v.4. pulse shaping cr-rc shaper with unequal time constantsspieler/physics_198_notes/...v.4. pulse...

Introduction to Radiation Detectors and Electronics Copyright 1998 by Helmuth SpielerV.4. Pulse Shaping

V.4. Pulse Shaping

CR-RC shaper with unequal time constants

For simplicity, the preceding discussion assumed equaldifferentiator and integrator time constants.

Furthermore, the detector capacitance was the only capacitancepresent at the input. In reality this is never the case, but additionalcapacitance can easily be accounted for by replacing CD by thetotal input capacitance Ci, i.e. the sum of all capacitances presentat the input node.

1. Noise voltage at shaper output

ωω dGvVk

knino )( 2

0

2,

2 ⋅= ∫ ∑∞

( )∫∞

⋅+++=0

222222 )( ωω dGvvvvV nanrnpnbno

ωτωτωπ

τωd

id

d )1)(1(2 2222

22

++

⋅

⋅

++

++= ∫

∞

0

222

2 4) (1

4

) (

2 naS

iP

P

i

beno vkTR

CR

kTR

C

IqV

ωω

++

+=

id

dbe

Pino Iq

R

kT

CV

τττ 2

22 2

4

4

1

+

++

++i

d

id

df

idi

dnaS AvkTR

ττ

τττ

ττττ

ln )(

)4( 22

22


2. Peak signal voltage at the shaper output

Assume the detector signal is a pulse of constant current withduration tc (i.e. the signal charges ramps up linearly until t= tc ).

At times t < tc

At times t > tc

The peak amplitude of the signal is

Assume that the shaper has no additional voltage gain. Then Vo isequal to the input signal voltage

In the limit that the collection time is negligible, tc<< τd , τi

( )

−

−−−= −−− )(

)(1 )( //

2/ icdi tt

idc

dt

c

doso ee

te

tVtV τττ

ττττ

( )

−

−−−

−= −− icicdcic tt

idc

idtt

idc

doso ee

tee

tVtV ττττ

ττττ

τττ ////

2

)1()(

1)(

)(

( )( ) )/( /

)/( /

1

1 idiic

idddc

t

t

c

doso

e

e

tVV ττττ

τττττ−

−

−

−=

i

sio C

QVV ==

),( idi

sso f

C

QV ττ=


Furthermore, since the peak signal depends only on the ratios oftime constants, i.e. it is independent of the absolute time scale

The noise at the output, however, does depend on the time scale(as it determines the noise bandwidth).

First, to simplify the expression for the output noise, neglect the 1/fterm and combine the terms for

current noise

and voltage noise

and introduce a characteristic time T, which initially will be setequal to the peaking time tp . With these conventions

)/( idsi

sso F

C

QV ττ=

Pben R

kTIqi

422 +=

22 4 naSn vkTRv +=

++

+=

id

dbe

Pino Iq

R

kT

CV

τττ 2

22 2

4

4

1

+

++

++i

d

id

df

idi

dnaS AvkTR

ττ

τττ

ττττ

ln )(

)4( 22

22

id

d

i

pn

id

d

p

d

ino

t

Tv

tTi

CV

τττ

τττττ

++

+=

1

4

1 22n2

2


Using the above results and reintroducing 1/f noise,

and

the equivalent noise charge

can be written as

where

sso

non Q

V

VQ =

+

++

++

=i

d

id

df

id

d

i

pn

id

d

p

d

ino A

t

Tv

tTi

CV

ττ

τττ

τττ

τττττ

ln 1

4

122

222

n22

)/( idsi

sso F

C

QV ττ=

vffivniinn FACFT

vCFTiQ 1

22222 ++=

)/(

1/1

14 idsdip

di Ft

Fττττ

τ+

≡

)/(1

ln)/(1

12

idsi

d

divf F

Fτττ

τττ

+

≡

)/(1

/11

idsdii

pv F

tF

τττττ +≡


The noise indices or “form factors” Fi , Fv and Fvf characterizea type of shaper, for example CR-RC or CR-(RC)4.

They depend only on the ratio of time constants τd /τi, rather thantheir absolute magnitude.

The noise contribution then scales with the characteristic time T.The choice of characteristic time is somewhat arbitrary. so anyconvenient measure for a given shaper can be adopted in derivingthe noise coefficients F.


CR-RC Shapers with Multiple Integrators

a. Start with simple CR-RC shaper and add additional integrators(n= 1 to n= 2, ... n= 8) with the same time constant τ .

With additional integrators the peaking time Tp increases

Tp = nτ

0 5 10 15 20T/tau

0.0

0.1

0.2

0.3

0.4

SH

AP

ER

OU

TP

UT

n=1

n=2

n=4

n=6

n=8


b) Time constants changed to preserve the peaking time(τn= τn=1 /n)

Increasing the number of integrators makes the output pulse moresymmetrical with a faster return to baseline.

⇒ improved rate capability at the same peaking time

Shapers with the equivalent of 8 RC integrators are common.Usually, this is achieved with active filters (i.e. circuitry thatsynthesizes the bandpass with amplifiers and feedback networks).

0 1 2 3 4 5TIME

0.0

0.2

0.4

0.6

0.8

1.0

SH

AP

ER

OU

TP

UT

n=8

n=1

n=2

n=4


Summary

Two basic noise mechanisms: input noise current ininput noise voltage vn

Equivalent Noise Charge:

where Ts Characteristic shaping time (e.g. peaking time)

Fi , Fv "Form Factors" that are determinedby the shape of the pulse.

Ci Total capacitance at the input node(detector capacitance + input capacitance of preamplifier + stray capacitance + … )

Typical values of Fi , Fv

CR-RC shaper Fi = 0.924 Fv = 0.924

CR-(RC)4 shaper Fi = 0.45 Fv = 1.02

CR-(RC)7 shaper Fi = 0.34 Fv = 1.27

CAFE chip Fi = 0.4 Fv = 1.2

Note that Fi < Fv for higher order shapers.Shapers can be optimized to reduce current noise contributionrelative to the voltage noise (mitigate radiation damage!).

Q i T F C vFTn n s i i n

v

s

2 2 2 2= +


The noise analysis of shapers is rather straightforward if thefrequency response is known.

On the other hand, since we are primarily interested in the pulseresponse, shapers are often designed directly in the time domain, soit seems more appropriate to analyze the noise performance in thetime domain also.

Clearly, one can take the time response and Fourier transform it tothe frequency domain, but this approach becomes problematic fortime-variant shapers.

The CR-RC shapers discussed up to now utilize filters whose timeconstants remain constant during the duration of the pulse, i.e. theyare time-invariant.

Many popular types of shapers utilize signal sampling or change thefilter constants during the pulse to improve pulse characteristics, i.e.faster return to baseline or greater insensitivity to variations indetector pulse shape.

These time-variant shapers cannot be analyzed in the mannerdescribed above. Various techniques are available, but someshapers can be analyzed only in the time domain.

Example:

A commonly used time-variant filter is the correlated double-sampler.

This shaper can be analyzed exactly only in the time domain.


Correlated Double Sampling

1. Signals are superimposed on a (slowly) fluctuating baseline

2. To remove baseline fluctuations the baseline is sampled prior tothe arrival of a signal.

3. Next, the signal + baseline is sampled and the previous baselinesample subtracted to obtain the signal


Noise Analysis in the Time Domain

What pulse shapes have a frequency spectrum corresponding totypical noise sources?

1. voltage noise

The frequency spectrum at the input of the detector system is“white”, i.e.

This is the spectrum of a δ impulse:

inifinitesimally narrow,but area= 1.

2. current noise

The spectral density is inversely proportional to frequency, i.e.

This is the spectrum of a step impulse:

const. =df

dA

fdf

dA 1∝


• Input noise can be considered as a sequence of δ and step pulseswhose rate determines the noise level.

• The shape of the primary noise pulses is modified by the pulseshaper:

δ pulses become longer,

step pulses are shortened.

• The noise level at a given measurement time Tm is determined bythe cumulative effect (superposition) of all noise pulses occurringprior to Tm .

• Their individual contributions at t= Tm are described by theshaper’s “weighting function” W(t).

References:

V. Radeka, Nucl. Instr. and Meth. 99 (1972) 525V. Radeka, IEEE Trans. Nucl. Sci. NS-21 (1974) 51F.S. Goulding, Nucl. Instr. and Meth. 100 (1972) 493F.S. Goulding, IEEE Trans. Nucl. Sci. NS-29 (1982) 1125


Consider a single noise pulse occurring in a short time interval dtat a time T prior to the measurement. The amplitude at t= T is

an = W(T)

If, on the average, nn dt noise pulses occur within dt, the fluctuation oftheir cumulative signal level at t= T is proportional to

The magnitude of the baseline fluctuation is

For all noise pulses occurring prior to the measurement

wherenn determines the magnitude of the noise

and

describes the noise characteristics of theshaper – the “noise index”

dtnn

[ ]∫∞

∝0

22 )( dttWnnnσ

[ ] dttWnT nn2

2 )( )( ∝σ

[ ] dttW∫∞

0

2 )(


The Weighting Function

a) current noise Wi (t) is the shaper response to a steppulse, i.e. the “normal” output waveform.

b) voltage noise

(Consider a δ pulse as the superposition oftwo step pulses of opposite polarity andspaced inifinitesimally in time)

Examples: 1. Gaussian 2. Trapezoid

current(“step”)noise

voltage(“delta”)noise

Goal: Minimize overall area

⇒ For a given pulse duration a symmetrical pulse provides the best noise performance.

)(')()( tWtWdt

dtW iv ≡=


Time-Variant Shapers

Example: gated integrator with prefilter

The gated integrator integrates the input signal during a selectabletime interval (the “gate”).

In this example, the integrator is switched on prior to the signal pulseand switched off after a fixed time interval, selected to allow theoutput signal to reach its maximum.

Consider a noise pulse occurring prior to the “on time” of theintegrator.

occurrence of contribution of the noise pulse noise pulse to

integrator output


For W1 = weighting function of the time-invariant prefilter

W2 = weighting function of the time-variant stage

the overall weighting function is obtained by convolution

Weighting function for current (“step”) noise: W(t)

Weighting function for voltage (“delta”) noise: W’(t)

Example

Time-invariant prefilter feeding a gated integrator(from Radeka, IEEE Trans. Nucl. Sci. NS-19 (1972) 412)

∫∞

∞−

−⋅= ' )'()'( )( 12 dtttWtWtW


Comparison between a time-invariant and time-variant shaper(from Goulding, NIM 100 (1972) 397)

Example: trapezoidal shaper Duration= 2 µsFlat top= 0.2 µs

1. Time-Invariant Trapezoid

Current noise

Voltage noise

Minimum for τ1= τ3 (symmetry!) ⇒ Ni2= 0.8, Nv

2= 2.2

3 )1( )]([ 31

20 0

2

3

2

2

1

221 2

1

3

2

τττττ

τ τ

τ

τ

τ

++=

++

== ∫ ∫ ∫ ∫

∞

dtt

dtdtt

dttWN i

31

2

30 0

2

1

22 111

1 )]('[

3

2

1

ττττ

τ

τ

τ

+=+

+

== ∫∫ ∫∞

dtdtdttWNv


Gated Integrator Trapezoidal Shaper

Current Noise

Voltage Noise

⇒ time-variant shaper Ni2= 1.4, Nv

2= 1.1

time-invariant shaper Ni2= 0.8, Nv

2= 2.2

time-variant trapezoid has more current noise, less voltage noise

∫ ∫ −=+

=

−T

I

TT

Ti

TTdtdt

T

tN

I

0

22

2

3)1( 2

∫ =

=

T

v Tdt

TN

0

22 21

2


Interpretation of Results

Example: gated integrator

Current Noise

Increases with T1 and TG ( i.e. width of W(t) )

( more noise pulses accumulate within width of W(t) )

Voltage Noise

Increases with the magnitude of the derivative of W(t)

( steep slopes → large bandwidth determined by prefilter )

Width of flat top irrelevant(δ response of prefilter is bipolar: net= 0)

∫∝ dttWQnv22 )]('[

∫∝ dttWQni22 )]([


Quantitative Assessment of Noise in the Time Domain

(see Radeka, IEEE Trans. Nucl. Sci. NS-21 (1974) 51 )

↑ ↑current noise voltage noise

Qn= equivalent noise charge [C]

in= input current noise spectral density [A/√Hz]

vn= input voltage noise spectral density [V/√Hz]

Ci= total capacitance at input

W(t) normalized to unit input step response

or rewritten in terms of a characteristic time t → T/ t

∫∫∞

∞−

∞

∞−

+= dttWvCdttWiQ ninn222222 )]('[

21

)]([ 21

∫∫∞

∞−

∞

∞−

+= dttWT

vCdttWTiQ ninn222222 )]('[

1

21

)]([ 21


Correlated Double Sampling

1. Signals are superimposed on a (slowly) fluctuating baseline

2. To remove baseline fluctuations the baseline is sampled prior tothe arrival of a signal.

3. Next, the signal + baseline is sampled and the previous baselinesample subtracted to obtain the signal


1. Current Noise

Current (shot) noise contribution:

Weighting function (T= time between samples):

Current noise coefficient

so that the equivalent noise charge

∫∞

∞−

= dttWiQ nni222 )]([

21

τ

τ

/)(

/

)( :

1)( : 0

0)( : 0

Tt

t

etWTt

etWTt

tWt

−−

−

=>

−=≤≤

=<

∫∞

∞−

= dttWFi2)]([

( ) ∫∫∞

−−− +−=T

TtT

ti dtedteF ττ /)(2

0

2/1

2

22/2/ τττ ττ +

−+= −− TT

i eeTF

( )

+−+= −− 1

2

2

1 /2/22 τττ TT

nni eeTiQ

+−+= −− 1

2

4

1 /2/22 ττ

ττ TT

nni eeT

iQ


Reality Check 1:

Assume that the current noise is pure shot noise

so that

Consider the limit Sampling Interval >> Rise Time, T >> τ :

or expressed in electrons

where Ni is the number of electrons “counted” during the samplinginterval T.

Iqi en 22 =

TIqQ eni ⋅≈2

ee

eni q

TI

q

TIqQ

⋅=⋅≈ 22

ini NQ ≈

+−+= −− 1

2

2

1 /2/2 ττ

ττ TT

eni eeT

IqQ


2. Voltage Noise

Voltage Noise Contribution

Voltage Noise Coefficient

so that the equivalent noise charge

∫∞

∞−

= dttWvCQ ninv2222 )]('[

21

∫∞

∞−

= dttWFv2)]('[

∫∫∞

−−−

+

=

T

TtT

tv dtedteF

2/)(2

0

2/

1

1 ττ

ττ

( )ττ

τ

21

1 21 /2 +−= − T

v eF

( )τ

τ/2222 2

41

1

Tninv evCQ −−=

( )τ

τ/22

21 T

v eF −−=


Reality Check 2:

In the limit T >> τ :

Compare this with the noise on an RC low-pass filter alone (i.e. thevoltage noise at the output of the pre-filter):

(see the discussion on noise bandwidth)

so that

If the sample time is sufficiently large, the noise samples taken at thetwo sample times are uncorrelated, so the two samples simply add inquadrature.

τ21222 ⋅⋅= ninv vCQ

τ41

)( 222 ⋅⋅= nin vCRCQ

2 )(

sample) double( =RCQ

Q

n

n


3. Signal Response

The preceding calculations are only valid for a signal response ofunity, which is valid at T >> τ.

For sampling times T of order τ or smaller one must correct for thereduction in signal amplitude at the output of the prefilter

so that the equivalent noise charge due to the current noise becomes

The voltage noise contribution is

and the total equivalent noise charge

τ/1 / Tis eVV −−=

2/

/2222

)1( 4

2

1 τ

τ

τ T

T

ninve

evCQ −

−

−−=

22nvnin QQQ +=

( )2 /

/2/

22

1 4

12

τ

ττ

ττT

TT

nnie

eeT

iQ−

−−

−

+−+=


Optimization

1. Noise current negligible

Parameters: T= 100 nsCd= 10 pFvn= 2.5 nV/√Hz

→ in= 6 fA/√Hz (Ib= 0.1 nA)

Noise attains shallow minimum for τ = T .

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3

tau/T

Eq

uiv

alen

t N

ois

e C

har

ge

Qni [el]Qnv [el]Qn [el]


2. Significant current noise contribution


→ in= 0.6 pA/√Hz (Ib= 1 µA)

Noise attains minimum for τ = 0.3 T .

0

1000

2000

3000

4000

5000

0 0.5 1 1.5 2 2.5 3

tau/T

Eq

uiv

alen

t N

ois

e C

har

ge




→ in= 0.2 pA/√Hz (Ib= 100 nA)

Noise attains minimum for τ = 0.5 T .

0

500

1000

1500

2000

0 0.5 1 1.5 2 2.5 3

tau/T

Eq

uiv

alen

t N

ois

e C

har

ge



3. Form Factors Fi, Fv and Signal Gain G vs. τ / T

Note: In this plot the form factors Fi, Fv are not yet corrected bythe gain G.

The voltage noise coefficient is practically independent of τ / T .

Voltage contribution to noise charge dominated by Ci /τ .

The current noise coefficient increases rapidly at small τ / T .

At large τ / T the contribution to the noise charge increases

because of τ dependence.

The gain dependence increases the equivalent noise charge withincreasing τ / T (as the gain is in the denominator).

0

5

10

0 0.5 1 1.5 2 2.5 3

tau/T

Fi

FvG


Summary

Two basic noise mechanisms: input noise current ininput noise voltage vn

Equivalent Noise Charge:

↑ ↑ ↑ ↑ front shaper front shaper end end

where Ts Characteristic shaping time (e.g. peaking time)

Fi, Fv "Form Factors" that are determinedby the shape of the pulse.

They can be calculated in the frequency or time domain.

Ci Total capacitance at the input node(detector capacitance + input capacitance of preamplifier + stray capacitance + … )

• Current noise contribution increases with T

• Voltage noise contribution decreases with increasing T

Only for “white” voltage noise sources + capacitive load

“1/f ” voltage noise contribution constant in T

s

vniisnn T

FvCFTiQ 2222 +=


1. Equivalent Noise Charge vs. Pulse Width

Current Noise vs. T

Voltage Noise vs. T

Total Equivalent Noise Charge


2. Equivalent Noise Charge vs. Detector Capacitance (Ci = Cd + Ca)

If current noise in2FiT is negligible

↑ ↑ input shaper stage

Zero intercept

1

)( 222

TFvCCTFiQ vnadinn ++=

1

)(

12

222

2

TFvCCTFi

TFvC

dC

dQ

vnadin

vnd

d

n

++=

2 T

Fv

dC

dQ vn

d

n ⋅≈

/ 0

TFvCQ vnaCnd

==


Noise slope is a convenient measure to compare preamplifiers andpredict noise over a range of capacitance.

Caution: both noise slope and zero intercept depend onboth the preamplifier and the shaper

Same preamplifier, but different shapers:

Caution: Current noise contribution may be negligible at highdetector capacitance, but not for Cd=0.

/ 2220

TFvCTFiQ vnainCnd

+==

v.4. pulse shaping cr-rc shaper with unequal time constantsspieler/physics_198_notes/...v.4. pulse...

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