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There must be something wrong. The fact is that, even if the transformer is ideal, the instantaneous input and output powers are not exactly equal. A not-so-correct current ratio has thus been derived. The fault is rooted in two errors. As long as the primary receives electrical energy from the source, V p I p is negative, so the power V p I p can only be understood as the negative of the input power.
Some texts and online resources are vague or careless in this respect [ 3 — 5 ].
Clarifying concepts and gaining a deeper understanding of ideal transformers - IOPscience
However, changing the power relationship to. The critical problem in equation 9 is that it misses a term corresponding to the temporal fluctuation of the total core magnetic energy. This term is necessary because the energy associated with the time-changing core flux must also be derived from the AC source. For this reason, the power balance must include such a term, for example,. Although some magnetic energy resides in the iron core, it is not dissipated via eddy currents, hysteresis loss, stray fields, etc, since the transformer is assumed to be ideal.
The magnetic energy that is more than the average will return to the source at a later time. Some texts state that the input and output powers are equal in their root-mean-square or averaged values, rather than being instantaneous, but little or inadequate explanation is provided alongside [ 6 , 7 ].
Later, we will ascertain what this constant actually is. The term P m in equation 10 is the rate of change of the stored magnetic energy, so. On the other hand, equations 1 and 2 allow us to express. By integrating equation 14 w. Substituting equations 14 and 16 into 13 and then P m into equation 10 , and after rearranging the terms, we get. Obviously, the current inside the square brackets in the last equation is identical to the primary current in equation 9. It must be emphasised that this negative current ratio is consistent with the whole theory see section 5.
Among the college-level physics texts surveyed by the author, only one derives a similar negative current ratio [ 8 ]. Next, I p,L is found by putting the last equation into equation 21 ,. Therefore, we obtain the final form of I p,M ,. Combining equations 16 and 24 , we find.
In the last equation, the relation. Furthermore, the two components, I p,L and I p,M in equation 26 can be combined mathematically to form. A formal mathematical approach to solving the ideal transformer is given in the appendix. At the end of this section, it is worth taking note of the distinctly different functions of the two primary currents. The current ratio of I p , L to I s equation 21 is negative—what does it mean? How does this negative sign eventually go to I s equation 22 , instead of I p , L equation 23?
Undoubtedly, this negative sign means the opposite of a direction, but what is this direction? Do I p , L and I s always flow in opposite directions?
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No, they may flow in the same or opposite directions, depending on how the two windings are coiled around and placed in the core see figure 4. Actually, this negative sign is related to the positive sense of rotations PSR of the two windings.
Magnetic flux through a coil is defined as the scalar product of the magnetic field passing through the coil and the area vector of the coil. Whenever an area vector of a coil is chosen, for example, upward or downward if the coil is horizontally placed, subsequently its PSR is determined. The method is the right-hand screw rule: Indeed, the sign of the final result calculated from Faraday's law of induction is relative to the PSR of the coil [ 10 ].
But if the two windings are wound on the same side of the core, their area vectors are parallel. The negative sign in the current ratio I p,L: N p is interpreted as follows: But it is not up to the primary's PSR to make the choice, it has already been predetermined in the conventions of physics. The primary winding of a no-load transformer is essentially an inductor, so the PSR of the primary winding must likewise be preset. Simply put, a positive negative current in the primary flows in the same opposite direction to the primary's PSR since the primary's PSR is so defined.
This is the reason why the negative sign in the current ratio eventually goes to I s. Unlike the primary's, the secondary's PSR cannot be established unless the relative position of the two windings is known. Nonetheless, using the PSR to determine a direction is somewhat complicated and surely not user-friendly.
There is a much better method. Thus, the negative sign in the current ratio suggests an alternative interpretation that seems to be more understandable and utilisable: This can serve as a simple method to find the direction of the secondary current see section 6. Further, the magnetic field inside a solenoid is proportional to the product of the number of turns and the current. They add up to exactly zero, a scenario that agrees and confirms our earlier findings. In circuit diagrams, occasionally two black dots are seen next to the terminals of a transformer symbol. One is at the primary, and the other is at the secondary.
They are also called phase dots, indicating which input and output terminals are simultaneously positive or negative w. There are three methods to ascertain where the two black dots should be placed. Let us consider the transformer shown in figure 2. In the figure, the PSRs of the two windings, the magnetic fields produced by the two currents I p , L and I s at this moment are all shown. With reference to figure 2 , the three methods are explained one-by-one in the following. The underlying physics of the first two has been elaborated in the previous section, so only their steps with brief reasons are stated.
Using this transformer, we explain the methods used to find the direction of the secondary current and the positions of the black dots. The third method is to use the in-phase relationship between V p and V s the positive voltage ratio. This relationship implies their two corresponding electric fields E p and E s , where and point in the same direction relative to their own PSR.
That is to say, the induced currents if any produced by V p and V s would flow in the same direction relative to their own PSR. To avoid confusion, one should be clear that I s is the induced current of V s , but neither I p,L nor I p,M is the induced current of V p. To use this method, the two connected PSRs are established by using the condition 'same flux along core' first.
By taking two assumed currents following these two PSRs, the in-phase terminals, for example, those from which the two currents leave are thus determined. Based on the prevailing models [ 1 , 3 ], we devise and put forth an equivalent circuit of an ideal transformer. Many major ideas discussed in this paper become obvious with the aid of this equivalent circuit. In figure 3 , the part inside the dashed line box is the ideal transformer.
The primary circuit is made up of a parallel-combination of a pure inductor and a pure resistor. The former has a presumably large inductance L p and number of turns N p while the latter has a resistance. The currents passing through these two branches are I p , M and I p , L , respectively; so their sum gives the primary current I p.
Since R ' and L p are connected in parallel, they do not affect each other. Obviously, no matter whether the transformer is loaded with any R or is even unloaded, I p,M is always present and the same. Figure 4 provides examples of the four cases. Whenever the overall sign is known, the two black phase dots can be placed accordingly, as shown in figure 4. In this equivalent circuit, the PSRs are no longer needed.
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Examples of the determination of the signs in equation The arrows around the circles show the coiling directions of the two windings. When the output circuit is open, R is removed; R ' in the input circuit is removed too. First, let us review some of the main findings in this paper. Unfortunately, this theory is a popular one and is commonly taught in schools. In section 4 , we clarify the concepts: Starting from this correct power relationship, the ideal transformer is completely solved. The theory thus built depicts a self-contained and comprehensive picture.
Clarifying Concepts in Physics
The main point is that there is always a magnetising current I p,M flowing in the primary winding to excite the core flux; only this current does this job. Irrespective of what the load R is, the core flux is always present and the same, and is I p,M. This explains why, in the argument in section 3 , a zero-flux results if I p,M is not taken into account.
Another noteworthy result is the negative current ratio I p,L: N p equation 21 , where I p,L is the primary current other than I p,M.