A Good Analog Computation Example

Introduction

I am always looking for real-world examples of analog computation and this blog post will discuss one of the best examples of analog computation that I found. I found this little gem in EDN magazines’ Design Ideas section, which is a great place to look for clever analog solutions for real problems.

The circuit that I am going to review here is shown in Figure 1. During my analysis, I will break the circuit down into sub-circuits and then analyze the sub-circuits.

Figure 1: EDN Circuit For Measuring Available Wind Power.

This circuit generates a voltage that is proportional to the wind power currently available. It does this using two sensors:

anemometer/wind turbine
I usually think of four rotating cups whose motion generates a signal with a frequency proportional to wind speed, which is how this circuit represents wind speed.
base-emitter junction of a transistor
The base emitter junction’s voltage variation with temperature provides an analog for the temperature variation of the air’s density.

This review needs to cover a lot of technical territory so let’s dig in …

Background

For background on windmills and how they work, see this web site. The key equation for computing the maximum normalized power from a windmill is given by Equation 1. The normalized power is defined as the available watts per unit area of wind turbine.

Eq. 1

$\displaystyle P=\frac{1}{2}\cdot A\cdot {{\rho }_{Air}}\cdot v_{_{Air}}^{3}\Rightarrow {P}'=\frac{P}{A}=\frac{1}{2}\cdot {{\rho }_{Air}}\cdot v_{_{Air}}^{3}$

where

ρ_Air is the density of air, which is a function of temperature and pressure.
v_Air is the air velocity.
A is the area of the wind turbine.
P′ is the watts per unit area of the wind turbine.

Our objective in this post is to analyze the circuit shown in Figure 1 and demonstrate how that circuit implements Equation 1.

Before we do any electronics design, we need to beat Equation 1 into a form that can be implemented using electrical components. Figure 2 goes through this derivation.

Figure 2: Rework of Equation 1 into am Electronics-Friendly Form.

Analysis

Requirements

The circuit designer (Woodward) appears to have worked to the following requirements:

The circuit is generate 1 V of output for every 1 kW/m²of available wind power per unit area.
The circuit can produce a wide range of values. A value needs to be chosen in order to determine concrete part values.
The circuit is to use a single power supply voltage.
This circuit provides a nice illustration of designing an analog circuit for single supply operation. A one-supply design is normally preferred over a multi-supply design because it is cheaper. The designer used parts based on the 4000 series of CMOS devices. This is a very old family, nonetheless, many designers have a fondness for this family of digital parts for analog applications. See Appendix B for details on using these parts in analog applications.
The anemometer measuring the wind speed generates a signal with a frequency variation of 10 Hz per 1 m/s of wind velocity.
The circuit can be adapted to various types of anemometers. We need to pick a specific conversion factor in order to pick specific components. Appendix C gives examples of anemometers that would work for this circuit.
The circuit will compensate for air density variations with temperature.
It turns out that this compensation is relatively simple. Appendix A contains a derivation of the calibration equation presented in the designer’s original article.

For this analysis, I will break the circuit up into three sub-circuits:

Forward-Biased Diode
The forward diode voltage drop will be shown to have a temperature variation very similar to that of air.
Frequency-to-Voltage Conversion
This circuit will be used to multiply the forward diode voltage drop times the frequency of the signal from an anemometer.
Level Shift and Amplify Stage
This circuit removes a DC bias and properly scales the output signal level.

Forward-Biased Diode Voltage and the Density of Air

Figure 3 shows how the density variation for air on a percentage basis is similar to the percentage forward voltage variation across a diode or base-emitter junction.

Figure 3: Variation of Air Density with Temperature Compared to a Diode’s Variation.

Note that the molecular weight of air is 28.97 gm/mol, which is computed at this web site.

Voltage-to-Frequency Converter Section Operation

Figure 4 summarizes how the frequency-to-voltage converter works.

Figure 4: Voltage to Frequency Converter Subsection Operation.

As shown in Figure 4, the frequency-to-voltage converter circuit generates an output with ripple on it. This ripple will be filtered out by the low-pass filter incorporated into the Level Shift and Amplify sub-circuit.

Figure 5 shows how I will represent the frequency-to-voltage converter as a circuit element.

Figure 5: Symbolic Representation of the Frequency-to-Voltage Converter.

The Ref pin shown in Figure 5 deserves some comment. It connects to the positive input pin of the operational amplifier. In a system with bipolar supplies, the Ref pin would be connected to ground. Because this is a single-power supply application, the Ref pin will be connected midway between ground and the supply voltage value. The single-supply setup will product a V_OUT with a DC bias. This bias is removed by the Level Shift and Amplify stage.

Level Shift and Amplify

Figure 6 shows the final stage of the circuit, which takes the output of the frequency-to-voltage converters and provides some amplification and removes the 2.5 V bias.

Figure 6: Output Circuit for Level Shift and Amplify Stage.

The component values can be selected as shown in Figure 7.

Figure 7: Component Selection for Output Circuit Stage.

Entire Circuit

Figure 8 shows the whole circuit from my point of view.

Figure 8: Whole Circuit from a Block Diagram Viewpoint.

We can determine the components required as shown in Figure 9.

Figure 9: Check of Final Component Values.

Conclusion

I went through this circuit in excruciating detail because I thought it does a nice job of illustrating the kind of interplay between physics and electronics that often occurs in analog sensor applications. Also, I have a circuit application that I am working on that will use a circuit related to this one and I wanted to review this work before I pressed on with my circuit.

Appendix A: Derivation of Calibration Equation

The original article contains an equation that is useful for calibration. I derive his expression in Figure 10.

Figure 10: Derivation of Circuit Calibration Equation.

Appendix B: Designing Linear Circuits with 4000 Series CMOS Parts

There are quite a few designers who still use 4000 series parts (in this case, 74HC4000 series). See this document for details on applying these digital parts in an analog application.

Appendix C: Example of an Anemometer with 10 Hz per m/sec Output

I thought it was worthwhile showing some anemometers with 10 Hz per m/sec output. Both examples are powered.

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