Development and construction: TestAS practice testlets

A measuring system can be described as a chain of blocks, each of which converts a signal into another form. Block diagrams show these blocks as boxes and the signals as arrows between them. For a digital thermometer in a weather station, the chain typically consists of a sensor, an amplifier, an analogue-to-digital converter (ADC) and a processor with a display.

The sensor converts the temperature into a small electrical signal. In the system described here, it delivers a voltage between 0 and 100 millivolts, which rises linearly with the temperature over a measuring range from −40 °C to +60 °C. This signal is too small to be processed directly, so an amplifier multiplies it by a constant factor, the gain, to produce a voltage between 0 and 10 V. The output of the amplifier therefore rises by 0.1 V for each degree, starting at 0 V at −40 °C.

gain = output voltage ÷ input voltage U_out = (T + 40 °C) × 0.1 V/°C resolution = range ÷ number of steps n-bit ADC: 2ⁿ steps

The ADC converts the continuous voltage into a number. An ADC with n bits divides its input range into 2ⁿ steps. An 8-bit converter has 256 steps, a 12-bit converter 4,096. The smallest change the system can detect, its resolution, is the measuring range divided by the number of steps. For a range of 100 °C, an 8-bit converter can therefore distinguish temperature differences of only about 0.4 °C, while a 12-bit converter reaches about 0.02 °C. The processor then converts the number back into a temperature and shows it on the display.

Block diagrams help to locate the limits of a system. The overall accuracy cannot be better than that of the weakest block: a very precise ADC is of no use if the sensor itself is inaccurate, and noise picked up in the cable before the amplifier is amplified together with the signal. For this reason, amplifiers are placed as close as possible to the sensor. Each block also has a limited range; a temperature outside the measuring range drives the amplifier to its maximum or minimum output, and the display shows a wrong but plausible-looking value unless the software detects the problem. Good designs therefore include checks that report such out-of-range readings as errors.

A bicycle transmits the rider's power from the pedals to the rear wheel by a chain that runs over a toothed chainring at the pedals and a smaller toothed sprocket on the rear wheel. Because the chain moves the same number of teeth past both wheels, the gear ratio is the number of teeth on the chainring divided by the number on the sprocket. It states how many times the rear wheel turns for each turn of the pedals.

gear ratio i = teeth on chainring ÷ teeth on sprocket wheel speed = cadence × i distance per crank turn = i × wheel circumference

A large ratio, with a small sprocket, is a high gear: each turn of the pedals moves the bicycle far, which is suitable for high speeds on flat roads, but the rider must push hard. A small ratio, with a large sprocket, is a low gear: each turn of the pedals moves the bicycle only a short distance, but the force needed on the pedals is smaller, which makes climbing hills easier. Ignoring friction, the chain transmits the same power in every gear, so what is gained in speed is lost in torque and vice versa: the torque at the rear wheel is the pedal torque divided by the gear ratio.

Cyclists usually try to keep their cadence, the number of pedal turns per minute, within a comfortable range, often between 70 and 100 turns per minute, and change gear so that they can maintain it as the road rises and falls. At a given cadence, the speed of the bicycle is proportional to the gear ratio. The distance travelled per wheel turn equals the circumference of the wheel, which for a typical road bike tyre is about 2.1 m.

Bicycles with derailleurs typically have one to three chainrings and eight to twelve sprockets, giving many combinations, although some of them overlap. Hub gears enclose the gear mechanism inside the rear hub and are common on city bikes because they need little maintenance. Electric bicycles add a motor that supports the rider's own effort, but the gear ratios still determine how the combined power is converted into speed or climbing force.