Page 377 - Mechatronics with Experiments

October 9, 2014 8:1
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JWST499-c06
JWST499-Cetinkunt
SENSORS 363 254mm×178mm
where the voltage due to back electromotive force is
̇
V bemf = K vw ⋅ (t) (6.77)
If we neglect the effect of inductance, L = 0, then the tachometer relationship can be
approximated
̇
R ⋅ i(t) =−K vw ⋅ (t) (6.78)
̇
V (t) =−K (t) (6.79)
R
vw
̇
(t) =− 1 ⋅ V (t) (6.80)
R
K vw
where V (t) is the voltage across the resistor which can be easily measured (but note that
R
resistance R varies with temperature), and K vw is known by design of the tachometer. Then
the speed can be calculated.
By design, tachometers have L and R parameters that result in a small current (i).
In steady-state, the output voltage is proportional to the shaft velocity. The proportionality
constant is a parameter determined by the size of the tachometer, type of winding, and
permanent magnets used in its construction. Ideally, the gain of the tachometer should be
constant, but in reality it varies with temperature (T) and rotor position ( ) due to a finite
number of commutators. Hence, the sensor output voltage and angular speed relation can
be expressed as,
̇
V (t) = K ⋅ (t) (6.81)
out vw
where
K = K (T, ) (6.82)
vw vw
The value of the gain at rated temperature and nominal rotor position is called the K vwo .
The gain changes as a function of temperature. In addition, due to the finite number of
commutators, the gain has a periodic ripple as a function of rotor position. The frequency of
the ripple is equal to the number of commutators. The ripple due to commutation is reflected
on the output voltage which is generally less than 0.1% of the maximum output voltage.
The parameters that specify the performance of a tachometer are
1. speed to voltage gain, K (V∕rpm),
vw
2. maximum speed, w max , limited either by the bearings or the magnetic field saturation,
3. inertia of the rotor,
4. maximum expected ripple voltage and frequency.
A typical dynamic time constant of a tachometer is in the range of 10–100 μs.
For very high speed applications, the winding and magnets are designed such that
K (V∕krpm) is a low value, whereas for very sensitive low speed applications, it is
vw
designed such that the K is a high value.
vw
Example Consider a tachometer with a gain of 2 V∕1000 rpm = 2V∕krpm. It is inter-
faced to a data acquisition system through an analog to digital converter (ADC) which
has 12-bit resolution and ±10 V input range. The sensor specifications state that the ripple
voltage due to commutators on the tachometer is 0.25% of the maximum voltage output.
1. Determine the maximum speed that the sensor and data acquisition system can
measure.
2. What are the measurement errors due to the ripple voltage and due to ADC resolution?
3. If the ADC was 8-bit, which one of the error sources is more significant: ripple
voltage or ADC resolution?