Motor Performance Analysis and Electromagnetism Fundamentals

Motor Performance Problem

A motor has a resistance of 250 Ω and operates at 100 V at 25 °C. The temperature coefficient of resistance (α) at 25 °C is 0.0018 /°C. We need to determine the following:

Questions for Analysis

1. The resistance value when the temperature is 50 °C.
2. The current absorbed from the network when the temperature is 25 °C and 50 °C.
3. The cost of the energy consumed in 120 minutes at 50 °C if the cost per kWh is $1.
4. At what temperature and why does the motor consume less energy?

Principles of Electromagnetism

Induced Electromotive Force (EMF)

When a conductor of length (l) moves with a constant velocity (v) within a magnetic field of induction (B), an electromotive force (EMF) is induced across its ends.

• Perpendicular Motion: If the conductor's trajectory is perpendicular to the magnetic field, the induced EMF (e) is given by:

  e = B × l × v

• Oblique Motion: If the conductor moves obliquely to the magnetic field lines, the component of velocity perpendicular to the magnetic field determines the induced EMF. If θ is the angle between the velocity vector and the magnetic field vector, the induced EMF is:

  e = B × l × v × sin(θ)

  The induced EMF is maximum (Emax) when sin(θ) = 1 (i.e., θ = 90°, perpendicular motion), so Emax = B × l × v. Therefore, e = Emax × sin(θ).

• Circular Motion: If a conductor moves in a circle with a constant angular speed (ω) within a uniform magnetic field, an electromotive force is continuously induced across its ends, varying with its instantaneous orientation relative to the field.

Magnetic Coulomb's Law

Magnetic Coulomb's Law describes the force between two magnetic poles. It states that the force of attraction or repulsion between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them. This force also depends on the medium in which they are situated.

• Attraction: Magnets of opposite polarity (North and South) attract each other.
• Repulsion: Magnets of like polarity (North-North or South-South) repel each other.

Magnetic Field

A magnetic field is a region in space where magnetic forces are exerted on magnetic materials or moving electric charges. It is represented by imaginary lines, known as magnetic field lines or lines of force. These lines are continuous and closed, emerging from the North Pole and entering the South Pole of a magnet.

The intensity of the magnetic field is directly proportional to the density of these lines; a higher number of lines crossing a unit area indicates a more intense magnetic field. The pattern of magnetic field lines can be visualized using iron filings, which align themselves along these lines, forming a magnetic spectrum.

Magnetic Permeability

Magnetic permeability (μ) is a measure of a material's ability to support the formation of a magnetic field within itself. It quantifies the extent to which magnetic field lines can pass through a substance.

• Relative Permeability (μr): This is the ratio of a material's permeability to the permeability of a vacuum (μ₀).
• Permeability of Vacuum (μ₀): A fundamental physical constant, approximately 4π × 10-7 H/m (Henry per meter).

Types of Magnetic Substances

Magnetic substances are classified based on their response to an external magnetic field:

• Ferromagnetic Substances:
  • These materials are strongly magnetized when placed in a magnetic field.
  • They significantly concentrate magnetic field lines within them.
  • Their relative permeability (μr) is much greater than 1 (e.g., iron, nickel, cobalt).
• Paramagnetic Substances:
  • These materials are weakly magnetized in the direction of the applied field.
  • They slightly concentrate magnetic field lines, allowing them to pass through.
  • Their relative permeability (μr) is slightly greater than 1 (e.g., aluminum, platinum, oxygen).
• Diamagnetic Substances:
  • These materials are weakly magnetized in the direction opposite to the applied field.
  • They tend to disperse magnetic field lines.
  • Their relative permeability (μr) is less than 1 (e.g., bismuth, copper, water).