EQUIVALENT CIRCUIT OF AN INDUCTION MOTOR BASIC INFORMATION AND TUTORIALS

The Equivalent Circuit of an Induction Motor

An induction motor relies for its operation on the induction of voltages and currents in its rotor circuit from the stator circuit (transformer action).  This induction is essentially a transformer operation, hence the equivalent circuit of an induction motor is similar to the equivalent circuit of a transformer. 

The Transformer Model of an Induction Motor

A transformer per-phase equivalent circuit, representing the operation of an induction motor is shown below:

	The transformer model or an induction motor, with rotor and stator connected by an ideal transformer of turns ratio aeff.

 

As in any transformer, there is certain resistance and self-inductance in the primary (stator) windings, which must be represented in the equivalent circuit of the machine.  They are - R₁ - stator resistance and

X₁ – stator leakage reactance

Also, like any transformer with an iron core, the flux in the machine is related to the integral of the applied voltage E₁.  The curve of mmf vs flux (magnetization curve) for this machine is compared to a similar curve for a transformer, as shown below:

The slope of the induction motor’s mmf-flux curve is much shallower than the curve of a good transformer.  This is because there must be an air gap in an induction motor, which greatly increases the reluctance of the flux path and thus reduces the coupling between primary and secondary windings.  The higher reluctance caused by the air gap means that a higher magnetizing current is required to obtain a given flux level.  Therefore, the magnetizing reactance X_m in the equivalent circuit will have a much smaller value than it would in a transformer.

The primary internal stator voltage is E₁ is coupled to the secondary E_R by an ideal transformer with an effective turns ratio a_eff.  The turns ratio for a wound rotor is basically the ratio of the conductors per phase on the stator to the conductors per phase on the rotor.  It is rather difficult to see a_eff clearly in the cage rotor because there are no distinct windings on the cage rotor. 

E_R in the rotor produces current flow in the shorted rotor (or secondary) circuit of the machine.

The primary impedances and the magnetization current of the induction motor are very similar to the corresponding components in a transformer equivalent circuit. 

WIND TURBINE BLADE TWIST BASIC INFORMATION AND TUTORIALS

Comparing a propeller turbine blade with an airplane wing, you will see that the airplane wing is flat relative to a turbine blade, which has a twist; that is, the tip of a blade is not parallel to the blade root. This twist is not much, and can be only a few degrees, depending on the blade length.

The reason why a blade must be twisted by an angle and not fl at can be seen from the fact that in order to have a good lift force on a blade the air fl ow must hit the blade at a proper angle.

When a blade rotates, the points at the tip side go faster than the points near the root. Since the wind speed is approximately the same for all the points on a blade, the relative speed of air fl ow with respect to the blade is different for points along the length of a blade.

Figure 4.4 illustrates the relative speed due to the combination of wind speed and the blade motion. Th e angle shown by φ (phi) in the fi gure is the angle of attack. This angle, as shown in Figure 4.5, is not the same for segments of the blade in the tip area, in the middle or at the blade root.

 FIGURE 4.4 Relative speed of air fl ow over different parts of a blade is the reason for blade twist.

FIGURE 4.5 If a blade is not twisted in design, a correct desired angle of attack cannot

be maintained.

In order to have more or less the same angle of attack for all the segments of a blade, these segments must encounter the wind at diff erent angles, as shown in figure 4.4. Otherwise some segments have very inappropriate angles of attack.

In fact, for the three segments shown in figure 4.4, the direction of the relative speed determines the twist angle. In figures 4.4 and 4.5 you are looking at a blade in a direction along the blade and toward the shaft.

Figure 4.5 shows a case where a blade has no twist. In such a case, the chord lines of all the segments of a blade are parallel. As shown, the angle of attack can be correct for certain parts of the blade, but is not correct for the other parts.

MECHANICAL POWER IN A TURBINE BASIC INFORMATION AND TUTORIALS

Mechanical power in a turbine is the amount of power that the turbine harnesses from wind.We can determine the available power based on the wind speed, air density, and the turbine size (blade diameter).

Out of this amount of power, the rotor of a turbine can harness up to a maximum theoretical value of 59%. In fact, a turbine rotor can harness between 0% and a maximum value, which cannot exceed 59%.

That maximum value depends on the quality of the rotor design. For instance, for a particular turbine it can be 52%. This is the energy on the shaft before the gearbox represents its efficiency.

Any device, such as a gearbox or a generator, has a definite efficiency. So, the final energy output from the turbine, aft er the gearbox and the generator, will be even less.

The efficiency determines how much of the input power is available on the device output. For example, in a generator, how many kilowatts of electrical energy can it deliver for each 100 kW of mechanical power input (certainly it is less than 100, since part of the power converts to heat during the process)?

Based on the preceding discussion, the energy grasped by the rotor of a turbine is further reduced in the succeeding components. In this section, nevertheless, we are not going to bring the effect of the efficiency of the components into picture.

The intention is to study the power grasped by a turbine rotor. In particular, we want to see if there is any difference between cases when a rotor rotates at different angular speeds.

When a turbine is stationary (it is not working), it grasps 0% of the wind energy. Th is is when a turbine is yawed out of wind and its blades are feathered. In this case, one wants a turbine not to grasp any power from wind.

The position when a turbine is yawed out of wind and its blades are feathered corresponds to the minimum power grasp. Any small amount of power that the rotor may grasp is canceled by the rotor brakes (in order to make sure that there is no rotor motion).

When a turbine is yawed into the wind, the blades capture the wind and a torque is created in the rotor shaft. In addition to the wind speed, the air density, the blade size, and the blade airfoil form, the magnitude of this torque depends on the pitch angle of the blades, if this angle can vary.

In fact, changing the pitch angle alters the design of the blade. So, we need to study
a. Th e effect of the angular speed change in a turbine, and
b. Th e effect of changing the blade pitch angle.

DARRIEUS TURBINE BASIC INFORMATION AND TUTORIALS

What is a Darrieus turbine?

A Darrieus turbine or Darrieus machine is more or less similar to an H-rotor in terms of having a vertical axis and working based on lift forces on the blades.

Th e difference is in the way the blades are attached to the shaft . A Darrieus turbine has the shape of an egg beater. Instead of the blades being attached to the shaft in the center, they are continued from both up and down and are curved.

Th e blades are attached to the shaft at both ends. Figure below shows a picture of a Darrieus machine. Th e curvature of the blades allows the vertical shaft to be supported by a number of guy wires. In this sense, the main tower supporting the turbine shaft does not need to be as solid and strong as it must be in an H-rotor.

A disadvantage of a Darrieus machine is that it does not have a good starting torque. Th is means that at low wind speeds it cannot easily start to rotate. Aft er it starts rotating, however, it has a good torque and can continue to generate electricity.

Despite the two main advantages of vertical-axis turbines (not being sensitive to wind direction change and most of the components being accessed from the ground), not many Darrieus machines have been built in the past, and none is made today. From practicality, preference is given to propeller turbines.

Darrieus turbine
A vertical-axis wind turbine that looks like an eggbeater.

WIND TURBINE CLASSIFICATIONS BASIC INFORMATION AND TUTORIALS

What's the difference between HAWT and VAWT?

When air flows around an object, two forces act on the object, drag and lift . Accordingly, we have turbines that work based on either of these forces. Th us, in general, we have lift -based (or lift -type) turbines and drag-based (or drag-type) turbines.

Th is categorization is based on the type of active force that makes the turbines turn. Turbines can also be classifi ed based on their axis, whether it is horizontal or vertical. Axis here refers to their main shaft about which the rotating parts revolve.

Certain turbine types can work only with a horizontal axis, while others can work with a horizontal axis or a vertical axis, and even they can be installed with their axis at an angle. In this sense, a wind turbine can be classifi ed as a horizontal-axis wind turbine (HAWT) or a vertical-axis wind turbine (VAWT).

Even without more details about any particular turbine, one can see a major diff erence between a horizontal-axis wind turbine and a vertical-axis wind turbine. Since in most cases wind blows horizontally, a wind turbine whose axis is horizontal (HAWT) is sensitive to the direction of wind.

Th is is not true for a turbine with vertical axis (VAWT), because no matter what the direction of wind, such a turbine can catch the wind.

Another advantage of a vertical-axis turbine is the fact that all the other equipment such as generator and gearbox do not need to be on the top of the tower, as is usually the case for a HAWT. So, they are easier to access when necessary.

Tip: A horizontal-axis wind turbine is sensitive to wind direction.

BELTZ LIMIT - WIND TURBINE ENERGY ABSORPTION

Any moving object has energy. This type of energy is called kinetic energy. For example, a car, a bicycle, or a ball, when moving, all have kinetic energy. If they stop, that energy is gone. The same is true for moving air, that is, the wind.

The amount of energy of a moving object depends on two factors, its mass and its speed. Using the proper units for measuring mass and speed, the relationship to determine the energy of a moving object is as follows:

Energy ½ (mass)(speed)2.

This equation implies that, for example, if mass is doubled (that is, if you have two objects the mass of one of which is two times the mass of the other, boTh having the same speed), the energy doubles.

But, if the speed is doubled (if you have two objects of the same mass, but one has a speed two times that of the other), then the energy is four times more. It is very important to understand the relationship given by an equation similar to that in.

This energy can be converted to work. In other words, it can be used to do work (see the text on energy and power in this chapter for the technical meaning of work).

As was mentioned earlier, for wind or any other moving fl uid it is more practical to fi nd the power rather than the energy. In this sense, in equation we can substitute for the mass of the air that fl ows in 1 sec, and that gives the power in the wind.

Note that
Mass (Density)(Volume) (2.2)

The power in a tunnel of wind is proportional to the air density, the cross-sectional area of the tunnel of wind, and the cubic power of the wind speed.

The cross-sectional area in this equation refers to the size of a turbine.

The following conclusions can be derived from equation:
1. For the same turbine and at the same time, if the wind speed doubles, the power in the wind increases by a factor of 8.

2. For the same turbine and the same wind speed, if the weather is cold (higher density), more power exists in the wind available for the turbine.

3. In the same weather conditions and for the same wind speed, a turbine that is two times larger in cross sectional area than a smaller one has twice as much wind power available to it.

From item 2 above it can also be concluded that a turbine at a given wind speed can produce more power in the winter than in the summer because the weather is colder. The same conclusion can be extended from day to night if the temperature diff erence is signifi cant. This diff erence due to temperature change is, nevertheless, not very much.

Power absorption by a turbine
In the previous section the available power in a stream of wind was defined in terms of the wind speed, turbine size, and the air density. This power, however, is the power that exists in the wind when it is blowing.

A turbine cannot necessarily capture all of this power; it can only absorb a portion of it. This depends on the type of turbine, the efficiency, and other conditions in the operation of a turbine.

In order to show the fraction of power in the wind that a particular wind turbine can harness from wind (i.e., wind harnessing), a coeffi cient is used in equation (2.4). Thus, we can say that

Power of a wind turbine (A coeffi cient)(Power in the wind) This coeffi cient must, obviously, be smaller than 1. It is called power coeffi cient.

Wind turbine power ½ (Power coeffi cient)(Density)(Cross–sectional area)(Speed)3 .The power coeffi cient depends on how good a turbine is in design and how well it can grasp the wind energy.

Thus, its value can be small or large. Nevertheless, there is a maximum value that no turbine in its best performance can exceed. It can be theoretically determined and is called the Betz limit. The value for Betz limit is =16/27 = 0.59.

Betz limit = 16/27 = 0.59 The power coeffi cients for certain turbines can reach values near the Betz limit. A value of 0.50 for a good design is acceptable. For others, this number can be smaller, say 0.25 or even 0.20. Any claim for coeffi cients greater than the Betz limit are groundless and can be rejected.