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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,
(d.) ZA
--," ~
-;,,...... ,.. .(,..."
--)
ZA
R
(el)
!.OiIJ:--------===;::::e====-------~., Vi
(d.)o.g
t 0·1 . ~ ;:=. 0.'2.1 'Fl 'l1S~-!V c.,
D" 'Lo c- /0 ')'JS~C ~Ir
fJ,3
0.2
0·1 o J
o ~ 4- , ~ 10 ,2. '4- /, !g"2.0 _2..'2. ~~ 2.' "2.8 ~o
i >- '>1S~c
=-;:::::"-==F-"'- - -
rtf, ~3 "Y}S'€C(6) -:D ::: 0::''3 Uf "7se £.-1..
o·f
FiClJo "3 (CLl b)
~
i·O - - ---
! o·tt 6.!
t)·1
0.' C!t.5'
D.lI
(c) -t: ~~'V]s~c.
1) z: O·Lf3'''I-YJ reC.i
7o~".1 'J1 se c.
03
0.2
0.1. o
0 1 Ji e 8 10 1'2. t~
14 '6 YlSec.
Ig 2.0 22. Zq. ~'2 g 30'
O·Cf
o·t 0.1
TO.{,
V 0.; 0.4 0., O.'L
O.!
7;~ lD Y)r--€!c..
(l) 1:> z: 0.21 'It ">ec.-I
O~..,.....-r----r--,r--~-,.---r---.----.-~-.-----r---.-----....--.........-
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ill'
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)< ",sec.
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