Circuitry coupled magnetiquements are electrical circuits wound around the same magnetic circuit. For example two rollings up of a transformer or an electric machine. One often shortens the expression in coupled Circuits
Parameters of a whole of two magnetically coupled circuits equations and diagrams
One in general represents two reels magnetically coupled using the following assembly: with L1 and L2 clean inductances of each reel and M: mutual inductance.
This modeling occults non-linearities completely, but it makes it possible to make an approximate and often sufficient analytical study of many devices of electrical engineering, such as the electric machines and the transformers. Resistances of the reels are not represented either, because they do not modify the demonstrations below.
For practical and ⁄ 2 or historical reasons, it is the assembly below which is used:
This second assembly does not reveal any more mutual inductance and it comprises four parameters instead of three. One of these parameters is thus arbitrarily selected and it is what makes the originality of each existing model. Conventionally the circuit of index 1 is called primary education circuit and that of index 2 circuit secondary, in reference to the transformers.
l1 and l2 are called inductances of primary and secondary escape
lµ is the inductance of magnetizing brought back to the primary education
α is the report ⁄ 2 ratio of transformation of the ideal transformer introduced into this modeling
A mathematical analysis of the two assemblies makes it possible to show that they are completely equivalent if the following relations are checked:
Lµ = M ⁄ α
l1 = L1 - M ⁄ α
l2 = L2 - αM
Usual models of the coupled circuits
Model with escapes added up with the primary education
In this model one affirms that the magnetic escapes do not exist for the secondary winding. The selected parameter is : l2 = 0 = L2 - αM
This has as a consequence which the parameters of this model are dependant with inductances by the relations:
Kp = α = L2 ⁄ M
Lp = Lµ = M² ⁄ L2
Lƒp = L1 - Lµ = L1 - M² ⁄ L2 = σL1
with : σ = 1 - M² ⁄ L1L2 : coefficient of escape or coefficient of Blondel.
This model is particularly interesting when one is interested in the effects of inductances of escape of the circuit coupled on the food of the assembly. For example for the dimensioning of the transformer in the food with cutting of the fly-back type.
Model with escapes added up with the secondary
In this model one affirms that the magnetic escapes do not exist for the primary winding. The selected parameter is : l1 = 0 = L1 - M ⁄ α
This has as a consequence which the parameters of this model are dependant with inductances by the relations:
Ks = α = M ⁄ L1
Lµ = L1
lƒs = L2 - M² ⁄ L1 = σL2
For reasons of convenience, it is frequent to bring back the impedance of escape on the primary education side
With : Ns : impedance brought back to the primary education of the secondary inductance of escape. This brought back impedance should not be confused with the primary education impedance of escape of the preceding model.
This model is very practical to calculate the influence of the magnetic circuit on the power supply when this one feeds the primary education. Is used it for example to model the asynchronous machine
Model with separate escapes
This model is usually used for the transformers.
is posed α = m = n2 ⁄ n1 equal to the report/ratio of the number of whorls of reel 2 by the number of whorls of reel 1.
One obtains :
Lµ = M ⁄ m
l1 = L1 - M ⁄ m
l2 = L2 - mM
One can also bring back the inductance of magnetizing to the secondary and to obtain the model are equivalent according to :
with : L2µ = L1µ ⁄ m²
Model in T one poses α = 1 what amounts making disappear the transformer from the model:
Caution: This model functions perfectly from a mathematical point of view but it is sometimes unrealistic to seek to find a direction physical with the three dipoles which constitute it.
For example the values of L1 - M or de L2 - M can be negative, which amounts saying, in sinusoidal mode of current, which inductance behaves like a condenser.
An electric transformer is a convertor being used to modify the values of tension and intensity of the current delivered by an alternative electric energy source, in an expander
Electric transformer - electric Component - Electrical engineering - passive Component - Distribution of electrical energy - electromagnetic Device - Processing of electrical energy
An electric transformer is a convertor which is used to modify the values of of electrical energy by lines with high voltage (as on a watch) between the primary and the secondary of the transformer (ex. (source)
An electric transformer is a convertor being used to modify the values of tension and intensity of the current delivered by an alternative electric energy source, in and a current expander of different values, but of the same frequency and of the same form. It carries out this transformation with an excellent output. It is similar to gears in mechanics (the couple on each toothed wheel being the analogue of the current and disk speed being the analogue of the tension).
One can distinguish the static transformers and the rotary convertors. In a static transformer, energy is transferred from the primary to the secondary via the magnetic circuit which form the carcass of the transformer. These two circuits then are magnetically coupled. This is used to carry out a galvanic insulation between the two circuits. In a rotary convertor, energy is transmitted in a mechanical way between a generator and an electrical motor.
Cross-section of a three-phase transformer.
The principles of the transformer were established in 1831 by Michæl Faraday, but this last only made use of it to show the principle of electromagnetic induction and did not envisage the practical applications of them.
Lucien Gaulard, young French electrician, introduces to the French Company electricians, in 1884, a "secondary generator", called since transformer.
In 1883, Lucien Gaulard and John Dixon Gibbs succeed in transmitting for the first time, on a distance from 40 km, AC current under a tension of 2000 volts with transformers with a core bar-shaped.
In 1884, Lucien Gaulard brings into service a looped connexion of demonstration 133 Hz) supplied with AC current under 2000 volts and energy of Turin at Lanzo and return 80 km). One ends then up admitting the advantage of the transformer which makes it possible to raise the tension delivered by an alternator and thus supports the transport of electrical energy by lines with high voltage. The recognition of Gaulard will intervene too tardily.
Meanwhile, of the patents were also taken by others. The first patent of Gaulard in 1882 was not even delivered in its time, under pretext which the inventor claimed to be able to do "something of nothing"! Gaulard tackles, loses its lawsuits, is ruined, and finishes its days in a lunatic asylum. The transformer of Gaulard of 1886 does not have large-thing to envy the current transformers, its closed magnetic circuit (the prototype of 1884 comprised an open magnetic circuit, from where a quite poor output) consists of a large variety of iron wire announcing the laminated circuit with isolated sheets.
Thus, in 1885, the Hungarians Károly Zipernowsky, Miksa Déry and Otto Titus Bláthy develop a transformer with an annular core marketed in the whole world by the Ganz firm in Budapest. In the United States, W.Stanley develops transformers.
Photographs rollings up of a three-phase transformer.
It consists of two principal parts, the magnetic circuit and rollings up.
The magnetic circuit
The magnetic circuit of a transformer is subjected to a variable magnetic field during time. For the transformers connected to the sector of distribution, this frequency is of 50 or 60 hertz. The magnetic circuit is generally laminated to decrease the losses by eddy currents, which depend on the amplitude of the signal and its frequency. For the most current transformers, the piled up sheets have the shape of E and I, giving the possibility thus of slipping a reel within the windows of the magnetic circuit thus made up.
Diagrams of sheets of a transformer single-phase current.
Diagram of the carcassed a transformer low end single-phase current.
The magnetic circuits of the "top-of-the-range" transformers have the shape of a core. The winding of the cores being more delicate, the price of the toroidal transformers is definitely higher.
Rollings up are generally concentric to minimise the escapes of flow.
The electric conductor used depends on the applications, but copper is the material of choice for the whole of the applications to strong powers. The electric wires of each turn must be isolated the ones from the others so that the current circulates in each turn. For small powers, it is enough to use magnetic conductors enamelled to ensure this insulation in the applications to stronger powers one surrounds the conductors of impregnated dielectric mineral oil paper. For the strongest powers one uses multistranded conductors to limit the effect of skin but also the losses by eddy currents.
Rollings up of the primary or the secondary can have external connexions, named taken, at intermediate points of rolling up to allow a selection of report/ratio of tension. The sockets can be connected to an automatic changer of assumptions of responsibility for the control of the tension of the distribution system. The transformers at audio frequencies, used for the distribution of the audio to loudspeakers, have sockets to allow the adjustment of the impedance of each loudspeaker. A transformer with median socket is frequently used in the amplifiers of audio power. The transformers of modulation in the transmitters with amplitude modulation are particularly identical.
In the field of electricity in low tension and in the field of electronics, the thermal dissipation of the transformers is carried out by simple natural convection of the air around the primary windings and secondary.
Within the framework of the electrical circuits with high voltage and strong power, the transformers can be equipped with various cooling systems:
metal wings fixed all around the tank of the transformer which evacuates heat by natural convection
fixed wings associated with a condensator with circulation forced of the oil of galvanic insulation of the transformer
for the most powerful transformers, for example those of broad outlines VHV of the RTE from 400 to 150 Kv, one uses devices of distribution forced of an important air flow associated or not with a heat exchange with oil with the tank. The cooling system is always coupled with a device of temperature gauges playing the part of thermostat (automatic control of the startup of distribution).
The oil contained in the tank plays a double part: coolant and dielectric. The PCB were used a long time, but since their prohibition in 1987 (decree 87-59 of 2 February 1987, NOR reference ENVP8700002D), one uses mainly mineral oil.
Finally let us announce that in the field of the broadcasting of strong power, the transformers of impedance and the transformers of agreement sometimes consist of an immense rigid hollow copper coil in which circulates of pure water (the pure water is insulator electrical). Sources of TDF with Allouis in the Expensive one as with St-Aoustrille close to Issoudun in Indre used this technology of thermal dissipation.
Operation of the transformer single-phase current
Perfect or perfect transformer
Transformer perfect single-phase current
It is a virtual transformer without any loss. It is used to model the real transformers. The latter are recognised like an association of a perfect transformer and various impedances.
If the whole of the losses and the escapes of flow are neglected, the report/ratio of the number of primary whorls on the number of secondary whorls completely determines the report/ratio of transformation of the transformer.
Example: A transformer whose primary comprises 230 whorls supplied with a sinusoidal tension of 230V of effective stress, the secondary which comprises 12 whorls will present to its limits a sinusoidal tension whose effective value will be equal to 12V. U2 ⁄ U1 = N2 ⁄ N1
As the losses are neglected, the power is transmitted completely, this is why the intensity of the current in the secondary will be in the report/ratio reverses is nearly 19 times more important than that circulating in the primary.
Equality of the apparent powers : S1 = S2 U1I1 = U2I2 U2 ⁄ U1 = I2 ⁄ I1
Losses of power of a transformer
Losses by Joule effect
The losses by Joule effect in rollings up are also named "losses coppers", they depend on the resistance of these rollings up and the intensity of the current which crosses them: with a good approximation they are proportional to the square of the intensity.
Pj = Σi.Ri.i²i with Ri resistance of the coil and I ii the intensity of the current passing through it.
These losses in the magnetic circuit, also named "losses iron", depend on the frequency and the supply voltage. At constant frequency one can regard them as proportional to the square of the supply voltage. These losses have two physical origins:
Losses by eddy currents. They are minimised by the use of varnished electrical sheets, consequently electrically isolated the ones from the others to form the magnetic circuit, it in opposition to a circuit massif.les losses hystereses, minimised by the use of a soft ferromagnetic material.
The method of the divided losses consists in placing the transformer in two states:
A state for which the Joule losses are high (strong current) and particularly weak magnetic losses (weak tension). The setting in short-circuit of the transformer (test in short-circuit) with a reduced tension supply is used to carry out these two conditions. The losses of the transformer are then almost identical to the Joule losses.
A state for which the magnetic losses are high (strong tension) and or the Joule losses are particularly weak (weak current). The no-load (no-load test), that is to say without receiver connected to the secondary, corresponds to this case. The consumption with the primary of the transformer is then almost equal to the magnetic losses.
It is said that one has two states which allow "a separation" of the losses from where the expression "method of the divided losses". They have also the advantage of allowing the measurement of the output with a consumption of reduced power, without making the real operational test. This is interesting when one carries out the tests of a transformer of strong power and that one does not have in the workshop the essential power to feed it with his nominal mode. Put aside for the platforms of test in the manufacturers, this method does not have consequently great interest for only knowing the output because, in this context, a direct measurement with rated power (normal) is frequently sufficient.
On the other hand, within the framework of theoretical electrical engineering, it is important because it is used to determine the elements being used to model the transformer.
Various types of transformers
These distinctions are frequently related to the particularly many possible applications of the transformers.
Symbol of an auto-transformer. 1 indicates the primary and 2 the secondary.
It is about a transformer without insulation between the primary and the secondary.
In this structure, the secondary is part of the primary winding. The current supplying the transformer traverses the primary entirely and a derivation at a given point of this last determines the output of the secondary. The relationship between the tension of input and the output voltage is similar to that of the isolated type.
With equal output, an auto-transformer occupies less place than a transformer that is due to the fact that there is one winding, and than the common part of single winding is traversed by the sum of the currents primary and secondary. The auto-transformer is not interesting that when the tensions of input and output are of the same order of magnitude: for example, 230V ⁄ 115V. One of its primary applications is to use in a country an electronic material planned for a country where the tension of the sector is different (the United States, Japan). It however presents the disadvantage of not presenting galvanic insulation between the primary and the secondary (that is to say the primary and the secondary are directly connected), which can present risks from the point of view of the security of the people.
In France, the auto-transformer is always used for connexion between the network 225kV and 400kV.
Variable transformer - variac - alternostat
It is about a variety of auto-transformer, insofar as he comprises one winding. The derivation of output of the secondary can move thanks to a contact slipping on the whorls of the primary.
Transformer of insulation
The transformer of insulation is only designed to create an electric insulation between several circuits for reasons frequently of security or techniques solution to problem. The whole of the transformers with primary winding isolated from (of) secondary (S) should be recognised like transformers of insulation, however, in practise, this name is used to indicate transformers of which the output voltage to the same effective value as that of the input.
The transformer of insulation comprises two rollings up almost similar to the primary and the secondary:
the number of whorls of the secondary is frequently particularly a little higher than the number of whorls of the primary to compensate for the weak voltage drop under operation,
the wire cross-sections to the primary and the secondary are similar because the intensity of the currents is the same one.
They, for example, are used much in the operating theatre suites: each room of the block is equipped with its own transformer of insulation, to prevent that a defect which appears there does not generate abnormal operations in another room.
Another interest is to be able to change mode of neutral (use case of computer material and/or sensitive electronics components in an installation IT).
Transformer of impedance
The transformer is always a transformer of impedance, but the electronics specialists give this name to the transformers which are not used in feeding circuits.
The transformer of impedance is primarily designed to adapt the output impedance of an amplifier to its load.
This kind of transformer was especially employed in the sound restitution, to adapt the output of an audio amplifier to lamps (high impedance), with the loudspeakers designed for the restitution of the sound and characterised by a low impedance.
In professional audio electronics, one always uses transformers for the inputs and outputs of top-of-the-range apparatuses, or in the manufacture of "Di-box" or box of direct. The transformer is then used, not only to adapt the impedance and the level of output of apparatuses (synthetizers, low electric, etc) to the audio inputs of the console of mixing but moreover to symmetrise the output of the connected apparatuses.
In technique of the high frequencies, one uses also transformers whose magnetic circuit is out of ferrite or without magnetic circuit (also named transformer without core) to adapt the output impedances of an amplifier, a line of transmission and an antenna. Indeed, for an optimal transfer of power of the amplifier towards the antenna, it is necessary that the standing wave ratio is equal to 1.
Such assemblies have moreover the advantage of making the apparatuses connected much more resistant to the electromagnetic interferences by a significant growth of the CMRR (Common Mode Rejection Ratio) or rate of rejection of the common mode.
The transmitters make the interface between the electrical communication and a measuring device. The power available to the secondary is defined according to the needs for the measuring device.
This type of transformer, named so transformer of current, is dedicated to the adaptation of the currents concerned in different but functionally interdependent circuits.
Such a transformer authorises the measurement of the high alternative courses. It has a whorl with the secondary primary, and several whorls: the report ⁄ ratio of transformation allows the use of a traditional ammeter to measure the intensity with the secondary, image of the intensity to the primary which can reach several kiloamperes (kA).
Transformer of tension
This transformer belongs to the means to measure high alternating voltages. It is a transformer which with the characteristic to have a report ⁄ ratio of transformation calibrated with precision, but designed to deliver only one particularly weak load with the secondary, corresponding to a voltmeter. The report ⁄ ratio of transformation is used to measure primary tensions being expressed in kilovolts (kV). One meets it in HTA and HTB. Other technologies exist, like that of the capacitive divider.
Transformer high frequency
Magnetic circuit of the transformers HF
The losses by eddy currents within the magnetic circuit are directly proportional to the square of the frequency but inversely proportional to the resistivity of the material which forms it. To limit these losses, the magnetic circuit of the transformers HF is carried out with insulating ferromagnetic materials:
ferrites soft: copper and iron mixed oxides or zinc and the materials nanocristallins.
Transformer of impulses
This type of transformer is used for the ordering of the thyristors, triacs and of the transistors. It presents, compared with the opto-coupler, the advantages following: possible operation at raised frequency, simplification of the assembly, possibility of bringing an important current, good behaviour in tension.
In the three-phase electrical communications, one could ideally plan to use 3 transformers, one by phase. In practise, the use of three-phase transformers (only one apparatus gathers the 3 phases) is generalised: this solution allows the design of transformers much less expensive, with especially of the economies on the level of the magnetic circuit. The transformers single-phase currents in fact are hardly used, except for very large apparent powers (typically higher than 500 Mva), where the transport of a large three-phase transformer is problematic and encourages with the use of 3 physically independent units.
For a three-phase transformer, there exist 3 types of coupling of rolling up:
The coupling star, defined by letter Y.
The coupling triangle, defined by the letter D or Δ.
The coupling zigzag, defined by letter Z.
List of possible couplings : Yy0, DD0, Dz0, Yd1, Dy1, YZ1, YD5, Dy5, Yz5, Yd6, Qd6, DZ6 The capital is always to the highest voltage.
Index of coupling
It is the characteristic of a three-phase transformer indicating the type of coupling carried out to the primary and the secondary but also dephasing between the primary expander and the secondary expander. The three-phase devices of tension are: "triangle" (D or d) and "star" (Y or there). The first letter of the index of coupling is always in capital letter and indicates the three-phase device to the highest tension, the second letter is into tiny and indicates the device to the lowest tension. In the device "star", the "neutral" (point central of star) can be left with the connector block the transformer: this is indicated by the presence of letter NR (or N) in the index of coupling. There exists also the coupling zigzag (Z), used especially with the secondary, there has a neutral. This coupling allows, at the time of the loss of a phase to the primary, to have with the secondary an almost similar tension on the three phases. Lastly, the index of coupling is supplemented by a "time index" which gives, by step of 30°, time dephasing in 12es of turn (as on a watch) between the primary and the secondary of the transformer (ex: 11 = 11×30° = 330° in time direction or 30° in anti-clockwise direction).
the three-phase device of tension high is in "triangle"
the three-phase device of low tension is in "star" with left neutral (indicated by "N")
is the shift between the two devices of 330° (=-30° or 11×30°).
The most used couplings are: Yyn0, Yyn6, Yzn5, Yzn11, Dyn5, Dyn11.
Choice of coupling
A coupling triangle is used to connect the winding of an engine because it does not require a neutral. It allows the connexion of a single-phase load a three-phase device. It is particularly used in the distribution of electricity.
A coupling star makes it possible to have access to two different tensions: the tension line with line and the tension line with neutral. It is particularly used in the electricity transmission.
A transformer with a coupling zigzag is sometimes used for earthings.
Transformer diphasic-three-phase current
Transformers of Scott
Diagram of the transformers of Scott
The assembly of Scott is used to transform three-phase tensions into diphasic and conversely. It finds its application in electronics but also in production, distribution and transmission of electricity where the diphasic one can be always used. The assembly of Scott is carried out thanks to two transformers. The first transformer has the limits of its primary connected to two phases of the three-phase current. The second transformer is connected between the central socket of the first transformer and the remaining phase of the three-phase current. The report/ratio of winding of the first transformer will be equal to 1 while for the second it will be equal to what is equivalent to about 0,866. The secondary of the two transformers will be of equal tension in standard and with a dephasing of 90 degrees.
feeding has cutting
A feeding with cutting (schematic diagram di below) is a power supply whose regulation is ensured by an electrotechnical component used in commutation (generally a transistor). This operating process is opposed to that of the feedings statics in which the electrotechnical component is used in static mode. The feedings with cutting strongly developed since the years 1980 to mitigate the disadvantages of the feedings statics: high weight and poor yield. They are used from now on in all the electronic devices "general public".
The feeding provides electric current to the whole of the computer components. The power supply must have a sufficient power to supply the various peripherals of this last.
It converts the alternating voltage of 230V into various continuous tensions used by the backplane and the peripherals (hard drive, CD player,).
In the United States the power supplies receive a tension with 110V and 60Hz, while in Europe the standard is of 230V at a frequency of 50Hz; this is why the power supplies have most of the time a switch allowing for choice of the type of received tension.
In general, the block feeding is a convertor of the feeding type to cutting, combining lightness, important output and compactness. On the other hand this type of feeding generates many parasites high frequency, filtered more or less well by the metal or metallized cases forming a Faraday screen room and the filters of input and output.
Tensions and interfacing
It is composed of two types of principal stitchings: AT and ATX of a switch to the back in order to avoid an unexpected short-circuit in the event of problems, and even quite simply in order to easily be able to put the computer on tension or not under tension.
Connexion of the peripherals
The two types of feeding comprise outputs to supply the peripherals. Three types of connectors are used:
diagram connector molex
connector sata and molex
connector atx, sata, molex
It is a format of feeding to cutting used in computers PC of the Pentium type and former. This type of feeding provides continuous output voltages of +5V, +12V and -12V. In these feedings, the switch of startup is directly connected on the electrical communication.
Its stitching is the following:
Plugging chart feeding AT
It is the format of feeding to cutting used in computers PC of the type Pentium II and posterior. The feeding provides the following output voltages :
In these feedings, the switch of startup is connected on the mainboard, the electrical communication is connected permanently, with sometimes a switch of security for maintenance.
There exist other less current formats:
BTX, developed by INTEL and Sony to replace the ATX, standard since many years; the main objective of the BTX is an optimal distribution of the processor. Some important defects nevertheless slowed down its expansion, and in 2007 it will be abandoned although declined in other format: the microBTX (26,4 × 26,6cm) and the picoBTX (20,3 × 26,6cm);
Baby AT, case similar to format AT, but less cumbersome;
NLX, format of case and backplane; it has the effect of separating the backplane in two distinct elements, one of them receiving the processor, the memory and the other essential components of the card, the other accomodating the expansion cards.
The output of a feeding is very important. It is about the relationship between the power delivered with the components and the power drawn from the electrical outlet. It must transform the alternate electric current of the sector into continuous electric current that the components of the PC can use. At the time of this transformation there is a loss of energy in the form of heat (it is well for that the feeding should be cooled). It is consequently important to choose a feeding with strong output, to have an electricity consumption, a less heat emission impacting a need for weaker and less noisy distribution.
While taking for comparison between the feedings and an output of 72% (output recommended by INTEL and its standard ATX) and of 80% (output recommended by the American grouping 80 more), "brought back to France for 2006, year at the time which 7850000 PC were sold, the economic gain is equivalent to more than 53 million euros in invoice of electricity for a gain of 667250000kWh
Tension of input Volts)
Storage of energy
Level of output
Up or down
Reversed output voltage
Up or down
Transformer + inductance
Transformer + inductance
Resonant, zero voltage switched
Capacitor + 2 inductances
Up or down
Negative tension for a positive input
Inverting load-pump (modified Cuk)
Negative amplitude and output voltage higher than the tension of input
Up or down
pump of load
Used to generate very high voltages voltage multipliers), or on integrated circuits low power (for example to polarise memories)