# Electromechanics: Theoretical basis

In the previous chapters we learned the operating principles of electric and magnetic fields. Elelctromagnetic forces are used to produce mechanical effects in the field of electromechanics. The applications of electromechanics are fairly wide like in electric motors and electromagnets. Electromechanics use two basic principles which we will discover below.

Magnets are very important in the field of elelctromechanics. Magnets can affect conductors, insulators and semiconductors without physical contact. As we know magnets are characterised by magnetic field, measured by the magnetic flux density B, Ts and magnetic field intensity H, A/m.

If the charge q is moving in the magnetic field with the speed v then there will be a force acting on the conductor perpendicular to the conductor and the magnetic field. The force magnitude is f = q[v,B], N. If the conductor speed (or charge) is directed with some angle to the field, then the force magnitude is f = qvBsinθ, where θ is the angle between the charge speed and the field.

Another characteristic of the magnetic field is the magnetic flux ϕ = ∫ BdA – that is integral to the magnetic flux density B over the cross-sectional area A. The magnetic flux ϕ, is measured in Webers (Wb).

Faraday’s Law states that the magnetic field B through the surface A, bounded by the conductor, induces the potential difference and the current through the conductor. If the magnetic flux ϕ varies in time, it can induce the electro-motive force:

Let’s consider a single-loop coil, closed by the resistor R. Let’s suppose that the magnetic field B flows through this coil, as shown in Figure 1. Then the current  will flow through this coil, creating the field in the opposite direction to the magnetic field B. This effect is known as Lenz Law. By the right-hand rule, there will be a current provoked by the generated field, moving clock-wise. And the voltage across the resistor will be negative.

In practice the voltage magnitude can be increased by increasing the quantity of coils (the bigger quantity of coils, the bigger surface the magnetic flux goes through). So for N coils:

$\epsilon =N\frac{\partial \varphi }{\partial t}$

Figure 2 shows the N-turns coil, linking the flux B, and as the coils are close to each other, we can then suppose the term of the flux linkage is:

So, how can we generally change the magnetic flux and provoke the electro-motive force ɛ? We can physically move the magnet in the vicinity of the coil, to provoke the electro-motive force. Or we can produce the magnetic field by electric current and just change the electric current. The voltages, generated by the moving in time voltages, are called transformer voltages. This effect is used in some electromechanical devices.

From the general course of elelctromagnetism the relationship between the linkage flux and current is λ = Li. So the current, changing in time, provokes the transformer voltage:

$v=L\frac{di}{dt}$

The L parameter is self-inductance. If these two circuits are close to each other, it will lead to the effect of magnetic coupling, when the second circuit will also experience the provoked magnetic flux by the current of first circuit. This effect lies in the base of the transformers.

Let’s consider two coils. One of them is connected to the current source i1, The inductance L1 generates voltage v1. The second coil L2 is not connected to the current source, but the excited magnetic field of the first coil induces the voltage in a second coil L2. This magnetic coupling happens because the proximity of two coils and is called the mutual inductance, which is characterised by the formula:

${v}_{2}={M}_{2}\frac{d{i}_{1}}{dt}$

The voltage polarity in the coils are shown in Figure 3. In real electromagnetic circuits self-inductance is not always constant. In practice the relationship between linkage flux and the current is not linear at all. And it is easier to operate in terms of energy in the magnetic circuit calculations.

Let’s consider the relationship between flux linkage λ and the current  as in Figure 4. Energy stored in the magnetic field of the magnetic system is Wm = ∫ Pidt = ∫ɛidt. The induced emf at the magnetic system is:

$\epsilon =N\frac{d\varphi }{dt}$

And the energy is:

${W}_{m}=\int \epsilon idt=N\int \frac{d\varphi }{dt}idt=N\int id\varphi =\int id\lambda$

This energy corresponds to the area above the curve (Figure 4). The area under the curve is called co-energy:

${W}_{m}^{co}=i\lambda –{W}_{m}$

Another important relationship for electromechanical devices is the Ampere Law, which states that the integral of the magnetic field H, closed by the loop l, is equal to the current carried by the loop: ∮Hdl = Σi. Another important term is magnetomotive force (MMF) that is described by the formula Ƒ = Ni.

In terms of applications, when the coil encloses the magnetic material, the greater magnetic flux is generated. The most frequent ferromagnetic materials are steel, iron and ferrites. The coiling of the magnetic material helps to keep the magnetic field with considered shape close to the magnetic material. The magnetic field in the area of the coil is depicted in Figure 5. The magnetic flux for the most practical inductors are:

Figure 7 shows the structure of a simple transformer. Let’s make the assumption that there is a mean path for a magnetic flux to exist, and that it corresponds to the mean magnetic flux density. The mean flux density is constant over the cross-sectional area of the magnetic structure:

$B=\frac{\varphi }{A}$

The cross-sectional area A is assumed to be perpendicular to the magnetic flux lines.

It does mean that for an N-turn coil winding the magnetic core, with current i, it’s the MMF Ƒ that generates the magnetic field flux ϕ, that is mostly concentrated within the core and is uniform in its cross-sectional area.

Let’s consider the magnetic structure and its equivalent circuit in Figure 8. We can represent the magnetic structure with an equivalent circuit with four legs, and MMF as a source. Two paths here have the mean path l1 with the cross-sectional area A1 = d1w, and two legs with the mean path l2 with cross sectional area A2 = d2w. Then the reluctance of the magnetic core is:

${R}_{total}=\frac{1}{{\mu }_{0}{\mu }_{R}S}\left({l}_{1}+{l}_{2}+{l}_{3}+{l}_{4}+{l}_{5}\right)+\frac{\delta }{{\mu }_{0}{S}_{\delta }}$

When the square of the gap area is bigger than the cross-section of the magnetic material, this phenomena is called fringing and happens because the magnetic field is spread more in the air gap. After the theoretical part we will consider some examples of real magnetic structures to calculate.

From the previous chapters of the electromagnetism course we know that μ permeability of magnetic material is not a linear function of flux density B and field intensity H, but has the following form (Figure 9).

The idea of this non-linear relationship between flux density and field intensity is grounded on the following basis. The big importance here is based on the spin of the electron. In most materials except ferromagnets, spin doesn’t constitute a big role.

Ferromagnets demonstrates the domain structure of the material, and the magnet field makes the domains oriented to an exact direction. At a certain moment all the domains will be oriented and any further magnetic field impacts won’t make sense.

This moment is called saturation. There are two reasons why the relationship of B and H are not linear – the eddy currents and hysteresis loop. The eddy currents are the closed electrical current loops that are induced by the time the varying magnetic fields are caused by Faraday’s Law of induction. The hysteresis loop happens when the conductor occurs in the external magnetic field, and its domains become aligned.

Transformers
Transformers are most frequent nowadays in the magnetic structures that are in use in many devices. The ideal transformer is a device that sets the AC voltage up or down with an exact coefficient. The ideal transformer is depicted in Figure 10. The magnetic flux is concentrated in the core, and that flux links all turns of the coils, and permeability of the transformer is infinite.

The transformer is operated by the AC voltage, that generates some level of the MMF, creating the time varying magnetic flux linking the L1 coils of the transformer. This time varying magnetic flux creates EMF at the coils L2 – these are the operating principles of the transformer. We must remember that the ideal transformer demonstrates lossless process.

Let’s consider the transformer in Figure 10. In accordance to the Faraday’s Law the EMF:

Here index 2 symbolises the output EMF, or output voltage. Then:

$\frac{{v}_{2}}{{v}_{1}}=\frac{{N}_{2}}{{N}_{1}},$

i.e. the output voltage can be measured as the input terminal voltage multiplied by the ratio of the coil turns of the transformer.

The current i2 induces the MMF Ƒ2 = N2i2, which is supposed to change the flux ϕ of the core. However, flux ϕ of the core does not change, as the MMF Ƒ2 is compensated by the induced input current iMMF Ƒ1 = N1i1. Then N1i1 = N2i2 and:

$\frac{{N}_{1}}{{N}_{2}}=\frac{{I}_{2}}{{I}_{1}}=a$

The constant a is called the transformer ratio.

An ideal transformer does not dissipate power, so v1i1 = v2i2. Real transformers do not behave in this way. Real transformers perform resistances of the wires, current leakages, power losses and other losses mechanisms.

Electromechanical devices can convert mechanical energy to electromagnetic energy. The name of the device that can perform this function is an energy transducer. If the transducer converts electromagnetic energy to the mechanical one, it is called an actuator. If the transducer converts mechanical energy to electromagnetic energy, it is called a sensor.

The mechanism that allows the conversion of electromagnetic energy to mechanical energy is a piezoelectric effect. This consists of the electrical or magnetic strains in the crystals of the materials, which leads to the electric charge generation in the electric field. The transformation of mechanical energy to electromagnetic energy can be made by coupling the energy stored in the magnetic field.

Let’s consider the electromechanical system that consists of the electrical part, mechanical part and the transforming part, making this transformation real. Figure 11 depicts how this energy coupling works. All three parts of the system are characterised by the losses – for the electrical system it’s resistance, for the mechanical system it’s friction, for the transforming part it is eddy currents and hysteresis losses.

Electromechanical systems can also supply energy or store it. One of the types of transducers are moving-iron transducers, like solenoids, electromagnets and relays. The scheme of electromagnetics is depicted in Figure 12. The electromagnet consists of two parts – the fixed and movable part. The movable part can be displaced by the magnetic force (as depicted in Figure 12). The equation of the energies in this system states that the small amount of energy is:

$d{W}_{m}=ℇidt–fdx=\frac{d\lambda }{dt}idt–fdx=id\lambda –fdx$

The flux in this system depends on the coil current and the displacement of the movable part of the electromagnet. It does mean that:

$f=i\left(\frac{\partial \lambda }{\partial i}di+\frac{\partial \lambda }{\partial x}dx\right)–\frac{\partial {W}_{m}}{\partial i}di–\frac{\partial {W}_{m}}{\partial x}dx$

Current and displacement are independent variables:

$f=\frac{\partial }{\partial x}\left(i\lambda –{W}_{m}\right)$

In order to simplify the analysis of an electromechanical system and calculate the energy stored in the magnetic field, let’s assume the system is magnetically linear. Then the energy of the magnetic structure is:

So from the formulas above we can calculate the force moving on the movable magnetic structure:

$f=–\frac{d{W}_{m}}{dx}=–\frac{{\varphi }^{2}}{2}\frac{dR\left(x\right)}{dx}$

Another class of electromechanical transducers are moving-coil transducers. This class of transducers unites devices like microphones, loudspeakers, electric motors and generators. Let’s start from the magnetic force acting on the electric charge in the magnetic field f = q[v,B]. Let’s consider the transducer in Figure 13. Here a conducting bar moves between the fixed frame. The magnetic field, the forces and the velocity vector of the conducting bar directions are depicted in Figure 13.

A moving–coil transducer can act as a motor. Here the motor is a moving coil-transducer, with an externally supplied current flowing through the conducting bar, displacing it to the given distance.

Let’s consider a small element  of the moving-coil transducer in Figure 14. Here the charge velocity u’ in the current i:

$idl=\frac{dq}{dt}u‘dt$

Then idl = u’dq, and il = u’q. Then considering the equation for the electric charge in the magnetic field f= i[l, B]. In case the l and B are perpendicular to each other, the magnitude of the force is f = ilB, if l and B have the angle γ between them, the magnitude of the force is f = ilBsinγ.

Another class is a generator case, where the generator is a moving-coil transducer, which displaces the moving coil by the external fields converted into the EMF and thus into the electric current through the conducting bar.

Specifically, potential difference will appear on the conducting bar, as positive and negative charges are oppositely directed. This potential difference is the EMF, that is equal to the magnetic field force acting on the charges. So ɛ = Blv.

This formula works if B, l, and v are mutually perpendicular. These vectors are not necessarily always perpendicular, and it makes the analysis a little more complicated, by including the correspondent angles to the equations. However, the most frequently used devices are constructed with the vectors B, l and v mutually perpendicular to each other.

If we will consider the device depicted in Figure 13, as a part of the system, there should be a current loop, formed by the conducting bar. The conductor moves to the right and generates the EMF, which then gives the current i. At the same time there is a force of the magnetic field, affecting the conductor, directed left (f = Blv). So there should be an additional external force, fadd, directed right.

The electrical and mechanical power for this ideal device is Pel = Blvi = ɛi, Pm – faddu = Bliu.

The considered transducer should be connected to the voltage source Vs, some resistance R, and inductance L as a part of the electric circuit. The transducer should also be characterised by the elastic force fadd (Figure 15). If the EMF is generated by the transducer ɛ > VS, then this device acts like a generator, if the EMF ɛ > VS, then the device works like a motor.

Practical transducers are characterised by the inertia, friction, elastic forces, and also inductance, resistance and capacitance. Let’s assume that the friction coefficient is d, the conducting bar mass is m and the elastic coefficient is k. Then from a mechanical point of view:

$f+m\frac{du}{dt}+du+\frac{l}{k}\int udt=Bli$

From the electrical point of view:

Then the conducting bar velocity and the current is:

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