-walk ) depends on ο¬uctuations of the signal amplitude, which cannot beminimised by design but only by dedicated signal processing; π dr ( delta-rays ) depends on the eο¬ectof longitudinally disuniformities in the energy deposit due to delta rays (Landau tail); π TDC dependson the digital resolution of the electronics (conversion error). The π un contribution corresponds tothe time dispersion caused by unevenness in the signal shapes, which are due to the diο¬erent possibledrift paths of the charge carriers in the sensor. The π un term depends only on the geometry of thesensitive volume. In order to minimise the π un term, maximum uniformity in the electric ο¬eld mustbe obtained by design [5]. The π ej (electronic jitter) term depends on the front-end electronics risetime and signal-to-noise ratio.In the present work, we will not be interested on the eο¬ect of the π tw and π TDC terms. π tw is considered a systematic uncertainty, which can be corrected by dedicated signal processingtechniques. Similarly, π TDC depends on the precision in time-to-digital conversion and not on thefront-end circuit.The π dr and π un terms decide the intrinsic sensor speed. Recent studies demonstrate that averagetime in charge collection distributions of 2-300 ps and standard deviations of 50-40 ps or less canbe obtained [4, 5]. Here we will not consider this aspect of the matter in any speciο¬c detail andwill tend to treat time distribution parameters of the sensors as System inputs. We will instead focusthe analysis on the π ej term, aiming to study the interaction of the front-end speciο¬cations and ο¬nalcharacteristics with the sensor behavior and performance. The aim is to understand which is theoptimal front-end to be designed for a given fast-timing sensor. The traditional textbook solution for the read-out of capacitive sensors is the well-known ChargeSensitive Ampliο¬er (CSA), possibly followed by a suitable number of diο¬erentiating (CR) andintegrating (RC) stages, realising a so-called
Shaper [6].Actually, the CSA circuit is a particular case of a more general conο¬guration, that is theTrans-Impedance-Ampliο¬er (TIA) with shunt-shunt feedback (FB-TIA), schematically shown ifο¬gure 1 (left). In this circuit, a fraction of the voltage is taken at the output of the inverting ampliο¬erand is converted to a current by the impedance of the feedback path that is subtracted from the input.This technique has the eο¬ect of lowering the input impedance of the ampliο¬er (Miller eο¬ect) leadingto a circuit that integrates the input signal. In the ideal case the input capacitance is β π΄πΆ π and isbig enough to make the system independent of the detector capacitance giving the output voltage: π ππ’π‘ = π ππ / πΆ π (3.1)A simpliο¬ed implementation of the TIA ampliο¬er can be realized by a common-source NMOSin a so-called self-biased topology (ο¬g 1, right).The ideal CSA behavior is achieved if the feedback resistance π π and load resistance π π· allowa high open-loop gain π΄ and an input impedance seen by the current generator that is given by: π ππ βΌ π πΆ π· (cid:4) π π π΄ ( + π π π ) (3.2)β 3 β igure 1 . General representation of the FB-TIA circuit (left). NMOS FB-TIA with a self-biased topology (right). Thecurrent generator πΌ π· ( π‘ ) and the πΆ π· capacitance model the operation of the capacitive sensor. where π π = π π πΆ π . if π π ββ β and π΄πΆ π >> πΆ π· , then we have π ππ βΌ π π΄πΆ π (3.3)If the input current is considered as a Dirac Delta we have the input voltage: π ππ βΌ β π ππ π΄πΆ π (3.4)and the output voltage: π ππ’π‘ = β π΄π ππ = π ππ πΆ π (3.5)Equation 3.5 describes the behavior of an ideal CSA, with the assumption of an inο¬nitely fastampliο¬er (inο¬nite bandwidth and slew-rate). A more realistic description can be obtain consideringthe small signal model of the circuit, as given in ο¬gure 2. Figure 2 . Small signal model of the TIA
The input capacitance is given by the sum of the detector capacitance and the one from gate andsource of the NMOS transistor ( πΆ ππ = πΆ π· + πΆ ππ ). The output capacitance πΆ πΏ represent the totalβ 4 βapacitance seen from the node π ππ’π‘ to ground with open loop conο¬guration and is given by thecapacitance πΆ ππ and the input capacitance of the following stage. The resistance π π· deο¬nes the gainof the common source which in given by π΄ βΌ β π π π π· , where π π is the NMOS trans-conductance.We can deο¬ne the impedances (see ο¬gure 2): π ππ = π πΆ ππ π π = π π + π π π π ππ’π‘ = π π· + π π π· (3.6)with the time constants: π π = π π πΆ π ed π π· = π π· πΆ πΏ . From the output node we ο¬nd the voltagegain πΊ π£ ( π ) : π ππ β π ππ’π‘ π π = π π π ππ + π ππ’π‘ π ππ’π‘ (3.7)and consequently: πΊ π£ ( π ) = π ππ’π‘ π ππ = π ππ’π‘ ( β π π π π ) π ππ’π‘ + π π . (3.8)By replacing back the impedances, we obtain: πΊ π£ ( π ) = π π· ( + π π π β π π π π ) π π ( + π π π· ) + π π· ( + π π π ) . (3.9)Introducing the parallel π β = π π (cid:4) π π· , the gain factor πΊ = ( π π π β β π β π π ) and the timeconstants π β π = π β πΆ π and π β πΏ = π β ( πΆ π + πΆ πΏ ) the voltage gain can be written as: πΊ π£ ( π ) = β πΊ β π π β π + π π β πΏ (3.10)The expression of πΊ π£ ( π ) shows a pole at frequency π π = ππ β πΏ and a zero at π π§ = πΊ ππ β π β π π ππΆ π .We can ο¬nd the input impedance from the current equation at the input node: πΌ π· + π ππ π ππ + π ππ β π ππ’π‘ π π = . (3.11)Using equation 3.10 we get: πΌ π· = β π ππ (cid:3) π ππ + π π ( β πΊ π£ ) (cid:4) . (3.12)The voltage at the input of the circuit is then given by: π ππ = β πΌ π· π π ( + π π β πΏ ) + πΊ + π ( π π ( πΆ ππ + πΆ π ( + πΊ )) + π β πΆ πΏ ) + π π β π π π (3.13)with π = πΆ πΏ πΆ ππ + πΆ πΏ πΆ π + πΆ ππ πΆ π . The input impedance is by deο¬nition: π ππ π = π ππ πΌ π· = π π ( + π π β πΏ ) + πΊ + π ( π π ( πΆ ππ + πΆ π ( + πΊ )) + π β πΆ πΏ ) + π π β π π π , (3.14)β 5 βrom where the DC value of the input impedance can be derived, setting π = π ππ π = π π + πΊ . (3.15)The output voltage is the product of the input voltage π ππ by the voltage gain πΊ π£ ( π ) . Theproduct cancels out the zero in π ππ (equation 3.13) but introduces the zero of the voltage gain πΊ π£ ( π ) (equation 3.10): π ππ’π‘ = πΌ π· π π ( πΊ β π π β π ) + πΊ + π ( π π ( πΆ ππ + πΆ π ( + πΊ )) + π β πΆ πΏ ) + π π β π π π (3.16)the trans-impedance of the circuit reads now: π ππ’π‘ πΌ π· = π π ( πΊ β π π β π ) + πΊ + π ( π π ( πΆ ππ + πΆ π ( + πΊ )) + π β πΆ πΏ ) + π π β π π π (3.17)This important formula can be rearranged to introduce the natural frequency of the circuit π π and the damping factor π (see also [7]). Ignoring the zero for now, we can write the trans-impedance π π ( π ) as: π π ( π ) = πΎπ + π π π π + π π (3.18)where πΎ = π π πΊ /( π π π β π ) . The natural frequency is then given by: π π = (cid:5) + πΊ π β π π π (3.19)since 1 + πΊ β π π π β , it follows that: π π β (cid:6) π π π π ( πΆ πΏ πΆ ππ + πΆ πΏ πΆ π + πΆ ππ πΆ π ) (3.20)which results independent of the load resistance π π· . The expression of the damping factor π is: π = ( π π ( πΆ ππ + πΆ π ( + πΊ )) + π β πΆ πΏ ) (cid:7) ( + πΊ ) π β π π π . (3.21)In general, if the damping factor 0 < π <
1, the poles are complex conjugated and we have anunder-damped system, which can lead to an oscillating behavior. This condition is therefore to beavoided. On the other hand, if π >>
1, we get real and distinct poles and an over-dumped system [7].Usually the system should be operated in a dumped condition, which is obtained at π β₯ critically dumped system ( π = π π ( π ) = πΎ ( π + π π ) . (3.22)Introducing the time constant: π = π π = π π (3.23)β 6 βnd simplifying as follows (being πΊ >> πΎπ π = π π πΊ π π π β π (cid:8) π π π β π + πΊ (cid:9) β π π , (3.24)we can write the transfer function of our TIA circuit as: π π ( π ) β π π ( + π π ) . (3.25)In order to explicit the β ππ΅ frequency, we can consider: π π ( π β ππ΅ ) = β π π = π π π π ( π π + π ) , (3.26)and therefore: π β ππ΅ = π π (cid:3) β β (cid:4) β . Β· π π (3.27)This frequency is about half of the natural frequency of the system π π ( π π π = ππ π ). If we donβtignore the zero at the numerator of equation 3.17, we have the trans-impedance: π π ( π ) = π π πΊ + πΊ ( β π π π§ )( + π π ) (3.28)where π π§ = π β πΆ π / πΊ (time constant corresponding to the TIA frequency of the zero) and πΊ = ( π π π β β π β π π ) (DC gain). Equation 3.28 is at the basis of our next analysis about the timingperformance of the TIA circuit. The trans-impedance in the s -domain π π ( π ) (equation 3.28) is the TIA transfer function ( π― ). Thisneeds to be convoluted with the detector current πΌ π· ( π ) in order to get the output voltage of the circuit.We consider here as an example the simpliο¬ed condition of a 3D-trench sensor operating with chargecarriers both at saturation velocities. In this case, the current has a shape that can be modeled asa simple rectangular pulse, having a width of duration π‘ π (where π‘ π is the charge collection time)and an amplitude πΌ , such that the product πΌ Β· π‘ π equals the total charge π ππ deposited by a particle(ο¬gure 3). This solution doesnβt take into account the diο¬erent drift velocities of the carries but it isstill a more realistic description compared to describing the current pulse as a simple Dirac delta.The current can then be expressed in the s -domain as: πΌ π· ( π ) = πΌ β π β π π‘ π π (4.1)in the time domain it can be written as the product of two Heaviside step functions: πΌ π· ( π‘ ) = πΌ π ( π‘ ) π ( π‘ π β π‘ ) (4.2)The output voltage π ππ’π‘ ( π ) from the circuit with π― = π π ( π ) can be written:β 7 β igure 3 . Current pulse πΌ π· ( π‘ ) (left) for a 3D pixel sensor with trench geometry (right). The simulated signalis obtained by TCoDe simulation [5]. The sizes of the pixel are 55 Γ Γ ππ . π ππ’π‘ ( π ) = πΌ β π β π π‘ π π π π πΊ + πΊ ( β π π π§ )( + π π ) (4.3)taking the inverse Laplace transform in the time domain we have the signal: π ππ’π‘ ( π‘ ) = L β ( π‘ ) (cid:10) πΌ β π β π π‘ π π π π πΊ + πΊ ( β π π π§ )( + π π ) (cid:11) (4.4)The solution is: π ππ’π‘ ( π‘ ) = πΌ π π πΊ + πΊ (cid:10) (cid:12) β π β π‘π (cid:3) + π‘π (cid:3) + π π§ π (cid:4) (cid:4) (cid:13) β π ( π‘ β π‘ π ) (cid:12) β π β ( π‘ β π‘π ) π (cid:3) + ( π‘ β π‘ π ) π (cid:3) + π π§ π (cid:4) (cid:4) (cid:13) (cid:11) (4.5)We now proceed to analyse the behavior of the TIA circuit by considering separately twodiο¬erent operating conditions, distinguished by the size of the circuit time constant π with respect tothe charge collection time π‘ π . As already discussed above, in both cases we choose to consider thesystem to operate in a critically dumped condition ( π =
1, see equations 3.21 and 3.22). π β and π >> π‘ π (CS-TIA). This condition is typical of a CSA where the value of the feedback resistor π π is maximized to havea better Signal over Noise ratio (SNR). This is an optimal conο¬guration when the precision in thesignal amplitude measurement is important at the expenses of time resolution. In any case, the use ofthe CSA conο¬guration often remains a convenient compromise between overall performance andpower consumption. β 8 βhe bandwidth of the TIA is kept much smaller compared to the bandwidth of the current pulseand consequently the shape of the current signal is not preserved. With a given trans-conductance π π of the input transistor, the output voltage reaches quickly the maximum achievable slope that thendecreases exponentially with time. When π‘ < π‘ π , we can ignore the factor π ( π‘ β π‘ π ) in equation 4.2,because this term is still not contributing. We therefore get the output signal: π ππ’π‘ ( π‘ ) π‘<π‘ π = πΌ π π πΊ + πΊ (cid:10) (cid:12) β π β π‘π (cid:3) + π‘π (cid:3) + π π§ π (cid:4) (cid:4) (cid:13) (4.6)which has derivative: π (cid:10) ππ’π‘ ( π‘ ) π‘<π‘ π = πΌ π π πΊ + πΊ (cid:10) π β π‘π π (cid:3) π‘π (cid:3) + π π§ π (cid:4) β π π§ π (cid:4) (cid:11) (4.7)This derivative equals zero at time π‘ : π (cid:10) ππ’π‘ ( π‘ ) π‘<π‘ π = ββ π‘ = ππ π§ π + π π§ (4.8)In the CSA, the zero frequency is much smaller than the one corresponding to its poles, therefore π >> π π§ and π‘ βΌ π π§ . Substituting the π‘ expression into equation 4.6 we get the voltage: π ππ’π‘ ( π‘ ) π‘<π‘ π β πΌ π π πΊ + πΊ (cid:3) β π β ππ§π ( π π§ + ππ π§ + π ) π (cid:4) (4.9)This value is negative for current pulses with πΌ >
0. A sinking current from the sensor leads toa negative voltage at the input of the circuit and, since we have an inverting ampliο¬er, the outputvoltage has a positive edge. As a consequence, the output voltage has to be negative before the totalcharge is collected and becomes positive only for π‘ > π‘ π .We now consider the second part of the equation 4.2, when π ( π‘ β π‘ π ) =
1. The output signalexpression becomes: π ππ’π‘ ( π‘ ) π‘>π‘ π = πΌ π π πΊ + πΊ π β π‘π (cid:3) π‘π ( π π‘ππ β ) ( π π§ + ππ ) + π π‘ππ ( π β π‘ π ( π β π π§ )) π β (cid:4) (4.10)We can deο¬ne: π΄ = πΌ π π πΊ + πΊ π΅ = ( π π‘ππ β ) (cid:3) π π§ + ππ (cid:4) πΆ = π π‘ππ ( π β π‘ π ( π β π π§ )) π β π ππ’π‘ ( π‘ ) π‘>π‘ π = π΄π β π‘π (cid:3) π΅ π‘π + πΆ (cid:4) (4.11)Taking the derivative of this expression, we ο¬nd the peaking time π ππππ : π ππππ = π΅ β πΆπ΅ π (4.12)β 9 β ππππ = π π‘ππ ( π π§ ( π β π‘ π ) + ππ‘ π ) β ππ π§ ( π π§ + π ) ( π π‘ππ β ) (4.13)since π >> π‘ π we can take the limit for π‘ π ββ
0, obtaining: π ππππ = π ( π π§ + π ) β π (4.14)The output signal π ππ’π‘ is plotted in ο¬gure 4. During charge collection, the signal is negative. At π‘ > π‘ π the signal becomes positive with positive derivative, reaching a maximum at π ππππ β π . Wecan anticipate here that this condition does not appear as the best possible one when the speed of thesensor is to be fully exploited. We will come back extensively on this point in section 6. We nowanalyze in further detail some relevant characteristics of the calculated circuit responses given inequations 4.11 and 4.10. π ππππ πΌ π π π π‘ π π π π πππ πΌπ· ( π‘ ) πππ’π‘ ( π‘ ) π‘ π πΌ π π π π‘ π π π π πππ πΌπ· ( π‘ ) πππ’π‘ ( π‘ ) Figure 4 . Calculated output voltage π ππ’π‘ in the CS-TIA conο¬guration. Right: Due to the high peaking time ( π ππππ β π )with respect to average collection time, the current signal can be approximated by a Delta function. Right: Detail of theunder-shoot during π‘ < π‘ π . π‘ π‘ ππππ π‘ πππ₯π Figure 5 . Output voltage π ππ’π‘ in the CS-TIA conο¬guration for current pulses with diο¬erent duration π‘ π and same charge π ππ = πΌ Β· π‘ π (left). Detail of the under-shoot and slope of the signal for diο¬erent π‘ π β 10 β .1.1 Slope of π ππ’π‘ in CS-TIA If the current pulse duration is much shorter compared to the time constant ( π >> π‘ π ), we havethat the slope of the signal after the induction is almost independent from π‘ π . As shown in ο¬gure 5,diο¬erent charge collection times lead to a delay of the signals but the initial slope is about the same.The maximum slope for every current is reached at time π‘ π and then decreases exponentially. We canconsider equation 4.10 to calculate the signal slope for π‘ > π‘ π . The signal derivative is: π (cid:10) ππ’π‘ ( π‘ ) π‘>π‘ π = β π ππ’π‘ ( π‘ ) π + π΄ Β· π΅π β π‘π π (4.15)setting π‘ = π‘ π and using π π‘ππ β π (cid:10) ππ’π‘ ( π‘ π ) β π΄ Β· π΅π ( β π‘ π π β πΆπ΅ ) β π΄ Β· π΅π ( β π‘ π π + π π§ π π§ + π ) (4.16)the terms π‘ π π and π π§ π π§ + π are small and with opposite sign, therefore we can write: π (cid:10) ππ’π‘ ( π‘ π ) β π΄ Β· π΅π = πΌ π π πΊ + πΊ ( π π‘ππ β ) (cid:3) π π§ + ππ (cid:4) β πΌ π π πΊ + πΊ π‘ π π (cid:3) π π§ + ππ (cid:4) (4.17) π (cid:10) ππ’π‘ ( π‘ π ) β πΌ π π πΊ + πΊ π‘ π π (cid:3) π π§ + ππ (cid:4) β πΊ + πΊ π ππ π π (cid:3) π Β· π ππππ (cid:4) (4.18)Being π ππππ β π (equation 4.14) and considering that πΊ + πΊ β π (cid:10) ππ’π‘ ( π‘ π ) β π ππ π π π (4.19)using equation 3.23 and the expression of π π found in equation 3.20: ππππ‘ β π ππ Β· π π π (4.20) ππππ‘ β π ππ Β· π π ( πΆ πΏ πΆ ππ + πΆ πΏ πΆ π + πΆ ππ πΆ π ) (4.21)The slope of the signal can be estimated just considering the input charge π ππ , the trans-conductance π π of the ο¬rst ampliο¬er stage and the circuit capacitances through the quantity π . π π in CS-TIA We can identify three noise sources in the TIA (ο¬gure 1, right). Two sources are due to the resistances π π and π π· , and a third source is due to the MOS transistor and more precisely depends on itstrans-conductance π π and πΎ factor ( πΎ βΌ but becomes higher for deep sub-micron technologies).The load can be provided by another MOS transistor and in that case the noise will also dependon the trans-conductance of the load element. Figure 6 shows the circuit with its identiο¬ed noisesources [7]. β 11 β igure 6 . noise sources of the Feedback TIA The calculation of the noise can be found in Appendix A, here we report the results: π π£,π = (cid:6) π π΅ π πΎ πΆ π + πΆ ππ π π π π π (4.22) π π£,π π· = (cid:6) π π΅ π π π· πΆ π + πΆ ππ π π π π π (4.23) π π£,π π = (cid:6) π π΅ π π π π π π (4.24)where π = πΆ π πΆ ππ + πΆ π πΆ πΏ + πΆ πΏ πΆ ππ .The transistor noise is proportional to the trans-conductance π π . It is known that increasing π π by an increase of the bias current leads to a smaller output resistance of the transistor that decreasesthe voltage gain and the output noise. In this case, the noise value is considered keeping the circuitin a condition where the damping factor π stays equals unity so that we are in a critically dampedsystem. Changing one parameter (for example π π ) leads to a change in both the natural frequencyand damping factor. The eο¬ect of this on noise must be carefully considered as it changes the powerspectral density of the noise sources (ο¬gure 7).The total noise is given adding in quadrature the three noise sources: π π£,π ππ = (cid:2) π π£,π + π π£,π π + π π£,π π· (4.25)The voltage peak can be calculated using the π ππππ expression given in equation 4.13 and isgiven by: π ππππππ’π‘ β π ππ’π‘ ( π ) π‘>π‘ π = π΄π β (cid:3) π΅ + πΆ (cid:4) (4.26)replacing π΄, π΅, πΆ with the values deο¬ned in equation 4.1 and considering π π‘ππ β ( β π‘ π π ) we get:β 12 β π»π§ ππ»π§ ππ»π§ ππ»π§ ππ»π§ ππ»π§
ππ»π§ πΊπ»π§ πΊπ»π§ π»π§ ππ»π§ ππ»π§ ππ»π§ ππ»π§ ππ»π§
ππ»π§ πΊπ»π§ πΊπ»π§ π π , π π’ π‘ [ π π / β π» π§ ] πΉππππ’ππππ¦ π π π π· π πππ‘ππ π π , π π’ π‘ [ π π / β π» π§ ] πΉππππ’ππππ¦ π π π π· π πππ‘ππ π»π§ ππ»π§ ππ»π§ ππ»π§ ππ»π§ ππ»π§
ππ»π§ πΊπ»π§ πΊπ»π§ π»π§ ππ»π§ ππ»π§ ππ»π§ ππ»π§ ππ»π§
ππ»π§ πΊπ»π§ πΊπ»π§ π π [ π π β π» π§ ] π π = π Ξ© π = . π π = π Ξ© π = . π π = . π Ξ© π = . π π = π Ξ© π = . π π = π Ξ© π = . π π = π Ξ© π = . π π = π Ξ© π = . π π = π Ξ© π = . π π [ π π β π» π§ ] π π = π Ξ© π = . π π = π Ξ© π = . π π = . π Ξ© π = . π π = π Ξ© π = . π π = π Ξ© π = . π π = π Ξ© π = . π π = π Ξ© π = . π π = π Ξ© π = . Figure 7 . Left: example of Noise Power Spectral Density contributions of π π , π π· and transistor π . Right:output noise for diο¬erent damping factor π obtained by changing the value of π π . π ππππππ’π‘ β πΊ + πΊ π ππ π π π (cid:3) π ( π β π‘ π ) β π‘ π π π§ π (cid:4) (4.27) π ππππππ’π‘ β π ππ π π ππ (4.28)As an example, considering only the noise contribution of π the SNR can be written: ππ π ππ’π‘,π β (cid:5) π π΅ π πΎ π ππ π ( πΆ ππ + πΆ π ) ( π π π π π ) (4.29)The expressions obtained in this section will be the main ingredients for our discussion ontiming performance of the CSA which we will cover below. Starting from equations 4.19 and 5.3, the CSA jitter can be written: π π = π π π π ππ π π (4.30)Using equation 4.21, we can make clear the trans-conductance π π and the circuit capacitancesthrough the quantity π = πΆ πΏ πΆ ππ + πΆ πΏ πΆ π + πΆ π πΆ ππ : π π = π π ππ ππ π π (4.31)Considering the noise calculated in subsection 4.1.2, we can calculate the contribution to thetime resolution given by the single noise sources: π π,π = (cid:6) π π΅ π πΎ πΆ π + πΆ ππ π ππ Β· π π ( π π π ) (4.32)β 13 β π,π π = (cid:6) π π΅ π π π ππ π π π π (4.33) π π,π π· = (cid:6) π π΅ π π π· πΆ π + πΆ ππ π ππ Β· π π (cid:8) π π π (cid:9) (4.34)summing in quadrature to ο¬nd the total jitter: π π,π ππ = (cid:2) π π,π + π π,π π + π π,π π· (4.35) π π [ π π ] π ππ [ π πΆ ] π½ππ‘π‘ππ π π π π [ π π ] π ππ [ π πΆ ] π½ππ‘π‘ππ π π Figure 8 . Jitter estimate for diο¬erent π ππ using equation 4.31. Equation 4.35 estimates the minimum jitter that can be reached in this topology. Since thederivative is taken at time π‘ = π‘ π , we are considering the maximum slope when the signal is still closeto zero. in the real case, we need to set a threshold with a value higher than the noise. Therefore, thejitter depends on the value chosen for the threshold ad since the slope decreases exponentially, ahigher threshold means higher jitter. Equation 4.31 is still useful to have a rough evaluation of theperformance achievable with a FB-TIA conο¬guration such that π >> π‘ π . Considering as an examplea circuit with having π = ππ , feedback resistance π π = π Ξ© , ππ π = π ππ = π πΆ the achievable jitter is in the order of 40 ππ (ο¬gure 8). In the present subsection, we use the jitter expressions above to suitably size the value of π π forminimum jitter, while still respecting the stability conditions of the amplifying stage.Supposing the most signiο¬cant noise source is given by the transistor π , then we can ο¬nd theoptimum jitter solving the following system: β 14 β ππ‘π‘ππ πππ‘ (cid:10) π π,π < π π,π πππππ‘ π = π½ππ‘π‘ππ πππ‘ β§βͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβ©(cid:2) π π΅ π πΎ πΆ π + πΆ ππ π ππ Β· π π ( π π π ) < π π,π πππππ‘ ( π π ( πΆ ππ + πΆ π ( + πΊ ))+ π β πΆ πΏ ) β ( + πΊ ) π β π π π = π π in the second equation we get: π½ππ‘π‘ππ πππ‘ β§βͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβ©(cid:2) π π΅ π πΎ πΆ π + πΆ ππ π ππ Β· π π ( π π π ) < π π,π πππππ‘π π ( πΆ π ( + π π π π· ))+ π π· πΆ πΏ +( π π + π π· ) πΆ ππ β ( + π π π π· ) π π· π π π = π π , the ο¬rst equation tells us that the jitterdecreases by a factor of 2 β .
68 while the second equation implies that the damping factorincreases by a factor of about β β .
414 ( π β β π π ). In this case we would have to decrease π π bythe same factor to have still a critically damped system (since π β (cid:7) π π ). If we double π π and halve π π the jitter becomes: π π,π ,πππ€ β / Β· / π π,π ,πππ = π π,π ,πππ πΆ π· , π π· , πΆ π , πΆ πΏ donβt change, we would endup with half the jitter due to π . Changing the biasing changes the output resistance of the transistor( π ) that is in parallel with π π· ( π β ππΌ ππππ , where π is the channel lenght modulation coeο¬cient).This has to be taken into account in the speciο¬c case. Following this method, it is possible forexample to make explicit the design parameters π, πΏ and optimize the size of the input transistor.Considering the other two jitter contributions we ο¬nd: π π,π π· ,πππ€ β / Β· / π π,π π· ,πππ = π π,π π· ,πππ (4.40)the jitter due to π π becomes: π π,π π ,πππ€ β / / Β· π π,π π πππ = π π,π π ,πππ β π π and halving the feedback resistance decreases all the three jitter contributions thatwe are considering. β 15 β .2 Condition II: π β and π β π‘ π (Fast-TIA). A possible solution for realizing a FB-TIA having a time constant of the same order of the chargecollection time π‘ π is using the same self-biased scheme of ο¬gure 1, but implemented with a high-bandwidth Si-Ge bipolar transistor stage, as illustrated in ο¬gure 9. It is thus possible to take advantageof the beneο¬ts of the Si-Ge devices that allow small input capacitances and high-frequency transitionsof the order of 100 GHz also for discrete-component circuit solutions. Figure 9 . Schematic of the FB-TIA with bipolar transistor NPN (left) and corresponding Small signal model (right).
The diο¬erence with the MOS solution is that now a bias current πΌ π ο¬ows through the feedbackresistance π π , giving a dynamic input resistance π π , deο¬ned as: π π = π π πΌ π (4.42)where π π = π π΅ ππ β ππ at 300 K. The circuit small signal model (ο¬gure 9, right) can be usedto solve the circuit in detail. We calculate the input voltage π ππ ( π ) , the natural frequency π π and thedamping factor π : π ππ ( π ) = β πΌ π· π π ( π π + π πΆ + π π π π πΆ ( πΆ π + πΆ πΏ ))( π π π πΆ π π π ) ( π + π π π π + π π ) (4.43) π π = (cid:5) π π ( + π π π πΆ ) + π πΆ + π π π π π πΆ π π π (4.44) π = (cid:10) π π πΆ ππ ( π π + π πΆ ) + π π πΆ π ( π π ( + π π π πΆ ) + π πΆ ) + π πΆ πΆ πΏ ( π π + π π ) (cid:7) ( π π ( + π π π πΆ ) + π πΆ + π π ) π π π πΆ π π π (cid:11) (4.45)The zero found in the voltage gain (equation 3.10) can be neglected due to the very hightrans-conductance π π of our Si-Ge BJT, which takes the frequency corresponding to the zero timeconstant π π§ to extremely high values. The voltage gain is then given by: πΊ π£ ( π ) = β ( π π π π β ) π πΆ π π + π πΆ + π π πΆ π π ( πΆ π + πΆ πΏ ) . (4.46)β 16 βlso in this case, we can simplify the transfer function considering a critically dumped behavior( π β π ππ’π‘ ( π ) = πΌ π· π π π πΆ ( π π π π β )( π π + π πΆ + π π ( + π π π πΆ )) ( + π π ) . (4.47)Deο¬ning the trans-impedance: π π = π π π π π πΆ π π β π π ( π π + π πΆ + π π ( + π π π πΆ ) , (4.48)that can be approximated using π β = π π || π πΆ and π ππ = π π ( π π + π πΆ )( π π + π πΆ + π π ( + π π π πΆ )) : π π β π π π β π ππ (4.49)we ο¬nally obtain the trans-impedance of the circuit: π π ( π ) = π π ( + π π ) (4.50)The output voltage is given by the convolution with the current pulse deο¬ned in equation 4.2,and reads: π ππ’π‘ ( π‘ ) = L β ( π‘ ) (cid:10) πΌ β π β π π‘ π π π π ( + π π ) (cid:11) (4.51)for π = π‘ π = π we obtain the voltage output given by: π ππ’π‘ ( π‘ ) = πΌ π π (cid:10) (cid:12) β π β π‘π (cid:3) + π‘π (cid:4) (cid:13) β π ( π‘ β π‘ π ) (cid:12) β π β ( π‘ β π‘π ) π (cid:3) + ( π‘ β π‘ π ) π (cid:4) (cid:13) (cid:11) (4.52) π ππ’π‘ in Fast-TIA When the time constant of the circuit π is of the same order of the charge collection time π‘ π , thefront-end output signal can reach the maximum slope before the charge is completely collected at thesensor electrodes. We can demonstrate this taking the derivative of the ο¬rst part of the solution 4.52which reads: π (cid:10) ππ’π‘ ( π‘ ) π‘<π‘ π = πΌ π π π π β π‘π π‘ (4.53)Taking the second derivative we get: π (cid:10)(cid:10) ππ’π‘ ( π‘ ) π‘<π‘ π = πΌ π π π π β π‘π (cid:8) β π‘π (cid:9) (4.54)the maximum slope is reached at time π‘ = π and since we are implying π‘ π = π we have the slope: (cid:8) ππππ‘ (cid:9) πππ₯ = π ππ π π ππ . (4.55)β 17 βsing the deο¬nitions of π and π π we ο¬nd: (cid:8) ππππ‘ (cid:9) πππ₯ = π ππ ( π π β π π π πΆ ) ππ . (4.56)Usually π π π πΆ << π π and we have a solution similar to the MOS transistor case with π >> π‘ π ,with the diο¬erence of the factor π βΌ .
71 at the denominator: (cid:8) ππππ‘ (cid:9) πππ₯ = π ππ π π ππ (4.57)To ο¬nd the peaking time π ππππ and voltage peak π ππππ we consider the solution for π‘ > π‘ π : π ππππ = π ππ β β . Β· π (4.58) π ππππ = πΌ π π π β (cid:18) ππ β (cid:19) ( π β ) (4.59) π ππππ β . Β· πΌ π π (4.60)The voltage at time π‘ = π , that is the maximum slope voltage, is: π πππ₯
πππππ = πΌ π π (cid:18) π β π (cid:19) β . Β· πΌ π π (4.61)taking the ratio between π πππ₯
πππππ and π ππππ we ο¬nd: π πππ₯
πππππ β . Β· π ππππ (4.62)In our condition of π‘ π = π , the maximum slope is reached at about 75% of the peaking value(ο¬gure 10). This means that for an ampliο¬er with short time constant the threshold has to be set to anhigher value compared to the CSA solution to minimize the electronic jitter (see also section 6). The noise introduced by a bipolar transistor is due to two correlated shot noise sources which causeο¬uctuations in the bias currents πΌ π and πΌ π . The Power Spectral Densities (PSD) for this two sourcesare: πΌ π,π = π π΅ ππ π πΌ π,π = π π΅ π π π (4.63)the other two noise sources are given by the resistor π π and π π· . For the calculation of thesingle noise contributions see appendix B, here we report the results: π π£,π = π π΅ π π π π ( π π π β π ππ ) (4.64)β 18 β igure 10 . Output signal and its derivative for a FB-TIA in Condition II ( π β π‘ π and π β Figure 11 . Small signal model for the noise sources of a self-biased circuit implemented with BJT π π£,π = π π΅ π π π π (cid:8) π ππ’π‘ + π β π ππ ( πΆ ππ + πΆ π ) π (cid:9) (4.65) π π£,π πΆ β π π΅ π ππ πΆ (cid:8) π ππ’π‘ + π β π ππ πΆ ππ πΆ πΏ (cid:9) (4.66) π π£,π π = π π΅ π π π ( + π π + π πΆ π π π π π πΆ ) π (4.67)As an example the signal to noise ratio ( ππ π ) considering only the source πΌ π,π and the expressionfor π ππππ (eq. 4.60) is given by: β 19 β π π β . Β· πΌ (cid:6) π π ππ π΅ π (4.68) In the case of π βΌ π‘ π and large bandwidth, usually the most signiο¬cant noise source is given by theο¬uctuations of the bias current πΌ π . The jitter given by this source can be written: π π,π = (cid:6) π π΅ π π π ππ ππ π (4.69)the dependence on π π ,(and therefore on power consumption), is implicit since the time constant π depends on π π . Also in this case we have that the jitter is proportional to π β π . however, forBJT-TIA implementation, the relationship with power consumption is more complicated than in theMOS case. For a bipolar transistor, higher trans-conductance π π means higher bias current πΌ π thatcauses the dynamic resistance π π to decrease (eq. 4.42). This means that the time constant changestogether with the damping factor π , thus changing the damping behavior of the circuit. Using Eq.4.69, we can obtain an estimate of the achievable jitter when: π βΌ π‘ π , π βΌ π πππ₯
πππππ . A clearer explanation of what obtained in the previous sections can be given using a circuit examplewith given speciο¬cations. Referring to ο¬gure 9, letβs consider a circuit with the following componentvalues: π π = π Ξ© π πΆ = Ξ© π π = . π π π = Ξ© πΆ π· = ππΉ πΆ ππ = . ππΉ πΆ π = π πΉ πΆ πΏ = ππΉπΆ ππ = . ππΉ π β = . Ξ© π ππ = . Ξ© π ππ’π‘ = . Ξ© with these values we have the following natural frequency and damping factor: π π = . Β· πππ / π π = ππ π = . π‘ π = ππ , π ππ = π πΆ , πΌ = π ππ π‘ π = . π π΄ , while the trans-impedance is π π = . π Ξ© . We have then the signal: π ππ’π‘ ( π‘ ) = (cid:10) πΌ π π ( β π β π‘π ( + π‘π )) if π‘ < π‘ π πΌ π π ( π β π‘ β π‘ππ ( + π‘ β π‘ π π ) β π β π‘π ( + π‘π )) if π‘ > π‘ π (4.70)Figure 12 (left) shows the output signal. Since we have π‘ π slightly smaller than π , the maximumslope is reached before the time π‘ = π and exactly at time π‘ = π‘ π . The voltage π ππππ equation 4.60 issmaller compared to the condition π‘ π = π : π ππππ β . ππ < . Β· πΌ π π = . ππ (4.71)β 20 β igure 12 . Output voltage π ππ’π‘ ( π‘ ) with current pulse πΌ π· ( π‘ ) (left), output voltage π ππ’π‘ ( π‘ ) and output voltagederivative π (cid:10) ππ’π‘ ( π‘ ) (right) . for the peaking time π ππππ we get: π ππππ = ππ < . Β· π = ππ (4.72)for the slope of the signal we have that, since the pulse is shorter than the time constant π , thederivative has a higher value: π (cid:10) ππ’π‘ ( π‘ π ) = πΌ π π π π‘ π Β· π β π‘ππ (4.73) (cid:8) ππππ‘ (cid:9) πππ₯ = π ππ π π π π β π‘ππ β . ππππ > π ππ π π π π β . ππππ (4.74) π»π§ ππ»π§ πΎβπ§
πΎπ»π§ ππ»π§ ππ»π§
ππ»π§ πΊπ»π§ πΊπ»π§ π π [ π π β π» π§ ] π π,πππ π π,π π π π,π πΆ π π,πΌπ π π,πΌπ Figure 13 . Power Spectral Density of the output noise π π ππ’π‘ Figure 12 (right) shows the output signal together with the derivative π (cid:10) ππ’π‘ . The derivative has aminimum corresponding to the condition of maximum descent. The noise of this circuit has the PSDβ 21 βhown in ο¬gure 13. The noise contributions can be speciο¬ed as: π π£,π = . πππ π£,π π = . πππ π£,π = . πππ π£,π πΆ = . ππ The total noise is: π π£,π ππ = . ππ The dominant term is π π£,π as can be seen from 13.Using the calculated noise and the slope of the signal we can estimate the jitter of the circuit as: π π = π π£,π ππππππ‘ π π = . ππ (4.75)The closed loop gain of the circuit is about 28 . ππ΅ with a frequency cut π π πΏ = ππ β ( πΆ πΏ + πΆ π ) = π π»π§ . The trans-impedance π π (equation 4.50) is shown in ο¬gure 14 and has a value of π π = . π Ξ© = . ππ΅ Ξ© , while the frequency cut is given by equation 3.27 and reads π β ππ΅ β π βπ§ . π»π§ ππ»π§ πΎβπ§
πΎπ»π§ ππ»π§ ππ»π§
ππ»π§ πΊπ»π§ πΊπ»π§ | πΊ π£ ( π ) | [ π π΅ ] πΊ π£ ( π ) π»π§ ππ»π§ πΎβπ§
πΎπ»π§ ππ»π§ ππ»π§
ππ»π§ πΊπ»π§ πΊπ»π§ | π π ( π ) | [ π π΅ Ξ© ] π π ( π ) Figure 14 . Voltage gain πΊ π£ ( π ) (left), and trans-impedance π π ( π ) (right) The transfer function π π ( π ) has to be convoluted with the current pulse πΌ π· ( π ) that in thefrequency domain has the spectrum of the ππππ ( π ) function (ο¬gure 15). πΌ π· ( π ) = πΌ πππ ( ππ‘ π π ) ππ‘ π π (4.76)The trans-impedance π π ( π ) acts as a ο¬lter by amplifying some frequencies while suppressingothers. The spectrum of the signal reach the ο¬rst zero at frequency π = π‘ π that for a pulse with π‘ π = ππ means π βΌ . πΊ π»π§ . To ο¬nd the optimal ampliο¬er time constant needed to process thesigniο¬cant (fast) part of the input current pulse, we can consider the β ππ΅ Bandwidth of the signaldeο¬ned as: | πππ ( ππ‘ π π )|| ππ‘ π π | = β
22 (4.77)β 22 β . . . . . . . . . . . . π»π§ ππ»π§ πΎβπ§
πΎπ»π§ ππ»π§ ππ»π§
ππ»π§ πΊπ»π§ πΊπ»π§ π π ( π )[ π Ξ© ] πΌ π· ( π ) πΌ π· ( π ) π π ( π ) Figure 15 . trans-impedance π π ( π ) and Fourier transform of πΌ π· ( π ) approximating | πππ ( ππ‘ π π )| at the second order we ο¬nd: π πΌ π· , β ππ΅ = (cid:8) ( β β ) (cid:9) ππ‘ π (4.78) π πΌ π· , β ππ΅ β . π‘ π (4.79)for π‘ π = ππ we have π πΌ π· , β ππ΅ βΌ πΊ π»π§ . Since the frequency cut of the trans-impedance π π ( π ) is given by equation 3.27 and is about half the one deο¬ned with the natural frequency of the circuit,we have that the time constant of the circuits needs to be at least: (cid:8) ( β β ) (cid:9) ππ‘ π = (cid:8) ( β β ) (cid:9) π π (4.80) π = π‘ π β π should be about a factor 3.4 smaller than the charge collection time π‘ π of the sensor. In the present section we analyze the performance in time resolution of the two conο¬gurationdescribed above, that is the CSA-like conο¬guration ( π >> π‘ π ) and the Fast conο¬guration ( π β π‘ π ).We are speciο¬cally interested in the relationship between the front-end timing characteristics and thenative time dispersion or speed of the sensor. This is of particular importance in case of very fastsensors, whose time distributions can have relatively small dispersion (standard deviations in therange of tens of ps). We can start considering an ideal 3D geometry with ο¬at parallel faces (ο¬gure 16). Such speciο¬cchoice is motivated by the high intrinsic speed of this kind of sensors [4]. In this case, chargeβ 23 βollection time π‘ π depends on the hit position of the impinging particle. For tracks closer to theelectrode at the higher potential, the sensor will collect electrons very quickly while holes wouldinduce for a longer time since they have to travel a longer distance and they move slower. The sameargument can be used in the opposite case with tracks close to the electrode at lower potential withshort current from holes and a longer pulse for electrons.When we have electric ο¬elds strong enough for both charge carriers to reach the respectivesaturation velocities π£ π and π£ β , the electrons and holes speeds becomes similar (ο¬gure 17). We willhave then a minimum π‘ ππππ and a maximum π‘ πππ₯π for the charge collection time π‘ π . Assuming that thedistance between the electrodes is π = ππ we obtain: π‘ π = (cid:10) π‘ ππππ = ππ£ π + π£ β βΌ ππ if π₯ = π£ π ππ£ π + π£ β π‘ πππ₯π = ππ£ β βΌ ππ if π₯ = π (5.1) π ( π‘ π ) π‘ π,πππ π‘ π,πππ₯ π‘ π,πππ₯ π ( π‘ π ) Figure 16 . Ideal parallel-plate sensor with 3D geometry (left), Charge collection time distribution (right)
For simplicity we can assume that the charge collection times generated by the two carriersare equally probable. In reality, shorter charge collection times are more probable, as electronsmove faster (ο¬gure 16, right). Following such assumption we will have a rectangular distributioncorresponding in the ideal case to a dispersion: π π‘ π βΌ | π‘ πππ₯π β π‘ ππππ |β βΌ ππ (5.2)Considering a more realistic description, we can refer to a 3D-trench pixel. Figure 18 showstwo distributions obtained by TCoDe simulation [5], referred to two diο¬erent pixel sizes. In thiscase, the sensor time behavior and associated time distribution can be characterized by its averagecollection time instead of a single π‘ π and by its standard deviation π π‘ π . We consider here the jitter contribution coming from the front-end circuit, depending on signal speedand SNR. The time resolution for a given signal can be estimated using the known equation: π π = π πππππ‘ (5.3)β 24 β igure 17 . Carrier velocities vs electric ο¬eld in silicon. 1) low ο¬eld region; 2) intermediate region; 3) saturation region.
Figure 18 . Charge collection time distributions for 3D-trench pixels (TCoDe simulation). Pixel sizes: 55 Γ Γ ππ (left) and 25 Γ Γ ππ (right). The simulated distributions take also into account the pixel dis-uniformities in theelectric and weighting ο¬elds. In a time measurement, equation 5.3 is evaluated at a given voltage threshold π π‘β . In the presentsection, exploiting the diο¬erent expressions calculated in section 4, we analyse the eο¬ect of suchoperation (called discrimination ) in the two cases of CS-TIA and Fast-TIA. We consider two kind oftypical discrimination techniques: the leading-edge ( LE ) discriminator and the constant fractiondiscriminator (CFD).It is important to point out that in what follows the speciο¬c amplitude-correction techniques(time-over-threshold and constant fraction discrimination), which are needed to cure the additionaldispersion due to time-walk (equation 2.1) are considered as already applied independently. In otherwords we limit our discussion to the intrinsic jitter contribution on the sensor and front-end sides,considering the time-walk as a mere systematic (and processing-recoverable) eο¬ect. With a leading edge we have a ο¬xed threshold at voltage π π‘β . If we set this value as low as possible,the time at which the threshold is crossed π‘ ππ would be small compared to the time constant of thecircuit π . Assuming this, the exponential term in equation 4.11 can be approximated as:β 25 β π‘β = π ππ’π‘ ( π‘ ππ ) π‘>π‘ π = π΄π β π‘πππ (cid:3) π΅ π‘ ππ π + πΆ (cid:4) β π΄ (cid:3) π΅ π‘ ππ π + πΆ (cid:4) (5.4)solving for π‘ ππ we ο¬nd: π‘ ππ = ππ΅ (cid:8) π π‘β π΄ β πΆ (cid:9) (5.5) π‘ ππ = π ( β π π‘ππ ) + π π‘ππ π‘ π ( π + π π§ )( π π‘ππ β ) ( π + π π§ ) + π ( π π‘ππ β ) ( π + π π§ ) π π‘β π‘ π π ππ π π + πΊ πΊ (5.6)The uncertainty on π‘ ππ can be found taking the derivative with respect to π‘ π and propagating the error: π π‘ ππ = (cid:8) ππ‘ ππ ππ‘ π (cid:9) π π‘ π (5.7)The term ππ‘ ππ,πΆπΉπ· ππ‘ π can be deο¬ned as the timing propagation coeο¬cient π« of time resolution fromsensor to electronics. In this speciο¬c case we have: π« = ππ‘ ππ ππ‘ π = π π‘ππ ( π ( π π‘ππ β ) β π‘ π ) π ( π π‘ππ β ) + π π‘ππ ( π β π‘ π ) β ππ ( π π‘ππ β ) π π π‘β ( + πΊ )( π + π π§ ) π ππ π π πΊ (5.8)We can consider two contribution to the time at the threshold: the ο¬rst is independent on π π‘β whilethe other gets smaller for low value of the threshold. The ο¬rst term of equation 5.8 becomes forlarge π while the second term has an oblique asymptote of value β π π‘β ( + πΊ ) π ππ π π πΊ as a function of π . Wecan write then: ππ‘ ππ ππ‘ π βΌ (cid:8) β π π‘β π ( + πΊ ) π ππ π π πΊ (cid:9) (5.9)deο¬ning the voltage π = π ππ π π π (with π βΌ π Β· π ππππ ) and using + πΊ πΊ β ππ‘ ππ ππ‘ π βΌ (cid:8) β π π‘β π (cid:9) (5.10)For time constant π >> π‘ π and low thresholds we have that the uncertainty on the time at thethreshold is: π π‘ ππ β π π‘ π π« β . .2.2 CFD time resolution in CS-TIA Using a CFD discriminator the threshold is always set at the same fraction of the maximum valueof the output voltage. This solution is useful to correct the time walk due to the fact the diο¬erentamplitudes would cross a ο¬xed threshold at diο¬erent times. The voltage at the threshold can bewritten as: π π‘β = πΌπ ππππππ’π‘ β πΌ π ππ π π ππ (5.12)where πΌ deο¬ne the chosen fraction. Using Eq. 5.8 we ο¬nd for the time at the threshold in the CFDcase: π‘ πΆπΉ π· = π ( β π π‘ππ ) + π π‘ππ π‘ π ( π + π π§ )( π π‘ππ β ) ( π + π π§ ) + πΌπ ππ‘ π ( π π‘ππ β ) ( π + π π§ ) (5.13)Taking the derivative we ο¬nd: ππ‘ πΆπΉ π· ππ‘ π βΌ (cid:8) + πΌ π ππ + π π§ (cid:9) (5.14)this approximation holds for low thresholds but also in this case we have a dominant term that isindependent of the chosen threshold so that we can estimate the time resolution using: π π‘ πΆπΉπ· = (cid:8) ππ‘ πΆπΉ π· ππ‘ π (cid:9) π π‘ π (5.15) π π‘ πΆπΉπ· β π π‘ π leading-edge and CFD , the time resolution is about half of the standard deviation of thecharge collection times distribution. This is a general result when π >> π‘ π as we will see in section 6 Using the expression of π ππ’π‘ ( π‘ ) π‘<π‘ π , we can ο¬rst use an approximation at the second order for theexponential term: π ππ’π‘ ( π‘ ) = πΌ π π ( β π β π‘π ( + π‘π )) (5.17) π π‘β = πΌ π π ( β π β π‘πππ ( + π‘ ππ π )) (5.18) π π‘β βΌ πΌ π π ( β ( β π‘ ππ π ) ( + π‘ ππ π )) (5.19) π‘ ππ = π (cid:5) π π‘β πΌ π π = π (cid:5) π π‘β π‘ π π ππ π π (5.20)propagating the ο¬uctuations we ο¬nd: π π‘ ππ = π‘ ππ π‘ π π π‘ π π π‘β to a small value according to a π‘ ππ < π‘ π condition, we can reduce the sensor jitter (seeequation 4.75 as an example). However, the approximation of equation 5.19 tends to underestimatethe ο¬uctuations, being valid only for very small threshold values. For a more complete understandingwe can consider the time π‘ as a function of π‘ π and derive both sides of the equation 5.18 as follows: πππ‘ π (cid:8) π π‘β π‘ π π ππ π π (cid:9) = πππ‘ π (cid:8) β π β π‘ππ ( π‘π ) π (cid:18) + π‘ ππ ( π‘ π ) π (cid:19) (cid:9) (5.22) π π‘β π ππ π π = π π‘πππ π‘ ππ π ππ‘ ππ ππ‘ π (5.23)solving for ππ‘ ππ ππ‘ π we can write the derivative as: ππ‘ ππ ππ‘ π = π π‘β π (cid:10) ( π‘ ππ ) π‘ π (5.24)the resolution at threshold becomes: π π‘ ππ = π π‘β π (cid:10) ( π‘ ππ ) π‘ π π π‘ π (5.25)Letβs suppose to set the threshold to the value corresponding to the maximum slope conditionwhen π‘ = π‘ π . In this case, we can use equations 4.53 and 4.61. The time resolution is then given by: π π‘ ππ β ( π β ) π π‘ π βΌ . Β· π π‘ π (5.26)Setting π π‘β = π πππ₯
πππππ leads to bring about 70% of the charge collection time ο¬uctuations to the timeresolution at the threshold. The total time resolution, in this case, would be dominated by the sensorcontribution with respect to the front-end one. Setting a lower π π‘β increases the electronic jitter butlowers signiο¬cantly the sensor contribution. Using the equation 5.18 and the deο¬nition of the slopewe can write the time resolution as: π π‘ ππ = πΌ π π ( β π β π‘πππ ( + π‘ ππ π )) πΌ π π π β π‘πππ π‘ ππ π π‘ π π π‘ π (5.27) π π‘ ππ = (cid:20) ( β π β π‘πππ ( + π‘ ππ π )) π β π‘πππ π‘ ππ π π‘ π (cid:21) π π‘ π (5.28)rearranging equation 5.28 and considering a ο¬xed value for π‘ π and π we can write the followingexpression as a function of π‘ ππ : (cid:22) ππ‘ ππ ππ‘ π ( π‘ ππ ) (cid:23) π‘ ππ <π‘ π = π ( π π‘πππ β ) β ππ‘ ππ π‘ ππ Β· π‘ π (5.29)β 28 βe will use this expression in section 6, when we will discuss the results obtained in the diο¬erentdiscrimination techniques for diο¬erent front-end solutions. To ο¬nd the derivative ππ‘ππ‘ π for the constant fraction case we can consider the following equation: π ππ’π‘ ( π‘ β ) π‘<π‘ π = πΌπ ππ’π‘ ( π ππππ ) π‘>π‘ π (5.30) π‘ β is the time at threshold, ο¬xed at the fraction πΌ of the voltage peak π ππππ = π ππ’π‘ ( π ππππ ) π‘>π‘ π . In thespecial case π = π‘ π the peaking time is given by Eq. 4.58, otherwise is given by: π ππππ = π π‘ππ π‘ π π π‘ππ β πΌ π π ( β π β π‘ β π ( + π‘ β π )) = πΌπΌ π π π β ππππππ ( π π‘ππ β ) (5.32)Taking the derivative with respect to π‘ π of both side we get: π β π‘ β π π‘ β π ππ‘ β ππ‘ π = πΌ πππ‘ π (cid:8) π β ππππππ ( π π‘ππ β ) (cid:9) (5.33)the fraction πΌ can be written using Eq. 5.32 as: πΌ = ( β π β π‘ β π ( + π‘ β π )) π β ππππππ ( π π‘ππ β ) (5.34)the derivative ππ‘ππ‘ π is then: ππ‘ β ππ‘ π = πΌ πππ‘ π (cid:8) π β ππππππ ( π π‘ππ β ) (cid:9) π β π‘ β π π‘ β π (5.35) ππ‘ β ππ‘ π = ( β π β π‘ β π ( + π‘ β π )) π β ππππππ ( π π‘ππ β ) πππ‘ π (cid:8) π β ππππππ ( π π‘ππ β ) (cid:9) π β π‘ β π π‘ β π (5.36)Using the value of π ππππ (Eq. 5.31) we can write the expression as: (cid:22) ππ‘ πΆπΉ π· ππ‘ π ( π‘ β ) (cid:23) π‘ β <π‘ π = π‘ π π π π‘ππ ( ππ π‘ β π β π‘ β β π ) π‘ β ( π π‘ππ β ) (5.37)β 29 βf we use the same method as eq. 5.30 but considering the signal for π‘ > π‘ π , we ο¬nd that the derivative (cid:22) ππ‘ πΆπΉπ· ππ‘ π ( π‘ β ) (cid:23) π‘ β >π‘ π is independent of π‘ β and can be written: (cid:22) ππ‘ πΆπΉ π· ππ‘ π (cid:23) π‘ β >π‘ π = π π‘ππ ( ππ π‘ππ β π β π‘ π ) π ( π π‘ππ β ) (5.38)which is exaclty the derivative of eq. 5.31: (cid:22) ππ‘ πΆπΉ π· ππ‘ π (cid:23) π‘ β >π‘ π = ππ ππππ ππ‘ π (5.39)Taking the limit for π ββ β (CS-TIA case) of Eq. 5.38 we ο¬nd:lim π ββ π π‘ππ ( ππ π‘ππ β π β π‘ π ) π ( π π‘ππ β ) =
12 (5.40)solving numerically equation 5.34 for π‘ β for diο¬erent πΌ , we can ο¬nd the value of the derivative ππ‘ππ‘ π asa function of the threshold. Starting from our analysis on the TIA characteristics of the previous sections, in section 5 we haveanalyzed in detail the eο¬ect of time measurement on the two basic TIA scheme: CSA-like, or chargesensitive TIA, and Fast-TIA, which could be also deο¬ned as a current-sensitive TIA.When the dependence of time resolution π π‘ with respect to the charge collection dispersion π π‘ π is considered, we have seen that for the CSA-TIA both the CFD and LE discrimination techniquesconverge to the same value (equations 5.11 and 5.16). In other words, for π >> π‘ π , the front-endelectronics is capable to reduce the sensor intrinsic time dispersion up to a factor 2 and the propagationcoeο¬cient is π« β .
5. Referring to ο¬gure 18, this means that a CSA-TIA based front-end could takethe time resolution of a suitable 3D-trench sensor to the range of approximately 26 ps (55 π m pitch)to 15 ps (25 π m pitch).However, as already stated at the beginning of this work, the diο¬erence between the CSA-TIAand the Fast-TIA is nothing of conceptually fundamental, and is based only on the relative size of the π βΌ π ππππ value with respect to the average charge collection time π‘ π in the sensor (see ο¬gures 16,right and 18). This important statement will be better clariο¬ed in a moment concerning the circuitbehaviour relatively to its timing performances.On the other hand, when the Fast-TIA case is concerned, the front-end becomes capable toappreciate the shape of the induced signals and the analysis is more complicated and diversiο¬ed,depending strongly on the threshold position which is possible to choose, according to the noise levelof the system. In the simpliο¬ed case of very low threshold, starting from equation 5.21, we have that: π π‘ ππ = π π‘ π (cid:5) π π‘β πΌ π π π π‘π β π π‘ π (cid:5) π π π ππππ π π‘ π = π« π π‘ π (6.1)β 30 β β β π π‘ π (cid:6) ππ (6.2)where we are assuming to place the threshold at 5 times the voltage noise π π . Equations 6.1 and 6.2show that for π < π‘ π and if a high SNR is obtained, the output from the front-end can reduceconsiderably the native signal time dispersion of the sensor π π‘ π . As an example, considering a casewith π = . π‘ π , SNR = 30, equation 6.2 would give π« β π‘ π , whose time integral corresponds to a given total deposited charge. Herethe amplitude ο¬uctuation are not considered, as the signals all correspond to the same charge amount.Therefore the plot can be interpreted as the output performance of the CFD and LE cases, once thetime-walk eο¬ect is compensated. It can be seen that for a π‘ π = π and low threshold, the π« coeο¬cientcan be very low, while when the threshold is increased the front-end tends to propagate entirely thesensor jitter contribution to the output and π« approaches unity. The Fast-TIA conο¬guration needs tooperate at suο¬ciently low threshold to be eο¬ective on time resolution, otherwise its performancebecomes rapidly worse than the CSA-TIA case. Figure 19 . Left: derivative ππ‘ππ‘ π for ο¬xed π and ο¬xed average π‘ π = ππ as a function of threshold,corresponding to the given fraction πΌ of π ππππ . Right: same plot for CFD with diο¬erent values of π . Figure 19 (right) is a clear demonstration of the fact that the CSA-TIA conο¬guration behaviouris the limit of the Fast-TIA one, when the ratio ππ‘ π increases. A CSA-like behaviour is alreadyobservable very early, when ππ‘ π β
2. This is an independent conο¬rmation of what already seen above(see equation 4.81).In a circuit having time constant π βΌ ππ , the signal reaches about 20% of π ππππ at time π‘ = π‘ π ,corresponding to already approximately 50% of the charge collection time ο¬uctuations. With a fasterfront-end electronics, even setting the threshold at a higher fraction, it is still possible to reduce thecharge collection time ο¬uctuations to β
20% of the total.A better estimate of the π« coeο¬cient can be found modeling the current signals of an ideal 3Dtrenched sensor with charge carriers both at saturation using Ramo theorem. The standard deviation π π‘ π is given by the charge collection time distribution (Fig. 16 right). In this case, in order to calculateβ 31 β , a numerical convolution of such current signals with several pulse responses at diο¬erent timeconstants π can be done. Figure 20 shows the behavior of diο¬erent front-end stages characterized bydiο¬erent π constants: a fast front-end can provide approximately a double reduction of the chargecollection time ο¬uctuations. The use of a FAST-TIA conο¬guration allows exploiting deeper andmore eο¬ectively the performance of intrinsically fast sensors, as for example those realized in 3Dtechnology and in particular in the trenched-electrode geometry. Figure 20 . Function ππ‘ππ‘ π of an ideal 3D detector with electrodes spaced 25 ππ for diο¬erent time constant π A Appendix A: Calculus of the noise contributions in the CSA-TIA conο¬guration
To calculate the output noise we can consider the input referred noise and use the transfer functionof the circuit [7]. The sum in quadrature of the three sources is: πΌ ππ,π ππ = πΌ ππ,π + πΌ ππ,π π· + πΌ ππ,π π (A.1) πΌ ππ,π ππ = π π΅ π πΎπ π π π + π π΅ ππ π π π π π· + π π΅ ππ π (A.2)The transfer function is given by eq. 3.28. The integral of the squared module can be calculatedusing: π π ( π ) = π π πΊ + πΊ (cid:3) ( + π π ) β π π π§ ( + π π ) (cid:4) (A.3) π π ( π ) = π π πΊ + πΊ (cid:3) ( + π ππ π ) β π ππ π§ ( + π ππ π ) (cid:4) (A.4) | π π ( π )| = π π πΊ + πΊ (cid:3) (cid:8) π π π π + π (cid:9) + (cid:8) π π π π π§ ( π π + π ) (cid:9) (cid:4) (A.5)β 32 β β | π π ( π )| ππ = (cid:8) π π πΊ + πΊ (cid:9) (cid:3) β« β π π ( π π + π ) ππ + β« β π π π π π§ ( π π + π ) ππ (cid:4) (A.6) β« β | π π ( π )| ππ = (cid:8) π π πΊ + πΊ (cid:9) (cid:8) π π π + π π π π π π π§ (cid:9) (A.7) β« β | π π ( π )| ππ = (cid:8) π π πΊ + πΊ (cid:9) π π π (cid:8) + (cid:8) π π§ π (cid:9) (cid:9) (A.8)the zero of the transfer function increases the bandwidth of the factor ( π π§ π ) which is normallynegligible. The output noise is then given by: π π,π ππ’π‘ = πΌ ππ,π ππ (cid:8) π π πΊ + πΊ (cid:9) π π π (A.9)since πΊ + πΊ βΌ π π,π ππ’π‘ β π π΅ π π (cid:25) πΎπ π + π π π π· + π π (cid:26) (A.10)On the other hand this method underestimate the contribution given by the two sources πΌ π,π and πΌ π,π π· . The correct value can be calculated considering the small signal model in ο¬gure 21: Figure 21 . Small signal model for the noise sources πΌ π,π e πΌ π,π π· Solving the circuit to ο¬nd π π,ππ’π‘ as a function of πΌ π we get: π π,ππ’π‘ = β πΌ π π ππ’π‘ + ( + π π π ππ ) π ππ’π‘ π ππ + π π (A.11)substituting the impedances we ο¬nd the transfer function for the source πΌ π :β 33 β π,ππ’π‘ = β πΌ π π π· ( + π π π ( πΆ ππ + πΆ π )) + π π π π· + π ( π π ( πΆ ππ + πΆ π ( + π π π π· )+ π π· ( πΆ ππ + πΆ πΏ ))+ π π π· π π π (A.12)This transfer function is similar to 3.28 but shows a zero at frequency π π§,π = ππ π ( πΆ ππ + πΆ π ) . It can beshown that multiplying by π π π π + π π· both numerator and denominator we can write eq. A.12 in thefollowing way: π π,ππ’π‘ = β πΌ π π π ( + π π π ( πΆ ππ + πΆ π ))( + π π ) (A.13)this expression leads back to the noise found in A.2 for π =
0. Considering π : π π£,π = πΌ π,π Β· π π Β· Ξ π = π π΅ π πΎπ π π π Β· Ξ π = π π΅ π πΎπ π Β· Ξ π (A.14)To ο¬nd the output noise we use eq. A.8 with π π§,π = π π ( πΆ π + πΆ ππ ) : π π£,π = πΌ π,π π π π π π (cid:8) + (cid:8) π π§,π π (cid:9) (cid:9) (A.15)in this case the ratio (cid:8) π π§,π π (cid:9) is bigger than 1 and not negligible : (cid:8) π π§,π π (cid:9) = (cid:8) π π ( πΆ ππ + πΆ π ) π π π π π (cid:9) (A.16)using the deο¬nition of π : (cid:8) π π§,π π (cid:9) = π π π π ( πΆ π + πΆ ππ ) πΆ π πΆ ππ + πΆ π πΆ πΏ + πΆ πΏ πΆ ππ (A.17)the dominant term can be written: π π£,π = (cid:6) π π΅ π πΎ πΆ π + πΆ ππ π π π π π (A.18)where π = πΆ π πΆ ππ + πΆ π πΆ πΏ + πΆ πΏ πΆ ππ . Substituing πΌ π,π with πΌ π,π π· , we ο¬nd the noise given by resistor π π· : π π£,π π· = (cid:6) π π΅ π π π· πΆ π + πΆ ππ π π π π π (A.19)while the one given by resistor π π is: π π£,π π = (cid:6) π π΅ π π π π π π (A.20)β 34 β Appendix B: Calculus of the noise contributions in the Fast-TIA conο¬guration
The noise introduced by the bipolar transistor is given by the two correlated sources πΌ π e πΌ π . ThePSDβs are: πΌ π,π = π π΅ ππ π πΌ π,π = π π΅ π π π (B.1) Figure 22 . Small signal model for the noise sources for the Fast TIA implemented with Bjt
The noise given by source πΌ π,π has to be convoluted with the same transfer function of the signalfrom the detector, therefore: π π£,π = π π΅ ππ π ( π π π β π ππ ) π π π (B.2) π π£,π = π π΅ π π π π ( π π π β π ππ ) (B.3)To ο¬nd the noise from source πΌ π,π , we can use eq. A.11 and π ππ = π π || πΆ ππ to get the transfer function: π π,ππ’π‘ = β πΌ π,π π πΆ ( π π + π π ) ( + π π π π ( πΆ ππ + πΆ π )) π π + π πΆ + π π ( + π π π πΆ )+ π (cid:18) π π πΆ ππ ( π π + π πΆ )+ π π πΆ π ( π π ( + π π π πΆ )+ π πΆ )+ π πΆ πΆ πΏ ( π π + π π ) (cid:19) + π π πΆ π π π π π (B.4)where π π π = π π || π π .Simplifying assuming π = π , we get: π π,ππ’π‘ = β πΌ π,π π πΆ ( π π + π π ) π π ( + π π π πΆ ) + π πΆ + π π ( + π π π,π§ )( + π π ) (B.5)β 35 βlso in this case we have a zero with time constant: π π,π§ = π π π ( πΆ π + πΆ ππ ) (B.6)integrating we get: π π£,π = π π΅ π π π (cid:3) π πΆ ( π π + π π ) π π ( + π π π πΆ ) + π πΆ + π π (cid:4) π π π (cid:8) + (cid:8) π π,π§ π (cid:9) (cid:9) (B.7)using the output resistance of the circuit π ππ’π‘ we have: π ππ’π‘ = π πΆ ( π π + π π ) π π ( + π π π πΆ ) + π πΆ + π π (B.8) π π£,π = π π΅ π π π π ππ’π‘ π (cid:8) + (cid:8) π π,π§ π (cid:9) (cid:9) (B.9)the noise from source πΌ π is given by two contributions: π π£,π ( ) = π π΅ π π π π ππ’π‘ π (B.10) π π£,π ( ) = π π΅ π π π π ππ’π‘ π (cid:8) π π,π§ π (cid:9) (B.11)making clear (cid:18) π π,π§ π (cid:19) we can write: π π£,π ( ) = π π΅ π π π π β π ππ ( πΆ π + πΆ ππ ) ππ (B.12)we can write it as a single term: π π£,π = π π΅ π π π π (cid:8) π ππ’π‘ + π β π ππ ( πΆ ππ + πΆ π ) π (cid:9) (B.13)If πΆ ππ >> πΆ π and π β πΆ ππ πΆ πΏ we get: π π£,π β π π΅ π π π π (cid:8) π ππ’π‘ + π β π ππ πΆ ππ πΆ πΏ (cid:9) (B.14)normally the second term is the dominant one. The same analisys can be done for the noise given byβ 36 βesistor π πΆ leading to: πΌ π,π = π π΅ π π π ββ πΌ π,π πΆ = π π΅ ππ πΆ (B.15) π π£,π πΆ β π π΅ π ππ πΆ (cid:8) π ππ’π‘ + π β π ππ πΆ ππ πΆ πΏ (cid:9) (B.16)For the feedback resistor π π we can ο¬nd the transfer function considering the small signal model inο¬gure 23 and the PSD: πΌ π,π π = π π΅ ππ π (B.17) Figure 23 . Small signal model for resistor π π π π,π π = πΌ π,π π π π π πΆ π πΉ + π πΆ + π π ( + π π π πΆ ) ( + π π π π + π π π πΆ ππ )( + π π ) (B.18)since π π π π >> π π π πΆ >> π π,π π = πΌ π,π π π π + π π + π πΆ π π π π π πΆ ( + π πΆ ππ π π )( + π π ) (B.19)the zero is normally at very high frequencies and we can ignore it: π π£,π π = π π΅ π π π ( + π π + π πΆ π π π π π πΆ ) π (B.20)β 37 β cknowledgments This work was supported by the Fifth Scientiο¬c Commission (CSN5) of the Italian National Institutefor Nuclear Physics (INFN), within the Project TIMESPOT. The authors wish to thank Angelo Loifor his help in providing the plots used in ο¬gure 18.
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