Study Material

Mixed electrical circuit

We explain what a Mixed electrical circuit is, its characteristics, parts, symbols and we give several examples

The mixed electrical circuit is one that contains elements connected both in series and in parallel, so that, when closing the circuit, different voltages and currents are established in each one of them. Mixed electrical circuit

Circuits are designed with a wide variety of objectives and their elements fall into two categories: active and passive.

The active elements of the circuit are the generators or sources of voltage or current, direct or alternating. On the other hand, the passive elements are the resistors, the capacitors or capacitors and the coils. Both of them admit serial and parallel connections, as well as combinations of these. Mixed electrical circuit

Mixed electrical circuit

Other possible connections, different from series-parallel associations, are delta (or triangle) and star, frequently used in electrical machines powered by alternating current.

In general terms, in a mixed circuit the following is observed:

  • The power supply of the circuit can be through a direct generator (battery) or alternate.
  • It is considered that the cables or wires that join the different elements do not offer resistance to current.
  • Both voltage and current can be constant or variable over time. Uppercase letters are used to denote constant values, and lowercase letters when they are variables.
  • In purely resistive mixed circuits, the current through series resistors is the same, while in parallel resistors it is generally different. To calculate the current and voltage across each resistor, the circuit is usually reduced to a single resistor, called an equivalent resistor, or R eq . Mixed electrical circuit

Series resistors

Mixed electrical circuit: Series resistors

Resistors in parallel

Mixed electrical circuit: Resistors in parallel
  • If the circuit consists of n capacitors, when they are associated in series the equivalent capacitance C eq results:

Series Capacitors Mixed electrical circuit

Series Capacitors

Capacitors in parallel

Capacitors in parallel
  • Coils or inductors follow the same association rules as resistors. Thus, when we want to reduce an association of coils in series to obtain the equivalent inductance L eq , the following formulas are used: Mixed electrical circuit

Inductors in series 

Inductors in series
Inductors in series
  • To solve mixed circuits with resistors, Ohm’s law and Kirchoff’s laws are used. In simple resistive circuits, Ohm’s law is enough, but for more complex networks it is necessary to apply Kirchoff’s laws in combination with Ohm’s law, in addition to the relationship between voltage and current for capacitors and coils, if these elements are also found present.

Relationship between voltage and current

Depending on the circuit element, there is a relationship between the voltage or voltage across the element with the intensity of the current that passes through it: Mixed electrical circuit

Resistance R

Ohm’s law is used:

R (t) = R ∙ i R (t)

Capacitor C

Capacitor C

Inductance L

Inductance L

Parts of a mixed circuit 

In an electrical circuit the following parts are distinguished:

Knot Mixed electrical circuit

Junction point between two or more conductive wires that connect some active or passive element of the circuit.

Branch

Elements, whether active or passive, that are between two consecutive nodes. Mixed electrical circuit

Mesh Mixed electrical circuit

Closed portion of the circuit traveled without going through the same point twice. It may or may not have a voltage or current generator. Mixed electrical circuit

Kirchoff’s laws or rules

Kirchoff’s rules apply whether the currents and voltages are constant or they are time dependent. Although they are often called laws, they are actually rules for applying conservation principles to electrical circuits.

First rule

It establishes the principle of conservation of charge, by pointing out that the sum of the currents entering a node is equivalent to the sum of the currents leaving it:

∑ I input = ∑ I output

Second rule Mixed electrical circuit

On this occasion, the principle of conservation of energy is established, when it states that the algebraic sum of the voltages in a closed portion of the circuit (mesh) is zero.

Symbols

To facilitate circuit analysis, the following symbols are used:

Mixed electrical circuit: Symbols

Examples of mixed circuits

Example 1 Mixed electrical circuit

Draw the mixed circuit of the figure at the beginning in compact form, using the symbols described above.

Answer Mixed electrical circuit

Mixed electrical circuit: Examples of mixed circuits

Example 2 Mixed electrical circuit

In the circuit of example 1, we have the following values ​​for the resistors and the battery: Mixed electrical circuit

1 = 50 Ω; R 2 = 100 Ω; R 3 = 75 Ω, R 4 = 24 Ω, R 5 = 48 Ω; R 6 = 48 Ω; ε = 100 V

For the circuit shown, the battery is considered ideal, that is, it has no internal resistance. Usually real batteries have a small internal resistance that is drawn in series with the battery and is treated the same as the other resistors in the circuit.

Calculate the following: Mixed electrical circuit

  • a) The equivalent resistance of the circuit.
  • b) The value of the current coming out of the battery.
  • c) The voltages and currents in each of the resistors.

Answer to

The first group of resistances: R 1 = 50 Ω; R 2 = 100 Ω; R 3 = 75 Ω are connected in series, therefore the equivalent resistance is R 123 :

123 = R 1 + R 2 + R 3 = 50 Ω + 100 Ω + 75 Ω = 225 Ω

Regarding the group of resistances R 4 = 24 Ω, R 5 = 48 Ω; R 6 = 48 Ω, they are connected in parallel and the corresponding formula must be applied: Mixed electrical circuit

Examples of mixed circuits

456 = 12 Ω

The simplified circuit that is obtained is shown in the following graph , consisting of two resistors in series with the cell or battery. These two resistors are added to find the equivalent resistance of the original circuit R eq :

Mixed electrical circuit: Examples of mixed circuits

eq = 225 Ω + 12 Ω = 237 Ω

Answer b Mixed electrical circuit

The current that leaves the battery (by convention it is always drawn leaving the positive pole) is calculated with the simplified circuit, which consists of the equivalent resistance R eq in series with the battery, to which Ohm’s law is applied:

ε = IR

I = ε / R = 100 V / 237 Ω = 0.422 A

Answer c

The voltages and currents in each of the resistors sa are calculated using Ohm’s law. The first thing that is observed is that the current that leaves the battery completely passes through the resistors R 1 , R 2 and R 3 and instead, it divides when it crosses R 4  , R 5 and R 6 .

The voltages V 1 , V 2 and V 3 are:

 = 0.422 A × 50 Ω = 21.1 V
 = 0.422 A × 100 Ω = 42.2 V
 = 0.422 A × 75 Ω = 31.7 V

On the other hand, the voltages V 4 , V 5 and V 6 have the same value, since the resistors are in parallel:

4 = V 5 = V 6 = 0.422 A × 12 Ω = 5.06 V

And the respective currents are:

4 = 5.06 V / 24 Ω = 0.211 A

5 = I 6 = 5.06 V / 48 Ω = 0.105 A

Note that adding I 4 , I 5 and I 6 , the total current coming out of the battery is obtained again.

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