JNTU B.Tech II ELECTRICAL MACHINES I Previous Question Paper (October - 2020 - R13)

In a doubly excited magnetic system, the energy stored in the magnetic field is a function of the currents in the two windings (i₁, i₂) and the mechanical displacement (θ). The field energy W_f is given by the area under the magnetization curves or the integral of the flux linkages with respect to currents.

The total energy balance equation for a magnetic system undergoing a virtual displacement dθ is:

Where dW_e is the electrical energy input, dW_f is the change in stored field energy, and dW_m is the mechanical work done by the system. The electrical energy input is given by:

The mechanical work done is defined as:

Substituting these into the energy balance equation:

Since W_f is a state function, its total differential is:

For a linear magnetic system, the flux linkages are λ₁ = L₁₁i₁ + L₁₂i₂ and λ₂ = L₂₁i₁ + L₂₂i₂. The field energy is W_f = 0.5L₁₁i₁² + L₁₂i₁i₂ + 0.5L₂₂i₂². Substituting the differentials of λ₁ and λ₂ into the energy balance equation and holding currents constant (di₁ = 0, di₂ = 0), the electrical input becomes:

Because the change in field energy at constant current is dW_f = 0.5i₁²dL₁₁ + i₁i₂dL₁₂ + 0.5i₂²dL₂₂, comparing the terms shows that the mechanical work T_edθ must equal the remaining change in field energy. Thus, the electromagnetic torque is:

In a singly excited magnetic system, the magnetic stored energy W_f is defined as the energy stored in the magnetic field when current flows through the coil. For a linear magnetic system, the energy stored is given by the area under the flux linkage (λ) versus current (i) curve.

The fundamental relation for magnetic stored energy is:

For a magnetic circuit with a core of reluctance R, the flux linkage λ is related to the current i by the number of turns N and the magnetic flux φ:

From the definition of reluctance, the flux φ is related to the magnetomotive force (MMF) by:

Substituting this expression for φ into the flux linkage equation:

Now, substitute this expression for λ back into the energy equation W_f = 1/2 λi:

Alternatively, since the MMF is defined as F = Ni, the stored energy can be expressed as:

This expression shows that the magnetic stored energy is directly proportional to the square of the MMF and inversely proportional to the reluctance of the magnetic path.