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IV. C O N C L U S I O N
This report show s that we are able to increase the output pow er
of the 87Rbmaser by about 5 dBm by us ing opt ical pumping at both
ends and to m ake the shor t- term frequency s tabil i ty improvement
of about a factor of 1.4. The short-term frequency stability might
be improved fu r ther by us ing tw o diode lasers with higher intens i ty
and by optimizing the ce ll temperature. Our measurements are con-
sistent with the density matrix treatment.
A C K N O W L E D G M E N T
The authors thank Tan Yongfang fo r technical support .
REFERENCES
[l ] P. Davidovits and R. Novick, The optically pumped rubidium
maser, Proc . IEEE vol. P-54, pp. 155-170, 1966.
[2]
G.
Busca,
R.
Brousseau and
J.
Vanier, A compact 87Rbmaser, IEEE
Trans. Ins t rum. Me a s . , vol. IM-24, pp. 291-295, 1975.
[3] M. Tetu, G . Busca and J. Vanier, Short-term frequency stability of
the 87Rbmaser, IEEE Trans. Ins t rum. Meas. , vol. IM-22, pp. 250-
257, 1973.
[4] M. Tessier and J Vanier, Theorie du maser au rubidium 87, Can.
[5] J . Vanier, R elaxation in rubidium-87 and the rubidium maser, Phys.
[6] G. Busca, M. Tetu and J. Vanie r, Light shift and light broadening in
J . Phys . ,
vol. 49, pp. 2680-2689, 1 971.
Rev . ,
vol. 168, pp. 129-149, 1968.
the 87Rbmaser, Can. J Phys . , vol.
51,
pp. 1379-1387, 1973.
A Simple Wide-Band Sine Wave Quadrature
Oscillator
Ralph Holzel
Abstract-A simple sine wave oscillator delivering two signals in
quadrature has been developed, covering the frequency range from
below
1
Hz to 100 MHz and being voltage-controllable over each fre-
quency decade. The circuit consists of two
90
phase shift stages using
active allpass filters and an inverter,
ll
arranged in a feedback loop.
Although general application is envisaged, the design is particularly
suitable in particle electrorotation studies.
I.
IN TR O D U C TIO N
Quadrature generators , i . e . , generators producing tw o s ignals of
ident ical f requency and a mpli tude but of dif ferent phase angle, are
used in a var iety of a ppl icat ions , as for exam ple in telecommuni-
cations f I quadrature mixers and s ingle-s ideband generators [11
or
for measurement purposes in vector generators
or
selective volt-
meters [2]. They have foun d furthe r utility in the physical charac-
terization of microscopic particles and biological cells using the
phenomenon of electrorotation, whereby rotational torques are in-
duced through application of rotating electric fields of variable fre-
quency [3]-[SI. The availability of wide bandwidth qu adrature
os-
cillators of simple design is important for such studies, and it is
Manuscript received August 21, 1992; revised October 20, 1992.
The autho r is with the Institute of M olecular and Biomolecular Electron-
IEEE Log Number 9207680.
ics, University of Wales, Bangor LL57 IUT, United Kingdom.
IEEE
TRANSACTIONS
ON INSTRUMENTATION
AND
MEASUREMENT,
VOL.
42, NO. 3 J UNE 1993
0018-9456/93 03.00 1993 IEEE
this consideration that has led to the circuit development described
here.
Rectangular quadrature signals (i.e. , of 90 phase shift) are
readily available up to some tens of MHz using ring oscillators
or
by dividing the f requency of a square wave by tw o and combining
this with the original signa l via logic gate s [SI. Sinusoidal voltages
can be generated using special commercially available ICs (e.g. ,
Burr Brown 4423 up to
20
kH z) . Higher f requencies can be
achieved by programming the differential equation of a sinusoidal
oscillation with the help of operational amplifiers [6]. For that two
integrator s tages are connected in ser ies with an inver ter , of which
the output is fed back to the input of the f i rs t s tage. How ever , the
upper f requency l imit of such an osci l lator
is
still limited by the
relatively low bandwidth of the integrators.
Here we describe a circuit that utilizes two allpass filters as 90
phase shif ters in ser ies with an inver ter operat ing up to 100 MHz.
11 D ESIG NC O N S I D E R A T I O N S
Ideal first-order allpass filters (Fig. 1) are characterized by a fre-
quency-dep endent phase shift varying from 80 at low frequen-
cies to 0 at high frequencies, given by
o
=
-2 arccot (27rfRC), (1)
while the am plificat ion equa ls uni ty for R I
=
R t all frequencies
[7]. Cascading two allpasses and an inverter and feeding back the
output to the first stage (Fig. 2) lead to an oscillation [8] at the
frequency where
o
= -90 i . e . , a t
f = 2 7 m - I .
Equat ions (1) and (2) only hold for moderately low frequencies ,
since real operational amplifiers exhibit an additional phase shift
increasing with f requency du e to paras it ic capaci tances . G eneral ly,
oscillation can only occur if
(oI
+ 02 +
pi
= - 3 6 0 .
As
long as the inver ter s tage does not add any addi t ional phase
error , i . e . , pi = 18 0, the sum of the al lpasses phase shifts must
amount to
o,
+ 02 = 3 6 0
pi
= 180 for osci l lat ion. For iden-
tical allpass stages (oI and 02are ident ical , even i f they d o not work
ideal ly, e.g. , at higher f requencies . W ith
o,
and
02
=
180 and
(oI
= 02 follows
o,
= 02 = 90. That means the output s ignals are
always in quadrature provided 1) the inverters phase error is zero
and 2) the al lpass s tages are ident ical. Therefore, the bandwidth of
the inver ter is more important fo r the circui t s f requency range than
the bandwidth o f the al lpasses .
In selecting suitable operational amplifiers account should be
taken of the fact that all stages work at amplifications of unity.
Therefore, many high-speed operational amplifiers, that are usually
not uni ty gain s table, cannot be used or only so with some restric-
tions. To cancel variations in amplification and to maintain low
dis tor t ions , the inver ters gain should also be controllable. How -
ever , inver ter gains deviat ing f rom unity wil l correspond to de-
viating allpass gains and thus lead to an inequality of the output
amplitudes.
In order to control the osci l lat ion f requency via a dc vol tage,
field effect transistors (FET), light dependent resistors (LDR)
or
capaci tance diode s can be used. Th e lat ter are less useful for f re-
quencies below 1 kHz because of their maxim um capaci tance val-
ues of about
1
nF and the typical input res is tance of the order 1
MQ
f fast operational amplifiers. FETs are available as matched
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I
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT,
VOL 42,
NO. 3, JUNE 1993
p.
Fig.
1.
First-order allpass filter.
U,
=
q s i n
wt
U
=
U,sn (cot+ )
t
t
O
cp
= -90 cp =-90 cp .
=
180
Fig.
2.
Block-diagram o f the quadrature oscillator.
pairs and thus have the adv antage over LDR's in achieving equal i ty
of both phase shift stages. Furthermore, the approximately recip-
rocal dependence of the drain-source-resistance o n the gate-source-
voltage
[9]
can be exploited to achieve a l inear relationship be-
tween controlling voltage and oscillator frequency.
111.
CIRCUITRY
ND
EXPERIMENTAL
ESULTS
To prevent instability caused by stray effects mainly of the two-
pole switch SI Fig. 3) only moderately fast operational amplifiers
(Analog Devices AD 847, ful l power bandwidth 10MH z) are used
in the allpass stages. The
RC
network at the inverter 's input per-
mits a s table function of the fas t op a mp , which otherwise would
need a gain of 4 or more. The osci l lation a mpli tude is l imited by
the nonlinearity of germanium diodes AA
112
and can b e adjus ted
by V, via FET B F 256, which works as a voltage-controlled resis-
tor. At f requencies below a few MHz the FET c an be replaced by
a fixed resistor, since the amplitude rem ains nearly constant. H ow-
ever, fo r minimizing signal distortion it is advisable to contro l the
gain automatically via
V,
and a PI-regulator, that measures the ac-
tual oscillation amplitude.
The f requency can be tuned by Vfiwhich co ntrols the drain-
source-res is tances of the F ET pair
2
N
5912.
Good linearity be-
tween controlling voltage and oscillation frequency was achieved
over a whole f requency decade as shown in Fig.
4 .
By switching
the frequency-determining capacitors a frequency range of more
than 7 decades f rom below 1 Hz up to 10 MHz could be covered.
On examining the use of different types of operational amplifiers
in the allpass stages it was found that, as a good measure of the
accessible frequency range, the full power bandwidth cited in the
semiconductor's datasheet can be used rather than the unity gain
or 3
dB bandwidth.
The s ignal 's harmonic content at
100
kH z am ount s to -24 d B
for the third and less than
-36
dB for al l other harmonics , i f the
gain control voltage V, is kept constant at a value that allows fre-
quency variations over several decades. If V , is opt imized and con-
trolled automatically by a PI-regulator the third harmonic is re-
duced to
-38
dB.
Even higher frequencies can be achived using faster amplifiers
and through a reduction of stray capacitances.
For
this purpose a
159
Fig. 3 . Circuit diagram of the voltage-controlled sine quadrature oscillator
operating up to
10
MHz.
controlling voltage
V, PI
Fig.
4.
Dependence o f the output frequency of the sine quadrature VCO
Fig. 3) on the controlling dc voltage.
Fig. 5 . Circuit diagram of the quadrature VCO for frequencies up to 100
MHz.
1 2 3
controlling voltage V [V
Fig.
6.
Dependence o f the output frequency of the quadrature VCO Fig .
5) on the controlling dc vo ltage.
second circuit was built employing surface mount compone nts (Fig.
5 ) , with the very fas t op amp O PA 621 (Burr Brown, ful l power
bandwidth 80MH z) being em ployed in the al lpass f i l ter stages and
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760
IEEE
TRANSACTIONS ON
INSTRUMENTATION AND MEASUREMENT, VOL 42,
NO. 3
J UNE
1993
with the relatively large FET p air being replaced by tw o capaci-
tance diodes. A dual-gate M OSFE T, fol lowed by a bipolar col lec-
tor s tage, operates as a vol tage-control led inver ter , and the L C net-
work at the MO SFETs d rain suppresses parasi t ic osci l lat ions at
230 M Hz caused by the o p amp ( that is usual ly only s table at gains
of at least 2). By these means the f requency range was extended to
100 MHz, al though somewhat higher operat ing f requencies are
possible, provided unequal amplitudes of the quadrature outputs
can be tolerated. The correlation between tuning voltage and fre-
quency is l inear over nearly a f requency de cade (Fig. 6) , w ith de-
creas ing s lop e above 80 MH z.
This quadrature osci l lator design has been successful ly em-
ployed to generate rotating electrical fields in studies of the ac
electrokinetics of biological cells [3], [4].
IV. CONCLUSION
A simple vol tage-controlled s ine wave osci l lator with quadrature
output has been developed, which is useful at f requencies f rom
below 1 Hz to about 100 MH z for appl ications such a s the s tudy
of particle electrorotation phenom ena. If a lower frequency limit is
required, this should be achieved by rapid switching of the fre-
quency-determining capacitors with analogue switches [lo], rather
than by using large electrolytic capacitors. A furth er expansion o f
the bandwidth to higher frequencies might be possible using even
faster hybrid operational amplifiers or discrete components, but this
would be at the expense of the circuits simplicity.
A C K N O W L E D G M E N T
Valuable advice and discuss ions with Prof . R. Pethig, Dr . K.
Winkler (Free University Berlin), and Dr. J. Burt are acknowl-
edged with grat i tude, as is the award by the Commiss ion of the
European Community.
R E F E R E N C E S
P. Horowitz and W. H ill,
The Art ofElec tronics.
Cambridge,
U.K. :
Cambridge University Press, 1991, ch. 5.16, p. 291.
U . Tietze and
C.
Schenk, Electronic Circu ifs: Design and Applic a-
tions. Berlin, Germany: Springer, 1991, ch. 25.3.4, pp. 795-796.
R. Holzel and 1. Lamprecht, Dielectric properties of yeast cells as
determined by electrorotation, Biochim. Biophys. Ac fa, vol. 1104,
Y Huang, R. Holzel, R. Pethig, and X.-B. Wang, Differences in
the AC electrodynamics of viable and non-viable yeast cells deter-
mined through combined dielectrophoresis and electrorotation stud-
ies, Phys. Med. Biol. , vol. 37, pp. 1499-1517, 1992.
J . Gimsa, R. Glaser, and
G .
Fuhr, Th eory and app licaton of the
rotation of biological cells in rotating electric fields (electrorota-
tion), in Physical Characterization of Biological Cells, W. Schutt,
H. Klinkman,
I .
Lamprecht and T. Wilson, Eds. Berlin, Germany :
Verlag Gesundheit, 1991, pp. 295-323.
U. Tietze and C . Schenk, Electronic Circuits: Design and Applica-
tions.
Electronic Circuits: Design and Applications. Berlin, Ger-
many: Springer, 199 1, ch. 14.10.2, pp. 394-395.
H. Schmidt, Hochwertiger durchstimmbarer Nf-Sinusoszillator,
Elektronik, vol29, H 19, pp. 113-1 14, 1980.
U. Tietze and C. Schenk, Electronic Circuits: Design and Applica-
tions.
H. Lemme, CMOS-Bausteine ersetzen Vielfach-Potentiometer
Elektronik,
vol. 29, H 19 , p. 114, 1980.
pp. 195-200, 1992.
Berlin, Germany: Springer, 1991, ch. 15.4, pp. 425-428.
Berlin: Springer, 1991, ch. 5.7, pp. 89-91.
An Algorithm for Solving Roberts-von Hippel
Equation: Separation
of
Close Solutions
T. P. Ig les ias , A. S eoane, and J . Rivas
Abstract-The proximity
of
the solutions
of
the transcendental equa-
tion tanh Z / Z =
c
on the com plex plane arising in the determination of
the comple x dielectric permittivity with the short-circuited line method
is studied. A numerical procedure is described which is capable of oh-
taining the required number of solutions, even those lying very close
to each other, from one single initial point. The method is applied to
measurements of various organic liquids.
I . IN TR O D U C TIO N
The shor t-circui ted l ine method for the determinat ion of the
complex dielectric permittivity [11 requires solving a transcen den-
tal equation of the form (tanh
z ) / z
= on the complex plane,
where is obtained experimen tally. Although this equation pos-
sesses infinite solutions, only one of them is physical. In order to
el iminate this ambigui ty i t is necessary ei ther to know in advanc e
an approximate value of the permittivity, to carry out two experi-
ments with samples of different thickness,
or
to perform m easure-
ments at two nearby frequencies [2], [3]. Before deciding which of
the solut ions is the physical one i t is necessary to be able to ident ify
and distinguish from each oth er all solutions which ca n be relevant.
In practice it is enough to c onsid er only a few of the infinite solu-
tions, with small values of the real part, given the range of dielec-
tric permittivities of usual substances. It is important to identify
and dis t inguish close roots w hen s tud ying, for ins tance , the effect
that the thickness of the sample has on the uncer tainty of the per-
mittivity [4]. It is also necessary when measuring dielectric per-
mit t ivi t ies with network analyzers , where the impedance of the
sample is measured for a cont inuous range of f requencies and a
fixed sample thickness [5]. An analogous situation arises in the
time-domain dielectric characterization technique which uses a
short-circuited sample.
Several methods for the resolut ion of the equat ion above hav e
been developed. One of them is Dakin-Works approximation [6] ,
which is not applicable for samples with loss tangent tan
6
>
0.1
due to the large errors that can ar ise, as i t is shown by S 0 Nelson
et al. [7] . There are also graphical methods such as Rober t-von
Hippels [l ] or Delbos-Demaus [8] . The lat ter is used when the
sample has constant thickness. Com puters have al lowed the use of
i terat ive numerical me thods , of which the m ore commonly used is
Newton-Raphsons [ 4 ] , [9] . This method converges f rom a good
ini tial approximate value to o ne s ingle roo t ,
so
that if there were
two roots close to each other one of them might be ignored. G el inas
et al. [lo] also use Netwons method for the case Re c) 0 s ince
there are no close roots corresponding to this region of the com plex
plane.
Manuscript received January 16, 1992; revised July 7, 1992.
T.
P
glesias is with the De partamento de Fisica Aplicada, Facultad de
Ciencias, Universidad de Vigo, 36200 Vigo, Spain.
A. Seoane is with the Departamento de Teoria de la Seiial, E.T.S.I.T.,
Universidad de Vigo, 36200 Vigo, Spain.
J Rivas is with the D epartamento de Fkic a Aplicada, Facultad de Fis-
ica, Universidad de Santiago de Compostela, Santiago de Compostela,
Spain.
IEEE Log Number 9207671.
0018-9456/93 03.00 993 IEEE