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This blog is all about system dynamics modelling and simulation applied in the engineering field, especially mechanical, electrical, and ...

### Theory of Energy Conversion in Wind Turbine

In wind turbine, the wind energy is converted to first mechanical energy, and then this energy is converted to electrical energy. Damping is an essential part for a generator. However, it is not the key factor for energy conversion. If there were no damping or loss, we would have 100% efficient conversion. This is not possible in real life, as we would have certain losses during the energy conversion process. These losses are represented by non-conservative forces, such as friction, viscous damping etc. which are the essential parts to be considered in the energy conversion process. I want to show the fundamentals behind energy conversion in DC motor.

DC motor converts electrical energy (input voltage) to mechanical energy (shaft rotation). This electromechanical conversion involves Faraday’s law of induction and Ampere’s law for force generated on the conductor moving in a magnetic field. In ideal situation, the torque (T) developed on the motor shaft is proportional to the input current (I) and the induced electromotive force (EMF) (V) or back EMF is proportional to the speed (W) of the motor. This can be expressed as [1];

T = K1 I ..........................................................(1)

V = K2 W .......................................................(2)

Where, K1 and K2 are the proportionality constant.
The electrical power (Pe) input to the motor is the product of the induced EMF and current.

Pe = VI = K2 W T / K1  ..................................(3)

And, the mechanical power output (Pm) is the product of the speed of the motor and torque.

Pm = T W .......................................................(4)

Now, by comparing equation (3) and (4), the following relation is obtained.

Pe = (K2 /K1) Pm  ...........................................(5)

From Ohm’s law, it is known that,

E - V = I R  ..................................................(6)

Where, E is the input voltage to the motor, and R is the resistance of the motor armature.
Moreover, we also know that torque produced at the motor shaft is equal to the product of the inertia of the load (J) and rate of change of angular velocity or angular acceleration.

T = J (dW/dt)  .............................................(7)

Now, from equations (1), (6) and (7), it is found that

J (dW/dt) = K1 I = K1 / R (E - V)  ...........................................(8)

Using equation (2) further, the following expression can be established.

dW/dt = (K1 K2 / J R) W + (K1 / J R) E  ..................................(9)

The above equation refers to the first order linear differential equation model where ‘W’ represents the state of the system and ‘E’ is the external control input. This first order equation is good enough to predict the output speed of the motor. However, in terms of measuring the position’ it is necessary to add the following equation.

W = dθ / dt  ...........................................................................(10)

Where θ is the output position of the DC motor and refers to another state of the system. Therefore, the model has one control input and two state variables (position and velocity).

By the above mathematical analysis, I want to specify, that the electrical energy is converted to mechanical energy by a gyrator which has a constant ratio K. Here, resistance of the armature is the energy loss during the conversion of electrical to mechanical energy which is also mechanical equivalence of a damper. A damper not only signifies the energy dissipation/loss from a system, but also helps to make a system stable by removing oscillations.

Reference
[1] Control system design : an introduction to state-space methods / Bernard Friedland.—Dover ed.