RPC simulations from a current stand point Diego González-Díaz (Zaragoza and Tsinghua University) Huangshan Chen, Wang Yi (Tsinghua University) Frascati, 07-02-09

Index Induced current vs induced charge. Which one to chose as the fundamental magnitude for evaluating the induction process? Introduction to the 'induction+transmission' model and assumptions. Circuit Theory (C-T) vs Multi-conductor Transmission Line Theory (MTL-T). Results for long strips. Structure of the solutions of the transmission line equations. Signal dispersion in long strips and how to compensate. Comparison with data for the main multi-strip characteristics.

1. Current vs charge. Which one to chose as the fundamental magnitude? - The fundamental input magnitudes needed for solving any electronic problem are voltage and/or currents generators. ? Cdetector Rscope however - The fundamental input magnitudes needed for solving an electromagnetic problem are the charges and potentials. ? q(t) fortunately - The Schockley-Ramo theorem (1938-39) provides a way to relate the currents induced by moving charges in grounded electrodes with a simple electrostatic problem: q(t) electronics electrostatics gas dynamics moreover

1. Current vs charge. Which one to chose as the fundamental magnitude? - Besides particular solutions derived through alternative techniques (e.g. energy conservation) a modern highly general set of theorems based on the Green theorems has been provided by Riegler (2002-2004) inspired partly by the work of Radeka et al. (1982) Cdetector ? q however εr, σ a) There is not a single (published) RPC simulation where the fundamental magnitude used to derive the basic RPC performances is the induced current i(t). b) The detector capacitance is never explicitly taken into account. The standard approach is to integrate the induced current i(t) in order to obtain an induced charge Qind(t) and evaluate the condition Qind(t)>Qth, where Qth is a threshold in charge units previously determined (implicitly assumed: under conditions of δ-impulse excitation... or not?) indeed

1. Current vs charge. Against all odds... charge wins... working with the 'induced charge' approach provides a) the relevant magnitude Qind(t) with reasonable accuracy for small capacitances and slow electronics, b) experimentally summarizes the electronics performance in a single quantity Qth (obtained under conditions of δ-injection... or not?); c) the maximum of Qind(t) represents with accuracy the charge measured by a QDC.

1. Current vs charge. But.... is the FE electronics really that slow? in absence of space charge I~exp(St), S=α-η D.Gonzalez-Diaz, doi:10.1016/j.nima.2010.09.067 S~1/50ps αg~30 (HADES) S~1/1ns αg~20 (ATLAS) parameters of C2H2F4 measured for the first time in RPC case A favorite charge pre-amplifier. Canberra -2004 (approximated by a 2-stage amplifying circuit, following specs) . fc~3GHz (BW~3kHz) fc~150MHz Philips BGM1013, Mini-circuits GALI55 (BW~2GHz) that's how is supposed to work: different signal shapes (ΔT) at equal charge when sent to the amplifier: yield different rise-times but same amplitude. In the δ-impulse excitation limit, the response is an universal property of the amplifier ATLAS, CMS FEE (BW=100-200MHz) MAXIM MAX3760, NINO, PADI, (BW~200-500MHz) perhaps these guys might be close to be 'slow' indeed

1. Current vs charge. And... are the RPC capacitances really that small? Most glass-based strip RPCs (HADES-TOF, STAR-MTD, CBM-TOF, R3B-neuLand) have (simulated) capacitances to ground in the range 1.5-3.5pF/cm (values for Bakelite-based RPCs are of the same order). For typical 1m-lengths (R3B-neuLand prototypes are 2m-long!) this can mean an RC constant of 17.5ns (correspondingly a bandwidth of barely 10MHz in a circuit model). And we expect to get a signal out (and it gets out) that we know it can intrinsically reach 3GHz (1.5GHz was measured directly in RPC structures (Fonte et al.))... that's very hard to take! does not look like the total capacitance is the main player here... we are anyhow missing something that looks important

1. Current vs charge. Some rather non –negligible pitfalls of the classical 'induced-charge approach'. No way to introduce the counter characteristic impedance Zc and/or its total capacitance C in a consistent way. Since Qind(t) is the integral of the current, its maximum is independent from the details of any mismatch between the counter impedance and the read-out chain impedance. A counter with a transmission coefficient (T=2Zc/(Zc+R)) of 1% (Zc~0.005R) will achieve the same maximum value of Qind(t) than having a transmission of 100%. More generally speaking, there is no way to include connections in a consistent way. Any connection not having ohmic leaks (at least this is to be expected) will yield, upon integration, the same maximum value of Qind(t) [this is essentially a result stemming from charge conservation]. No way to describe the signal rising-slope in a consistent way. The slope is always coming from the integral of the current, that is clearly not the case for broad-band amplifiers matched to the signal time-components. Since we have already given up on connections, there is no way out anyhow. No way to describe capacitive cross-talk in a consistent way. Bipolar cross-talk might integrate down to zero or little charge, except if there is charge sharing due to extended weighting field profiles. No hope to describe system performances besides a physics parameterization. In view of the previous limitations, describing a system with several strip/pad conductors (the real thing) seems a daunting task.

The price to pay for abandoning the 'induced-charge approach'. 1. Current vs charge. The price to pay for abandoning the 'induced-charge approach'. No way to characterize the FE electronics with a single quantity, Qth. The minimal functional description requires of knowing the FE input impedance, the frequency response and the discriminator threshold. Nevertheless, Qth is still well defined in the 'δ-impulse excitation' limit. Of course noise is also important, if it can be determined properly. The role of the detector capacitance must be included in a proper way.

The bricks needed for abandoning the 'induced-charge approach'. 1. Current vs charge. The bricks needed for abandoning the 'induced-charge approach'. - The Schockley/Ramo/Radeka/Riegler theorems have a strong limitation: one must assume that, for every instant of time, the whole electrode surface is equi-potential (a low-frequency concept). In terms of a common high-frequency concept, this means that the detector has to be electrically small (L, w, h < λc) in all 3-dimensions. h w L signal propagation velocity ~1/2 cutoff frequency (3dB drop in frequency-domain, i.e. 1/√2 drop) if you are not familiar just take this (in typical structures, vprop~1/2c): -An exponential signal with a rise-time of 100ps has λc=5cm. -A step signal with a width of 1.5ns has λc=130cm. - The general way to overcome this problem is through a 3D FEM simulation. This approach (Pestov et al in late 90's) even nowadays, seems to be unacceptably slow for detector characterization/optimization. Besides, it is very difficult to debug without analytic tools.

The bricks needed for abandoning the 'induced-charge approach'. 1. Current vs charge. The bricks needed for abandoning the 'induced-charge approach'. - Intuitively, the most critical dimension is the longitudinal one, L (the transverse coupling declines with distance anyway, while the anode-cathode distance is usually small). Modern RPCs in multi-strip configuration can reach electrical lengths (Λe = L/λc) up to Λe = 40 with some 50-100 densely-packed strips (R3B-NeuLand). This makes them, a priori, some of the harshest conceivable systems from the point of view of signal transmission. How might the problem look like for a 2x2 strip case?: circuit theory transmission-line theory ? Cdetector Rscope Cmutual Cdetector Rscope

2. Introduction to the 'induction+transmission' model and assumptions. We propose an 'obvious' model based on a separation of the induction (solved through Ramo theorem, for instance) and transmission (solved through transmission-line techniques). Avalanche model depends on use case. At the moment we use a 1D model. 1 q(t) -> 1D model (not the main purpose of this work) 3 weighting field x y z direcction across the strip 2 I(t) = Ew(x) vd q(x, t) (from Ramo-theorem)

2. The 'induction+transmission' model at present. Assumptions. theoretical assumptions The structure is electrically short in the transverse dimensions. (possibly ok in most cases, since the transverse coupling decays relatively fast with transverse distance. Note: going beyond this assumption requires abandoning MTL theory) The excitation mechanism can be considered as taking place localized at a cross-section (this assumption is difficult to justify, except because the model agrees well with data. We are working on an approach where this assumption is not needed). technical assumptions (not necessary, but very handy) The Ramo theorem can be applied assuming the electrode is infinite in length (possibly ok, except around the last 1-2cm close to the detector ends). The material electrical properties are homogeneous and isotropic. (Seems to be the case for the most common materials used in multi-gap timing RPCs). For practical purposes, the material electrical properties show little variability from sample to sample (seems to be the case). An overall high mechanical accuracy well below 1mm-precision is achievable (usually ok, since this is pre-requisite of timing RPCs in any case). So the strip can be considered to be uniform along its length. The dielectric losses take place only with respect to ground (seems to be enough in practice. The assumption can be removed, but requires some effort). The connection between transmission lines (for instance cable-RPC) is ideal (seems to be enough in practice. The assumption can be partly avoided, but not straight-forward).

3. Circuit Theory vs Multi-conductor Transmission Line Theory. But after all... which theory to chose?. Circuit theory or transmission line theory?. Should we use the total capacitance or should we think in terms of MTL (i.e, characteristic impedance) or both... or are we double-counting then? Let's try it out, take an exponential signal, up to a multiplication factor of ~107, then assume a drift for about half a 0.3mm gap. Then send the signal through a 2GHz-BW amplifier. And take the published values for C2H2F4 at 100kV/cm. Change the width and length of your 1-gap structure and compare the circuit response against the MTL response for the case of the signal induced right at the middle of the structure.

RPC length Λe = L/λc RPC width

3. Circuit Theory vs Multi-conductor Transmission Line Theory. in 2D

3. Circuit Theory vs Multi-conductor Transmission Line Theory. in 1D

(although it seems to make some sense). This is not a proof that we should use transmission line theory for proper calculation of induced currents (although it seems to make some sense). Let's take our induction/transmission assumption for granted and go into the wild.

4. Cross-talk and signal integrity: here the transmission problem in one of its most discouraging incarnations The telegraphers' equations rule the phenomenon: V(t)? Cm Lo Lm +V Cg -V D~1m D~1m x z y D. Gonzalez-Diaz, H. Chen, Y. Wang, doi:10.1016/ j.nima.2011.05.039

4. Solve the telegrapher's equations + C and L matrixes in order to calculate signal transmission the telegrapher's equations for loss-less transmission lines for instance, for 2-strips they take the familiar form The weighting field maps and electrostatic matrixes C, L can be calculated with a 2D electrostatic solver (we are using the FEM-solver MAXWELL-2D from Ansoft). The solutions to eqs. (1), (2) are then used.

4. The main goal might be to go from the left situation to the right one transmission cross-talk 1st cross-talk 2nd distance from the read-out point

4. The general solution for transmission of a signal induced at any position –y- along an N-strip transmission line neglecting losses the modern point of view is that this is actually a trivial problem (just in case you thought it might not be the case) The matrix M and system velocities come from the solution to the diagonalization problem: the signal is a superposition of N modes, each traveling at a different velocity-> dispersive system! Remember this important theorem: if the 2 capacitance matrixes have this form and are proportional, then all velocities become equal, modal dispersion disappears!. Application of such a mathematical condition implies that the capacitive and inductive coupling are balanced/compensated.

4. Experimental verification in 2-strip (2m-long) RPCs (I) in time-domain in frequency-domain compensated under-compensated over-compensated dielectric losses What makes the trick? In this particular configuration... increasing the coupling !

4. Experimental verification in 2-strip (2m-long) RPCs (II) within ±0.2mm cross-talk increases by a factor 2-3!

4. 'Literal' solutions of the transmitted and cross-talk currents under approximations valid for typical RPCs when it is compensated ~ ~ the 2-strip parameters reflections neglected

4. Demystifying the characteristic impedance -> An N-strip RPC is characterized by an NxN C-matrix, NxN L-matrix, and NxN G-matrix (losses), if we neglect skin effect (a valid assumption for typical strip widths). All these matrixes are symmetric. -> This means 3(N2+N)/2 independent parameters. A structure like the one used by the FOPI experiment at GSI (the only operational wall in a physics experiment based on multi-strip timing RPCs) requires of 408 independent parameters in order to be fully characterized, due to its 16 strips (actually the obvious symmetry left-right allows for usage of only 408/2=204). This makes the simulation of this type of structure neither very easy, nor particularly intuitive, nor computationally very cheap. -> Out of this, in publications/presentations the FOPI 'characteristic impedance' is regarded as Zc=50±2Ω (after a very careful design, 100μm by 100μm...), the signal propagation as v/c=0.5, and this statement closes the topic on signal transmission. I am not saying that it is not enough. The final physics performance at high multiplicities is the main factor to decide here. And this is not yet available.

4. Here are the parameters we propose (7, out of which 5 are critical) D. Gonzalez-Diaz, H. Chen, Y. Wang, doi:10.1016/ j.nima.2011.05.039 (for details) coupling coefficient characteristic impedance strength of modal dispersion tangent loss long-range coupling strength

5. Comparison with data for the main multi-strip characteristics. (Includes avalanche model) data from Fonte 2002 D.Gonzalez-Diaz, doi:10.1016/j.nima.2010.09.067