Efficiency of mechanisms: calculation, formula + examples

Description and principle of operation of the transformer

A transformer is a device used to lower or increase voltage, change the number of phases and, in rare cases, change the frequency of alternating current.

The following device types exist:

power;
measuring;
low power;
pulse;
peak transformers.

A static device consists of the following main structural elements: two (or more) windings and a magnetic circuit, which is also called a core. In transformers, voltage is supplied to the primary winding and removed from the secondary in a converted form. The windings are connected inductively by a magnetic field in the core.

Along with other converters, transformers have an efficiency factor (abbreviated as efficiency ), with the symbol . This coefficient represents the ratio of energy effectively used to energy consumed from the system. It can also be expressed as the ratio of the power consumed by the load to the power consumed by the device from the network. Efficiency is one of the primary parameters characterizing the efficiency of the work performed by a transformer.

Efficiency of heat engines

In the last lesson, we started getting acquainted with heat engines. Let's remember that this is the name of a device that performs mechanical work using the internal energy of fuel.

The simplest heat engine is a cylindrical vessel containing gas under a piston. When the gas is heated, its pressure and volume increase, and the piston begins to move, raising the load to a certain height.

Any heat engine consists of three main elements: a heater, a working fluid (usually gas) and a refrigerator (usually atmosphere or water at ambient temperature).

The energy released during fuel combustion in the heater is transferred to the working fluid through heat transfer. When a gas expands, part of its internal energy goes towards doing work. And some amount of heat is inevitably transferred to the refrigerator. Thus, it turns out that complete conversion of the internal energy of a gas into work is impossible. This is due to the irreversibility of processes in nature. If heat could return spontaneously from the refrigerator to the heater, then the internal energy could be completely converted into useful work by any heat engine. But the second law of thermodynamics prohibits this: it is impossible to create a perpetual motion machine of the second kind, that is, an engine that would completely convert heat into mechanical work.

The energy balance per cycle can be obtained based on the first law of thermodynamics.

For an ideal heat engine, the change in internal energy is zero, since the working fluid has returned to its original state. From here we find that the useful work done by the heat engine is equal to the difference between the amount of heat received from the heater and the amount of heat given to the refrigerator:

The ratio of useful work to the amount of heat that the working fluid received from the heater is called the coefficient of performance of a heat engine (abbreviated as efficiency):

Since part of the heat received from the heater is transferred to the refrigerator, the efficiency of any heat engine is always less than unity:

To obtain the highest possible efficiency, it is necessary to cool the working fluid before compression.

This can be done by adiabatic expansion of the gas, in which its temperature drops to the temperature of a refrigerator. Further, during isothermal compression, the working fluid will transfer a certain amount of heat to the refrigerator. And the most effective way to complete the cycle of a heat engine is by adiabatically compressing the gas to its original temperature. This cycle was first proposed by the French engineer Sadi Carnot, which is why it is also called the Carnot cycle.

You now see the formula for determining the efficiency of the Carnot cycle on the screen:

The efficiency of any real heat engine cannot exceed the efficiency of an ideal Carnot cycle. Carnot's formula gives a theoretical limit for the maximum value of the efficiency of heat engines. It shows that the greater the temperature difference between the heater and refrigerator, the more efficient the engine.

And the efficiency of an ideal heat engine could be equal to unity only if it were possible to use a refrigerator with a temperature equal to absolute zero. But, as we know, this is impossible even theoretically, because absolute zero temperature cannot be reached.

To consolidate the new material, we will solve the problem with you. Task 1.

Each of the four engines of a jet aircraft develops an average thrust force of 0.11 MN over a 5000 km journey. Determine the volume of kerosene consumed along this route if the engine efficiency is 24%. The density and specific heat of combustion of kerosene are respectively 800 kg/m3 and 43 MJ/kg.

To conclude the lesson, we note that the invention of the steam engine, and subsequently the internal combustion engine, by the French engineer Etienne Lenoir in 1860 was extremely important.

Now it is difficult to imagine our life without cars, planes, ships and other devices in which the internal energy of burned fuel is partially converted into mechanical work.

The use of heat engines in energy and transport is of greatest importance. Heat engines - steam turbines - are installed at thermal and nuclear power plants, where steam energy is converted into mechanical energy in the rotors of electric current generators.

Internal combustion engines are installed in cars, motorcycles, helicopters and airplanes, tractors and heavy vehicles. The creation of a jet engine made it possible to lift aircraft to greater altitudes, increase the speed and range of their flights.

However, the intensive use of heat engines in the energy sector and transport has a negative impact on the environment. During operation, heat engines emit huge amounts of hot steam or gas into the atmosphere, which leads to thermal pollution of the atmosphere.

The widespread use of various types of fuel entails an increase in carbon dioxide in the atmosphere, which, when combined with water vapor in the atmosphere, forms carbonic acid and falls in the form of acid rain.

Combustion of fuel at thermal power plants leads to the accumulation of carbon monoxide in the atmosphere, which is poisonous to living organisms. For example, the combustion of a ton of gasoline produces about 60 kg of carbon monoxide.

Scientists and designers see solutions to the problems that arise when burning fuel:

· in the purification of gas emissions into the atmosphere;

· increasing the efficiency of heat engines, in particular, by creating conditions for the most complete combustion of fuel;

· replacing heat engines with more environmentally friendly engines, for example, electric ones;

· use of alternative energy sources.

Types of losses in a transformer

The process of transferring electricity from the primary winding to the secondary is accompanied by losses. For this reason, not all energy is transferred, but most of it.

The design of the device does not include rotating parts, unlike other electric machines. This explains the absence of mechanical losses in it.

So, the device contains the following losses:

electrical, in copper windings;
magnetic, in steel core.

Energy diagram and the Law of Conservation of Energy

The principle of operation of the device can be schematically represented in the form of an energy diagram, as shown in Image 1. The diagram reflects the process of energy transfer, during which electrical and magnetic losses are generated.

According to the diagram, the formula for determining the effective power P2 is as follows:

P2=P1-ΔPel1-ΔPel2-ΔPm (1)

where, P2 is useful, and P1 is the power consumed by the device from the network.

Denoting the total losses ΔP, the law of conservation of energy will look like: P1=ΔP+P2 (2)

From this formula it is clear that P1 is spent on P2, as well as on the total losses ΔP. Hence, the efficiency of the transformer is obtained in the form of the ratio of the supplied (useful) power to the consumed power (the ratio of P2 and P1).

What is the efficiency of a transformer and what does it depend on?

Efficiency (the full interpretation of this abbreviation) is the ratio of useful electricity to that supplied to the device.

In addition to energy, the efficiency indicator can be determined by calculation based on power indicators based on the ratio of the useful value to the total value. This characteristic is very important when choosing a device and determines the effectiveness of its use.

The magnitude of the efficiency depends on the energy losses that are allowed during the operation of the device. These losses are of the following types:

electrical - in the conductors of the coils;
magnetic - in the core material.

The magnitude of these losses when designing a device depends on the following factors:

overall dimensions of the device and shape of the magnetic system;
compactness of the coils;
density of assembled sets of plates in the core;
diameter of wire in coils.

Reducing losses in the unit is achieved during the design process of the device, using soft magnetic ferromagnetic materials for the manufacture of the core. Electrical steel is assembled into thin plates, insulated relative to each other by a special layer of applied varnish.

Determination of efficiency by direct measurements

The formula for calculating efficiency can be presented in several versions:

(3)

This expression clearly reflects that the efficiency value of the transformer is not more than one, and is also not equal to it.

The following expression determines the net power value:

P2=U2*J2*cosφ2, (4)

where U2 and J2 are the secondary load voltage and current, and cosφ2 is the power factor, the value of which depends on the type of load.

Since P1=ΔP+P2, formula (3) takes on the following form:

(5)

Electrical losses of the primary winding ΔPel1n depend on the square of the current flowing in it. Therefore, they should be defined this way:

(6)

In its turn:

(7)

where rmp is the active winding resistance.

Since the operation of an electromagnetic device is not limited to the nominal mode, determining the degree of current load requires the use of a load factor, which is equal to:

β=J2/J2н, (8)

where J2н is the rated current of the secondary winding.

From here, we write down expressions for determining the secondary winding current:

J2=β*J2н(9)

If we substitute this equality into formula (5), we get the following expression:

(10)

Note that determining the efficiency value using the last expression is recommended by GOST.

Summarizing the information presented, we note that the efficiency of a transformer can be determined by the power values of the primary and secondary windings of the device at rated mode.

The golden rule of mechanics

It must be remembered that winning at work cannot be achieved using a simple mechanism.

Let us express each of the works in formula (3) as the product of the corresponding force and the path traveled under the influence of this force, then we transform formula (3) to the form:

\[F_1s_1\approx F_2s_2\left(4\right).\]

Expression (4) shows that using a simple mechanism, we gain in strength as much as we lose in travel. This law is called the “golden rule” of mechanics. This rule was formulated in ancient Greece by Heron of Alexandria.

This rule does not take into account the work of overcoming friction forces, therefore it is approximate.

Determination of efficiency by indirect method

Due to the large efficiency values, which can be equal to 96% or more, as well as the uneconomical nature of the direct measurement method, it is not possible to calculate the parameter with a high degree of accuracy. Therefore, its determination is usually carried out by an indirect method.

Summarizing all the obtained expressions, we obtain the following formula for calculating the efficiency:

η=(P2/P1)+ΔPm+ΔPel1+ΔPel2, (11)

To summarize, it should be noted that a high efficiency indicator indicates the efficient operation of the electromagnetic device. Losses in the windings and core steel, according to GOST, are determined by testing no-load or short circuit, and measures aimed at reducing them will help achieve the highest possible efficiency values, which is what we need to strive for.

Why do you need to calculate efficiency?

How to calculate electrical energy consumption

A clear example of an insufficiently efficient device is the classic incandescent lamp. Passing current through a tungsten coil increases the temperature of the conductor. In operating mode, a significant amount of power consumption is spent on generating radiation. However, only a small part of the spectrum belongs to the visible part of the spectrum. Since the generated heat does not perform a useful effect, the corresponding energy costs should be recognized by the excess ones.

If we express efficiency in terms of power in this case, we must also take into account durability. This technique increases the accuracy of the assessment, since it implies the need to periodically replace a damaged emitter.

In typical operating mode, an incandescent lamp heats the filament to 2600-2800K. With this value, the service life is 900-1200 hours, efficiency is from 5 to 7%. The efficiency can be increased by 2-5 times by increasing the temperature to 3400-3600K. However, in this option, durability is reduced to 5-6 hours. Such practical characteristics cannot be considered satisfactory.

Comparison of efficiency and other parameters of different types of lamps

This table demonstrates the superiority of economical light sources. The service life of modern LEDs is measured in tens of thousands of hours. Even at the final stages of operating cycles, high brightness and high-quality distribution of spectral components are ensured.

Finding the current in a complete circuit

How to calculate amperes

To study the efficiency of energy consumption in electrical engineering, you can use basic formulas. A complete circuit, by basic definition, considers a current source (I) with an internal resistance (r). The connected load consumes a certain amount of power. It is characterized by electrical resistance R.

The passage of current through such a circuit is ensured by the energy of the source, which is determined by the value of the electromotive force (EMF - E). It can be expressed as the ratio of the work (A) performed by external forces to move a charge (q) with a positive sign along the corresponding contour. Taking into account the well-known formula I = q / t , it is easy to determine the relationship between the quantities under consideration:

A = E * I * t,

where t is the control time interval.

Separately, you can consider areas with internal and external resistance. Each of them releases a quantity of heat determined by the Joule-Lenz law Q = I2 * R * t. Since the energy does not disappear without a trace, we can correctly conclude that Q = A. Substituting the values into the original expression, we get:

E = I*R + I*r.

The EMF of the complete circuit is calculated by adding two voltage drops in the internal and external sections. An elementary transformation allows you to find out the current strength in the corresponding conductor:

I = E/ (R+r).