Integrated silicon flip-flop sensors

INTEGRATED SILICON

FLIP-FLOP SENSORS

Geïntegreerde silicium flip-flop sensoren

Proefschrift

ter verkrijging van de graad van doctor aan

jieJje.chnische-U-ni-ver-si-tei-t-Del-ft—op-gezag ~vairde

P.A. Schenck, in het openbaar

Rector Magnificus prof.drs.

te verdedigen ten overstaan van een commissie aangewezen door

het College van Dekanen, op donderdag 27 april 1989, te 16.00 uur

door

Weijian Lian

elektrotechnisch ingenieur

geboren te Kanton, China

TR ólss

1717

%

ft

* •

1. INTRODUCTION 1

1.1 SILICON SENSORS 3 1.2 SMART SENSORS 5 1.3 FLIP-FLOP SENSORS 7

2. THEORY OF THE FLIP-FLOP SENSOR

TECHNIQUE is

2.1 INTRODUCTION 13

2.2 PRINCIPLES OF THE "FLIP-FLOP SENSOR" U 2.2.1 Inpjit/mitpjüLLcharacteristics—15 — ~2X2 ~ISrror analysis 17

2.2.3 Offset 22

2.2.4 Literature research 22 2.2.5 Discussions 23

2.3 ELECTRICAL REALIZATION OF THE FLIP-FLOP SENSOR 24

2.3.1 Realization of the unstable states 24 2.3.2 Flip-flop circuitry 26

2.3.3 Flip—flop sensing elements 28 2.3.4 Measurement setup 28

2.4 CALCULATION OF THE FLIP-FLOP SENSOR CHARACTERISTICS 30

2.4.1 Modeling of the effects of current sources, voltage sources and mismatches of resistors and

transistors 31

2.4.2 Thermal behavior of the flip-flop sensor 37 2.4.3 Limitations of the DC model 38

2.4.4 Noise of the switched flip-flop 38 2.5 TRIANGLE WAVE AND FEEDBACK

TECHNIQUES 50

2.5.1 Triangle wave technique 50 2.5.2 Feedback technique 52 2.6 CONCLUSIONS 52

3.1 INTRODUCTION 55

3.2 FABRICATION TECHNOLOGY 57 3.3 OPTICAL FLIP-FLOP SENSORS 60

3.3.1 Phototransistor flip—flop sensors 60 3.3.2 Bipolar photodiode flip—flop sensors 67 3.3.3 NMOS photodiode flip-flop sensors 71 3.4 THERMAL FLIP-FLOP SENSORS 75

3.4.1 Transistor thermal flip-flop sensors 76 3.4.2 Thermocouple flip—flop sensors 78 3.5 MAGNETIC FLIP-FLOP SENSORS 81 3.5.1 Hall plate flip-flop sensors 81

3.5.2 NMOS split—drain—transistor magnetic flip—flop sensors 84

3.6 MECHANICAL FLIP-FLOP SENSORS 85

3.6.1 Bipolar piezoresistor flip—flop sensors 86 3.6.2 NMOS piezoresistor flip-flop sensors 88 3.7 CONCLUSIONS 91

4. FLIP-FLOP SENSOR ARRAY 95

4.1 INTRODUCTION 95

4.2 CIRCUITRY AND OPERATION OF THE ARRAY 96 4.2.1 Flip—flop sensors 96

4.2.2 Sense amplifiers 97

4.2.3 Selection and bias circuitry 101 4.2.4 Flip—flop sensor array 103 4.3 EXPERIMENTAL RESULTS 106 4.3.1 Measurement accuracy 106

4.3.2 Bipolar optical flip—flop sensor array 110 4.4 DISCUSSIONS 113

SUMMARY 117 SAMENVATTING 121 ACKNOWLEDGEMENT 125 BIOGRAPHY 127

1. INTRODUCTION

Information is becoming the -key word in many issues. Our work and our life are increasingly dependent on information. It is not surprising that many people are talking about the "information society".

In fact, the human society has always been an information-oriented society. Communication is an essential human characteristic. The ability to handle information effectively was the key which led to the great success of human beings — besides the biological evolution, cultural development is thus possible.

The amount of information to be handled has significantly increased over the past decade. The transformation of information can no longer be accomplished through mechanical means. Electronics, which provide us with, among other things, computers, is the future means of information processing.

Processes in which the transfer of information is involved can be divided into three stages [1.1]. Information must originally be obtained

suitable for stage two. Second, the observed information must be processed. Computers have excellent performance records in information processing when a large amount of numbers is involved. Finally, the results of the first two stages must be presented to human beings or machines.

The subject of this thesis belongs to stage one, namely information observation. To be more specific, it belongs to the research area of transducers. Transducers transform input energy signals into electrical energy signals. Information can be carried by different forms of energy including electromagnetic—radiant energy, gravitational energy, mechanical energy, thermal energy, electrostatic and electromagnetic energy, molecular energy, atomic energy, nuclear energy and mass energy. Information carried by these energies — signals — first has to be transformed into electrical signals in order to be further processed in computer systems. The electrical signal, which has been chosen to be that signal, to which all other signals are converted, is not a human invention. Biological evolution has long made use of the convenience in which electrical signals can be handled. As a result, our eyes, ears, noses, tongues and hands all convert signals into electrical pulses.

The word "transducer" is given to those elements which perform the signal transformation. There are input transducers and output transducers (Fig. 1.1). Input transducers are used for information observation. They transform nonelectrical signals into electrical ones. Output transducers are employed to present the final results to human beings and they transform electrical signals into nonelectrical signals such as light and sound. A widely used word for the concept of input transducer is sensor.

input

transducer modifier

output transducer

Fig. 1.1 Block diagram illustrating an information transfer process by the three units; input transducer, modifier and output transducer.

The functions of sensors can be illustrated as shown in Fig. 1.2. A

sensor is a device, which converts signals from the radiant, mechanical,

thermal, magnetic or chemical field into electrical signals. This signal

conversion is made possible by virtue of the physical characteristics of

certain materials. One example of such a material is quartz crystal, which

produces a voltage when subjected to a mechanical strain. Thus, in this

example a mechanical signal is converted into a voltage, which is an

electrical signal. The materials that are widely used for sensor applications

include polymers, metal oxides, thick- and thin-film materials,

piezoelectric materials, III-V and II-VI semiconductors and silicon.

Fig. 1.2 Diagram indicating the five possible signal conversions in input

transducers.

from sensor research laboratories all over the world. The reason for this originates from the microelectronic revolution, which is also based on silicon. Because of the enormous research efforts spent on the development of the electronic industry, one has gained a great degree of expertise in silicon process technology. No other sensor material has yet obtained such a large investment in capital and time, and not one is likely to obtain it in the next ten years (although there are indications that some III—V semiconductors are more suitable materials for certain sensor applications than is silicon).

The following advancements made in the silicon microelectronics industry have also promoted silicon sensor technology. Developments made in the microelectronics field include:

— A highly developed fabrication technology with which it is possible to integrate devices with physical sizes on the order of micrometers. The use of this technology allows the integration of tens of millions of devices in an area as small as one square centimeter. Many process steps such as lithography, diffusion, ion implantation and etching, which were developed for integrated circuit fabrication, can be used for silicon sensor fabrication. The integrated circuit process technology has not only created many technical possibilities for sensor fabrication, but has also allowed silicon sensors to be fabricated at a low cost.

— The integration of sensors and electronic circuitry on the same chip, which creates possibilities of: improvement of the signal/noise ratio by amplifying the signal before transmission; temperature compensation by integrating a thermal sensor close to the sensor under design in order to generate a signal counteracting the thermal dependence of the sensor; impedance matching through on—chip impedance transformation; on-chip analog/digital conversion, which yields digital signals immune to disturbances; and construction of sensor arrays consisting of large numbers of sensors connected and multiplexed through electronic circuitry. Sensors incorporating these functions are called smart sensors.

These advancements, which are the results of heavy investments in research and development, justify the current broad interest in silicon sensors. Moreover, the material characteristics of silicon are highly suitable for sensor application. A large number of physical and chemical effects, which transform nonelectrical signals into electrical ones, have been identified in silicon [1.2]. These effects form the underlying principles

of sensor operations. In fact, for each of the five signal domains shown in Fig. 1.2, silicon possesses more than one effect, which can be employed for sensor application. As a result, silicon sensors can be used in most situations.

1.2 SMART SENSORS

Within the large field of silicon sensors, the subject smart sensors has attracted great attention. There is confusion in the sensor research literature as to what extent electronics should be incorporated into a sensor design in order for the device to be called a smart sensor. In theory it is possible to integrate highly complex data—processing and control functions together with the sensors. In practice such an integration is often not feasible on the device level. Practically, it can only be realized on the system level; using separate instrumentation units. This thesis focuses on sensor devices; sensor systems are not within the scope of this ——work—Therefore^in-this^casej-the^term^smart^sensor^LiS-given^to^hose.

sensor devices that incorporate basic electronic circuits.

Inside the large smart sensor family, there is a class of sensors, which has the distinguishing characteristic of yielding a digital output. As most signal processing is carried out in the digital form, signals from sensors have to undergo analog/digital conversion. On-chip analog/digital (A/D) conversion results in the following advantages [1.3, 1.4]:

— A digital output does not need a precise electrical match in the signal magnitude between the sensor and the data—processing unit. Particularly, in cases where the electrical side of the sensor is dimensionless, as in the case of measuring ratios, this is the most accurate way to convert a ratio directly into a digital number. On the other hand, all necessary electrical references must be generated on-chip.

— Digital signals are relatively immune to disturbances. On-chip A/D conversion eliminates interference and noise in the transmission lines.

— Once a digital signal is present on the chip, it can be digitally modified on—chip for purposes of calibration, linearization, multi—variable calculation, signal averaging, signal recognition and decision making.

Direct A/D converters can be integrated into the sensor design. However, because of the complexity, the yield of the fabrication would be

therefore often available in alternative forms, such as frequency or pulse count outputs. The digital outputs can be obtained either by employing an intrinsic sensing principle, such as in the case of a quartz Crystal oscillator, or by implementing electronic circuitry, such as a voltage controlled oscillator, within the sensor design.

Digital sensors can be categorized into the following types:

— Resonator structures consisting of a piezoelectric material, such as quartz crystal or ZnO. The oscillating frequency is directly sensitive to physical and indirectly sensitive to chemical parameters. These effects are employed for sensor construction. Temperature sensors, pressure sensors and gas sensors have been realized using resonating structures [1.5, 1.6].

— Sensors based on periodic geometrical structures. A disc containing a grid or a gray code is used in combination with a light source and a silicon photodetector, for the measurement of lateral displacement and angle changes [1.7].

— Current or voltage—to—frequency converters. Electronic oscillators can be designed in such a way that their oscillating frequency depends on a current or a voltage [1.8]. Frequency output can thus be realized by using sensors that have a current or a voltage as an output to control the current or voltage of the oscillator [1.9].

— Resistance and capacitance—to—frequency converters. Oscillators have been designed in which the oscillating frequency depends on certain resistance and capacitance values. Sensors that have outputs, which take the form of changes in resistance or capacitance values, such as piezoresistors and capacitor sensors, can be incorporated within the oscillator designs to produce frequency outputs [1.10, 1.11].

The above-mentioned digital sensors are representative for the field. Some of these sensors are already being used in measurement systems. However, the majority is still in the research stage, being investigated for optimal circuit design, for compatible processing and for their merits and deficits.

1.3 FLIP-FLOP SENSORS

The flip-flop sensor is part of a class of silicon sensors with a digital output. The idea originated from the magnetic research area. A magnetic dipole in a material with a uniaxial anisotropic characteristic has two stable states: it aligns itself along one of the two directions of the axis. A magnetic sensor could be realized by setting the dipole in a position perpendicular to the axis, so that the dipole could be set free. The dipole would then go to one of the two stable states. Which one of the two stable states would be the final stable state of the dipole, depends on any perturbations present, such as a magnetic field. By designing the dipole system in such a way that its final stable state is sensitive to the magnetic field, and by detecting the final stable state of the dipole, a magnetic field sensor could be constructed.

This idea is based on the deliverance of systems with stable states into an unstable state. It can be applied to any system or device with

-stable-states.-The-electronic-flip=flop-is-sueh-a--system,—A—flip—flop—is-depicted in Fig. 1.3. It has two stable states, namely the "one" and the "zero" state. The flip—flop can be brought into an unstable state by switching off the bias current, and, after a short time, switching the bias current back on. During the switch-on period, the flip-flop is in the unstable state. Once it is on, it will decide either to switch to a "one" or to a "zero". The decision of switching over to a "one" or to a "zero" can be influenced by external parameters. For instance, the transistors in the flip—flop can be designed to be sensitive to light. Then, when illuminating only one of the two transistors of the flip—flop, an asymmetry will be created in the flip-flop. The flip—flop will thus switch to a given stable state.

The process in which a flip-flop is brought into an unstable state, with the subsequent switching to a stable state, is repeated periodically with a high frequency. The number of the resulting "ones" is then a measure of the parameter.

Such a system, which employs a flip—flop that incorporates sensitive elements, and which is brought into an unstable state many times, is called the flip—flop sensor and is the subject of this thesis.

The idea of using unstable states to perform the sense action for weak electrical signals was published by some authors [1.12, 1.13]. The flip-flop sensors presented here are based on integrating sensing elements,

which are all designed to measure nonelectrical signals, within the

flip—flop sensors themselves.

I

1

1

c

T

b

Fig. 1.3 Circuit diagram oj'a flip-flop.

The flip-flop sensor has the following features:

— A digital output is realized. The output of the flip—flop sensor is

a series of "ones" and "zeros". The number of "ones" can be counted. This

number is a digital representation of the measurand.

— A low threshold-sensitivity, which is defined as the smallest

quantity that can be detected [1.14], is obtained. The flip-flop sensor

performs measurements under the system noise level. Signals below the

noise level can be measured. However, there are other measurement

techniques which yield even lower threshold-sensitivities than the

flip-flop sensor technique. One example of such a measurement technique

includes the use of a sensor, an amplifier, which is designed for optimum

signal/noise ratio, and a comparator. However, such a measurement

technique is more difficult to realize in the form of a smart sensor.

- Simplicity is achieved. As shown in Fig. 1.3, a flip—flop is a very simple circuit. It consists of two transistors and two resistors. These transistors and resistors can be used as sensing elements, i.e. as phototransistors and piezoresistors, which will result in a compact structure. When necessary, other sensing elements can be added to the flip—flop.

- A large sensor array is feasible. Since flip—flops are small circuits, a very large number can be integrated onto one single chip. Currently, some laboratories can integrate more than one million flip—flops on one memory chip. When advanced memory fabrication technology is applied for flip-flop sensor fabrication, a very large flip—flop sensor array can be realized.

These interesting features justify the necessity of further investigation into the flip—flop sensor technique. The objectives of the study are:

- To establish a theory for the purpose of understanding the underlying principles of the flip—flop sensor operation, and to provide analytical expressions to describe the flip-flop sensor characteristics. Chapter 2 of this thesis deals with these problems.

- To investigate the design, the process technology and the characteristics of flip—flop sensors through experiments. Optical, thermal, magnetic and mechanical flip—flop sensors based on phototransistors, photodiodes, thermocouples, Hall plates and piezoresistors respectively, are presented in Chapter 3.

- To find the applications. Though there are many possible applications, a very interesting one is the flip—flop sensor array. This is presented in Chapter 4.

and pitfalls," J. Phys. E: Sci lustrum., Vol. 20, pp. 1080-1086, 1987.

1.3] S. Middelhoek, P.J. French, J.H. Huijsing and W. Lian, "Sensors with digital or frequency output," The 4th

International Conference on Solid—State Sensors and Actuators,

pp. 17-24, Tokyo, Japan, June 2-5, 1987.

1.4] J.H. Huijsing, "Signal conditioning on the sensor chip," Sensors & Actuators, Vol. 10, pp. 219-237, 1987.

1.5] A. Kindlund, H. Sundgren and I. Lundstrom, "Quartz crystal gas monitor with a gas concentrating stage," Sensors &

Actuators, Vol. 6, pp. 1-17, 1984.

1.6] R.M. Landon, "Resonator sensors — A review," J. Phys. E: Sci.

Instrum., Vol. 18, pp. 103-115, 1985.

1.7] J.C. Culey, Transducers for microprocessor systems, MacMillan, London, 1985.

1.8] B. Gilbert, "A versatile monolithic voltage—to—frequency converter," IEEE J. Solid-State Circuits, Vol. SC-11, pp. 852-864, 1976.

1.9] A.J.M. Boomkamp and G.C.M. Meijer, "An accurate biomedical temperature transducer with on—chip microcomputer

interfacing," 11th European Solid—State Circuits Conference, pp. 420-423, Toulouse, France, Sept. 16-18, 1985.

1.10] S. Sugiyama, M. Takigawa and I. Igarashi, "Integrated piezoresistive pressure sensor with both voltage and frequency output," Sensors k Actuators, Vol. 4, pp. 113-120, 1983.

1.11] A. Hanneborg, T.E. Hansen, P.A. Ohlckers, E. Carlson, B. Dahl and O.Holwech, "An integrated capacitive pressure sensor with frequency—modulated output," Sensors & Actuators, Vol. 9, pp. 345-351, 1986.

1.12] K.U. Stein, A. Sihling and E. Doering, "Storage array and sense refresh circuit for single—transistor memory cells," IEEE J.

Solid-State Circuits, Vol. SC-7, pp. 336-340, 1972.

1.13] D. van Willigen and L. M. van Berkel, "A new, highly sensitive, low—power, integrated sample—and—hold polarity detector for Loran—C and Omega navigation receivers," 4th International

pp. 9-14, Bangor, U.K., July 1-4, 1986.

[1.14] IEEE Standard Dictionary of Electrical and Electronics Terms.

The IEEE Inc., New York, 1984.

2. THEORY OF THE FLIP-FLOP SENSOR

TECHNIQUE

2.1 INTRODUCTION

The theoretical basis of the flip-flop sensor technique is developed in this chapter. The major objectives are to provide the physical and mathematical foundations of the operating principles of the flip-flop sensor and to establish a well-organized set of theoretical predictions, which will aid in the design of flip-flop sensors.

The difficulty in developing such a theoretical basis for the flip—flop sensor arises due to the presence of the unstable state. An unstable state is associated with uncommon characteristics. Therefore, a special approach is required for its analysis. Moreover, many problems concerning the flip—flop sensor were new when this investigation began, so few references were available on which to lay the foundation for further research. However, these problems reflect the importance of the creation of a correct theoretical model.

This chapter contains four main sections. Section 2.2 presents the concept of the "Flip—Flop Sensor", i.e. what is meant by the term "flip—flop sensor" and what underlying principles are involved in its operation. Section 2.3 depicts the physical realization of the flip-flop sensor in integrated—electronic technology. Section 2.4 develops the theoretical model and derives formulae for the calculation of the flip-flop sensor characteristics. Section 2.5 discusses the use of a triangle wave and the use of the feedback technique in performing measurements on the flip—flop sensor.

2.2 PRINCIPLES OF THE "FLIP-FLOP SENSOR"

In general, physical systems, which are brought into an unstable state, are very sensitive to perturbations. A standard flip-flop consisting of two transistors and two resistors (Fig. 2.1) is characterized with two stable states. Under certain conditions, such a flip—flop can be brought into an unstable state. When, for instance, the supply current is switched on and off, the flip—flop is first brought into an unstable state. The flip-flop will subsequently transfer to one of the two stable states. Which stable state is chosen is dependent on internal and external excitations, the magnitude of which may be infinitely small.

To create a sensing device from a standard flip—flop, existing circuit components are replaced by elements that are sensitive to certain physical or chemical parameters. As described above, the flip—flop is first brought into an unstable state. The presence of the parameter to be measured will then generate an excitation in the sensitive element, thus bringing the flip—flop into a certain stable state.

By observing which stable state is chosen, the presence or absence of the parameter can be determined. However, the magnitude of the parameter can not be inferred. A reference is required in order to quantify the physical or chemical parameter. The threshold—sensitivity of any given measurement is set by the total system noise, which is always present. The total system noise has therefore been used as a reference for measurements aimed at low threshold—sensitivity. In the unstable state, not only are external parameters able to generate a perturbation in the flip-flop, but the total system noise is also able to create an excitation with random characteristics. To infer the parameter from the noise, the flip-flop measurements are repeated many times, and the number of "ones" and "zeros" in the output is counted. If the parameter to be

Fig. 2.1 The circuit diagram of a standard flip-flop.

measured has zero magnitude, the number of "ones" will account for 50%

of the number of unstable states, because the noise in this case is

symmetrically distributed with a zero mean. When the parameter is

present, the resulting number of "ones" will not be 50% of the total

unstable states, but instead will depend on the magnitude of the

parameter. In other words, the number of the resulting "ones" is the

measure of the parameter [2.1].

In summary, the operation of a flip—flop sensor is based-on:

1. Replacing some components by sensitive elements;

2. Bringing the flip-^lop into an unstable state and detecting the final

stable state;

Probability

Density

Probability

Density

Noise Value

Noise+ Signal

Fig. 2.2 a) Probability density distribution of Gaussian noise, b) Combined probability density distribution of the Gaussian noise and the measurand s.

2.2.1 Input/output characteristics

The total system noise of a flip—flop sensor is derived from two main sources: the thermal noise of the resistors and the shot noise of the transistors, both of which are characterized by a Gaussian distribution [2.2]. Figure 2.2.a shows the probability density function of Gaussian noise, and Fig. 2.2.b is the combined probability density function of the Gaussian noise and the parameter. Each time the flip—flop is brought into the unstable state, the probability of obtaining a "one" equals the integral of the density function from zero to positive infinity. This probability, noted as P here, is equal to

s/u 1

P = 0 . 5 ±

V(27T) dx (1)

where s is the signal generated by the parameter and u is the standard deviation of the noise. The sign of the integral is determined by the sign of

s. When the signal is smaller than the noise, P can be approximated by

P = 0 . 5 +

into the unstable state many times. The percentage of the number of unstable states, which result in a "one", is an approximation of P. Thus,

P = ^ = 0 . 5± X

S/W- - £ 2

N """ V(2TT) 2 dx (2)

0

where Ni is the number of "ones" and N is the number of unstable states. This equation describes the relationship between the number of "ones", A'l, the number of unstable states, TV, the signal 5 and the noise u. The signal is the input, the number of "ones" is the output and the number of unstable states and the noise are given or chosen parameters of the flip—flop sensor. In practice, both the number of "ones", N\, and the ratio of the number of "ones" to the number of unstable states, Ni/N, can be used as the output. Moreover, in theoretical analysis, it is more -convenient Jo_uae^5/^,Jaste^4i^,_^5jheJnput.

2.2.2 Error analysis

The accuracy in measuring P depends on how many measurements (i.e. unstable states) have been taken. This accuracy is given by ([2.3])

a = jP(l-P)/N (3)

where a is the standard deviation — or the error — in measuring P. The flip—flop sensor input/output characteristic, determined by eq. (2), is depicted in Fig. 2.3. The error function given by eq. (3) is shown in Fig. 2.4. When JVi is chosen as the output, the flip—flop sensor output is

sfu 1

1

N,/N

0.5

s / u

Fig. 2.3 Flip-flop input/output characteristic. The output is the number of "ones", N\, divided by the number of unstable states, N. The input is the signal, s, divided by the noise value, u.

k

10"

10

-N

Fig. 2.4 The error, a, which occurs in the measurement of P, versus the number of unstable states, N.

The relative error is JV aN

K = p S = * / M ) / A '

The numerical values for Ni and N1 have been calculated from (4) and (5)

and are shown in Table 2.1.

Table 2.1 The number of "ones", Ni, and the error number,

N', in the output, versus the input, s/u. The number of unstable states, N, is 104. The quantization error is ignored.

s/u 0.0

Ni 5000

M 50

Ni 7580

43

0.1

5398

50

0.8

7881

41

0.2

5793

49

0.9

8159

39

0.3

6179

49

1.0

8413

37

0.4

6554

48

1.1

8643

34

0.5

6915

46

1.2

8849

32

0.6

7257

45

1.3

9032

30

When the output "one" is counted Ni times from a total of N unstable states, how large will the signal s, which is generated by the parameter to be measured, be, supposing the noise value u is known? It is not possible to derive an analytical expression for 5 from eq. (4), but numerical solutions for eq. (4) are rather simple. One practical way is to use tables with precalculated values. Table 2.2 presents the values of

Ni/N, calculated from eq. (4), for s/u is 0.00, 0.01, 0.02, through 1.29.

Therefore, s can be extracted from Table 2.2.

Error in the input — The output is within the range of M ± N*. The error in the output, TV, will be transferred into an error in s when s is to be extracted. For convenience we consider s/u instead of s; the error in

s/u is defined here as the error in the input. The error is analyzed from its

absolute and relative values. There are two factors — the number of unstable states TV and the input signal s - which determine the error. The error JV can be translated into an error in the input s/u in a similar manner as translating Nl: i.e. using Table 2.2. Numerical methods could

also be utilized in solving equations (1) through (5). Only the results are presented here, no details are given.

Table 2.2 Values of (Ni/N)*Kfl, versus s/u. s/u 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0 5000 5398 5793 6179 6554 6915 7257 7580 7881 8159 8413 8643 8849 1 5040 5438 5832 6217 6591 6950 7291 7611 7910 8186 8438 8665 8869 2 5080 5478 5871 6255 6628 6985 7324 7642 7939 8212 8461 8686 8888 3 5120 5517 5910 6293 6664 7019 7357 7673 7967 8238 8485 8708 8907 4 5160 5557 5948 6331 6700 7054 7389 7703 7995 8264 8508 8729 8925 5 5199 5596 5987 6368 6736 7088 7422 7734 8023 8289 8531 8749 8944 6 5239 5636 6025 6406 6772 7123 7453 7764 8051 8314 8554 8770 8962 7 5279 5675 6064 6443 6808 7157 7486 7794 8078 8340 8577 8790 8980 8 5319 5714 6103 6480 6844 7190 7517 7823 8106 8365 8599 8810 8997 9 5359 5753 6141 6517 6879 7224 7549 7852 8132 8389 8621 8830 9015

Absolute error versus the normalized input — The absolute error,

A(s/w), in the input is calculated for ./V=104, and is shown in Fig. 2.5. For

A^IO6, the figure has also been calculated and appears to be the same as

that for iV=104, but with magnitudes of exactly 10 times smaller.

Absolute error versus the number of unstable states — Using a

greater number of unstable states would obviously yield a higher accuracy. This effect has been calculated and the results are shown in Fig. 2.6.

Relative error versus the normalized input — It is often easier to

work with the relative error instead of the absolute error. Therefore, in Fig. 2.7 the relative error, A(s/u)/(s/u), is plotted versus s/u.

Relative error versus the number of unstable states — The effect of

the number of unstable states on the absolute error is given in Fig. 2.6. The relative error, which is equal to the absolute error divided by (s/u), has a curve which can be obtained from a vertical shift of Fig. 2.6.

Through the use of the above results, the question: how many times should the flip—flop be brought into an unstable state, if, within the measurement range of 0.1<s/u<1.3, the relative measurement error must be less than 1%, can be answered. The answer is: 107 times is sufficient,

< ]

185

165

-145

125

Fig. 2.5 The absolute error, A (s/u), in the normalized input, versus the magnitudeoj'the-normalized-input,-(s/u),-for^N=_l_(£.

A(s/u)

Fig. 2.6 The absolute error, A (s/u), as a junction of the number of unstable states, N, when s/u is equal to 1. Note that the y-axis is depicted in the reverse direction.

A ( s / u )

(s/u)

1 2

8

4

0

0 0.2 0.4 0.6 0.8 1.0 1.2

s/u —

Fig. 2.7 The relative error, A (s/u)/(s/u), in the normalized input, versus (s/u). N is equal to 104.

2.2.3 Offset

Another factor that influences the unstable states is the asymmetry in the two sides of the flip—flop, which is caused by mismatches in the elemental components. A flip—flop with two unequal resistors tends to go into a certain stable state. This problem is called the offset of the flip—flop sensor. While performing measurements of the flip—flop sensor characteristics, the asymmetries were compensated by the addition of a small DC voltage, which maintained the flip-flop sensor in the 50% state.

2.2.4 Literature research

It is interesting to know which scientific and technical disciplines are related to the flip—flop sensor technique.

A related subject is that involving the sense amplifiers in dynamic random access memories. These sense amplifiers also explore the high sensitivity of the flip—flops in the unstable states. The purpose in this case is to detect a very small charge from a storage cell [2.4, 2.5]. These sense amplifiers, which are in fact NMOS flip-flops, are brought into an unstable state. The stored charge is then transferred to the flip-flops so as

used in hardlimiters [2.6]. This device is used to detect the polarity of an electrical signal, which can be obtained through a radio receiver and enhanced by an amplifier. A third subject related to the flip—flop sensor is the so-called metastable state of synchronizers [2.7]. Here the synchronizer is a flip—flop, which outputs a "one" or a "zero" depending on the input pulse. The input pulse can by chance have an energy that exactly brings the flip—flop into the totally balanced unstable state, in which the flip—flop can stay for a relatively long time before going into a stable state. The problem is that if this time is too long the flip—flop will create an error in the synchronizer operation.

From these publications, many interesting details can be learned. However, there were two major theoretical problems concerning the flip—flop sensor, of which the answers were not available. As stated in Section 2.2.1, the flip—flop sensor characteristic is described by four parameters, namely the number of "ones", Ni, the number of unstable ^ t l ï t ê s ^ A ^ h e - s i g n a ^

parameters; the problems concern s and u. In the above discussion, the analysis is based on the dimensionless quantity s/u. For further analysis, s and u must be determined separately.

s is defined as a voltage source, which is equivalent to the sum of the total effects with DC characteristics, such as DC currents generated by phototransistors, DC voltages generated by Hall plates or mismatches in the load resistors of the flip—flops, u is defined as the noise voltage of the flip-flop. The two problems are then:

1. How can s be calculated? 2. How can a be calculated?

2.2.5 Discussions

In summary, the principles of the flip—flop sensor are presented. The flip—flop characteristic is determined by the four parameters Ni, N, s and u. The accuracy of the technique is discussed in relation to N, the number of unstable states. Because N is itself related to time, the relation between the accuracy and the measurement time is established. However, this thesis focuses on measuring parameters with a DC characteristic. The time aspect has been given a lower priority. Therefore, the accuracy is

analyzed in relation to the number of unstable states. What time is needed to realize that number of unstable states, will not be discussed in detail. The two major problems are pointed out. However, before further consideration of these two problems in Section 2.4, the realization of the flip-flop sensor is first presented in the following section.

2.3 ELECTRICAL REALIZATION OF THE FLIP-FLOP SENSOR

The flip-flop circuitry consists in fact of two inverter amplifiers (Fig. 2.1). The output of inverter 1 is connected to the input of inverter 2, and the output of inverter 2 is connected to the input of inverter 1. In this way a positive feedback is formed. When the supply current is switched on, the currents through the flip-flop transistors will increase and, as thus, the current gain will also increase. When the currents through the transistors are small the amplification of the inverters will be less than 1. When the currents reach a certain value, the amplification of the inverters will be slightly larger than 1. Then, any perturbation, either from the parameter to be measured or from the noise, will be amplified by the first inverter and passed to the second inverter. This process is repeated and regenerated again, until the differential output voltage is so great that one of the transistors is cut off, resulting in a stable state. This is the basis of the operation of the flip—flop circuit, in which it is first brought into an unstable state and subsequently switches to a stable state.

2.3.1 Realization of the unstable states

Before the flip-flop is brought into an unstable state, the two sides of the flip—flop should be discharged and balanced, namely to clear all the stored charge, so as to ensure that the past will have no influence on the future unstable state. Such a reset process can be performed in many ways. One example involves switching off the supply current of the flip—flop (Fig. 2.8.a). The same effect can be realized by switching off the supply voltage (Fig. 2.8.b). Forcing the two sides of the flip—flop to exist at the same electrical potential by shortening these two sides through two diodes (Fig. 2.8.c) also brings the flip-flop into an unstable state. The criteria for judging the best method include: sensitivity, speed, chip area and number of components. One disadvantage of the circuit in Fig. 2.8.C is the existence of the two extra diodes, which consume additional chip area, and furthermore, create additional mismatches beyond those within the

b)

v.M

c)

Fig. 2.8 Illustrations of methods used to bring a flip-flop into an unstable state; a) switching the current source, b) switching the voltage source and c) switching through two diodes.

flip-flop itself. Moreover, this circuit requires one more voltage input than the circuits of Fig. 2.8.a and Fig. 2.8.b. Namely, in addition to the upper node and the lower node, which are sufficient for the circuits in Fig. 2.8.a and Fig. 2.8.b, the circuit in Fig. 2.8.c needs a third node connected to the

diodes. The circuit with voltage switching as shown in Fig. 2.8.b has one disadvantage. It has a low speed due to the saturation of the conducting transistor. When considering the sensitivity, the circuits in Fig. 2.8.a and Fig. 2.8.b are similar. Therefore, the method of realizing unstable states as shown in Fig. 2.8.a is recommended for sensor application.

2.3.2 Flip-^lop circuitry

As the flip-flop circuitry forms the basis for the flip-flop sensor, and as there are many different flip—flop circuit designs, it is interesting to find out which flip—flop design is most suitable for sensor application. A collection of flip-flop circuit diagrams is shown in Fig. 2.9.

The circuit in Fig. 2.9.a presents the elementary circuitry of the flip—flop, consisting of two load resistors and two cross—coupled transistors. Other flip-flops are in fact modified forms of this circuit. Figure 2.9.b shows a flip-flop with two additional base resistors. These resistors are used when a high output voltage of the flip—flop is required, as the elementary flip-flop shown in Fig. 2.9.a provides an output of less than 0.7V. Figure 2.9.c shows a flip-flop with diodes parallel to the load resistors. These diodes could be pn—junction diodes or Schottky diodes. Their function is to provide a shunt for the currents in the switching mode. A high speed and a low power consumption can be achieved with this design and as thus, it is widely used as the memory cell of the ECL (emitter—coupled logic) RAM [2.8]. The same idea of.changing the load resistance is also applied in the circuit of Fig. 2.9.d. Additional load resistances consisting of transistors and resistors are implemented to reduce the load resistances of the flip-flop during the switching process. Figure 2.9.e and Fig. 2.9.f show flip—flops with the load resistors replaced by pnp transistors in order to accomplish low standby DC currents.

Figure 2.9.g is an NMOS flip-flop with normally-on transistors employed as the loads. Figure 2.9.h is the flip-flop widely used as the sense amplifier in DRAM's, with a pass transistor connected between the two outputs to bring the flip—flop into balance before it is brought into the unstable state. Flip-flops made using other processing techniques are not considered here, as the activities of our laboratory and IC workshop are concentrated on bipolar and NMOS processing. Although all flip-flops shown in Fig. 2.9 can be used for sensor applications, the theoretical research presented in this chapter is carried out on the circuit of Fig. 2.9.a, since this is the elementary circuit of all flip-flops and its

VTt\

c) ï

T

H

h H

P H

i r

T

r n

1

* > r

theoretical model can be generalized to all other flip-flop circuits.

2.3.3 Flip-^lop sensing elements

The flip—flop in an unstable state needs some internal and/or external perturbation to direct it to a stable state. By incorporating sensing elements within the flip-flop circuitry, the parameter to be measured can generate a perturbation in the flip-flop circuit, which can cause it to switch to a specific stable state. The question now is: which sensor elements can be incorporated into the flip—flop circuitry in the realization of a flip—flop sensor? In fact, any sensor element can be attached to or incorporated within a flip-flop circuit, resulting in a flip—flop sensor. As smart sensors appear to have interesting characteristics [2.9], we will consider here only those sensing elements that can be fabricated with standard silicon IC processing. Due to the very special material characteristics of silicon, elements for sensing signals from the optical, magnetic, mechanical, thermal and chemical fields can all be fabricated in silicon [2.10], thus providing a broad application area for the flip—flop sensors. For optical application, phototransistors and photodiodes can be used to measure light intensity. Hall plates, magnetotransistors and split-drain NMOS transistors can be used for magnetic—field strength measurements. Piezoresistors and NMOS transistors with piezoresistive channels can be used for mechanical stress measurements. Resistors, diodes, transistors and thermocouples, which display temperature dependence, can be used to measure the absolute temperature and temperature differences. Finally, ISFET's can be used as the sensing elements in flip—flop sensors to measure pH values or gas concentrations.

The sensing elements mentioned above can be attached to the flip—flop, or, more efficiently, they can replace existing elements of the flip-flop. For instance, the two load resistors of a flip—flop can be replaced by two piezoresistors. Such a replacement yields a very compact construction and the whole device requires only a small chip area.

2.3.4 Measurement setup

The complete test setup is shown in Fig. 2.10 for an ECL flip—flop.

Rz and R^ are the load resistors of the flip—flop and usually range from a

r

fl*

I

R,

R-.Q

counter

pulse

generator

Fig. 2.10 The experimental setup for measuring the flip-flop sensor characteristics.

offset compensation and their value is normally two orders of magnitude smaller than R% and R4. The voltage [/is attenuated by the ratio R5/R2

(R5»R2) and is fed to the flip-flop. By adjusting U, the offset due to

asymmetries in the flip—flop components can be compensated, thus bringing the flip-flop into the 50% state. The two outputs of the flip-flop are connected to a buffer amplifier and the buffer output is connected to the counter. The bias current of the flip-flop is switched on and off by a pulse generator. The behavior of the flip-flop output voltage is shown in Fig. 2.11.

The speed of a flip-flop sensor is determined by the load resistances, the parasitic capacitances and the bias condition. This speed can be found using the circuit simulation program SPICE. The flip-flop transistors fabricated in our IC laboratory with a standard bipolar process have a cut-off frequency of 500 MHz. Flip-flops with 2 kfi load resistances demonstrated a switch-on time of <40 ns and a switch-off time of 56 ns. Therefore, the maximum operating frequency of this flip-flop is on the order of 10 MHz.

100

ns

t

Fig. 2.11 Switching behavior of the flip-flop of Fig. 2.1. v\, V2 and I are the two output voltages and the bias current, respectively.

2.4 CALCULATION OF THE FLIP-FLOP SENSOR CHARACTERISTICS

A well—organized set of theoretical predictions on the operating principles of a flip-flop sensor is necessary to establish. At the time this research was begun, such theoretical predictions could not be made simply because it was not known how to deal with the unstable state. The final solutions for this problem were inspired by the way in which the threshold level is calculated in Schmitt triggers [2.11] and by the manner in which a linear differential equation with stochastic input is solved [2.12].

In a switching process, the flip-flop decides either to go to a "one" or to a "zero" depending on the mismatches in the load resistances and transistor characteristics, on the signal generated by the measurand and on the noise. The uncommon nature of the unstable state, which represents a sudden rapid change, makes the theoretical calculation difficult. Unknown phenomena concerning the flip-flop operation include the loop gain in the unstable state, the effect of the transistor current

effect of the measurand. A theoretical model describing the flip-flop switching process is derived here. This model has proven to be able to provide satisfactory answers to quantitative and qualitative questions about the flip-flop sensor behavior. The model consists of two parts. Part one, presented in Section 2.4.1, deals with the effects associated with the DC characteristics, such as the asymmetries of the resistances and the transistor characteristics and the effects of DC current and voltage sources [2.13]. In part two, Section 2.4.4, the noise is calculated [2.14].

2.4.1 Modeling of the effects of current sources, voltage sources and mismatches of resistors and transistors

When one of the two load resistances is slightly larger than the other, the flip-flop is forced to go to a certain stable state. To qualify this effect, a DC voltage was introduced into the flip-flop as shown in Fig. 2.12.a. There exists a certain value of this voltage at which the effect of the asymmetry in the resistors is fully compensated by the addition of this DC voltage. This voltage is called the equivalent voltage. In fact, resistors have always appeared to be asymmetric and a DC voltage must be used in practice to compensate this asymmetry. Other effects due to transistor characteristics, current sources and voltage sources generated by the measurand can be transformed into equivalent voltages in the same manner.

The next step is to derive the magnitude of these equivalent voltages. The derivation is based on the fact that in the unstable state the change of the current is very rapid (infinite when the parasitic capacitance is zero), thus the derivative d/c/d/0 has a singularity, where Ic is the

collector current and I0 is the bias current being switched. This technique

has also been used in the determination of the threshold points of a Schmitt trigger [2.11]. The flip-flop in Fig. 2.12.a is described by the equations

(h + I2/p)Ri + Ube2 = {h+h/0)R2 + Ubei + U (6)

h = IsexV(Ubel/Vt) (?)

h = hexp(Uhe2/Vt) (8)

C)

Fig. 2.12 a) Small DC voltage U introduced into the flip-flop for the compensation of asymmetries; b) and c) voltage sources U\, U2 and U3, caused by sensing elements, can be placed in positions as shown; d) current source I created by sensing elements can be placed as shown.

current gain, Vt is equal to kT/q and I

is the current source being

switched. By taking the derivatives of equations (6) through (9) it follows

(d/i + dh/P)Ri + dU

= (d/

+ dh/p)R

+ dt/

ei

dh = (Ii/V

)dU

dI

= (h/V

)dU

2

dl

= d/i + dhjp + dl

+ dhjp

Eliminating dU

\ and d£/

e2 results in

dh_ Vt/h-R2+Ri/P

v

d/

~ (1+1//J)[Vt/h+V

/Iv-Ri (l-l/fl-R2(l-l/0)]

dh_ V

/h-R

+R

/l3

.

d/

(l+l/^){V

/h+V

/I

-R

(l-l/0)-R

(l-l/P)]

Here, dii/d/

and dl

/dl

are the changes in the collector currents versus

the changes in the current source, I

. In the switching process of the

flip-flop, as I

increases, the collector currents also increase, until the

moment when the unstable state is reached, at which dli/dl

and dl

/dl

approach infinity. This singularity is given by a zero in the denominators

of d/i/d/o and dl

/dl

, thus

(1+1/P)[V

/h+ V

/I

- R^l-l/P) - /i

(l-l/^)]=0 (12)

The fact that equation (12) represents the unstable state is also illustrated

by modifying eq. (12) to

R

(l-l/0)+R

(l-l/0)

,

V

/h+V

/I

thus

R

(l-l/P)+R

(\-l/0)

where r

i=Vt//i and r

2=Vt/'h. The above equation shows that the

flip—flop has a unity loop gain. A system with a unity loop gain is in an

unstable state [2.15].

Once the flip—flop is in the unstable state three things can happen:

the flip-flop can move to "one" state, to the "zero" state or it can remain

in the unstable state if the asymmetry is exactly compensated. In order to

remain in the unstable state the numerators in dh/dl

and dh/dlo must

equal zero because the denominators are equal to zero. The definition of

the equivalent voltage assumes an exact compensation, that leads to a

zero in the numerators

7 * - £2 + ^ = 0 (13)

%-R

f=0 (14)

These two equations determine the values of the collector currents, upon

which a total balance is realized. The voltage U, which is required for such

a total balance, can be derived through (6), (13) and (14). The result is

v

=

v

^w^ê

(15)

In the above derivation only the two resistors were assumed to be

unequal, while other parameters were assumed symmetric. The effects of

the transistor parameters are dominated by the saturation currents and

the current gains [2.16]. The equivalent voltages for the saturation

currents and the current gains can be calculated in the same way

CfcV,ln(ii) (16)

M*k^Hk

(17)

Equations (16) and (17) present the equivalent voltages of the

mismatches in saturation currents and in current gains, respectively.

Sensing elements, which generate a voltage or a current, can be

sources can be added to the flip-flop in the manners shown in Fig. 2.12.b and Fig. 2.12.C, or a current source can be added to the flip-flop between the two bases (Fig. 2.12.d). Their equivalent voltages can be calculated.

In Fig. 2.12.b

U= Ui= U2= Uz (18)

In Fig. 2.12.d

U= 727. (19)

The formulae (15) through (19) were verified by the circuit simulation program SPICE and through experiments. In SPICE, up to 30 parameters were used to model the bipolar transistor. To determine the equivalent voltage of the saturation currents, a DC voltage source was used in the simulations. If this voltage was too great, which_reaul.ted_in_.the_

situation that the simulated flip-flop went to a "one", then, in the next simulation this voltage was given a smaller value, until the flip-flop went to zero. Through such an iteration, a small voltage range could be found, within which the equivalent voltage resided. In this way formulae (15) through (19) were verified and very good agreements were found.

Examples: in formula (15), for 7_.=28 kfi and R2—U kfi, theoretically, according to (15), U will be equal to 18.23 mV, while SPICE simulation predicted that U would lie between 18.20 and 18.30 mV. In formula (16), for 7S_/7S2=4, t/will be equal to 35.85 mV according to (16),

and SPICE predicted that U would lie between 35.90 and 36.00 mV. In formula (17), for /?i=150 and /?2=135, £7 will be equal to 19.3 /zV according to (17), whereas SPICE predicted that U would lie between 20.8 and 20.9 [N. For formula (18), SPICE showed that the inaccuracy must be less than 0.5%. Finally, in formula (19), for 72=20 kfi and 7=2.5 nA, U will be equal to 100.00 fiV according to (19), and SPICE predicted that U would lie between 99.96 and 100.00 /xV.

Experiments were carried out to test the formulae. In a test circuit, one of the load resistances was set to 20.00 kfi, and the other resistance was subsequently given several different values. The equivalent voltages were measured, and, for comparison, the results of the experiments along with those from formulae and SPICE simulations are shown in Table 2.3.

Table 2.3 A comparison of the measured, simulated and

calculated equivalent voltages U (m V) due to unequal resistances.

#2=20.00 kJl Ri= Exp. SPICE MODEL 22.02 U 2.46 U 2.51-U 2.53 2.52 30.01 10.88 10.77-10.78 10.75 39.99 (kO) 18.67 18.41-18.42 18.39

The deviations, defined by (CW—#for)/#for,

ê

fr°

1-2% to

2.8% for the data of Table 2.3. Another experiment was performed for the

verification of formula (16). A value of 2 for the ratio of hi/hi was simply

realized by using two transistors connected in parallel on one side of the

flip-flop, while on the other side, there was only one transistor. The

results of the measurement, the SPICE simulation and the theoretical

calculation are shown below:

Experiment U= 17.76 mV

SPICE [7=17.75-17.76 mV

Formula {7=17.74 mV

In conclusion, a theoretical model has been developed, which can

predict the effects of any DC changes in the parameters of the transistors

and resistors. The effects of voltage sources and current sources can also

be calculated using the same model. Beyond the calculation of the

sensitivity of the flip—flop sensor, this model can also be used to calculate

the flip-flop offset voltage caused by mismatches in resistors and

transistors. For instance, a 1% mismatch in the load resistances will

produce a 0.261 mV offset, and a 1% mismatch in the saturation currents

will create a 0.257 mV offset voltage.

The above results have been obtained for the flip—flop using a

switched bias current source, as shown in Fig. 2.1. Calculations have

shown that current switching is not essential. The same formulae have

been obtained when voltage switching is used.

As an application of the flip—flop model, the thermal drift of the flip-flop offset voltage is calculated. Assume that the two transistors are subjected to the same temperature. The offset voltage change versus the temperature change is calculated below for mismatches in resistances and for mismatches in transistor saturation currents and current gains respectively.

The offset voltage due to mismatches in the resistances is given by eq. (15). Under the conditions that AR/R«l and /?>>1, the change in the offset is then

AU=-AT^

q R

where AU is the change of the offset, A T is the change of the temperature, AR=Ri—R9 and

fl=(fli-hJk)/2.^It_is_clear—that—AU—is-^pvoport\ona]lx^ARjR. FO7~AR[R=1%, Al/=0.87(/zV/C°)AT.

The offset due to AIS/IS is given by eq. (16). The change in offset is

then

AU=-AT^fy

It is clear that AU is proportional to the mismatch of the saturation currents. For A/S//S=l%, A£/=0.87(/iV/C°)Ar.

The offset due to mismatches in the transistor current gains is given in eq. (17). The offset change is then

AU=

-AT\nP^-q 1-1/02

For A/?//?=10%, Af/=0.064(//V/C°)AT. It is clear that the effect of mismatches in the transistor current gains is negligible compared with the effects of mismatches in resistances and transistor saturation currents.

2.4.3 Limitations of the DC model

As with all mathematical models, this model is only valid under certain conditions. Since capacitances were not included in the model, the validity of the formulae at high speeds is not ensured. SPICE includes most of the capacitances of the transistors in its model parameters, so it provides a better accuracy at high speeds. Simulations at several operating speeds have been done to verify the model. The results are shown in Table 2.4.

Table 2.4 The equivalent voltage values for mismatches in resistances and

transistor saturation currents and current gains are calculated using the formulae and shown in the first row. For different rise times, the equivalent voltage values are simulated using SPICE and shown in the last four rows. The unit is n V. Formulae 20 fjs 2 fjs 200 ns 20 ns AR/R=l% 261 262-263 290-300 530-540 670-680 A/S//S=l% 257 261-262 259-260 264-265 269-270

A0/p=lO%

It can be seen that the equivalent voltage for mismatches in the load resistances is frequency dependent. At a rise time i—20 ^s, the formula agrees well with SPICE. At higher speeds, the formula failed to accurately predict the equivalent voltage. The equivalent voltages due to mismatches in the transistor saturation currents and current gains are nearly insensitive to frequency and remain valid at high speeds, as shown in Table 2.4. The fact that the equivalent voltage due to AR/R is frequency dependant, but that the equivalent voltages due to AIS/IS and

A/?//? are not, can be explained by the RC time. Different resistances in a flip—flop create unequal RC times in the two sides of the flip—flop, making one transistor charge more quickly than the other. However, mismatches in saturation currents and current gains have no influence on the RC time and are thus not frequency dependent.

1

output

0

noise __

1

offset

J

input

Fig. 2.13 The flip-flop stable output after a switch operation versus an

input signal.

2.4.4 Noise of the switched flip-flop

The flip—flop circuitry can perform both sensing and latching

operations and is thus used in the interface circuitry between the analog

signal and the digital signal. It compares an input signal — preamplified or

not - with a threshold or a reference. Depending on the results of the

comparison, the flip—flop will output a "one" or a "zero". In fact, the

flip—flop is one of the most elementary circuits in electronics. The

flip—flop operation is as follows: in order to obtain a high sensitivity the

flip—flop is first reset by reducing the bias current. Then, the flip—flop is

activated by increasing the bias current, thus bringing the flip—flop into

the regenerative state, also called the unstable state, which is very

sensitive to any stimulus. Finally the flip—flop goes to a "one" or a "zero".

The stable output voltage after a sampling versus the input signal (current or voltage), is depicted in Fig. 2.13. The offset in this input/output figure is determined by the mismatches of the components and is studied in the previous section. This section deals with the question: to what extent is the input/output characteristic (Fig. 2.13) determined by the noise of the load resistors and the transistors?

When the input signal is very great, it is almost certain that the output will be a "one". When the input is great, but with a negative sign, it is also certain that the output will be a "zero". In between, there is a certain value of the input such that the probability of obtaining a "one" in the output is equal to 84%. This input value is defined here as the equivalent input noise voltage value. The probability of 84% is used in the definition, because that is the probability at which the mean of a Gaussian process has a shift equal to the standard deviation.

The difficulty in solving this problem is due to the uncommon nature of the unstable state. Unanswerable phenomena include the bandwidth of the flip—flop in the unstable state, the influence of the noise before the unstable state and after the unstable state, the DC gain of the flip-flop in the unstable state, the influence of the initial (just before switching) noise and the effect of the switching speed of the current source. Moreover, the currents are not constant and therefore the spectral density of the current noise is time variant, making the calculation difficult. To the best of the author's knowledge, no publication dealing with the noise of a switched flip-flop has been reported. In this section an analytical expression for the noise is derived.

The equivalent circuit for the flip—flop of Fig. 2.1 is shown in Fig. 2.14, where ia is the stochastic variable representing the white noise

current from the load resistors and the transistors in the flip—flop, i is the DC current source equal to the equivalent input noise current defined above; G\1--{Ic\+h'i)l V\. (where Ici and IC2 are the collector currents, Vt

is equal to kT/q with k is the Boltzmann constant, T is the absolute temperature and q is the charge of an electron); G is the conductance of the two load resistors: G—1/(2R); Cis the total capacitance between the two collector nodes and is assumed to be constant, and v(t) is the voltage between the two collector nodes. For a better understanding of

Gt=—(Ici+Ic2)/ Vu a schematic representing the cross—coupled transistor

pair is shown in Fig. 2.15, where w is a small voltage change, i is the change of current caused by u and 7ci and IC2 are the original collector

in

è ' è

Q

>

V(t)

Fig. 2.14 Equivalent circuit of a flip-flop used in noise modeling.

Fig. 2.15 The schematic used in the calculation of the conductance of the cross-coupled transistor pair.

currents. Changes in the base—emitter voltages are given by u

U = A C/be2 - A t^bel

By the first order approximation, u becomes

u = dt^be2/ -s d[/bel, _{d /}c l v "' d/C2V

Because Ic-Is exp(qUbe/kT), which leads to

it follows

J c 2 ^Cl ^ c 2 J c l

and

n _ i _ kl+Ic2

The "—" sign in Gt shows that if u is positive (as defined in Fig. 2.15), i will have a direction opposite to the direction given by the arrow in Fig. 2.15. This effect is called negative resistance or conductance and it is caused by the feedback of the cross—coupled transistor pair. Further, it was assumed that /ci+/c2=^ in which the base current is ignored because

its small effect. Therefore, the switching operation of the flip—flop, through the bias current I, can be mathematically described by

Gt=-(Icl+Ic2)/Vt=-I/Vt.

The noise current can be represented by the spectral density. The noise current due to the two resistors has a spectral density of 4kT/(2R). The noise of a transistor is well known and, to simplify the model here, the shot noise of the collector current, which is dominant for the transistor noise behavior, is considered to be the only noise in the transistor. The other noise sources in the transistors are ignored. The resulting noise circuit of the flip-flop is shown in Fig. 2.16. According to the Blakesley transformation [2.17], the equivalent noise source between the two collector nodes is (l/2)[ii(t)—i2(t)] due to the transistors. Moreover, z'i(Z), i2(t) and k(t) are correlated through h{t)+h{t)=h{t)- In order to calculate

the spectral density of (l/2)[ii(t)—i2(t)], it is recalled that the spectral

density of h(t), i2(t) and z3(i) are Si{w)=2q(I/2), 52(w)=2?(//2) and

S3(w)=2ql respectively. The spectral density of (l/2)[ii(t)—i2(t)\ can be

derived from its covariance and the correlation

cov{jj[ti(*) - h(t)], £[ti(r) - i2(T)]}

= \siu>)6(t-T) + \s2(»)6(t-r) - \2cov[k(t), i2(r))

where 6(t—r) is the impulse function. To calculate cov[ii(t), «2(7")], it follows from ii(t) + i2(t)=i3(t)

[kit))2 + 2h(t)h(t) + [i2(t)}2 = [i3(t)}2 2h(t)i2(t) = [i3(t)}2-[il(t)}2-[h(t)}2

Thus,

2cov[h(t), i2{r)}

=S3(u)6(t-T) ~ Si{u)S{t-r) - S2(u>)S(t-r) =6(t-T)[S3(u>) - S^u) - S2{u)]