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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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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT,

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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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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