handbook of modern sensors

176 5 Interface Electronic Circuits
Table 5.1. Integer and Fractional Binary Codes
Decimal Fraction Binary MSB Bit2 Bit3 Bit4 Binary Decimal
Fraction x1/2 x1/4 x1/6 x1/16 Integer Integer
0 0.0000 0 0 0 0 0000 0
1/16 (LSB) 0.0001 0 0 0 1 0001 1
2/16 = 1/8 0.0010 0 0 1 0 0010 2
3/16 = 1/8 + 1/16 0.0011 0 0 1 1 0011 3
4/16 = 1/4 0.0100 0 1 0 0 0100 4
5/16 = 1/4 + 1/16 0.0101 0 1 0 1 0101 5
6/16 = 1/4 + 1/8 0.0110 0 1 1 0 0110 6
7/16 = 1/4 + 1/8 + 1/16 0.0111 0 1 1 1 0111 7
8/16 = 1/2 (MSB) 0.1000 1 0 0 0 1000 8
9/16 = 1/2 + 1/16 0.1001 1 0 0 1 1001 9
10/16 = 1/2 + 1/8 0.1010 1 0 1 0 1010 10
11/16 = 1/2 + 1/8 + 1/16 0.1011 1 0 1 1 1011 11
12/16 = 1/2 + 1/4 0.1100 1 1 0 0 1100 12
13/16 = 1/2 + 1/4 + 1/16 0.1101 1 1 0 1 1101 13
14/16 = 1/2 + 1/4 + 1/8 0.1110 1 1 1 0 1110 14
15/16 = 1/2 + 1/4 + 1/8 + 1/16 0.1111 1 1 1 1 1111 15
Source: Adapted from Ref. [4].
Additional bits simply provide more fine structure without affecting the full-scale
range. To illustrate these relationships, Table 5.1 lists 16 permutations of 5-bit’s worth
of 1’s and 0’s, with their binary weights, and the equivalent numbers expressed as
both decimal and binary integers and fractions.
When all bits are “1” in natural binary, the fractional number value is 1 − 2 −n ,
or normalized full-scale less 1 LSB (1 − 1/16 = 15/16 in the example). Strictly
speaking, the number that is represented, written with an “integer point,” is 0.1111
(= 1 − 0.0001). However, it is almost universal practice to write the code simply as
the integer 1111 (i.e., “15”) with the fractional nature of the corresponding number
understood: “1111” → 1111/(1111 + 1), or 15/16.
For convenience, Table 5.2 lists bit weights in binary for numbers having up to
20 bits. However, the practical range for the vast majority of sensors rarely exceeds
16 bits.
The weight assigned to the LSB is the resolution of numbers having n bits. The
dB column represents the logarithm (base 10) of the ratio of the LSB value to unity
[full scale (FS)], multiplied by 20. Each successive power of 2 represents a change
of 6.02 dB [i.e., 20 log 10 (2)] or “6 dB/octave.”
5.4.2 V/F Converters
The voltage-to-frequency (V/F) converters can provide a high-resolution conversion,
and this is useful for the sensor’s special features as a long-term integration (from
seconds to years), a digital-to-frequency conversion (together with a D/A converter),
a frequency modulation, a voltage isolation, and an arbitrary frequency division and