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Sergej Korshak

Faculty of Fire Safety, Radio Engineering and Information Protection

Department of Radio Engineering and Information Protection

Program Radio engineering

Exponential voltage to frequency converter

Scientific adviser: Candidate of Engineering Sciences, Associate professor Vladimir Paslen

Resume Abstract

Abstract

Content

• Introduction
• 1 Theme revelence
• 2 Goal and tasks of the research
• 3 Exponential converter work study
• Conclusion
• References

Introduction

The human sense of hearing is amazingly broad and accurate. It can detect pitches of sound as low as 20 Hz, and as high as 20 kHz — again, a wide range of 1,000 times. The amazing part is that the relative accuracy of each of these perceptions does not change much as we move through these ranges. For this reason, it is often useful to express these quantities in terms of exponents (or logarithms). For example, we talk of pitch in terms of octaves, or a doubling of frequency, which follows an exponential: F = 2(n=12) where n is the note of interest, and F is the frequency. We also talk of volume in decibels, which is the logarithm of the air pressure we experience: dB — 20log10(P1=P0) [1]. So, for electronic music systems, an exponential converter or logarithmic compressor is a useful tool. It allows us to move smoothly over a large range of values without losing resolution at any part of the range. The inability to have this level of control can be quite frustrating, as anyone who has used a linear potentiometer on a volume or attack/decay knob can attest to. The parameter being controlled seems to move too quickly through the values, and can never be set quite right.

1. Theme revelence

Analog exponential converters have been used for quite some time in the sound synthesis audio complex, and quite a great popularity. The volts per octave control of Voltage Controlled Oscillators (VCOs) was a great innovation, and allowed them to be far more stable as control voltages (CVs). But, the limiting factor in their use comes down to stability, as they often drift with temperature.

2. Goal and tasks of the research

This paper will discuss the causes of VCO temperature drift, and explore some of the solutions used to compensate for it. The main goal of this work is to see what the limits are for creating a very accurate, and temperature stable exponential converter for VCOs. VCOs are chosen as they are a very demanding application, with stricter tolerances than other applications (e.g. volume control). A well trained ear can hear a change in pitch down to a few cents (a cent is 1 = 100 th the distance to an adjacent note) [1]. As the change in pitch required to move up to the next note is only 6 %, this is a very small band that the exponential converter must be kept within (1 cent is therefore 0.06 %). And, this tolerance needs to be held over 10 octaves. It is a difficult task indeed.

3.Exponential converter work study

It will be useful to have a mathematical representation of the exact exponential characteristic we want, so that all converter topologies can be compared to a standard. Since we are looking at VCOs, the 1 V per octave CV standard will be a good reference. Assuming an oscillator that produces a frequency proportional to an input current, we are looking for a relationship as follows:

where Iout is the current to the oscillator, Iref  is a fixed reference current, CV is the input control voltage in volts, and ln and e are the natural log and its base. If you are not familiar with the natural log, do not fret, as it’s just another way of expressing a logarithm, but with a base of ~ 2.71828 (which is a fixed number abbreviated as e) instead of the usual base of 10. One of the useful properties of logarithms is that they can be used to easily do multiplication by simply adding them together, and then exponentiating them (its how slide rules work). If you don’t know much about them, there are plenty of resources to learn more, but it is not critical to understanding the work done here.

Most exponential converters are implemented with Bipolar Junction Transistors (BJTs). This is because they are inherently exponential. The current they produce is exponentially related to the base voltage applied [2]. A simple model of this relationship is as follows:

where Ic is temperature dependent parameters [3], Is is a device specific parameter, Vbe is the voltage between the base and emitter, and Vt is the thermal voltage (~26 mV). There are more accurate models, but as it turns out, we can not compensate for the drifts that this model predicts, so there is no need to go to lower levels just yet.

Both the Is and Vt terms are temperature dependent. Is can be mostly canceled out by using a transistor with a very similar Is parameter as a reference transistor. This is the matched pair that is often talked about. In this case, the base voltage applied is actually a differential voltage between the bases of the two transistors (the reference transistor, and the exponentiating transistor). A typical implementation is shown in Figure 1.

The current through the reference transistor is Iref, and is held constant by the action of the op-amp. The current through the exponentiating transistor is our output, and controlled by the differential base voltage dV. Since the reference transistor’s base is held at ground, this difference is merely the voltage at the base of the exponentiating transistor. Using math we can show that this cancels out the Is term (assuming Is is identical between the two transistors) [3]:

But, you can also intuitively think of the reference transistor as canceling out any base voltage variations that might occur over temperature, as both transistors will increase by the same amount, and the applied voltage (the difference between the two transistors) stays the same [4].

This just leaves us with the pesky Vt term. Vt is called the thermal voltage because it is proportional to absolute temperature:

where K is Boltzman’s constant, T is absolute temperature (in Kelvin (K<>/var)), and q is the fundamental charge of an electron. So, at room temperature (300 K) it’s around 26 mV, but it increases with temperature. How bad this drift is depends on two things: the difference in temperature and the difference in base voltages between the reference and exponentiating transistors. To illustrate this effect Figure 2 shows the error produced as a function of temperature for various applied CVs.

Two very important things to note are that the error is out of any usable range for almost any temperature drift, and that this error increase linearly with the CV. This means that it is easier to maintain accuracy if you limit the applied CV. We can figure out the maximum dV required at a 10 V CV [5]:

So, for a 0 V to 10 V CV, 180 mV is required. But, for a ±5 V CV, which would cover the exact same range, only half of that (±90 mV) would be needed. This has the double benefit of halving any Vt related errors, and minimizing errors in the most commonly used portion of the musical range (the middle). For example, if a 100 nA reference current was used, and the full output range was 100 nA to 100 uA, then the middle of the keyboard would be at 3 uA (+5 V CV), and would have all the drift associated with that large CV value. But, if the reference current was 3uA, and the output swung from 100 nA to 100 uA via a ±5 V CV, that same middle of the range would have negligible error, and the extremes of the keyboard would only have half the maximum error of the previous example. So a system that allows for bipolar CVs will inherently have less drift.

There are a few limitations which are common to all exponential converters. for example, the Early effect, which causes a small variation in output current with changes in Vce (the collector to emitter voltage). A more accurate model of transistor behaviour would look like:

where Va is the Early voltage (usually 50 V to 100 V). The Early effect is important to keep in mind for other parts of an exponential converter as well [5]. Many of the active gain compensation techniques use discrete multipliers comprised for accurate results. It is not only good to keep Vce constant, but also as small as possible. Ideally, Vcb (the collector to base voltage) would be 0 V. This helps reduce collector to base leakage, which increases exponentially with temperature. Topologies which hold Vcb at 0V on all transistors will perform better than those that do not.

Another common source of error is fluctuations in op-amp parameters with temperature [7]. For example, a more accurate version of the exponential core showing error sources is shown in Figure 4. The current through the reference transistor is set

by the value of the supply voltage, the value of the resistor R1, and the values of the offset voltage and bias current in the op-amp:

So, Rref must move less than 0.06% over the temperature range of interest. This means less than 10ppm/°C metal film resistors (10ppm/°C × 50°C = 500ppm = 0.05%). Likewise, Vref must be a constant source with less than 10ppm/°C drift. But, Vos and Ib do not need to be as carefully controlled, as long as they are kept significantly smaller than Vref and Iref . For example, if Vref is 5 V and Vos is 1mV, the voltage across the resistor is 4.999 V. A 0.05% decrease in this voltage would give 4.9965 V, which is the equivalent of a 2.5mV increase in the offset voltage. For a 50°C temperature change, that would be a +50uV/°C drift. The case for Ib is similar, although for JFET input op-amps the bias current increases exponentially with temperature, making it a bit more difficult to control. It is often easier to just ensure that for all operating temperatures Ib < 0.0005 × Iref , rather than make sure its change is within specification. This is also an argument for keeping Iref as large as possible (within reason) so that op-amp bias currents are not as detrimental.

It is important to keep in mind that the above offset voltage and bias current calculations are for just one portion of the exponential converter, and that other parts might have tighter constraints. For example, any parameter that affects the exponent will have a much a larger effect than one that effects the reference current, as the reference current is linear to the output, whereas the exponent is, as it sounds, exponential to the output. Also, it is good to keep variations well below the 0.06% mark, as all of the errors will add together, and could easily push things ten times past that limit. [6]

Conclusion

Ultimately, to compensate for the variation in Vt, either the temperature must be held constant, or the voltage applied to the base must be scaled in proportion to Vt. If it is not, the exponential current will drift. These two techniques are applied in a number of different ways, each with their relative benefits. The following topologies will be discussed, each in their own section:

1. Thermal oven compensation (constant Vt technique)
2. Thermistor compensation (passive gain compensation)
3. OTA Vt multiplier (active gain compensation)
4. Inverted OTA Vt multiplier (active gain compensation)
5. Gilbert cell Vt multiplier (active gain compensation)
6. Feedback Vt multiplier (active gain compensation)

They will be compared by cost, frequency response, linearity, and temperature drift. Some of them were built and tested, and ones experimental data is given, and partial schematics of the circuit is shown. Full schematics is not shown to improve clarity of their basic functionality, and in some cases because of poor documentation.

References

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2. Шило В. Л. Линейные интегральные схемы / В. Л. Шило. — М.: Сов. радио, 1979. — 368 с.
3. Пейтон А. Дж., Волш В. Аналоговая электроника на операционных усилителях / Пейтон А. Дж., Волш В. — М.: БИНОМ, 1994. — 352 с.
4. Linden T. Harrison Current sources & voltage references / Linden T. Harrison. — Oxford OX2 8DP, UK. — 2005. — pp. 260
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6. А. И. Новиков Экспоненциальные преобразователи с логарифмическим законом преобразования / А. И. Новиков. — М.:Автомат. и телемех., 1960. —12 с.
7. И. В. Новаченко Микросъемы для бытовой радиоаппаратуры / И. В. Новаченко. — М.:КУБК., 1996. — 384 с.