Transformation ratio

Transformer transformation ratio is a value that expresses the scaling (converting) characteristic of the transformer relative to any parameter of the electric circuit (voltage, current, resistance, etc.).

For power transformers, GOST 16110-82 defines the transformation coefficient as "the ratio of the voltages at the terminals of two windings in idle mode" and "is taken equal to the ratio of the numbers of their turns" ^[1] .

Content

1 General
- 1.1 Voltage Scaling
- 1.2 current scaling
- 1.3 Resistance Scaling
- 1.4 Summary remarks
2 Additional Information
- 2.1 Feature accounting turns
3 notes

General information

The term “scaling” is used in the description instead of the term “conversion” in order to focus on the fact that transformers do not convert one type of energy into another, and not even one of the parameters of the electrical network into another parameter (as they used to say about conversion, for example, voltage to current by step-down transformers). A transformation is just a change in the value of any of the circuit parameters in the direction of increasing or decreasing. And although such transformations affect almost all the parameters of the electrical circuit, it is customary to isolate the most “main” from them and associate the term transformation coefficient with it. This selection is justified by the functional purpose of the transformer, the connection circuit to the supply side, etc.

Voltage Scaling

For transformers with parallel connection of the primary winding to an energy source, scaling in relation to voltage is usually of interest, which means that the transformation coefficient n expresses the ratio of primary (input) and secondary (output) voltages:

n={\frac {U_{1}}{U_{2}}}={\frac {\varepsilon \cdot W_{1}+I_{1}\cdot R_{1}}{\varepsilon \cdot W_{2}-I_{2}\cdot R_{2}}}

{\ displaystyle n = {\ frac {U_ {1}} {U_ {2}}} = {\ frac {\ varepsilon \ cdot W_ {1} + I_ {1} \ cdot R_ {1}} {\ varepsilon \ cdot W_ {2} -I_ {2} \ cdot R_ {2}}}}

,

Where

$U_{1}$ ${\ displaystyle U_ {1}}$ , $U_{2}$ ${\ displaystyle U_ {2}}$ - input and output voltages, respectively;
$\varepsilon$ ${\ displaystyle \ varepsilon}$ - EMF induced in each coil of any winding of this transformer;
$W_{1}$ ${\ displaystyle W_ {1}}$ , $W_{2}$ ${\ displaystyle W_ {2}}$ - the number of turns of the primary and secondary windings;
$I_{1}$ ${\ displaystyle I_ {1}}$ , $I_{2}$ ${\ displaystyle I_ {2}}$ - currents in the primary and secondary circuits of the transformer;
$R_{1}$ ${\ displaystyle R_ {1}}$ , $R_{2}$ ${\ displaystyle R_ {2}}$ - active resistance of the windings.

If we neglect the losses in the windings, that is, $R_{1}$ ${\ displaystyle R_ {1}}$ , $R_{2}$ ${\ displaystyle R_ {2}}$ considered equal to zero then

n={\frac {U_{1}}{U_{2}}}={\frac {W_{1}}{W_{2}}}

{\ displaystyle n = {\ frac {U_ {1}} {U_ {2}}} = {\ frac {W_ {1}} {W_ {2}}}}

.

Such transformers are also called voltage transformers .

Current Scaling

For transformers with serial connection of the primary winding to the energy source, scaling is calculated in relation to the current strength, that is, the transformation coefficient n expresses the ratio of the primary (input) and secondary (output) currents:

n={\frac {I_{1}}{I_{2}}}

{\ displaystyle n = {\ frac {I_ {1}} {I_ {2}}}}

In addition, these currents are connected by another dependence

I_{1}\cdot W_{1}=I_{2}\cdot W_{2}+I_{0}

{\ displaystyle I_ {1} \ cdot W_ {1} = I_ {2} \ cdot W_ {2} + I_ {0}}

,

Where

$I_{1}$ ${\ displaystyle I_ {1}}$ , $I_{2}$ ${\ displaystyle I_ {2}}$ - currents in the primary and secondary circuits of the transformer;
$W_{2}$ ${\ displaystyle W_ {2}}$ , $W_{1}$ ${\ displaystyle W_ {1}}$ - the number of turns of the primary and secondary windings;
$I_{0}$ ${\ displaystyle I_ {0}}$ - current "idling", consisting of the magnetization current and active losses in the magnetic circuit.

If we neglect all the losses of magnetization and heating of the magnetic circuit, i.e. $I_{0}$ ${\ displaystyle I_ {0}}$ considered equal to zero then

I_{1}\cdot W_{1}=I_{2}\cdot W_{2}

{\ displaystyle I_ {1} \ cdot W_ {1} = I_ {2} \ cdot W_ {2}}

=>

{\frac {I_{1}}{I_{2}}}={\frac {W_{2}}{W_{1}}}

{\ displaystyle {\ frac {I_ {1}} {I_ {2}}} = {\ frac {W_ {2}} {W_ {1}}}}

n={\frac {I_{1}}{I_{2}}}={\frac {W_{2}}{W_{1}}}

{\ displaystyle n = {\ frac {I_ {1}} {I_ {2}}} = {\ frac {W_ {2}} {W_ {1}}}}

Such transformers are also called current transformers .

Resistance Scaling

Another application of transformers with parallel connection of the primary winding to an energy source is resistance scaling.

This option is used when the change in voltage or current itself is not directly interested, but it is required to connect a load with an input impedance significantly different from the values presented by this source to the energy source.

For example, the output stages of sound power amplifiers require a load resistance higher than that of low-impedance speakers . Another example is high-frequency devices, for which the equality of the wave impedances of the source and the load allows you to get the maximum power output in the load. And even welding transformers , in fact, are converters of resistance to a greater extent than voltages, since the latter serves to increase the safety of work, and the former is a requirement for the load resistance of electrical networks. Although it may not matter to the welder how the required thermal energy was obtained from the network for heating the metal, it is understandable that practically a “short circuit” in the network is not welcomed by the energy supplying party.

Accordingly, we can say that resistance scaling is designed to transfer power from a source to any load in the most “civilized” way, without “shock” modes for a source and with minimal losses (for example, if we compare transformer scaling and a simple increase in load resistance using a sequential ballast resistance , which will “eat” a significant fraction of the energy at the source).

The principle of calculating such scaling is also based on the transfer of power, namely, on conditional equality of powers: consumed by the transformer from the primary circuit (from the source) and given to the secondary (load), neglecting the losses inside the transformer.

S_{1}=S_{2}+\Delta S

{\ displaystyle S_ {1} = S_ {2} + \ Delta S}

,

Where

$S_{1}$ ${\ displaystyle S_ {1}}$ , $S_{2}$ ${\ displaystyle S_ {2}}$ - power respectively consumed and given by the transformer;
$\Delta S$ ${\ displaystyle \ Delta S}$ - losses in the transformer itself (on average 1-2% of $S_{1}$ ${\ displaystyle S_ {1}}$ ), which can be neglected in this case.

S_{1}=U_{1}\cdot I_{1}={\frac {U_{1}^{2}}{Z_{1}}}

{\ displaystyle S_ {1} = U_ {1} \ cdot I_ {1} = {\ frac {U_ {1} ^ {2}} {Z_ {1}}}}

... ..

S_{2}=U_{2}\cdot I_{2}={\frac {U_{2}^{2}}{Z_{2}}}

{\ displaystyle S_ {2} = U_ {2} \ cdot I_ {2} = {\ frac {U_ {2} ^ {2}} {Z_ {2}}}}

,

Where

$Z_{1}$ ${\ displaystyle Z_ {1}}$ , $Z_{2}$ ${\ displaystyle Z_ {2}}$ - the input resistance of the transformer together with the load relative to its primary circuit and the input resistance of the load in the secondary circuit, respectively (that is, the first is the load for the energy source in the presence of the transformer, the second in the absence);

S_{1}=S_{2}

{\ displaystyle S_ {1} = S_ {2}}

=>

{\frac {U_{1}^{2}}{Z_{1}}}={\frac {U_{2}^{2}}{Z_{2}}}

{\ displaystyle {\ frac {U_ {1} ^ {2}} {Z_ {1}}} = {\ frac {U_ {2} ^ {2}} {Z_ {2}}}}

=>

{\frac {U_{1}^{2}}{U_{2}^{2}}}={\frac {Z_{1}}{Z_{2}}}=n_{Z}=n_{U}^{2}

{\ displaystyle {\ frac {U_ {1} ^ {2}} {U_ {2} ^ {2}}} = {\ frac {Z_ {1}} {Z_ {2}}} = n_ {Z} = n_ {U} ^ {2}}

As can be seen above, the transformation coefficient of resistance is equal to the square of the coefficient of transformation of voltage.

Such transformers are sometimes called matching transformers (especially in radio engineering).

Summary Comments

Despite the differences in the switching schemes, the principle of operation of the transformer itself does not change and, accordingly, all the dependences of the voltages and currents inside the transformer will be as shown above. That is, even the current transformer, in addition to its “main” task, to scale the current strength will have the dependences of primary and secondary voltages the same as if it were a voltage transformer, and introduce into the serial circuit into which it is connected, the resistance of its load, changed according to the principle matching transformer.

It should also be remembered that currents, voltages, resistances and powers in alternating circuits have, in addition to absolute values, a phase shift, therefore, in calculations (including the above formulas) they are vector quantities. This is not so important to take into account for the transformation coefficient of transformers of general technical purpose, with low requirements for conversion accuracy, but it is of great importance for measuring transformers of currents and voltages.

For any scaling option, if $n<1$ ${\ displaystyle n <1}$ , then the transformer can be called boost; in the opposite case - lowering ^[2] . However, GOST 16110-82 ^[1] does not know such a distinction: “In a double-winding transformer, the transformation coefficient is equal tohigh voltage to lower ”, that is, the transformation coefficient is always greater than unity.

Additional Information

The feature of accounting for turns

Transformers transfer energy from the primary circuit to the secondary circuit through a magnetic field. With the rare exception of the so-called "air transformers", the magnetic field is transmitted through special magnetic cores (made of electrical steel, for example, or other ferromagnetic substances) with a magnetic permeability much greater than that of air or vacuum. This concentrates the magnetic lines of force in the body of the magnetic circuit, reducing magnetic scattering, and in addition, enhances the magnetic flux density (induction) in this part of the space occupied by the magnetic circuit. The latter leads to an increase in the magnetic field and less consumption of the “idle” current, that is, less loss.

As is known from the course of physics, magnetic lines of force are concentric and self-enclosed “rings” that enclose a conductor with current. A direct conductor with current is enveloped by rings of a magnetic field along its entire length. If the conductor is bent, then the magnetic field rings from different sections of the length of the conductor approach each other on the inside of the bend (like a coil spring, curved to the side, with pressed coils inside and stretched outside the bend). This step allows you to increase the concentration of lines of force within the bend and, accordingly, strengthen the magnetic field in that part of space. It is even better to bend the conductor with a ring, and then all the magnetic lines distributed along the circumference of the circle will "pile together" inside the ring. This step is called the creation of a coil of conductor with current.

All of the above is very suitable for coreless transformers (or other cases with a relatively uniform magnetic environment around the turns), but it is absolutely useless in the presence of magnetic closed cores, which, unfortunately, for geometric reasons cannot fill the entire space around the transformer winding. And therefore, magnetic lines of force covering the coil of the transformer winding are in unequal conditions around the perimeter of the coil. Some power lines are “luckier" more, and they pass only along the facilitated route of the magnetic conductor, while others have to go part of the path along the core (inside the turn), and the rest through the air, to create a closed power "ring". The magnetic resistance of the air almost extinguishes such field lines and accordingly levels the presence of that part of the coil that generated this magnetic line.

From all of the above and shown in the figure, there is a conclusion - not only the entire coil, but only a small part that is completely surrounded by this magnetic circuit takes part in the operation of the transformer with a closed ferromagnetic core. Or in other words - the main magnetic flux passing through the closed core of the transformer is created only by that part of the wire that passes through the “window” of this core. The figure shows that to create 2 "turns" it is enough to pass the wire with current through the "window" of the magnetic circuit twice, while saving on the winding.

Notes

↑ ¹ ² Power transformers. Terms and Definitions. GOST 16110-82 (ST SEV 1103-78) (neopr.) (Inaccessible link) . Date of treatment February 10, 2017. Archived on August 9, 2016.
↑ Such a definition of step-up and step-down transformer can be found in various educational materials of the school level: [1] , [2] .

[ГОСТ_16110-82-1] ¹ ² Power transformers. Terms and Definitions. GOST 16110-82 (ST SEV 1103-78) (neopr.) (Inaccessible link) . Date of treatment February 10, 2017. Archived on August 9, 2016.

[2] Such a definition of step-up and step-down transformer can be found in various educational materials of the school level: [1] , [2] .

Transformation ratio

Content

General information

Voltage Scaling

Current Scaling

Resistance Scaling

Summary Comments

Additional Information

The feature of accounting for turns

Notes

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