Introduction of RF Communication
Instructional guidelines for laboratory works
Laboratory work 1
RC Filters
Brief information from the theory
Observe the following circuit
R
C Vin
Vout
Fig. 1.1 RC filter
From fig. 1.1 it is obvious that
C
C
in out
X R
X
V V
+
1.1!
where
C j
X
C
1."!
Is the reactive resistance of the ca#acitor. $he am#lification coefficient of the circuit will be
C
C
in
out
X R
X
V
V
K
+
1.%!
&utting 1."! in 1.%! the following will be obtained'
RC j
K
+
1
1
1.(!
2 2 2
1
1
C R
K
+
1.)!
*sing 1.)!+ observe the following cases'
a! 0 + 1 K
b!
RC
1
+ 7 . 0 K
c!
+ 0 K
$he de#endence of ,m#lification coefficient-s module on the fre.uency is called am#litude/
fre.uency characteristic ,FC!.
$he ,FC for fig. 1.1 has the following view' fig. 1.".!.
Fig. 1." ,FC of low fre.uency RC filter
From fig. 1." it is seen that this filter am#lifies low fre.uencies+ and re0ects high fre.uencies+
that is why this filter is called low fre.uency filter.
$he range of fre.uencies for the filter when 7 . 0 > K is called band#ass+ 7 . 0 1 . 0 K
transition and 1 . 0 < K bandre0ect layer.
In ideal case
transition
0
$he fre.uency corres#onding to 7 . 0 K is called cutoff fre.uency.
Let-s observe another circuit of the filter.
Vin
Vout
R
C
Fig. 1.%. RC filter
From fig. 1.% it is obvious that
C
in out
X R
R
V V
+
1.1!
C j
R
R
K
1
+
1.2!
2 2
2
1
C
R
R
K
1.3!
*sing 1.3!+ observe the following cases'
a! 0 + 0 K
b!
RC
1
+ 7 . 0 K
c! + 1 K
From the above mentioned im#ressions it is clear that this filter am#lifies low fre.uencies+ and
re0ects high fre.uencies+ that is why this filter is called low fre.uency filter.
$he study of RC filter
1. Ob0ective
1.1. Introduction to o#erating #rinci#le of RC filters
1.". 4imulation of RC filters by 54&IC6 and obtaining its ,FC.
". 7escri#tion of testing circuits
".1. $he following circuits are used for the study of RC filters.
a! Low fre.uency filter b! 5igh fre.uency filter
Fig. 1.( RC filters
". 5omework
".1. &resent the circuits of RC filters
".". 7efine the cutoff fre.uency of RC filter if 10 R kOhm+ 1 C nF.
".%. 7efine RC
a! low fre.uency+
b!
transition
layer of high fre.uency filters use the obtained results in #oint "!.
%. $esting #art
%.1. Collect the circuit shown in fig. 1.( a by the following #arameters'
a! 10 R kOhm
b! 1 C nF
c! In#ut voltage ,C81
%.". 9et the ,FC.
Below is the 54&IC6 netlist:dec of the given circuit and its simulation results. fig. 1.)!.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
cc<ca#acitor1 out gnd c 8 1n cty#e 8 =
rc<resistor1 in out R 8 1=k
vc<vgeneric1 in gnd dc 8 = ac 8 1
.global gnd
.ac dec 1== 1k 1meg
.o#tions #ost #robe
.#robe >;!
.end
Fig. 1.).,FC of low fre.uency filter
%.%. Collect the circuit shown in fig.( b by the following #arameters'
a! 10 R kOhm
b! 1 C nF
c! In#ut voltage ,C81
%.(. 9et the ,FC.
Below is the 54&IC6 netlist:dec of the given circuit and its simulation results. fig. 1.1!.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
vc<vgeneric1 in gnd dc 8 = ac 8 1
cc<ca#acitor1 in out c 8 1n cty#e 8 =
rc<resistor1 gnd out R 8 1=k
.global gnd
.ac dec 1== 1k %==?
.o#tions #ost #robe
.#robe >;!
.end
Fig. 1.1. ,FC of high fre.uency filter
(. Re#ort
$he re#ort must include'
(.1. 4imulation circuits of RC filters
(.". Basic calculation formulas
(.%. Results of homework calculation
(.(. Oscillograms obtained during testing
(.). 6stimation of results and conclusions.
). Checking .uestions
).1. @hat is called transition layer of RC filterA
).". @hat is called cutting fre.uency of layer of RC filterA
Resonance ,m#lification and Fre.uency Bulti#lication
1. Brief theoretical introduction
4ingle stage resonance am#lifier-s circuit is de#icted in Fig. ".1.
M2
M1
R C
L
VDD
VSS
u(t)
Fig. ".1. 4ingle stage resonance am#lifier
Consider the case when the in#ut voltage is a##lied on nonlinear element of the am#lifier
( ) t cos U U t u U ) t ( u
m 0 0 0 1
+ + + ".1!
@here+
0
U
is the initial bias voltage which decides the location of the functional #oint on
transistor IC> characteristics+
m
U is the am#litude of t U ) t ( u cos
m 0
in#ut voltage+
0
is the in#ut signal-s fre.uency.
IC> characteristics of the active element+ which is a##roDimated by two liner e.uations is
introduced by the following system'
'
<
1 1
1
); (
; 0
U u U u S
U u
i
when
when
"."!
where *
1
is the threshold voltage of the nonlinear element+ 4 is the slo#e of the
characteristics.
Fig. ".". ,##roDimated IC> characteristics of the transistor.
Eonlinear resonance am#lifier-s functional state is eD#lained by a #lot de#icted in Fig. ".%.
where out#ut current-s view is obtained by using #ro0ection method.
*
=
IE
O*$ *
"
t!
0 U
1
u
i
7uring t
1
F t
"
+ u
1
G *
1
+ the transistor is OFF and there is no current flow. 7uring t
"
F t
%
+ u
1
> *
1
the transistor is OE and the drain conducts discrete im#ulse current.
$he current-s view is #resented by dote line. If the IC> characteristics were always linear
during gate voltage-s switching #eriod the transistor would conduct sinusoidal current with I
am#litude.
$he #ro#ortion of a #eriod when a drain current declines from its maDimal value to = is
called drain current-s breaking angle and is re#resented by . Consecutively+ Ht
%
FHt
"
= "
. From Fig. ".% it is obvious that breaking angle is calculated by the following
e.uations'
I
I I
cos
m
or
m
U
U U
cos
0 1
".%!
where I
m
is the maDimal value of current-s im#ulse+ I is the im#ulse-s value which is due while
2 / .
Fig. ".%. &lot illuminating nonlinear functional mode of the resonance am#lifier
7rain-s current is defined by the following e.uation'
) I I ( t cos I ) t ( i
m
0
".(!
Introducing I/I
m
value in ".%! will result
) cos (cos ) cos (cos cos cos ) (
0 0 0
t SU t I I t I t i
m
".)!
Because+
cos / ) I I ( I
m
".1!
It will result
) cos 1 /( I I
m
".2!
and
cos 1
cos cos
) (
0
t
I t i
m
".3!
,s the current flowing through the drain is #eriodic+ i
d
t! can be transformed to Fourier series+
i.e.
0
t
0
t
i i
0
t
3
t
1
I
m
I
2
2
U
m
0 U
0
U
1
u
t
2
u
1
+ + + +
0
0 0 0 2 0 1 0
, cos cos ... 2 cos cos ) (
n
n n
t n I t n I t I t I I t i
".I!
where the current am#litudes I
n
are found from Fourier formula
t d ) t ( i
0 0
2
1
+ ".1=!
t td n cos ) t ( i
n 0 0
1
' ".11!
4ubstituting formula for
) t ( i
from ".3! after integration this is obtained'
) ( I
) cos 1 )( 1 n ( n
) sin n cos n cos (sin 2
I I
); ( I
) cos 1 (
cos sin
I I
); ( I
) cos 1 (
cos sin
I I
n m
2
m n
1 m m 1
0 m m 0
".1"!
In these eD#ressions
) ( ,..., ) ( ), (
n 1 0
are coefficients of Fourier transformation of
cosine im#ulse Berg coefficients!+ which only de#end on . For counting it is reasonable to
eD#ress I
m
and with #arameters of in#ut signal
m m
m
U
U U
SU
) U U ( S
I
I I
cos
0 1 0 1
+ ".1%!
( ) [ ]
0 1
1 1 U U U S ) cos ( SU ) cos ( I I
m m m
' ".1(!
In Fig.( gra#hs of
) ( ),..., ( ), (
5 1 0
coefficientsJ de#endence on cross angle is
de#icted+ when varies in the range of
0 0
180 0 . *sing these gra#hs am#litudes of current
harmonics are found.
) ( I I
n m n
' ".1)!
) (
n
coefficients reach their maDimal value when cross angle is e.ual to o#timal value
0
180
opt
Cn' @hen 8 = current is e.ual = Eon/linear device is turned off during all #eriod!.
@hen 813=
=
the current crossing is absent and o#eration regime becomes linear.
= (= 3= "= 1"= 11=
1= 1== 1(=
=.)
=.(
=.%
=."
=.1
=
Fig. ".(. 9ra#hs of ) (
n
functions
4#ectrum of #ulsing current for some values of angle is de#icted in fig. ".).
I
m
<I I
m=
I I
m
>I
Fig. ".). 4#ectrum of #ulsing current for some values of angle
In crossing mode current
) t ( i
in the out#ut circuit of am#lifier is #ulse sha#ed and besides dc
com#onent and useful first harmonic+ contains also higher order harmonics ".I!+ which must be
attenuated filtered!. It is im#lemented by #arallel tank fig. ".1!+ which is tuned for the in#ut
fre.uency of H
=.
In case of resonance e.uivalent resistance of tank between nodes 1/1J is very
high and a##ears as a load resistance for am#lifier. But in res#ect to high harmonics of current
) t ( i
tank+ which has enough high .uality+ can be considered as a shorted circuit. ,s a result
harmonic voltage a##ears in tank as well as in linear am#lifier. 4uch an am#lifier is called
resonance. 7rain current which is eD#ressed by formula ".I!+ flowing through #arallel tank
connected in drain circuit+ leads to the following voltage in the out#ut
1
0 0
cos ) ( ) (
n
n
t n n I t u
2
+ ".1(!
2
2
0
0
0
Q
n
n
1
R
) n (
,
_
+ ".1)!
where
2 0 n n
U I ) n ( / am#litude of n Fth harmonic of out#ut voltage+ ) n (
0
/ resistance
of #arallel tank for fre.uency
0
n + LC /
R
1 / resonance fre.uency of tank+
Q
/ .uality
factor+
C / L
R R
1
/ characteristic resistance of tank+
R
K / resistance of tank during
resonance.
By tuning tank for different harmonics of drain current one can get different o#eration modes of
nonlinear resonance am#lifier. For eDam#le+ when
0
R
+
t cos R I t cos ) ( I ) t ( U
0 1 0 0 1 2 1
+ ".11!
where
R ) (
0
/ resistance of tank in case of resonance.
Bore fre.uently one takes
, 2 /
5 , 0 ) (
1
in case of which
2
12
/ R SU U
m
.
9ain in case of resonance
2
2
/ SR U / U !
m
+ ".12!
i.e. gain in nonlinear mode is twice smaller than in linear resonance am#lification+ but here the
#erformance indeD is much higher.
9ain of nonlinear resonance am#lifier is relative to the first harmonic
m m
U / R I U / U !
1 2 1 1
+ ".13!
I
n
/I
m
4
=120
0
0,2
0,4
=60
0
=90
0
0 2 4 n 0 2 n 0 2 4 n
where
( ) / cos sin SU I
m 1
' ".1I!
7ividing this e.uation by
m
U the #arameter is obtained
( ) / cos sin S U / I S
m
1 1
+ "."=!
which can be considered as an average slo#e of characteristic for the first harmonic. In contrast
to the linear mode+
1
S
de#ends on and therefore de#ends on in#ut voltageJs
m
U
am#litude.
7rain current
1
I first harmonic or out#ut voltage
2 1
U first harmonic de#endence on am#litude
of in#ut voltage
m
U ' I
1
*
m
!+ *
1"
*
m
! is called oscillating characteristics.
7e#endence of
1
! gain on am#litude of in#ut voltage
m
U '
) U ( !
m 1
is called am#lifier
am#litude characteristic.
7uring fre.uency multi#lication in the out#ut of the circuit+ oscillation is obtained the fre.uency
of which is n times higher than the fre.uency
0
of in#ut signal n Fis integer!+ i.e. am#lifier
which works in nonlinear mode can be used as a fre.uency multi#lier. For that it is only needed
to tune the tank of resonance am#lifier to fre.uency
0
n
of the n/th harmonic.
In order to have maDimal am#litude
n
I of the desired harmonic for the given value
m
I of
out#ut current im#ulse+ it is needed to choose cross angle so that the corres#onding
coefficient ) (
1
is maDimal by
n
opt
/ 180
0
formula!.
,s maDimal values of coefficients
n
decrease ra#idly when increasing n+ such multi#liers are
used only for multi#lying by "/%.
". 7escri#tion of testing circuit
Before circuit design it is needed to se#arately design contour of studied circuit with elements-
#lacement netlist+ deck!.
$o study the o#eration modes of nonlinear resonance am#lifier the circuits de#icted in fig. ".1
are used which are simulated with 54&IC6 tool.
In the circuit se#arate offset voltage source *
=
is missing+ which is given as dc com#onent of
sinewave voltage source offset!.
a! @ith the out#ut active resistance
b! @ith the oscillating tank
Fig. ".1. 4imulation circuits of a nonlinear resonance am#lifier
Laboratory work "
Resonance ,m#lifier
1. Ob0ective
1.1. 4imulation of resonance am#lifier by 54&IC6 and study of its fre.uency and gain
characteristics.
1.". Com#arison of calculation and simulation results for am#lifierJs oscillating characteristics.
". 5omework
".1. ,##roDimate IC> characteristics of a transistor by linear #arts and find value of turn on
voltage *
1
.
Fig. ".2. IC> characteristics of a transistor
".". Find 4 slo#e of IC> characteristics of a transistor from a##roDimate IC> characteristics.
%. $esting #art
%.1. 7esign circuit shown in fig. ".1.a+ with the following #arameters of signal source
a! 4ha#e of in#ut signal' sinewave
b! am#litude of in#ut signal 6ffective value! 1== m>
c! Fre.uency of in#ut signal 1==?5L
d! 7C com#onent of in#ut signal offset! =.=I>
e! 7C su##ly voltage 1>
%.". 4tudy the o#eration of nonlinear resonance am#lifier.
%.".1. 9et oscillograms of in#ut and out#ut signals.
$he following is the 54&IC6 netlist:deck of the circuit and its simulation results Fig. ".3!'
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
cc<ca#acitor1 out1 out c 8 ".2n
rc<resistor1 out gnd R8)k
>
g
I
d
mnmos(" vdd vdd out1 gnd E l 8 =.1u w 8 %u m 8 1
mnmos(1 out1 in gnd gnd E l 8 =.(u w 8 1==u m 8 1
vc<v<sin1 in gnd dc 8 = 4IE =." =.=) fre. !
vc<vgeneric1 vdd gnd dc 8 1
.global gnd
.model n nmos
.#aram fre.81==k
.tran J1C1==;fre.!J J)=;1Cfre.J
.o#tion #ost #robe
.#robe >;!
.end
Fig. ".3. 4imulation results
%.".". Change offset voltage
0
U
and get nonlinear am#lification mode for various values of
cross angle
2 <
+
2
+
2 >
+ ' 4ee oscillograms of out#ut signal in
those modes.
$he following is the 54&IC6 netlist:deck of the circuit and its simulation results fig. ".I!'
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
cc<ca#acitor1 out1 out c 8 ".2n
rc<resistor1 out gnd R8)k
mnmos(" vdd vdd out1 gnd E l 8 =.1u w 8 %u m 8 1
mnmos(1 out1 in gnd gnd E l 8 =.(u w 8 1==u m 8 1
vc<v<sin1 in gnd dc 8 = 4IE offset =.=) fre. !
vc<vgeneric1 vdd gnd dc 8 1
.global gnd
.model n nmos
.#aram fre.81==k
.tran J1C1==;fre.!J J)=;1Cfre.J swee# offset =.1) =.% =.=)
.o#tion #ost #robe
.#robe >;!
.end
Fig. ".I. 4imulation results
".%. 7esign circuit shown in fig.1.b+ with the following #arameters of signal source'
a! 4ha#e of in#ut signal' sinewave
b! ,m#litude of in#ut signal 6ffective value! 1== m>
c! Fre.uency of in#ut signal 1==?5L
d! 7C com#onent of in#ut signal offset! =.=I>
e! 7C su##ly voltage 1>
".%.1. Change the fre.uency of in#ut signal for the given values of
m
U and
m
U U U +
1 0
this
case corres#onds to linear regime of am#lification when 813=
=
! and find resonance
fre.uency of tank by getting it from the maDimum of out#ut signalJs oscillogram.
9et ,FC of am#lifier in linear am#lification mode.
$he following is the 54&IC6 netlist:deck of the circuit and its simulation results fig. ".1=!'
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
cc<ca#acitor1 out gnd c 8 ()=# cty#e 8 =
cc<ca#acitor" out1 out c 8 1)=# cty#e 8 =
vc<vgeneric1 vdd gnd dc 8 1
vc<v<sin1 in gnd 4IE =.=I =.1 fre. !
mnmos(1 out1 in gnd gnd E l 8 =.1u w 8 1==u m 8 1
mnmos(" vdd vdd out1 gnd E l 8 1u w 8 1u m 8 1
lc<inductor1 out gnd l 8 1))u
.global gnd
.model n nmos
.#aram fre.81==k
.tran J1C1==;fre.!J J)=;1Cfre.J swee# fre. (==k 2==k 1=k
.o#tion #ost #robe
.#robe >;!+ i;!
.end
Fig. ".1=. 4imulation results
".%.". 7efine resonance fre.uency and am#lifier gain at that fre.uency with the hel# of ,FC.
$he following is the 54&IC6 netlist:deck of the circuit and its simulation results fig. ".11!'
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
cc<ca#acitor1 out gnd c 8 ()=# cty#e 8 =
cc<ca#acitor" out1 out c 8 1)=# cty#e 8 =
vc<vgeneric1 vdd gnd dc 8 1
vc<v<sin1 in gnd 4IE =.=I =.1 fre. !
mnmos(1 out1 in gnd gnd E l 8 =.1u w 8 1==u m 8 1
mnmos(" vdd vdd out1 gnd E l 8 1u w 8 1u m 8 1
lc<inductor1 out gnd l 8 1))u
.global gnd
.model n nmos
.#aram fre.81==k
.tran J1C1==;fre.!J J)=;1Cfre.J
.o#tion #ost #robe
.#robe >;!
.end
Fig. ".11. 4imulation results
%. Re#ort
$he re#ort must include'
1. 4imulation circuit of resonance am#lifier
". Basic calculation formulas
%. Results of homework calculation
(. &lots of characteristics obtained from calculations and testing.
). Oscillograms obtained during testing
1. 6stimation of results and conclusions.
(. $esting .uestions
(.1. 6D#lain the o#eration of nonlinear resonance am#lifier.
(.". @hat is the de#endence of drain current on cross angle and am#litude of in#ut signalA
(.%. 7raw transient diagrams of drain current in nonlinear am#lification mode for different
values of cross angle.
Laboratory work E %
Fre.uency Bulti#lication
1. Ob0ective.
1.1. 4tudy and simulation of fre.uency multi#lier based on resonance am#lifier by 54&IC6.
". 5omework
".1. Calculate e.uivalent resistance of tank in case of resonance for the given f
R
+ C+ K!
#arameters use results of laboratory work "!.
".". Calculate am#litudes of in#ut voltage
m
U + which corres#onds to the following values of
cross angle and offset voltage a! 813=
=
+
0
U
8 =.3)> + b! 81"=
=
+
0
U
8=.3)>+ c!
8 1=
=
+
0
U
8=.1)>+ using results of laboratory work ".
%. $esting #art
%.1. 7esign circuit shown in fig. ".1.a+ with the following #arameters of signal source
a! 4ha#e of in#ut signal' sinewave
b! am#litude of in#ut signal 6ffective value! 1== m>
c! Fre.uency of in#ut signal 1==?5L
d! 7C com#onent of in#ut signal offset! =.=I>
e! 7C su##ly voltage 1>
%.". 4tudy the nonlinear resonance am#lifier in fre.uency multi#lying mode.
%.".1. &ut
1 0
U U
. 7ecrease fre.uency of in#ut signal twice in order to have the same value
of fre.uency as for resonance fre.uency of tank. Changing offset voltage find maDimal
value of out#ut voltage of am#lifier in doubling mode.
$he following is the 54&IC6 netlist:deck of the circuit and its simulation results fig. %.1!'
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
cc<ca#acitor1 out gnd c 8 ()=# cty#e 8 =
cc<ca#acitor" out1 out c 8 1)=# cty#e 8 =
vc<vgeneric1 vdd gnd dc 8 1
vc<v<sin1 in gnd 4IE =.=I =.1 fre. !
mnmos(1 out1 in gnd gnd E l 8 =.1u w 8 1==u m 8 1
mnmos(" vdd vdd out1 gnd E l 8 1u w 8 1u m 8 1
lc<inductor1 out gnd l 8 1))u
.global gnd
.model n nmos
.#aram fre.8%==k
.tran J1C1==;fre.!J J)=;1Cfre.J
.o#tion #ost #robe
.#robe >;!
.end
Fig. %.1. 4imulation results
%.".". Find the gain of am#lifier in fre.uency doubling mode.
$he following is the 54&IC6 netlist:deck of the circuit and its simulation results fig. %."!.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
cc<ca#acitor1 out gnd c 8 ()=# cty#e 8 =
cc<ca#acitor" out1 out c 8 1)=# cty#e 8 =
vc<vgeneric1 vdd gnd dc 8 1
vc<v<sin1 in gnd 4IE =.=I =.1 fre. !
mnmos(1 out1 in gnd gnd E l 8 =.1u w 8 1==u m 8 1
mnmos(" vdd vdd out1 gnd E l 8 1u w 8 1u m 8 1
lc<inductor1 out gnd l 8 1))u
.global gnd
.model n nmos
.#aram fre.8%==k
.tran J1C1==;fre.!J J)=;1Cfre.J
.o#tion #ost #robe
.#robe >;!
.end
Fig. %.". 4imulation results
%.".%. For the circuit in fig. ,1.b see oscillograms of out#ut voltage in fre.uency doubling mode.
$he following is the 54&IC6 netlist:deck of the circuit and its simulation results fig. %.%!'
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
cc<ca#acitor1 out1 out c 8 ".2n
rc<resistor1 out gnd R8)k
mnmos(" vdd vdd out1 gnd E l 8 =.1u w 8 %u m 8 1
mnmos(1 out1 in gnd gnd E l 8 =.(u w 8 1==u m 8 1
vc<v<sin1 in gnd dc 8 = 4IE =." =.=) fre. !
vc<vgeneric1 vdd gnd dc 8 1
.global gnd
.model n nmos
.#aram fre.8%==k
.tran J1C1==;fre.!J J)=;1Cfre.J
.o#tion #ost #robe
.#robe >;!
.end
Fig. %.%. 4imulation results
(. Re#ort
$he re#ort must include'
(.1. Circuits of fre.uency multi#lier based on nonlinear resonance am#lifier
(.". Basic calculation formulas
(.%. Results of homework calculation
(.(. &lots of characteristics got from calculations and testing
(.). Oscillograms get during testing
(.1. 6stimation of results and conclusions.
). Checking .uestions
).1. 5ow can am#litudes of first+ second and third harmonics of drain current be found if the
am#litude of drain current im#ulse and slo#e of transistorJs a##roDimate IC> characteristic
are knownA
).". 7raw drain current and voltage transient diagrams in doubling and tri#lication cases.
