Measurement and Instrumentation - Deflecting Torque (Td) of Dynamometer Type Instruments in Terms of Mutual Inductance
1 Answer
In dynamometer-type instruments, deflecting torque (Td) is the torque applied to the moving coil of the instrument due to the current flowing through it. The deflecting torque is responsible for positioning the pointer or indicator of the instrument against the restraining torque to indicate the measurement being taken.
The expression for the deflecting torque (Td) in terms of mutual inductance (M) in a dynamometer-type instrument can be derived as follows:
The general expression for the torque in an electromechanical system is given by:
Torque = Force × Lever Arm
In the case of a dynamometer-type instrument, the torque exerted on the moving coil is due to the interaction of the magnetic field produced by the current in the coil with the external magnetic field (usually produced by a permanent magnet) in which the coil is placed. This interaction results in a force on the coil that tries to align it with the external magnetic field.
The force (F) on a coil carrying current (I) placed in a magnetic field (B) is given by the formula:
F = BIL
Where:
B is the magnetic field strength
I is the current flowing through the coil
L is the length of the coil's active conductor in the magnetic field
The lever arm (L) is the perpendicular distance from the axis of rotation to the line of action of the force (which is the radius of the coil). Let's denote it as R.
Substituting the force formula into the torque formula, we get:
Torque = BIL × R
Now, the mutual inductance (M) between the moving coil and the external magnetic field is a measure of how strongly the coil's magnetic field interacts with the external magnetic field. It's defined by the formula:
M = k × √(Lm × Le)
Where:
k is a constant
Lm is the self-inductance of the moving coil
Le is the self-inductance of the external magnetic field (usually produced by a permanent magnet)
The self-inductance of the moving coil (Lm) is related to the coil parameters.
Substituting the expression for M into the torque formula:
Torque = BIL × R = M × I
This equation relates the deflecting torque (Td) to the mutual inductance (M) and the current (I) flowing through the coil.
It's important to note that this derivation provides a simplified explanation and doesn't account for certain factors like the angle between the coil axis and the magnetic field direction. Also, real-world instruments might have additional complexities and factors that influence their behavior.
The expression for the deflecting torque (Td) in terms of mutual inductance (M) in a dynamometer-type instrument can be derived as follows:
The general expression for the torque in an electromechanical system is given by:
Torque = Force × Lever Arm
In the case of a dynamometer-type instrument, the torque exerted on the moving coil is due to the interaction of the magnetic field produced by the current in the coil with the external magnetic field (usually produced by a permanent magnet) in which the coil is placed. This interaction results in a force on the coil that tries to align it with the external magnetic field.
The force (F) on a coil carrying current (I) placed in a magnetic field (B) is given by the formula:
F = BIL
Where:
B is the magnetic field strength
I is the current flowing through the coil
L is the length of the coil's active conductor in the magnetic field
The lever arm (L) is the perpendicular distance from the axis of rotation to the line of action of the force (which is the radius of the coil). Let's denote it as R.
Substituting the force formula into the torque formula, we get:
Torque = BIL × R
Now, the mutual inductance (M) between the moving coil and the external magnetic field is a measure of how strongly the coil's magnetic field interacts with the external magnetic field. It's defined by the formula:
M = k × √(Lm × Le)
Where:
k is a constant
Lm is the self-inductance of the moving coil
Le is the self-inductance of the external magnetic field (usually produced by a permanent magnet)
The self-inductance of the moving coil (Lm) is related to the coil parameters.
Substituting the expression for M into the torque formula:
Torque = BIL × R = M × I
This equation relates the deflecting torque (Td) to the mutual inductance (M) and the current (I) flowing through the coil.
It's important to note that this derivation provides a simplified explanation and doesn't account for certain factors like the angle between the coil axis and the magnetic field direction. Also, real-world instruments might have additional complexities and factors that influence their behavior.