Blum W., Riegler W., Rolandi L. Particle Detection with Drift Chambers

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6.5 Electronics for Time Measurement 237

surface towards the cathode and the ampliﬁer output signal v

(t) for such an input

pulse:

(t)=

1+t/t

(t)=g



h(t −t



)dt



. (6.147)

From Fig. 6.16 and Eq. (6.44) we know that the peak v

and the peak time T

(t) are approximately given by

≈ gI

ln(1+t

/2t

) T

≈t

(6.148)

for both the unipolar and the bipolar shaper.

6.5.1 Inﬂuence of Electronics Noise on Time Resolution

In order to isolate the effect of electronic noise from chamber intrinsic ﬂuctuations

we ﬁrst assume that m electrons arrive at the wire at exactly the same time and that

all electrons experience the same avalanche multiplication G. The induced current,

the output signal and the peak of the output signal are then

i(t)=mGi

(t) v(t)=mGv

(t) v

≈ gmGI

ln(1+t

/2t

) (6.149)

Applying a discriminator with a threshold v

to the ampliﬁer output signal results

in a measured time t

which, in the absence of electronic noise, is strictly related to

the arrival time of the electrons (Fig. 6.45). If electronic noise with an r.m.s of

is superimposed on the output signal v(t), the edge time of the discriminator output

will vary, as illustrated in Fig. 6.46a. From the ﬁgure it is evident that the r.m.s. of

the discriminator output time jitter is

gENC

k = v



), (6.150)

where k is the slope of v(t) at the threshold level. The slope k is proportional to

,sowehave

Fig. 6.45 Deﬁnition of the

slope k at the threshold

crossing, the signal peak time

, and the signal peak v

5 10 15 20

0.2

0.4

0.6

0.8

Threshold

Slope k

Time (ns)

238 6 Electronics for Drift Chambers

5 10 15 20

Time (ns)

(a) (b)

Signal (mV)

Threshold

noise

time

5 10 15 20 25 30

0.25

0.5

0.75

1.5

(ns)

r.m.s (ns)

1.25

Fig. 6.46 (a) Electronic noise resulting in time jitter. (b) Dependence of

on the ampliﬁer peaking

time for a cathode readout channel and the parameters C

= 20 pF, R

= 10 k

, R

= 150

/2ln(100),m = 10,G = 10

= 1ns,c

= 2, unipolar shaper with n = 4

k ∝

= c

gmGI

ln(1+t

/2t

)

ENC

mG I

ln(1+t

/2t

)

, (6.151)

where c

is a constant with a value of c

≈ 1.5 −2.5. For faster signal rise times,

i.e., for smaller values of t

, the time jitter

is reduced. We must, however, keep

in mind that the ENC also depends on the ampliﬁer peaking time t

as given by

Eq. (6.115). Neglecting 1/ f noise we found ENC(t



A/t

+ Bt

for a typical

cathode pad readout channel. Thus, for the time jitter we get



A/t

+ Bt

mG I

ln(1+t

/2t

)

A = 4kTR

B = 4kTK

. (6.152)

For small values of t

we ﬁnd

∝ 1/

√

; for large values of t

the expression

increases as t

3/2

/lnt

. We therefore have an ampliﬁer peaking time t

for which the

slope-to-noise ratio

/k and, therefore, the time jitter

become minimal. It can

be shown that the above expression has a minimum at t

≤ 8t

for any choice of

parameters. Figure 6.46b shows an example of the time jitter versus peaking time

for some typical parameters of a cathode channel.

Formally, the problem of minimizing the noise-to-slope ratio, or maximizing the

slope-to-noise ratio, can be studied in the same way as the question of the maximum

achievable signal-to-noise ratio treated earlier. The formulation of the problem is the

following: A signal F(i

) with a superimposed noise of power spectrum w(

) is

processed by an ampliﬁer with transfer function H(i

). Which transfer function

H(i

) maximizes the slope to noise ratio?

The slope of the ampliﬁer output signal g(t) is dg(t)/dt = g



(t), which in the

frequency domain corresponds to a multiplication of G(i

) with i

. The slope-to-

noise ratio is









)





∞

−∞

F(i

)H(i



∞

−∞

)|H(i

. (6.153)

6.5 Electronics for Time Measurement 239

Proceeding as before we ﬁnd the maximum achievable slope-to-noise ratio





≤



∞

F(i

)

(6.154)

with the optimum transfer function H(i

) and the output signal G(i

) being

H(i

)=i

∗

)

−i

G(i

)=i

|F(i

)

−it

. (6.155)

We realize that H(i

) and G(i

) differ only by the factor i

from the ﬁlters op-

timizing the signal-to-noise ratio that we found earlier. Since a multiplication with

in the frequency domain corresponds to the derivative in the time domain we can

conclude:

For a given signal f (t) and noise power spectrum w(

), the ampliﬁer delta

response that maximizes the slope-to-noise ratio is equal to the derivative of the

ampliﬁer delta response that maximizes the signal-to-noise ratio.

Since the ampliﬁer output signal for maximizing the signal-to-noise ratio is sym-

metric around t = 0, the ampliﬁer output signal optimizing the slope-to-noise ratio

is antisymmetric around zero and the maximum slope-to-noise ratio is found at the

zero crossing timer of the signal (Fig. 6.47).

6.5.2 Inﬂuence of Pulse-Height Fluctuations on Time Resolution

We now consider the same situation as before, where m electrons arrive at the same

time at the sense wire, but we let the number m ﬂuctuate from event to event accord-

ing to a distribution p(m) with an average of

m and an r.m.s. of

(Fig. 6.48a).

Although all the electrons arrive at the same time, we ﬁnd different threshold

5 10 15 20 25

0.5

Time (ns)

Fig. 6.47 Example for the output signal of two optimum ﬁlters. The symmetric curve is the output

signal for a ﬁlter that maximizes the signal-to-noise ratio. The antisymmetric curve is the output of

a ﬁlter that maximizes the slope-to-noise ratio. The maximum slope-to-noise ratio is found at the

zero crossing. The delta response of the ﬁlter that maximizes the slope-to-noise ratio is equal to

the time derivative of the delta response for the ﬁlter that maximizes the signal-to-noise ratio. The

same is true for the output signals of the two ﬁlters

240 6 Electronics for Drift Chambers

Time (ns)

Threshold

(a)

5 10 15 20 25

–1

–0.5

0.5

Time (ns)

(b)

5 10 15 20

0.2

0.4

0.6

0.8

Fig. 6.48 (a) Variations of the pulse height result in different threshold crossing times. The effect

is called ‘time walk’. (b) The zero crossing time of a bipolar pulse is independent of the pulse

height

crossing times owing to the pulse-height variations of the signal, an effect that is

called ‘time walk’. The threshold crossing time and its variance are

(m)=

k(m)

≈

(m)

m=m

mGI

ln(1+t

/2t

)





(6.156)

We see that the time walk decreases linearly with the threshold and the peaking

time and we therefore want fast electronics and the lowest possible threshold for

achieving the best time resolution. However, the lowest possible threshold is again

determined by the noise. The frequency f

of the noise threshold crossings for a

threshold v

is given by

= f

zero

−v

, (6.157)

where f

zero

is the number of zero crossings from Eq. (6.83) [RIC 44]. Setting the

threshold v

to 1,2, 3, 4, 5 times the noise r.m.s. results in a threshold crossing fre-

quency of 0.6, 0.14, 0.01, 3.3 ×10

−4

,3.7 ×10

−6

times f

zero

. With f

zero

being in the

range of 10–100 MHz we have to set the threshold v

to at least ﬁve times the noise

r.m.s. in order to arrive at a noise count rate of less than 100 Hz. We use v

= 5

in the following. The discriminator threshold v

is typically expressed as a charge

according to v

= gQ

in order to arrive at expressions independent of the ampli-

ﬁer sensitivity. The ampliﬁer output noise is

= gENC, and we therefore have a

threshold of Q

= 5ENC:



A/t

+ Bt

mGI

ln(1+t

/2t

)





. (6.158)

This expression is equal to the expression for the time jitter up to a factor 5

/m.

Therefore the peaking time that minimizes the jitter due to electronic noise also

minimizes the jitter due to pulse-height variations. The combined effect of electronic

noise and pulse-height ﬂuctuations becomes

jitter

walk







A/t

+ Bt

mGI

ln(1+t

/2t

)

. (6.159)

6.5 Electronics for Time Measurement 241

If the relative pulse-height variations

/m are greater than one, which is usually

the case in wire chambers, the time resolution is dominated by the pulse-height

variations and not by the jitter from electronic noise. On the other hand, it is the

noise that sets the lower limit on the threshold level, so it is the noise alone that

determines the time resolution owing to jitter and walk.

There are numerous ways of eliminating this pulse-height dependence of the

threshold crossing time. One can, for instance, exploit the fact the zero crossing

time of a bipolar-shaped pulse is independent of the signal amplitude, which follows

directly from the linearity of the system, as shown in Fig. 6.48b. Another possibility

is the use of a so-called constant fraction discriminator that uses a threshold which

is always set to a certain fraction of the signal peak. These techniques are applicable

when the arrival time spread of the electrons is negligible compared to noise and

pulse-height variation effects. In the next section we investigate how variation in the

electron arrival time inﬂuences the time resolution.

6.5.3 Inﬂuence of Electron Arrival Time Fluctuations

on Time Resolution

In the previous section we assumed that the number of electrons arriving at the wire

varies from event to event, but that all the electrons arrive at the same time. Now we

include the effect that the m individual electrons arrive at different times t

, t

....t

The arrival times of the electrons as well as the number of electrons m ﬂuctuate from

event to event. In Sect. 7.4 it is shown that the average electron arrival time gives

the best timing only when the diffusion dominates the arrival spread, as for distant

tracks in a TPC.

For short drift distances, the primary ionization ﬂuctuations together with effects

such as track inclination and path length variations dominate, and the ﬁrst arriving

electrons show the smallest arrival time ﬂuctuations. For geometries like the drift

tube, the average electron arrival time shows a very large r.m.s. even for long drift

distances because the last arriving electrons always originate close to the tube wall

and therefore have no correlation with the drift distance (Fig. 6.49). To study the

effect of the arrival time spread we ﬁrst investigate the bipolar shaping scheme,

Fig. 6.49 Electron drift lines in a drift tube. The distance of the track from the wire is determined

by the ﬁrst arriving electrons. The last arriving electrons originate from the tube wall and have no

correlation with the track distance

242 6 Electronics for Drift Chambers

5 10 15 20

Time (ns)

20 25 30

0.5

1.5

2.5

12.5 13 13.5 14 14.5 15

0.3

0.4

0.5

9.5

10 10.5 11 11.5 12

0.4

0.2

0.4

Time (ns)

(a)

(b)

Time (ns)

Fig. 6.50 Pulses from three electrons arriving at the wire at different times. The solid lines show

the pulses of the individual electrons; the dotted lines correspond to the sum of the pulses giv-

ing the measured chamber signal. Parts (a) and (b) show the bipolar-shaped signals where the

time measurement is given by the zero crossing of the chamber signal. Parts (c) and (d) show

the unipolar-shaped signals where the time measurement is given by the threshold crossing of the

chamber signal. The two schemes have distinctly different timing properties

where the zero crossing of the signal determines the measurement time. Figure 6.50a

shows the signals from three individual electrons arriving at different times and the

resulting chamber signal which is the sum of the individual electron signals. Around

the zero crossing the electron signals can be approximated by a straight line with a

slope −k (Fig. 6.50b), and the individual electron signals v

(t) and the resulting

signal v(t) can be approximated by

(t)=−k(t −t

) v(t)=

∑

i=1

(t)=−kmt−k

∑

i=1

. (6.160)

The zero crossing time t

of the signal v(t) is

v(t)=0 → t

∑

i=1

= t (6.161)

and equal to the average arrival time of the m electrons. This scheme is therefore

applicable for the ‘diffusion dominated’ cases mentioned above.

If we want the nth electron to give the timing information we can imagine an

ampliﬁer that produces a step function output signal of amplitude v

for a single

6.5 Electronics for Time Measurement 243

electron. The step functions of the individual arriving electrons would therefore pile

up and by setting the threshold to v

= nv

we would measure the arrival time of the

nth electron. However, true step function requires an ampliﬁer of inﬁnite bandwidth,

and a realistic step function will take a certain time to rise from zero to v

. This time

must be shorter than the time between arriving electrons such that the pulses are

really piling up. Since the time between the ﬁrst arriving electrons is typically much

less than a nanosecond, the bandwidth of such an ampliﬁer must be extremely broad,

and the corresponding large noise might not allow a threshold that is low enough to

record the early electrons. In some rare cases this might still be a viable option.

Typically the ampliﬁer peaking time is much longer than the time intervals between

arriving electrons and it is not the pulse heights but the slopes of the individual

electrons that pile up as shown next.

Figure 6.50c shows the signals from three individual electrons and the resulting

chamber signal. The time information is given by the threshold crossing time of the

leading edge of the chamber signal. The signals v

(t) from the individual electrons

can again be approximated by ‘straight lines’, as indicated in Fig. 6.50d, but in

contrast to the bipolar shape from before, these straight lines do not cross the zero

line, but start from zero at t = t

. This is illustrated in Fig. 6.51. A single electron

arriving at time t

produces the signal

(t)=k(t −t

)

(t −t

) v(t)=

∑

i=1

k(t −t

)

(t −t

). (6.162)

The chamber signal is therefore

v(t)=0 t < t

v(t)=k(t −t

) t

< t < t

v(t)=k(t −t

)+k(t −t

) t

< t < t

etc.

(6.163)

Fig. 6.51 Four single

electrons arriving at different

times and producing signals

with slope k. The sum of the

individual signals gives the

chamber signal, which

changes the slope by k

whenever a new electron

arrives

0.5 1 1.5 2 2.5

0.5

1.5

2.5

3.5

Time (ns)

Threshold v

244 6 Electronics for Drift Chambers

and in general we get

v(t)=

∑

i=1

k(t −t

)=knt−k

∑

i=1

< t < t

n+1

. (6.164)

Whenever a new electron arrives the slope changes by k, and after the arrival of the

nth electron the signal slope is k

= nk. The threshold crossing time t

v(t)=v

→ t

∑

i=1

if t

< t

n+1

. (6.165)

We see that the threshold crossing time is equal to the average arrival time of the n

electrons that arrived before the threshold crossing plus another term that is inversely

proportional to n. Figure 6.51 shows an example in which four electrons arrive at

times t

, and t

. The chamber signal crosses the threshold before the arrival of

the fourth electron, so we have n = 3 and the threshold crossing time is

. (6.166)

If all the electrons arrive at the same time t = 0 we are left with only the second

term, which is equal to the expression for the time walk discussed in the previous

section. We can therefore state that the ﬁrst term is due to the electron arrival time

spread and the second term to the varying number of electrons that contribute to the

signal at the threshold crossing time. However, the two expressions correlate in a

complex way.

If we want to explicitly calculate the threshold crossing time distribution and its

r.m.s., it is not sufﬁcient to know the average and r.m.s of the arrival distribution for

the nth electron. The threshold crossing time depends on the arrival times of all n

electrons, so we need the probability density

P(t

...)dt

..., (6.167)

which gives the probability that the ﬁrst electron arrives in the time interval t

, the second arrives in the time interval t

+dt

, etc. The distribution P(t

...)

is typically generated with a Monte Carlo simulation program. Instead of ﬁnding an

analytic parametrization of the Monte Carlo generated distributions and evaluating

the integrals of the probability distribution it is much easier to record the arrival

time pattern t

,...t

for each Monte Carlo event and to ﬁnd the chamber signal by

superimposing the single-electron pulse v

(t) according to

v(t)=

∑

i=1

(t −t

). (6.168)

By adding the expected noise signal to v(t) we ﬁnd a set of pulses from which

we can calculate the threshold crossing times and therefore the time r.m.s. This

6.5 Electronics for Time Measurement 245

procedure is very fast and by varying the threshold and electronic parameters it is a

straightforward task to ﬁnd the optimum electronic parameters.

We can, on the other hand, get a good qualitative idea about the optimum elec-

tronic parameters. If we assume that the ﬁrst electron arrives at t = 0 and the

following electrons arrive at regular time intervals

t, the chamber signal v(t)

becomes

v(t)=

∑

k(t −i

(t −i

t) ≈

. (6.169)

The threshold crossing time t

and the number of electrons n having arrived at the

wire before t

are

≈



n ≈



. (6.170)

If the electrons are separated by 2

t instead of

t the threshold crossing time is a

factor

√

2 later. The two ‘events’ with an electron separation

t and 2

t are shown

in Fig. 6.52. The time difference

between the two threshold crossings for the

two signals is

√

2 −1)



. (6.171)

The variation of the threshold crossing time therefore decreases for lower threshold

and large slope (faster electronics) as was the case for pulse height ﬂuctuations and

time walk. The lower threshold limit is again determined by the noise level. The

effects of pulse-height variation, discussed in the previous section, and the arrival

time ﬂuctuations cannot be separated because they are highly correlated. A larger

primary ionization will also result in shorter time intervals between arriving elec-

trons. The only effect that can be added in square is the jitter from electronic noise

that is independent of the electron statistics.

On the one hand, lowering the threshold v

and increasing the signal slope k

(broader bandwidth, i.e., smaller peaking time) will reduce the effect of arrival time

2 4 6 8 10 12 14

0.5

1.0

1.5

2.0

2.5

3.0

Threshold v

Time (ns)

Fig. 6.52 Two signals where the electrons arrive at the wire at regular time intervals

t and 2

respectively

246 6 Electronics for Drift Chambers

ﬂuctuations and related pulse-height ﬂuctuations. On the other hand, a broader band-

width will increase the noise and a lower threshold will increase the noise jitter

because of the parabolic shape of the signal. The compromise between these two

tendencies will deﬁne the optimum ampliﬁer peaking time, which will again be of

the order of a few times t

Finally we note that the slope of the signal at the threshold crossing time k

= nk

measures the number of electrons that have contributed to the signal, and by using

this information we can improve the time resolution. The slope can, for example,

be measured by a second threshold (Fig. 6.52) or by a device that measures the

signal charge in a short time interval following the threshold crossing time. This

technique is known as amplitude and rise time compensation and allows extraction

of the optimum time information from the signal.

6.6 Three Examples of Modern Drift Chamber Electronics

In this last section we present some front-end electronics examples that are being

used in large detector systems. We discuss an ampliﬁer for time measurement used at

rates up to 15 MHz, an ampliﬁer for charge measurement in a cathode strip chamber,

and a front-end system for a TPC.

6.6.1 The ASDBLR Front-end Electronics

The ASDBLR (ampliﬁer shaper discriminator baseline restorer) chip [BEN 96] is an

eight-channel front-end chip developed for readout of the ATLAS [ATL 94] transi-

tion radiation tracker [AKE 04]. The chip is implemented in a radiation hard bipolar

process. The tracker consists of almost half a million drift tubes of 4 mm diame-

ter with a 30-

m-diameter anode wire. A xenon-based gas mixture at a gas gain

of ≈ 2 ×10

is used. The goal is a spatial resolution of better than 150

matex-

treme rates of up to 15 MHz. To cope with this high rate, unipolar signal shaping

was chosen. The ASDBLR circuit is based on a front-end described in [FIS 85], and

the equivalent block diagram is equal to the one shown in Fig. 6.27 top. It consists

of a large bandwidth preampliﬁer with 1 ns rise time followed by a unipolar shaper

(n = 3) that provides a peaking time of t

= 8ns. The preamp input impedance of

= 295

is matched to the characteristic impedance of the drift tube, which rep-

resents a capacitance of C

≈ 12pF. A double pole-zero network cancels the signal

tail. In order to provide the baseline stability at the very high rates, the shaper is

followed by a fast (nonlinear) baseline restoration circuit using diodes. For very

small signals, the BLR acts like a CR differentiation circuit and produces a bipolar

output. As the shaper signals grow in magnitude, the exponential behaviour of the

diodes causes the undershoot to rise logarithmically with the pulse height, which

results in a unipolar signal shape. With this signal processing chain, the return to