A.3 Formulae for the Phenomenological Analysis
A.3.2 Approximate Analytical Expressions
4.1.3 Bimetallic Movement
Angle ~ i2
The conductor exists of two laminated materials with different thermal expansion coefficients. By measuring a current the conductor is warmed up and curves. The re-sulting angle is the value of the current.
Fig. 4.1.4: Bimetallic Movement
In practice the bimetallic band is wrapped to a spiral where the pointer is fastened.
For the compensation of the variations in environment temperature serves a second spiral with an opposed direction of winding. Bimetallic movements are used switchboards and are frequently equipped with a trailing pointer to show the maxi-mum value.
Advantage: Simply, durably, high torque
Disadvantage: low measuring accuracy, big thermal slowness
4.2 Forces on Electric Charge
If you bring an electric charge into a magnetic field, so a force is exerted on these par-ticles which depend on the size and the speed of the electric charge as well as on the strength of the magnetic induction at the position of the particle. The direction of the resulting force is normal to the movement of the electric charge and to the direction of the induction.
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Fig. 4.2.1: Coil in a Magnetic Field
Vectorial writing shows:
(
v B)
Q
F
⋅ x
= (4.2.1)
with the absolute value:
( )
v BB v Q
F = ⋅ ⋅ ⋅sin , (4.2.2)
replacing now
= ⋅
⋅
=
⋅ i
Q t v
Q (4.2.3)
And if you assume that the angle is a right one between the direction of the current and the induction, you get the force:
B i
F = ⋅ ⋅ (4.2.4)
with sin
( )
i,B =1If you now consider a rectangle coil with the length L and w sinuosity in the homo-geneous magnetic field and the axis of rotation of the coil stands normal to the induc-tion lines, so you find the torque in dependence to the projected distance r⋅cosα.
α cos 2⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= L w i B r
Mel (4.2.5)
with A= 2⋅r⋅Lthe surface of the rectangle coil it also applies:
α
⋅cos
⋅
⋅
⋅
=w A i B
Mel (4.2.6)
Technical applications of this effect are moving-coil measuring instruments and elec-trodynamic measuring mechanism.
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4.2.1 Moving-Coil Measuring Instrument with a Permanent Magnet
In moving-coil measuring instruments the coil moves in the air gap of a permanent magnet, so that the resulting moment is a function of the current to be measured.
i k
Mel = i⋅ (4.2.7)
With a spiral spring the mechanical counter moment, the deflecting force MR, is estab-lished in relation to the torsion angle.
α
⋅
= R
R k
M (4.2.8)
Now the coil twists itself so far, till that the sum of the moments becomes zero. Then it applies:
If the current i is an alternating one, the pointer should oscillate around the zero posi-tion, but as the result of its inertia you get an average value. Therefore it applies i.e., however, to a sinusoidal current:
⋅( )
⋅ ⋅Therefore you can not directly measure an alternating current with a moving coil measuring instrument.
Nevertheless doing so you add a full wave rectifier. Now the movement of the pointer is the arithmetic average value formed about the half oscillation (average ab-solute value or mean modulus).
⋅( )
⋅ ⋅The arithmetic average value corresponds to a direct current which has the same electrolytic effect, how a half oscillation of the considered alternating current (electric rectifier technology).
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Moving-coil measuring instruments are damped electro-dynamically, so that me-chanical attenuation elements are obsolete. By the movement of the moving coil a voltage is induced. The so-called shut down resistor RS allows a current flow in op-position to the measuring current.
Moving-coil measuring instruments are suitable to measure currents up to 10 mA di-rectly. According to direction of the axis the smallest measuring range will be 0 to 1 μA (vertically) or 0 to 10 μA (horizontal).
As long as the current is measured directly, the temperature dependence of the coils resistance has no influence on the measuring accuracy, because the whole measuring current I0 flows through the coil. If, however, you must extend the measuring range, you can do this by switching a resistor in parallel to the measuring element. How-ever, now the current divides by the resistance values therefore a heating of the coil by the measuring current I0 the displayed value is incorrect. Therefore, the compen-sation is done by switching a " temperature independent "resistance RVW in series with the coil. Its value should amount a multiple of the coils resistance. Then for the shunt RN to extent the working rang you will find:
R0
R
RN = S − (4.2.12)
with R0 =RM +RVW .
The extension is mostly not limited to one rang, but it encloses several ranges with the associated shunts. In practice the shunts for several current measuring ranges are divided by the method of Ayrton, so that apart from the external measuring circle the resistors RM, RVW and RN form a constant shut down resistor RS of the measuring ele-ment which is given by the needed damping.
Fig. 4.2.2: Measuring Range Extension by Ayrton
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A measuring element with its resistor RM = 200Ω and its current for full-scale deflec-tion I0 =200μA is given. The shut down resistor should be RS = 500Ω. The combina-tion of shunts has to be calculated for the measuring ranges 1,0 mA, 10 mA, 0,1 A and 1,0 A.
The measuring element should be extended to a voltmeter with the measuring ranges 100 mV, 1,0 V, 10 V and 100 V and the smallest current measuring range should be used.