Current resistance: formula. Electrical resistance of conductors

We are starting to publish materials in a new section “” and in today’s article we will talk about fundamental concepts, without which not a single electronic device or circuit can be discussed. As you may have guessed, I mean current, voltage and resistance😉 In addition, we will not ignore the law that determines the relationship of these quantities, but I won’t get ahead of ourselves, let’s move gradually.

So let's start with the concept voltage.

Voltage.

A-priory voltage is the energy (or work) that is expended to move a unit positive charge from a point with a low potential to a point with a high potential (i.e., the first point has a more negative potential compared to the second). We remember from the physics course that the potential of an electrostatic field is a scalar quantity equal to the ratio of the potential energy of a charge in the field to this charge. Let's look at a small example:

There is a constant electric field in space, the intensity of which is equal to E. Consider two points located at a distance d from each other. So the voltage between two points is nothing more than the potential difference at these points:

At the same time, do not forget about the connection between the electrostatic field strength and the potential difference between two points:

And as a result, we get a formula connecting stress and tension:

In electronics, when considering various circuits, voltage is still considered to be the potential difference between points. Accordingly, it becomes clear that voltage in a circuit is a concept associated with two points in the circuit. That is, to say, for example, “voltage in a resistor” is not entirely correct. And if they talk about voltage at some point, then they mean the potential difference between this point and "earth". This is how we smoothly arrived at another most important concept in the study of electronics, namely the concept "Earth":) So here it is "earth" in electrical circuits, it is most often accepted to consider the point of zero potential (that is, the potential of this point is equal to 0).

Let's say a few more words about the units that help characterize the quantity voltage. The unit of measurement is Volt (V). Looking at the definition of the concept of voltage, we can easily understand that to move a charge of magnitude 1 pendant between points having a potential difference 1 Volt, it is necessary to do work equal to 1 Joule. With this, everything seems to be clear and we can move on 😉

And next in line we have one more concept, namely current.

Current, current strength in a circuit.

What is it electricity?

Let's think about what will happen if charged particles, for example, electrons, come under the influence of an electric field... Consider a conductor to which a certain voltage:

From the direction of the electric field strength ( E) we can conclude that title="Rendered by QuickLaTeX.com" height="16" width="60" style="vertical-align: -4px;"> (вектор напряженности всегда направлен в сторону уменьшения потенциала). На каждый электрон начинает действовать сила:!}

Where e is the charge of the electron.

And since the electron is a negatively charged particle, the force vector will be directed in the direction opposite to the direction of the field strength vector. Thus, under the influence of force, particles, along with chaotic motion, also acquire directional motion (velocity vector V in the figure). As a result, there arises electricity 🙂

Current is the ordered movement of charged particles under the influence of an electric field.

The important point is that current is assumed to flow from a point with a more positive potential to a point with a more negative potential, even though the electron is moving in the opposite direction.

Not only electrons can act as charge carriers. For example, in electrolytes and ionized gases, the flow of current is primarily associated with the movement of ions, which are positively charged particles. Accordingly, the direction of the force vector acting on them (and at the same time the velocity vector) will coincide with the direction of the vector E. And in this case, no contradiction will arise, because the current will flow exactly in the direction in which the particles are moving :)

In order to estimate the current in a circuit, they came up with such a quantity as current strength. So, current strength (I) is a quantity that characterizes the speed of movement of an electric charge at a point. The unit of current is Ampere. The current strength in the conductor is equal to 1 Ampere, if for 1 second charge passes through the cross section of the conductor 1 pendant.

We have already covered the concepts current and voltage, now let's figure out how these quantities are related. And for this we have to study what it is conductor resistance.

Conductor/circuit resistance.

The term “ resistance” already speaks for itself 😉

So, resistance– physical quantity characterizing the properties of a conductor to hinder ( resist) the passage of electric current.

Consider a copper conductor of length l with a cross-sectional area equal to S:

Conductor resistance depends on several factors:

Specific resistance is a tabular value.

The formula with which you can calculate the resistance of a conductor is as follows:

For our case it will be equal 0.0175 (Ohm * sq. mm/m)– resistivity of copper. Let the length of the conductor be 0.5 m, and the cross-sectional area is equal to 0.2 sq. mm. Then:

As you already understood from the example, the unit of measurement is resistance is Ohm 😉

WITH conductor resistance everything is clear, it's time to study the relationship voltage, current and circuit resistance.

And here the fundamental law of all electronics comes to our aid - Ohm's law:

The current in a circuit is directly proportional to the voltage and inversely proportional to the resistance of the section of the circuit in question.

Let's consider the simplest electrical circuit:

As follows from Ohm's law, voltage and current in a circuit are related as follows:

Let the voltage be 10 V and the circuit resistance be 200 ohms. Then the current in the circuit is calculated as follows:

As you can see, everything is not difficult :)

Perhaps this is where we will finish today’s article, thank you for your attention and see you soon! 🙂

Elements of an electrical circuit can be connected in two ways. A series connection involves connecting elements to each other, and in a parallel connection, the elements are part of parallel branches. The way the resistors are connected determines the method for calculating the total resistance of the circuit.

Steps

Serial connection

Determine if the circuit is in series. A serial connection is a single circuit without any branches. Resistors or other elements are located one behind the other.

Add up the resistances of the individual elements. The resistance of a series circuit is equal to the sum of the resistances of all elements included in this circuit. The current strength in any part of the series circuit is the same, so the resistances simply add up.

For example, a series circuit consists of three resistors with resistances of 2 ohms, 5 ohms and 7 ohms. Total circuit resistance: 2 + 5 + 7 = 14 ohms.

If the resistance of each element of the circuit is not known, use Ohm's law: V = IR, where V is voltage, I is current, R is resistance. First find the current and total voltage.

Substitute the known values into the formula describing Ohm's law. Rewrite the formula V = IR to isolate the resistance: R = V/I. Plug the known values into this formula to calculate the total resistance.
- For example, the voltage of the current source is 12 V and the current is 8 A. The total resistance of the series circuit is: R O = 12 V / 8 A = 1.5 ohms.
Parallel connection
1. Determine whether the circuit is parallel. A parallel chain branches at some point into several branches, which are then connected again. Current flows through each branch of the circuit.
  
  Calculate the total resistance based on the resistance of each branch. Each resistor reduces the amount of current flowing through one leg, so it has little effect on the overall resistance of the circuit. Formula for calculating the total resistance: where R 1 is the resistance of the first branch, R 2 is the resistance of the second branch and so on until the last branch R n.
  - For example, a parallel circuit consists of three branches whose resistances are 10 ohms, 2 ohms and 1 ohm.
    Use the formula 1 R O = 1 10 + 1 2 + 1 1 (\displaystyle (\frac (1)(R_(O)))=(\frac (1)(10))+(\frac (1)(2))+ (\frac (1)(1))) to calculate R O
    Reduce fractions to a common denominator: 1 R O = 1 10 + 5 10 + 10 10 (\displaystyle (\frac (1)(R_(O)))=(\frac (1)(10))+(\frac (5)(10))+ (\frac (10)(10)))
    1 R O = 1 + 5 + 10 10 = 16 10 = 1 , 6 (\displaystyle (\frac (1)(R_(O)))=(\frac (1+5+10)(10))=(\ frac (16)(10))=1.6)
    Multiply both sides by R O: 1 = 1.6R O
    R O = 1 / 1.6 = 0,625 Ohm.
2. Calculate the resistance from the known current and voltage. Do this if the resistance of each circuit element is not known.
  
  Substitute the known values into the Ohm's law formula. If the total current and voltage in the circuit are known, the total resistance is calculated using Ohm's law: R = V/I.
  - For example, the voltage in a parallel circuit is 9 V and the total current is 3 A. Total resistance: R O = 9 V / 3 A = 3 ohms.
3. Look for branches with zero resistance. If a branch of a parallel circuit has no resistance at all, then all the current will flow through such a branch. In this case, the total resistance of the circuit is 0 ohms.
Combined connection
1. Divide the combination circuit into series and parallel. A combination circuit includes elements that are connected both in series and in parallel. Look at the circuit diagram and think about how to break it up into sections with elements connected in series and in parallel. Trace each section to make it easier to calculate the total resistance.
  - For example, a circuit includes a resistor whose resistance is 1 ohm and a resistor whose resistance is 1.5 ohms. Behind the second resistor, the circuit branches into two parallel branches - one branch includes a resistor with a resistance of 5 Ohms, and the second with a resistance of 3 Ohms. Trace two parallel branches to highlight them on the circuit diagram.
2. Find the resistance of the parallel circuit. To do this, use the formula to calculate the total resistance of a parallel circuit: 1 R O = 1 R 1 + 1 R 2 + 1 R 3 + . . . 1 R n (\displaystyle (\frac (1)(R_(O)))=(\frac (1)(R_(1)))+(\frac (1)(R_(2)))+(\ frac (1)(R_(3)))+...(\frac (1)(R_(n)))).
  - In our example, the parallel circuit includes two branches, the resistances of which are R 1 = 5 Ohms and R 2 = 3 Ohms.
    1 R p a r = 1 5 + 1 3 (\displaystyle (\frac (1)(R_(par)))=(\frac (1)(5))+(\frac (1)(3)))
    1 R p a r = 3 15 + 5 15 = 3 + 5 15 = 8 15 (\displaystyle (\frac (1)(R_(par)))=(\frac (3)(15))+(\frac (5 )(15))=(\frac (3+5)(15))=(\frac (8)(15)))
    R p a r = 15 8 = 1 , 875 (\displaystyle R_(par)=(\frac (15)(8))=1.875) Ohm.
3. Simplify the chain. Once you have found the total resistance of the parallel circuit, you can replace it with one element whose resistance is equal to the calculated value.
  - In our example, get rid of the two parallel legs and replace them with a single 1.875 ohm resistor.
4. Add up the resistances of resistors connected in series. By replacing the parallel circuit with one element, you get a series circuit. The total resistance of a series circuit is equal to the sum of the resistances of all elements that are included in this circuit.
  - After simplifying the circuit, it consists of three resistors with the following resistances: 1 ohm, 1.5 ohm and 1.875 ohm. All three resistors are connected in series: R O = 1 + 1, 5 + 1, 875 = 4, 375 (\displaystyle R_(O)=1+1.5+1.875=4.375) Ohm.

By assembling an electrical circuit consisting of a current source, a resistor, an ammeter, a voltmeter, and a switch, it can be shown that current strength (I ) flowing through the resistor is directly proportional to the voltage ( U ) at its ends: I-U . Voltage to current ratio U/I - there is a quantity constant.

Consequently, there is a physical quantity that characterizes the properties of the conductor (resistor) through which electric current flows. This quantity is called electrical resistance conductor, or simply resistance. Resistance is indicated by the letter R .

(R) is a physical quantity equal to the voltage ratio ( U ) at the ends of the conductor to the current strength ( I ) in him. R = U/I . Resistance unit – Ohm (1 ohm).

One Ohm- the resistance of a conductor in which the current is 1A with a voltage at its ends of 1V: 1 Ohm = 1 V / 1 A.

The reason that a conductor has resistance is that the directional movement of electrical charges in it prevented by ions of the crystal lattice making erratic movements. Accordingly, the speed of directional movement of charges decreases.

Electrical resistivity

R ) is directly proportional to the length of the conductor ( l ), inversely proportional to its cross-sectional area ( S ) and depends on the conductor material. This dependence is expressed by the formula: R = p*l/S

R - this is a quantity characterizing the material from which the conductor is made. It is called conductor resistivity, its value is equal to the resistance of a conductor of length 1m and cross-sectional area 1 m2.

The unit of conductor resistivity is: [p] = 1 0m 1 m 2 / 1 m. Often the cross-sectional area is measured in mm 2, therefore, in reference books the conductor resistivity values are given as in Ohm m so in Ohm mm2/m.

By changing the length of the conductor, and therefore its resistance, you can regulate the current in the circuit. The device with which this can be done is called rheostat.

In his work, an electrician often encounters the calculation of various quantities and transformations. So, to select the cable correctly, you have to select the required cross-section. The logic for choosing the cross-section is based on the dependence of the resistance on the length of the line and the cross-sectional area of the conductor. In this article we will look at how the resistance of a wire is calculated based on its geometric dimensions.

Formula for calculation

Any calculation starts with a formula. The basic formula for calculating conductor resistance is:

R=(ρ*l)/S

Where R is the resistance in Ohms, ρ is the resistivity, l is the length in m, S is the cross-sectional area of the wire in mm2.

This formula is suitable for calculating the resistance of a wire by cross-section and length. It follows from it that the resistance changes depending on the length; the longer, the greater. And on the contrary, depending on the cross-sectional area, the thicker the wire (large cross-section), the lower the resistance. However, the quantity designated by the letter ρ (Po) remains unclear.

Resistivity

Specific resistance is a tabular value; it is different for each metal. It is needed for calculations and depends on the crystal lattice of the metal and the structure of the atoms.

The table shows that silver has the lowest resistance; for a copper cable it is 0.017 Ohm*mm 2 /m. This dimension tells us how many ohms there are for a cross section of 1 square millimeter and a length of 1 meter.

By the way, silver coating is used in contacts of switching devices, circuit breakers, relays and other things. This reduces, increases service life and reduces. At the same time, gold-plated contacts are used in contacts of measuring and precision equipment due to the fact that they are slightly oxidized or do not oxidize at all.

Aluminum, which was often used in electrical wiring in the past, has a resistance 1.8 times greater than copper, equal to 2.82 * 10 -8 Ohm * mm 2 /m. The greater the resistance of the conductor, the more it heats up. Therefore, with the same cross-section, an aluminum cable can transmit less current than a copper cable, this has become the main reason why all modern electricians use. For nichrome, which is used in heating devices, it is 100 times greater than for copper 1.1 * 10 -6 Ohm * mm 2 /m.

Calculation by diameter

In practice, it often happens that the cross-sectional area of the core is unknown. Without this value, nothing can be calculated. To find out, you need to measure the diameter. If the wire is thin, you can take a nail or any other rod, wind 10 turns of wire around it, use a regular ruler to measure the length of the resulting spiral and divide by 10, this way you will find out the diameter.

Well, or just measure it with a caliper. The cross section is calculated using the formula:

Are calculations required?

As we have already said, the cross-section of the wire is selected based on the expected current and the resistance of the metal from which the wires are made. The logic of the choice is as follows: the cross-section is selected in such a way that the resistance at a given length does not lead to significant voltage drops. In order not to carry out a series of calculations, for short lines (up to 10-20 meters) there are quite accurate tables:

This table shows the typical cross-sectional values of copper and aluminum conductors and the rated currents through them. For convenience, the load power that this line will withstand is indicated. Please note the difference in currents and power at a voltage of 380V; naturally, this assumes a three-phase power supply.

Calculating wire resistance comes down to using a couple of formulas, and you can download ready-made calculators from the Play Market for your smartphone, for example, “Electrodroid” or “Mobile Electrician”. This knowledge will be useful for calculating heating devices, cable lines, fuses, and even today’s popular coils for electronic cigarettes.

Materials

Electrical resistance refers to any opposition that detects current passing through a closed circuit, weakening or inhibiting the free flow of electrical charges.

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Measuring resistance with a multimeter

Physical concept of resistance

Electrons, when current flows, circulate in a conductor in an organized manner according to the resistance they encounter along the way. The lower this resistance, the greater the existing order in the microworld of electrons. But when the resistance is high, they begin to collide with each other and release thermal energy. In this regard, the temperature of the conductor always increases slightly, by a greater amount, the higher the electrons find resistance to their movement.

Materials used

All known metals are more or less resistant to the passage of current, including the best conductors. Gold and silver have the least resistance, but they are expensive, so the most commonly used material is copper, which has high electrical conductivity. On a smaller scale, aluminum is used.

The greatest resistance to the passage of current is nichrome wire (an alloy of nickel (80%) and chromium (20%)). It is widely used in resistors.

Another commonly used resistor material is carbon. Fixed resistances and rheostats are made from it for use in electronic circuits. Fixed resistors and potentiometers are used to regulate current and voltage values, such as when controlling the volume and tone of audio amplifiers.

Resistance calculation

To calculate the value of the load resistance, the formula derived from Ohm’s law is used as the main one if the values of current and voltage are known:

The unit of measurement is Ohm.

For a series connection of resistors, the total resistance is found by summing the individual values:

R = R1 + R2 + R3 + …..

When connecting in parallel, the expression is used:

1/R = 1/R1 + 1/R2 + 1/R3 + …

How to find the electrical resistance for a wire, taking into account its parameters and material of manufacture? There is another resistance formula for this:

R = ρ x l/S, where:

l – wire length,
S – dimensions of its cross section,
ρ – specific volume resistance of the wire material.

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Resistance formula

The geometric dimensions of the wire can be measured. But to calculate the resistance using this formula, you need to know the coefficient ρ.

Important! Beat values volumetric resistance has already been calculated for different materials and summarized in special tables.

The value of the coefficient allows you to compare the resistance of different types of conductors at a given temperature in accordance with their physical properties without taking into account dimensions. This can be illustrated with examples.

An example of calculating the electrical resistance of a copper wire 500 m long:

If the cross-sectional dimensions of the wire are unknown, you can measure its diameter with a caliper. Let's say it's 1.6 mm;
When calculating the cross-sectional area, the formula is used:

Then S = 3.14 x (1.6/2)² = 2 mm²;

Using the table, we found the value of ρ for copper equal to 0.0172 Ohm x m/mm²;
Now the electrical resistance of the calculated conductor will be:

R = ρ x l/S = 0.0172 x 500/2 = 4.3 Ohm.

Another example– nichrome wire with a cross section of 0.1 mm², length 1 m:

The ρ indicator for nichrome is 1.1 Ohm x m/mm²;
R = ρ x l/S = 1.1 x 1/0.1 = 11 Ohm.

Two examples clearly show that nichrome wire, a meter long and with a cross-section 20 times smaller, has an electrical resistance 2.5 times greater than 500 meters of copper wire.

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Resistivity of some metals

Important! Resistance is influenced by temperature, with increasing temperature it increases and, conversely, decreases with decreasing temperature.

Impedance

Impedance is a more general term of resistance that takes into account reactive loads. Calculating resistance in an AC circuit involves calculating impedance.

While a resistor provides active resistance to perform certain tasks, reactive component is an unfortunate by-product of some circuit components.

Two types of reactance:

Inductive. Created by coils. Calculation formula:

X (L) = 2π x f x L, where:

f – current frequency (Hz),
L – inductance (H);

Capacitive. Created by capacitors. Calculated using the formula:

X (C) = 1/(2π x f x C),

where C is capacity (F).

Like its active counterpart, reactance is expressed in ohms and also limits the flow of current through the circuit. If there is both a capacitance and an inductor in the circuit, then the total resistance is equal to:

X = X (L) – X (C).

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Active, inductive and capacitive reactance

Important! Interesting features follow from the reactive load formulas. As the frequency of alternating current and inductance increases, X(L) increases. And, conversely, the higher the frequencies and capacitance, the smaller X (C).

Finding impedance (Z) is not a simple addition of active and reactive components:

Z = √ (R² + X²).

Example 1

The coil in the circuit with industrial frequency current has an active resistance of 25 Ohms and an inductance of 0.7 H. You can calculate impedance:

X (L) = 2π x f x L = 2 x 3.14 x 50 x 0.7 = 218.45 Ohm;
Z = √ (R² + X (L)²) = √ (25² + 218.45²) = 219.9 ohms.

tan φ = X (L)/R = 218.45/25 = 8.7.

The angle φ is approximately 83 degrees.

Example 2

There is a capacitor with a capacity of 100 μF and an internal resistance of 12 ohms. You can calculate impedance:

X (C) = 1/(2π x f x C) = 1/ 2 x 3.14 x 50 x 0.0001 = 31.8 Ohm;
Z = √ (R² + X (C)²) = √ (12² + 31.8²) = 34 Ohm.

On the Internet you can find an online calculator to simplify the calculation of the resistance and impedance of the entire electrical circuit or its sections. There you just need to enter your calculation data and record the calculation results.