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Since the input to ADC saturates at 10 V, the maximum output from tachometer
should be 10 V. Hence, the maximum speed that can be measured is
10 V
w = = 5krpm = 5000 rpm (6.83)
max
2V∕krpm
The measurement error due to ripple voltage is
0.25 0.25
E = ⋅ 10 V = 0.025 V or ⋅ 5000 rpm = 12.5 rpm (6.84)
r
100 100
The measurement error due to the ADC resolution is one part in 2 12 since ADC has 12-bit
resolution over the full range of measurement (±10 V = 20 V range),
20 V 20 1
E ADC = = = 0.00488 V or ⋅ 10 000 rpm = 2.44 rpm (6.85)
2 12 4096 4096
If the ADC is 12-bit resolution, the measurement error is dominated by the ripple voltage.
Measurement error due to ADC resolution is negligable. If the ADC is 8-bit, then the
measurement error due to the ADC resolution is
20 V 20 1
E ADC = = = 0.078 V or ⋅ 10 000 rpm = 39.0625 rpm (6.86)
2 8 256 256
In this case, the measurement error due to ADC resolution is larger than the error due to
the ripple voltage of the sensor.

6.5.2 Digital Derivation of Velocity from Position Signal

In most motion control systems, velocity is a derived information from position mea-
surements. Accuracy of the derived (estimated) velocity depends on the resolution of the
position sensor. The velocity can be derived from the position measurement signal either by
analog differentiation using an op-amp circuit or digital differentiation using the sampled
values of the position signal.
The ideal op-amp differentiator has the following input signal to output signal rela-
tionship,
d
V (t) ≈−K (V (t)) (6.87)
out in
dt
where V (t) represents the position sensor signal, V out (t) represents the estimated velocity
in
signal.
Clearly, digital derivation is more flexible since it allows more specific filtering and
differentiation approximations,
ΔX
V = (6.88)
T
sampling
Velocity Estimation using Encoders at Very Low Speed Motion Appli-
cations An interesting method of velocity estimation in low speed applications using
encoders is to determine the time period between two consecutive pulse transitions in order
to increase velocity estimation accuracy. In very low velocity applications, the number of
pulse changes in an encoder count within a given sampling period may be as low as one
or two counts. If one of the count changes occurs at the edge of the sampling period, the
velocity estimation can be different by 50% whether that count occured right before or
right after the sampling period ended. Notice that typically, T sampling = 1 ms, and ΔX is 1 or
2 counts in low speed applications. One count variation in the position change, depending