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UPR - 0071E

University of Pennsylvania

June 1979

PULSE SHAPING PASSIVE FILTER DESIGN.

Manolis A. Dris

University of Pennsylvania

Philadelphia Pennsylvania 19104

Calculations are given for the construction of a two port

network to slow down fast pulses.

Work supported by the United States Department of Energy.

FERMILAB-PUB-79-106-E

I. -2­

Introduction.

Sometimes there is the need for networks that take pulses

of small rise time and produce output pulses of larger rise time.

The rise time should be the same, within a few percent,. independent " " t" (1)of the lnput pulse rlse lme.

Calculations are given for a passive filter of the following

properties, a) its input impedance is resistive,and independent

of frequency, b) a step pUlse of the form, •

) i~o J

tLoJ ) 'i

in its input produces a pulse at its output of the form,

'·1

In what follows all time functions will be considered to be zero

for (time) t <o.

Analysis.

The Laplace transform of the above input step function is given by~4)

The Laplace transform of the output voltage is,

.S+p

A two port network to have this response

have a transfer function,

T(5):::' S-o(~) ::.(!!::- _~)f5.)[c"CS) s s+~J!l-s ; Dr

to the above

b

input, should

(2)

~I

-) ­

A) Symmetric network. Equal input and output impedances.

For a (Fiq .-1 t;f, ) above IIS.J it is easylattice network . Wloth the -r-(.\

(~),) to proove that,

()T(s)-= _R-~t\~.. R +'-A(t;)

~i (s) and --_.. == 'R if

J:i (S)

ZA (5) • 2a(~) -:::. 1<2.. (4) t

R is ohmic resistance.

The network then if connected to a load R has an input impedance R

independent of frequency.\ It is obvious that its output impedance

is the same, R, if the circuit is fed by a source of impedanceR.

From eqs (2) and () we get,

bR.--ZA($)

, R-+ -z.." (~)

From this one gets,

1 ·L 1. "'%6- + en

ZA(S) - - [Ji. ) ·5 ~.

;2.1",

•

+-R

-4­

B) Asymmetric netwokk~

For the asymmetric network of Fig. (-:a,)

2a it is easy to prove that

if,

• then R

r{s)~--­

R+2Brr)

and

T (S) : lv_If again the transfer function is the same, b R s-+b

then, - '= _ which gives, Sfh fZ + ~-etS)

(8)

Synthesis.

A) Asymmetric case.

'I'he synthesis of the network is now trivial with the use

of eqs (5) and (6).

Equation (5) suggests that ~~ is a parallel connected inductance

L-==­

and Ohmic resistance R. see Fig. lb.

Equation (6) suggests that ~~is a seriqlly connected ohmic

resistance R and a capacitance.

C­- .... i: W¢

(iO ).

See Fig.1 b.

-5­

R, obviously, is the terminating resistance (and the network~s

r.haracteristic impedance).

One may need to put a 1:1 pUlse transformer at the input of this

circuit to avoid braid effects from the grounding of the coaxial

cable, if such a cable is used to feed the filter.

B) Asymmetric case.

For the asymmetric case the synthesis is again easy from eqs (7)

and (8). Eq. (7) suggests an inductance,

L~ (11)

see Fig. 2b, and eq. (8) a capacitance,

1 -

and an ohmit resistance R in series, see fig. 2b.

More general case input voltage.

Let us consider now an input of the more general form,

(2.) Its Laplace transform is,

i _ ) S~j)

For the lr($) found above, one gets,

;}D) ·

-6­~.

This can be written as,

+

The inverse Laplace transform of the last expression gives,

It is easy to find from eq. 1~ what happens when D~OO (step

function result), or when D~ 0 (zero input).

When D...... b. by using de L' Hosp i taL" s rule. one finds.

Numeric~l estimates.

Let the rise time of the transmitted pUlse (for the step pulse case)

be tt.=10 nsec (defined as the time from 10% to 90% voltage).

Then b =:. .ev,,/ nsec-i= 0.2197 nsec"1 •'0 A) Symmetric case,

From eqs (9) and (10) we find. if R=5052,

L=O.1138 rHand C=45.51 pF.

B) Asymmetric case.

From eqs (11) and (12) we find, L=0.2276 r» and C=91.0J pF.

Fig. J shows the input and output voltages for various inputs

(circuits of Figs 1b and 2b). Fig. 4 is a plot of output rise time 'To

versus the corresponding input rise time '"C, •. Lo was calculated

graphically.

·. .. .~...

.' -7­

References.

1) W. Selove, private communication. W. Selove suggested the use

of a pulse shaping filter for scintillator calorimeter pulses.

2) Charles M. Close, The Analysis of Linear Circuits, pages ]24-]25,

Harcourt, Brace & World, pUblishers, Chicago-San Francisco-

Atlanta (1966).

3) Anatol I. Zverev, Handbook of Filter Synthesis, page 6,·-John W"iley &

Sons pUblishers, NY-London-Sydney (1967) •

. r­

-8­ ... Figure captions.

Fig. 1. a) A lattice network.

b) A lattice filter with characteristics as described

in the text.

Fig.2. a) An asymmetric network.

b) An asymmetric filter with characteristics as described

. in the text.

Fig. 3. a) to d). Input (Vi(t)) and output (Yo (t)) voltages

of the filters of Figs ib and 2b. The general input ..1)·t .

form is, 1- e • The general output form is given by eq. 14.

Fig, 4. Output rise time (defined as the time. between 10% to

90% voltage) versus input rise time, for the filters

of Figs lb and 2b as calculated in the text. The output

rise time was calculated graphically,

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--," ~

-;,,­...... ,.. .(,..."

--)

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(el)

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