Electric Potential theory and study

This chapter is included in the items below. If you want to go directly to an item, click on it.

2 - Coupling and Mutual Inductance Click here!

3 - Equivalence of Inductances Click here!

4 - Coupling Coefficient Click here!

5 - Linear Transformer Click here!

5.1 - Reflection of Impedances Click here!

5.2 - "T" Model Click here!

5.3 - "Pi" Model Click here!

6 - Ideal Transformer Click here!

6.1 - Reflection of Impedance Click here!

6.1.1 - Reflection from Secondary to Primary Click here!

6.1.2 - Reflection from Primary to Secondary Click here!

In this chapter we will study the behavior of magnetically coupled circuits. Coupled coils are found in many applications, such as power systems, communications, professional audio systems, etc ... The developed concepts lead to a new element in circuits called transformer. Through of this new element we can use it for impedance matching, electrical insulation and also provide changes in voltage levels in power systems. We are currently more interested in developing a good knowledge base on coupled circuits.

2. Coupling and Mutual Inductance

"Mutual inductance is the ability of an inductor to induce voltage in a neighboring inductor and is represented by letter M . Its unit of measure is the henry."

Mutual inductance, M, is always a positive value. To determine the voltage value induced from one winding to another, we should know the physical orientation of each winding and, applying Lenz's law together with the right hand rule, we would define its value. However, showing this on an electrical diagram is not an easy task. Therefore, was created the so-called "dot convention". This rule establishes that a point will be placed at each end of each of the two coils magnetically coupled. Thus, if the current enters the side marked with a dot, this will determine the direction of the electrical flow.

So, we have two possible cases: the coils are in the additive or subtractive configuration. Therefore, we can establish a rule to determine the configuration of the coils.

"Mutual inductance is additive when both currents are entering or leaving the points, otherwise it is subtractive, as long as the two points are on the same side of the coils."

In Figure 78-01, we can see two possible arrangements for the additive configuration. Note that wherever the current enters, the polarity of the voltage in the winding is positive. And where it leaves the winding, negative. Another possibility would be to pass the two points to the bottom of the windings, maintaining the direction of the currents.

acopla 78-1J.png — Figure 78-01
additive configuration

In Figure 78-02, we can see two possible arrangements for the subtractive configuration. Note that wherever the current enters, the polarity of the voltage in the winding is positive. And where it leaves the winding, negative. Another possibility would be to pass the two points to the bottom of the windings, maintaining the direction of the currents.

acopla 78-2J.png — Figure 78-02
configuração subtrativa

In Figure 78-03, we can see two other possible arrangements for the subtractive configuration. Here, the dots are on opposite sides. Pay attention to the direction of the currents.

acopla 78-3J.png — Figure 78-03
alternative to the subtractive configuration

On the other hand, we know of the possibility that a positive rate in the current variation in one inductor will produce a negative voltage in another inductor. In this way, additive and subtractive couplings include the sign ± in the term of the mutual inductance, M. So, the equations that we use to determine the voltage on L₁ and L₂ are:

eq. 78-01

eq. 78-02

So when we apply these equations to the elements of a circuit, we must determine whether we are going to use the plus sign or the minus sign. On this site we choose to use, whenever possible, the additive configuration. In some cases, we must reverse the current direction. In doing so, we guarantee that the induced voltages will always have the positive polarity facing the side of the winding that does not have the point. The use of this technique can be seen Here!

Example 78-1

To exemplify the use of these equations we will use example 15-1 from the book The analysis and Design of Linear Circuits, Thomas, Roland, page 793 where we have L₁ = L₂ = 10 mH, M = 2 mH. Let's calculate the value of v₂ when i₂ = 0 , knowing that v_i = 200 sen (400 t) V. Look at Figura 78-04.

Solution

Through the circuit we see that v₁ = v_i. It is also possible to observe that we are facing an additive configuration and for this reason we will use mutual inductance with a positive sign . Let's abbreviate the notation and use the derivative of the variable in relation to time as i'₁. Note that as i₂ = 0, the second part of eq. 78-01 and eq. 78-02 cancel each other out. So we can calculate i'₁ (t) as:

i'₁ (t) = v₁ (t) / L₁ = 200 sen (400 t) / 0.01

Carrying out the calculation, we find:

i'₁ (t) = 20 000 sen (400 t)

Substituting this value in eq. 78-02, let's get:

v₂ (t) = 2 x 10^-3 x 20 000 sen (400 t) = 40 sen (400 t) V

3. Equivalence of Inductances

The two mutually coupled coils shown in Figure 78-05 can be interconnected in four different ways. Let's look at each case.

Case 1

The two mutually coupled coils shown in Figure 78-06 have the following equation:

V = jω L₁ I + jω M I + jω L₂ I + jω M I

This equation can also be written as:

V = jω I ( L₁ + L₂ + 2 M ) = j ω I L_eq

In this way, we easily recognize that we can replace the entire circuit with an equivalent inductance given by:

eq. 78-03

Case 2

The two mutually coupled coils shown in Figure 78-07 have the following equation:

V = jω L₁ I + jω M I + jω L₂ I - jω M I

This equation can also be written as:

V = jω I ( L₁ + L₂ - 2 M ) = j ω I L_eq

In this way, we easily recognize that we can replace the entire circuit with an equivalent inductance given by:

eq. 78-04

Case 3

The two mutually coupled coils shown in Figure 78-08, are actually in parallel, as we see on the right in the figure above. The two equations that govern the circuit are:

V = jω L₁ I₁ + jω M I₂

V = jω M I₁ + jω L₂ I₂

Of these two equations it is possible to solve for I₁ and I₂. So, let's get:

I₁ = V ( L₂ - M ) / jω ( L₁ L₂ - M² )

I₂ = V ( L₁ - M ) / jω ( L₁ L₂ - M² )

Using LKC on the circuit shown in Figure 78-08, we easily conclude that I = I₁ + I₂. With this information we can write

I = I₁ + I₂ = V / jω L_eq

Where L_eq is given by eq. 78-05.

eq. 78-05

Case 4

Note that in case 4 the coils remain in parallel, but there was an inversion in the points. Thus, the current I₂ enters the coil L₂ on the side without point . Thus, the equations that govern the circuit are:

V = jω L₁ I₁ - jω M I₂

V = - jω M I₁ + jω L₂ I₂

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:

I = I₁ + I₂ = V / jω L_eq

Where L_eq is given by eq. 78-06.

eq. 78-06

4. Coupling Coefficient

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

eq. 78-07

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

eq. 78-08

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

eq. 78-09

5. Linear Transformer

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 ...

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:

( R₁ + jω L₁ ) I₁ + jω M I₂ = V

eq. 78-10

jω M I₁ + ( R₂ + jω L₂ + Z_L ) I₂ = 0

eq. 78-11

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.

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.

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.

eq. 78-13

eq. 78-14

eq. 78-15

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.

eq. 78-16

eq. 78-17

eq. 78-18

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.

6. Ideal Transformer

We define ideal transformer when it has the following characteristics:

The coils have very large reactances (L₁, L₂ e M → ∞ ).
The coupling coefficient is unitary (k=1).
Primary and secondary coils show no losses ( R_p = R_s = 0 ).

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.

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:

eq. 78-20

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

Where the variables are:

V₁ - Electrical voltage applied to transformer primary
V₂ - Electrical voltage removed from the transformer secondary
I₁ - Electrical current in transformer primary
I₂ - Electrical current in transformer secondary
N₁ - Number of turns of transformer primary
N₂ - Number of turns of transformer secundary
Z_L - Load connected to secondary
P₁ - Power delivered to transformer primary
P₂ - Power delivered to the load

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.

6.1. Reflection of Impedances

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:

Z_s = V₂ / I₂ = Z_L

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:

Z_p = V₁ / I₁ = a V₂ / (I₂ / a) = a² Z_L

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:

eq. 78-21

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:

Z_s = V₂ / I₂ = (V₁ / a) / a I₁ = V₁ / a² I₁ = Z_p / a²

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:

eq. 78-22

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