Physical properties often present as analog quantities in the real world. Computerized measurement of these properties entails some form of analog to digital transformation. Let us examine conventional as well as unconventional analog to digital techniques.

From Ohm's Law, V = I R, we observe a duality between voltage V and current I. If we can measure voltage we can measure current, and vice versa. Similarly, the charge Q stored in a capacitor C is given as Q = CV. Thus the same duality holds for Q and V.

Problem 1(a): Derive the formula for the voltage V(t), across a capacitor C being charged with a constant current I. Problem 1(b): Derive the formula for the voltage V(t), across a capacitor C being charged with a constant voltage V through a series resistor R. |

Frequency and period is another example of duality. In this case, period T
is the reciprocal of frequency f, that is, T = 1/f. Hence if we can measure
one, we can determine the other. While time is continuous (analog), the measurement
of frequency or period is a natural analog to digital transformation. Frequency
is easily recorded using a digital counter, either in hardware or in software.
The number of discrete events or pulses is recorded over a given period. Inversely,
the duration of an unknown period can be recorded as the number of clock pulses
appearing during the unknown period. In either case, a constant reference clock
or **time-base** is required.

Physical properties, for example, temperature, pressure, humidity, elasticity,
opacity, colour etc. can be converted to a current or voltage using appropriate
sensors or **transducers**. Very often, the transducer itself is a resistor
or capacitor. Either of the two can be measured knowing the basic formula for
**time-constant**, tau = RC.

Problem 2: Derive the formula, tau = RC, for a parallel circuit consisting of resistance R and capacitance C. |

Problem 3: When a capacitance C is charged or discharged through a resistor R, the time-constant is given as tau = RC. Show in tabular form the normalized voltage V as a function of N x tau for N = 1 to 10. If the residual voltage (i.e. the difference between the present voltage and the final steady state voltage) respresents the resolution of an n-bit ADC, show how N varies with n, for n = 1 to 24. |

The pulse-duration of a monostable multivibrator is based on the basic time-constant, tau = RC. If we can measure the width of the output pulse, this becomes a simple technique for determining R or C.

Problem 4: Devise a simple circuit for converting a monostable multivibrator into a source of continuous clock pulses. |

Another simple circuit for converting R or C to a period or frequency is the 555 Timer IC.

The leakage current at the input of a **CMOS** (Complementary Metal-Oxide-Semiconductor)
gate is very low. In the case of the MC9S08 MCU, this is less that 1µA.
Hence it is practical to use a bi-directional input/output port of a CMOS microprocessor
in order to measure R or C. Simply enable the pin as an output port and discharge
the capacitor to 0 or charge it to 1. Turn the pin into an input port and measure
the time it takes for the capacitor to charge or discharge through the resistor
R, until the digital logic threshold is crossed. Using the frequency versus
period duality, software can be configured to measure either frequency or period.
Note, however, that because of quantization, frequency and period measurements
do not give the same resolution.

2007.01.04