Compensation of reactive power

illustration of the compensation electric on lines high voltage
The transport of the reactive power by the electric lines causes losses, a reduction in the stability of the network and a voltage drop at its end. In order to avoid that, of the compensation of reactive power, series or shunt according to the cases, is used to limit this transport of reactive power. Various electricals appliance can be used to carry out this compensation: synchronous machines, capacitor batteries, inductance or FACTS. The passive compensations are distinguished, which function in all or anything and those active which are gradual.
Active power and reactivates in an electric line without loss
The active power depends mainly on the angle of transport
The reactive power depends mainly on the amplitude of the tensions
The power activates P and reactivates Q transported in an electric line in alternative course
express themselves as follows for a line without loss:
P = V1 * V2 ⁄ X sinδ
Q = V1 * (V1 - V2cos(δ)) ⁄ X
Where δ is the angle of transport. For summary 3 parameters are important: the amplitude of the tensions, the angle of transport and impedance. For the networks in alternative course, control binds the active power to the frequency on the one hand and the reactive power with the control of the tension of the other.
Initial problem
Fall of tension

Fall of tension of a line modelled by a resistance R and an inductance X
If a line made up of elements considers only resistive and inductive
the voltage drop is worth:
|ΔU| = RlineI cos(Φ) + XligneI sin(Φ)
One can also write that
I = P2 - jQ2 ⁄ U2
ΔU + jδU = ]Zline * I = (R + jX)(P2 - jQ2 ⁄ U2)_
ΔU + jδU = ]RlineP2 + XlineQ2 ⁄ U2 + j * XlineP2 - RlineQ2 ⁄ U2_
ΔU = RlineP2 + XlineQ2 ⁄ U2
δU = XlineP2 - RlineQ2 ⁄ U2
The voltage drop thus depends at the same time on the active and reactive power.
However the resistance of the line being much smaller than its inductance, the expression can be simplified.
ΔU = XlineQ2 ⁄ U2
δU = XlineP2 ⁄ U2
The transfer of active power creates a voltage drop in squaring with V1. If one supposes, as it is the case in practice, than || V2 - V1|| is weak in front of V1, one can conclude that the transport of power activates induced mainly a dephasing of the tensions. The transfer of reactive power creates a voltage drop in phase with V1. One can conclude from it that the transport of power reactivates induced mainly a fall of the tensions.
In general with more the power which transit is raised, with more this voltage drop is important. In other words, without adjustment, in the event of strong electric charge, the tension will be lower than in the event of weak load. The control of this voltage drop is essential for the piloting of the electrical communication, it must maintain it in an interval ± 10% approximately. An overpressure is of insulation dielectric of the material ones, an undervoltage obliges an increase in the current forwarding to maintain the power constant and can lead to a collapse of the network.
Relation tension power without approximation

Relation between reactive, active power and tensions
For an adapted line the transportable active power, called natural power or virtual power of the line, is equal to : Pnat = U12 ⁄ Zw
Note in this case : U1 = U2ejBl
The active power and reactivates consumed by load Z are noted P2 et Q2. On a U2 = Z * I2
The circulation of reactive power also causes overloads on the level of the transformers of power, the heating of the electric cables and losses. Indeed the losses of the electric lines are equal to: Pperte = LP ⁄ κ WITH the ² cos (Φ), where L is the length of the line, P transported active power, κ the conductivity of the driver, U the interlinked voltage and cos (phi) the power-factor.
It is thus advisable to limit the transport of reactive power to use the network to the maximum of its capacities. In other words to produce the reactive power where it is consumed.
Principle of the compensation
Principle of the compensation series
When the active power transported by a line is not equal to the natural power, an excess or a lack of reactive power is created. This reactive power must be transported by the line, limiting its capacity to transport active power, it is thus appropriate to limit it to the maximum.
If the transported active power is too low, in other words if the line has a too capacitive behavior, typically for a cable, two possibilities propose to restore a neutral behavior for the reactive power: either to increase inductance series of the line or to decrease the capacity shunt of this one. The first solution poses the problem to increase the angle of transport: it is equal to Β L = ω √ The EC * L, which decreases the stability of the network. The privileged solution is thus to decrease the capacitance shunt by connecting a reel in parallel to the line. One speaks about compensation shunt.
The same if the transported active power is too high, in other words if the line has a too inductive behavior, for the long air lines typically, 2 possibilities also propose: to increase the capacity in parallel or to decrease inductance. For the same reasons of stability that previously, the reduction in the parameters is to be privileged. One speaks about compensation series.
Value of the compensation
The reactive power consumed by an inductance in a three-phase system is :
QL = 3LI²ω
The reactive power produced by a capacity in a three-phase system is :
Qc = 3U²Cω
In the case of a parallel compensation, one defines kp as follows the coefficient of compensation :
Coptimal = Cline(1 - kp
(1 - kp) = Coptimal ⁄ Cline
kp = 1 - Coptimal ⁄ Cline
For the compensation series :
Loptimal = Lline(1 - ks
ks = 1 - Loptimal ⁄ Lline
ks = ωLlinel - ]ωLline - (l ⁄ ωCcompensation)_ ⁄ ωLlinel
Consumers of reactive power
The principal consumers of reactive power apart from the lines themselves are:
ordinary asynchronous motors
magnetic lamps with ballast with fluorescence or discharge
arc and induction furnaces
machines to be welded
stations with D.C. current LCC
Producers of reactive power
The principal producers of power reactivates are the electric cables. The installations with D.C. current VSC, the FACTS and the generating engines ⁄ synchronous can also produce some but are adjustable, they thus do not pose a problem and do not require normally a compensation.
The electric generators produce reactive power, however their contribution is not important enough in the current networks. Various electricals appliance are used to carry out electric compensation: synchronous machines, capacitor batteries and inductances, FACTS. The passive compensations are distinguished, which function in all or anything and those active which are gradual.
The synchronous machine before was used but its reaction speed is rather slow and requires an important maintenance.
The static reels have the defect to be heavy and expensive. The capacities are on the contrary relatively not very expensive. On the other hand they bring power by stage réactivement and thus while following a function staircase, their connection or disconnection is ordered by circuit breakers. They are adjustable and produce little loss. They are adapted to the variation of consumption of slow reactive power, but not for the defects. They can be installed in stations THT ⁄ HT, but also in stations HT ⁄ MT, in this last case their dimensioning must correspond to the local load and its consumption in reactive power.
The use of electronics of power makes it possible to carry out the more economic compensation of manner. Thus the static compensators are consisted the whole of condensers and inductances ordered by thyristors, assembled in head-digs in each phase. Each one of them being thus conducting during a half-period. The reactive power absorptive by inductance varies by controlling the effective value of the current which crosses it by action on the angle of starting of the thyristors. They appeared in the years 1970. The FACTS have the advantage of being at the same time flexible and fast, thus making it possible to quench the oscillations in the network.
The stations of the lines with D.C. current said in source of tension can also produce reactive power.
The phase-shifting transformers do not influence the reactive power and are thus not compensations. They influence on the other hand the transfer of active power, just like the FACTS.

Currents of short-circuit

Symbol short-circuit, breaking capacity
PdC Capacity breaking Scc Power of short-circuit
S Section of the drivers Sn Power connects transformer
α Angle of interlocking C Factor of tension
cos φ Power-factor e Instantaneous electromotive force
E Electromotive force (effective value) φ Angle of dephasing
I Current instantaneous I Intensity (effective value)
iCC Component continues current instantaneous
iCA component sinusoidal alternative of the current instantaneous
iρ Value maximum of the current Ib Current of cut short-circuit
icc Steady short-circuit current Ik Steady short-circuit current
I"K SymmetrICAl current of short-circuit Ir Running assigned alternator
IS Current of service k Constant of correction
K Factor of correction of the impedances Ra Resistance are equivalent of the network upstream
RL Linear electrICAl resistance of the lines U Instantaneous tension
λ Factor depend on the inductance of saturation of the alternator
ucc Tension of a short-circuit U Tension made up of the network except load
Un Nominal voltage in load of the network X reactance in % of the revolving machines
Xa Equivalent reactance of the network upstream XL Linear reactance of the lines
Xt Subtransitory reactance of the alternator Za Equivalent impedance of the network upstream
Zcc Impedance upstream of the network known three-phase defect Zd or Z1 Direct Imédance
Zi or Z2 Opposite impedance Zo or Z0 Homopolar impedance
ZL Impedance of connection G Generator
K or k3 Three-phase short-circuit k1 Short-circuit single-phase current
k2 Two-phase short-circuit k2E ⁄ kE2E Two-phase short-circuit with the ground
S Group with on-load tap changer SO Group sanc changer on-load tap

Location of wire of phase and the neutral

It is very important to be able to distinguish wire from phase of wire of neutral and ground. Although the ground is generally located by dominant of green, or a naked driver, the uses all over the world saw being born for various wire, of the combinations of varied colors. Some attempts at standardization were born, according to the areas, in particular by the writing of standards.
The table below gathers a certain number of combinations of colors met in various countries.
Color code three phase
Country Phase 1 (L1) Phase 2 (L2) Phase 3 (L3) Neutral (N) Ground (T ⁄ G)
European Union, the United Kingdom
  Brown Black Gray Blue Green ⁄ Yellow
Europe (old)
  Black Red White Blue Green ⁄ Yellow
France (old before 1970)
  Green Yellow Blue gray White Black Red
The United Kingdom (old), South Africa, Malaysia
  Red Yellow Blue Black Green ⁄ Yellow
The United States (commun run)
  Black Red Blue gray White Green ⁄ Yellow
The United States (alternative)
  Brown Orange Yellow gray White Green
Canada (official)
  Red Black Blue White Green
Canada (isolated installations)
  Orange Brown Yellow White Green
Australia, New Zealand
  Red White Yellow Blue Black Green ⁄ Yellow
Popular republic of China
  Yellow Blue ⁄ Green Red Brown Black Green ⁄ Yellow
It is good to recall that a code color is viable only if it is respected by all, in the contrary case, one risks large damage
In Europe and in the United Kingdom, the standard is from now on to employ the Brown-Black-Gray triplet for the phases and to hold Blue for the neutral, while the wire of ground can be either Green edging of Yellow or to be with naked. However, one still finds old installations making use of Red for the phases, even of White, White being sometimes also used for the neutral. To avoid any confusion, the standards prohibit the use of White wire now.
In certain cases (old installations of the Scandinaves countries, the exit of the transformers of the United Kingdom, and some other cases) the two cables of a domestic catch can be phases either come from the three-phase network, or at exit of transformer single-phase current (if it is not connected to a neutral potential). That is to be disadvised

Section of the drivers by circuits

Type of circuit Protection Remarks Section More infos
Circuit breaker Fuse
Lighting 10A
10A 8 points max
by circuit
1,5 mm² Standards and recommendations lightings
Ordered catches 10A
10A 1 inter for 2 catches max and in the same part 1,5 mm² Diagram and scpécificitées of the ordered catch
(not specialized)
16A / 5 catches max 1,5 mm² Standards socket-outlet
20A 16A 8 catches max 2,5 mm²
One-way circuits
Washing machine, lava crockery, freezer, furnace, water-heater
20A 16A 1 catch or base and 1 protection by apparatus 2,5 mm² Example of table with one-way circuits
Hotplates 32A
32A Single-phase current 6 mm²  
20A 16A Three-phase current 2,5 mm²
VMC 2A One-way circuit 1,5 mm² VMC simple flow, VMC double flow, VMC hygroadjustable
Travelling shutters 16A 10A One-way circuit 1,5 mm²  
mono 230V
10A 2250 W 1,5 mm² The sum of the powers of the apparatuses
of the same circuit does not have to exceed
values indicated (in Watts).
Diagrams and standards heating with wire pilots
16A 3500 W
16A 2,5 mm²
20A 4500 W
 ⁄  20A 4 mm²
25A 5750 W
25A 6 mm²
32A 7250 W
Heating floors
mono 230V
16A 1700 W 1,5 mm²  
25A 3400 W 2,5 mm²
32A 4200 W 4 mm²
40A 5400 W 6 mm²
50A 7500 W 10 mm²

Standard UTE C 18-510 gathers a whole of regulations relating to the security concerning the operations and actions on or near wiring.
Collection UTE C18-510 is the technical paper of reference for the control of the operations near an electric risk. It definite obligations and responsibilities for the building owners, the chiefs of establishment and the speakers. It describes the titles of enablings necessary for each type of intervention according to the fields of tension.
All the personnel, which within the framework of their work has access or approaches the electrical circuits, must follow a specific training. This formation is intended to make known to them the dangers of electricity like learning how to them to secure itself some. The electricians are of course the first concerned, but also all those which them pleasing work to mix with closely wiring.
This formation is sanctioned by the delivery of a proposal for an enabling. With this proposal for an enabling the employer can deliver an enabling with the personnel which must intervene or work on or near wiring under tension or carry out operations on the electrical circuits.
This enabling is not to in no case a command of work, the personnel do not have to take the initiative of an intervention.
This collection is approved like collection of general instructions of security of signal by the decree of January 17th, 1989 (OJ January 26th, 1989). It can thus be used as collection of general instructions of security of signal of in accordance with article 4 of the Decree n°82-167 of February 16th, 1982 relating to the particular measures intended to ensure the safety of the workers against the dangers of electric origin during, exploitation and the maintenance building work of the works of electric energy distribution.

TBT, definition

TBTS: very low working stress: by principle, assured security in any circumstance, using a feeding of security to double insulation
The instruments used in instrumentation must preferentially be produced in this mode. Mode TBTS is necessary as soon as there is possibility of contact with naked conductors: it is recommendé for all and the measurement control circuits of instrumentation.
TBTP: very low tension of protection: same device that in TBTS but with an additional connection with the ground. Replace the TBTS by functional need, in particular in electronics when an earthing is essential to the good performance of the circuits.
No precaution is to be taken in TBTS and TBTP with respect to the rules of electrification. It is necessary toutefoisde to secure risk of short-circuit and burns.
TBTF: very functional low tension: used in the absence of TBTP, when the hardware does not answer any particular specification of BT must be applied.
category tension limits U L (Veff) transformer (UUL) connection with the ground
active parts
cutting and protection
against the short-circuit
protection against
indirect contacts
protection against
direct contacts
wet dryness
TBTS 25 60 50 120 TR of security
security standard CEI 742
prohibited all active conductors not not  
TBTP 12.5 30 25 60 TR of security
security standard CEI 742
yes all active conductors not not  
TBTF 50 120 50 120 transformer
yes all active conductors yes yes
(IP 2x)

Principle of superposition

Application to the electrical circuits
In the case of the electrical circuits made up exclusively of linear elements (resistances, capacities, inductances, generators of tension or current independent or dependant linearly on a current, of a tension, etc), the answer in a branch is equal to the sum of the answers for each independent generator taken separately, by decontaminating all the other independent generators (generating of tension replaced by short-circuit and generators of current by open circuits).

Illustration of the principle of superposition.
In (A): The tension out of P compared to the common mass is of 6,11 volts. This value was calculated by applying the principle of the superposition. The following stages make the demonstration of it.
In (b): Short-circuit of V1 to find the influence of V2. The tension between P and the mass become equal to the terminal voltage of R1. One calculates this tension with the formula of the tension divider
VR1 = R1 ⁄ R1 + R2 * V2 = 2,77 Volt
In (c): Short-circuit of V2 to find the influence of V1. The formula of the tension divider is again employed
VR2 = R2 ⁄ R2 + R1 * V1 = 8,88 Volt
The addition (superposition) of the values obtained, gives well us the tension at the point P of our circuit
- 2,77 Volt + 8,88 Volt = 6,11 Volt
One can apply the same principle to circuits using more than two sources. Also, each tension divider can include ⁄ understand an unspecified number of resistances in series.

Relation of Butler-Volmer
In electrochemical kinetics, one can treat an elementary stage of transfer of load while following the model of Butler-Volmer which one owes in John Alfred Valentine Butler and max Volmer. The law speed is given by the relation of Butler-Volmer.
j = j0 * {exp ](α * z * F ⁄ R * T) * (E - Eeq)_ - exp ]- (1 - α * z * F ⁄ R * T) * (E - Eeq)_}
j : density of current (in A.m-2)
j0 : density of trade flow including the constant speed
E : Potential of the electrode
Eeq : Potential of balance
T : temperature (in K)
z : many electrons intervening in the stage determining the reaction speed
F : constant of Faraday (in C.mol-1 )
R : constant of perfect gases (in J · K-1 · mol-1)
α : coefficient of transfer of load
j0 = i0 ⁄ S = n * F * K° * ]Cox_αSol * ]CRed_1-αSol


The zero-sequence phase of a quantity is one of the three components of the decomposition by the method of symmetrical components:
Homopolar (Index 0 is used to identify it as such 0 V
Direct (The index 1 is used to identify it as such V 1
Indirect (index 2 is used to identify it as such V 2
Relations Database
The zero-sequence voltage and current of a three-phase system (a, b and c) is calculated through the matrix to Fortescue
V0 = 1 ⁄ 3(Va + Vb + Vc)
I0 = 1 ⁄ 3(Ia + Ib + Ic)
Thus a balanced system:
V0 = 0
I0 = 0
neutral current
In = (Ia + Ib + Ic)
in a star connection of a load is related to the fault current by the relation:
In = 3 I0
zero sequence impedance
symmetrical components of the impedance
Either the matrix Fortescue
  1 1 1
A = 1 a
  1 a

relations matrix is following:
Vabc = AV012
Vabc = ZabcIabc
Knowing the impedances in a three phase system can be represented by a 3x3 matrix elements and is expressed by the relation:
Z012 = A-1ZabcA
Then the corresponding matrix in the theory of symmetrical components is:
V012 = Z012I012
This gives an equivalent of our system phase governed by the equation:
Z0 = 1 ⁄ 3 (Zaa + Zbb + Zcc + 2Zab + 2Zac + 2Zbc)
case of symmetrical load
A balanced load is a load or the self-impedance is the same for all three phases and the mutual impedance is the same among the three phases.
Zaa = Zbb = Zcc
Zab = Zac = Zbc
Thus, the power of symmetrical components
reveal here because the impendance transformed by Fortescue is diagonal with diagonal components: zero sequence impedance
Z0 = Zaa + 2Zab
impedance direct and indirect Z1 = Z2 = Zaa - Zab
Case load balanced star with neutral grounded
Tensions are expressed relative to the 0 voltage of the earth. The impedance between neutral and earth is Znand the impedance is a phase Zy example:
Va = Van + Vng = ZYIa + ZnIn = (ZY + Zn)Ia + ZnIb + ZnIc
This case is actually a case of symmetrical load with:
Zaa = Zbb = Zcc = ZY + Zn
Zab = Zac = Zbc = Zn
homopolar impedance Z0 = ZY + 3 Zn
Impedance direct and indirect Z1 = Z2 = ZY
If the neutral is not grounded, Z0 = ∞
which is represented by a switch opened in the schematic representation of symmetrical components.
Flux Homopolar
fault currents create flux components magnétqiues homopolar say within a magnetic circuit.
homopolar generator
The homopolar generator was invented by Michael Faraday in 1831 and is also called Disk Faraday. It was the first dynamo, the electric generator works by using a magnetic field. It was very inefficient and has not been used as a viable source of energy, but he has shown the ability to generate electric power using magnetism.
Homopolar Motor
The crucial piece is the small cylindrical magnet glued under the screw head. The more powerful the better because it is the Lorentz force that drives the engine: this force in green in the figure is perpendicular to the magnetic field and electric current blue violet.

Faraday screen room.

A Faraday screen room is an enclosure used to protect from the harmful effects electric and subsidiarily electromagnetic external or conversely to prevent an equipment from polluting its environment. A Faraday screen room is often used when one wishes to take precise measurements in electronics or electricity.
Such as she was studied by Michael Faraday during her work on the conductors, the Faraday screen room (that is to say a conducting enclosure which is connected to the ground in order to maintain its potential fixed) is tight with the electric fields (created by the simple presence of a potential difference, without a current not being necessary) and this, whether the disturbing source is inside or outside the enclosure.
This structure can also have an indirect effect of protection against the disturbances of electromagnetic origin (due to a current). One speaks then rather about electromagnetic shielding. For this use, it is more necessary only the structure is connected to the ground but the effectiveness is strongly influenced by the frequency of the disturbance and the magnetic permeability of material.
The experiment of the Faraday screen room to the palate of the discovery. The person in the cage does not feel the electric arc: she is protected there
The metal enclosure must be closed on each side in theory. But it can also be made up of openwork netting (from where the name of cage). A netting with a mesh of a few centimetres acts like a mirror on a decimetre wave, as that is used in the primary mirrors of the radio telescopes (Effelsberg, Nançay). The higher the frequency of the wave is (thus more its longueur d'onde is short), plus the mesh must be small.
There exist three principal techniques of realization of the industrial Faraday screen rooms:
Modular cages:
They are carried out using folded steel containers or panels out of wooden covered on the two faces of a steel sheet. The vats are assembled between them using bolts. The panels out of wooden are assembled using profiles of assembly out of steel. Favors vats: insensitivity to moisture and the hygrometrical variations. Good behavior in the time of the geometry. Favors panels: It can be recut. Dimensions of the room can be modified (in the event of removal for example). The modular cages make it possible to reach superior performance with 100 dB with 100 MHz.
Architectural copper cages:
They are carried out using a copper strip iron of 2 or 3 tenth of mm (delivered in roller) which is posed in recovery and is brazed uninterrupted with tin. This technique adapts well to the large-sized buildings and makes it possible to adapt to the complex geometries (corners, setbacks, beams, pillars), which is more complicated, even impossible using a modular cage. There is no loss of place, copper applying directly to the walls. However, it is necessary to envisage a doubling for decoration. The cages coppers make it possible to reach superior performance with 100 dB with 100 MHz.
Architectural metallized fabric cages:
The cage is built using a metallized tapestry posed using adhesive, like a traditional wallpaper. This technique has the same advantages as the cages coppers. The performances reached are higher than 60 dB with 100 MHz. These performances are enough in a majority to application. The interest of this technique is that it is possible to associate windows there. Indeed, the principal disadvantage of the Faraday screen rooms is that to preserve the performances, it is impossible to install there windows (80 max. dB). It is thus difficult to consider a permanent work station out of modular Faraday screen room or copper.
It is necessary to recall:
That the performances (and the cost) of a cage hold essentially in its accessories: gates, windows, passages for distribution (honeycombs), and fluids (waveguides).
That all the conductors penetrating and outgoing of the cage must be provided with radioelectric filters (if not, conductors comprise like antennas and very strongly decrease overall efficiencies of the cage).
The visitors of the Palate of the discovery, in Paris, can observe a Faraday screen room and its operation.
Sample applications
The car is a current Faraday screen room, which although imperfect cheek often well its role. The use of nonconducting composite materials as well as the glazed openings make that according to the model, it is only seldom a good Faraday screen room.
The metal case of the computers also constitutes a Faraday screen room. If this case is nonmetal (plastic), it is, to meet the standards of radio-compatibility, doubled at the strategic places, of a fine metal sheet connected to the electric ground of the machine.
In general, much with appareils électroménagers are equipped with forming shielding interns of the Faraday screen rooms at least for the significant parts. Very often for requirements of costs of construction, the metal sheets of shielding are replaced by a layer of a conducting material applied by injection to the interior of the material body made insulating.
the apparatuses of IRM are surrounded by a Faraday screen room to isolate the part from the waves being able to interfere with the waves of radio frequency emitted by the generator of radio waves.
the equipment of electrophysiology is always surrounded by a Faraday screen room, to maintain unwanted noise weak, thus increasing the signal ratio on noise.
The maisons individuelles ones are sometimes provided with a Faraday screen room what makes it possible to protect all those which are inside.

The installation of the house automation and the automatism in the dwellings makes increasingly sensitive to the transitory phenomena such as the lightning. These disturbances can be at the origin of trading losses Indeed, it proves that in the case of a direct blasting, the energy applied to lightning protectors BT can be largely energy than in the event of indirect phenomena. For this reason a wave of test 10/350 is to use for the validation of lightning protectors Type 1. The lightning protectors Type 2 and 3 are used for all the other cases, only or downstream from a Type lightning protector.

The lightning protector

The lightning protectors are intended to limit the level of overpressures to an acceptable level by the electrical material.
The level of behaviour to the shocks is defined by the coordination of insulation, normalises CEI 664-1.
The lightning protector behaves in so much normal like an open circuit. At the time of the passage of the current of the lightning, it is transformed into a short-circuit, thus making it possible to limit the dangerous potential difference between the various circuits of the installation.
The lightning protector can be made up of spark-gaps, varistors or bidirectional zener diodes. The lightning protectors are in general connected downstream from the general circuit breaker of the installation, between each conductor and the principal terminal of ground by connexions as short as possible.
Characteristics of a lightning protector
A lightning protector is designed according to:
Configuration of the installation, (capacity to dissipate energy).
Behaviour with the shocks of the system to be protected, (capacity to chop overpressure).
There exist lightning protectors dedicated to the high tension currents (Energy) and others intended for the low currents (Measurement, Commande, Régulation, Télécommunication, etc).
Constituted of spark-gaps, varistors, bidirectional diodes or a combination of the latter, the lightning protector answers an application always well defined.
The lightning protector is characterised by its acceptable tension CPU, its capacity of Imax discharge and In, like its Up level of protection.
CPU: Maximum tension with 50Hz which the lightning protector can support permanently. In mode TT (or TN) this value must be higher or equal to 1,45.Uo; this value is higher or equal to 1,732.Uo in mode IT.
Up: Level of protection of the lightning protector, this value must be lower or equal to the tension of behaviour of shock UTC of the electrical material to protect.
Iimp: Impulse impulse current which the lightning protector without damage can run out only once. This value is measured starting from the wave of test 10 ⁄ 350µs.
Imax: Maximum current of discharge which the lightning protector without damage can run out only once. This value is measured starting from the wave of test 8 ⁄ 20µs.
In: Nominal current of discharge which can run out the lightning protector 20 times. This value is measured starting from the wave of test 8 ⁄ 20µs.
Iimp: Maximum current of the lightning which the lightning protector can run out. This value is measured starting from the wave of test 10 ⁄ 350µs.


level keraunic "NK" determines the number of days per year or the lightning strikes the ground, according to what it is necessary according to the areas and of the regulations in force to make install a lightning protector on the electric line and the line of telecommunication


measuring unit of electromotive force international which is equal to the difference of potenciel between two conductors transporting 1Ampère when the power to diffuse is of 1Watt, the symbol is V
Peak value, average and effective of a tension AC
The effective value (RMS) of an alternating voltage represents its potential of average power: for example a tension AC of 220 V produces the same power (average) in a given resistance that a tension cd. of 220 V. the power evolving/moving with the square of the tension, a measuring device must thus be able to form the quadratic average of the tension AC.
The modern multimeters easily measure the true effective value (TRMS - true rms) thanks to the integrated functions of calculation whereas the analogical multimeters (almost disappeared from our laboratories) cheat in form the median value of the tension detected by integration then multiply it per 1,11 (scale). This easy way passes unperceived for a signal sine but, when the signal to be measured is presented in another form, the error of measurement to become very important little.
The effective value thus indicates the capacity of an alternating signal to produce an average power. For example, a tension of 220V EFF AC produces in the same resistance same calorific energy (in median value) that a continuous tension of 220V.
Detection of peak value
The apparatuses equipped with a detector of peak value measure the maximum value of the tension applied. That is obtained thanks to a condenser which takes care with the peak value and preserves this load so that the reading can be carried out. One distinguishes the detectors from positive value (Uc+), the detectors of negative value (CPU, just as the detectors of peak value to peak.
Detection of median value
The median value (Um) of a rectified alternating voltage (rectified value) is expressed by the integral of the absolute value (module) of the tension according to time, that corresponds to the value of the surface limited by the curve on the one hand and line zero on the other hand divided by the duration T, of the period.

Vmoy = Integral 1 ⁄ T pi |V|dt
When one cumulates the instantaneous amplitudes of a sinusoidal signal of 0 to pi by very weak increments and one forms then the arithmetic mean, one obtains a result very close to the median value obtained by integration (the surface under a sine of 0 with pi = Integral sin X dt = 2; average = 2 pi = 0,6367).

Sinusoidal signal Uridge = 1V:
The average height of the 314 samples (from 0,1 to 3,14) is of 0,635, that is to say very near to 2 ⁄ pi
Effective detection of value
The effective value (Ueff) is obtained starting from the square of the instantaneous tension U (T)² integrated over one period and divided by the duration T of the period. A circuit of extraction of square root must be used to obtain a linear scale.

Synoptic of a voltmeter AC with calculation of the true effective value
The effective value of a tension AC corresponds to the value of a continuous tension producing the same thermal power in an identical resistance.
The red curve (sin²) represents the thermal power produced in a resistance by a sinusoidal signal. By folding up surfaces being with the top of the line of 0,5 V, one can fill empty surfaces and form a rectangle of 0,5 * 2 pi = pi.
When one observes a signal sine², one notes that the produced average power is of 0,5 W for Ucc = 2V, R = 1 ohm).
The tension of the sine of 2V (ueff = root (0,5) = 0,707) produced the same thermal power (in same resistance) as a tension cd. of 0,707 V.
To obtain the effective value of a sinusoidal tension : to divide V by 2 * racine (0,5) = Vridge ⁄ 1,41).
The peak factor corresponds to effective the peak value report ⁄ ratio ⁄ value of an alternating voltage and constitutes an important criterion for the measurement in particular of nonsinusoidal alternating voltages characterized by short impulses of great amplitude, separated by very long periods, extent to which the peak value is high and the low effective value. The measuring device to be used must be able to transmit the amplitude of the peaks correctly in order to avoid the errors of measurement.
The factor of form F corresponds to the effective report ratio value ⁄ median value and plays a big role in the apparatuses comprising a detector of median value. For the sinusoidal signals, the factor of form is 0,707 ⁄ 0,637 = 1,11
It corresponds to the potential difference electric which exists between two points of a circuit traversed by a constant current of 1 amp when the power dissipated between these two points is equal to 1 Watt.
1V = 1 * W ⁄ has = 1 * J ⁄ C = 1 * (NR * m) ⁄ (* S A) = 1 * (kg * m²) ⁄ C * S²)
It can be defined starting from the basic units
1V = (1 kg * m²) ⁄ (1 A * S³)
W: in Watt
A: in amp
J: in joule
NR: in newton
m: in meter
S: in second
kg: in kilogram
C: in Coulomb
Multiples of the volt
10N Prefixed Symbol Number
1024 yottavolt YV Quadrillion
1021 zettavolt ZV Trilliard
1018 exavolt EV Trillion
1015 pétavolt PV Billiard
1012 téravolt TV Billion
109 gigavolt GV Milliard
106 mégavolt MV Million
103 kilovolt kV Mille
102 hectovolt hV Cent
101 décavolt daV Dix
100 volt V Un
10-1 décivolt dV Tenth
10-2 centivolt cV Hundredth
10-3 millivolt mV Thousandth
10-6 microvolt μV Millionth
10-9 nanovolt nV Billionth
10-12 picovolt pV Trillionth
10-15 femtovolt fV Quadrillionth
10-18 attovolt aV Trillionième
10-21 zeptovolt zV Trilliardième
10-24 yoctovolt yV Quadrillionième


One can define the electrical resistance of a circuit as being the difficulty which this circuit presents in the passing of the electric current. The unit of resistance is the OHM. We can also declare that when the length (L) of a conductor increases, its resistance grows proportionally. In the same way when the length decreases, its resistance decreases. Lastly, the law of Pouillet gives us the following relation
The resistance of a conductor is directly proportional to its resistivity and its length (L).
it is inversely proportional to the section (S) of this conductor.
R = rho X L S
With R the electrical resistance of the conductor in ohm
resistivity of the matter in ohm ⁄ mm ² ⁄ m
L length of the conductor in m
S section of the conductor in mm ²
For recall, the section of a cable following its diameter is calculated as follows:
S = 3,14 x d² ⁄ 4 Or according to its radius: S = 3,14 X R²
aluminium : 0.028
constantan : 0.5
copper : 0.017
however : 0.024
iron : 0.1
lead : 0.22
money : 0.016
bronze : 0.067
mailleschort : 0.25
platinise : 0.1
brass : 0.07
tin : 0.07
tungsten : 0.055
carbon : 0.4

Resistivity of metals with 0°

aluminium : 0.004
cuivre : 0.004
argent :0.00377
bronze : 0.0005
mailleschort : 0.0036
tungstène : 0.0065
carbone : -0.0007
Nichrome : 0.0004

In electromagnetism, the theorem of Gauss makes it possible to calculate the flow of an electric field through a surface taking into account the burden-sharing.
It is due to Carl Friedrich Gauss.
The flow of the electric field through a surface S closed is equal to the sum of the loads contained in the volume V delimited by this surface divided by (the permittivity of the vacuum).
∫∫sE . dS = 1 ⁄ ε0∫∫∫v pdt = (Σ Q int) ⁄ ε0

Method of Boucherot

The method of Boucherot allows, in sinusoidal mode of tension and current, to calculate the total power consumed by an electrical installation comprising several electric dipoles of various power-factor, as well as the called total intensity.
This method developped at the point by Paul Boucherot, makes it possible to make calculations according to a formalism of the vectorial type without using the too heavy representation of Fresnel when one is in the presence of many dipoles.
Utility of the method
Within the framework of a study of an installation, it is necessary to calculate:
Consumed total power : it is what one pays.
Absorptive intensity : for the dimensioning of the cables, circuit breakers, disconnecting switch and choice of the subscription.
The total power-factor when it is useful (installations fed in high voltage, generally industrial).
The value of the condensers if it is necessary to improve the power-factor.
For each dipole I one calculates Pi and Qi (P indicates the active power and Q the reactive power). The installation consumes Pt = ΣPi and Qt = ΣQi.On from of deduced St = √t + Q²t from where the total intensity It = St ⁄ U
Sinusoidal modes single-phase currents
It is rare that these installations of low powers require to make calculations of power-factors. However it is sometimes useful to be able to calculate the absorptive total intensity.
The active power either known, is indicated by the maker badge of the receiver, or obtained using the relation:
active power: P = U * I * Cosφ
apparent power: S = U * I
reactive power: Q = U * I * Sinφ
Three-phase sinusoidal modes
The method applies same manner but one uses the following relations with the root of 3 which is worth 1,732:
active power: P = √³ * U * I * Cosφ
apparent power: S = √³ * U * I
reactive power: Q = √³ * U * I * Sinφ
Installations fed in sinusoidal tension and absorbing nonsinusoidal currents
If the absorptive current is not sinusoidal, the problem is more complex: even if the current is in phase with the tension, the power is not equal to the product of the effective values
The tension, sinusoidal, can be written
U(t) = U √2 * Cosωt
The current, nonsinusoidal, can break up into series, known as Fourier series:
I(t) = I1√2 * Cos (ωt + φ1) + I2√2 * Cos (ωt + φ2) + etc.....
The calculation of the active power gives like result:
P = U * I * Cosφ1
In addition the apparent power can be written:
S = √P² + Q² + D²
With the definitions of the following intermediaries of calculation:
reactive power
Q = U * I1 * Sinφ1
deforming power
D² = U²1 (I²2 + I²3 + I²n) = U²1 * I²h
I1 = the effective value of the current fundamental
Ih = the effective value of the whole of the harmonic of higher row has 1 of the current
φ1 : the value of the dephasing of fundamental I1(t) compared to the tension
Cosφ : the factor of displacement

Pont de Wheatstone

A Wheatstone bridge is an measuring instrument invented by Samuel Hunter Christie in 1833, then improved and democratized by Charles Wheatstone in 1843. This is used to measure an unknown electrical resistance by balancing of two branches of a lattice network, with a branch containing the unknown component.
Let us consider the figure above. The bridge consists of two known resistances, R 1 and R 2, of a variable resistor of precision, R 3, and of a galvanometer or sensitive voltmeter, V G.
The potential at the junction point between R 1 and R 2 (noted D) is obtained thanks to the theorem of Millman and is worth V.R 2 ⁄ (R 1 + R2), where V is the potential difference at the boundaries of the pile. If we place between R 3 and masses it an unknown resistance, R X, the tension at the junction point between R 3 and R X is worth V.R X ⁄ (R 3 + R X).
Let us adjust R3 in order to cancel the current in the galvanometer; the potential difference at the boundaries of this one is thus null. By equalising the two tensions calculated above, one finds:
Rx = (R3 * R2) ⁄ R1
In practise, the Wheatstone bridge comprises a whole of gauged resistances, in order to be able to measure a broad range of values of R X with only one precision resistor, it is enough to change report ⁄ ratio R1 ⁄ R2.
In addition, the same technique can be used to measure the value of capacitors bridge of Sauty) or of inductances One replaces the source of continuous tension by a source of alternating voltage and the precision resistor by a capacitor or an inductance of precision. With the balance of the bridge (running no one in the galvanometer), the report ⁄ ratio of the impedances in the reactive branch is equal to the report ⁄ ratio of resistances.
The Wheatstone bridge is also used at the time of the placement of gauges of deformation.
A gauge of deformation is based on the property which have certain materials to see their conductibility varying when they are subjected to such known constraints, pressures or deformations piézorésistance under the name of barrorécepteurs. It makes it possible to manufacture pressure pick-ups, acceleration, etc As the variations of resistance are too weak to be directly measurable, it is necessary to call upon an assembly in Wheatstone bridge.
Supplied with a source of tension the bridge has, with balance, a null tension V, but the variation of one or the other of resistances reveals a nonnull tension. In practise, several of these resistances are gauges.
The interest of this assembly is that two adjacent resistances act as opposite direction and two opposite resistances act in the same direction.
One can thus reduce the parasitic variations (like the temperature) and to have a better precision.
A sensor with four gauges makes it possible to have still a better precision than a sensor with a gauge. In practise, the number of gauges is often dictated by the geometry of the part.


the loie of matthiessen shows that the more important the temperature of an ohmic concuctor is and the more resistance of that Ci increases by using the following formula: Rt=R0. (1+a.t)

Steinmetz formula

The formula used to calculate approximately Steinmetz hysteresis losses in the magnetic circuit: k * V * F * Bnm
k : a constant without unit, equal to 0.02
V : volume of the magnetic circuit m³
F : frequency of the magnetic field Hz
Bm: maximum magnetic induction in the magnetic circuit by T
n : power value between 1.6 and 2

Law of Paschen

The law of Paschen, stated by the German physicist Friedrich Paschen in 1889, indicates that the appearance of an electric arc in a gas, with a certain electric field of breakdown (known as disruptive field), is a generally nonlinear function of the product of the pressure p of gas by the distance D between the electrodes divided by the temperature T of gas:
V = F (p * d ⁄ T)
Curve of Paschen
Curve of Paschen, in X-coordinate produces it pressure time outdistances, in ordinate the tension
The theoretical relation of the appearance of the electric arc between two plane and parallel electrodes immersed in a gas, function of the pressure and temperature of this gas and electrode spacing, is described with the curve of Paschen. The term p X D ⁄ T is in fact proportional to the mass of gas contained between the electrodes, because the tension known as disruptive (from which a breakdown intervenes) is directly connected to this gas mass whose ionization is necessary to obtain the electric shock.
Minimum of Paschen
This relation indicates that there exists always a minimal electric tension for a certain electrode spacing (minimal field disruptive, which is an electric tension per unit of length (Volt ⁄ m) being expressed in this case classically in kilovolt per millimetre) with a given pressure, making it possible the electric current to discharge in gas: this value easily shown by the experiment is called the minimum of Paschen.
With the atmospheric pressure with the sea level, the air is insulating having a high tension of breakdown. There are not enough free electrons and their mean free path is too weak so that they accelerate sufficiently between two collisions: their kinetic energy is insufficient to ionize gas.
But the more the pressure of the air decreases and the more the electric shock occurs with weak tensions, the curve of Paschen reaches a minimal value called the minimum of Paschen (some torrs for the air, where the tension to be applied is minimal with approximately 330 volts, for very weak distances about the millimetre).
A credible minimum for the air is for example 350V at the point of X-coordinate 0.73 kPa*mm. For the SF6 (gas used in electrical installations) the minimum is for 500V with 0.35 kPa*mm approximately.
On the other hand, if the pressure continues to descend under this minimum from Paschen then the tension required increases again (the curve of Paschen goes up). The mean free path of the electrons becomes this time too large: there is no more enough of atoms on their way to start, by collisions with those, the effect of avalanche which transforms gas into plasma.

Theorem of Amp

Into magnetostatic the theorem of Amp allows to determine the value of the magnetic field thanks to the data of the electric currents. This theorem is an integral form of the equation of Maxwell-Amp. He was discovered by Andre-Marie Ampère, and constitutes the magnetostatic equivalent of the theorem of Gauss. To be applied analytically in a simple way, the theorem of Amp requires that the problem considered is of high symmetry.
Statement of the theorem of Amp

An electric current I product an electromagnetic induction field B.
In quasi-static or permanent mode, in the vacuum, the theorem of Amp states that circulation, along a closed circuit, magnetic field generated by a distribution of current is equal to the algebraic sum of the currents which cross the surface defined by the directed circuit, multiplied by the permeability of the vacuum (µ0 = 4π * 10-7 H ⁄ m).
τB * dl = µ0 * ∑itraversant
τ represents the curvilinear integral on contour closed τ
B is the magnetic induction field
dl is the infinitesimal element of displacement along contour τ
µ0 is the permeability of the vacuum
itraversant is the algebraic sum of the intensities of the currents intertwined by contour τ
By the theorem of Stokes, one obtains the expression of the law of Amp in local form which establishes a relation between the B field in a point of space and the density of J current in this same point, ∇ ∧ B = µ0J
Intertwined intensity
one can distinguish several cases concerning the intensity intertwined by the circuit
if the circuit intertwines a voluminal current J, then the intertwined intensity will have the following form
Itraversant = ∫∫S j * dS
if the circuit intertwines a surface current K, then the intertwined intensity will have the following form
Itraversant = ∫l k * dl
if the circuit intertwines several thread-like circuits then one can say that the intertwined intensity will be written
Itraversant = ∑Ii
with Ii intensity of a wire of the thread-like circuit
attention, it acts of an algebraic sum: it is necessary to direct the contour of Amp, and thus to give a normal to surface, from where a convention of sign concerning the intertwined currents, counted positively or negatively according to their direction.

Theorem of Poynting

The theorem of Poynting, stated by John Henry Poynting, relates to the conservation of energy in an electromagnetic field. It establishes a relation between electromagnetic energy, Joule effect and the flow of the vector of Poynting.
- ∫∫∫ (∂Wem ⁄ ∂t) * dτ = ∫∫∫ div π * dτ + ∫∫∫ j * E * dτ
maybe, in local form, for a volume dτ
- (∂ ⁄ ∂t) [(∈0E² ⁄ 2) + (B² ⁄ 2µ0)] = div (E Λ B ⁄ µ0) + j * E
maybe in the general case
- ∂ ⁄ ∂t (E * D ⁄ 2 + B * H ⁄ 2) = div (E Λ H) + j * E
with :
E : electric field
H : magnetic field
B : magnetic induction
D : electric flux density
J : density of current
0 : permittivity in the vacuum
µ0 : permeability in the vacuum
In abstract terms, one can say that the flow of the vector of Poynting through a closed surface is equal to the sum of the electromagnetic variation of energy and the Joule effect in interior volume on the surface.
Demonstration starting from the Maxwell's equations
div II = div E * B ⁄ µ0 = - 1 ⁄ µ0 E * rot B + 1 ⁄ µ0 B * rot E
by using the formula of vectorial analysis
div II = - 1 ⁄ µ0 E * [µ0j + µ0ε0 * (∂E ⁄ ∂t)]
by using the Maxwell - Amp and Maxwell - Faraday equations
div II = - 1 ⁄ µ0 E * [µ0j + µ0ε0 * (∂E ⁄ ∂t)] + 1 ⁄ µ0 B * (- ∂B ⁄ ∂t)
div II = -j * E -∂u ⁄ ∂t
u = ε0E² ⁄ 2 + B² ⁄ 2µ0
voluminal density of electromagnetic energy.

Vector of Poynting

The vector of Poynting, noted Π, S, N, or R, is a vector whose direction indicates, in an isotropic medium, the direction of propagation of an electromagnetic wave and whose intensity is worth the density of power conveyed by this wave. The module of this vector is thus a power per unit of area, i.e. a flow of energy.
General expression of the vector of Poynting
The vector of Poynting is expressed in Watt per square meter
Are E and B the electric field and the magnetic field. Then, the vector of Poynting is defined by
π = (E Λ B ⁄ µ0)
où µ0 are the permeability of the vacuum. In an unspecified material of magnetic permeability µ, it is appropriate to take into account the magnetic excitation H defined by the relation B = µ H. The more general expression of the vector of Poynting is thus
π = (E Λ H
Temporal average in complex notation
In the case of a harmonic progressive plane wave electromagnetic, one has
E = E0 cos (ωt - φ)
B = B0 cos (ωt - ψ)
One can thus associate complex sizes with the fields
E and B by posing with i the number i² = -1
E = E0eiωt = E0e-iφeiωt

B = B0eiωt = B0e-iψeiωt
The temporal average of the vector of poynting is worth then
〈πt = 1 ⁄ 2µ0 Re (E Λ B→*)
B→* indicate combined B
Electromagnetic power crossing a surface Σ
A consequence of the theorem of Poynting is that the electromagnetic power crossing a surface is given by the flow of the vector of Poynting through this surface.
PS = ∫∫Σπ * dS
Equation of the energy of an electromagnetic field
That is to say Uem the energy of the electromagnetic field
Uem = ∫∫∫V Wem
with W density voluminal of energy
one defines the quantity of energy leaving a Τ volume during a time δt
- dUem ⁄ dt = - d ⁄ dt ∫∫∫V (Wem dΤ) = - ∫∫∫V (∂Wem ⁄ ∂t) * dΤ
that is to say P, vector flow of energy of the field. According to the theorem of Green-Ostrogradsky, one can say that the outgoing flow of volume V is
∫∫Σ P * ndS
with n normal vector on the surface. Σ of the volume, directed towards outside
One can clarify the loss of energy of volume in the following way
Losses due to frictions of the live loads
Losses due to the outgoing electromagnetic radiation of volume
One can thus say that
- ∫∫∫V (∂Wem ⁄ ∂t) * dΤ = ∫∫∫V + ∇ * PdΤ work provided by the field to the matter
let us calculate this work
FElectrique = q(E + v * B)
WElectrique = F * dr = qE * dr
it is seen easily that the magnetic force does not work
Let us pass to the power provided by the field
∂WElectrique ⁄ ∂t = F * v = qE * v→for a load.
one is in the case of N loads
∂WElectrique ⁄ ∂t = NqE * v
Nqv = j
∂WElectrique ⁄ ∂t = j * E
this loss of power is equal to the loss of energy of the field per unit of time and of volume thus one writes finally
- ∫∫∫v (∂Wem ∂t) * dΤ = ∫∫∫v * PdΤ + ∫∫∫v j * E
thus finally one has
- ∂Wem ∸ ∂t = ∇ * P + j * E, eq. energy of the electromagnetic field

Theorem of sampling of Nyquist-Shannon

The theorem of Nyquist-Shannon, named according to Harry Nyquist and Claude Shannon, states that the sampling rate of a signal must be equal or higher than the double of the maximum frequency contained in this signal, in order to conGreen this signal of a continuous form to a discrete form (discontinuous in time). This theorem is at the base of the analog-to-digital conversion of the signals.
The best illustration of the application of this theorem is the determination of the sampling rate of an Audio CD, which is of 44,1 Khz.
Elementary considerations
If one wants to use a sampled signal, it is necessary to be sure that this one contains all the information of the analog signal of origin. It is often convenient to regard this one as a sum of sinusoids. However it is intuitively obvious that a loss of information occurs if the step of sampling is too large by comparison with the periods in question, the sampling rate being too weak compared to the frequencies considered.

That is to say a sinusoidal signal of amplitude α and frequency ƒ:
x (t) = α cos (2παt)
By sampling it with a step T is a frequency 1 ⁄ T
the continuation of digital assets is obtained
xn = α cos (2πnƒT)
Now let us consider the signal of amplitude β and frequency 1 ⁄ T - ƒ:
y (t) = β cos (2π(1 ⁄ T - ƒ)t)
Once sampled at the same frequency, it becomes
yn = β cos (2πn(1 ⁄ T - ƒ)T) = β cos (2πn(1 - ƒT))
elementary trigonometry leads to
yn = β cos (2pnƒT)
Thus, in the sum xn + yn, it is impossible to distinguish what belongs to the signal of frequency ƒ and with that from frequency 1 ⁄ T - ƒ. This result led to the effect of folding up of spectrum or aliasing, which indicates that one takes a sinusoid for another alias.
If more the high frequency of a signal is ƒM, the frequency 1 ⁄ T - ƒM should not belong to the spectrum of the signal, which leads to the inequality : 1 ⁄ T ≥ 2ƒM
So that a signal is not disturbed by sampling, the sampling rate must be higher than the double of the more high frequency contained in the signal. This limiting frequency is called the frequency of Nyquist.
Precise details
One can interpret the preceding result by considering a transitory signal X (T), therefore provided with a transform of Fourier X (F).
Let us consider the distribution obtained by multiplying signal X (T) by a comb of Dirac function, summons deltas of intensity T distant of T.
x* (t) = Tx(t) . δT(t)
the transform of Fourier of x* (T) is the convolution of the TF of X (T) by the TF of the comb of Dirac function:
X*(ƒ) = X(ƒ)* ]∞ ∑ n=-∞_δ(ƒ - n ⁄ T)
The Dirac function being the neutral element of the convolution, one obtains:
X*(ƒ) = ]+∞ ∑ n=-∞_ X(ƒ - n ⁄ T)
The bringing together of the two results shows that the calculation of the transform of a signal sampled with the step T by the method of the rectangles gives the sum of the true transform and all relocated this one with a step equal to the sampling rate 1 ⁄ T.
All the useful information is contained in the interval ]-1 ⁄ (2T), ⁄ (2T)_.
If the frequencies present in the signal do not overflow of this interval, that is to say if the sampling rate is higher than the double of the more high frequency, the true transform is obtained. In the contrary case, the relocated close ones come to be superimposed. This phenomenon is called recovery of the spectrum
Because of symmetry, all occurs as if the true spectrum were folded up, energy associated with the frequencies higher than half of the sampling rate is transferred in lower part from this frequency. If one wants to avoid the Franglais one in general uses the folding up term preferably with aliasing.
These results apply without modification to a signal to finished variance.
Formulate of Shannon
Since the transform X* (F) of the correctly sampled signal contains, in the interval ]-½T,½T_, the transform of the signal of origin X (T), one can reconstitute this one by calculating the opposite transform, integration being limited with this interval.
One obtains thus
x(t) = ]+∞ ∑ n=-∞_ x (nT) . sinc (π ⁄ T (t-nT))
with: sinc which is the noted cardinal sine:
sinc (X) = sin (X) /x

Theorem of Kennelly

Presentation of the assemblies in the form of triangle (on the left) and of star (on the right).
The theorem of Kennelly, or transformation triangle-star, or transformation Y-Δ, or T-∏ transformation, are a mathematical technique which makes it possible to simplify the study of certain electrical communications.
This theorem, named thus in homage to Arthur Edwin Kennelly, makes it possible to pass from a configuration triangle (or Δ, or ◷, according to the way which one draws the diagram) to a configuration star (or, in the same way, Y or T). The diagram opposite is drawn in the form triangle-star, the diagrams below in the T-∏ form.
This theorem is sometimes used in electrical engineering or electronics of power in order to simplifer of the three-phase systems.
Transformation star towards triangle

Table of the formulas of transformation (star towards triangle)
With the impedances With the admittances
The sum of the products of the impedances

divided by the opposite impedance.
The product of the adjacent admittances

divided by the total sum of the admittances.




Table of the formulas of conversion (triangle towards star)
With the impedances With the admitances
The product of the adjacent impedances

divided by the total sum of the impedances.
The sum of the products of the admittances

divided by the opposite admittance.





Theorem of Millman

The theorem of Millman is a particular form of the law of the nodes expressed in terms of potential. It is thus named in the honor of the American electronics specialist Jacob Millman.

Illustration of the theorem of Millman
In an electrical communication of branches in parallel, including ⁄ understanding each one a perfect generator of tension in series with a linear element, the terminal voltage of the branches is equal to the sum of the electromotive forces respectively multiplied by the admittance of the branch, the whole divided by the sum of the admittances.
Vm = (k=1NEk * Yk) / (k=1NYk) = (k=1N Ek / Zk) / (k=1N 1 / Zk)
In the particular case of an electrical communication made up of resistances:
Vm = (k=1NEk * Gk) / (k=1NGk) = (k=1N Ek / Rk) / (k=1N 1 / Rk)
One can also generalize it with generators of currents. If there is, always in parallel, from the known Ik currents injected towards the same point M, then one can write
Vm = (k=1NEk * Gk) + k=1NIk / (k=1NGk) = (k=1N Ek / Rk + k=1PIk) / (k=1NEk 1 / Rk)
With G, conductance. It is noticed that the presence of generators of currents does not modify the denominator.

Theorem of Tellegen

In electricity, the Theorem of Tellegen is a direct consequence of the laws of Kirchhoff which translates in particular the conservation of energy in a isolated electrical circuit. This theorem owes its name in Bernard Tellegen, a Dutch researcher with whom one owes in particular the invention of the pentode, and which formulated it for the first time in a publication of 1952.
If an unspecified electrical circuit has NR branches, individually subjected to a tension the U.K. and traversed by a current ikmais respecting all unit same convention generator or receiver, then :
k=1N Uk * Ik = 0
The formulation of this theorem makes it possible to note that it does not depend on the linear aspect and the material constitution of the circuits which use it or, more generally, of the relation of dependence between the tension and the current in each one of their branches. In practice, with a given circuit, it is enough just that the two distributions considered, of the currents on the one hand and the tensions on the other hand, that they are dependant between them or not, obey respectively the law of the nodes and the law of the meshs to be assured the applicability of the theorem there. More precisely, with the same topology of circuit where the laws of Kirchhoff generally are respected, if there exist two possible situations where the currents and the tensions are distributed differently then one has, for the first:
k=1N uk * ik = 0
and, for the second:
k=1N u’k * i’k = 0
but also
k=1N u’k * ik = 0
k=1N uk * i’k = 0
From a physical point of view, independently of the contents of an electrical circuit, this theorem indicates that a circuit respecting the laws of Kirchhoff has a total assessment of power which is null. This is only the translation of the assimilation of the electrical circuit to an isolated thermodynamic system.
From a mathematical point of view, this theorem shows that the subspaces vectorial Vi and Vu made up of all the vectors which satisfy the equations of Kirchhoff, for respectively the currents and the tensions, are orthogonal in Rn
The validity of this theorem is already simple to establish, thanks to the law of the meshs, for a circuit containing only one mesh. When the circuit is more complex, that is to say if it consists of several meshs, it is enough to consider that this assembly is only the agglomerate of several circuits with only one mesh to extend its validity. In this last stage, the law of the nodes is then used to break up the currents of the whole circuit into those of each mesh taken individually.

Theorem of Norton

In the same way one can replace any linear network, not comprising ordered sources, taken between two of his terminals by a power source I0 in parallel with a resistance R0.L' I0 intensity is equal to the current of short-circuit, the two terminals being connected by a perfect conductor. R0 resistance is that of the circuit seen of the two terminals when all the sources are extinct.
Equivalence between representations of Thévenin and Norton
The respective application of the theorems of Thévenin and Norton makes it possible to show the equivalence of two following circuits

Theorem of Thévenin

A linear network, including/understanding only sources independent of tension, current and resistances, taken between two terminals behaves like a generator of E0 tension in series with a R0 resistance. The f.e.m. E0 of the equivalent generator is equal to the tension existing between the two terminals considered when the network is in open circuit. R0 resistance is that of the circuit seen of the two terminals when all the sources are extinct.

with: E0 = R0 I0

Kirchhoff (meshs)

The law of the meshs expresses the fact that when a load traverses a closed circuit, the energy which it loses while crossing part of the circuit is equal to the energy which it gains in the other part. As follows:
the algebraic sum of the tensions along a mesh is null
mailleU = 0
For that, it is necessary to arbitrarily choose a direction of course of the mesh and to be appropriate that the tensions whose arrow points in the direction of the course are counted like positive and others like negative.

NOTE: A mesh consists of tensions forming a closed course. Each tension is present of a point of the course at another without there being necessarily a current which circulates between them.

Kirchhoff (nodes)

The law of the nodes expresses the conservation of the load which means that the sum of the currents leaving a node is equal to the entering sum of the currents. In other words:
the algebraic sum of the currents is null in any node of a circuit
noeudI = 0
For that, it is necessary to choose a sign for the currents entering and the contrary sign for the currents leaving. In general, one chooses the positive sign for the entering currents.


Röntgen (symbol R) is an old unit cgs allowing to quantify the exposure to the ionizing rays gamma, originally definite like radiation inducing an electrostatic unit of load in one cubic centimeter of dry air with pressure and temperature normals. He is named in the honor of the German physicist Wilhelm Röntgen, discoverer of x-rays.
He was supplanted by Coulomb per kilogram (C/kg).
The natural amount of background cosmic rays, especially is from approximately 10 µR per hour, it increases with altitude.
Röntgen is not a unit of amount but of exposure. Röntgen expresses the capacity of ionization of x-rays or there in the air and corresponds to the formation of even 2,1×109 of ions in 1cm3 of air, which leads to an absorptive amount of 83 ergs per gram.


The Rayleigh is a unit of luminous intensity of old System CGS, symbol R, it corresponds to the brightness of a source emitting in all the directions a luminous photon million a second per square centimeter.
Thus, 1 R = 795.774.716 photons (m²
The name of Rayleigh was allotted to him in 1956, in homage to the British physicist John William Strutt Rayleigh, this unit was used in astronomy and physics to measure the brightness, more particularly in the case of a monochromatic source.


measuring unit of the intensity of the electrical signal of the international system, is equivalent to 1 Coulomb a second, that is to say approximately 6,28x10 elementary charge, symbol has


the law of ohm is a rule making it possible to connect the intensity crossing a dipole and the tension to measure has these terminals, symbol Z out of AC and R in cd.


The oersted (Oe symbol) is electromagnetic unit CGS with three dimensions of magnetic excitation or magnetic field.
The oersted, named thus in the honor of Hans Christian oersted, cannot be compared strictly with the corresponding unit of the international system (SI), the amp per meter, because IF is with four dimensions when one limits oneself to the mechanical magnitudes and electric. However, the oersted corresponds to 10 ³ ⁄ 4p A.M-1 ˜˜79,577471545947 A.M-1.


Coulomb is an electric unit of charge in the international system, a these derived unit, these quantity of electricity crossing a section of a conductor traversed by an intensity of 1A during 1 second, symbole = C


the Tesla is the unit derived from induction magnétiue international system which distributes uniformly on a surface of 1 square metre, produces a magnetic induction of 1 weber, symbole = T


the weber is the measuring unit of the flow of magnetic induction of the system international, symbole = WB


the hertz is the electric unit of frequency of the international system, it is equivalent to an oscillation a second, symbole=Hz


the farad is the electric unit of capacity of the system international, these electric capacitance of a capacitor between the reinforcements of which a difference of potenciel of 1 volt appears when it is charged with 1 Coulomb, symbole = F


the lumen is the unit of luminous flow derived from the international system, 1 lumen corresponds to the flow emitted in a solid angle of 1 steradian by a uniform point source located at the node of the solid angle whose intensity is worth 1 candela, symbole = Ln


the candela is the unit of the luminous intensity derived from the international system, the candela is the luminous intensity in a given direction of a source which emits a monochromatic radiation of frequency 540X10exposant12 hertz and whose radiant intensity in this direction is of 1 ⁄ 683 Watt by steradian, symbole = CD


difference of potenciel electric able to make circulate electric current in a circuit

law of joule

the joule is the quantity heat to release during one second by a resistance of one ohm to cross by an intensity of an amp

Law of Planck

The law of Planck defines the distribution of energy luminance monochromatic of the thermal radiation of the black body according to the thermodynamic temperature.
λ = 2hc² ⁄ λ 5 * 1 ⁄ ehc ⁄ (λ k b T - 1) with L°λ in
where C is speed of light in the vacuum, H is the Planck's constant and KB is the Boltzmann constant.

Law of Wien

The maximum of this spectrum is given by the law of Wien :
with λmax meters and T in Kelvins. This last law expresses the fact that for a black body, the product of the temperature and wavelength of the peak of the curve are always equal to a constant. This very simple law thus makes it possible to know the temperature of a body compared to a black body by the only form of its spectrum and the position of its maximum.

Law of Stefan-Boltzmann

According to the law of Stefan-Boltzmann, density flux of energy or density of power or radiant emittance M O (T) (out of W m2) emitted by the black body according to the absolute temperature T varies (expressed in Kelvin) according to the formula : M° (T) = σ T 4
where S is the constant of the Stefan-Boltzmann who are worth approximately 5,67.10-8 Wm-2K-4.

Law of Nernst-Einstein

The law of Nernst-Einstein is a law which intervenes in the migration of the species in the crystalline solids, when the species are subjected to a force. By species, one understands crystalline defects.
This law makes it possible to calculate the speed of migration of the species according to the intensity of the force and the coefficient of diffusion of the species in the crystal.
In the absence of force
In the absence of force, the defects migrate by chance, by jumps of a site to a nearby site. These jumps are possible thanks to thermal agitation.
Per unit of time, a species has a Γi probability of making to a jump towards a site I neighbor. The mean velocity of the particles is null, the quadratic average of displacements <X²> during a time t is nonnull and one a : < X²> = t * ∑1n Γi * δξi.
if δξi is the positive or negative algebraic length according to the direction of reference of jump I.
Effect of a force
When the species is subjected to a force, that breaks the symmetry of the jumps, the probabilities of two opposite jumps are not more equal. To simplify, one considers only one species and a movement in a given direction. If Γ+ is the probability that the particle moves of a length +δx per unit of time and Γ- the probability that it moves a length -δx, then the range <X> after a time T is worth : <X> = t * (Γ+ - Γ-), which makes it possible to define the mean velocity v : v = <X> ⁄ t = (Γ+ - Γ-) * δx.
This movement under the effect of a force creates a gradient of concentration. However, the random diffusion tends to level the concentrations and thus is opposed to the forced migration.
there are thus two flows:
a flow J 1 created by the force J 1 = v * c, where C is the concentration of the species
a flow J 2 opposite which follows the law of Fick J2 = -D * ∂c ⁄ ∂x where D is the coefficient of diffusion of the species
Total flow is thus worth
J = v * c -D * ∂c ⁄ ∂x
Stationary mode
If one waits sufficiently a long time, one reaches a stationary mode : flows J 1 and J 2 are compensated, one has a constant gradient of concentration. There are thus j = 0, that is to say, if c(x) is this constant concentration : v * c∞ = D * ∂c∞ ⁄ ∂x, let us suppose now that the force is conservative. It thus derives from a potential η : F = - ∂η ⁄ ∂x, with dynamic balance, the particles are distributed according to statistics of Maxwell-Boltzmann : C∞(x) = c0 * exp (-η ⁄ kT), where K is the Boltzmann constant and T is the absolute temperature. By introducing this into the preceding equation, one obtains : v * c0 * e-η ⁄ kT = - D ⁄ kT * ∂η ⁄ ∂x * c0 * e -η ⁄ kT, which gives us the law of Nernst-Einstein : v = DF ⁄ kT.
This law resembles a law of fluid friction. At the time of a movement at low speed in a nonturbulent fluid, one can estimate that the force of friction is proportional to the speed and thus that one reaches a stationary mode where speed is proportional to the force : v = B * F, where B is the mobility of the species.
The law of Nernst-Einstein thus gives us : B = D ⁄ kT, from where one deduces the law from Einstein : D = B * kT.

specific heat

the specific heat symbolise by the letter C is the coefficient making it possible to calculate the time necessary and the power requirement to raise 1° centigrade 1 gramme of unspecified matter
Air = 1,01
Aluminium = 0,90
Ethanol = 2,45
Or = 0,18
Granite = 0,80
Fer = 0,45
Oil olive = 2,00
Argent = 0,24
Cuivre = 0,39
Zinc = 0,38
Acier = 005
Steel inoxydable = 0,51
Eau = 4,18
Bois = 1,76
Argile-Brique = 0,92
Masonry courante = 0,84 at 1,05
Verre = 0,77

Magnetic permeability

B: the magnetic field, H : the magnetic field of excitation
has magnetic permeability is the capacity of a body to modify a magnetic field, this value depends thus on the medium in which it evolves ⁄ moves, the magnetic field varies linearly with the excitation of the magnetic field

Effect of point

The effect of point, also called to be able of the points, is a physical phenomenon explaining the influence of a pointed metal object on the surrounding electric field. The typical applications of this phenomenon are the lightning conductors.
Let us consider two conducting balls.
A large ball of ray R and small of ray r = R ⁄ 2.
If one carries them to the same electric potential V, the load of each ball can be calculated according to this formula :
qR = V * 4πε0R
qr = V * 4πε0r = (qR = V * 4πε0R) ⁄ 2 = qR ⁄ 2
The small ball will thus have an electric charge half less important than the large ball.
The module of the electric field is expressed then as follows :
ER = V * R ⁄ d²
Er = V * r ⁄ d² = (V * R ⁄ 2) ⁄ d² = ½ * (V * R ⁄ d²) = ER ⁄ 2
With the same given distance D, it will be thus twice less important for the small ball.
However we can approach their vicinity D = R for the small one, D = R for the large one, because it is the distance compared to their center:
ER = V * R ⁄ R½
Er = V * r ⁄ r½ = (V * R ⁄ 2) ⁄ (R ⁄ 2)½ = (V * R ⁄ 2) ⁄ (R½ ⁄ 4) = 2 * (V * R ⁄ R½) = 2ER
Finally, we obtain that the electric field in the vicinity of the small ball will be twice more important than in the vicinity of the large ball. If a point is considered as a ball whose diameter is small, one obtains an electric field whose value tends towards the infinite one to its vicinity. This very important electric field will contribute to the ionization of the air and thus the starting of a possible electric arc.

Visualization of the intensity of the electric field on a point.
This phenomenon is very present in the daily life: it is, for example, at the origin of fires of Saint-Elme and makes it possible to explain why the lightning generally falls on pointed objects (bell-tower, tree, top of an umbrella and of course lightning conductor).
Under the action of the intense electric field at the end of the point, electrons of metal can be emitted (mode of emission of field per tunnel effect). It is the Fowler-Nordheim effect.
The effect of point explains why one takes the current by touching a car: in particular by friction on the air, the metal carcass of the car is carried to a potential not no one, and as it in general is not directly connected to the ground (the tires are not conductive), it preserves this potential, until a person approaches the fingers, those forming of the points, the loads accumulate there and a transitory electric arc is formed between the car and the fingers. It would be enough to initially touch the car with the dish of the hand to discharge it without the electric field being sufficient to create a spark and to avoid this nuisance.

Effeit meissner

The Meissner effect results from expulsion of the magnetic fields by a superconductor.
The Meissner effect is the total exclusion of any magnetic flux of the interior of a superconductor. He was discovered by Walther Meissner and Robert Ochsenfeld in 1933 and is often called perfect Diamagnétisme or the Meissner-Ochsenfeld effect the Meissner effect is one of the properties defining supraconductivity and its discovery made it possible to establish that the appearance of superconductivity is a transition from phase. The exclusion of the magnetic flux is due to electric currents of écrantage which circulate on the surface of the superconductor and which generate a magnetic field which cancels exactly the field applied. These currents of écrantage appear when a superconductor is subjected to a magnetic field. Indeed, if one cools a superconductor in the presence of a magnetic field, the field is expelled at the time of the superconductive transition! While a hypothetical material presenting only the property of null resistance would maintain the intensity (and it direction) of the magnetic field, that it would have had at the time of the transition, constant in its centre as long as this property would be maintained. The Meissner effect is thus a property of the superconductors which is distinct from infinite conductivity. In fact, the Meissner effect or perfect diamagnetism is the property main feature of a superconductor. But, that cannot be understood only by the fact that the electrical resistance of a superconductor is null: the eddy currents induced by the later movements of material in the magnetic field, are not attenuated. F. London could describe the Meissner effect while postulating that in a superconductor there exists current proportional to the electromagnetic potential vector This equation is not invariant of gauge, it thus should be specified that one considers the gauge of Coulomb.En using the equation of Maxwell-Amp where in the case of a superconductive medium extending in the half space X 0. The length? is the length of penetration of the magnetic field. This equation shows that the magnetic fields penetrate only the surface of the superconductors. Another consequence of the Meissner effect is since the electric currents (superconductor) generate magnetic fields such as they cancel the external field, its electric currents run out primarily in its immediate surface. The equation of London can result from the Theory of Ginzburg-Pram

Effeit josephson

In physics, the Josephson effect appears by the appearance of a current between two superconductors separated by a layer made from an insulating material or metal not-superconductor.
In the first case, one speaks about "Josephson junction LOCATED" (superconductor-insulator-superconductor) and in the second about "junction S-M-S".
One distinguishes moreover two types of effect Josephson, the "continuous" effect josephson (D.C. Josephson effect in English) and "alternate" the Josephson effect (A.C. Josephson effect). These two effects were predicted by Brian David Josephson in 1962 starting from the theory BCS.Ces work were worth to him the Nobel Prize of physics in 1973, with Leo Esaki and Ivar Giaever.
Although the pairs of Cooper cannot exist in insulating or a metal not-superconductor, if the layer which separates the two superconductors is sufficiently thin, they can cross it by the tunnel effect and keep their coherency of phase. It is the persistence of this coherency of phase which gives place to the Josephson effect.

Alternate Josephson effect

Because of the tunnel effect of the pairs of Cooper, the superconductive current through the barrier separating the superconductors is:
Is = Ic sin (∅1 - ∅2)
where I C is current a characteristic of the junction and f1,2 is the superconductive phases of the two superconductors.
In addition, the superconductive phase being canonically combined with the number of particles, it obeys the equation of the movement:
h * ]d (∅1 - ∅2)_ ⁄ dt = 2e (V1 - V2)
where E is the electron charge. and V 1 - V 2 is the potential difference existing between the two superconductors. It results from it that:
I (t) = Ic sin ]2e ⁄ h (V1 - V2) t + φ0_
In other words, the application of a potential difference involves oscillations of the superconductive current to a pulsation
2e ⁄ h (V1 - V2)
Alternate the Josephson effect thus provides a means of measuring the ratio E ⁄ H or of connecting the standards of the Volt and Second

Continuous Josephson effect

The equation of the paragraph above, binding the current to the difference in tension applied to the junction, can be completely written with null tension. One obtains a D.C. current Ic then characteristic of the junction and called "critical current". Known as differently, a junction subjected to a difference in null tension is the seat of a D.C. current of pairs of Cooper.
Continuous the Josephson effect is often observed by applying a magnetic field to a Josephson junction. The magnetic field causes a dephasing between the pairs of Cooper which cross the junction in a way similar to the effect Aharonov-Bohm.Ce dephasing can produce destructive interferences between the pairs of Cooper, which involves a reduction of the maximum current being able to cross the junction. If F is the magnetic flux through the junction, one with the relation:
Ismax = Ic ]sin (πφ ⁄ φ0)_ ⁄ (πφ ⁄ φ0)
The Josephson junctions : a device with high efficiencies
The Josephson junctions, by their physical properties, constitute a device of choice for several scopes of application:
It is the constituent elementary one (Superconducting Quantum Device Interference), the finest detector of magnetic field (and thus of current). A SQUID consists of 2 junctions in parallels in a loop.
It is also constituent basic fast logic known as RSFQ (Rapid Single Flow Quantum) where they play the part of the transistor and would authorise rates in hundreds of Ghz.
It is also one of the detectors of the most powerful photons. One speaks then about superconductive junctions with tunnel effect (English STJ). These devices combine an ultimate sensitivity to the single photons in broad band spectral X-rays to the close relation with a good resolution in energy.

Hall Effect

Illustration of the Hall effect for different current direction and magnetic field.
Legend: 1-electrons (in the sense unconventional!)
2: element or Hall effect sensor
3: Magnets
4: Magnetic field
5: Power Source
In the drawing A, a negative charge appears at the top border of the element (blue), and a positive charge at its bottom border (red color). In B and C, the reversal of direction of current or magnetic field that causes the reversal of this polarization. D, the double inversion of the electric current and magnetic field give the element in the same polarization A.
called classical Hall effect was discovered in 1879 by Edwin Herbert Hall: an electric current through a material immersed in a magnetic field generates a voltage perpendicular to them.
Under certain conditions, the voltage increases in steps, characteristic effect of quantum physics is the integer quantum Hall effect and the fractional quantum Hall effect.

Principle of the Hall effect in a threadlike conductor of rectangular section .
When current flows through a bar of semi-conductor or conductor and if a magnetic field induction B is applied perpendicular to the direction of current flow, voltage, called Hall voltage proportional to magnetic field and current appears on the sides of the bar.
This voltage is proportional to the speed of movement of the charge carriers which is considerably greater in the semiconductor materials in the metallic conductors.
Classical physics of the Hall effect

magnetic Lorentz Force
Hall and electric force
in a conductor carrying a current
and subjected to a magnetic field.
A magnetic field acts on moving charges. The current through the conductive material is produced by charges, free electrons moving with a speed which will be denoted V.
These electrons are then subjected to a force F = -e.v ∧ B Lorentz force, where-e is the charge of a electron. This results in a displacement of electrons and a concentration of negative charges on the one side of the material and a deficit of negative charges on the opposite side. This charge distribution gives rise to the Hall voltage Vhall as well as an electric field EH.
This electric field is responsible for an electric force acting on the electrons: Fe = -e.EH Coulomb force. Equilibrium is reached when the sum of two forces is zero, Newton's second law.
Hall effect sensors to measure:
magnetic fields (Teslamètres)
The intensity of electric currents: current sensors Hall effect.
They also allow the production of sensors or position sensors without contact, used especially in the automobile, for detecting the position of a rotating shaft.
There are also Hall sensors in systems for measuring speed rail equipment.
There are also Hall sensors under the keys of keyboard musical instruments modern avoiding wear that underwent the traditional electrical contactors.
The Hall effect is sometimes used in the field of artificial satellites, specifically in the design of the engines of these satellites.

The kirk effect

The kirkest effect a parasitic effect of the bipolar transistor which consists of the widening of the basic zone to the collecting detriment of the zone following a strong density of majority carrier injected of the base towards the collector. This effect was discovered in 1962, by C.T Kirk, following its work with MIT.
This effect is met in the normal operating process of the transistor, on the level of the junction bases collecting polarized in reverse.
Dependant on an injection of majority carriers towards the collector, the kirk effect meets in situations of strong injection. To a threshold of injection of carriers superior to the collecting doping of the zone, we will observe a progressive obliteration of the zone of space charge between base and collector. One thus speaks about widening of the basic zone ωB.
This widening of the basic zone leads thus to a reduction in the profit of the transistor β = 2 * (lnB ⁄ ωB) ²
Attenuation of the kirk effect
To limit the Kirk effect in the bipolar structures results in first approach in increasing the doping of the collector. This solution will generate an increase in the capacity bases collecting (CBC). There will be then a lowering of the speed of commutation and a reduction in the allowed tension of breakdown to the junction bases collecting.
Thus, the bipolar transistors in technology RF (Radio frequency) and electronics of power will be in particular affected. A compromise is thus necessary between profit of the transistor and behavior in tension and ⁄ or speed of commutation.

The Early effect

The Early effect is a physical phenomenon which appears in the linear area of operation of a bipolar transistor. This effect was discovered in 1952 per James Mr. Early.
Evolution of the current of the collector of a bipolar transistor according to the tension collector-transmitter for various values of the basic current.
Theoretically, in its zone of linear operation, the collector current Ic of a bipolar transistor should not be influenced by the tension Vce collector-transmitter. Actually, the rise of this tension (Vce) modifies the collector current slightly. It is what is called the Early effect. When the basic current Ib is weak, the effect is less made feel. On the other hand, more the basic current is large, more the Early effect appears.
The bipolar transistor being essentially an amplifier of current: a basic current induced a collector current according to the profit specific to the transistor. An increase in the potential difference present between the collector and the transmitter causes to modify the thickness of the base slightly and thus, for the same basic current, to amplify the current of collector slightly.

Miller effect

One names Effet Miller the influence of the profit of an amplifier of tension reverser on his own characteristics of entry. In the case of an amplifier not-reverser, the same effect led to the generation of negative impedances.
Explanation of the Miller effect
The phenomenon can be explained simply thanks to the following diagram.
For the explanation, one considers only current IM circulating in the resistance of Miller.
The assembly is an amplifier reverser of profit AV = - R2 ⁄ R1. The current circulating in the resistance of Miller is : Im = {Ve - [Ve * (-R2 ⁄ R1)]} ⁄ RM soit 1M = Ve * (1 - Av) ⁄ RM
The equivalent resistance seen of the Ve source is : Req = Ve ⁄ IM = RM ⁄ (1 - Av).
Rm resistance seen of the Ve source thus seems having a value 1 - weaker AV time.
The same demonstration is applicable to a capacity placed between the entry and the exit, with for result multiplying its value by 1 - AV
This effect in the case of explains inter alia the increase in the input capacitance of an amplifying stage reverser due to the capacity base-collector a bipolar transistor out of common transmitter, grid-drain for a field-effect transistor in common source or grid-anode for a vacuum tube in common cathode.
Consequences of the Miller effect
The Miller effect has for first consequence a reduction of the impedances of entry of the electronic assemblies. The second induced consequence is a reduction of the band-width of the amplifiers reversers very sensitive in high frequency and the generation of harmful dephasings to the stability of the assembly.
Various techniques make it possible to compensate for, reduce or cancel the Miller effect
The neutrodynage active compensation
The amplifying assembly bases common or roasts common, not Miller effect
The assembly cascode stage of entry of profit in unit tension + stage in common base
The Miller effect however finds applications sometimes unsuspected although directly induced in particular in the realization of active filters by allowing the realization of variable capacities several microfarads or extra-light inductances of several Henrys.
It is also the Miller effect which made it possible to fix in a reliable way the Gain.bande product of the operational amplifiers since the advent of the µA741.

Curve and classification of a circuit breaker

Current of employment IB : it is about the rated current or maximum of the load.
Rated current of the safety device In : is the gauge in Amps of the fusible cartouche.
Acceptable current in the Iz drain : it is about the maximum intensity authorized in the line.
Running assigned In : is the maximum value of the closed-circuit current which a circuit breaker equipped with a release to a room temperature can carry specified by the manufacturer, by respecting the prescribed heat limits.
Current of adjustment Ir : is the current maximum one that the circuit breaker can support, without release. This value must be higher than the current of Ib employment and lower than the acceptable current in the Iz drain. The thermal releases are in general adjustable from 0,7 to 1 time In whereas in electronic technology the beaches are generally broader (usually from 0,4 to 1 time In).
Operating current Im : current which causes release for the strong overcurrents. It can be fixed or adjustable and can vary between 1,5 In and 20 In.
Capacity breaking Icu or Icn : is the greatest intensity of current of short-circuit (running supposed) that a circuit breaker can stop under a given tension. It is expressed in general in kA effective symmetrical and is indicated by Icu (capacity breaking ultimate for the industrial circuit breakers and by Icn (capacity breaking assigned) for the circuit breakers of domestic or assimilated use.
To be able of limitation : it is the capacity of a circuit breaker to let pass only one current lower than the current of supposed short-circuit.
Assigned tension of Ue employment : is the tensions to which the apparatus can be used.
Polarity of a circuit breaker : is the number of poles being crossed during a release and the number of poles being surveilléspar a thermal relay.
Curve of release
Blue: Curve B
Red: Curve C
Curve B
Threshold of release of magnetic between 3 and 5 or 3,2 and 4,8 times the nominal intensity according to the apparatuses, and conforms to standard NF C 61-410, INTO 60898 and IEC 947.2
Standard thermal release
Curve C
Threshold of release of magnetic between 5 and 10 or 7 and 10 times the nominal intensity according to the apparatuses, and conforms to standard NF C 61-410, INTO 60898 and IEC 947.2
Standard thermal release
Curve D
Threshold of release of magnetic between 10 and 14 times nominal intensity in accordance with standard IEC 947.2
Standard thermal release
Curve MY
Threshold of release of magnetic to 12 (± 20%) times nominal intensity in accordance with standard IEC 947.2
No the thermal release
Curve K
Threshold of release of magnetic between 10 and 14 times nominal intensity in accordance with standard IEC 947.2
Standard thermal release
Curve Z
Threshold of release of magnetic between 2,4 and 3,6 times nominal intensity in accordance with standard IEC 947.2
Standard thermal release
Burst of a circuit breaker
1-lever being used to cut or rearm the circuit breaker manually. It also indicates the state of the circuit breaker (opened or closed). The majority of the circuit breakers are conceived to be able to trip even if the lever is manually maintained in closed position.
2-mechanism related to the lever, separates or approaches the contacts.
3-contacts allowing the current to pass when they are touched.
5-thermal switch (2 welded blades with different dilation coefficients): thermal relay (overload surcharge)
6-screw of calibration, makes it possible to the manufacturer to adjust the instruction of current with precision after assembly.
7-wind or solenoid: magnetic relay (protection against the short-circuits)
8-room of cut of the electric arc

Insulating envelope Core
massive conducting core in cuivre.b- conducting core has cable length (multistranded) out of copper.
Sheath Conductors (soul + envelope)
A cable is composed of one or several conductors electrically, distinct and mechanically interdependent, under coatings of protections (sheath, braid, armour).
Code colours:
Two-tone location Green ⁄ Yellow is reserved exclusively for the function of protection PEN, the BLUE conductor is intended for the circuit of the neutral NR.
The conductors MAROON BLACK RED are intended for the circuit of L.Ou phase in another colour other than BLUE and Green ⁄ Yellow.
reference of a cable,
Reference of a power cable for enclosures of computer H03VVH2-F 2x0,75 mm ².
Designation harmonised:
Code designation:
H : Harmonized cable
A : Cables standardised recognised
Nominal voltage :
03 : 300 volts maximum
05 : 300 ⁄ 500 volts maximum
07 : 450 ⁄ 750 volts maximum
Symbol of the mixture insulating :
R : Rubber
S : Silicone rubber
X : Crosslinked polyethylene
Symbol of the mixture sheaths :
J : Braid glass fibre
NR : Polychloroprene
R : Rubber
T : Braid textile
V : Vinyl polychloride
Special construction :
Cable round
H : Cable flat "divisible"
H2 : Cable flat "nondivisible"
Symbol of the conducting core (flexibility) :
F : flexible core, class 5 (Flexible device)
H : flexible extra core, class 6
K : flexible core (installation fixes)
R : rigid core cable length (Rigid with joined together strands)
S : rigid core cable length, section sectorale,
U : massive core (single)
W : massive rigid core, section sectorale
Y : core with wire rivet washer
Nature of the metal of the core :
Composition of the cable :
Many conductors
X : cable without conductor V ⁄ J
G : cable with conductor V ⁄ J

An electric wire 10.000 times finer than a hair

A wire dix-mille time finer than a hair of an exceptional electric conductivity, opening the way with branches between electronics components on an atomic scale of the quantum computers of the future.
To be able to carry out wire branches on this microscopic scale will be essential for the development of the future electronic circuits of atomic size, underlines Bent Weber, of the University of New South Wales in Australia, principal author of this work published in the American review Science of 6 January.
This wire was created with chains of phosphorus atoms inside a silica crystal, explain these Australian and American researchers.
They discovered that the electrical resistance of their wire, a measurement of conductivity, did not depend its thickness, as that is described by the law of Ohm taught in the schools.
It is extraordinary to note that such an elementary law of physics still applies at the atomic level, underlines Bent Weber.
Excellent electric conductivity
In spite of their surprisingly tiny diameter, just four atoms of width on an atom height, these wire show properties of exceptional electric conductivity, identical to copper.
They thus make grow the hope that one day these wire will be able to feed in electricity of the components of atomic size in the computers of tomorrow, add these researchers.
These discoveries prove that the branches in silica can be reduced to lossless atomic dimensions of electric conductivity, adds Michelle Simmons, director of the Australian Centre for Quantum Computation and Communication Technology at the University of New South Wales, person responsible for this work.
The electronics components continue to see their size being reduced making it possible to build increasingly small computers and more powerful, she adds.
We are about to manufacture transistors of the size of an atom but to build a functional quantum computer it is also necessary that the wire and branches between the components and the circuits are also of atomic size, continues Michelle Simmons.

resistivity according to the nature of the ground
nature of the ground average resistivity
marshy ground s3 -30
silt s20-100
humus s10-150
wet peat s5-100
plastic clay s50
marnes and hardpans s100-200
marnes of the Jurassic one s30-40
clayey sand s50-500
siliceous sand s200-3000
stony ground s1500-3000
stony ground and grass s300-500
calcareous ground to tend s100-500
compact calcareous ground s1000-5000
schist s50-300
mica schist s800
granite and sandstone s1500-10000
faded granite and sandstones s100-600

resistance of an earth electrode according to the type of ground and the technique of installation
technique used arable wet fat, fill arable thin coarse fill dry stony ground sands dry
loop trench bottom s3-10 s30-60 s100-200
1 Greenical stake of 2 metres s2-75 s220-300 s750-1500
4 Greenical stakes with the angles s6-18 s60-120 s300-600
sliced of 10 metres 3-8 23-45 s120-220

letter of location of electrical material according to DIN
old letter of reference mark example of electrical material new letter of reference mark
transducers of measurement
devices of memorizing
electric filters
thermal releases
fuses (end, hh, signal)
frequency conGreeners
starters progressive
feeding without interruption
optical and acoustic signal devices
indicator of indication
auxiliary relay
relay contactor
contactors with semi conductors
contactor of power
timing relays
amplifiers of separation, anplificateurs reversers
measuring device
disconnecting switches has cut in load
overload switch
driving circuit breaker
switches star triangle
disconnecting switches
resistances of adjustment
resistances of measurement
resistances of heating
control switches
switch of position
transformer of tension
transformers of current
frequency conGreener
free diode of wheel
filter CEM
devices of suppression and attenuation

Table of the densities of the electric drivers

The values are in kilograms per DM³
Metals Mass
ρ(Kg ⁄ dm³)
Aluminum 2,70 660 Al With Driver of electric cable
Antimony 6,70 630 Sb R Manufacture of the transistors
Money 10,50 960 Ag   Wire, welding, fuse, contact
Cadmium 8,64 320 Cd Cd Accumulators (not ROHS)
Carbon/graphite 2,25 3700 C   Electronic resistance
Chromium plate 7,20 1800 Cr C Electrical resistance
Cobalt 8,90 1495 Co K  
Constantan 8,91 1300 - - Thermal thermal switch of relay
Copper 8,90 1083 Cu U Driver of electric cable
Tin 7,28 232 Sn E Welding
Iron 7,88 1535 Fe Fe Electrical resistance
Iridium 22,42 2739 Ir   Paratonner, contact of spark plug
Lithium 0,535 453 Li   Piles, accumulators
Magnesium 1,738 923 Mg G Lamp flash
Manganese 7,47 1517 Mn M Crush saltworks
Mercury 13,59 -39 Hg   Not ROHS
Nickel 8,90 1455 Ni NR Resistance
However 19,30 1063 With   Electrical contact
Platinize 21,46 2041 Pt   Magnet, resistance, thermocouple
Lead 11,34 327 Pb Pb Accumulator battery
Silicon 2,33 1687 If S Transistor and microprocessor
Germanium 5,32 1211 Ge   Diode of detection
Tantalum 16,65 3290 Your   Condenser
Titanium 4,5 1941 Ti T Reflectors infra-red
Tungsten 19,25 3695 W   Filament of incandescent lamp
Uranium 19 1405 U   Nuclear fuel
Zinc 7,14 419 Zn Z Piles
Zirconium 6,40 2128 Zr Zr Superconductive component

Specific mass of the unit of volume (density)

The specific mass of a body (solid, liquid or gas) is expressed in kilogram per unit of volume (kg ⁄ dm ³ or kg ⁄ m ³).

Electrical insulator density

ρ(Kg ⁄ dm ³)
ABS 1,04 Case box for electronics
Email 2,70 Wire for winding
Bakelite 1,35 - 1,40 Connector blocks and printed circuits
Rubber 3,51 Insulation and flexible sheath of cable
Ceramics 7,65 Condenser
Cotton 0,20 Insulation of wire
Diamond ⁄ Carbon 3,51 Cutting tools
Oil ⁄ mineral 3,51 Transformer and condenser
Mica 2,825 Condensers of precision and high-quality
Paper 2,25 Condenser for the Hi-fi
Polyamide 1,14 Clamp for cable
Polyethylene 1,38 - 1,41 Insulator of electric cable
Polypropylene 0,85 - 0,92 Electric case
Polystyrene 1,04 Condensers and cases for electronics
Porcelain 2,50 Electronic tube socket
PVC 1,38 - 1,41 Insulation of the drivers and cases
Quartz 2,65 Electronic oscillator
Steatite 2,98 Support for heating resistance
Teflon 2,15 Insulator of wire and printed circuit
Glass 2,53 Insulator high voltage
Epoxy glass 8,64 Printed circuit
Vacuum - Electron and cathodic tube

Gas density used in electronics

Gas with 0°C Symbol
ρ(Kg ⁄ dm³)
Air - 0,0012 Variable capacitator
Argon Ar 0,0017832 Device of lighting, bulb
Helium He 0,001785 Tube LASER
Krypton Kr 0,00374 Lamp
Neon   0,0009 Fluorescent tube
Ozone 0,0024 Ionized air

execution time customer :
runtime server : 0.006 seconds