Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks
This article points out the erroneous calculations of power loss and voltage loss in long-distance transmission lines in high school physics textbooks; Based on the basic circuit structure of long-distance power transmission, provide a series of correct functional relationships between power transmission power, line loss power and loss voltage, transmission line current and transformer ratio, load resistance, transmission line resistance, and the relationship between power supply voltage and transmission voltage, and discuss related issues based on these functional relationships.
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Long-Distance Power Transmission, Loss of Power, Loss of Voltage, Incorrect Calculation, Power Factor
1. Background of the Problem
Regarding the calculation and analysis of power loss and voltage loss in long-distance transmission lines, physics textbooks in high school and various versions have always adopted similar methods, which analyze or calculate conclusions based on the relationship between transmission power, transmission current, and transmission voltage. During the 1980s, textbooks
LIU Kehuan, Xing Huilan, Ma Dongling, Du Min. Senior middle School Textbook Physics (B) Volume 2 [M]. Beijing: People's Education Press. 1984. 12(1): 173-175.
[1, 2]
only used this as a qualitative analysis without quantitative calculations, and there seemed to be no abnormal phenomena in the teaching process. However, after entering this century, some textbooks
[3]
Liao Boqin. Curriculum Standard Experimental Textbook Physics Elective 3-2 for ordinary high school [M]. Shandong: Shandong Science and Technology Press. 2011. 7(4): 75-78.
[4]
Shu Bingru, He Runwei. Curriculum Standard Experimental Textbook Physics Elective 3-2 for ordinary high school [M]. Shanghai: Shanghai Science and Technology Education Press. 2007. 2(2): 62-65.
[5]
Research Center of Physics Curriculum Materials, Institute of Curriculum Materials, People's Education Press. Senior high school curriculum Standard Experimental Textbook Physics Elective 3-2 [M]. Beijing: People's Education Press. 2010. 4(3): 45-47.
[3-5]
almost all provided specific cases for quantitative calculations, but due to the lack of consideration of the matching relationship between transmission line resistance and load and transmitted power in the calculation process, the calculation results seriously violated the law of conservation of energy, resulting in a ridiculous phenomenon.
In this regard，Earlier literature
[6]
Huang Shaoshu, Zhou Huahai. Analysis of the relationship between transmission Power and line loss power in long-distance transmission [J]. Physics Teaching. 2017. 39(4): 13, 6.
[7]
Huang Shaoshu, Wu Shouchong. Basic Research on Power loss of non-ideal lines in long-distance transmission [J]. Asia-Pacific Journal of Physics. 2019. 1(2): 25-29.
[6, 7]
pointed out this erroneous phenomenon and discussed in detail the issue of power loss in transmission system lines as a function of load and other factors. However, the commonly used new textbooks
[8]
LIAO Boqin. Secondary School Textbook Physics Optional Compulsory Volume 2 [M]. Shandong: Shandong Science and Technology Press. 2019. 12(1): 69-70.
have not corrected the errors of previous textbooks in response to this phenomenon, but have also made the same mistakes.
2. Serious Errors in High School Physics Textbooks
The example cases attached on page 70 of the Physics Selective Compulsory Volume 2 of the first edition of the general high school textbook published by Shandong Science and Technology Press in December 2019, as shown in Figure 1, the analytical thinking of this kind of case exposes the errors of high school physics textbooks in the calculation of transmission power, line loss power and loss voltage in long-distance transmission system.
[Example] A certain power station transmits power , and uses a resistance transmission line to transmit electrical energy outward. What is the power loss on the transmission line if a voltage of is used for power transmission? What is the voltage loss? If we switch to high voltage transmission with , what would be the power and voltage loss on the transmission line?
[Analysis] Based on the transmission power and voltage, the transmission current can be calculated, and then the lost power and voltage can be determined.
[Solution] According to , the transmission current when using voltage transmission
(1)
Power loss of transmission lines
(2)
Lost voltage
(3)
When switching to high-voltage transmission, the transmission current
(4)
Power loss of transmission lines
(5)
Lost voltage
(6)
Obviously, when transmitting a certain amount of power, the higher the transmission voltage, the smaller the current in the transmission line, and the less energy is lost due to heat generation in the wire, resulting in less voltage loss on the line.
2.2. Error Exposure and Root Causes
From the analytical approach to the above problems, it can be concluded that the power loss of a transmission line is directly proportional to the square of the transmission power of the power source and inversely proportional to the square of the transmission voltage; The loss voltage of transmission lines is directly proportional to the transmitted power of the power source and inversely proportional to the transmitted voltage. On the surface, this analysis may seem to have no problem, but with a little deeper research, its errors become apparent. To clarify the issue, a question is added on the basis of the original question: What is the power and voltage loss on the transmission line if voltage transmission is used instead?
At this point, according to the above idea, it is easy to calculate the transmission current
(7)
Power loss of transmission lines
(8)
Lost voltage
(9)
It is not difficult to see that the power loss of transmission lines is much greater than the transmission power, and the voltage loss is also much higher than the transmission voltage. In fact, according to this approach, as long as the transmission voltage is lower than a certain value (for example, in this problem, it can be calculated that the transmission voltage is lower than ), the loss power of the transmission line is greater than the transmission power, and the loss voltage is also higher than the transmission voltage. Moreover, the lower the transmission voltage, the greater the power loss of the transmission line compared to the transmission power, and the higher the voltage loss compared to the transmission voltage. Obviously, this seriously violates the law of conservation of energy and is a rather absurd result.
The root cause of this error lies in the fact that the current in the transmission line is not unilaterally determined by the transmitted power and voltage, but is closely related to factors such as the resistance, load, and transformer ratio of the transmission line.
3. Determining Factors of Transmission Power
In order to have a clear understanding of the determining factors of the transmission power and line loss power of the transmission system. In this paper, the ideal transmission line (the transmission line has no resistance) and the non-ideal transmission line (the transmission line has resistance) are respectively analyzed and discussed through the ideal transformer as the transmission link.
Firstly, it should be clarified that a transformer is an electrical energy transmission component that integrates self inductance and mutual inductance based on the principle of electromagnetic induction. According to literature
[12]
Huang Shaoshu, Chen Hai. Mathematical and logical Explanation of the doubts in Transformer Teaching [J]. Physics Teaching. 2019. 41(4): 18-20.
[12]
, the working process of transformers has a rigorous mathematical logic relationship; Under no-load conditions, the current in both the primary and secondary coils of an ideal transformer is zero.
3.1. Ideal Transmission Line
The basic structure of an ideal transmission line is shown in Figure 2, which includes three circuits. The current and related physical quantity symbols of each circuit have been indicated in the diagram. Let the transformation ratio of the step-up transformer be ( ), the transformation ratio of the step-down transformer be ( ), and the load be a pure resistor component.
Due to the fact that ideal transformers belong to pure inductive components, the power factor of their primary and secondary coils is 0, while the power factor of pure resistive loads is 1. Therefore, the transmission power of the generator (i.e. active power, the same below) is equal to the power consumed by the load, and there is no loss in the transmission line. According to the law of conservation of energy, there are
(10)
According to the transformation ratio relationship of the transformer, 、 、 、 ，So, equation (10) can be rewritten as
(11)
This indicates that in an ideal transmission line, the output power of the generator is determined by the generator (power source) voltage or transmission voltage, transformer ratio, and load size.
According to equation (11), it can be seen that under ideal transmission line conditions, the transmission power of the generator (power source) is proportional to the square of the transmission voltage (or power source voltage), proportional to the square of the transformer ratio, and inversely proportional to the resistance value of the load resistor; When the transmission system is in an unloaded ( ) state, the output power of the generator is equal to zero.
3.2. Non Ideal Transmission Lines
The basic structure of a non ideal transmission line is shown in Figure 3, which differs from Figure 2 only in the presence of resistance in the transmission line between two ideal transformers. For ease of comparison, the physical quantity symbols are labeled with the same symbols as in Figure 2.
In this non ideal transmission line, the output power (active power) of the generator is equal to the power consumed by the load and the power lost by the resistance of the transmission line. According to the law of conservation of energy, there are
(12)
Similarly, based on the transformation ratio relationship of the transformer, 、 、 、 ，Therefore, equation (12) can be rewritten as
(13)
This expression indicates that in non ideal transmission lines, the output power of the generator is determined by the generator (power source) voltage or transmission voltage, transformer ratio, and the resistance of the load and transmission line.
According to equation (13), it is not difficult to understand that in non ideal transmission lines, the transmission power of the generator (power source) is proportional to the second power of the transmission voltage (or power source voltage), proportional to the second power of the step-up transformer ratio, monotonically increases with the increase of the second power of the step-down transformer ratio, and monotonically decreases with the increase of the resistance of the transmission line and load resistance (or monotonically increases with the decrease of the resistance of the transmission line and load resistance); When the transmission system is in an unloaded ( ) state, the output power of the generator is equal to zero.
4. Correct Calculation of Power Loss and Voltage Loss
4.1. Loss of Power
In order to facilitate the discussion of the dependence between the power loss of transmission lines and the transmission power and voltage, we use to represent the power loss of transmission lines in (13). According to equation (13), we can obtain
(14)
Obviously, combining equations (13) and (14), it can be concluded that the power loss of a transmission line is not only related to the transmission power or transmission voltage, power supply voltage, but also to the transmission line resistance, load resistance, and transformer ratio, and the power loss cannot be greater than the transmission power. Under the condition of constant load, transmission line resistance, and transformer ratio, the loss power of the transmission line is proportional to the square of the transmission voltage (or the square of the power supply voltage).
Figure 5. Variation of Line Loss Power with Line Resistance.
It is not difficult to see from equation (14) that if the resistance of the transmission line remains constant, when the load resistance (short circuit in the transmission system), the power loss of the line is equal to the transmitted power; When the load resistance (transmission system unloaded) is applied, the power loss of the line is equal to zero, as shown in Figure 4. If the load resistance remains constant, when the transmission line resistance , the line loss power is equal to zero. When the transmission line resistance is constant, the transmission line loss power is equal to the transmitted power, as shown in Figure 5.
4.2. Voltage Loss
The calculation of the loss voltage of the transmission line is also not difficult to obtain according to the ratio relationship of the transformer
(15)
This indicates that the loss voltage of transmission lines is related to the resistance of transmission lines, load resistance, and transformer ratio, and the loss voltage cannot be greater than the transmission voltage.
From equation (15), it is easy to see that if the resistance of the transmission line remains constant, when the load resistance (short circuit in the transmission system), the line loss voltage is equal to the transmission voltage; When the load resistance (transmission system unloaded) is present, the line loss voltage is equal to zero, as shown in Figure 6. If the load resistance remains constant, when the transmission line resistance , the line loss voltage is equal to zero. When the transmission line resistance is reached, the line loss voltage is equal to the transmission voltage, as shown in Figure 7.
Figure 7. Variation of Line Loss Voltage with Line Resistance.
4.3. Transmission Current
Further discuss the relationship between current and power supply voltage or transmission voltage in transmission lines. According to equation (15), the current of the transmission line, i.e. the transmission current, can be obtained as
(16)
This is an expression that determines the magnitude of current in a transmission line.
The formula (16) indicates that under the condition of constant load, transmission line resistance, and transformer ratio, the current of the transmission line is proportional to the transmission voltage (or power supply voltage).
5. Conclusion
(1) In long-distance transmission systems, the transmission power of generators (power sources), the loss power and voltage of transmission lines, and the current of transmission lines are determined by equations (13), (14), (15), and (16), respectively. Any speculation that violates science is not acceptable.
(2) Using equations (1) to (9) to calculate the transmission current, loss power, and loss voltage of a transmission line violates the law of conservation of energy and is an extremely absurd and erroneous measure. The reason for this serious error is that the deployment relationship of the transformer ratio is not considered, that is, the essential factor that the current of the transmission line is determined by the formula (16).
(3) From equations (13) or (14) and (15), it can be seen that when the transmission system is in a short-circuit ( ) state, the power or voltage transmitted by the generator (power source) is completely lost on the transmission line, which is not only meaningless but also a very dangerous phenomenon.
(4) In long-distance transmission engineering and teaching, it is usually said that "the line current and loss power of the transmission line decrease with the increase of the transmission voltage", which is not contradictory to the statements in equations (16) and (14) that "the current of the transmission line is proportional to the transmission voltage (or power supply voltage)" and "the loss power of the transmission line is proportional to the square of the transmission voltage (or power supply voltage)", respectively. Among them, the former is a comparison between two or more sets of transmission systems with different transformer ratios under equal transmission power conditions, while the latter is a comparison of the same set of transmission systems with different transmission power conditions determined by the transformer ratio. When comparing the power loss or voltage loss of two or more transmission systems, their transmission currents must be calculated separately according to equation (16) and cannot be arbitrarily assumed.
6. Explanation and Reflection
(1) In transmission engineering, an important indicator of power grid renovation is to change the load capacity and transformation ratio structure of transformers by replacing them based on the regional power consumption (load matching) situation.
(2) Given the complexity of power loss issues in transmission lines, related topics can only be qualitatively analyzed and discussed in high school physics textbooks, college entrance exams, and simulation preparations, and should not be quantitatively calculated. However, such propositions can be fully considered in university independent enrollment, strong foundation programs, and Olympic competitions.
(3) The textbook is the model for teaching and learning activities, and its preparation and review must be careful and rigorous to avoid small mistakes, let alone cause serious mistakes in knowledge.
Author Contributions
Huang Shaoshu: Writing – original draft
Feng JunJie: Writing – original draft
Conflicts of Interest
The author declares that there is no conflict of interest.
LIU Kehuan, Xing Huilan, Ma Dongling, Du Min. Senior middle School Textbook Physics (B) Volume 2 [M]. Beijing: People's Education Press. 1984. 12(1): 173-175.
[3]
Liao Boqin. Curriculum Standard Experimental Textbook Physics Elective 3-2 for ordinary high school [M]. Shandong: Shandong Science and Technology Press. 2011. 7(4): 75-78.
[4]
Shu Bingru, He Runwei. Curriculum Standard Experimental Textbook Physics Elective 3-2 for ordinary high school [M]. Shanghai: Shanghai Science and Technology Education Press. 2007. 2(2): 62-65.
[5]
Research Center of Physics Curriculum Materials, Institute of Curriculum Materials, People's Education Press. Senior high school curriculum Standard Experimental Textbook Physics Elective 3-2 [M]. Beijing: People's Education Press. 2010. 4(3): 45-47.
[6]
Huang Shaoshu, Zhou Huahai. Analysis of the relationship between transmission Power and line loss power in long-distance transmission [J]. Physics Teaching. 2017. 39(4): 13, 6.
[7]
Huang Shaoshu, Wu Shouchong. Basic Research on Power loss of non-ideal lines in long-distance transmission [J]. Asia-Pacific Journal of Physics. 2019. 1(2): 25-29.
[8]
LIAO Boqin. Secondary School Textbook Physics Optional Compulsory Volume 2 [M]. Shandong: Shandong Science and Technology Press. 2019. 12(1): 69-70.
Shaoshu, H., JunJie, F. (2024). Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks. Engineering Physics, 7(1), 10-16. https://doi.org/10.11648/j.ep.20240701.12
Shaoshu, H.; JunJie, F. Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks. Eng. Phys.2024, 7(1), 10-16. doi: 10.11648/j.ep.20240701.12
Shaoshu H, JunJie F. Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks. Eng Phys. 2024;7(1):10-16. doi: 10.11648/j.ep.20240701.12
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author = {Huang Shaoshu and Feng JunJie},
title = {Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks
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journal = {Engineering Physics},
volume = {7},
number = {1},
pages = {10-16},
doi = {10.11648/j.ep.20240701.12},
url = {https://doi.org/10.11648/j.ep.20240701.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ep.20240701.12},
abstract = {This article points out the erroneous calculations of power loss and voltage loss in long-distance transmission lines in high school physics textbooks; Based on the basic circuit structure of long-distance power transmission, provide a series of correct functional relationships between power transmission power, line loss power and loss voltage, transmission line current and transformer ratio, load resistance, transmission line resistance, and the relationship between power supply voltage and transmission voltage, and discuss related issues based on these functional relationships.
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T1 - Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks
AU - Huang Shaoshu
AU - Feng JunJie
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PY - 2024
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DO - 10.11648/j.ep.20240701.12
T2 - Engineering Physics
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PB - Science Publishing Group
SN - 2640-1029
UR - https://doi.org/10.11648/j.ep.20240701.12
AB - This article points out the erroneous calculations of power loss and voltage loss in long-distance transmission lines in high school physics textbooks; Based on the basic circuit structure of long-distance power transmission, provide a series of correct functional relationships between power transmission power, line loss power and loss voltage, transmission line current and transformer ratio, load resistance, transmission line resistance, and the relationship between power supply voltage and transmission voltage, and discuss related issues based on these functional relationships.
VL - 7
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Shaoshu, H., JunJie, F. (2024). Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks. Engineering Physics, 7(1), 10-16. https://doi.org/10.11648/j.ep.20240701.12
Shaoshu, H.; JunJie, F. Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks. Eng. Phys.2024, 7(1), 10-16. doi: 10.11648/j.ep.20240701.12
Shaoshu H, JunJie F. Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks. Eng Phys. 2024;7(1):10-16. doi: 10.11648/j.ep.20240701.12
@article{10.11648/j.ep.20240701.12,
author = {Huang Shaoshu and Feng JunJie},
title = {Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks
},
journal = {Engineering Physics},
volume = {7},
number = {1},
pages = {10-16},
doi = {10.11648/j.ep.20240701.12},
url = {https://doi.org/10.11648/j.ep.20240701.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ep.20240701.12},
abstract = {This article points out the erroneous calculations of power loss and voltage loss in long-distance transmission lines in high school physics textbooks; Based on the basic circuit structure of long-distance power transmission, provide a series of correct functional relationships between power transmission power, line loss power and loss voltage, transmission line current and transformer ratio, load resistance, transmission line resistance, and the relationship between power supply voltage and transmission voltage, and discuss related issues based on these functional relationships.
},
year = {2024}
}
TY - JOUR
T1 - Interpretation of the Calculation Problem of Power Loss in Long-distance Transmission Lines——Starting from the Serious Errors in Calculation Methods in High School Physics Textbooks
AU - Huang Shaoshu
AU - Feng JunJie
Y1 - 2024/12/19
PY - 2024
N1 - https://doi.org/10.11648/j.ep.20240701.12
DO - 10.11648/j.ep.20240701.12
T2 - Engineering Physics
JF - Engineering Physics
JO - Engineering Physics
SP - 10
EP - 16
PB - Science Publishing Group
SN - 2640-1029
UR - https://doi.org/10.11648/j.ep.20240701.12
AB - This article points out the erroneous calculations of power loss and voltage loss in long-distance transmission lines in high school physics textbooks; Based on the basic circuit structure of long-distance power transmission, provide a series of correct functional relationships between power transmission power, line loss power and loss voltage, transmission line current and transformer ratio, load resistance, transmission line resistance, and the relationship between power supply voltage and transmission voltage, and discuss related issues based on these functional relationships.
VL - 7
IS - 1
ER -
, and uses a resistance 
transmission line to transmit electrical energy outward. What is the power loss on the transmission line if a voltage of 
is used for power transmission? What is the voltage loss? If we switch to high voltage transmission with 
, what would be the power and voltage loss on the transmission line? 
, the transmission current when using 
voltage transmission 
(1) 
(2) 
(3) 
high-voltage transmission, the transmission current 
(4) 
(5) 
(6) 
voltage transmission is used instead? 
(7) 
(8) 
(9) 
), the loss power of the transmission line is greater than the transmission power, and the loss voltage is also higher than the transmission voltage. Moreover, the lower the transmission voltage, the greater the power loss of the transmission line compared to the transmission power, and the higher the voltage loss compared to the transmission voltage. Obviously, this seriously violates the law of conservation of energy and is a rather absurd result. 
( 
), the transformation ratio of the step-down transformer be 
( 
), and the load be a pure resistor component. 
(10) 
、 
、 
、 
，So, equation (10) can be rewritten as 
(11) 
) state, the output power of the generator is equal to zero. 
in the transmission line between two ideal transformers. For ease of comparison, the physical quantity symbols are labeled with the same symbols as in Figure 2. 
(12) 
、 
、 
、 
，Therefore, equation (12) can be rewritten as 
(13) 
) state, the output power of the generator is equal to zero. 
to represent the power loss of transmission lines in (13). According to equation (13), we can obtain 
(14) 
(short circuit in the transmission system), the power loss of the line is equal to the transmitted power; When the load resistance 
(transmission system unloaded) is applied, the power loss of the line is equal to zero, as shown in Figure 4. If the load resistance remains constant, when the transmission line resistance 
, the line loss power is equal to zero. When the transmission line resistance 
is constant, the transmission line loss power is equal to the transmitted power, as shown in Figure 5. 
(15) 
(short circuit in the transmission system), the line loss voltage is equal to the transmission voltage; When the load resistance 
(transmission system unloaded) is present, the line loss voltage is equal to zero, as shown in Figure 6. If the load resistance remains constant, when the transmission line resistance 
, the line loss voltage is equal to zero. When the transmission line resistance 
is reached, the line loss voltage is equal to the transmission voltage, as shown in Figure 7. 
(16) 
) state, the power or voltage transmitted by the generator (power source) is completely lost on the transmission line, which is not only meaningless but also a very dangerous phenomenon. 
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