Inductance

Called inductance, self-inductance also, self-inductance and self-inductance L is a characteristic of electrical circuits, particularly of coils. The self inductance of a circuit is the rate of change of electrical current I to the electric voltage U in relation

The symbol for inductance is L, is the unit of inductance in the SI unit system, the Henry, named after the U.S. physicist Joseph Henry. Is abbreviated unity with H.

In the important case of wire loops, one can understand the relationship between voltage and current directly with Ampèreschem law and law of induction: an electric current is generated due to the Ampère law a magnetic field, and " induced " the change of the magnetic field due to the law of induction in the same circuit (and others) an electrical voltage. It is due to its also clear that the inductance of a loop with N turns is proportional to N2.

Changes in current in a circuit m also induce voltages in other nearby circuits n This " reciprocal induction " to describe with coefficients Lmn = Lnm the mutual inductance.

Self and mutual inductance and the corresponding inductors play an important role in transformers, electric motors and electronics.

• 4.1 Inductive reactance

• 6.1 inductance of a toroid
• 6.2 Inductance of a solenoid
• 6.3 Determination of the inductance by AL value

• 8.1 Inductance of a coaxial

Scope

/ Dt is the first prerequisite for the validity of the equation u = Ldi that the magnetic field is generated ' quasi-static ' from the electricity. There must be no phase differences between the magnetic field and flow and no radiation of electromagnetic waves, ie the frequencies must be sufficiently small.

Further, the current distribution must be independent of frequency. This is the case at low frequencies, where the current density is constant over the conductor cross-section, as well as at high frequencies at full skin effect where the current flows in the conductor surface (such as superconductors ). At frequencies with partial skin effect, the inductance is frequency dependent. In the case of thin conductor, it makes little difference whether the current in the conductor cross-section or in the printed surface flows - the frequency dependence is small.

Finally it is assumed that magnetic materials in the field of magnetic fields have constant permeability. This is not the case, then one has to deal with non-linear inductance.

In this context, however, the relationship u = Ldi / dt for conductor loops with arbitrarily extended ladders applies. , And the inductances are determined solely by the location and extent of the electrical conductor and magnetizable materials.

Induction in electrical engineering

In the case of conductor loops in the electrical inductance is often defined by the conductor loop comprised by the linked magnetic flux Ψ according

Includes the ladder the same magnetic flux Φ several times, eg all turns of a coil with N equal size is obtained for the concatenated magnetic flux of the special case NΦ and the self to

If only magnetic materials with constant permeability are present in the surroundings of the circuit, then it follows from Ampere's law, the magnetic flux density B is proportional to the current I in a conductor loop. Therefore, the total magnetic flux generated by the current I Φ is directly proportional to the instantaneous value of the current i The proportionality factor thereby occurring at N turns is called inductance L (probably in dignitaries of Emil Lenz) referred to:

Magnitude of an inductance of 1 H is one in which when at a uniform current change of one ampere per second results in a self-induction voltage of one volt along the conductor.

Note: Assign the magnetic substances such as iron in the vicinity of the electrical conductor is not a constant permeability ĩr on ( it depends, for example, of the magnetic flux density B ), then the inductance is not a constant proportionality factor, but a function of the magnetic flux density. This is also referred to as the saturation magnetization. This results in non-linear inductors, which are analytically much more difficult to treat.

Is a non-linear inductance be controlled to an operating point around, so the change in the concatenated flux vary with respect to the change in the current value of the static inductance. For infinitesimally small changes about the operating point is obtained from the slope of the tangent to the curve, the dynamic inductance

Outdated unit

Until the mid-20th century, the inductance of coils was sometimes labeled with the unit cm '. This in centimeters is because the inductance is expressed in today practically hardly used electromagnetic CGS system of units in the length dimension.

An inductance of 1 cm in the electromagnetic CGS units corresponds to 1 nH in SI units.

Induction

According to the law of induction results in the circulation voltage ui a stationary magnetic flux guided to the loop from the time rate of change of the light passing through this loop magnetic flux and from the induced electric field strength Ei along the loop:

When the conductor loop of the magnetic flux loops around the N- times, as is the case for example with a coil, approximately applies:

The negative sign of this occurring is directly derived from the Maxwell equations ( law of induction ), and Lenz's law, and is based on the convention that the magnetic field is right-handed and the path of integration to the electric field.

The induced voltage is a voltage source that seeks to drive a current which counteracts the initial change in current. If current and voltage for the inductance defined as passive component in the same direction is obtained for u: . With the above definition of the inductance can be the relationship of the terminal voltage u as a function of the current i expressed as a differential equation:

In most cases, the inductance does not change in time and the current-voltage relationship of the inductance can be set to

Of this occurring is that of magnetic flux flow that is generated only due to the current i through the conductor loop. A change in this flux induces a voltage in each of the N conductor loops and becomes effective as of linked magnetic flux. Other external magnetic fluxes through the conductor loops are assumed in this case as absent or constant over time. The voltage u is called the self-induction occurs in terminal voltage.

The sign in the above equation depends on the counting direction of current and voltage. Is the direction of the voltage u to the direction of the current along the conductor loop, respectively, as shown in the diagram below which one speaks of the so-called consumer reference system, and we have:

Are the directions of voltage u current opposite in direction i along the conductor loop, one speaks of the so-called generator reference, and we have:

It is called voltage drop resulting from current change then u as a self-induction voltage or inductive voltage drop. The above differential equation is the element equation, with the can -linear inductors, such as coils in electric circuits, describe.

Network model

Based on the law of induction, produce externally applied, time-varying magnetic fluxes in constant time conductor loop time-varying electrical voltages. But also the magnetic flux generated by a current through the coil itself, acts on the spool. Changes the current through the coil, so changes the magnetic field generated by it and thereby induced in the coil itself a voltage that is opposite to the change of current. This circumstance is generally called a self-induction. The faster and higher the magnetic field varies, the higher the induced voltage generated. Basically, the self-induction can be completely described by the induction law and does not require formal amendments or adjustments.

However, it comes with the usual in electrical network theory, which, for example, for the description of electrical machines such as transformers is used, sometimes a source of confusion, because the network theory knows no field quantities as the magnetic flux.

Instead, working with time-varying voltages and currents in equivalent circuits with passive components such as coils and electrical resistances. The induced voltages can be modeled as voltage sources that historically as electromotive force (EMF ) can be referred to. However, since this is no force in the physical sense in induced voltages, the term EMF should be avoided to avoid misunderstandings and are referred to as self-induced voltage source.

In the network model, as represented by, among others, schematics, working with Zählpfeilen and specific orientations, as is the adjacent figure. To illustrate the acting externally on the conductor loop magnetic flux Φext and it caused induced voltages is Uext with the index provided ext. When loading an externally closed loop current i flowing generates a magnetic flux? I, which is marked with the index I. The self-induced voltage source can be used as a voltage source with the amount ui model, as shown in Figure first, and is the driving voltage to the coil current i in the opposite direction Uext. They are therefore referred to as reverse voltage. Applications of this illustration, for example, in the description of so-called magnetizing current in a transformer.

The model of the inductive voltage drop, as shown in the second figure, does not need any additional power source. The voltage appearing at the shown coil L has in this case in the same direction as the current i which is caused by the Uext of external driving voltage. This representation has the advantage that the relationships in the network model at harmonic processes by Ohm's law can be described with reactances easier. The major in electrical engineering special case of harmonic processes in the sizes occurring reduces the time derivatives in the induction law on multiplications with j? ( D.phi / dt ≡ jωΦ ), which corresponds to the complex plane, a rotation of 90 ° and represents the access to the complex AC bill. Here denotes the imaginary unit.

Inductive reactance

Transforming the differential equation:

In the Laplace domain with the independent variables, then from the differential operator of the factor, and we have:

The sign denotes the angular frequency. Similar to Ohm's law can be used for the coil from an AC resistance:

Define, also referred to as a (complex) impedance.

If, for example, for very simple circuits, the phase shift between current and voltage is not considered at the inductor, one can perform AC bills without complex numbers. An inductor to an alternating voltage is applied from the amount U, the magnitude I of the current according to the formula:

Are calculated, where f is valid for the inductive reactance XL at the frequency:

• Voltage and current waveform

Coil with an iron core

However, this applies only for inductors have a constant and not dependent on the modulation permeability, as is the case for example with an air coil.

The idle current of air coil may be shifted in phase by less than 90 degrees to the voltage. At the low frequency of 50 Hz in the example of measurement of the ohmic resistance is dominant. The idle current of the coil with an iron core extends, the hysteresis curve due to entirely different from an air-core coil.

Self-induction applications

The self-induction is used, among other things, to produce the required high ignition voltage at the spark plugs in the fluorescent lamp or a gasoline engine. It is possible to generate voltages of a few 1000V. In the electric fence and the spark coil high voltage pulses are generated in this way.

Switching each inductor provides for switch a load represents the resulting high voltage for switches, in particular electronic switches such as transistors, dangerous, because when you turn off the magnetic field changes very abruptly, thus generating the high self- induction voltage. To avoid the destruction of the switch or to limit the voltage, a capacitor, or a free-wheeling diode is connected in anti- parallel with the coil.

The self-induction is, among other things, the reason for the inductor, which is required to describe the behavior of the coil in AC circuits. The inductance results in a phase shift between current and voltage, which is applied in the context of complex alternating current calculation in the computation of inductive resistors.

Induktivitätsbestimmung different conductor arrangements

By applying the methods for calculating magnetic fields, in particular the Biot - Savart law can be determined analytically for some simple geometric conductor arrangements the external inductances. More complicated conductor arrangements are in the field calculation usually only by numerical calculation methods available.

The circuit arrangements illustrated in more detail below also have technical meanings: in the manufacture of inductors (designation for electrical parts having a defined inductance as a main property) such standard form is often used. Such devices are called coils or chokes.

Inductance of a toroid

An annular coil, also referred to as toroidal coil consists of a ring with the mean radius R and the cross -sectional area A, the magnetic properties are significantly. It is often magnetically highly conductive and has a high relative permeability ĩr, such as ferrite. The thereby occurring μ0 is the magnetic constant, a fundamental constant.

This annular core is wound with a thin layer of wire with N turns. The inductance L is then added to approximation of the form:

A better approximation of the inductance of a ring coil, which observes the dependence of the magnetic field strength as a function of the radius, as follows:

In this case a rectangular cross section of the ring with the height h, it is assumed. The outer radius of the core is designated R and the inner radius r.

In all cases, these equations provide good approximate results only for sufficiently thin bewickeltem ring.

The advantage evenly wound coils toroidal coils lies in the field of freedom outside of the coil, which is why they emit no magnetic interference fields and are hardly susceptible to such.

Inductance of a solenoid

In a cylindrical coil, the length L is very large relative to the diameter of the cross section A, the inductance can be determined as follows:

In this case, the magnetic resistance of the outer space in addition to the condition of a sheet-like member closed winding neglected and only accepted Rm of the core. Thus, this equation is also valid only in approximation. For shorter solenoids approximation formulas that take into account the finite length of the coil, and thus the " poorer " magnetic field guidance in their interior exist. For a coil whose length is at least 0.6 times the radius (rw: winding radius ) applies:

The cylindrical coil filled with a material whose relative permeability is greater than 1, the external magnetic field space is relevant, and the above simple formulas can not be applied. Such is filled with a magnetic material with open solenoid magnetic circuit are widely used, and despite the prevailing magnetic fields around them a number of advantages: they are difficult due to the air proportion in the magnetic circuit in which magnetic saturation hardly and are easy to manufacture.

Determining the inductance by AL value

In practice, often finished cores are used frequently by the manufacturer for a Inductance AL (Al- value, mostly in nH per Quadratwindung ) is indicated and corresponds to the reciprocal value of the magnetic resistance Rm. In this value, all material constants and the specific geometry of the array are already summarized as an approximation. If the core is wound with N turns, to obtain a coil having the inductance:

This applies only when the core material is operated in a linear region of its characteristic curve of induction B and magnetic field strength H or remains below the saturation induction.

Field energy

Inductive components such as coils store energy in the form of the magnetic field. A solenoid magnetic field of the inductance L, which is flowed through by the current value of the current I, which contains the energy W:

In a sudden interruption of the circuit, the energy stored in the coil in a very short time has to implement and yields at the terminals a very high self-induction voltage which can cause damage to the insulation or other circuit components. To avoid this, before the switching off inductive components are generally short-circuited to a load resistor, in the converting the stored energy in the magnetic field heat. However, this high voltage may also supply to the electrical components of the high voltage requirements, are such as a spark plug may be used.

Other methods for the thermal conversion in switching the protection diodes used in DC circuits.

Internal and external inductance

The term external inductance is used for the contribution of the occurs in the space outside of the electrical conductor magnetic flux to the inductance. In the above examples for the determination of the inductance of conductors of different geometric cross-sections of the electrical conductors have been assumed to be negligibly thin. In this case, the determination of the inductance of the inductor determine the outer shape or an idealized field may limit.

Has the electric conductor ( wire ), however, a non-negligible spatial extent, a corresponding cross-sectional area occurs, a magnetic flux density distribution within the conductor. The derived inductance is called the internal inductance. In the simplest case of a uniform current distribution over the cross section of the conductor of length l can be the internal inductance using the following equation to determine:

Remarkable thing is, that the internal inductance is not dependent on the specific geometrical dimensions such as the diameter of the cross-sectional area of the conductor. That expression is only valid for a uniform current distribution, which is only in direct current, and only when the cross-sectional area of the conductor has no internal boundary. If the current distribution due to the skin effect at higher frequencies no longer uniform, there are other, more complex expressions for the frequency-dependent internal inductance. The internal inductances are strongly frequency dependent because of the skin effect in the conductor and decrease with increasing frequency.

The internal inductance is especially in the determination of the total inductance of the electric cables of importance, since in these low-frequency (eg mains frequency), the conductor cross-sections are often can not be neglected.

Inductance of a coaxial

To determine the inductance of a coaxial cable of length L ( so-called inductance ) have to be considered at low frequencies, the self inductance of the inner conductor and the outer conductor Lia Lii. But mainly affects the inductance La of the concentric space between the two conductors. The total inductance of a coaxial line of length l is derived from the sum of the individual partial inductance:

In the case of direct current can be used for the inner conductor with a diameter d of the above expression for the internal inductance:

Which are also highly frequency dependent internal inductance of the outer conductor with the thickness S and the inner diameter D of which is arranged as a circular ring concentric with can be determined in the direct current in the case to a good approximation:

The frequency-independent external inductance in the dielectric is in coaxial conductors:

At higher frequencies, up from 10 kHz, the two terms of the internal inductance can be neglected because of the skin effect. Therefore, only the summand La is the external inductance is essential for the determination of the characteristic impedance of a coaxial cable in typical frequencies.

Mutual inductance

The mutual inductance featuring mutual magnetic influence between two or more spatially adjacent circuits. It is also referred to as a magnetic coupling. The most important industrial application is the mutual inductance in a transformer.

Nonlinear Inductance

The relative permeability depends on the material constant ĩr not only by the particular material used but also depends on many materials, the magnetic flux density. At high magnetic flux densities, there is a so-called magnetic saturation of the material and as a consequence a reduction in the relative permeability ĩr until the first result, the inductance is directly related to the magnetic flux density, which in turn mostly a function of the current flowing through the coil electric current is. Thus, the inductance of coil changes depending on the instantaneous value of the current flowing through the coil.

The result is that the dynamic inductance can vary in very small modulations to the working of the static inductance. For larger modulations of the linear Arbeitspunktnäherung addition, additional harmonics may occur as a non-linear distortion with non-linear inductors in AC applications. Also, the simple methods (linear) complex AC circuit analysis is no longer directly applicable to non-linear inductors in calculations.

The non-linearity of inductors may be desirable, eg inductors in switch controllers to better suit different load cases, or in the deflection circuits of television receivers to counteract the nonlinear current in the deflection coils. Even with the so-called saturation chokes for suppression of thyristor nonlinearity is desired.

Meters

Inductance can not be measured directly. It can only be measured their impact.

By switching of a known AC voltage and measuring the current flowing through the inductor alternating current (or vice versa ), the inductance can be determined via the reactance. These amplitude and phase are determined. This method is applied in simple laboratory instruments and supplies the inductance value, the quality and the value of an equivalent series or parallel resistor.

By connecting a known capacitance to the inductance of a resonant circuit is obtained. Determined to its resonant frequency, it can be close to the inductance. This method can also be performed without special equipment and therefore more popular among hobbyists and amateurs. The accuracy is quite high.

For high accuracy, a bridge is used: Maxwell bridge. This method is very accurate and is used inter alia in the automated production of coils.

In determining the real coil inductors must be noted that at very high frequencies, the capacitive coupling between the turns and layers to take effect, depending on the coil design out. The impedance curve rises to a maximum value and gets resonant circuit character to sink to even higher frequencies again - the coil then provides a capacity dar.

Inductance as annoying feature

Each electric current causes a magnetic field ( Electromagnetism ) is stored in the magnetic energy. So each piece of an electrical conductor having a low inductance. On printed circuit boards can be expected as a rough calculation with an inductance of about 1.2 nH per millimeter line length. In summary, this result, the parasitic inductances Aufbauinduktivität an electrical circuit.

The magnetic fields of closely spaced conductive elements influence each other through the magnetic coupling then each other. Lying, for example, return line of a circuit very close together, their magnetic fields cancel each other partially, which greatly reduces the total inductance of this arrangement. Hence, often current paths are guided closely together and twisted cables.

If the current in an inductive conductor loop change must be effective for a current change di / dt proportional voltage U ind:

Frequently for switching loads with inductive behavior used switches and relays have resulted in significant signs of wear on the contacts, which may impair their function strongly: when switching off the current flows due to the inductance further and forms an arc ( see circuit arc ), in which the energy stored in the inductor is discharged. Even more critical are current flow changes that are caused by a semiconductor switch. Semiconductor components are destroyed by such high voltages often. Therefore, care must be taken in the design of circuits with high rates of current change to a low-inductance structure. In addition, the snubber networks are often attached to the semiconductor. If possible and necessary, even free-wheeling diodes are used. Newer semiconductor switches can often also switch without protection inductive loads by the energy dissipation takes place in a controlled avalanche breakdown.

Another problem is the interaction of parasitic inductances with parasitic capacitances. Period, the resulting resonant circuit can generate problems as voltage oscillations can damage the semiconductor components and deteriorate the electromagnetic compatibility and signal transmission properties.

In computers, the power requirements of individual integrated circuits may change in the nanosecond clock. Because this corresponds to a frequency in gigahertz range, the inductance of the power supply lines can not be ignored, even if they are only a few inches short. The inductive resistance of the wire increases the internal resistance of the voltage source the frequency increases considerably. As a result, the voltage at current changes, for example, between 2 V and 10 V can vary and disrupt the IC, possibly even destroy it. As antidote low inductance capacitors are soldered directly to the IC terminals, which ensure a very low dynamic internal resistance.

Calculation techniques

In the most general case, power distribution and magnetic field from the Maxwell equations are to be determined. In the case of conductor loops of thin wires, the current distribution on the other hand is at least approximately determined, skin effect and shielding currents arise, however, even here complications and case distinctions.

Mutual inductance of two conductor loops

The mutual inductance of two " thready " conductor loops m and n can be with the Neumann line integral

Receive. The symbol μ0 denotes the permeability of free space, Cm and Cn, the plane spanned by the conductor loop curves. The formula is a good approximation to actual wire loops applicable when the radii of curvature of the loops and the spacing between the wires is greater than the wire radius.

Self-inductance of a wire loop

Neumann 's formula can not be used for calculation of self-inductances, since m = n, the two curves coincide and the integrand 1 / | x- x '| is diverging. It is necessary to take account of the finite wire of radius a and the current distribution in the conductor cross section. There remains, the contribution of the integral over all point pairs with | x- x ' |> a / 2, and a correction term,

There are A and L here for the radius and length of the wire, Y is a dependent of the current distribution constant: Y = 0, if the current in the wire surface flows (skin effect), Y = 1/ 2, if the current density in the wire cross section is constant. The error is small when the wire loop is large relative to the wire radius.

Method of mirror currents

In some cases, different current distributions generate a magnetic field identical in a room area. This fact can be exploited, in order to put self-inductances related to each other (see also image-charge ). An example is the two systems:

• A wire at a distance d / 2 in front of a perfectly conductive wall ( the current returns to the back wall )
• Two parallel wires a distance d, the opposite stream

The magnetic field of the two systems is the same ( within a half-space ). Magnetic field energy, and inductance of the second system are thus twice as big as that of the first.

Relationship between inductance and capacitance

In the case of a conductor pair consisting of two parallel ideal conductors arbitrary constant cross section between inductance per unit length L 'and capacitance per unit length C' the relationship

There are ε and μ here for the dielectric permittivity and the magnetic permeability of the surrounding medium. In conductors there is no electric and no magnetic field (full skin effect, high frequency). Outside the conductors are everywhere perpendicular electric and magnetic field. Signals propagate. At the lines along at the same speed as free from electromagnetic radiation in the surrounding medium

Self-inductances of simple circuits

The self-inductances of many types of circuits can be exactly or specify to a good approximation.

At w << 1   at w >> 1

Μ0 is the symbol for the constant magnetic field ( 4π × 10-7 H / m). At high frequencies, the current flows in the conductor surface (skin effect), and depending on the geometry must sometimes distinguish high-frequency and low-frequency inductors. This is the constant Y: Y = 0, if the power is distributed uniformly over the wire surface (skin effect), Y = 1/2 if the current is uniformly distributed over the conductor cross section. In case of skin effect is also to be noted that at a small distance between the conductors additional shielding currents are induced, and the Y -containing expressions then be inaccurate.