**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 =
√resistance^{2} + resistance^{2}. 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.

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