# Rate-of-change indicator

**PARTS AND MATERIALS**

- Two 6 volt batteries
- Capacitor, 0.1 µF (Radio Shack catalog # 272-135)
- 1 MΩ resistor
- Potentiometer, single turn, 5 kΩ, linear taper (Radio Shack catalog # 271-1714)

The potentiometer value is not especially critical, although lower-resistance units will, in theory, work better for this experiment than high-resistance units. I've used a 10 kΩ potentiometer for this circuit with excellent results.

**CROSS-REFERENCES**

*Lessons In Electric Circuits*, Volume 1, chapter 13: "Capacitors"

**LEARNING OBJECTIVES**

- How to build a differentiator circuit
- Obtain an empirical understanding of the
*derivative*calculus function

**SCHEMATIC DIAGRAM**

**ILLUSTRATION**

**INSTRUCTIONS**

Measure voltage between the potentiometer's wiper terminal and the "ground" point shown in the schematic diagram (the negative terminal of the lower 6-volt battery). This is the input voltage for the circuit, and you can see how it smoothly varies between zero and 12 volts as the potentiometer control is turned full-range. Since the potentiometer is used here as a voltage divider, this behavior should be unsurprising to you.

Now, measure voltage across the 1 MΩ resistor while moving the potentiometer control. A digital voltmeter is highly recommended, and I advise setting it to a very sensitive (millivolt) range to obtain the strongest indications. What does the voltmeter indicate while the potentiometer is *not* being moved? Turn the potentiometer slowly clockwise and note the voltmeter's indication. Turn the potentiometer slowly counter-clockwise and note the voltmeter's indication. What difference do you see between the two different directions of potentiometer control motion?

Try moving the potentiometer in such a way that the voltmeter gives a steady, small indication. What kind of potentiometer motion provides the *steadiest* voltage across the 1 MΩ resistor?

In calculus, a function representing the rate of change of one variable as compared to another is called the *derivative*. This simple circuit illustrates the concept of the derivative by producing an output voltage proportional to the input voltage's *rate of change over time*. Because this circuit performs the calculus function of differentiation with respect to time (outputting the time-derivative of an incoming signal), it is called a *differentiator* circuit.

Like the *averager* circuit shown earlier in this chapter, the differentiator circuit is a kind of analog computer. Differentiation is a far more complex mathematical function than averaging, especially when implemented in a digital computer, so this circuit is an excellent demonstration of the elegance of analog circuitry in performing mathematical computations.

More accurate differentiator circuits may be built by combining resistor-capacitor networks with electronic *amplifier* circuits. For more detail on computational circuitry, go to the "Analog Integrated Circuits" chapter in this Experiments volume.