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Inductors and the Farady emf An inductor is usually a coil of wire. In an ideal inductor, the resistance of this wire is negligibile, as is its capacitance. The voltage that appears across an inductor is due to its own magnetic field and Faraday's law of electromagnetic induction. Remembering that the derivative is the local slope of the curve the purple line , we can see in the next animation why voltage and current are out of phase in an inductor.

Again, there is a difference in phase: the derivative of the sinusoidal current is a cos function: it has its maximum largest voltage across the inductor when the current is changing most rapidly, which is when the current is intantaneously zero. The animation should make this clear. Note how this is represented on the phasor diagram. This is shown in the next animation: when the frequency is halved but the current amplitude kept constant, the current is varying only half as quickly, so its derivative is half as great, as is the Faraday emf. For an inductor, the ratio of voltage to current increases with frequency, as the next animation shows.

Impedance of components Let's recap what we now know about voltage and curent in linear components. The impedance is the general term for the ratio of voltage to current.

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The table below summarises the impedance of the different components. It is easy to remember that the voltage on the capacitor is behind the current, because the charge doesn't build up until after the current has been flowing for a while.

The same information is given graphically below. It is easy to remember the frequency dependence by thinking of the DC zero frequency behaviour: at DC, an inductance is a short circuit a piece of wire so its impedance is zero. At DC, a capacitor is an open circuit, as its circuit diagram shows, so its impedance goes to infinity. RC Series combinations When we connect components together, Kirchoff's laws apply at any instant. This should be clear on the animation and the still graphic below: check that the voltages v t do add up, and then look at the magnitudes.

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The amplitudes and the RMS voltages V do not add up in a simple arithmetical way. Here's where phasor diagrams are going to save us a lot of work. Play the animation again click play , and look at the projections on the vertical axis. Because we have sinusoidal variation in time, the vertical component magnitude times the sine of the angle it makes with the x axis gives us v t.

So v t , the sum of the y projections of the component phasors, is just the y projection of the sum of the component phasors.

Alternating current

So we can represent the three sinusoidal voltages by their phasors. While you're looking at it, check the phases. We'll discuss phase below. Now let's stop that animation and label the values, which we do in the still figure below. So we can 'freeze' it in time at any instant to do the analysis. The convention I use is that the x axis is the reference direction, and the reference is whatever is common in the circuit.

In this series circuit, the current is common. In a parallel circuit, the voltage is common, so I would make the voltage the horizontal axis. Be careful to distinguish v and V in this figure! Careful readers will note that I'm taking a shortcut in these diagrams: the size of the arrows on the phasor diagrams are drawn the same as the amplitudes on the v t graphs. The reason is that the peak values V mR etc are rarely used in talking about AC: we use the RMS values , which are peak values times 0. Phasor diagrams in RMS have the same shape as those drawn using amplitudes, but everything is scaled by a factor of 0.

The phasor diagram at right shows us a simple way to calculate the series voltage. The components are in series, so the current is the same in both. The voltage phasors brown for resistor, blue for capacitor in the convention we've been using add according to vector or phasor addition, to give the series voltage the red arrow.

At high frequencies, the capacitive reactance goes to zero the capacitor doesn't have time to charge up so the series impedance goes to R. We shall show this characteristic frequency on all graphs on this page. That simple result comes about because the two voltages are both in phase with the current, so their phasors are parallel. Ohm's law in AC. We can rearrange the equations above to obtain the current flowing in this circuit.

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So far we have concentrated on the magnitude of the voltage and current. We now derive expressions for their relative phase, so let's look at the phasor diagram again. So, by chosing to look at the voltage across the resistor, you select mainly the high frequencies, across the capacitor, you select low frequencies.

This brings us to one of the very important applications of RC circuits, and one which merits its own page: filters, integrators and differentiators where we use sound files as examples of RC filtering. The resulting v t plots and phasor diagram look like this. It is straightforward to use Pythagoras' law to obtain the series impedance and trigonometry to obtain the phase. We shall not, however, spend much time on RL circuits, for three reasons. First, it makes a good exercise for you to do it yourself. Second, RL circuits are used much less than RC circuits. If you can use a circuit involving any number of Rs, Cs, transistors, integrated circuits etc to replace an inductor, one usually does.

The third reason why we don't look closely at RL circuits on this site is that you can simply look at RLC circuits below and omit the phasors and terms for the capacitance. In such circuits, one makes an inductor by twisting copper wire around a pencil and adjusts its value by squeezing it with the fingers.

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Among his many experiments Professor Crookes shows some performed with tubes devoid of internal electrodes, and from his remarks it must be inferred that the results obtained with these tubes are rather unusual. If this be so, then the writer must regret that Professor Crookes, whose admirable work has been the delight of every investigator, should not have availed himself in his experiments of a properly constructed alternate current machine - namely, one capable of giving, say 10, to 20, alternations per second.

His researches on this difficult but fascinating subject would then have been even more complete. It is true that when using such a machine in connection with an induction coil the distinctive character of the electrodes -which is desirable, if not essential, in many experiments - is lost, in most cases both the electrodes behaving alike; but on the other hand, the advantage is gained that the effects may be exalted at will.

When using a rotating switch or commutator the rate of change obtainable in the primary current is limited. When the commutator is more rapidly revolved the primary current diminishes, and if the current be increased, the sparking, which cannot be completely overcome by the condenser, impairs considerably the virtue of the apparatus.

No such limitations exist when using an alternate current machine as any desired rate of change may be produced in the primary current. It is thus; possible to obtain excessively high electromotive forces in the secondary circuit with a comparatively small primary current; moreover, the perfect regularity in the working of the apparatus may be relied upon. The writer will incidentally mention that any one who attempts for the first time to construct such a machine will have a tale of woe to tell.

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He will first start out, as a matter of course, by making an armature with the required number of polar projections. He will then get the satisfaction of having produced an apparatus which is fit to accompany a thoroughly Wagnerian opera.

click here It may besides possess the virtue of converting mechanical energy into heat in a nearly perfect manner. If there is a reversal in the polarity of the projections, he will get heat out of the machine; if there is no reversal, the heating will be less, but the output will be next to nothing. He will then abandon the iron in the armature, and he will get from the Scylla to the Charybdis. He will look for one difficulty and will find another, but, after a few trials, he may get nearly what he wanted.

Among the many experiments which may be performed with such a machine, of not the least interest are those performed with a high-tension induction coil. The character of the discharge is completely changed. The arc is established at much greater distances, and it is so easily affected by the slightest current of air that it often wriggles around in the most singular manner. It usually emits the rhythmical sound peculiar to the alternate current arcs, but the curious point is that the sound may be heard with a number of alternations far above ten thousand per second, which by many is considered to be, about the limit of audition.

In many respects the coil behaves like a static machine. Points impair considerably the sparking interval, electricity escaping from them freely, and from a wire attached to one of the terminals streams of light issue, as though it were connected to a pole of a powerful Toepler machine. All these phenomena are, of course, mostly due to the enormous differences of potential obtained.

As a consequence of the self-induction of the coil and the high frequency, the current is minute while there is a corresponding rise of pressure. A current impulse of some strength started in such a coil should persist to flow no less than four ten-thousandths of a second. As this time is greater than half the period, it occurs that an opposing electromotive force begins to act while the current is still flowing. As a consequence, the pressure rises as in a tube filled with liquid and vibrated rapidly around its axis.