THE SIGMA-DELTA MODULATOR FOR MEASUREMENT OF THE THERMAL CHARACTERISTICS OF THE CAPACITORS

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1 MEASUREMENT SCIENCE REVIEW, Volume 2, Section 3, 22 THE SIGMA-DELTA MODULATOR FOR MEASUREMENT OF THE THERMAL CHARACTERISTICS OF THE CAPACITORS Martin Kollár Department of Electronics and Multimedia Telecommunications, Tecnical University of Košice, Park Komenskéo 13, 41 Košice, Slovakia Abstract. Te paper presents a simple and successful design of te sigma-delta modulator wit flip-flop sensor for measurement of te termal caracteristics of te capacitors. Te teoretical considerations are verified by a laboratory experiment. 1 Introduction A capacitance of te real capacitors is a function of te temperature and oter influences. In Fig.1 can be seen a termal caracteristic of a capacitor. C/C T [ o C] Figure 1: A termal caracteristic of a capacitor Te cange of a capacitance, if we assume an influence of te temperature only, can be calculated by formula [6]: ( T ) C T C = α (1) were α(t ) is a termal coefficient and C is a capacitance in te point T, T =T -T is a termal cange and C is a capacitive cange. Te termal coefficient α(t ) in equation (1) can be calculated by using formula [6]: dc α ( T ) = in te point T (2) C dt were (dc/dt ) is a capacitive derivative to temperature and C is capacitance in te point T. Te termal coefficient α is non-linear function of te temperature T in practice [6]. 1

2 Measurement of Pysical Quantities M. Kollár 2 Sigma-delta modulator wit flip-flop sensor Te key element of te sigma-delta modulator is flip-flop sensor [1]. Te circuit in Fig.2 was introduced in reference [1] as a flip-flop sensor. i R1 K i R2 U N R 1 i 2 i 1 R 2 β 2 β 1 i(t) C 1 i 1 i 2 C 2 u 1 u 2 T 1 T 2 Z Figure 2: Flip-flop sensor Te flip-flop sensor is part of a class of silicon sensors wit a digital output. A standard flipflop consisting of two transistors and two resistors (see Fig.2) is caracterized wit two stable states. One of te autors of te patent flip-flop sensor was Lian [1] wo sowed tat flip-flop sensor can be used for measurement of non-electrical quantity and derived formula for calculation of equivalent voltage of te flip-flop sensor controlled by slow-rise control pulse. Te principle of measurement is based on tis tat measured non-electrical quantity will break te value symmetry of te inverters relative to te morpological symmetry axis passing troug points K and Z (see Fig.2). However it can be compensated by a voltage U N =U NE in suc way tat by repeated connection to a source i(t) te 5% state [1] is restored, so tat te magnitude of te measured non-electrical quantity will be reflected into te voltage U NE, wic we will call te equivalent voltage. If needed, owever, it is not necessary to stick to te custom of using sensor elements in Fig.2. It sould be noted tat in current control we also distinguis between te pulses wit a vertical or slow-rise segment of te control pulse (see Fig.3). i(t) I m δ 1 T/2 δ 2 t Figure 3: Current control pulse Te control wit a vertical-rise segment of te control pulse is caracterized by te ratio I m /δ 1 being suc tat te currents passing troug te capacitors are not negligible compared to te transistors currents of te flip-flop sensors. Te notion negligible sould be understood in its relative sense. In practice te condition is satisfied if δ 1,δ 2 <<R 1 C 1 and δ 1,δ 2 <<R 2 C 2 at te same time. In te case of control wit te vertical rise segments of te control pulse te unequal values of capacitances C 1, C 2 will break te value symmetry of te inverters of te flip-flop sensor but it can be compensated by voltage U N =U NE [2]. Final formula for te equivalent voltage as te form [2]: 2

3 MEASUREMENT SCIENCE REVIEW, Volume 2, Section 3, 22 U R C = (3) 2C NE I m were I m is an amplitude of te current control pulse (see Fig.3), C 1 =C+ C, R=R 1 =R 2 and C 2 =C. Te flip-flop sensor wit feedback is sown in Fig.4a. i R1 i R2 u N R K R K R source of eat R 1 R 2 u 1 i 1 i 2 T 1 T 2 C 2 i(t) u 2 R i R i + C i C i u C`1 C 1 Figure 4a: Flip-flop sensor wit feedback R 1 and R 2 are te load resistors of te flip-flop and usually range from a few kω to tens of kω. R k is small resistor its value is normally two orders of magnitude smaller tan R 1 and R 2. Te voltage u is attenuated by te ratio R /R k (R >>R k ) and is fed to flip-flop sensor. P(t) R T1 R T C T1 T C`1 + flip-flop sensor S/H bitstream termal part C 2` U electrical part ±U m Figure 4b: Equivalent block diagram By adjusting u, te asymmetry due to components in te flip-flop can be compensated, tus bringing te flip-flop sensor into 5% state [1]. Te two outputs of te flip-flop are connected to an integrator. C`1 represents a measured capacitor. Equivalent block diagram is sown in Fig.4b, were on te left side can be seen an equivalent termal diagram. In Fig.4b R T is a termal resistance of te environment, R T1 is a termal resistance and C T1 is a termal capacitance of te capacitor C`1 [5]. In our case was as te source of eat used a resistor R T (see Fig.5). 3

4 Measurement of Pysical Quantities M. Kollár Heat, wic flows between R T and C`1. u AM (t) Amplitudemodulated signal TR Ω R T C`1 TK pf Flip-flop sensor wit feedback output Figure 5: Resistor R T as te source of eat In Fig.5 can be seen, tat te source of te amplitude-modulated signal u AM is connected to te resistor R T on wic arises an electrical power P(t) as function of te time. If we assume R T <<(R T1 C T ), so ten te equivalent termal diagram from te Fig.4b as te form sown in Fig.6. ~ P(t)R T R T1 C T1 T C`1=α(T ) T C`1 Figure 6: Equivalent termal circuit Te termal circuit in Fig.6 can be solved by using analogy wit te solving of te electrical circuits [5] and for te termal cange T we ave: and for ωr T1 C T1 >>1 it follows ( jω ) ( jω ) P R T = (4) ( j ) ( ω ) ( ) ( ) ( ) T 1+ jωrt 1CT1 P ( jω ) jωt jωt were P j = P t e dt and T jω = T t e dt. R T T ω = (5) jωrt1ct1 Te final capacitive cange can be derived troug equations (1), (5). Te result is α ( T ) P( jω ) R C ` ` T 1 C1( jω ) = (6) jωrt 1CT1 Te final block diagram of te sigma-delta modulator wit flip-flop sensor is ten sown in Fig.7 [3], were C 2 represents a cange of te capacitance C 2 of te flip-flop sensor [3]. 4

5 MEASUREMENT SCIENCE REVIEW, Volume 2, Section 3, 22 P(t) α(t ) R T C`1 P C bitstream digital S/H filter output (R T1 C T1 ) -1 termal part (R i C i ) -1 C 2` U -R K 2C (R +R K )RI m ±U m electrical part Figure 7: Final block diagram of te sigma-delta modulator wit flip-flop sensor Te sigma-delta modulator wit flip-flop sensor is in more detail described in reference [3]. 3 Principle of te metod Te principle of te measurement of te termal caracteristic of te given capacitor is based on te measurement of te capacitive cange as te function of te temperature. Te termal cange can be acieved by using te resistor R T as source of eat (see Fig.5). In our case was source of te amplitude-modulated signal connected to te resistor R T. If we assume te modulation signal wit square sape so ten te electrical power, wic arises on te resistor R T, as te square sape too [3]. From te equation (5) ten it follows a triangular sape of te termal cange T as a function of te time [5]. T C`1max T 2 T max T 1 T C`1 C`1 t P(t Figure 8: Principle of te measurement of te termal caracteristic of a capacitor Te principle of te measurement of te termal caracteristic of a capacitor is obvious from te Fig.8. 4 Experiments Proposed metod was tested on te capacitor, wic catalog termal caracteristic is sown in Fig.9 and its value was 2.1 pf. 5

6 Measurement of Pysical Quantities M. Kollár C/C T [ o C] Figure 9: Catalog termal caracteristic of te capacitor As it was described above te source of te amplitude-modulated signal was connected to te resistor R T (see Fig.5). Te amplitude-modulated signal can be described by formula: u () t U [ 1+ mx() t ] sin( πf t = ) AM N 2 N (7) were U N is an amplitude of te carrier signal, m is an index of te amplitude modulation, f N is a value of te carrier frequency and x(t) is a modulation signal in our case a periodical signal wit te square sape. In te case of our experiment U N =3.5 V, m=.3, f N =35 khz. Te experimental test circuit was realized by Fig.4a, so tat te values of te parameters were set as follow: R=R 1 =R 2 =6.8 kω, R k =1 Ω, R =1.8 kω, C i =1 nf, R i =1 kω and C =387 pf. Te flip-flop sensor was controlled by a current pulse according to Fig.3, wile δ 1 =δ 2 =1 ns, I m =1.17 ma and T=7 µs. Te termal caracteristic of a capacitor can be measured by using two metods. First metod is based on te measurement of te voltage u (see Fig.4a) as a function of te time and te capacitive cange C`1 can be calculated by formula: C ` 1 = u R K 2C ( R + RK ) Ri I m i (8) because u N =- u R K /(R +R K ). Te measured voltage u as function of te time for te capacitor, wic catalog termal caracteristic is sown in Fig.9, can be seen in Fig.1a and in more detail is sown in Fig.1b. Te final capacitive cange C`1 it is ten possible to calculate from te equation (8). u AM (t) u (t) u omax a) Figure 1: Measured voltage u as te function of te time b) 6

7 MEASUREMENT SCIENCE REVIEW, Volume 2, Section 3, 22 It is clear tat te voltage u is te function of te time, but from te equation (5) it follows tat for ωr T1 C T1 >>1te termal cange T is given by equation: t RT T () t = P( τ ) dτ (9) R C T1 T1 From te equation (9) it is obvious tat te time axis x can represent te termal axis, but te termal parameters R T, R T1, C T1 must be known..12 C`1/C` (resolution) T [ o C] Figure 11: Final termal caracteristic of te capacitor Te principle of te second metod is based on te processing of te bitstream from te output of te flip-flop sensor (see Fig.4b). Te final termal caracteristic given capacitor measured by second metod is sown in Fig.11. A reverse counter was used in tis case as digital filter, so tat its output information was processed in MATLAB. In Fig.11 can be seen te final termal caracteristic as function of te temperature T, so tat te test circuit was calibrated for given type of te resistor R T and measured capacitor C`1. Quantization error in Fig.11 can be reduced by iger sampling frequency f=1/t (see Fig.3). For te verification of te teoretical considerations was made an experimental circuit by SMT (see Fig.12). C`1 flip-flop sensor Figure 12: A potograpy of te experimental circuit Some important parameters of te experimental circuit are sown in Tab.1. Parameters Supply voltage Power consumption Frequency of te control pulses Resolution (see Fig.11) Values ±9 V 75 mw 14.3 khz.1 Table 1: Some important experimental parameters 7

8 Measurement of Pysical Quantities M. Kollár 5 Conclusions A new metod for te measurement of te termal caracteristics of te capacitors as been presented, wic is based on using of te flip-flop sensor in te structure of te sigma-delta modulator. Te main property of tis system is its ability to measure te termal caracteristics of te capacitors in range a few pf. Te validity of te teoretical considerations was proved by laboratory experiments. It is true tat te proposed system will be ard to calibrate as a board-level product, owever, it may well prove an interesting tecnique to assist wit process caracterisation test-masks, were suc a tecnique could be used on-cip, and were te limitations will be muc less of an issue. Acknowledgments I would like to tank for stimulation and advice Prof. Ing. Viktor Špány, DrSc., wit wom I ave ad many interesting discussions. Te work presented in tis paper was supported by Grant of Ministry of Education and Academy of Science of Slovak republic VEGA under Grant No.1/93/2. References [1] LIAN, W. Integrated silicon flip-flop sensor. Doctoral Tesis, Delft: Tecnise Universitet Delft, 199, 14 p. [2] KOLLÁR, M., ŠPÁNY, V., GABAŠ, T. Autocompensative system for measurement of te capacitances. Radioengineering. 22, Vol. 11, No. 2, pp [3] KOLLÁR, M. New model of te sigma-delta modulator wit flip-flop sensor. Slaboproudy obzor, Prague, 21, Vol. 58, No. 4, pp.1-4. [4] MICHAELI, L. Modeling of te analog-digital interfaces. Mercury-Smékal Press, 21, 168 p. (In Slovak). [5] LEVICKÝ, D., MICHAELI, L., ŠPÁNY, V., PIVKA, L., KALAKAJ, P. Autocompensative systems wit flip-flop sensor. Proceeding of IMEKO 1 t Int.Symposium on Development in Digital Measuring Instrumentation, Vol.II., Napoli 1996, pp [6] GALAJDA, P., LUKÁČ, R. Electrical devices. Mercury-Smékal Press, 21, 25 p. (In Slovak). 8

Modified Sigma-Delta Converter and Flip-Flop Circuits Used for Capacitance Measuring

Modified Sigma-Delta Converter and Flip-Flop Circuits Used for Capacitance Measuring MILAN STORK Department of Applied Electronics and Telecommunications University of West Bohemia P.O. Box 314, 30614