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

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6.1 Linear Signal Processing 187

i(t)

I(s)

w(t)

W(s)

o(t) = ∫w(t-t')i(t')dt'

O(s) = W(s) I(s)

Fig. 6.3 A linear device is characterized by the transfer function W (s) in the Laplace domain or

the delta response w(t)=L

−1

[W (s)] in the time domain

The step function

(t) is deﬁned in Eq. (6.16). For k = 0wehaveL

−1

[1]=

(t).

If a

> 0, the delta response tends to inﬁnity for t → ∞, so the criterion for stability

of a linear system is given by the requirement that the real parts of all poles of W(s)

must be negative. In the case where a

+ ib

is a root of a polynomial, the complex

conjugate a

−ib

is also a root of the polynomial. Thus the terms in Eq. (6.13)

always appear in complex conjugate pairs, the imaginary part is cancelled, and the

delta response is always real.

Figure 6.4 shows an example of the terms from Eq. (6.13) for k = 2,4,6 and b

0. In this book we mainly discuss transfer functions with real poles. The imaginary

parts of poles arise from inductances. (As these cannot be implemented physically as

coils in integrated devices, one would make them feedback loops using operational

ampliﬁers).

6.1.3 CR, RC, Pole-zero and Zero-pole Filters

In this section we discuss four elementary ﬁlters that illustrate the formalism out-

lined in the previous section. They are also elementary in the sense that by cascading

these four ﬁlter types we can construct any desired transfer function W(s) with real

poles. If we wanted to construct the most general transfer function W(s) also in-

cluding complex poles we would have to add a few elementary circuits containing

an inductance L. The CR and RC ﬁlters are shown in Fig. 6.5. The output voltages

of the two ﬁlters for an input voltage signal V (s)=L [v(t)] are

Fig. 6.4 The delta response

w(t) of a linear network with

real poles consists of the sum

of shapes indicated in this

ﬁgure. In the case where

W(s) has complex poles we

ﬁnd terms with an oscillatory

behaviour of frequency b

and a damping factor of

exp(−a

1 2 3 4 5

0.2

0.4

0.6

0.8

k = 6

t/t

k = 4

k = 2

188 6 Electronics for Drift Chambers

v(t)

(t) v

(t)

Fig. 6.5 A CR (left) and RC (right) ﬁlter consisting of a resistor R and a capacitor C. A voltage

signal v(t) is transformed into the output voltage signals v

(t) and v

(t)

(s)=

sRC

1+ sRC

V(s)=

1+ s

V(s)=W

(s)V(s) (6.14)

(s)=

1+ sRC

V(s)=

1+ s

V(s)=W

(s)V(s). (6.15)

The value

= RC is the characteristic time constant of the ﬁlters. In the following

we frequently use the step function

(t), which is deﬁned by

v(t)=

(t)=

0 t ≤0

1 t > 0

V(s)=L [v(t)] =

. (6.16)

Applying this voltage step of amplitude V

at the inputs of the two ﬁlters yields the

output signals

(s)=

1+ s

(s)=

s(1+ s

)

. (6.17)

Returning to the time domain, we ﬁnd

(t)=V

−

(t) v

(t)=V

(1−e

−

)

(t) (6.18)

The two responses are shown in Fig. 6.6. The output of the CR ﬁlter decays exponen-

tially from the value V

to 0, while the output of the RC ﬁlter increases exponentially

from 0 to V

. We can also calculate the output signal in the time domain by ﬁnding

the delta response of the circuit w(t)=L

−1

[W(s)] and convoluting it with the input

signal:

(t)=

(t)+

−

(t) w

(t)=

−

(t) (6.19)

1 2 3 4 5

0.2

0.4

0.6

0.8

1 2 3 4 5

0.2

0.4

0.6

0.8

/τ t

/τ

Fig. 6.6 Output of the CR and RC ﬁlter for a voltage step function input

6.1 Linear Signal Processing 189

(t)=





(t −t



) −

−

t−t



(t −t



)





)dt



= V

−

(t)

(t)=





−

t−t



(t −t



)





)dt



= V

(1−e

−

)

(t) . (6.20)

We ﬁnd the same result as before. If we replace s by i

we have the transfer function

in the Fourier (frequency) domain:

ωτ

1+ i

ωτ

√

exp [i arctan1/

ωτ

1+ i

ωτ

√

exp [i arctan

ωτ

(6.21)

The absolute values and the phases of the transfer functions are shown in Fig. 6.7.

At the frequency

= 1/

, the input voltage is attenuated by 1/

√

2 ≈ 0.707 by

both ﬁlters. The CR ﬁlter attenuates frequencies

while high frequencies

are passing the circuit, so the CR circuit is called a high-pass ﬁlter. The RC circuit

passes low frequencies and attenuates high frequencies and is called a low-pass

ﬁlter. At the frequency

, the phase shift between the input and the output voltage

is arctan(1)=

/4 = 45

From Eq. (6.4c,d) we know that the differentiation in the time domain corre-

sponds to a multiplication with s in the Laplace domain and that the integration in

the time domain corresponds to division by s in the Laplace domain. The transfer

function of a ‘differentiator’ or ‘integrator’ is therefore s or 1/s. The RC ﬁlter there-

fore becomes an integrator for large values of

, or more precisely if the frequency

is much lower than all the frequencies contained in the signal. The CR circuit

approaches a differentiator if

is small, i.e., if the frequency 1/

is much higher

than the frequencies contained in the input signal.

To conclude this section we investigate the pole-zero and zero-pole ﬁlters shown

in Fig. 6.8, which are used extensively for signal shaping. The transfer functions of

the two ﬁlters are given by

0.01 0.1 1 10 100

Phase deg

CRRC

0.01 0.1 1 10 100

0.2

0.4

0.6

0.8

CRRC

ωτ

Fig. 6.7 Absolute value and phase of the transfer function for an RC (low-pass) and a CR (high-

pass) ﬁlter. At the frequency

= 1/

the input voltage is attenuated by 1/

√

2 ≈ 0.707 and the

phase shift between the input and the output signal is

/4 = 45

190 6 Electronics for Drift Chambers

v(t)

(t)

Fig. 6.8 Pole-zero (a) and zero-pole (b) ﬁlters used for unipolar shaping of exponential signals

(s)=

s + 1/

= R

+ R

(6.22)

and

(s)=

s + 1/

= R

=(R

+ R

. (6.23)

Equations (6.22) and (6.23) are direct applications of the concepts presented in the

last paragraph of Sect. 6.1.1. The ﬁrst ﬁlter is called pole-zero because the trans-

fer function has a single pole and a single zero, and the pole s = −1/

is smaller

than the zero s = −1/

of the transfer function. For the zero-pole ﬁlter the zero is

smaller than the pole. The transfer functions of both ﬁlters are shown in Fig. 6.9a.

The pole-zero ﬁlter passes low frequencies and attenuates high frequencies by

k = R

/(R

+ R

). The zero-pole ﬁlter attenuates low frequencies by a factor k and

passes high frequencies. At a frequency

= 1/

√

the input voltage is attenu-

ated by

√

k for both ﬁlters, which is the geometric mean of the transfer functions at

zero frequency and inﬁnite frequency.

Applying an exponential signal with time constant

to the input of both ﬁl-

ters with time constants

and

results in an exponential output signal with time

constant

(Fig. 6.9b) [NIC 73]:

V(s)=L [e

−

(t)] =

s + 1/

(s)=W

(s)V (s)=

s + 1/

. (6.24)

0.01 0.1 1 10 100

0.2

0.4

0.6

0.8

Pole Zero Zero Pole

1 2 3 4 5

0.2

0.4

0.6

0.8

Zero Pole

Pole Zero

t / τ

ω √(τ

)

peak

Fig. 6.9 (a) Transfer functions of the pole-zero and zero-pole ﬁlter. (b) Output of both ﬁlters for

an input signal with the form exp(−t/

). The output signal is an exponential with time constant

, so the pole-zero ﬁlter shortens the exponential tail while the zero-pole ﬁlter lengthens the tail

6.1 Linear Signal Processing 191

Therefore the pole-zero ﬁlter is used for shortening exponential signal tails in

electronic circuits, and we also use it later to shorten the tail of wire chamber

signals.

For later use we give the delta response of the pole-zero ﬁlter, which evaluates to

(t)=L

−1

(s)] =

(t) −

−

(t). (6.25)

As mentioned above, we only discuss transfer functions with real poles in this

book. A transfer function of this kind can be represented by a sequence of CR,

RC, pole-zero and zero-pole ﬁlters because the form given in Eq. (6.10) can

be interpreted as a product of suitably chosen transfer functions W

, W

and W

6.1.4 Cascading of Circuit Elements

Let us assume two RC ﬁlters which are connected as shown in Fig. 6.10. An ap-

plication of the combination rules of the four impedances (Sect. 6.1.1) yields the

transfer function

(s)=

V(s)

(k + sRC)(1/k + sRC)

=

V(s)

(1+ sRC)

(6.26)

with k =(3 −

√

5)/2. The transfer function of two cascaded circuit elements is

therefore not simply equal to the product of the two individual transfer functions.

The reason is that the second RC ﬁlter takes current out of the ﬁrst ﬁlter so that the

voltage at its output is no longer V

= 1/(1+ sRC)V(s).

In order to decouple the two circuits we must introduce a so-called voltage buffer

between them, which is an active device of inﬁnite input impedance, inﬁnite band-

width, and voltage gain G. Such a buffer produces an output signal which is an exact

copy of the input signal scaled by G. Owing to the inﬁnite input impedance, no cur-

rent is taken out of the ﬁrst RC circuit and the transfer function becomes the product

of the individual transfer functions. A realization of such a buffer was shown earlier

in Fig. 6.2a.

v(t)

(t)

v(t)

Fig. 6.10 (a) Series connection of two RC ﬁlters. As the second ﬁlter is ‘loading’ the ﬁrst, the

transfer function is different from the product of the individual transfer functions. (b)TwoRC

ﬁlters separated by a voltage buffer of inﬁnite input impedance. For this setup the transfer function

is given by W(s)=W

(s)W

(s)

192 6 Electronics for Drift Chambers

We conclude that a cascade of circuit elements with individual transfer functions

(s),W

(s)...W

(s), which are decoupled by ideal voltage buffers, has a transfer

function equal to the product

W(s)=W

(s) ×W

(s) ×...×W

(s). (6.27)

6.1.5 Ampliﬁer Types, Bandwidth, Sensitivity, and Ballistic Deﬁcit

The practical realizations of linear signal processing systems discussed above are

ampliﬁers and ﬁlter circuits connected to the detector electrodes. Parameters that

characterize these devices, such as gain and bandwidth, have a range of differing

deﬁnitions in the literature on microelectronics. We therefore review and deﬁne the

vocabulary for the discussion of detector electronics.

Bandwidth Limit: In the frequency domain, an ampliﬁer is characterized by the

gain |W(i

)| and the phase shift arg[W (i

)] for each frequency. The bandwidth

limit of an ampliﬁer is deﬁned as the frequency at which the signal transmission has

been reduced by 3 dB from the central or midrange reference value. Since the power

level is deﬁned as 10log(P/P

re f

) dB and the voltage level as 20log(V /V

re f

)dB,

a 3dB reduction corresponds to a power level of ≈ 0.5 and a voltage level equal

to ≈ 0.708 ≈ 1/

√

2 of the value at the centre frequency reference [MOT 93]. The

bandwidth limit of the RC low-pass ﬁlter is thus given by f

= 1/2

πτ

= 1/2

RC.

Rise Time, Peaking Time: The rise time t

of a pulse is deﬁned as the time taken

for its leading edge to rise from 10 to 90% of the peak height. The peaking time t

of a pulse is deﬁned as the time taken for its leading edge to rise from zero to peak

height. When we talk about the peaking time of an ampliﬁer we mean the peaking

time of its delta response.

Voltage, Current, and Charge Ampliﬁers: A voltage ampliﬁer processes a

voltage signal presented at its input and is characterized by high input impedance.

A current ampliﬁer processes the current signal ﬂowing into the ampliﬁer and is

characterized by low input impedance. The signals in wire chambers are induced

current signals, as discussed in the previous chapter. The example of the drift tube in

Sect. 5.6.1 shows that the detector capacitance C

det

together with the input resistance

forms an integration stage with bandwidth limit of f

= 1/2

det

, which is

undesirable if one wants to preserve the fast signal. In order to preserve the chamber

signal shape, the input impedance of the ampliﬁer must be small compared to all

other impedances in the detector or in the ideal case, equal to zero, which means

that we use current ampliﬁers for readout of wire chambers. If the bandwidth of

the current ampliﬁer is such that it integrates a signiﬁcant fraction of the chamber

signal, or the entire chamber signal, it is usually called a charge ampliﬁer.

Sensitivity of Current and Charge Ampliﬁers: The ﬁlters discussed in the

previous sections transform an input voltage signal to an output voltage signal, and

therefore the dimension of the transfer function is [W(s)] = 1. The transfer function

of a current ampliﬁer that transforms a current input signal into a voltage output

signal can be written as V(s)=kW (s)I(s), where k has dimensions of V/A=

and

6.1 Linear Signal Processing 193

W(s) keeps the dimension 1. In the time domain the relationship between the input

and output signals is

v(t)=k



w(t −t



)i(t



)dt



= kw(t

)



w(t −t



)

w(t

)

i(t



)dt



= g



h(t −t



)i(t



)dt



(6.28)

The value w(t

) is the peak of the delta response w(t), and the dimensionless func-

tion h(t) is the normalized delta response. An input current pulse of i(t)=Q

(t)

results in an ampliﬁer output peak voltage of v(t

)=gQ. We call g which has the

dimension V/C, the sensitivity of the ampliﬁer, Typical wire chamber ampliﬁers

have sensitivities in the range of 1–50 mV/fC.

Ballistic Deﬁcit: If a current ampliﬁer has a peaking time t

which is much

longer than the duration of the input current signal i(t), the convolution integral can

be approximated and the output pulse is given by

v(t)=g



h(t −t



)i(t



)dt



≈ gh(t)



i(t



)dt



= gh(t)Q

tot

. (6.29)

The peak of the output signal is gQ

tot

. The output pulse height is therefore propor-

tional to the total signal charge and such an ampliﬁer is called a charge ampliﬁer.

For timing purposes or due to high signal rates, one typically wants to preserve the

signal speed and keep the pulse width to a minimum. Although the ampliﬁers used

for wire chambers will always have peaking times shorter than the maximum drift

time of the ions, an ampliﬁer that integrates a signiﬁcant fraction of the induced

charge is traditionally still called a charge ampliﬁer.

As long peaking times limit the rate capability of a detector one usually has to

live with a compromise between pulse-width and charge measurement accuracy.

The incomplete measurement of the signal charge is characterized by the so-called

ballistic deﬁcit, which is deﬁned as the difference between the ampliﬁer output pulse

height for the input signal i(t) and the output pulse height if the entire input signal

charge Q

tot



i(t)dt is contained in a delta current pulse Q

tot

(t).

Equivalent Block Diagram: In Sect. 6.1.1 we saw that the transfer function

W(s) of a linear electronic circuit is speciﬁed by an expression of the form

W(s)=A

(s −z

)(s −z

)...(s−z

)

(s −p

)(s −p

)...(s−p

)

. (6.30)

Let us now assume that we have deﬁned such a transfer function and we want to

ﬁnd an electronic circuit realization of it. As an example we assume that we want to

transform a voltage signal V

(s) by the transfer function

(s)=W(s)V

(s)=G

s +

(s +

)(s +

)

(s), (6.31)

where

, and

are positive and real. We can interpret W (s) as the product of a

pole-zero ﬁlter and an RC ﬁlter, so a possible realization of this transfer function

can be seen in Fig. 6.11, where a pole-zero and an RC ﬁlter are coupled by an

194 6 Electronics for Drift Chambers

Fig. 6.11 Cascade of an RC

ﬁlter and a pole-zero ﬁlter

v(t)

(t)

ideal voltage buffer of gain G. Such a representation is of course not unique and

there are several ways to realize a transfer function. The important point is that any

transfer function can be represented by a cascade of simple, elementary circuits,

which facilitates the circuit analysis and serves as a basis for the actual design.

Up to now we have investigated ﬁlters that convert voltage input signals into volt-

age output signals. However, the signals induced on chamber electrodes are current

signals, so for our block diagram representation we need an idealized device which

converts a current input signal into a voltage signal. We assume an idealized device

with inﬁnite bandwidth such that for a current input signal i(t) we get a voltage

output signal v(t)=ki(t). With this device we can complete our block diagram rep-

resentation of the detector readout electronics (Fig. 6.12). In principle this kind of

block diagram can also be used for a practical realization, where the voltage buffer

and current to voltage converter are devices with a bandwidth that must be broader

than the bandwidth of the other circuit elements. Although most designs of chamber

electronics consist of cascaded blocks with intermediate buffers, there are numerous

different ways of realizing the individual transfer functions. As outlined earlier, the

goal of this chapter is not the design a practical realization of the readout electronics

but its precise speciﬁcation. The block diagram deﬁned above serves as a basis for

the circuit realization.

6.2 Signal Shaping

The transfer function W (s) of the readout electronics must be chosen in order to

optimize the measurement of the quantity of interest, such as signal charge or signal

time. For this discussion the electronics noise plays a key role, and we have to

characterize noise sources and apply the theory of optimum ﬁltering. Parameters

(t)

i(t)

v(t)

i(t)

Fig. 6.12 An ideal current to voltage converter followed by a CR and pole-zero ﬁlter

6.2 Signal Shaping 195

to be considered for this optimization process are rate capability (pulse width) and

crosstalk as well as the practical aspects for the realization of the electronics.

In this section we ﬁrst discuss two common examples of ampliﬁer transfer func-

tions, a unipolar and bipolar shaper, and we discuss their characteristics as outlined

in the last section. After that we describe various techniques to remove the long

1/(t + t

) tail of wire chamber signals in order to avoid pileup of signals at high

rates.

6.2.1 Unipolar and Bipolar Signal Shaping

Two commonly used transfer functions for signal shaping, a so-called unipolar

shaper and bipolar shaper, are deﬁned by

uni

(s)=kA

(1+ s

)

n+1

bip

(s)=kA

(1+ s

)

n+1

. (6.32)

The equivalent block diagram of the unipolar shaper (Fig. 6.13a) consists of an

ideal current-to-voltage converter with equivalent resistance k followed by n + 1

RC ﬁlters. The ﬁlters are separated by n ideal voltage buffers with voltage gains

...G

, which results in a gain factor of A = G

...G

. The bipolar shaper

uses n RC ﬁlters followed by a single CR ﬁlter, as shown in Fig. 6.13b. The delta

responses w(t) of the two shapers are calculated by performing the inverse Laplace

transforms of Eqs. (6.32), which are tabulated in standard textbooks [OPP 97]. Writ-

ing w(t)=gh(t), with the peak of h(t) normalized to unity, we ﬁnd

uni

(t)=e





−

(t) h

bip

(t)=

√



n −





n−1

−

(t) (6.33)

i(t)

k i(t)

....

v(t)

n+1

....

k i(t)

Fig. 6.13 (a) Unipolar shaping circuit consisting of n + 1 identical RC ﬁlters which are separated

by ideal voltage buffers. (b) Bipolar shaping circuit consisting of n identical RC ﬁlters and a single

CR ﬁlter with the same time constant

196 6 Electronics for Drift Chambers

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

(a) (b)

2 4 6 8

–0.4

–0.2

–0.4

–0.2

0.2

0.4

0.6

0.8

Fig. 6.14 Delta response for an unipolar shaper consisting of ﬁve RC ﬁlters and a bipolar shaper

consisting of four RC ﬁlters and a single CR ﬁlter. (a) The plot gives the delta responses for the

same value of

and shows that the bipolar response is the derivative of the unipolar response.

(b) The plot shows the two delta responses with the same peaking time

with the sensitivity

uni

bip

√

r = n−

√

n. (6.34)

Figure 6.14a shows the two delta responses for n = 4. The unipolar shaper has a

peaking time of t

= n

. Mathematically the bipolar delta response is the derivative

of the unipolar response, because W

bip

(s)=

uni

(s) and a multiplication with s

in the Laplace domain corresponds to a derivative in the time domain [Eq. (6.4c)].

Thus the bipolar delta response crosses the zero line at t = n

. The peak and the

minimum of the bipolar delta response are situated symmetrically around the zero

crossing time at t

(n−

√

n) and t

(n+

√

n). If we want the peaking time t

of the two shapers to be equal we have to use the RC time constant

= t

/n for the

unipolar shaper and

= t

/r for the bipolar shaper.

uni

(t)=





n(1−t/t

)

(t), (6.35)

bip

(t)=

√



n −





n−1

r(1−t/t

)

(t), (6.36)

uni

(s)=

(n + st

)

n+1

bip

(s)=

√

(r + st

)

n+1

. (6.37)

Here we have used the symbols H(s) to represent the Laplace transforms of the

delta responses h(t) normalized to unity at the peak. Figure 6.14b shows the two

delta responses with identical peaking times and n = 4.

The absolute value of the transfer function for the two shapers with a peaking

time t

is given by

uni

)| =

n+1

bip

)| =

n+1

. (6.38)