How to calculate resistance: parallel, series and combined circuit

Solving problems in the field of electronics and electrical, you have to deal with various calculations. Most often they are associated with the simplification of electrical circuits. For this, the equivalent method is used, when a part of the circuit is replaced by one element with characteristics similar to it. But in order to do this, you need to know how to calculate the resistance of a circuit section and what types of connections are.

Determination of the value

Current is the ordered movement of charge carriers under the influence of an electric field. The ability of a substance to conduct current is called electrical conductivity. The more carriers of particles a material has, the more conductivity it has. Depending on this characteristic, all substances are divided into three types:

1. Conductors. They are characterized by good electrical conductivity. These include metals and their alloys, as well as electrolytes.
2. Dielectrics. Substances that are practically non-conductive. These are mainly gases, rubber, mineral oils, plastics.
3. Semiconductors. Materials with two types of conductivity simultaneously - hole and electronic. These are substances with a covalent bond: silicon, germanium, selenium.

The reciprocal of electrical conductivity is called electrical resistance. That is, it is a physical quantity that prevents the passage of current. In addition to the ability of any material to limit the number of charges passing through it, there is a special radio element that limits the current strength - a resistor.

Thus, there are two concepts of resistance: a radioelement and a physical quantity.

Radio element resistance

The term "resistor" comes from the Latin word resisto - "resistance". All resistors are divided into fixed and variable. The latter allow you to change your resistance. On the diagrams and in the literature, such a radio component is signed with the Latin letter R. The unit of measurement is Ohm. Graphically, the resistor is designated as a rectangle with two leads from the middle of the edges. In addition to the nominal resistance, it is characterized by power dissipation and accuracy class.

In essence, it is a passive radioelement that converts part of electrical energy into heat. Thus, it limits the current, linearly converting its strength into voltage and vice versa. The main parameter describing resistance, is found according to Ohm's law for a section of the circuit according to the following formula: R = U / I, where:

• R - electrical resistance, Ohm.
• U is the potential difference applied to the element, V.
• I is the current passing through the resistor, A.

But here it should be noted that this law is valid only for resistive circuits. That is, for those in the calculation of which capacitance and inductance are neglected. If this formula is applied to reactive elements, then for an inductor the resistance will be zero, and for a capacitor it will be infinite. But this is true for DC current and voltage.

With variable values, the voltage across the inductor will not be zero, nor will the current flowing through the capacitor. Such cases are no longer described by resistance, since it assumes constant values ​​of current and voltage.

Specific parameter of the substance

To distinguish between a concept and an element, the name resistivity was introduced. It is designated by the Greek symbol ρ. In the International System of Units, this value is measured in Ohms multiplied by a meter. It depends solely on the properties of the material.

To calculate the electrical resistance of a homogeneous substance, the formula is used: R = ρ * l / S, where:

• l - conductor length, m;
• S - cross-sectional area, m².

Therefore, in the physical sense, the resistivity of a material is the reciprocal of the specific conductivity, which is the resistance of a uniform conductor of unit length and area cross section. This means that it is numerically equal to the impedance of a section of an electrical circuit made of a substance one meter long and a cross-sectional area of ​​one square meter.

For each substance, the resistivity is known and is a reference value. For example, for copper - 0.01724 Ohm * mm²/ m, aluminum - 0.0262 Ohm * mm²/ m, bismuth - 1.2 Ohm * mm²/ m, nichrome - 1.05 Ohm * mm²/м. These data were obtained at a temperature of t = 20 ° C, since materials have the property of changing their specific characteristics with changes in temperature. So, the conductivity of metals increases with decreasing temperature, and semiconductors - decreases.

Equivalent circuit

When calculating the resistance of electrical circuits, the concept of "equivalent equivalent circuit" is widely used. Its purpose is to simplify a complex circuit to a form consisting of a minimum of elements. In other words, each complex radio element can be represented in the form of equivalent simple radio components corresponding to it: resistor, capacitance, inductance, current and voltage sources. This allows not only to describe mathematically any circuit, but also to calculate its parameters.

In this case, radioelements are usually idealized, that is, their parasitic parameters are not taken into account. Similarly, for calculating the resistance of a circuit, each component is represented as an ideal resistor. After that, the circuit is redrawn, and as a result, only resistors connected in different ways to each other remain on it.

There are two types of connection:

• consistent;
• parallel.

The main elements of an electrical circuit are a node, a branch and a circuit. A node is where two or more branches meet. A branch is a sequential section of a circuit between two nodes, and a loop is any closed circuit. A series connection consists of elements in which all the components of the chain are connected so that the section of the chain formed from them does not have a single knot. And with a parallel connection, all components of the electrical circuit are in contact with each other in two nodes. However, these nodes are not directly connected.

Impedance calculation

The methods for calculating the total resistance depend on how the resistors are connected. When calculating the total impedance, Kirchhoff's laws are taken as a basis.

So, his first law says: the sum of currents in a node is equal to zero. Or, to paraphrase it, the value of the current flowing into a node is equal to the sum of the currents flowing out of that node. The second law is associated with electromotive force, and its formulation sounds like this: the sum of the potential difference in the circuit is equal to the sum of the potential difference drops across each resistor in the circuit.

With a serial connection, all elements are located one after another without branches. Since, according to the Kirchhoff rule, the current strength is the same at any place of the branch I = I1 = In, then the voltage drop across the first element is: U1 = I * R1, and on n: Un = I * Rn, where:

• In is the current flowing through the resistor, A.
• Un is the value of the voltage drop across the resistor, V.
• Rn is the resistance value of the element, Ohm.

The total potential difference is equal to the sum of all voltages, so you can write: U = U1 +… + Un = I * (R1 +… + Rn) = IRo.

As a result, the formula for calculating the resistance of the circuit in this case will look like this:

Ro = R1 +… + Rn, where:

• Ro is the total resistance of the branch.
• R1 is the impedance value of the first element.
• Rn is the resistance value of the n-th element.

If the chain is parallel, then this means that in this section several branches diverge, and then they are connected again. It turns out that the current strength in each branch will be different, and the voltage value is the same. Therefore, Uo = U1 =… = Un, and Io = I1 +… + In. Using Ohm's law, you can write:

Uo / Ro = U1 / R1 +… + Un / Rn, or

1 / Ro = 1 / R1 +… 1 / Rn.

As a result, the equivalent resistance in parallel connection is calculated as the product of the resistor values ​​divided by the sum of their products. For two resistors, the formula for finding the total resistance can be written as: Ro = (R1 * R2) / (R1 + R2).

Browser online calculator

If there are few elements in the circuit, then, simplifying the circuit, it is quite easy to calculate, using the formulas for parallel and series connection of resistors, the total impedance of the circuit. But if there are a lot of elements in the diagram, moreover, it is such that it contains both connections (combined), it is easier to use browser-based online calculators.

They are based on the same formulas for calculating the equivalent resistor, but all calculations are performed automatically. There are a huge number of offers of such calculators. But they all work the same way. Online calculation is a program code that contains a calculation algorithm. The consumer only needs to indicate in special cells what type of connection is used, how many elements are in the circuit and the resistances of the resistors. Then you need to press the "Calculate" button and get an answer in a matter of seconds.

It should be noted that, even if this is not indicated in the program, all values ​​are entered only in the International System of Units, current strength - ampere, voltage - volt, resistance - ohm. Then the answer will be in Omah.

A bonus is that many of these programs immediately calculate the power of the element. For this, the formula is used: P = U²/ Ro = I²* Ro, W.

Practical use

Most often, in practice, the calculation of the total resistance of the circuit is performed in order to find out the power consumption of a particular circuit. At the same time, knowing the total resistance, you can find such important circuit parameters as current and voltage. Therefore, an equivalent circuit is drawn. Simple circuits consist only of serial or parallel sections, but combined connections are more common.

Before proceeding with the calculation of the equivalent resistance, the entire electrical circuit is divided into simple circuits. As soon as the impedance of each such circuit is calculated, the circuit is redrawn, but instead of the circuits, a resistor is drawn. Then everything is repeated, and this happens until one element remains.

Simple connection

Let a circuit be given, consisting of three resistors connected in series. In this case, the resistance of R1 and R2 is the same and equal to 57 ohms, and the resistance of R3 is one kiloohm. To calculate the total resistance of the circuit, you first need to bring the value of R3 according to the International System of Units.

R3 = 1kΩ = 1000Ω.

Since the connection is serial, the formula is used: Ro = R1 + R2 + R3. Substituting the known values, the equivalent value is calculated: Ro = 57 + 57 + 1000 = 1114 ohms.

If the same resistors are located parallel to each other, then a different expression is already used to calculate the total resistance:

1 / Ro = 1 / R1 + 1 / R2 + 1 / R3.

Ro = R1 * R2 * R3 / (R1 * R2 + R2 * R3 + R1 * R3).

Substituting the initial data into this formula, we get:

Ro = 57 * 57 * 1000 / (57 * 57 + 57 * 1000 + 57 * 1000) = 3249000/117249 = 27.7 Ohm.

Combined circuit

It is necessary to calculate the power and the equivalent resistance of a mixed circuit consisting of four resistors. Resistor R1 = R2 = 5 ohms, R3 = 10 ohms, R4 = 3 ohms. The circuit is powered by five volts.

Initially, you will need to simplify the circuit. Resistors R3 and R4 are connected in parallel with respect to each other. Therefore, their united resistance is found:

Rp = (R3 * R4) / (R3 + R4).

Rp = (10 * 3) / (10 + 3) = 2.3 ohms.

Now the circuit can be redrawn in the form of three series-connected resistors and find the total resistance by adding their values:

Ro = R1 + R2 + Rp = 5 + 5 + 2.3 = 12.3 Ohm.

Knowing the equivalent resistance, using Ohm's law, it is easy to calculate the current in the circuit and the power of the equivalent resistor:

I = U / R = 5 / 2.3 = 2.2 A.

P = I * U = 2.2 * 5 = 11 W.

Thus, by gradually simplifying the circuit, it is possible to reduce a circuit of series and parallel connected resistors to one element. And then calculate its resistance and the required power.