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# Alternating current circuits: types, applications, examples

The circuit AC or AC circuits consist of combinations of resistive elements, inductive and capacitive, combined with a source of alternating voltage, which is usually sinusoidal.

By applying the voltage, a variable current is established for a short time, called a transient current, which gives way to the sinusoidal standing current.

The sinusoidal current has values ​​that alternate between positive and negative, changing at regular intervals determined by a previously established frequency. The shape of the current is expressed as:

I (t) = I m sin (ωt − φ)

Where I m is the maximum current or current amplitude, ω is the frequency, t is the time and φ the phase difference. Commonly used units for current are the ampere (A) and its submultiples, such as the milliamp and microampere.

For its part, time is measured in seconds, for frequency there are hertz or Hertz, abbreviated Hz, while the phase difference is an angle that is generally measured in radians, although it is also sometimes given in degrees. Neither these nor the radians are considered units.

Often the alternating voltage is symbolized by the wave inside the circle, to differentiate it from the direct voltage, symbolized by the two unequal and parallel lines.

## Types of alternating current circuits

There are many kinds of alternating current circuits, starting with the simplest circuits shown in the following figure. From left to right you have:

-Circuit with resistance R

-Circuit with coil L

-Circuit with capacitor C.

### Circuit with resistive element

In the circuit with a resistor R connected to an alternating voltage source, the voltage across the resistor is V R = V m sin ωt. By Ohm’s law, which is also valid for purely resistive alternating current circuits:

R = I R ∙ R

Therefore the maximum current I m = V m / R.

Both current and voltage are in phase, which means that they reach their maximum values, as well as 0, at the same time.

### Inductive element circuit

In the coil L, the voltage is V L = V m sin ωt and is related to the current in the inductor by the equation:

Integrating:

By properties of trigonometric ratios, I L is written in terms of sin ωt as:

L = I m sin (ωt – ½ π)

Then, the voltage and the current are out of phase, the latter lagging ½ π = 90º with respect to the voltage (the current starts earlier, with t = 0 s the starting point). This is seen in the following figure that compares the sinusoid of I L and that of V L :

#### Inductive reactance

Inductive reactance is defined as X L = ωL, it increases with frequency and has dimensions of resistance, therefore, in analogy with Ohm’s law:

L = I L ∙ X L

### Circuit with capacitive element

For a capacitor C connected to an alternating current source, it is true that:

Q = C ∙ V C = C ∙ V m sin ωt

The current in the capacitor is found by shifting the charge with respect to time:

C = ωC ∙ V m cos ωt

But cos ωt = sin (ωt + ½ π), then:

C = ωCV m sin (ωt + ½ π)

In this case, the current leads the voltage by ½ π, as can be seen from the graph.

#### Capacitive reactance

Capacitive reactance can be written X C = 1 / ωC, it decreases with frequency and also has units of resistance, that is, ohms. In this way, Ohm’s law looks like this:

C = X C .I C

## Applications

Michael Faraday (1791-1867) was the first to obtain a current that periodically changed its meaning, through his induction experiments, although during the early days, only direct current was used.

At the end of the 19th century the well-known war of the currents occurred, between Thomas A. Edison, defender of the use of direct current and George Westinghouse, supporter of alternating current. Finally, this was the one that won due to economy, efficiency and ease of transmission with lower losses.

For this reason, to date, the current that reaches homes and industries is alternating current, although the use of direct current has never completely disappeared.

Alternating current is used for almost everything, and in many applications, the constant change of direction of alternating current is not relevant, such as light bulbs, the iron or the stove for cooking, since the heating of the resistive element it does not depend on the direction of movement of the charges.

On the other hand, the fact that the current changes direction with a certain frequency is the foundation of electric motors and various more specific applications, such as the following:

### Phase shifting circuits

Circuits consisting of an alternating source connected to a resistor and a capacitor in series are known as RC series circuits and are used to eliminate unwanted phase shifts in another circuit, or to add some special effect to it.

They also serve as voltage dividers and to tune in to radio stations (see example 1 in the next section).

### Bridge circuits

Bridge-type circuits fed with alternating current can be used to measure capacitance or inductance, in the same way that the Wheatstone bridge is used, a known direct current circuit capable of measuring the value of an unknown resistance.

## Examples of alternating current circuits

In the previous sections the simplest alternating current circuits were described, although of course, the basic elements described above, as well as others a little more complex such as diodes, amplifiers and transistors, to name a few, can be combined to obtain different effects.

### Example 1: Series RLC circuit

One of the most common ac circuits is one that includes a resistor R, a coil or inductor L, and a capacitor or capacitor C placed in series with an alternating current source.

Series RLC circuits respond in a particular way to the frequency of the alternating source from which they are fed. That is why one of the most interesting applications is as radio tuner circuits.

A radio signal with frequency ω generates a current with that same frequency in a circuit specially designed to serve as a receiver, and the amplitude of this current is maximum if the receiver is tuned to that frequency, through an effect called resonance.

The receiver circuit serves as a tuner because it is designed so that signals of unwanted frequencies generate very small currents, which are not detected by the radio speakers and therefore are not audible. On the other hand, at the resonant frequency, the amplitude of the current reaches a maximum and then the signal is heard clearly.

The resonant frequency occurs when the inductive and capacitive reactances of the circuit equalize:

L = X C

1 / ωC = ωL

ω 2 = 1 / LC

The radio station with the frequency signal ω is said to be “tuned in”, and the values ​​of L and C are chosen for that particular frequency.

### Example 2: RLC circuit in parallel

Parallel RLC circuits also have certain responses according to the frequency of the source, which depends on the reactance of each of the elements, defined as the ratio between voltage and current.

## Exercise resolved

In the series LRC circuit of Example 1 in the preceding section, the resistance is 200 ohms, the inductance is 0.4 H, and the capacitor is 6 μF. For its part, the power supply is an alternating voltage of amplitude equal to 30 V, with a frequency of 250 rad / s. It is asked to find:

a) The reactances of each element

b) The value of the module of the impedance of the circuit.

c) The amplitude of the current

### Solution to

The respective reactances are calculated with the formulas:

C = 1 / ωC = 1 / (250 rad / sx 6 x10 -6 F) = 666.67 ohm

L = ωL = 250 rad / sx 0.4 H = 100 ohm

And the reactance of the resistance is equal to its value in ohms:

R = R = 200 ohm

### Solution b

Impedance Z is defined as the ratio of voltage to current in the circuit, either in series or in parallel:

Z = V m / I m

The impedance is measured in ohms, the same as a resistance or a reactance, but it refers to the opposition to the passage of current of the inductances and capacitors, considering that in addition to its particular effects, such as delaying or advancing the voltage, it also they have a certain internal resistance.

It can be shown that for the series RLC circuit, the impedance modulus is given by:

When evaluating the values ​​given in the statement , we obtain:

### Solution c

From:

Z = V m / I m

It has to;

m = V m / Z = 30V / 601 ohms = 0.05 A.

## Themes of interest

Differences between alternating and direct current

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