Ballistic Galvanometer
Ballistic galvanometers
are the measuring instruments which are used for measuring the quantity of electric
charges obtained from magnetic flux. Its construction is similar to
the moving coil galvanometer and it consists of two additional
properties.
- It consists of extremely small electromagnetic damping.
- It consists of undammed oscillations.
Working Principle of Ballistic Galvanometer
The working principle of ballistic galvanometer is that the charge measured by the ballistic galvanometer must be passed
through the coil. So, the coil starts moving. When the charge flows through the
coil, it gives rise to a current due to the torque produced in the coil. This
torque acts for a short time. The product of the torque and the time period provides
a force to the coil and the coil starts rotating. When the initial kinetic
energy of the coil is completely used in doing work, the coil starts moving
back to its original position. Thus, the coil oscillates in the magnetic field
and the deflection is noted from which charge can be calculated.
Construction of Ballistic Galvanometer
In the construction of ballistic galvanometer, a ballistic galvanometer consists of a circular or a rectangular coil of a copper wire of almost 10
to 15 turns. This coil is suspended in a radial field between the concave pole
pieces of a strong magnet. when the coil rotates in the magnetic field,
an EMF is induced across the coil according to the lenz’s law it
opposes the motion of the coil and this is known as electromagnetic damping. To
minimize the electromagnetic damping the coil should be wound on a
wooden frame and the whole suspension is enclosed in a metal case provided with
glass faces.
Theory of Ballistic Galvanometer
The torque developed by the coil at any point of time is:
Where L is the length, W is the width, n is
the number of turns of the coil and B is the air gap flux density.
The torque of acceleration is:
Where J is the moment of inertia of the coil and w is the
angular velocity. If the coil is closed to its zero point then the discharge
takes place and the torque of suspension is zero. The value of the driving
torque is equal if the damping torque is neglected. During the short
discharge period:
By integrating:
Where the subscript zero refers the conditions at the end of the
discharge time. The integral form of the Eq. (4) is the amount of the charge
that has passed through the coil. Therefore:
The above equation indicates the velocity of the coil acquires
from the pulse is proportional to the quantity of charge that passed through
it.
During the actual motion, the deflection torque is zero and the
equation of motion is:
Where D is the damping constant, S is the control constant and
is
the deflection in radians. Thus,
Where A and B are constant m1 and m2 are imaginary, Then the
initial conditions are:
Under this condition the solution may be written as:
Where
The deflections in Eq. 9 are proportional to
and from Eq. 12 the deflections are
proportional to Q.
The amplitude of the first swing
is:
The ratio of successive swings is found by exponential
multiplier for time interval
t = π/β. The ratio of successive swings is:
The natural logarithm of this ratio is:
The third swing in the same manner is as follows:
In general:
In case of critical damping, Eq. (9) will be written as:
and
Maximum deflection is found for
or
t = 2J/D.Substitute this value in Eq. 15 and call the deflection
, then:
or
Summarizing the results in the following equation of the charge
passing through the galvanometer:
The working units of the Eq. (19) are:
K2 = galvanometer sensitivity in millimeter deflection
Θ = deflection in millimeters
Q = charge in micro coulombs
Measurement of Electric Flux by Ballistic
Galvanometer
To measure the magnetic flux of a bar magnet, the bar magnet is
surrounded by a coil connected in series with a variable resistor and a
galvanometer. The series resistor provides critical damping and it is used to
control the sensitivity of the magnetic flux. This sensitivity is controlled by
adjusting the number of turns in a coil. When the magnet is suddenly withdrawn
from the coil, an impulse is produced in a coil for few seconds and the
deflection of the galvanometer is taken as a measure of the flux. The induced
voltage in the coil are:
Where flux is measured in webers and N is the number of turns in
a coil. If R is the total resistance of the circuit including series resistor
and a galvanometer then the current flowing to the circuit is:
or
The quantity of charge passed through the galvanometer is:
Deflection of the galvanometer is:
or
Where K2 is the sensitivity factor and it must be properly evaluated
for the resistance used in the test measurements.
Calibration of Ballistic Galvanometer
The calibration of ballistic galvanometer can be done in so many
different ways. Some methods of calibration are as follows:
By Capacitor:
In this method, a capacitor is charged through the
voltage and is discharged by the galvanometer. The resistor and a switch S2
is used to bring the galvanometer to its zero position quickly after a
deflection. The capacitor is charged through the upper position of the switch S1
and is discharged by the contacts of this switch S1 in the lower
position. The discharged quantity of electricity and the capacitance of the
capacitor is calculated so the constant K2 is divided by the observed
deflection. This is the undamped sensitivity because of the infinite resistance
of the galvanometer. A shunt is added in the parallel to the series resistor
and a galvanometer. This shunt provides damping and if the shunt is in critical
value then the action is sluggish and the damping conditions are improved with
the combination of shunt and series resistances.
This method is not used commonly because it is difficult to
measure the exact amount of capacitance of the capacitor and the damping of the
galvanometer is different during the operation of test.
By Standard Solenoid:
This method is mostly used for calibration purposes. In this
method, a standard solenoid of a long coil is wound on a cylinder of a
nonmagnetic material. The length of a solenoid is at least 1 meter and the
diameter is of 10cm.the winding should be uniform and its length must be that
its field strength H is of 10000A/m or more when maximum current is applied on
a coil. The calibration is done by means of a known flux. The flux linking to
the coil is:
Where N1 are the primary turns/meter, I1 is the primary current
in amperes, A is the cross-section area of the coil m2.
This arrangement creates a flux change twice, so by
substitution:
Where N2 are the turns of the coil, R is resistance of the coil
and galvanometer circuit.
The calibration for flux measurements is in convenient form,
once sensitivity factor K2 is evaluated. If the galvanometer is used for the
measurement of unknown flux, then it will be written as:
where
is
the unknown flux change,
is
the deflection in millimeters and
is
the number of turns in the coil.
By Mutual Inductance:
This method is used to measure the large range of calibration. It
consists of a mutual inductor and is very small in size as compared to a
solenoid. In this method, if the mutual inductance is known in the
circuit then a deflection φ1 is produced by the reversal of a known primary
current I is observed. By changing primary current:
Let R be the total resistance of the galvanometer circuit then the
galvanometer current is:
By integration
So,
and
