Ohm's law for an alternating current circuit: the formula for the relationship between electrical quantities, the calculation procedure

The fundamental position describing the dependence of current, resistance and voltage from each other is Ohm's law for an alternating current circuit. Its main difference from the position of the same name for a section of the circuit is to take into account the impedance. This value depends on the active and reactive components of the line, that is, it takes into account the capacitance and inductance. Therefore, it will be more difficult to calculate the parameters for the complete chain compared to the section.

Basic concepts

The whole science of electrical engineering is built on the operation of such concepts as charge and potential. In addition, electric and magnetic fields are important phenomena in the circuit. In order to understand the essence of Ohm's law, it is necessary to understand what these quantities are, and on what certain electromagnetic processes depend.

Electricity is a phenomenon caused by the interaction of charges with each other and their movement. This word was coined by William Gilbert in 1600 after his discovery of the ability of certain bodies to become electrified. Since he conducted his experiments with pieces of amber, then the property of attracting or repelling they called other substances "amber", which in translation from Greek sounds like electricity.

Later, various scientists such as Oersted, Ampere, Joule, Faraday, Volt, Lenz and Ohm discovered a number of phenomena. Thanks to their research, concepts appeared in everyday life: electromagnetic induction and field, galvanic cell, current and potential. They discovered the connection between electricity and magnetism, which led to the emergence of a science that studies the theory of electromagnetic phenomena.

In 1880, the Russian engineer Lachinov theoretically indicated what conditions are necessary for the transmission of electricity over distances. And after 8 years, Heinrich Rudolf Hertz registered electromagnetic waves during experiments.

Thus, it was found that electric charges are capable of creating electric radiation around them. They were conventionally divided into particles with positive and negative charge signs. It was established that charges of the same name attract, and charges of the opposite - repel. For their movement to occur, it is necessary to apply some kind of energy to the physical body. When they move, a magnetic field is generated.

The property of materials to ensure the movement of charges is called conductivity, and the inverse value is resistance. The ability to pass charges through itself depends on the structure of the crystal lattice of the substance, its bonds, defects and the content of impurities.

Determination of voltage

Scientists have found that there are two types of movement of charges - chaotic and directional. The first type does not lead to any processes, since the energy is in a balanced state. But if a force is applied to the body, forcing the charges to follow in one direction, then an electric current will arise. There are two types:

1. Constant - the strength and direction of which remain constant over time.
2. Variable - having a different value at a certain point in time and changing its movement, while repeating its change (cycle) at regular intervals. This variability is described by the harmonic sine or cosine law.

A charge is characterized by such a concept as potential, that is, the amount of energy that it possesses. The force required to move a charge from one point of the body to another is called voltage.

It is determined with respect to the change in the charge potential. The strength of the current is determined by the ratio of the amount of charge that has passed through the body per unit of time to the value of this period. Mathematically, it is described by the expression: Im = ΔQ / Δt, measured in amperes (A).

With respect to the alternating signal, an additional value is introduced - the frequency f, which determines the cyclicity of the signal passage f = 1 / T, where T is the period. Hertz (Hz) is taken as its unit of measurement. Based on this, the sinusoidal current is expressed by the formula:

I = Im * sin (w * t + Ψ), where:

• Im is the current strength at a certain point in time;
• Ψ - phase determined by the displacement of the current wave with respect to the voltage;
• w - circular frequency, this value depends on the period and is equal to w = 2 * p * f.

Voltage is characterized by the work that an electric field does to transfer charge from one point to another. It is defined as the potential difference: Um = φ1 - φ2. The work expended consists of two forces: electrical and external, called electromotive (EMF). It depends on the magnetic induction. The potential is equal to the ratio of the interaction energy of the charge of the surrounding field to the value of its magnitude.

That's why for a harmonic change in the signal, the voltage value is expressed as:

U = Um * sin (w * t + Ψ).

Where Um is the peak value of the voltage. AC voltage is measured in volts (V).

Circuit impedance

Every physical body has its own resistance. It is due to the internal structure of the substance. This value is characterized by the property of the conductor to prevent the passage of current and depends on the specific electrical parameter. It is determined by the formula: R = ρ * L / S, where ρ is the resistivity, which is a scalar quantity, Ohm * m; L is the length of the conductor; m; S - cross-sectional area, m². This expression determines the constant resistance inherent in passive elements.

At the same time, impedance, impedance, is found as the sum of the passive and reactive components. The first is determined only by the active resistance, which consists of the resistive load of the power supply and resistors: R = R0 + r. The second is found as the difference between capacitive and inductive reactance: X = XL-Xc.

If an ideal (lossless) capacitor is placed in an electrical circuit, then after an alternating signal arrives at it, it will be charged. The current will begin to flow further, in accordance with the periods of its charge and discharge. The amount of electricity flowing in the circuit is equal to: q = C * U, where C is the capacity of the element, F; U is the voltage of the power source or on the capacitor plates, V.

Since the rates of change of current and voltage are directly proportional to the frequency w, the following expression will be true: I = 2 * p * f * C * U. Hence it turns out that capacitive impedance is calculated using the formula:

Xc = 1/2 * p * f * C = 1 / w * C, Ohm.

Inductive resistance arises due to the appearance in the conductor of its own field, called the EMF of self-induction EL. It depends on the inductance and the rate of change of the current. In turn, the inductance depends on the shape and size of the conductor, the magnetic permeability of the medium: L = Ф / I, measured in tesla (T). Since the voltage applied to the inductance is equal in magnitude to the EMF of self-induction, then EL = 2 * p * f * L * I is true. In this case, the rate of change of the current is proportional to the frequency w. Based on this, the inductive reactance is:

Xl = w * L, Ohm.

Thus, the impedance of the circuit is calculated as: Z = (R ² + (X c-X l) ²) ^½, Ohm.

That is, it depends on the frequency of the alternating signal, the inductance and capacitance of the circuit, as well as the active resistance of the source and the electric line. In this case, parasitic quantities are most often used as the reactive component.

AC law

The classical law was discovered by German physicist Simon Ohm in 1862. Through experiments, he discovered the relationship between current and voltage. As a result, the scientist formulated the statement that the current strength is proportional to the potential difference and inversely proportional to the resistance. If the current in the electric circuit decreases several times, then the voltage in it will also decrease by the same amount.

Ohm's law has been described mathematically as:

I = U / R, A.

This expression is valid for both sinusoidal and direct current. But such a dependence of the quantities corresponds to an ideal situation in which parasitic components and the resistance of the current source are not taken into account. In the case of a harmonic signal, its passage is influenced by the frequency, due to the presence of a capacitive and inductive component in the electric line.

That's why Ohm's law for alternating current is described by the formula:

I = U / Z, where:

• I - AC current, A;
• U is the potential difference, V;
• Z is the total resistance of the circuit, Ohm.

The impedance depends on the frequency of the harmonic signal and is calculated using the following formula:

Z = ((R + r)² + (w * L - 1 / w * C)²)^½ = ((R + r)²+ X²)^½.

When a variable current passes, the electromagnetic field does work, while heat is released due to the resistance in the circuit. That is, electrical energy is converted into heat. Power is proportional to current and voltage. The formula describing the instantaneous value looks like: P = I * U.

At the same time, for an alternating signal, it is necessary to take into account the amplitude and frequency components. That's why:

P = I * U * cosw * t * cos (w * t + Ψ), where I, U - amplitude values, and Ψ - phase shift.

To analyze the processes in AC electric circuits, the concept of a complex number is introduced. This is due to the phase displacement that appears between the current and the potential difference. This number is denoted by the Latin letter j and consists of the imaginary Im and real Re parts.

Since power is transformed into heat on an active resistance, and on a reactive resistance it is converted into energy of an electromagnetic field, its transitions from any form to any are possible. You can write: Z = U / I = z * e^{j *Ψ.}

Hence the total resistance of the circuit: Z = r + j * X, where r and x are the active and reactance, respectively. If the phase shift is taken equal to 90⁰, then the complex number can be ignored.

Using a formula

Using Ohm's law allows you to build the temporal characteristics of various elements. With the help of it, it is easy to calculate the loads for electrical circuits, choose the required wire cross-section, choose the right circuit breakers and fuses. Understanding the law makes it possible to apply the correct power source.

The use of Ohm's Law can be applied in practice to solve the problem. For example, let there be an electrical line consisting of series-connected elements such as: capacitance, inductance, and resistor. In this case, the capacitance is C = 2 * F, the inductance is L = 10 mH, and the resistance is R = 10 kΩ. It is required to calculate the impedance of the complete circuit and calculate the amperage. In this case, the power supply operates at a frequency equal to f = 200 Hz and outputs a signal with an amplitude of U = 12 0 V. The internal resistance of the power supply is r = 1 kΩ.

First, you need to calculate the reactance in the AC circuit. So, the capacitive resistance is found from the expression: Xc = 1 / (2 * p * F * C) and at a frequency of 200 Hz it is equal to: Xc = 588 Ohm.

The inductive reactance is found from the expression: XL = 2 * p * F * L. At f = 200 Hz and it leaves: X * L = 1.25 ohms. The total resistance of the RLC circuit will be: Z = ((10 * 10 ³ +1*10 ³ ) ² + (588−1,25) ² ) ^½ = 11 kΩ.

The potential difference, changing according to the harmonic sine law, will be determined: U (t) = U * sin (2 * p * f * t) = 120 * sin (3.14 * t). The current will be equal: I (t) = 10 * 10 ⁻³ + sin (3.14 * t + p / 2).

Based on the calculated data, you can build a current graph corresponding to a frequency of 100 Hz. For this, the current versus time is displayed in the Cartesian coordinate system.

It should be noted that Ohm's law for an alternating signal differs from that used for the classical calculation only by taking into account the impedance and frequency of the signal. And it is important to take them into account, since any radio component has both active and reactance, which ultimately affects the operation of the entire circuit, especially at high frequencies. Therefore, when designing electronic structures, in particular impulse devices, it is the complete Ohm's law that is used for calculations.