Inductance: the formula. Measurement of inductance. Inductance of the circuit

Who at school did not study physics? For someone, it was interesting and understandable, and someone pored over textbooks, trying to learn complex concepts by heart. But each of us has remembered that the world is based on physical knowledge. Today we will talk about such concepts as current inductance, loop inductance, and find out what kind of capacitors are and what a solenoid is.

Electrical circuit and inductance

Inductance serves to characterize the magnetic properties of the electrical circuit. It is defined as the proportionality coefficient between the current electric current and the magnetic flux in a closed loop. The flow is created by this current through the contour surface. Another definition is that the inductance is a parameter of the electrical circuit and determines the EMF of self-induction. The term is used to indicate the element of the chain and it is necessary to characterize the effect of self-induction, which was discovered by D. Henry and M. Faraday independently of each other. Inductance is associated with the shape, size of the contour and the value of the magnetic permeability of the environment. In the SI unit of measurement, this value is measured in henry and is denoted as L.

Self-inductance and inductance measurement

Inductivity is a quantity that is equal to the ratio of the magnetic flux passing through all the turns of the circuit to the current:

• L = N × F: I.

The inductance of the circuit depends on the shape, size of the circuit, and on the magnetic properties of the medium in which it is located. If an electric current flows in a closed loop, then a changing magnetic field arises. This will subsequently lead to the emergence of EMF. The birth of an induction current in a closed loop is called "self-induction." According to the Lenz rule, the value does not allow the current in the circuit to change. If self-inductance is detected, then an electrical circuit can be used, in which a resistor and a coil with an iron core are connected in parallel. In series with them are connected and electric lamps. In this case, the resistance of the resistor is equal to the resistance at the direct current of the coil. The result will be a bright burning of the lamps. The phenomenon of self-inductance occupies one of the main places in radio engineering and electrical engineering.

How to find the inductance

The formula, which is the simplest for finding the value, is as follows:

• L = F: I,

Where F is the magnetic flux, and I is the current in the circuit.

Through the inductance, we can express the EMF of self-induction:

• Ei = -L x dI: dt.

The formula suggests the inference about the numerical equality of induction with EMF, which arises in the circuit with a change in the current strength by one ammeter per second.

Variable inductance makes it possible to find the energy of the magnetic field:

• W = LI ² : 2.

"Spool of thread"

The inductor is a wound insulated copper wire on a solid base. As for insulation, the choice of material is wide - this is varnish, wire insulation, and fabric. The magnitude of the magnetic flux depends on the area of the cylinder. If the current in the coil is increased, the magnetic field will become larger and vice versa.

If an electric current is applied to the coil, a voltage opposite to the current will appear in it, but it suddenly disappears. This type of stress is called the electromotive force of self-induction. When the voltage is applied to the coil, the current changes its value from 0 to a certain number. The tension at this moment also changes, according to Ohm's law:

• I = U: R,

Where I characterizes the current strength, U - shows the voltage, R - coil resistance.

Another special feature of the coil is the following fact: if the "coil-current source" circuit is opened, the EMF will be added to the voltage. The current will also grow in the beginning, and then it will decline. This implies the first commutation law, which states that the current in the inductor does not change instantaneously.

The coil can be divided into two types:

1. With a magnetic tip. Ferrites and iron act as the heart material. The cores serve to increase inductance.
2. With non-magnetic. Used in cases where the inductance is not more than five milligrams.

Devices differ in appearance and internal structure. Depending on such parameters, the coil inductance is located. The formula in each case is different. For example, for a single-layer coil, the inductance will be:

• L = 10μ0ΠN2R2: 9R + 10l.

And now for a multi-layered formula:

• L = μ0N ² R ² : 2Π (6R + 9l + 10w).

The main conclusions related to the work of coils:

1. On cylindrical ferrite, the greatest inductance arises in the middle.
2. To obtain the maximum inductance, it is necessary to closely wind the coils on the coil.
3. Inductance is less, the smaller the number of turns.
4. In the toroidal core, the distance between the turns does not play the role of a coil.
5. The value of the inductance depends on the "turns in the square".
6. If the inductors are connected in series, then their total value is equal to the sum of the inductances.
7. When connecting in parallel, care should be taken to ensure that the inductors are spaced apart on the board. Otherwise, their readings will be incorrect due to the mutual influence of magnetic fields.

Solenoid

This term refers to a cylindrical winding made of wire that can be wound in one or more layers. The length of the cylinder is much larger than the diameter. Due to this feature, when a current is applied to the solenoid cavity, a magnetic field is produced. The rate of change of the magnetic flux is proportional to the change in the current. The inductance of the solenoid in this case is calculated as follows:

• Df: dt = L dl: dt.

Another type of coil is called an electromechanical actuator with a retractable core. In this case, the solenoid is supplied with an external ferromagnetic magnetic yoke.

Nowadays, the device can combine hydraulics and electronics. On this basis four models are created:

• The first is able to control the line pressure.
• The second model differs from the other by forced control of the clutch locking in torque converters.
• The third model contains in its composition pressure regulators responsible for the work of switching the speeds.
• The fourth is hydraulically controlled or valves.

Necessary formulas for calculations

To find the solenoid inductance, the formula applies as follows:

• L = μ0n ² V,

Where μ0 indicates the magnetic permeability of the vacuum, n is the number of turns, and V is the volume of the solenoid.

It is also possible to calculate the inductance of a solenoid using another formula:

• L = μ0N ² S: l,

Where S is the cross-sectional area, and l is the length of the solenoid.

In order to find the inductance of a solenoid, the formula is applied to any one that fits the solution to this problem.

Work on direct and alternating current

The magnetic field, which is created inside the coil, is directed along the axis, and is equal to:

• B = μ0nI,

Where μ0 is the magnetic permeability of the vacuum, n is the number of turns, and I is the current value.

When the current moves along the solenoid, the coil stores energy, which is equal to the work required to establish the current. To calculate the inductance in this case, the formula is used as follows:

• E = LI ² : 2,

Where L shows the value of the inductance, and E - the storing energy.

EMF of self-induction occurs when the current in the solenoid changes.

In the case of AC operation, an alternating magnetic field appears. The direction of the force of attraction can change, or it can remain unchanged. The first case occurs when a solenoid is used as an electromagnet. And the second one, when the anchor is made of soft magnetic material. The AC solenoid has a complex resistance, which includes the winding resistance and its inductance.

The most common application of solenoids of the first type (direct current) is in the role of a progressive power drive. Strength depends on the structure of the core and body. Examples of use are the work of scissors when cutting checks in cash registers, valves in engines and hydraulic systems, locks of locks. Solenoids of the second type are used as inductors for induction heating in crucible furnaces.

Oscillatory contours

The simplest resonant circuit is a sequential oscillatory circuit consisting of the included inductors and a capacitor through which an alternating current flows. To determine the coil inductance, the formula is used as follows:

• XL = W x L,

Where XL indicates the reactance of the coil, and W is the circular frequency.

If the reactance of the capacitor is used, the formula will look like this:

Xc = 1: W x C.

Important characteristics of the oscillatory circuit are the resonance frequency, wave resistance and Q-factor of the circuit. The first characterizes the frequency, where the resistance of the circuit is of an active nature. The second shows how the reactance at the resonant frequency passes between such quantities as the capacitance and inductance of the oscillatory circuit. The third characteristic determines the amplitude and width of the amplitude-frequency characteristics of the resonance and shows the dimensions of the energy reserve in the circuit in comparison with the energy losses in one oscillation period. In the technique, the frequency characteristics of the circuits are estimated using frequency response. In this case, the circuit is considered as a four-terminal network. When plotting graphs, the value of the transmission coefficient of the voltage (K) is used. This value indicates the ratio of the output voltage to the input voltage. For circuits that do not contain energy sources and different amplifying elements, the value of the coefficient is not greater than unity. It tends to zero when, at frequencies other than resonant, the resistance of the circuit is high. If the resistance is minimal, the coefficient is close to unity.

With a parallel oscillatory circuit, two reactive elements with different reactivity are included. The use of this kind of contour implies knowledge that when parallel inclusion of elements it is necessary to add only their conductivity, but not resistance. At the resonant frequency, the total conductivity of the circuit is zero, which indicates an infinitely large resistance to alternating current. For a circuit in which capacitance (C), resistance (R) and inductance are in parallel included, the formula combining them and Q (Q) is:

• Q = R√C: L.

When a parallel loop operates in one oscillation period, an energy exchange occurs between the capacitor and the coil twice. In this case, a loop current appears, which is much larger than the current value in the external circuit.

Condenser operation

The device is a two-terminal network of low conductivity and with a variable or constant capacitance value. When the capacitor is not charged, its resistance is close to zero, otherwise it is equal to infinity. If the current source is disconnected from this element, it becomes this source until it is discharged. The use of a capacitor in electronics is the role of filters that remove noise. This device in power units on power circuits are used to make up the system at high loads. This is based on the ability of an element to pass an alternating component, but a non-constant current. The higher the frequency of the component, the lower the resistance of the capacitor. As a result, all the interferences that go over the DC voltage are silenced through the capacitor.

The resistance of the element depends on the capacity. Proceeding from this, it will be more correct to put capacitors with different volumes in order to catch various kinds of interference. Due to the ability of the device to transmit direct current only during the charge period, it is used as a time-consuming element in generators or as a shaping link of a pulse.

Condensers come in many types. In general, the classification is based on the type of dielectric, since this parameter determines the stability of the capacitance, insulation resistance and so on. The systematization of this value is as follows:

1. Condensers with gaseous dielectric.
2. Vacuum.
3. With a liquid dielectric.
4. With a solid inorganic dielectric.
5. With a solid organic dielectric.
6. Solid state.
7. Electrolytic.

There is a classification of capacitors by purpose (general or special), by the nature of protection from external factors (protected and unprotected, insulated and not insulated, sealed and sealed), in mounting techniques (for hinged, printed, surface, screwed terminals, with snapping terminals ). Also, the devices can be distinguished by the ability to change the capacity:

1. Constant capacitors, that is, whose capacitance is always constant.
2. Trimming. Their capacity does not change with the operation of the equipment, but it can be adjusted once or periodically.
3. Variables. These are capacitors that allow a change in capacitance during the operation of the equipment.

Inductance and capacitor

Current-carrying elements of the device are capable of creating its own inductance. These are such structural parts as masonry, connecting buses, current leads, terminals and fuses. You can create additional inductance of the capacitor by attaching the busbars. The operating mode of the electrical circuit depends on the inductance, capacitance and active resistance. The formula for calculating the inductance, which appears when approaching the resonant frequency, is as follows:

• Ce = C: (1 - 4Π ² f ² LC),

Where Ce determines the effective capacity of the capacitor, C indicates the actual capacitance, f is the frequency, L is the inductance.

The value of the inductance must always be taken into account when working with power capacitors. For pulsed capacitors, the value of intrinsic inductance is most important. Their discharge falls on the inductive circuit and has two types - aperiodic and oscillatory.

The inductance in the capacitor is dependent on the connection scheme of the elements in it. For example, with parallel connection of sections and buses, this value is equal to the sum of the inductances of the main bus and pin package. To find such an inductance, the formula is as follows:

• Lk = Lp + Lm + Lb,

Where Lk shows the inductance of the device, Lp-package, Lm-main bus, and Lb-inductance of the terminals.

If, with a parallel connection, the bus current varies along its length, then the equivalent inductance is determined as follows:

• Lk = Lc: n + μ0 lxd: (3b) + Lb,

Where l is the length of the tires, b is its width, and d is the distance between the tires.

To reduce the inductance of the device, it is necessary to arrange the current-carrying parts of the capacitor so that their magnetic fields are mutually compensated. In other words, current-carrying parts with the same current motion must be removed from each other as far as possible, and with the opposite direction brought together. When combining the current collectors with a decrease in the thickness of the dielectric, the inductance of the section can be reduced. This can be achieved even by dividing one section with a large volume into several with a smaller capacity.