In the same way as in case 3 it is possible to solve the system for I₁ and I₂. And, using LKC, the relation between the currents is given by I = I₁ + I₂. Developing, we can write the following relation:

Where L_eq is given by eq. 78-06.

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Based on the fact that the energy stored in a coupled circuit cannot be negative, since the entire circuit is passive, we can establish an upper limit for mutual inductance. Thus, it can be concluded that the mutual inductance cannot be greater than the geometric mean of the coil autoinductances. The degree to which the mutual inductance M approaches the upper limit is specified by the coupling coefficient, k, given by

It should be noted that 0 ≤ k ≤ 1 because it represents the fraction of the total magnetic flux that emanates from one coil and passes through the other coil. If every flow produced by one coil passes through the other coil, then k = 1 and we say that we have a 100% coupling or that the coils are perfectly coupled. If k ≤ 0.5 we say that the coils are loosely coupled and, if k > 0.5, they are said to be tightly coupled .

In many problems M is not given, but k is given. So, from the value of k we should find the value of M. For this we use the eq. 78-04, or

It may also happen that the values of L₁ and L₂. And neither does the value of ω. Only the values of X_L1, X_L2 and k are provided. Well, there is no need to provide the value of ω. How we are interested in the value of ωM , then we can make the following transformation in eq. 78-04, where ω M = ω k √ (L₁ L₂ ). Putting ω into the radical we can write the following: ω M = k √ (ω² L₁ L₂ ) = k √ (ω L₁ ω L₂). But ω L₁ is the inductive reactance of L₁ and ω L₂ is the inductive reactance of L₂. So we can write that

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We can consider the transformer as a device that contains two or more coils magnetically coupled. Normally, the power supply is connected to the primary and the load connected to the secondary. We call the transformer linear if the magnetic permeability, μ of the paths through which flows flow is constant. In transformers built without high permeability material in the core, the coupling coefficient, k, it is typically very small. Thus, we can use air core, plastic, bakelite, wood and others, as these materials have a constant permeability. This type of transformer has varied applications as in oscillators and radiofrequency amplifiers in radio receivers, television sets, measuring equipment, etc ...

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    5.1   Reflection of Impedances

Often, in solving a problem, it is necessary to know the impedance that the entire circuit represents for the power supply. In this situation, it is interesting to reflect all the impedance from the secondary to the primary. In this way, we easily calculate this impedance as we eliminate the presence of the transformer. So, let's analyze the circuit that appears in Figura 78-09.

Following the methodology we have adopted, we will write the circuit equations shown in Figure 78-09 . Therefore:

To determine the impedance that the circuit represents for the source is the same as determining the input impedance of the circuit. For this, we can define Z_in = V / I₁. This relationship can be found by solving the eq. 78-10 such that, we find I₂ as a function of I₁ and substitute this value in eq . 78-11. Thus, we obtain the input impedance of the circuit being expressed by eq. 78-12.

By the eq. 78-12 , it is noticed that the input impedance is composed of two terms: the first term ( R₁ + jω L₁ ) represents the primary impedance; the second term is due to the coupling between the two windings. It is interpreted as a reflection of the impedance from the secondary to the primary. Normally, in the technical literature it is known as reflected impedance and represented by Z_R.

It should be noted that the result of eq. 78-12 is not affected by the position of the transformer points, as we achieve the same result when we replace M by - M.

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    5.2   "T" model

Often in coupled circuits, the construction of the solution equations can be quite complex. So, sometimes, it is perfectly understandable to want to replace a magnetically coupled circuit with one without a magnetic coupling. So, let's study how we can transform a magnetically coupled circuit into a circuit "T" or "Star", that does not have mutual inductance, as shown in Figura 78-10.

Starting from a matrix equation that translates the voltage-current relationships for the primary and secondary windings and, later, find the inverse matrix, we can relate them to the corresponding equations for the "T" circuits. Thus, using the "T" circuit of the Figure 78-10 and making the appropriate comparisons, we find the equivalence according to the equations below.

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    5.3   "Pi" Model

We will study how we can transform a magnetically coupled circuit into a "Pi" circuit, which is represented in Figure 78-11, which does not have a magnetic coupling.

In the same way as was done for the "T" circuit, we will now compare it with the equations that represent the "Pi" circuit and we will obtain the equivalence between the circuits, as shows the equations below.

Looking closely at the equations for the "T" and "Pi" circuits, it can be seen that depending on the circuit configuration with magnetic coupling, we can find a negative value for some inductance. Although this is physically unattainable, the mathematical equivalent model is valid.

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We define ideal transformer when it has the following characteristics:

We can approximate a transformer to an ideal one, replacing the air core used, in general, in linear transformers, with an iron core.

When a sinusoidal voltage is applied to the primary winding, a magnetic flux, Φ, passes through both windings. This flow will induce a tension in the secondary winding. The relation between the voltage applied to the primary and the voltage that appears on the secondary will depend on the number of turns that each winding has. We normally represent the number of turns of the primary by N₁ and the number of turns of the secondary by N₂. It should also be noted that, in the ideal transformer, the output power is equal to input power, that is, there are no losses. At this point, we define the so-called transformation relation, which is expressed by eq. 78-19.

And as previously mentioned, the input and output powers in the ideal transformer are the same, that is,   P₁ = P₂ = V₁ I₁ = V₂ I₂.  So, this allows us to write that:

In figure Figure 78-12 we present the schematic of an ideal transformer

Where the variables are:

To represent an ideal transformer, notice the parallel vertical lines separating the coils as shown in Figure 78-12 . This indicates that the core is made of material with high magnetic permeability, such as iron.

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In the study of transformers we can work with the so-called impedance reflection . That is, we can reflect the impedance from the primary to the secondary and vice versa. It depends on the convenience of one or the other. Let's look at how these reflections are made.

    6.1.1   Reflection from Secondary to Primary

The secondary impedance can be calculated as the ratio of the secondary voltage and current. Referring to the circuit shown in Figure 78-12, we can write that:

But by eq. 78-20 knowing that V₁ = a V₂ e  I₁ = I₂ / a. Thus, calculating the impedance that the circuit offers to the primary, we find:

That is, when we reflect the impedance of the secondary to the primary, we must multiply the impedance of the secondary by the square of the transformation relation. In short:

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    6.1.2   Reflection from Primary to Secondary

Just as we reflect the impedance from the secondary to the primary, we can reflect that from the primary to the secondary. From eq. 78-20 we concluded that I₂ = a I₁ and also  V₂ = V₁ / a. Thus, calculating the impedance that the circuit offers the secondary, we find:

We conclude that when we reflect the impedance from the primary to the secondary, we must divide the impedance of the primary by the square of the transformation relation. In short: