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What is Efficiency?
The meaning of “efficient” in statistics isn’t much different from the dictionary definition of efficient:
“…capable of producing desired results without wasting materials, time, or energy. ~ Miriam Webster.
In other words, an efficient procedure produces results that maximize your use of materials, time and energy. You’ll use less energy if you have smaller sample sizes, for example. So a procedure that can work with a smaller sample is usually more efficient than one that requires a larger sample.
Efficiency can refer to any procedure you want to optimize. For example, an efficient experimental design is one that produces your desired experimental results with the minimum amount of resources (e.g. time and money). An efficient hypothesis testing procedure also conserves resources in the same way. Generally, the most efficient hypothesis test, experimental design or estimator is going to be the one with the fewest observations.
What is Relative Efficiency?
Relative efficiency has a couple of meanings, depending on the context:
- As another name for relative precision in sampling. If you use simple random sampling to find the mean of a large population, relative precision and relative efficiency are equal. In other cases, they may not be equal.
- As a reference to a gold standard — a “best possible” procedure. It is the ratio of the potential procedure against the gold standard. for example, 75/100 indicates the prospective procedure is 75% as efficient as the theoretically best possible procedure.
What is an Efficient Estimator?
An estimator is a statistic that estimates some fact about the population. for example, the X̄(the sample mean) is an estimator for the population mean, μ. Your options for finding X̄ are limitless: you could have a sample of ten, fifty of three hundred and one. You could use different classes, ages, or heights (depending on what you are trying to estimate). You could use a simple random sampling technique, or a more complex one like stratified sampling. Out of all these possible scenarios, an efficient estimator is one that has small variances (the estimator with the smallest possible variance is also called the “best” estimator). In other words, the estimator deviates as little as possible from the “true” value you are trying to estimate.
The more formal definition of an efficient estimator is covered in the next article :Cramer-Rao Lower Bound (CRLB) Simple Introduction
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Transformer Efficiency & Energy Loss2020-12-18 10:00:49Transformer Efficiency & Energy Loss Transformer efficiency is of great importance in the world ofElectrical Transformers. Our study in this article will analyze and understand the importance ...
Transformer Efficiency & Energy Loss
Transformer efficiency is of great importance in the world of Electrical Transformers.
Our study in this article will analyze and understand the importance of studying the transformer’s efficiency. I will discuss the subject in a practical and scientific in-depth to get the desired goal.
[This video can help you understand better, although we will explain it in detail later.]
Before listing the Five Facts in detail, we must first clarify the transformer’s efficiency.
Primarily, any transformer on earth has input power and output power. We know that the Output Power is smaller than the Input Power due to the transformer’s power losses.
Therefore, the efficiency can be defined as the Ratio between the output power and the input power, as is evident from the following equations:
we can represent efficiency as
And also may be
So, we can obtain the transformer’s efficiency from any equation above according to the given data.
Let us now turn to essential facts in turn.
The First Fact
Is there any transformer that has an efficiency of 100 %?
The efficiency of the ideal transformer Is 100%. For those who do not know the ideal transformer, the input power is equal to the output power. However, this transformer is not available in reality. It is only a case for Scientific and Theoretical Studies.
Looking at most of the topics that speak about the transformer’s efficiency does not address this fact.
The Second Fact
Why do we analyze and study the efficiency of the transformer?
Of course, there are thousands but millions of learners worldwide chanting the transformer efficiency equations every day, but without a real understanding of the goal of studying and analyzing it.
In fact, we study and analyze transformers’ efficiency to determine the transformer’s performance, which is, of course, reflected on the Economic Side.
And all of us know that the Higher The Transformer’s Efficiency, the less power lost in the transformer, the costs of periodic maintenance of the transformer, and the transformer’s cost of Energy Consumption.
The Third Fact
Is loading the transformer have any effect on the transformer’s efficiency?
It means that the efficiency may be Variable depending on the load of the transformer.
Many articles speak about calculating the transformer’s efficiency but do not mention that the transformer is more than efficient and where the disagreement occurs.
And Here Comes The Vital Question: Is The Efficiency Of The Transformer Fixed Or Variable?
Let me clarify it with the equations as it is the best evidence to answer this question.
By studying the losses inside the transformer, we find two types of losses: Copper Loss and Iron Loss.
Suppose a transformer has a power of 1000 kVA, and the iron losses are 1.2 kVA. In the case of non-loading (no-load condition), the value of the copper losses is zero.
- So, Compensation in equation no. 4 :
- In the case of the load condition, copper losses have a value suppose that is 9.2 KVA.
Therefore, the higher the load, the greater the copper losses, and the lower the efficiency.
And through this fact, we can deduce the maximum efficiency that the transformer can reach.
So, We Can Derive The Equation Of Maximum Efficiency Of Transformer:
Suppose that (X) is the fraction of the full load then compensation in equation no.3:
The copper losses vary according to the transformer’s loading condition (X fraction of the load).
Also, efficiency will be maximum if the denominator concerning the variable copper losses is equated to zero.
We can conclude that we will obtain maximum efficiency if copper losses (Pc) are equal to iron losses (Pi).
At Maximum Efficiency So,
Also copper losses
(Where Pc Is The Full Load Copper Losses)
For maximum efficiency — >
Final Equation At Maximum Condition.
The Fourth Fact
Does the power factor of the load effect on the efficiency of the transformer?
Again, I will clarify in numbers as is the best evidence to prove the facts:
- Suppose there is a transformer with a power of 100 kVA and the iron losses has 0.2 kVA and the current in the secondary coil is 8 amperes with a lag power factor of 0.8, then :
- For the same transformer and the same load but the current in the secondary coil is 8 amperes with a lag power factor of 0.6, then:
Therefore, the Higher The Load power factor, the Less The Efficiency when the load is constant.
The Fifth Fact
Is the efficiency of the transformer calculated and analyzed only momentarily?
After identifying the above facts, we can understand that the transformer efficiency is Variable all the time, but will the efficiency be analyzed and recorded every hour, every day, or every month?
As we explained earlier, this efficiency has many factors. Most importantly, the download and the loading are not fixed; we have to conduct a comprehensive study Every Hour to produce a daily report to analyze the transformer’s performance.
But how is the transformer efficiency calculated throughout the day? The answer is straightforward and lies in the following Equations:
Note that: iron loss is fixed all day, and Copper loss changes according to the loading condition
10 Secrets About Losses In Transformer
Previously, we talked about the transformer’s efficiency and how this efficiency affects the transformer’s lifetime. Here, we will discuss more on losses in the transformer and how to reduce them.
The losses of any machine in a simple scientific way are the amount of Difference between the Input Power to the machine and the Output Power coming out of it.
But we will not let things go this way and as we would go deeper into the transformer to see what exactly happens to consume this power within the transformer and the factors affecting it.
The First Secret
We previously understood the ideal transformer and its characteristics. It has become clear to us that the ideal transformer is found only in the World Of Theories and has no practical presence on the ground.
So, the ideal transformer’s efficiency is 100%, so the ideal transformer’s losses are 0%.
That means the Input Power is equal to the Output Power, and this is theoretically only.
The Second Secret
Most of the scientific sites that deal with this subject have classified the transformer’s losses into two parts, namely Iron Loss and Copper Loss. Still, The secret here is that the losses within the transformer are divided into 6 categories.
- Hysteresis losses.
- Eddy current losses.
- Dielectric losses.
- Copper losses.
- Stray losses.
- Conductor eddy current losses.
Now I will reveal in detail the set of secrets for each type of loss in a transformer.
The Third Secret
The third secret about No-Load Losses in details are below:
When we disconnect the transformer’s loads while still connecting it to the source, the transformer continues to draw power from the source. We consider this power as a loss because there is No Output.
But this power has a Key Benefit that is making the transformer energized and ready for service.
This power withdraws when the transformer is not loaded; it also remains a lost power while loading the transformer with the same value.
We can divide these losses into three main types: A lot of sites classified no-load losses as hysteresis losses and eddy current losses only … but no-load losses divide into 3 Types Of Losses:
a. Hysteresis Losses.
b. Eddy’s current Losses.
c. Dielectric losses.
The Fourth Secret
The fourth secret about Hysteresis Losses in details:
This loss is part of the core losses and is also a type of no-load losses, that is, whether the transformer is connected to a load or not.
The idea of this type of loss is is that “every time the magnetic field is reflected, a fraction of the power is lost as a result of hysteresis.”
So, what’s the meaning of hysteresis? This word means the tendency of Magnetism to stay in its old state.
And the fact that a part of the magnetism remains in the iron core is called “Residual Flux.” Whiles, a lost part of the magnetic power within the iron core, is called “Hysteresis Losses.” Their effect is represented in the circuit equivalent to the real transformer in the form of a winding with an Impedance Xm.
When passing the sinusoidal current in the primary winding, it produces the Magnetic Flux. This magnetic flux is quite similar to the created voltage, and it has the same sinusoidal curve.
In the first half of the sinusoidal cycle, this magnetic flux causes magnetization of the core’s Magnetic Material. We can consider being internally composed of a group of domains aligned in one direction because of the magnetic field.
When the magnetic field direction is Reversed in the second half of the second cycle of the sinusoidal curve, the domains inside the core must change their direction. The poles must align parallel in the new direction of the field.
Thus we can consider hysteresis loss as the Lost Power during the friction of the particles of matter and the movement of its poles in each cycle of the magnetic field.
In fact, the energy consumed in the row of poles of the magnetic material comes from the input power. Of course, it does not transfer this energy to the secondary side, which does not move to the load and then considers it Lost Energy.
Calculation Of Hysteresis Losses
We conclude from this that the Stronger the magnetic field and the Higher the frequency, the higher the energy consumed in the row and the change of the direction of the electrodes inside the material.
We can calculate the value of hysteresis losses from the following equation:
We can calculate the value of hysteresis loss in another way, through the area within the loop known as the hysteresis loop as in the following figure:
So, the larger the area within the loop, the greater the hysteresis losses.
From the previous figure:
It is clear that we can reduce hysteresis losses by improving the quality of the core material.
The Fifth Secret
The fifth secret in details about Eddy Current Losses:
We all know that magnetic materials such as iron also have the ability to connect the electrical current and not only to pass the magnetic flux. When the flux lines cut off the wires of copper winding, they generate a great e.m.f under the Law Of Faraday.
When they cut off these winding, they also cut off Iron core, which generates an electric current called Eddy Current.
Of course, these eddy currents are not desirable. They do not reach the load and cause the iron core to warm unnecessarily. Usually, eddy currents cause about 50% of the iron core loss.
This type of loss depends mainly on the type of magnetic material, frequency, and magnetic field density.
We can calculate according to the following formula:
Such that :
Ke is the constant that depends on the type and thickness of the material.
Reduce Eddy Current Losses
The value of the thickness of the Magnetic Material has a strong effect on the Eddy current loss. So, the less thickness, the greater the electrical resistance and the less the current passing through the iron core.
Therefore all the iron cores of all electrical transformers should be in the form of Lamination separated from each other and compressed together.
It involves transformer lamination core1 and transformer lamination core2.
In terms of magnetism, the lamination set gives the area of the cube of the iron core ability to withstand the passage of the flux.
In the electric sense, it is High Resistance to the section’s small area in each slice. Usually does not exceed the thickness of the single lamination is 0.35 mm. There is only a small current, and this can be a way to reduce the eddy current.
The Sixth Secret
The seventh secret in detail is about Dielectric Losses:
The dielectric losses are classified under No-Load Losses… they are present during loading and the transformer’s no-load condition. The insulation materials used to isolate the conductors inside the transformer causes a kind of capacitor known as the Stray Capacitors.
They are capacitors that are not visible to the eye but do the same work as the real capacitor and cause some kind of Energy Loss.
In fact, we can represent the ideal capacitor by Capacitance only without resistance… where there is no loss in power, it charges by the first half of the sine wave and then discharge in the second half. And then charged and discharged and so on…
And therefore does not lose anything of the active power, so the angle between the voltage and the current 90 Degrees. But this is only Theoretically Possible… There is no capacitor, in fact, only capacitance, but always with small resistance.
The Severity Of Dielectric Loss
So, the angle between voltage and current is less than 90 degrees by a small angle called Delta (Δ).
Tan δ is a standard measure to indicate the magnitude of the capacitor spacing as an ideal capacitor. The smaller the angle, the closer the capacitor is to the Ideal Capacitor.
Thus, dielectric losses are proportional to the other with tan δ, and this amount called “Dissipation Factor”… and we can calculate the dielectric losses from the following equation:
Although dielectric loss is a small fraction of the transformer’s losses, it is one of the Most Dangerous types of losses in the transformer because temperature strongly affects Tan Δ. “The higher the temperature, the more tan δ.”
The Seventh Secret
The seventh secret in details about Load Losses:
These load losses appear only during the transformer’s loading as a result of the passage of the load current through the Winding. Therefore, it consists mainly of Copper Losses in winding resistance, either in the primary or secondary winding.
The importance of calculating load losses is a fundamental element when Estimating The Size of the transformer. The heats resulting from the passage of the current in the winding raises the temperature to dangerous degrees.
So it is necessary to work to reduce these losses, which is often by reducing the Value Of Resistance winding.
We can calculate load losses through the Short Circuit Test of the transformer.
Load losses divided into three main types:
a. Copper losses.
b. Stray losses.
c. Leakage flux losses.
The Eight Secret
The eighth secret about Copper Losses in details:
These losses are the First Type Of Load Loss because it does not appear as an effective value until after loading the transformer. The more the transformer loads, the higher power losses are.
It is known that the copper winding (primary and secondary) has a certain resistance, and therefore the passage of a current which Causes Loss Of Power is calculated from the following equation:
For accuracy, the copper winding loss is at No-Load Condition because of the passage of no-load current. The no-load current is the current that passes through the Primary Winding only at the rated voltage with the fact that the other winding open and often within 1% to 2% of full load.
Effect Of Temperature On Copper Losses
As the temperature rises, the value of winding resistance increases in contrast to the eddy current loss, decreasing with temperature.
Statistically, each 1 ° C Increase can cause a rise in Losses Of 0.4%.
It is known that the value of the resistance is calculated from the law (R = ρ *L /A). But this law is absolutely true in the case of DC Current only if the current AC, which is a kind of approximation acceptable and not accurate. Because the AC tends to pass at the ends of the connector away from its center, it causes a skin effect.
The concentration of the current in the sides makes the area of the actual section of the connector less than the conductor’s engineering area, and then Rac increases the value of Rdc, and this is the skin effect.
Therefore, the copper losses increase more if the current of AC and, of course, show this effect clearly whenever the area of the section of the conductor is larger … We can neglect this effect in small conductors.
The Ninth Secret
The ninth secret about Stray Losses has the following details:
These stray losses produce a result of the Leakage Flux … We know that the flux generated when the current passes in a winding is not completely coupled to the other winding, but there is a missing part.
We can express this missing part of the flux as leakage flux. This leakage flux may cut off the External Iron parts of the transformer and create the Eddy Current.
The Tenth Secret
The tenth secret about Conductor Eddy Current losses in details:
This eddy current causes the hotness of these Non-Current Metal Parts. This is a type of energy loss that shows an effect on large transformers only.
This type of loss is generated in the Copper Conductors of the winding due to leakage flux. Still, it has small values, and its effect is weak … its value is calculated by Experimentation and Measurement and not by equations.
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