Key words:

quartz units, cryogenic modeling, quartz circuit analysis

Franciszek BALIK* Andrzej DZIEDZIC**

QUARTZ CIRCUIT ANALYSIS ENHANCED AT CRYOGENIC TEMPERATURES BY USING NOVEL

QUARTZ UNIT MODEL

The aim of this paper was to investigate the influence of new quartz crystal electrical model de-

veloped for the temperature range from 83.15 K (-190oC) to 303.15 K (+30oC) on fundamental

characteristics of quartz crystal circuits, through simulation analysis. The examination was per-

formed using two kinds of circuits - Colpitts oscillator (transient analysis) and Tchebyshev filter

(ac analysis). Functional dependence of parameters of both time and frequency characteristics

were determined and compared with those obtained for standard SPICE quartz model. It should

be noticed, that the relative frequency generated by Colpitts oscillator was changed even up to

500 ppm in relation to that at 200C, while the Fourier analysis of generated signals in case of

standard SPICE quartz model showed that the frequency is not changed. In case of Tchebyshev

band-pass passive filter the frequency characteristics were shifted towards lower frequencies.

Their fundamental parameters have been also changed, but surprisingly in case of standard SPICE

quartz model the frequency characteristics were completely unchanged.

1. INTRODUCTION

The temperature properties of quartz crystals are important for designing the electronic

circuits for military, medical or outer space applications [1]. Though the thermal prop-

erties of quartz crystals are well recognized in standard application temperature range

(-40oC < T < +150

oC) [2,4,7], but their electrical models at cryogenic temperature

range are not sufficiently finished up. Hitherto, the most of electrical models of quartz

resonators in PSPICE computer program are temperature independent [5], but practi-

cal experience shows that electronic circuits consisting quartz units change their prop-

* Wroclaw University of Technology, Faculty of Electronics, Institute of Telecommunication,

Teleinformatics and Acoustics, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland

** Wroclaw University of Technology, Faculty of Microsystem Electronics and Photonics, Wyb.

Wyspianskiego 27, 50-370 Wroclaw, Poland

(This work was supported by National Science Centre, Grant No 2011/01/B/ST7/06564)

Franciszek Balik and Andrzej Dziedzic 2

erties at low temperatures [1,3]. For PSPICE computer program the temperature prop-

erties of some quartz units were described by the electrical models exploiting first or

second order temperature coefficients, which characterize the quartz crystal sufficient-

ly well in narrow range of temperature, only. Standard SPICE models were deter-

mined based on data taken from [6] and e.g. for BT-cut type the quadratic temperature

dependence is taken into account. The AT-cut is an exception, and has cubic tempera-

ture dependence, which is not included in PSPICE models. Unfortunately, many con-

temporary electronic circuits need to work in wide temperature range including cryo-

genic range, too [1,14,15]. Therefore, a need to elaborate models of electronic ele-

ments for this scope of temperatures appears. The new temperature-related model of

quartz crystal unit, valid for temperature range from 83.15 K (-190oC) to 303.15 K

(+30oC), was presented in [3]. Now, in this work the analyses in the time and the fre-

quency domains of quartz crystal electronic circuits at the cryostatic temperature range

is carried out taking this model into consideration.

2. THE QUARTZ UNIT TEMPERATURE–DEPENDENT MODEL

2.1. FUNDAMENTAL MEASUREMENTS

In practice, the quartz unit is modeled by the fundamental PSPICE electrical equiva-

lent circuit (Fig. 1) [2,5,7]. Therefore, the low-temperature properties of this funda-

mental model and its influence on circuits’ parameters were the subject of our investi-

gations.

Fig. 1. The quartz crystal unit electrical model

The electrical model of AT-cut type quartz crystal resonator had to be temperature-

related i.e. its elements should be expressed as some functions of temperature: Ck =

Ck(T) – series (motional) capacitance, Lk = Lk(T) – series (motional) inductance, rk =

rk(T) – series resistance, C0 = C0(T) – parallel-plate (static) capacitance. The resonant

frequencies of series (fs) and parallel (fp) resonances are given by the well-known rela-

tionships:

kk

sCL

f2

1, (1a)

)(2 00 CC/CCLf

kkk

p

1. (1b)

Both the instrument set-up and measurement methods needed to collect the necessary

Quartz circuit analysis enhanced at cryogenic temperatures… 3

data have been worked out. The cryostat system shown in Fig. 2, which allowed us to

perform characterization of electronic components and circuits at low temperature,

exploits the continuous gas-flow type N2/He cryostat and works under Lab View com-

puter program control, was used. The principle of its operation and measuring setup

were exactly described in [3,8]. All measurements were performed with accuracy as

high as possible. To avoid parasitic effects the short- and open-circuit operations were

made at the beginning of measurements. The first element, which had to be measured,

was the parallel capacitance C0. This element represents the shunt capacitance result-

ing from stray capacitance between the terminals and capacitance between the elec-

trodes. This static capacitance was measured far from resonance at 100 kHz and 1 V

amplitude. Because the cable capacitance CL is connected in parallel with C0 and is

placed in cryostat its value also is depended on temperature. Therefore, we measured

the temperature characteristic of the resonator with cable and the cable capacitance

itself. Knowing C0 and CL the values of motional parameters Ck(T) and Lk(T) were

calculated from (2a) and (2b), having measured the frequencies of series fs and parallel

fL resonances (including the cable capacitance CL) as a function of temperature.

Fig. 2. Physical principles of operation of continuous gas-flow type cryostat

)()(2

0 L

s

sLk CC

f

ffC

, (2a)

)()(π8

1

0

2

LsLs

kCCfff

L

. (2b)

Fig. 3. The measuring set up for quartz resonator impedance measurements

Franciszek Balik and Andrzej Dziedzic 4

Lk(T) = (((((((((((2.74160197995312e-024T + 2.64194454143027e-021) T + 1.11773132250230e-018) T

+ 271.829095157317e-018) T + 41.5279952041439e-015) T + 4.07192990371513e-012) T +

248.602505679450e-012) T + 8.26140081802474e-009) T + 56.1421221668988e-009) T –

8.65457153438779e-006) T -117.737531061562e-006) T + 22.9612714058108e-003) T +

12.8870352956 Ck(T) = ((((((- 4.23970376353478e-018T – 3.41140641696408e-015) T – 1.13953013964350e-012) T –

59.8686114144432e-012) T + 19.8469480395879e-009) T + 328.437266262630e-009) T –

97.0880817640254e-006) T + 54.6541059267272e-003

C0(T) = (((((211.886351286049e-015T + 121.338832687354e-012) T + 23.5335077288204e-009) T +

1.44478655729065e-006) T – 55.8950342670289e-006) T – 10.0879506711649e-003) T +

5.60621283177 rk(T) = -0.6164T + 71.003

Having values of Ck(T) and Lk(T) the resonant frequency fp is calculated from Eq. (1b).

Having voltages U1 and U2 measured in the set-up, shown in Fig. 3, at series resonant

frequency, the series resistance rk(T) was calculated from the expression [2]:

)1(2

11

U

URrk

.

(3)

2.2. THE SPICE BEHAVIORAL MODEL

Each curve Lk(T), Ck(T), C0(T) and rk(T) can be fitted by some polynomial of degree

depending on demanded approximation accuracy. Evidently, for the same resonant

frequency determination accuracy, different curves have different orders. In aim of

realization of this approximation task, the computer program was written in Matlab,

using Polyfit Matlab library function. Appropriate orders of polynomials ensure good

accuracy of approximation. As a result of these calculations we received the following

approximations, which fulfill the demanded accuracy (computer printout):

(4)

where: temperature T is in oC, Lk(T) in mH, Ck(T) and C0(T) – in pF, rk(T) – in .

Assuming that accuracy of frequency determination should be better than 0.02%, the

maximal relative errors of polynomial approximations for each polynomial reached

values: δLk = ~13.65e-003%, δCk = ~ 37.25e-003%, δC0 = ~ 201.57e-003%, δrk = 1e-

003%. The elements of quartz crystal resonators used in standard SPICE models li-

brary offer only maximally quadratic temperature coefficients. It is possible to over-

come this limitation in PSPICE ver. 8 and higher, applying the Analog Behavioral

Modeling (ABM) approach [9,10]. PSPICE extensions allow arbitrary equations

and/or table lookup. The ABM sources allow us accessing the global variable tempera-

ture TEMP. Having in disposal the functional symbolic description (4) for electrical

model elements, we were able to compose the SPICE behavioral model working at

cryogenic temperature range, which consists of four sub-models.

The temperature–related inductor can be replaced by ABM model, which uses

voltage-controlled current source (Gvalue) and voltage–controlled voltage source

(Evalue). In Fig. 4 this model consists of the following elements: G1, E1, L1, R1 and

V_Isec1. To realize this modeling task, the Ben-Yaakov and Peretz method has been

Quartz circuit analysis enhanced at cryogenic temperatures… 5

applied [10]. The basic idea of the proposed inductor modeling is to reflect the

behavior of a linear reference inductor (L1) via nonlinear “transformer” to the

Fig. 4. Behavioral PSPICE temperature-related model of the quartz crystal resonator working at low

temperature

input port (Lin). It is realized by using the depended sources, mentioned above (E1,

G1). The coefficient of reactance transformation is

KL = Xin/XL1 = Lin/L1. (5)

The V_Isec1 is zero–valued voltage source for current measurement.

The temperature-related capacitor can be replaced by ABM model using the same

method as for inductor. In Fig. 4 the elements: G2, E2, C1 and V_Isec2 constitute this

Franciszek Balik and Andrzej Dziedzic 6

model. The capacitance transformation coefficient KC is

KC = Xin/XC1 = C1/Cin . (6)

For the parallel capacitor C0(T), temperature behavioral model is obtained in simi-

lar way like for series capacitor. Another appropriate function in (4) for C0(T) was

applied in KC expression. In Fig. 4 the elements: G3, E3, C2 and V_Isec3 constitute

this model. The temperature dependence of series resistance rk can be achieved by

using Gvalue voltage–controlled current source with short circuited input and output

ports (Vin = Vout) [11]. In such configuration the current in expression describing this

source can be written as

I = Vin/rk(T). (7)

In Fig. 4 the source G4 constitutes this model, only. The Lk(T), Ck(T) and C0(T) func-

tions are replaced by the appropriate expression in (4) and variable T must be substi-

tuted by global temperature variable TEMP. The resistances R1, R8, R9 and R10 are

inserted to avoid the floating point errors. The quartz crystal resonator temperature

behavioral model shown in Fig. 4 was obtained as a composition of these partial mod-

els. The result of simulations of this model confirmed appropriative resonator tem-

perature characteristics. It was stated that accuracy of this model is between 0.0001%

and 0.022% [3]. It should be remarked that very high accuracies of polynomial ap-

proximations are necessary to achieve such good result.

3. QUARTZ CRYSTAL CIRCUIT ANALYSIS 3.1. COLPITTS’ QUARTZ OSCILLATOR

Let’s consider the temperature dependence of the frequency and amplitude of the sine

wave generated by quartz Colpitts oscillator [12,13], of which electrical scheme is

shown in Fig. 5. The R, C elements were supposed to be temperature independent,

because it is well known, that their influence on frequency generated by quartz oscilla-

tor can be neglected. The investigated signals have been obtained as the result of tran-

Fig. 5. Quartz crystal Colpitts oscillator

Quartz circuit analysis enhanced at cryogenic temperatures… 7

α = 0.0003T5 - 0,0096T4 + 0.1063T3 - 0.4488T2 + 0.9706T - 4.3425 -6

-4

-2

0

2

-150 -140 -130 -120 -100 -80 -60 -40 -20 0 20

Temp [0C]

sient analysis performed using PSPICE computer program. The analysis covers the

temperature range from -1500C to +20

0C. The quartz unit X4 was modeled first by

standard SPICE library model. The Fourier analysis of generated signals showed that

the frequency is not changed (Fig. 6) – what does not agree with experience [1,13,14].

Therefore, this result enforced us to apply more exact model of the quartz unit, tem-

perature dependent model described in Chapt. 2. In this case the obtained results are

presented in Fig. 7.

Fig. 6. Frequency of generated waves; temperature as parameter: from -1500C (lower characteristic) up to

+200C (upper characteristic)

Fig.7.Relative frequency change vs. temperature: δ = 106(f - f20)/f20

Fig. 8. Relative amplitude: α = log10(Vamp/V20) vs. temperature ; V20 – output voltage at +20oC

As we see, the relative generated frequency can be changed up to 500 ppm in relation

to that at 200C and has almost linearly decreasing character at low temperature.

δ = 0.0749T6 - 2.3664T5 + 27.701T4 - 151.13T3 + 404.69tT2 - 424.87T - 352.45

-1000

-500

0

500

-150 -140 -130 -120 -100 -80 -60 -40 -20 0 20

δ [ppm] Temp [0C]

9.998MHz 9.999MHz

10.000MH

z

10.002MHz 10.004MHz

10.005MHz

10.006MHz ...

200mV

400mV

600mV

767m

V

10.001MHz 10.003MHz 9.997MHz V(C4:2)

α

Franciszek Balik and Andrzej Dziedzic 8

The amplitude dampening (Fig. 8) is also affected by the transistor amplification

coefficient diminishing at low temperature [15].

3.2. QUARTZ – CRYSTAL TCHEBYSHEV BAND-PASS FILTER

Now we will investigate the temperature dependence of the frequency characteristics

of the quartz crystal band-pass filter with electrical scheme shown in Fig. 9. This is the

example of the 4-th order Tchebyshev band-pass passive filter [12,13].

Fig. 9. Schematic of band pass Tchebyshev filter

Fig. 10. Temperature dependence of frequency characteristics in case of standard quartz model; tem-

perature as parameter: from -1500C up to +200C

Fig. 11. Three characteristics in case of novel quartz model at -180oC (green), 0oC (red) and +20oC

Frequency

5.9700 5.980

0

5.9900 6.000

0

6.0100 6.020

0

6.0300 6.0396

[MHz] db(V(R1:2)/ V(V1:+))

-25.00

-20.00

-15.00

-10.04

(6.0000M,-14.374) (6.0096M,-14.374)

(5.9957M,-25.919)

(6.0028M,-16.862) (5.9961M,-

16.838)

(5.9935M,-26.613)

5.960

3

Frequency

9.9700 9.9800 9.990

0

10.0000 10.0100 10.020

0

10.0300 10.040

0

10.050

0

10.060

0

10.070

0

10.0800

[MHz]

9.9602

-15.00

-10.00

9.961M,-20.738

(10.012M,-8.3270)

db(V(R1:2)/ V(V1:+))

-6.00 (10.016M,-7.5079)

(6.0009M,-11.380)

Quartz circuit analysis enhanced at cryogenic temperatures… 9

Resistors R3, R4 and R6 are inserted to avoid floating point errors and they don’t have

influence on the numerical results of calculation. The lasting R, C elements are sup-

posed to be temperature independent. When standard SPICE quartz model was used

for elements X1 and X2, the frequency characteristics were completely independent

on temperature (Fig. 10). All characteristics are coinciding. This was not real result at

cryogenic temperature range [1,13,14]. Taking the new quartz unit model, we obtained

radically different results (Fig. 11). The frequency characteristics were shifted towards

lower frequencies. Their fundamental parameters have been changed, too. While tem-

perature decreases, the dampening coefficient Dmin is growing almost linearly (Fig.

12), the filter bandwidth B is also narrowing (Fig. 13).

Fig. 12. Minimal dampening Dmin vs. temperature

Fig. 13. Filter bandwidth B vs. temperature

Fig. 14. Relative centre frequency change vs. temperature: δ = 106(f -f20)/f20

Dmin = 0.006T2 + 0.1374T - 14.056

-15

-13

-11 -190 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20

Dmin [dB] Temp[0C]

B = -8E-05T6 + 0.0032T5 - 0.0466T4 + 0.2703T3 - 0.3964T2 - 0.3425T + 7.3189

5 6 7 8 9

10 11 12 13 14 15

-190 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20

Temp [0C]

B[kHz]

δ = 0.1427T5 - 4.4358T4 + 48.988T3 - 254.25T2 + 865.36T - 1971.9

-1500

-1000

-500

0

500

-190 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20

δ [ppm]

Temp [0C]

Franciszek Balik and Andrzej Dziedzic 10

The relative centre frequency change exceeds 1000 ppm at -190oC (Fig.14). The most

interesting is fact, that while temperature decreases we observe the transition of the

filter characteristic shape from two to one resonant at -60oC – -80

oC.

4. CONCLUSIONS

In this paper the influence of novel temperature-related quartz unit model on circuits

characteristics at cryogenic temperature range has been examined. The results

achieved look promising and provide useful information for analog circuit designers

developing circuitry for extreme low temperature ranges. The examination was per-

formed using two kinds of circuits: Colpitts’ oscillator (transient analysis) and

Tchebyshev filter (ac analysis). The comparison of obtained results with those for

standard SPICE quartz model confirmed better accuracy and appropriateness usage of

this novel model. Get results of analyses are showing the greater conformity to practi-

cal results. Taking into account the results mentioned above, the newly elaborated

quartz unit model should be preferred when designing quartz electronic circuits for

cryogenic temperature range.

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