# How do solenoids work

## The sink

In signal processing electronics, a coil as a passive component with its inductive property determines the frequency range of high-pass, low-pass and filter circuits. Coils are manufactured with the smallest possible manufacturing tolerances. In the field of communications technology, they often have a magnetizable coil core made of ferrite materials. Coils without a core are drawn as air-core coils. In series or parallel circuits combined with capacitors, LC resonant circuits are created. In oscillator circuits, they generate sinusoidal signals in a narrow frequency range or preferably filter this resonance frequency out of frequency bands for targeted use.

Choke and transformer coils

In the energy sector, coils are used in transformers or as inductors. Your manufacturing tolerances can be larger and almost always have coil cores. They are composed of laminated ferromagnetic sheets. In transformers, the iron cores are magnetically closed, while in reactors they have a defined air gap. The air gap creates a magnetic resistance. In filter circuits, the high magnetic resistance of the choke coil with its self-induction dampens alternating voltage components or electrical interference pulses on subsequently connected circuit components. In the variously used switched-mode power supplies, inductors store magnetic energy in the coil core during the switch-on time and release it again in the form of electrical energy during the switch-off time.

### General properties

A coil is created when an insulated wire is wound onto a non-conductive carrier. The turns form individual conductor loops connected in series. A distinction is mainly made between a cylinder coil open on both sides and a closed circular ring coil. Coils made of thick wire and a few turns can also be self-supporting. Coils without additional or non-magnetizable core material are called air-core coils. If there is a ferromagnetic core in the coil body, it is electrically isolated from the coil winding. The most important property of a coil is its inductance.

The picture on the left shows that each conductor loop forms a magnetic field perpendicular to the current flow and interacts with the neighboring fields. The field forces between the conductor windings balance each other out. A cumulative field whose polarity is determined by the direction of the current is effective to the outside.

If a clockwise screw is turned in the direction that the current takes from plus to minus through the coil, it moves towards the magnetic north pole of the coil.

If the right hand grips the coil in such a way that the fingertips point in the technical direction of the current from plus to minus, then the splayed thumb shows the direction of the magnetic field lines in the coil interior and points to the north pole.

If the left hand grasps the coil in such a way that the fingertips follow the direction of the physical current flow from minus to plus, then the splayed thumb shows the direction of the magnetic field inside the coil and points to the north pole.

If electrical current flows through a wire, a wire loop or a coil, a magnetic vortex field is created perpendicular to the direction of the current. Since the magnetic field lines are self-contained, the field effect is effective inside and outside the coil. Cylinder coils and flat coils have a relatively large outwardly effective magnetic field, while that of toroidal or toroidal coils is small. The maximum use of the magnetic field is achieved with this design. Their external magnetic stray fields are small and neighboring coils and components are only slightly affected.

Magnetizable core material concentrates the magnetic field lines on the core area and strengthens the magnetic properties there. With the same design, a coil with a ferromagnetic core has a greater inductance. The magnetizable core should be laminated or made of ferrite material. This minimizes the eddy current losses occurring in the core and the resulting thermal load on the system. If a coil is exposed to a magnetic field that changes over time, it generates an electrical voltage at its connections.

### Inductance of a coil

The main property of a coil is its inductance. In addition, it always has an ohmic resistance, which is determined by the conductor material, the length and the cross-section. When electric current flows through a coil, it creates a magnetic field. A cylinder coil connected to DC voltage ultimately acts like a bar magnet. The magnetic force can be increased by ferromagnetic core materials, as they concentrate the magnetic field lines and the field runs preferentially there. Relay coils and lifting magnets work according to this principle.

If electrical voltage is applied to an ohmic resistor, the current that can be calculated using Ohm's law flows immediately. If the ohmic (wire) resistance of a coil is known, the maximum current can be calculated for an applied DC voltage. The inrush current generates a magnetic field in the coil and takes the necessary energy from the voltage source during this time. In the chapter on the physical quantities of a magnetic field, the relationships between the formation of the magnetic field are discussed in more detail. The magnetic field depends on the core material of the coil, the number of turns and the coil geometry and influences the inductance of a coil.

If the inductance is interpreted as a resistance that changes over time against the change in current acting from the outside, then the change in current generates a counter voltage (self-induction voltage) on the coil according to Lenz's rule. Like any voltage drop, it is directed against the source voltage. Immediately after switching on, the change in current is greatest and thus the induction (counter) voltage is also greatest. The voltage difference, which is small in this first time segment, remains for the current flow. As the magnetic field builds up, the assumed resistance behavior of the inductance decreases. According to exponential functions, the current increases, the induction voltage decreases and the magnetic field of the coil reaches its maximum after five time units equal to seven half-lives. In the chapter on the coil in a direct current circuit, precise descriptions of the behavior of a coil in the event of changes in current and the mathematical relationships can be found.

The symbol for inductance is L. The unit has been given the abbreviation H in honor of the physicist Joseph Henry. The exact mathematical derivation of inductance with vector quantities is complicated. The simplified form with scalar values applies in the frequency range below 1 MHz for cylinder coils whose length is large compared to the radius. Strictly speaking, instead of the equal sign in the formulas, there should be approximately equal (≈) and the formula should only be used for coils without a magnetizable coil core. For practical applications, the formula for inductance can almost always also be used for coils with a ferromagnetic core. The permeability number μr depends on the field strength H and is only conditionally a constant.

In the formula for the magnetic field strength H, the product of the current I and the number of turns N is the flux Θ, which is referred to as the magnetic voltage. It causes a changing magnetic flux Φ that is proportional to the inner coil area A. The constant of proportionality is the flux density B. The magnetic permeability μ is the ratio B / H. After a few adjustments, the partial equations are followed by a simplified determining equation for the inductance L. The core factor AL has the meaning of a magnetic conductance. It is often given by the manufacturer as a numerical value in the order of nH. There is hardly any precise calculation of the inductance, as scatter factors and the inhomogeneous field profile at the end faces are not taken into account.

The inductance of an optimally single-layer wound solenoid is halved if the distances between the turns are doubled, i.e. the entire coil is pulled apart to twice the length l. If all other parameters such as the number of turns and the wire diameter remain unchanged, the length of the magnetic field lines inside the coil doubles and the core factor AL is halved.

The inductance of a toroidal core or toroidal coil is calculated using a slightly different formula. In the case of a coil core with a circular or square cross-section, the diameter or the surface height must be taken into account. Another factor is the natural logarithm for the ratio of the outer to the inner coil radius, which is related to the mean field line length inside the coil.

Scientifically precise considerations are not necessary for practical use, and they are also quite complicated mathematically. With the formulas given above, the inductance for solenoid coils can be calculated relatively easily and with a small error. If the coil length l is 5 ... 10 times greater than the coil radius R, the error is less than 1%. Strictly speaking, this only applies to coils without a magnetizable core or with a closed ferromagnetic core. For a toroidal coil, the mean core circumference lm is to be used as the length and the mean cross-section for the coil cross-section in multi-layer coils.

It is not possible to determine the influence l / R on the inductance of a cylinder coil in a mathematically simple manner. If the number of turns with its quadratic influence is to remain the same and if the coil length or the coil radius is varied, the wire diameter must change. This has an influence on the magnetic field and thus on the inductance. If the wire diameter is also to remain constant, the distance between the windings must change and this also influences the magnetic field and the inductance.

There is a comparative comparison of the quantities in the electric and magnetic circuit. The induction voltage depends on the change in current in a certain time interval. The unit of inductance can also be specified in Ω · s. In addition to the ohmic wire resistance, the coil thus has a further special resistance behavior.

A coil has the inductance L = 1 henry if the current intensity changes evenly by I = 1 ampere in s = 1 second, and a voltage of U = 1 volt is induced.

### The real coil

There is no such thing as the ideal inductive reactance. The main property of a coil is its inductance L, which determines the reactance XL. In addition, there is the ohmic wire resistance R, which converts active power into thermal energy. If the coil has a magnetizable core and an alternating current flows through it, eddy currents occur in the core material during the periodic reversal of magnetization, which heat the core. This energy conversion is assigned to the so-called core or iron loss resistance RFe. In the equivalent circuit of a coil shown below, it is represented by an effective resistance connected in parallel to the reactance. If necessary, it can be converted into an equivalent series resistance.

### The loss factor and figure of merit

The equivalent circuit diagram of the real coil is a mixed series and parallel circuit that can be converted into an equivalent series circuit. The current is therefore smaller than it would be with the ideal reactance. The quality of a coil is primarily determined by the ohmic wire resistance. In telecommunications, coil cores are mostly made of ferrite, as they minimize eddy currents more than laminated metal sheets. The parallel core loss resistance is then negligibly large or, as a converted series resistance, is negligibly small compared to the wire resistance.

In circuits with high operating frequencies, coils have a higher ohmic resistance due to the skin effect. Inside the conductor, the current is displaced so that the current can only flow in a smaller outer ring area. The higher the frequency, the smaller the effective wire cross-section that can be used by the current and the resistance of the conductor increases.

Usually the frequency range where the coil is used as an inductive resistor is well below its natural resonance, which is discussed in the next section. The determination of the quality is then reduced to the series connection of an ideally assumed inductance and the equivalent series resistance ESR. It is the sum of the ohmic direct current resistance and the frequency-dependent increase in resistance due to the skin effect and the converted core loss resistance.

The lower the ESR value of the coil, the lower the losses and the coil has a high quality factor. In the data sheets for the fixed inductors, additional information on the ohmic DC resistance, the maximum current and the natural resonance frequency can often be found. The information on the minimum quality is given within the individual decades of inductance at fixed measuring frequencies. The values for Q are in the range from 20 ... 70.

### Self-capacitance of a coil

The equation for the reactance of a coil shows with XL = ω · L = 2 · π · f · L a direct proportionality to the frequency. If this relationship is examined more closely, it no longer applies at very high frequencies. The linear dependency changes into the resonance curve of a parallel oscillating circuit. The coil must therefore also have capacitive properties.

With a properly wound coil, turn is next to turn. The voltage measured parallel to the coil is the sum of all partial voltages between adjacent turns. As a potential difference, voltage is always connected to an electric field. The coil wires lying in parallel thus have the property of small capacitors, the partial capacities of which together determine the intrinsic capacitance of a coil. In the case of small designs, the outer solder caps of the coil also form a capacitor. In the equivalent circuit diagram, the parasitic capacitance is parallel to the inductance. Both reactive components together form a parallel resonant circuit. Since the LC ratio is sufficiently large, each coil has at least one pronounced natural resonance.

The self-capacitance of a coil can be reduced with special winding methods. In the case of coils with a few turns, a slight gap is kept between each turn. This reduces the capacity, as it is inversely proportional to the distance between the pads. Coils with a high number of turns are divided into partial windings, called disc windings. Even smaller winding capacities can be achieved using the special method of cross winding.

The following diagram shows the frequency behavior of two real coils from the laboratory. Both coils have identical values with L = 10 mH and the same copper wire resistance of 3 Ω. One coil is wound in layers in a ferrite pot core and the other is made using the cross-winding technique and has a rod ferrite core. The Bode diagram was created for both coils under the same test conditions.

The layer wound coil has inductive behavior in the frequency range from 100 Hz to 60 kHz. After that, the red curve is no longer linear. With the measured natural resonance of 167 kHz, a self-capacitance of 89 pF is calculated.

The self-resonance of the coil in cross-wound technology is 330 kHz. This results in a smaller self-capacitance of 23 pF. The range of the linear inductive behavior extends up to 100 kHz.

Stray magnetic fields from a coil on surrounding components can be dampened by electrically conductive sheaths. The stray field induces eddy currents in the shielding, which are short-circuited to the circuit ground. The shielding in turn acts on the self-capacitance of the coil, so that in filter circuits the inductance of the coil has to be corrected by trimming the core.

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