## Wednesday, September 8, 2021

### SIMPLE A.C CIRCUIT

THE  A.C  CIRCUIT

EFFECT OF RESISTANCE

Let us see what happens when an alternating current is passed through a resistance. The circuit will be as shown below, in which the symbol  represents the source of current, say an a.c. generator.

To find the current at any moment, we divide the voltage by the resistance in accordance with ohm’s law.

As the resistance remains the same, the current is proportional to the voltage, and as the latter is continually changing in strength and direction, the current changes in a similar manner.

The e.m.f and current curves, therefore, rise and fall together, as shown in the diagram. It does not matter which curve rises to the greater height; this depends upon the vertical scales when choose to employ for volts and amperes. What does matter is that both curves are of the same frequency, and that when one of them reaches a maximum or minimum, the other does the same. When the e.m.f. and current reach their maximum and minimum values at the same time simple means in step with each other.

The power (watts) in the circuit  at any moment is found as usual by multiplying volts by amperes. If we do this at different points along the curves and plot the results, we shall obtain a curve such as that shown at on the diagram. This curve relates to the same period of time as the curves on the left, but is shown separately to avoid confusion.

It is clear that the power reaches a maximum at the same time as the voltage and current. Note, however, that the power curve lies wholly above the zero lines; we can explain this mathematically by saying that whether volts and amperes are both negative and both positive, their product is positive. We had a similar case in the last article when we were finding the effective value of an alternating current.

In the diagram total energy (power*time) expended during the cycle in represented by the shaded area, and as this is half the total area under the horizontal dotted line, the average power is half the maximum.

EFFECT OF INDUCTANCE

Now suppose that the source of alternating current is connected to an inductive circuit, the resistance of which is small enough to be neglected. We saw in the previous article that when a current change in an inductive circuit an e.m.f is induced, and it will be remembered that the value of this e.m.f. at any instant is dependent upon the rate of change of current and that its direction is such as to oppose the change.

Let the current be represented by curve, the steepness of this curve at any point indicates the rate at which the current is changing, and can be represented by a similar curve is placed horizontally by 90 or one-quarter of a cycle. Such a curve is shown, and its represents the induced e.m.f.

The tendency of the induced e.m.f. is to prevent the current from changing, and as the current does change, the applied e.m.f. must be opposing the induced e.m.f. at every instant. In the present case this is all the applied e.m.f. has to do, since we have assumed that the resistance of the circuit is negligible. The applied e.m.f. can therefore be represented by a curve, drawn equal and opposite to the induced e.m.f.

Note that in this circuit we have the curious condition of a current reaching its maximum when the voltage which causes it to flow is at a minimum, and vice versa.

The current is always one-quarter of a cycle behind the e.m.f. and is said to lag by this amount. Another way of expressing the same thing is to say that the voltage and current are 90 out of phase. Do not be misled because it reaches its maximum and minimum value after the voltage, and in the diagram, this means that the current curve is displaced towards the right.

The applied e.m.f. and current are shown again on the left of the diagram. In this figure, the curves start at the moment when the e.m.f. is zero, but the phase relationship (i.e, the relative positions of the two curves) is just the same. Let us see how the lagging current affects the power (watts) in the circuit.

To obtain the power curve, we again multiply volts by amperes at different points on the curves and plot the results. Evidently, the power will be zero and as they are out of phase, this occurs four times during each cycle.

The power curve does not lie wholly above the centerline, as it did when the circuit was non-inductive. This is because there are times when the e.m.f. is positive and the current negative, and vice versa. The mathematical result of multiplying a positive by a negative quantity is negative, and at such times the power curve is below the line. The energy is again represented by shading, and it will be observed the areas below the line are equal to those above.

The shaded areas above the line represent energy being taken from the source to build up the magnetic field; note that they coincide with the times when the current is growing. The shaded areas below the line represent energy being restored on the source during the collapse of the field; note that they coincide with the times when the current is falling. We have already met the storage of energy in the magnetic field in the previous article.

As the positive parts of the power curve are equal to the negative parts, the average value is zero. It follows that although there are both current and e.m.f. no power (watts ) is being expended in the circuit. The current which flows under these conditions is sometimes said to wattle.

EFFECT OF CAPACITANCE

The effect of capacitance in an alternating current circuit can be most readily appreciated by considering again the flexible diaphragm seepage in a water circuit. Clearly, such a diaphragm would not prevent the flow of an alternating water current; it would simply stretch for one way and then the other as the pressure changed. Note, however, that the maximum flow of water would occur when the diaphragm was midway between one stretched position and the other, while the pressure will be at its maximum when the stretched positions were actually reached.

Now suppose that a source of alternating current(see page) is connected to a capacitor, the resistance of the circuit being negligible. The current flowing at any instant (amperes, or coulombs per second) is evidently equal to the rate at which the charge on the capacitor (coulombs)is changing, whether the charge is increasing or decreasing. As the charge is proportional to the voltage, the current is dependent upon the rate of change of voltage.

The diagram in which the curve e once more represents e.m.f. and the curve current, illustrates these conditions. Starting on the left, the e.m.f rises from zero to a maximum. The capacitor is then fully charged and the current momentarily zero. As the e.m.f. falls, the capacitor discharges, the current reaching a maximum when the e.m.f. is zero. The current continues to flow in this direction as the e.m.f. builds up in its new direction. It falls as the capacitor charges until when the e.m.f. has reached its maximum in the new direction, the capacitor is again fully charged and the current has stopped to zero. These operations continue indefinitely.

As in the case of other purely inductive circuit, the current and voltage are 90 out of phase, but this time the current is leading. The power curve for the same interval of time is shown on the right as before, and it will be observed that the average power is again zero. This is explained by the fact that the energy is taken from the source to charge the capacitor is given up again when it discharges. In the meantime it has been stored, as we saw in the previous article, in the electric field set up in the capacitor dielectric.

RESISTANCE WITH CAPACITANCE OR INDUCTANCE

If resistance is connected in series with the capacitor in the last example, the result is somewhere between the case of pure resistance  and pure capacitance  consequently the current leads the voltage by some amount less than 90.

This condition is shown in a note that the power curve now encloses a greater area above the centerline than below it so that the average power is no longer zero. This is to be expected since some of the energy is being converted into heat in the resistance.

Similarly, if the circuit includes inductance and resistance, the result is somewhere between the case of pure resistance and pure inductance. Consequently, the current lags behind the voltage by some amount less than 90, and again the average power is more than zero. It is left to the reader to draw the curves for this case.

FACTOR POWER

Since in an alternating current circuit it is possible to have current and e.m.f without any power, it is clearly not possible to find the power in watts simply by multiplying volts by amperes. To find the true power, the product of volts by amperes must be multiplied by a figure which takes the phase difference into account. This figure is known as the power factor. It varies from zero in a circuit comprising pure inductance or pure capacitance to unity in a circuit comprising pure resistance. We may say, therefore, that

Watts = volts X Amperes X power factor.

The abbreviations p.f. is used for power factor.

Example – what is the true power in a circuit connected to a 230-volt a.c. mains, the current being 10 amperes and the power factor 0.85?

Power = (230 x 10 x 0.85)watts = 1955watts.

Those who are familiar with elementary trigonometry may note that in all the cases we have been considering the power factor is equal to the cosine of the angle  representing the phase difference between current and voltage.

The product of volts and amperes (neglecting the power factor) is called the apparent power and measured in volt-amperes, or kilovolts-amperes for large loads, in order to distinguish it from the true power measured in watts or kilowatts. The abbreviation kVA is used for kilovolt-amperes.

REACTANCE

Inductance and capacitance, like resistance, impose a limit on the current which a given voltage will cause to flow in a circuit, but their effect, unlike that of resistance, varies according to the frequency of the current. The frequency of a steady direct current is zero, and to it an inductance acts as an ordinary conductor and a capacitance as a complete break. As the frequency increases, the inductance offers a less and less easy path, and the capacitance a more and more easy one.

For current of any given frequency, the effect of either an inductance or capacitance may be compared with that of a resistance and expressed in ohms. This value is termed the reactance of the inductance or capacitance at that frequency. It follows that if there is no real resistance in the circuit, current=voltage/reactance

Evidently the reactance of an inductance increases as the frequency increases, while that of a capacitance diminishes. The actual relations are, for an inductance reactance (ohms)= 2π×frequency×inductance(henrys)

And for a capacitance,

Reactance (ohms) = 1/2π×frequency×capacitance

The greek letter π (pi) represents the ratio between the  circumference and  diameter of a circle, its value being rather more than 3.14. it comes into the equations because the reactance is dependent upon the number of radians through which the conductor generating the current moves in one second. A radian is an angle obtained by measuring along the circumference of a circle a length equal to its radius, and one revolution is therefore equivalent to 2π radians.

IMPEDANCE

Since real circuits usually possess appreciably resistance in addition to their inductance or capacitance, this must be combined with the reactance in order to find the total equivalent resistance at any given frequency. The total equivalent resistance is called the impedance and, like the reactance, can be expressed in ohms. We may therefore write:

Current = voltage/Impedance

The impedance is not the simple arithmetical sum of the reactance and resistance, but the square of the impedance can be found by adding the square of the reactance to the square of the reactance, from which it follows that

Impedance = √resistance2 + resistance2. The ratio of resistance to impedance is another way of expressing the power factor, so that

Power factor = Resistance/Impedance

It is obvious from this relation that in a circuit with no resistance the power factor is zero, while in one with nothing but resistance (impedance = resistance) it is unity. This is in accordance with the results already obtained.

RESONANCE

A consideration of the combined effects of inductance and capacitance would be discussed later, but we may note that they can in some cases counteract each other. For example, it is possible to have a combination of inductance and capacitance in series in which the current is in phase with the total applied voltage. The reactance is then zero, and the current, being limited only by the resistance of the circuit is a maximum. For any given combination of inductance and capacitance there is a definite frequency at which this will occur, and the circuit is said to be resonant et that frequency.