Impedance

The electric impedance measures the opposition of an electrical circuit to the passage of a sinusoidal alternative course. The definition of impedance is a generalization of the law of Ohm in the study of the circuits in alternative course.
The word impedance was invented by Oliver Heaviside in July 1886.
According to the vocabulary and the notations standardized by the international electrotechnical Commission (CEI), the impedance of a passive linear dipole of terminals has and B is defined in sinusoidal mode of current and tension by the quotient of the complex number image of phasor U_AB, representing the tension between the terminals, by the complex number image of the phasor I_ representing the electric current through the dipole.
Z = U_AB ⁄ I_
The formalism of the impedances lays down some rules of calculations of the potentials and the intensities of the current in any point of a circuit supplied with various sources and comprising inductive and capacitive elements idealized. The methods of calculating are then similar to those used for the circuits in D.C. current.
Definitions and notations
Representation of the signals by complex numbers
Impedance, resistance and reactance

Chart, in the complex plan, of impedance Z
resistance R
and reactance X
The impedance is a complex number, in general noted Z.
That is to say an electric component or a circuit supplied with a sinusoidal current I0 cos(ωt + φi). If the tension on its terminals is V0 cos(ωt + φv), the impedance of the circuit or the component is defined as the complex number whose module is equal to the report/ratio V0 ⁄ I0 and whose argument is equal to φ = φv - φi..
The module of the impedance is homogeneous with a resistance and is measured in ohms.
Often, one uses by abuse language the impedance term to indicate his module.
An impedance can be represented as the sum of a real part plus an imaginary part:
Z_ = R + jX
R : is the real part known as resistive
X : is the imaginary part known as reactive or reactance.
An impedance of which the imaginary part is positive will be described as inductive. If the imaginary part is negative, one speaks about capacitive impedance.
Rules of calculation of circuits with the impedances
With what has just been known as, one can calculate circuits including ⁄ understanding of the impedances in a way similar to that used for calculation with resistances in D.C. current. The result of the calculation of a tension or a current is, in general, a complex number.
This complex number is interpreted in the following way:
The module indicates the value of the tension or the calculated current. If the values used for the sources were peak values, the result will be also a peak value. If the values used were effective values, the result will be also an effective value.
The argument of this complex number gives dephasing compared to the source used like reference of phase. If the argument is positive, the calculated tension or the current will be in phase lead.
Calculations of equivalent impedances
The calculation of the equivalent impedance of a whole of impedances is treated like resistances with the law of Ohm.
Impedances in series

Together of N impedances in series.
The impedance equivalent to a whole of impedances in series is equal to the sum of the impedances of the elements. Attention that functions only with the impedances in their complex form, after one transformed cissoidal : Z = Z1 + Z2 + ... + Zn
Impedances in parallel

Together of N impedances in parallel.
When the dipoles are placed in parallel, in fact the admittances are added. The admittance equivalent to a whole of admittances in parallel is equal to the sum of the admittances of the elements : Y = Y1 + Y2 + ... + Yn
The admittance being the reverse of the impedance it results from it that:
1 ⁄ Z = 1 ⁄ Z1 + 1 ⁄ Z2 + ... + 1 ⁄ Zn
Thus the impedance equivalent to a whole of impedances in parallel is equal contrary to the sum of their opposite:
Z = 1 ⁄(1 ⁄ Z1 + 1 ⁄ Z2 + ... + 1 ⁄ Zn)
Law of Ohm generalized
The terminal voltage of an impedance is equal to the product of the impedance by the current : Vz = ZIz
As well the impedance as the current and the tension are, in general, complexes.
Laws of Kirchhoff
The laws of Kirchhoff apply same manner as in mode of tension and D.C. current: the sum of the currents arriving on a node is null and the sum of the tensions around a mesh is null, but the currents and the tensions are represented by complex numbers.
Validity of the rules of calculation
These rules are valid only
In established sinusoidal mode, thay is to say with sources of sinusoidal tension and current and once the transitory phenomena starting disappeared.
With presumedly linear components, thay is to say components whose characteristic equation (relation between the tension on their terminals and the intensity of the current which crosses them) is comparable with a differential equation with constant coefficients. Nonlinear components as the diodes are excluded. The reels with ferromagnetic core will give only approximate results and this, with the proviso of not exceeding the values of intensity above which them operation cannot be regarded as linear any more following the saturation which intervenes in these materials.
If all the sources do not have the same frequency or if the signals are not sinusoidal, one can break up calculation into several stages with each one which one will be able to use the formalism of impedances.
Impedance of the elementary dipoles
In order to model reality one uses three types of elementary ideal components.
Ideal resistance
The impedance of an ideal resistance R is equal to R:
ZR = R
It is the only component to have a purely real impedance.
Ideal reel
The impedance of an inductance coil L is :
ZL = jωL
ω is the pulsation of the signal. Contrary to the preceding case, this impedance is purely imaginary and depends on the frequency of the signal.
Ideal condenser
The impedance of an ideal condenser of capacity C is :
ZC = 1 ⁄ jωC
Real components
The real components are approached by models built in the form of complex impedances of expressions which generally depend on the frequency and of the amplitude of the current which crosses them. To take account of the effects of the frequency it is generally necessary to add to the model of the elementary dipoles in series or parallel. For example, a real resistance presents, in general, an inductance in series with its resistance. A wire-wound resistor resembles an inductance and it presents a value of significant inductance. In high frequency, it is necessary to associate with this model a condenser in parallel to take account of the existing capacitive effects between two contiguous whorls.
In the same way, a condenser and a reel realities can be modelled by adding a resistance in series or parallel with the capacity or inductance to take account of the defects and losses. Even sometimes it is necessary to add inductances to the model of a real condenser and capacities for the model of a reel. These modelings of the variations compared to the simple model used in low frequency can become prevailing beyond of a certain value of the frequency.
In certain cases, it happens that one adds a resistance whose value depends on the frequency in order to take account of the evolution of the losses with the latter.
Generally, it should be remembered that a model always has a field of validity.
Impedance interns of a generator
To give an account of the voltage drops and the internal heating noted during the use of an electric generator of tension or sinusoidal current, generally an alternator or a transformer, one models this real generator by the association of an ideal generator and an impedance called internal impedance of the generator.
Model of Thévenin
The model of Thévenin is most frequent in electrical engineering (model of Kapp of the transformer or model of Behn Eschenburg of the alternator) because it at the same time makes it possible to give an account of the losses by Joule effect and the fall of charging voltage. One associates in series an ideal generator of tension with an inductive impedance.
Model of Norton
The model of Norton associates in parallel a generator of ideal current with an impedance or an admittance. This model does not have the same losses by Joule effect as the assembly that it models, this is why it is mainly used as intermediary of calculation to analyze the parallelization of real generators.
More complex models
Certain generators must be modelled by more complex associations of sources of power and impedances. For example, the simplest model of the asynchronous machine used out of generator is consisted of the association of a negative resistance - it thus provides power - whose value is related to the slip, in series with an inductance. This unit is in parallel with a comparable inductive impedance with a pure inductance. This model must be weighed down if one wishes to take into account the magnetic losses and the losses by Joule effect with the stator.
Impedance and quadripoles
Quadripole.
Impedances of entry and exit
A quadripole in charge at exit by an impedance of load given behaves, seen since its terminals of entry, as a passive dipole which one can define and measure the impedance. This one is called impedance of entry of the quadripole.
Line of transmission
The impedance characteristic of a line of ideal transmission is defined by : Zc = √L ⁄ C
L and C are respectively the coefficient of car induction or inductance and the capacity per unit of length of the line.
It is indicated in the catalogs of the manufacturers. It depends:
dimensions of the drivers, and their spacing
permittivity of insulator, in the coaxial line
Typical values of the characteristic impedance:
50 or 75 ohms for a coaxial line
300 ohms for a two-wire line
The use of a line of transmission is mainly the transmission of electrical energy which by a suitable modulation supports information. The good transmission of this information supposes the good transfer of the energy what supposes a good adaptation of the impedances to the entry and the exit of the cable. This good adaptation occurs when the impedance of the terminations is equal to the impedance specification the wire. In the contrary case the transfer of energy is not total and not transferred energy makes half turn what presents disadvantages compared to the sought-after goal. Therefore some values of characteristic impedances were selected to facilitate the work of the originators in the use of the coaxial cables and their termination.

Diagrammatic representation of the elementary components of a line of transmission.
For a line of real transmission, the characteristic impedance is a complex number :
Zc = √R + jωL ⁄ G + jωC
where R and G are respectively the resistance and the conductance of losses per unit of length.
It is noticed that high frequency (ω rather large) R and G are negligible in front of jωL and jωC from where the good approximation on a high frequency real line of Zc = √L ⁄ C
Determination practices characteristic impedance : it depends on the physical parameters of the line. For example, for a coaxial cable, Zc depends on the report/ratio of the diameters of the interior driver and the outer main, as well as permittivity of insulator. The standardized values were adopted because they minimize the losses by Joule effect. For a printed line microphone-ribbon, the impedance characteristic depends on the relationship between the width of the ribbon (W), the thickness (H) of insulator between the ribbon and the plan of mass, and of the permittivity of insulator.
Adaptation of impedances
The adaptation of impedances is a technique in electricity making it possible to optimize the transfer of an electric output between a transmitter (source) and an electric receiver (load):
In the case of quadripoles in cascade, the impedance of the receiver must be very large compared to that of the transmitter. The output is thus optimized when one has a maximum loss of adaptability.
The theory of the maximum capacity determines that the impedance of the load must be the combined complex of the impedance of the generator.
In the presence of a line of transmission, the impedance of the receiver must be equal to the impedance characteristic of this one to avoid the reflections.
Methods of measurement of an impedance
Assemblies in bridge
There exist very many assemblies in bridge, similar to the Wheatstone bridge used for the measurement of resistances, which make it possible to measure impedances: bridge of Sauty, bridge of Maxwell
Electronic measuring devices
There exists a large variety of more or less sophisticated commercial apparatuses making it possible to measure the impedance of a component or a dipole. One finds them under the names of analyzer of impedance, bridge RLC, electronic bridge of measurement.
They consist of a generator of sinusoidal current created using the output voltage of an adjustable oscillator. This current crosses the dipole to be measured and the decomposition of the tension on its terminals in a component in phase and the other in squaring with the tension delivered by the oscillator make it possible to determine the parts real and imaginary of the impedance. The pulsation being known it is possible starting from the reactance to post L or C.
The simplest apparatuses function with an oscillator delivering a tension of amplitude and fixed frequency. A continuous component is sometimes added for the measurement of the electrochemical condensers. Although the technical notes indicate precise details about pourcent, it is necessary to keep in mind that the modeling of a dipole by a model series (Rs resistance and Xs reactance) or parallel (resistance RP and Xp reactance) depends much on the amplitude and the frequency of the tension on its terminals. Certain automatic bridges of impedance make it possible to regulate these parameters.
As for the ohmmeters, it is sometimes necessary to take precautions of wiring: assembly 4 wire or use of a guard.
Experimental techniques based to the measure of the impedance
Spectroscopy of impedance - general information
The spectroscopy of impedance is a general term which recovers the whole of the techniques consisting to measure and analyze the electric impedance of a sample, in general according to the frequency, so as to draw some from information on its physico-chemical properties. One also speaks about dielectric spectroscopy, when it is applied to dielectric materials.
Electrochemical spectroscopy of impedance
Principle of calculation of a faradic impedance
Stationary mode
For a reaction redox R ↔ O + e which is held with an interface electrode | electrolyte in absence of gradient of concentration of the électroactives species R and O, the relation density of current of transfer vs. overpressure of electrode is given by the relation of Butler-Volmer : jt = j0 (exp(αoƒη) - exp(- αrƒη))
j0est the density of trade flow, αo and αr are the factors of symmetry in the direction of oxidation and the reduction with αo + αr = 1,η the overpressure of electrode with η = E - Eeq et ƒ = F ⁄ (RT)

Graph density of current of transfer of load vs. overpressure of electrode
The graph of jt vs. E is not a line
the behavior of the reaction redox is not that of a linear system
Dynamic mode
Faradic impedance
Let us suppose that the law of Butler-Volmer correctly describes the dynamic behavior of the reaction redox of transfer of electron
jt (t) = jt(η(t)) = jo (exp(αoƒη(t)) - exp(- αrƒη(t)))
The dynamic behavior of the reaction redox is characterized then by its differential resistance defined by
Rt = 1 ⁄ (∂jt ⁄ ∂η) = 1 ⁄ (ƒ j0o exp (αoƒη) + αr exp (- αrƒη)))
Differential resistance whose value depends on the overpressure of electrode
In this simple case the faradic impedance Zf is reduced to the resistance of transfer Rt and one notes in particular that
Rt = 1 ⁄ ƒ j0
when overpressure η is null.
Condenser of double-layer
An interface electrode | electrolyte behaves in dynamic mode like a condenser called condenser of double-layer interfacial and noted Cdc. This electric or electrochemical double-layer is described by the model of Gouy-Chapman-Stern. The behavior in dynamic mode of a reaction redox in absence of gradient of concentration of the électroactives species is thus similar to that of electrical circuit Ci below.

Electrical circuit are equivalent of a reaction redox in absence of gradient of concentration.
The electric impedance of this circuit is calculated easily.
Zdc(ω) = 1 ⁄ iωCdc
where ω is the pulsation, in rd⁄s and i = √ -1.
One finds
Z(ω) = Rt ⁄ (1 + RtCdciω)
The graph of Nyquist of the electrochimists who carry the changed imaginary part of sign of the impedance according to his real part is, in a orthonormé reference mark, a half-circle of Rt diameter and pulsation at the top equal to 1 ⁄ (RtCdc).

Graph of Nyquist of the electrochimists of a parallel circuit RC.
The arrow indicates the direction of the increasing frequencies.
Resistance of electrolyte
When the resistance of the portion of electrolyte lain between the electrode of work and the electrode of reference is not negligible the equivalent circuit of the reaction redox includes/understands moreover Rω resistance connected in series. The graph of impedance is then relocated Rω value
Measure parameters of the impedance
The layout of the graph of the impedance of a reaction redox using a potentiostat and of an analyzer of impedance, included in the majority of the modern potentiostats, thus allows the measurement of the resistance of transfer of the reaction, of the interfacial capacity of the condenser of double-layer and the resistance of electrolyte. When this layout is carried out for a null overpressure it is possible to determine the density of trade flow j0.
For electrochemical reactions more complex than the reaction redox and in the presence of gradients of concentration of the électroactives species, the graphs of the electrochemical impedances consist of several arcs.
Medical applications
The balances impedancemeters make it possible to separate the lubricating mass and the thin mass during a weighing. Today available commercially, these apparatuses exploit the differences in conductivity between these various fabrics to go back to this information.
The pneumography of impedance is a technique making it possible to follow the respiratory movements (variations of volume of the lungs) by the variations of impedances between electrodes placed in a suitable way. The electric tomography of impedance is a technique of imagery of the human body by which one reconstitutes an image in three dimensions starting from multiple measurements of impedances between electrodes placed on the skin.

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