Why is voltage generated by a rotating loop

More elaborate arrangements of multiple coils and split rings can produce smoother DC, although electronic rather than mechanical means are usually used to make ripple-free DC. Figure 4. Split rings, called commutators, produce a pulsed DC emf output in this configuration. All other quantities are known. The maximum emf is greater than the average emf of V found in the previous example, as it should be.

In real life, electric generators look a lot different than the figures in this section, but the principles are the same. The source of mechanical energy that turns the coil can be falling water hydropower , steam produced by the burning of fossil fuels, or the kinetic energy of wind. Figure 5 shows a cutaway view of a steam turbine; steam moves over the blades connected to the shaft, which rotates the coil within the generator.

Figure 5. The steam produced by burning coal impacts the turbine blades, turning the shaft which is connected to the generator. Generators illustrated in this section look very much like the motors illustrated previously. This is not coincidental. In fact, a motor becomes a generator when its shaft rotates. Certain early automobiles used their starter motor as a generator.

In Back Emf , we shall further explore the action of a motor as a generator. Calculate the peak voltage of a generator that rotates its turn, 0. In order to find the parametric equation of the curve when , we use the primed coordinate system which is received through rotation of the unprimed one by angle with respect to axis defined in Figure 1 b. The polar equation of the loop is now given by , while the two azimuthal angles are connected obviously via the relation.

Therefore, we take the Formulas 1 - 3 replacing by and the unprimed variables by the primed ones. In this sense, the parametric set of the primed coordinates with respect to unprimed azimuthal angle after trivial algebraic manipulations , is written as follows:. The parametric equations of the arbitrarily rotated wire loop expressed in the unprimed coordinate system are denoted by and are determined from the transformation relation below [13]:.

An extra time argument has been added to the dependencies of the parametric equations for better comprehension. Once the boundary of the rotating wire is rigorously specified, the field quantities related to the induced voltage can be computed.

The magnetic vector potential of a -polarized infinite dipole into vacuum at distance from its axis is given by [14]:. The transverse distance of the representative loop point with azimuthal angle , at an arbitrary time , from the axis , is found apparently equal to:. It is well-known [15] that the magnetic flux through a closed loop is defined from the line integral of the magnetic potential on the curve, that is. In our case, this formula is particularized [16] to give:.

It should be noted that the magnetic flux is independent from the reference distance. Figure 2. The transversal distance between the axis of the dipole and a representative point of the rotated loop projected on x-z plane. In this way, the generated voltage on the coil due to the self-rotation, in the presence of a dipole posed along the axis , is given by:. In case the structure is excited by an axial surface current distributed according to the law measured in , along the line , the induced electromotive force on the wire loop, is expressed in terms of the following line integral:.

Such an expansion is permissible because the operator applied on to find , is linear. Similarly, if the excitation is a volume axial current measured in , across the area , the voltage is obtained through the double integral below:. Indicative Results In this section, we examine the produced voltage by rotating loops of variable shape and the effect of their geometrical parameters on it. The described method is applied to two families A, B of wire boundaries possessing the following polar equations:.

The arguments have length dimension and the argument is a positive integer number. For brevity, we chose to excite the structure exclusively through point sources avoiding surface or volume distributions described through 13 , The following graphs will exhibit the dependencies of the produced voltage and the corresponding RMS value:.

A single wire loop is not practically used for voltage generation as clusters of coils are utilized instead. We are more concerned for the qualitative description of the output quantity than measuring the exact magnitude in Volts of the produced voltage which is extremely small. Each figure below is divided into two subfigures, the first of which is a diagram depicting the variation of the investigated quantities for various shapes of the loop, while the second is a polar plot of the corresponding wire boundaries.

In Figure 3 a , we represent the time dependence of the developed normalized voltage for various shapes of loops corresponding to different parameters of the curve family A with constant and. The observation duration is a single period as from that point on the procedure and the results are repeated unchanged. Even though the shape of the wire is arbitrary, the waveforms exhibit typical sinusoidal behavior, a fact that can be attributed to the horizontal and vertical symmetry of the metallic loop boundary.

Also, the amplitude of the oscillation gets greater, as the area of the closed wire increases. This is a natural conclusion because by keeping fixed the rotation frequency and the axis parallel to the dipole, larger loops lead to more significant magnetic flux variance and implicitly to more substantial output voltage.

Therefore, the oscillation amplitude of the investigated quantity is decaying with increasing because the occupying area of the loop is greater for than for as shown in Figure 3 b. We calculate the magnetic fluxes in both orientations of the loop and divide the magnitude of their difference by the time that the rotation took. Thus we obtain the average induced electromotive force.

It was pointed out already in the analysis that an electromotive force is induced in a conductive loop when the magnetic flux through the area of the loop changes. We use the absolute value of the difference because we are interested in the magnitude of the induced electromotive force only:.

Our aim is to assess the course of the induced eletromotive force in addition to its average value. For this we need to assess the course of the magnetic flux through the loop in time because the magnitude of the electromotive force induced in the conductive loop is equal to the speed of change i.

First, we determine the course of the angle between the loop and the magnetic field. This is alternating current or ac. The output of an alternator can be represented on a potential difference-time graph, with potential difference on the vertical axis and time on the horizontal axis.

The graph shows an alternating sine curve. The maximum potential difference or current can be increased by:.