handbook of modern sensors

induced in the liquid. The magnitude of the e.m.f. is defined by
11.5 Electromagnetic Sensors 371
v = e − e ′ = 2aBv, (11.21)
where a is the radius of the tube of flow and v is the velocity of flow.
By solving Maxwell’s equations, it can be shown that for a typical case when the
fluid velocity is nonuniform within the cross-sectional area but remains symmetrical
about the tube axis (axisymmetrical), the e.m.f generated is the same as that given by
Eq. (11.21), except that v is replaced by the average velocity, v a [Eq. (11.3)]:
v a = 1 ∫ a
πa 2 2πvrdr, (11.22)
where r is the distance from the center of the tube. Equation (11.21) can be expressed
in terms of the volumetric flow rate:
0
v = 2B
πa . (11.23)
It follows from Eq. (11.23) that the voltage registered across the pickup electrodes is
independent of the flow profile or fluid conductivity. For a given tube geometry and
the magnetic flux, it depends only on the instantaneous volumetric flow rate.
There are two general methods of inducing voltage in the pickup electrodes. The
first is a dc method where the magnetic flux density is constant and induced voltage is a
dc or slow-changing signal. One problem associated with this method is a polarization
of the electrodes due to small but unidirectional current passing through their surface.
The other problem is a low-frequency noise, which makes it difficult to detect small
flow rates.
Another and far better method of excitation is with an alternating magnetic field,
which causes the appearance of an ac voltage across the electrodes (Fig. 11.11).
Naturally, the frequency of the magnetic field should meet a condition of the Nyquist
rate; that is, it must be at least two times higher than the highest frequency of flowrate
spectrum variations. In practice, the excitation frequency is selected in the range
between 100 and 1000 Hz.
Fig. 11.11. Electromagnetic flowmeter with synchronous (phase-sensitive) demodulator.