of Microprocessors

230 MUSICAL ApPLICATIONS OF MICROPROCESSORS
Digital
Analog x Vref
00000000 0.000000
00000001 0.003941
00000010 0.007882
01111110 0.496586
01111111 0.500527
10000000 0.499473
10000001 0.503414
11111110 0.996059
11111111 1.000000
As can be seen, an increase in the digital input from 01111111 to
10000000 results in a slight decrease in analog output, a classic manifestation
of nonmonoionicity. Except for this step, the rest are 0.003941Vre! high.
Some additional calculation will reveal that the maximum allowable value of
1R for monotonic performance is (1+1/128)R, at which point the voltage
levels for 01111111 and 10000000 are the same. If 1R were too small by the
same amount, this step would be twice the size of the others, which still
gives a lLSB differential linearity error but preserves monotonicity. It can
also be easily determined that the allowable percentage error for less significant
resistors doubles for each bit toward the least significant end. In general,
though, all of the resistors will have some error, so individual resistors will
have to be more precise to guarantee differential linearity better than lLSB.
A rule of thumb that will always work is that the resistor corresponding to
bit N should have a tolerance better than 1/2 N + 1 . Thus, the most significant
resistor of a 12-bit DAC should have a tolerance of ±0.024% or better.
•
Even if the resistors were perfect, the analog switches used have a finite
on resistance, which adds to the effective resistance of each bit. Ifall switches
have the same internal resistance, proper ratios are destroyed and linearity
suffers again. The effect ofswitch resistance can be minimized by making the
weighted resistors very large but then speed suffers. Also, stable, tight
tolerance resistors in the megohm range are difficult to find. Sometimes the
switches are scaled in size, and therefore resistance in proportion to the bit
significance to maintain proper ratios in spite of high switch resistance.
Generally, this is practical only for the most significant few bits because of
the wide range in resistor values. In any case, it is usually necessary to trim
the most significant few bits with a potentiometer or high-value parallel
"trimming" resistors.
Note that a finite output load has no effect on the linearity of the
circuit. If a load of value R was connected from the output to ground in the
example in Fig. 7-5, the four voltage levels would be altered to 0, 0.2Vre!,
0.4Vre!, and 0.6Vref. Even a short circuit load would provide output currents
of 0, 0.5Vre!/R, 1.0Vref/R, and 1.5Vre!/R. Thus, the equivalent circuit of the
converter can be represented as a variable-voltage generator in series with a