Quartz crystal unit modeling at cryogenic temperatures

Materials Science and Engineering B 177 (2012) 1254–1260

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Quartz crystal unit modeling at cryogenic temperatures a ´ F. Balik a,∗ , A. Dziedzic b , T. Swietlik a b

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

a r t i c l e

i n f o

Article history: Received 13 September 2011 Received in revised form 19 November 2011 Accepted 23 December 2011 Available online 21 January 2012 Keywords: Quartz unit Low temperature measurements Modeling

a b s t r a c t The aim of this paper was to develop the new quartz crystal electrical model including its temperature properties in the temperature range from 83.15 K (−190 ◦ C) to 303.15 K (+30 ◦ C) through experimental set-up and simulation analysis. Both the methodology of the quartz resonator measurements, the instrument setup, and the measurement methods needed to collect the necessary data as well as polynomial approximation of temperature dependence were described. The electrical model of AT-cut type quartz crystal for cryogenic temperatures was developed, in which its elements were expressed as functions of temperature. Using these polynomials, the behavioral model for PSPICE computer program has been worked out. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The temperature properties of quartz crystals are important for designing the electronic circuits for military, medical or outer space applications. Though the thermal properties of quartz crystals are well recognized in standard application temperature range (−40 ◦ C < T < +150 ◦ C) [1–3], 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 [4]. For some of them the temperature properties of the electrical models of quartz units were described by first or second order temperature coefficients, which characterize the quartz crystal sufficiently well in narrow range of temperature. They were determined based on data taken from [5] and for some of them (the BT-cut type) the quadratic temperature dependence is taken into account. The AT-cut is an exception, and has cubic temperature dependence, which is not included in PSPICE models. In paper [6] the quartz crystal model was considered as three-nodal circuit in which only the series inductance was temperature-depended. This model could not be accurate at cryogenic temperature range, where even small changes of lasting elements of model have significant influence on resonant frequencies. In this paper the electrical model of AT-cut type quartz crystal unit in a wide temperature range was elaborated in which all its

∗ Corresponding author. Tel.: +48 604 821 538; fax: +48 717 858 633. E-mail addresses: [email protected] (F. Balik), ´ [email protected] (A. Dziedzic), [email protected] (T. Swietlik). 0921-5107/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2011.12.047

elements were expressed as functions of temperature. Of course, we should be conscious that although Grupp and Goldman [7] showed that strontium titanate (SrTiO3 ) exhibits a rapidly increasing piezoelectric response with decreasing temperature below 50 K (the magnitude of its response around 1 K is comparable to that of the best materials at room temperature), this piezoelectric material is not yet competitive on market with the quartz crystal resonators at higher cryogenic temperature range. Although the Q- factor of quartz crystal resonators exhibits serious decline at temperature range of 73.15–293.15 K, they can still work in electronic circuits, thanks to their big nominal Q value. Moreover the AT-cut crystal resonators occupy about 70% of market today. Therefore the preparation of good model for this range seems to be well-founded. In our work the polynomials approximating functional temperature dependence of the model elements in a wide temperature range were presented. Next, applying these symbolic functions the behavioral model of AT-cut type quartz unit was created. In Section 2 the theoretical description of electrical model of the quartz resonator was described. Section 3 concerns the measuring set-up description and presents results of measurements. The method of the behavioral model creation is explained in Section 4. 2. Quartz unit electrical model Though the parasitic elements exist in many electrical models of quartz resonators [2], they can be eliminated by appropriate measurement methods in practically used frequency range. In practice, the quartz unit is modeled by the fundamental PSPICE electrical model [1,4,8]. Therefore, the low-temperature properties of this fundamental model (Fig. 1) were the subject of our

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Fig. 1. The quartz crystal unit electrical model.

Fig. 3. Physical principles of operation of continuous gas-flow type cryostat.

investigations. We need the electrical model of AT-cut type quartz crystal resonator 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. Moreover, Zk (T, ω) = Rk (T, ω) + jXk (T, ω) is the quartz resonator impedance, where ω = 2␲f, f – frequency. The resonant frequencies of series and parallel resonances are given by the well-known relationships: fs =

fr =

1

2



2

(1a)

Lk Ck 1



Lk Ck C0 /(Ck + C0 )

(1b)

Typical shape of the absolute value of resonator impedance as a function of frequency is shown in Fig. 2. The minimum corresponds to the series resonant frequency while maximum is related with the parallel resonant frequency. This fact will be exploited in further part of our work. Volume of the quartz plate and the angle of cut determine the thermal properties of the resonator [8]. We are interested in resonators made by so called AT cuts; they are made at about 35◦ of cut angle with respect to the optical axis. Their temperature dependence represents a cubic curve, the slope of which changes dramatically with slight variations in the cut angle.

Fig. 2. Absolute value of the resonator impedance versus frequency.

3. Low temperature measurements 3.1. The measuring setup Both the instrument set-up and measurement methods needed to collect the necessary data have been worked out. The cryostat system, which allowed us to perform characterization of electronic components and circuit in low-temperature conditions, was used [9]. This system exploits the continuous gas-flow type N2 /He cryostat working under Lab View computer program control. The principle of its operation is shown in Fig. 3. The cooling liquid might be nitrogen or helium. The source of the cooling liquid is Dewar vessel from which the liquid is transported to cryostat chamber through the siphon with siphon bulb. In chamber the liquid fumes are transported to heater, sample, flow meter and finally to the outlet of fumes. The resistance temperature sensor and PID regulator for temperature control were applied. The device under test (DUT), in this case quartz resonator, was placed inside the cryostat in the special holder and connected with instruments by the appropriate cables and connectors. The 6 MHz AT – cut crystal resonator (type HC-49U/HC-49T made by Fronter Electronics Co. [10]) was investigated. 3.2. The measurements 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 resulting from stray capacitance between the terminals and capacitance between the electrodes. This static capacitance was measured far from resonance at 100 kHz and 1 Vdc, using HP4263A LCR meter, in the measuring setup shown in Fig. 4. This instrument has 6 significant digits, which ensures 10−6 accuracy at each range. To minimize the measurement noises the result of measurement is calculated as average value of 64 measurements taken at one point. As we see, the cable capacitance CL is connected in parallel with C0 . The cable is placed in cryostat and its capacitance also depends on temperature. Therefore, we measured the temperature characteristic of the resonator with cable (a), the temperature characteristic of the cable capacitance itself (b) and then subtract it from the results obtained in (a). 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 fr resonances as a function of temperature. These measurements were performed in the measuring set-up shown in Fig. 5, which consists of HP33120A function generator (8 significant digits) and HP54503A oscilloscope (500 MHz). The resonant frequencies (see Fig. 2) were determined

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Fig. 4. Measuring setup for C0 measurement.

automatically using golden search algorithm. Such method allowed us to minimize error up to ±10 Hz, which corresponds to the relative error equal to ±10−5 . More details concerning the measuring set up one can find in work [9]. The following relationships for Ck (T) and Lk (T) are valid: Ck =

2(fL − fs ) (C0 + CL ) fs

(2a)

Lk =

1 82 fs (fL − fs )(C0 + CL )

(2b)

Fig. 6. delfs /fs and delfr /fr versus temperature in range of 73.15 K (−200 ◦ C) to 223.15 K (50 ◦ C).

Eqs. (2a) and (2b) are derived from the well-known relationship concerning the resonant frequency increment caused by load capacitance connected in parallel to the crystal [2,8]. The parallel fL resonance is measured including the cable capacitance CL . Having values of Ck (T) and Lk (T) the resonant frequency fr is calculated from Eq. (1b). Having voltages U1 and U2 measured in the measuring set-up, shown in Fig. 5, at series resonant frequency, the series resistance rk (T) was calculated from the expression [1]: rk = R1

U

1

U2



−1

(2c)

All measurements were performed automatically under Lab View program. Such measuring system allowed us to perform 64 measurements (or more) at one point and this way to improve the accuracy of measurements. 3.3. Results of measurements The measured resonator, like most contemporary produced crystal resonators, has partly temperature-compensated frequency characteristic in the narrow temperature range, which was achieved by appropriate cut angle (Fig. 6). This resonator is sufficiently well compensated at temperature range of 233.15–303.15 K (−40 ◦ C to +30 ◦ C), where the relative changes of series resonant frequency delfs /fs and parallel resonant frequency delfr /fr do not exceed

Fig. 5. The measuring set up for quartz resonator impedance measurements.

Fig. 7. delfr /fr versus temperature in range of 193.15 K (−80 ◦ C) to 303.15 K (+30 ◦ C).

30 ppm. But at wide temperature range this barrier is exceeded many times. Let us look at relative changes of parallel resonant frequencies (delfr /fr ) at temperature range of 193.15 K (−80 ◦ C) to 303.15 K (+30 ◦ C), which are exactly shown in Fig. 7. Investigating the shape of the parallel resonant characteristics precisely towards decreasing temperature, we can see, that it first is going slightly down, next a little bit up and finally strongly down. The measured Lk (T), Ck (T), C0 (T) and rk (T) temperature characteristics of the components of the electrical model are given in Figs. 8–11. As can be seen, the temperature dependence of the series capacitance and inductance have mutually opposite character, that somewhat compensates the resonant frequency change. The temperature dependence of series resistance is very small and almost linear. However, the parallel capacitance exhibits two slopes with decreasing temperature - middle slope at higher temperature and very small and almost constant slope at low temperature.

Fig. 8. Series inductance Lk (T) versus temperature.

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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 Fig. 9. Series capacitance Ck (T) versus temperature.

C0(T) [pF]

− 003

6.4

C0 (T ) = (((((211.886351286049e − 015T + 121.338832687354e

6.2

− 012)T + 23.5335077288204e − 009)T + 1.44478655729065e

6

− 006)T − 55.8950342670289e − 006)T − 10.0879506711649e

5.8 5.6

− 003)T + 5.60621283177

5.4 o -200 C (73.15 K)

o o -150 C (123.15 K) -100 C (173.15 K)

5.2 o o -50 C (223.15 K) 0 C (273.15 K)

50

Fig. 10. Parallel capacitance C0 (T) versus temperature.

rk (T ) = −0.6164T + 71.003

(3c)

(3d)

in ◦ C

Fig. 11. Series resistance rk (T) versus temperature.

3.4. Polynomial approximations Each curve Lk (T), Ck (T), C0 (T) and rk (T) can be fitted by some polynomial of degree depending on demanded accuracy of approximation. Evidently, for the same resonant frequency determination accuracy, different curves have different orders. In aim of realization of this approximation task, the computer program in Matlab was written. The selection of polynomial was made applying this Matlab program, which was used in the following way: - first, orders for all polynomials were established at maximal possible values (in this case 16) - next, polynomial orders were decreased one by one (but not lower than 1), calculating at each step one of the polynomials rk (T), C0 (T), Ck (T), and Lk (T), using Polyfit Matlab library function. This process was repeated until the differences between calculated and measured resonant frequencies were greater than expected values (0.02%). Appropriate orders of polynomials ensure good accuracy of approximation. As the result of these calculations we received the following approximations, which fulfill the accuracy demanded: 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

where T is given and Lk (T) is calculated 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: ıLk = 13.6500214182246e −003% (∼13.65e − 003%), (∼37.25e − 003%), ıCk = 37.2520922624475e − 003% ıC0 = 201.575233818110e − 003% (∼201.57e − 003%), ırk = 1e − 003%. The demanded high accuracies of approximations needed sufficiently high orders of polynomials: 12, 7, 6, 1. Moreover, basing on these polynomials we are able to determine the universal curves as relative polynomials: Lk r (T) = Lk (T)/Lk0 , Ck r (T) = Ck (T)/Ck0 , C0 r (T) = C0 (T)/C00 , rk r (T) = rk (T)/rk0 . Having the element values measured at 293.15 K (+20 ◦ C) (for example Lk0 = 0.013287659 H, Ck0 = 52.98521e − 15 F, C00 = 5.4 pF, and rk0 = 58.7 ) we can calculate their values at whole temperature range. These universal curves can be applied to all specimens of the same type of resonators. 4. The SPICE behavioral model The elements of quartz crystal resonators used in standard SPICE models library offer only maximally quadratic temperature coefficients. It is possible to overcome this limitation in PSPICE version 8 and higher, applying the Analog Behavioral Modeling (ABM) approach. Behavioral Modeling is the process of developing a model for a device or system component from the viewpoint of externally observed behavior rather than from a microscopic description [11]. PSPICE extensions allow arbitrary equations and/or table lookup. The ABM sources allow us accessing the global variable temperature TEMP (variable TEMP should be substituted instead of T in derived expressions (3)). Having in disposal the functional symbolic description (3) 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. 12 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 applied [12], in which the Evalue and Gvalue sources are described by equations:

− 006)T − 117.737531061562e − 006)T + 22.9612714058108e − 003)T + 12.8870352956

(3b)

(3a)

Vsec =

Vin and Isec = Iin K

(4)

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Fig. 12. Behavioral PSPICE temperature- related model of the quartz crystal resonator working in low temperature.

Fig. 13. The absolute value of the input impedance of quartz crystal resonator model versus frequency; temperature as parameter.

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Fig. 14. The input reactance of quartz crystal resonator model versus frequency; temperature as parameter.

Fig. 15. The minima of absolute value of the input impedance of quartz crystal resonator model versus frequency; temperature as parameter.

The basic idea of the proposed inductor modeling is to reflect the behavior of a linear reference inductor (L1) via nonlinear “transformer” to the input port (Lin ). It is realized by using the depended sources, mentioned above (E1, G1). The coefficient of reactance transformation is Xin L = in XL1 L1

(5)

Relative changes of Quality factor

relative change Q(T) in [%]

KL =

The resistances R1, R8 are inserted to avoid the floating point errors. The Lk (T) function is replaced by the expression (3a), where variable T must be substituted by global temperature variable TEMP. 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. 12 the elements: G2, E2, C1 and V Isec2 constitute this model. The resistance

0 -10 -20 -30 -40 -50 -60 -180

-160

-140

-120

-100

-80

-60

-40

-20

0 20 Temperature [Celsius]

Fig. 16. The relative changes of the quality factor in regard of this at 293.15 K (20 ◦ C).

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Table 1 Accuracy of series and parallel resonant frequency realizations by the quartz crystal behavioral model. Temperature

Calculated

Measured

Relative error

Calculated

Measured

Relative error

fs calc [Hz]

fs [Hz]

|fs |/fs [%]

fr calc [Hz]

fr [Hz]

|fr |/fr [%]

5994125.0 5995054.0 5995968.0 5996328.0 5996660.0 5996897.0 5997233.0 5997603.0 5997889.0 5998096.0 5998159.0 5998163.0 5998173.0

0.0001334 0.0001000 0.0001167 0.0004836 0.0006336 0.0006169 0.0030013 0.0033345 0.0001500 0.0061852 0.0125205 0.0191058 0.0036010 ∼0. 00010% ∼0.019%

6008440.0 6008415.6 6010600.0 6010616.1 6014440.0 6014372.2 6016543.0 6016476.3 6018597.0 6018570.0 6020573.0 6020593.1 6022300.0 6022408.0 6025481.0 6025264.5 6026851.0 6026880.0 6027161.0 6027532.5 6028246.0 6027443.6 6026155.0 6027435.1 6027316.0 6027522.2 Minimal relative error Maximal relative error

T [K]

T [◦ C]

93.15 113.15 133.15 143.15 153.15 163.15 173.15 193.15 213.15 233.15 253.15 273.15 293.15

−180 5994117.0 −160 5995060.0 −140 5995975.0 −130 5996357.0 −120 5996622.0 −110 5996860.0 −100 5997053.0 −80 5997803.0 −60 5997880.0 −40 5997725.0 −20 5998910.0 0 5997017.0 20 5997957.0 Minimal relative error Maximal relative error

R9 is inserted to avoid the floating point error. The capacitance transformation coefficient KC is KC =

Xin C1 = XC1 Cin

(6)

The Ck (T) function is replaced by the expression (3b), where variable T must be substituted by global temperature variable TEMP. For the parallel capacitor C0 (T), temperature behavioral model is obtained in similar way like for series capacitor. Another appropriate function (3c) for C0 (T) was applied in KC expression. In Fig. 12 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) [13]. In such configuration the current in expression describing this source can be written as I=

Vin rk (T )

(7)

In Fig. 12 the source G4 belongs only to this model. The resistor R10 is inserted to avoid the floating point error. First, all these partial models were simulated separately, using PSPICE computer program and results of these simulations confirmed their proper characteristics. Next, the quartz crystal resonator temperature behavioral model shown in Fig. 12 was obtained as a composition of these partial models. This model was simulated using PSPICE computer program. As result of these simulations, the absolute value of its input impedance and input reactance with temperature as parameters, are shown in Figs. 13–15. These simulations confirmed temperature properties of quartz resonator measured in cryogenic temperature range. When temperature is moving down from 293.15 K to 73.15 K (+20 ◦ C to −200 ◦ C), first we observe slight shift the resonant frequencies down, next little bit up and finally constantly down. The model characteristics shown in Figs. 13–15, confirm strictly those theoretically anticipated (Fig. 2). The maxima of the absolute value of input impedance are situated at parallel resonant frequencies, while the minima at series resonant frequencies. The resonance damping is caused mainly by temperature-dependent resistance rk , which is slightly decreasing function (Fig. 11). This element has main influence on quality factor at low temperatures and can significantly damp amplitude of oscillations of oscillators working in these circumstances. The plot of relative changes of the quality factor is shown in Fig. 16. As we see, the quality factor at considered temperature range can be decreased up to 50% in regard of this at 293.15 K (20 ◦ C).

0.0004060 0.0002678 0.0011272 0.0011086 0.0004486 0.0003338 0.0017933 0.0035932 0.0004811 0.0061633 0.0133124 0.0212378 0.0034209 ∼0.00026% ∼0.021%

Comparing these values with measured ones it can be stated, that the presented behavioral model has very high accuracy. The relative errors calculated at each measured temperature are presented in Table 1. As we see, the resonant frequencies of this model can be determined with high accuracy, which ranges between 0.0001% and 0.019% for series resonant frequency, and between 0.00026% and 0.021% for parallel resonant frequency. Such good result was achieved thanks to very accurate polynomial approximations.

5. Conclusions In this paper the measurement methods as well as the measuring setup for quartz crystal unit measurements at cryogenic temperature have been described. Applying these arrangements the fundamental AT-cut quartz resonator characteristics have been measured. Basing on obtained results, the new behavioral PSPICE model has been developed, which appears to be very accurate. It can be stated that accuracy of this model is between 0.0001% and 0.022%. It should be remarked that very high accuracies of polynomial approximations are necessary to achieve such good result. The method delivered in this paper can be directly extended to modeling of other type quartz crystal resonators working in cryogenic temperature range, too. The created model can be used during designing of electronic circuits working in cryogenic temperature range.

References [1] V.E. Bottom, Introduction to Quartz Crystal Unit Design, Van Nostrand Co, 1982. [2] W.G. Androsova, W.N. Bankov, et al. Spravotchnik po kwarcevym rezonatoram, Ed. P.G. Pozdniakov, Moscow 1978 (in Russian). [3] J.A. Kosinski, J.G. Gualtieri, A. Ballato, Proc. 45th Annual Symposium on Frequency Control, 29–31 May, 1991, pp. 22–28. [4] MicroSim PSpice A/D, Circuit Analysis Reference Manual, MicroSim Co., 1995. [5] L.J. Giacoletto (Ed.), Electronics Designers’ Handbook, Second ed., McGraw-Hill Co., 1977. [6] D. Goehrig, J. Haffelder, Electron. Des. (November) (1998) 106–108. [7] D.E. Grupp, A.M. Goldman, Science 276 (April) (1997) 392–394. [8] D.R. Abel, The Sentry Technology Manual, Sentry Manufacturing Co, 1977. [9] F. Balik, W. Sommer, Environment for Automated Low-temperature Measurements of Electronic Circuits, vol. 52, Elektronika, Poland, 2011, pp. 84–89, No. 3. [10] Fronter Electronics Data Sheet, http://www.chinafronter.com/product440.html. [11] B. Hirasuna, Analog behavioral modeling using PSPICE, ORCAD Application note, Ed. D. Busdeicker, 2000. [12] S. Ben-Yaakov, M. Peretz, Simulation Bits: A SPICE Behavioral Model of NonLinear Inductors, IEEE Power Electronics Society NEWSLETTER, (Fourth Quarter 2003), pp. 9–10. [13] S. Hamilton, An Analog Electronics Companion, Cambridge Univ. Press, Cambridge, 2003.