All electronic devices contain resistors,which are their main element. With its help, change the value of the current in the electrical circuit. The article describes the properties of resistors and methods for calculating their power.

## Resistor assignment

To adjust the current in the electrical circuits used resistors. This property is defined by Ohm's law:

I = U / R (1)

It is clearly seen from formula (1) that the smallerthe higher the current, and vice versa, the smaller the value of R, the greater the current. It is this property of electrical resistance that is used in electrical engineering. Based on this formula, current divider circuits are widely used in electrical devices.

In this circuit, the current from the source is divided into two, inversely proportional to the resistances of the resistors.

In addition to current regulation, resistors are used in voltage dividers. In this case, Ohm's law is again used, but in a slightly different form:

U = I ∙ R (2)

It follows from formula (2) that as the resistance increases, the voltage increases. This property is used to construct voltage divider circuits.

From the circuit and formula (2) it is clear that the resistor voltages are distributed in proportion to the resistances.

## The image of resistors on the circuits

According to the standard, resistors are representedrectangle with dimensions of 10 x 4 mm and are denoted by the letter R. Often the power of the resistors on the circuit is indicated. The image of this indicator is executed by oblique or straight lines. If the power is more than 2 watts, the designation is made in Roman numerals. This is usually done for wire resistors. In some states, for example in the USA, other conventions are used. To simplify the repair and analysis of the circuit, the power of the resistors is often given, the designation of which is fulfilled in accordance with GOST 2.728-74.

## Technical characteristics of devices

The main characteristic of the resistor is the nominal resistance R_{Mr.}, which is indicated on the diagram near the resistor andon his body. The unit of resistance is ohm, kilo and mega. Resistors with resistances from ohms to hundreds of megaohms are produced. There are many technologies for manufacturing resistors, they all have advantages and disadvantages. In principle, there is no technology that would make it possible to produce an exact resistor with a specified resistance value.

The second important characteristic is deviationresistance. It is measured in% of the nominal R. There is a standard deviation range of resistance: ± 20, ± 10, ± 5, ± 2, ± 1% and further down to ± 0.001%.

The next important characteristic is thepower resistors. During operation, they are heated by the current flowing through them. If the power dissipation exceeds the permissible value, the device will fail.

When heated, the resistors changeTherefore, for devices operating over a wide range of temperatures, one more characteristic is introduced: the temperature coefficient of resistance. It is measured in ppm / ° C, i.e. 10^{-6 }R_{Mr.}/ ° C (millionth part of R_{Mr.} at 1 ° C).

## Serial connection of resistors

Resistors can be connected in three different ways: sequential, parallel and mixed. With a series connection, the current alternately passes through all the resistances.

With such a connection, the current at any point in the chain is the same, it can be determined by Ohm's law. The total resistance of the circuit in this case is equal to the sum of the resistances:

R = 200 + 100 + 51 + 39 = 390 Ohm;

I = U / R = 100/390 = 0.256 A.

Now it is possible to determine the power at a series connection of resistors, it is calculated by the formula:

P = I^{2}∙ R = 0.256^{2}∙ 390 = 25.55 W.

Similarly, the power of the remaining resistors is determined:

P_{1}= AND^{2}∙ R_{1}= 0.256^{2}∙ 200 = 13.11 W;

P_{2}= AND^{2}∙ R_{2}= 0.256^{2}∙ 100 = 6.55 W;

P_{3}= AND^{2}∙ R_{3}= 0.256^{2}∙ 51 = 3.34 W;

P_{4}= AND^{2}∙ R_{4}= 0.256^{2}∙ 39 = 2.55 W.

If you combine the power of the resistors, you get a complete P:

P = 13.11 + 6.55 + 3.34 + 2.55 = 25.55 W.

## Parallel connection of resistors

With a parallel connection, all the beginnings of resistorsconnect to one node of the circuit, and the ends to the other. With this connection, the current branches and flows through each device. The magnitude of the current, according to Ohm's law, is inversely proportional to the resistance, and the voltage across all resistors is the same.

Before finding the current, it is necessary to calculate the total conductivity of all the resistors according to the well-known formula:

1 / R = 1 / R_{1}+ 1 / R_{2}+ 1 / R_{3}+ 1 / R_{4}= 1/200 + 1/100 + 1/51 + 1/39 = 0.005 + 0.01 + 0.0196 + 0.0256 = 0.06024 1 / Ohm.

Resistance is the reciprocal of the conductivity:

R = 1 / 0.06024 = 16.6 Ohm.

Using Ohm's law, one finds a current through a source:

I = U / R = 100 ∙ 0.06024 = 6,024 A.

Knowing the current through the source, find the power of the parallel connected resistors according to the formula:

P = I^{2}∙R = 6,024^{2}∙ 16.6 = 602.3 W.

According to Ohm's law, the current is calculated through the resistors:

AND_{1}= U / R_{1}= 100/200 = 0.5 A;

AND_{2}= U / R_{2}= 100/100 = 1 A;

AND_{3}= U / R_{1}= 100/51 = 1.96 A;

AND_{1}= U / R_{1}= 100/39 = 2.56 A.

A little by another formula you can calculate the power of the resistors in parallel connection:

P_{1}= Y^{2}/R_{1}= 100^{2}/ 200 = 50 W;

P_{2}= Y^{2}/R_{2}= 100^{2}/ 100 = 100 W;

P_{3}= Y^{2}/R_{3}= 100^{2}/ 51 = 195.9 W;

P_{4}= Y^{2}/R_{4}= 100^{2}/ 39 = 256.4 W.

If you put it all together, you get the power of all the resistors:

P = P_{1}+ P_{2}+ P_{3}+ P_{4}= 50 + 100 + 195.9 + 256.4 = 602.3 W.

## Mixed compound

Schemes with a mixed connection of resistorscontain a serial and simultaneously parallel connection. This circuit is easy to convert, replacing the parallel connection of the resistors in series. To do this, replace the resistance R_{2} and R_{6} on their common R_{2.6}, using the formula given below:

R_{2.6}= P_{2}∙ R_{6}/R_{2}+ P_{6.}

Similarly, two parallel resistors R_{4}, R_{5 }one R_{4,5:}

R_{4,5}= P_{4}∙ R_{5}/R_{4}+ P_{5}.

The result is a new, simpler scheme. Both schemes are given below.

The power of resistors on a circuit with a mixed connection is determined by the formula:

P = U ∙ I.

To calculate this formula, first findvoltage at each resistance and the magnitude of the current through it. You can use another method to determine the power of the resistors. To do this, use the formula:

P = U ∙ I = (I ∙ R) ∙ I = I^{2}∙ R.

If only the voltage across the resistors is known, then another formula is used:

P = U ∙ I = U ∙ (U / R) = U^{2}/ R.

All three formulas are often used in practice.

## Calculation of the circuit parameters

Calculation of the parameters of the scheme consists in findingunknown currents and voltages of all branches on the sections of the electrical circuit. With this data, you can calculate the power of each resistor included in the circuit. Simple methods of calculation have been shown above, in practice the situation is more complicated.

In real schemes, a compound is often encounteredresistors with a star and a triangle, which creates significant difficulties in the calculations. To simplify such schemes, methods for converting a star into a triangle have been developed, and vice versa. This method is illustrated in the diagram below:

The first scheme has a star in its composition,connected to nodes 0-1-3. Resistor R1 is connected to node 1, to node 3 - R3, and to node 0 - R5. In the second scheme, the resistors of the triangle are connected to the nodes 1-3-0. To node 1, resistors R1-0 and R1-3 are connected, to node 3 - R1-3 and R3-0, and to node 0 - R3-0 and R1-0. These two schemes are completely equivalent.

For the transition from the first scheme to the second one, the resistance of the resistors of the triangle is calculated:

R1-0 = R1 + R5 + R1 = R5 / R3;

R1-3 = R1 + R3 + R1 ∙ R3 / R5;

R3-0 = R3 + R5 + R3 ∙ R5 / R1.

Further transformations are reduced to computingparallel and series-connected resistances. When the impedance of the circuit is found, the Ohm's law finds the current through the source. Using this law, it is easy to find currents in all branches.

How to determine the power of the resistors after finding all the currents? To do this, use the well-known formula: P = I^{2}∙ R, applying it for each resistance, we find their power.

## Experimental determination of characteristics of circuit elements

For the experimental determination of the requiredcharacteristics of the elements it is required to assemble a given scheme from real components. After this, all the necessary measurements are performed with the help of electrical measuring instruments. This method is time consuming and expensive. Developers of electrical and electronic devices for this purpose use modeling programs. With the help of them, all necessary calculations are made, and the behavior of the elements of the circuit is modeled in different situations. Only after this is the prototype of a technical device assembled. One of these popular programs is the powerful Multisim 14.0 simulation system from National Instruments.

How to determine the power of resistors using thisprograms? This can be done in two ways. The first method is to measure the current and voltage using an ammeter and a voltmeter. Multiplying the measurement results, we obtain the required power.

From this scheme we determine the power of resistance R3:

P_{3}= U ∙ I = 1,032 ∙ 0,02 = 0,02064 W = 20,6 mW.

The second method is direct measurement of power with a wattmeter.

** **

From this diagram it is seen that the power of the resistance R3 is P_{3}= 20.8 mW. The discrepancy due to the error in the first method is greater. Similarly, the powers of the remaining elements are determined.

## comments