In this chapter, a review on linear generators is presented. The discussion focuses on generators that are tubular and equipped with moving permanent magnets. Details on the machine construction and its components are briefed. This is followed by a study on the fundamentals of the magnetic field analysis which forms the basis of the mathematical model and the FEA.
Introduction
A linear generator is an electromechanical energy converter driven by a prime mover which converts the kinetic energy of linear motion into electrical energy. The motion can be in one direction or reciprocating. In a system where the motion is in one direction and has a long stroke such as in the magnetic levitated train (MAGLEV), the principle of operation of the machine is identical to that of a rotary machine. Thus it can produce a three-phase electrical power. In a reciprocating linear generator, the three-phase output cannot be produced since the voltage will reverse as the mover reverses its direction of the motion after reaching the end point of the stroke. Indeed, the reciprocating machine can only generate a single phase electrical power. However, the three-phase term is used in the linear generator to describe a three sequential flux linkage as a function of motion displacement generated by the machine.
A linear generator converts the mechanical energy to electrical energy using electromagnetic induction. The source of the mechanical energy is a prime mover which can be a steam engine, water falling through a turbine or waterwheel, a Stirling engine, an internal combustion engine, a wind turbine, a hand crank, tidal wave, compressed air or any mechanisms that produce linear motion.
Another term commonly used for a linear generator is a linear alternator. The simplest type of a linear alternator is the Faraday flashlight or the shake torchlight. It contains a coil and a permanent magnet. When the appliance is shaken back and forth, the magnet oscillates through the coil and induced an electric current. This current charges a capacitor that stores the electrical energy. The capacitor is discharged when used to light up a light emitting diode and it can be re-charged by further shaking.
Figure 2. 1. Hand shake flashlight [Wikipedia]
Most linear generators are proposed to be used as a stand alone power supply [Parviz Famouri, William R. Cawthorne, 1998] and as a part of a special purpose equipments like a gun [J.Li, et. al ,1999; Peter Mongeau, 1997; Duan Xiaojun, 2005] or a hybrid application [William R. Cawthorne,1999]. An example of a linear alternator used for remote electrical power generation is presented in the work of Famouri et. Al. [Parviz Famouri, William R. Cawthorne, 1998]. The machine is directly coupled to an internal combustion engine with dual pistons to form a free piston generator. Each end of the translator is connected directly to a piston. The combustion process at each end moves the pistons in the opposite directions alternately thus producing a reciprocating motion.
Machine Construction
An imaginary description of the transformation of a rotary machine into a linear machine is shown in Figure 2. 2 and Figure 2. 3. In principle, for every rotary electric machine configuration a linear counter part can be imagined. The rotary machine is cut and unrolled to obtain a flat or tubular linear induction machine. The same process may be imagined for flat and tubular linear synchronous PM motors. [Boldea, Nasar, 1997].
Figure 2. 2. Double and single sided flat linear machine
[Boldea, Nasar, 1997]
Figure 2. 3. Flat and tubular linear machine
[Boldea, Nasar, 1997]
Construction-wise, a linear generator consists of a static part and a movable part. The static part, known as the stator is fixed at a certain position, while the moveable part, known as the translator moves along the machine axis. An example of linear generator is shown in Figure 2. 4. The stator comprises the armature windings, the back iron, and the winding spacers. The translator is the moving part of the machine which is formed by permanent magnets, non-magnetic spacers and a shaft.
Figure 2. 4. The free-piston generator
[Cawthorne, 1999]
Most linear generators are tubular or planar, although other shapes are possible depending on the applications and requirements of the system. Tubular machines are symmetric in the radial direction and the flux leakage in tubular machines occurs only at the machine's ends.
Unlike rotary machines which produce angular torque, a linear generator produces a high moment when the translator stops at the stroke ends before moving back for a complete cycle. Thus, it requires a minimal translator weight. If the machine is running at high speed, the weight of the translator will be critical. In general, the translator weight should be kept as light as possible to reduce mechanical vibration.
Current rare-earth permanent magnets produce high flux density with less weight. The permanent magnets can be mounted on the translator surface. The flux of an axial permanent magnet flows parallel with the machine axis, so that a permeable medium is needed to channel the flux radially to link with the windings. On the other hand, no medium is needed to connect the flux of a radial permanent magnet to the windings.
It is found that the surface mounted permanent magnet is the most commonly used configuration due to the application of the radial magnetized permanent magnets. The tube shaped permanent magnet offers a surface mounted construction for RMPM or embedded construction for AMPM. The AMPM construction is actually surface mounted except for the need of spacer or permeable material insertion between permanent magnets.
A special construction known as the Halbach array permanent magnet can be considered as a surface mounted. The machine offers a high power density, and a high efficiency and low moving mass [Jiabin, et all, 2005]. Details of different types of constructions for linear machines will be given in chapter 3.
Fundamentals of the Magnetic Field Analysis
In electromagnetic field analysis, all fundamental fields are governed by the Maxwell's equations. In this chapter, single scalar parameters are represented as normal characters, single vector parameters are represented as boldface characters and matrices as characters in brackets. The four Maxwell's equations that govern electromagnetic and electrostatic fields are given by
where
[H] : magnetic field intensity vector,
[J] : total current density vector,
[Js] : applied source current density vector,
[Je] : induced eddy current density vector and
[Jv] : velocity current density vector.
[D] : electric flux density vector.
[E] : electric field intensity vector,
[B] : magnetic flux density vector and ρ is electric charge density.
By applying the conservation law, the divergence of both sides of Equation gives the continuity equation,
The behavior of electromagnetic materials also contributes to a constitutive relation in the field equations. The constitutive relation for the magnetic fields involving saturable materials is given by
where [μ] is the magnetic permeability matrix which may be a function of temperature or field, or even both. If [μ] is only a function of field, then
where μh is the permeability derived from the input B versus H curve. This equation is valid for a system without permanent magnets. When permanent magnets exist, the constitutive relation becomes:
where [Mo] is the remanent intrinsic magnetization vector. Rewriting the general constitutive equation in terms of reluctivity it becomes:
where [ν] = [μ]-1 is reluctivity matrix. The reluctivity of free space is.
The solution of the magnetic field problems is commonly obtained using potential functions. Two kinds of potential functions, the magnetic vector potential and the magnetic scalar potential are used depending on the problem to be solved. Factors affecting the choice of potential include: field dynamics, field dimensionality, source current configuration, domain size and discretization. The applicable regions for all kind of materials are shown in Figure 4. 1. These will be referred to with each solution procedure discussed in the following sections.
Figure 4. 1. Electromagnetic Field Regions
where:
Ω0 : free space region
Ω1 : non-conducting permeable region
Ω2 : conducting region
μ : permeability of iron
μo : permeability of air
Mo : permanent magnets
S1 : boundary of Ω 1
σ : conductivity
Ω = Ω1 + Ω2 + Ω0
Magnetic Scalar Potential
The magnetic scalar potential B is defined in a region of space in the absence of currents. In the magnetostatic problems, the time varying effects are ignored. This reduces the Maxwell's equations for magnetic fields to Equation and Equation below:
Solution Strategies
Referring to Figure 4. 1, in the domain of Ω0 and Ω1 (excluding Ω2) a solution is required which satisfies the relevant Maxwell's Equation and Equation , and the constitutive relation Equation in the following forms:
where [Hg] is known as the preliminary or initial magnetic field and φg is the generalized potential. The development of [Hg] varies depending on the problem and the formulation. Fundamentally [Hg] must satisfy Ampere's law in Equation so that the remaining part of the field can be derived as the gradient of the generalized scalar potential φg. This ensures that φg is single valued. In addition, the absolute value of [Hg] must be greater than that of Δφg. In other words, [Hg] should be a good approximation of the total field to avoid difficulties with cancellation errors.
As mentioned above, the selection of [Hg] is important to the development of any of the following scalar potential strategies. The development of [Hg] involves the Biot-Savart field [Hs] which satisfies Ampere's law and is a function of the source current [Js]. [Hs] is obtained by evaluating the integral:
where [Js] is current source density vector at dv, [r] is position vector from current source to node point and V is volume of current source.
The above volume integral can be reduced to the following surface integral:
where S is the surface of the current source
The solution of this integral should be suitable for the initial condition. The values of [Js] are obtained either directly from the current sources or as the result of an external electric field calculation.
Magnetic Vector Potential
The vector potential method is implemented for both 2-D and 3-D electromagnetic fields. Considering the static and dynamic fields and neglecting displacement currents (quasi-stationary limit), the following subset of Maxwell's equations apply.
In the entire domain Ω of an electromagnetic field problem, a solution is required which satisfies the above Maxwell's equations (Equation through Equation ).
Since the magnetic field density B is divergenceless it can be expressed as the curl of the magnetic potential A as
Thus, replacing B with -xA in Equation 4.37, the electric field [E] can be written as
where:
[A] is magnetic vector potential
V is electric scalar potential
Electromagnetic Field Evaluations
The basic magnetic analysis results include magnetic field intensity, magnetic flux density, magnetic forces and current densities. Whereas, the basic electric analysis results include electric field intensity, electric current densities, electric flux density, Joule heat and stored electric energy.
Magnetic Scalar Potential Results
The magnetic field intensity can be divided into two parts [Hg] and [Hφ] which are the generalized field and as the gradient of the generalized potential -φg respectively. [Hφ] is evaluated at the integration points using the element shape function as:
where [N] is shape functions and [ï¦g] is nodal generalized potential vector .
Then the magnetic field intensity is defined as
where [H] is magnetic field intensity (output as H)
The magnetic flux density is calculated from the field intensity as
where [B] is magnetic flux density (output as B)
Nodal values of the field intensity and flux density are computed from the integration point's values.
Magnetic Vector Potential Results
As noted earlier the magnetic flux density is defined as the curl of the magnetic vector potential. This evaluation is performed at the integration points using the element shape functions:
where [B] is the magnetic flux density, [NA] is the shape functions and [Ae] is the nodal magnetic vector potential
Then the magnetic field intensity is computed from the flux density as
The current densities are also related to the vector potential and is given as
where: [Jt] is total current density
where:
[Je] : current density component due to [A] ,
[NA] : element shape functions for [A] evaluated at the integration points,
[Ae] : time derivative of magnetic vector potential,
[Js] : current density component due to V,
[Ve] : electric scalar potential
[N] : element shape functions for V evaluated at the integration points
[Jv] : velocity current density vector
[v] : applied velocity vector
Magnetic Forces
The magnetic forces are computed using the vector potential method or the scalar potential method. Three different techniques are usually used to calculate the magnetic forces at the elemental level.
Lorentz forces
The Lorentz forces are the magnetic forces generated in current carrying conductors. They are calculated using numerical integration by the following formula
Where Fjb is the Lorentz force.
For 2-D analysis, the corresponding electromagnetic torque Tjb about +Z is given by
where [Z] is the unit vector along +Z axis and [r] is the position vector in the global Cartesian coordinate system.
Maxwell Forces
The Maxwell stress tensor is a second rank tensor used to represent the interaction between electric or magnetic force and mechanical momentum []. The tensor is used to compute forces on ferromagnetic regions. This force calculation is performed on the surfaces of the air material elements which have a nonzero face loading. For the 2-D applications, this method uses extrapolated field values and results in the following numerically integrated surface integral given below.
where
Virtual Work Forces
Another method used for computing the magnetic force is known as the virtual work force. The magnetic forces calculated using the virtual work method are obtained as the derivative of the energy versus the displacement of the movable part. This calculation is valid for a layer of air elements surrounding a movable part. To determine the total force acting on the body, the forces in the air layer surrounding it is summed. The basic equation for the force of an air material element in the s direction is:
where Fs is the force in the element in the s direction, is the derivative of the field intensity with respect to displacement, s is the virtual displacement of the nodal coordinates taken alternately in the X, Y, Z global directions and dv is the elemental volume of the surrounding air.
Conclusion
Ring permanent magnets and ring coil are often used in tubular linear machine. Taking advantage of its symmetry, a tubular machine can be represented by 2D layout. The two-dimensional simulation of the machine is advantageous since it is simpler and faster than the normal 3D simulation.
Like rotary machines, all fundamental fields that govern the behavior of linear machines are given by the Maxwell's equations. These include the vector potential A and the flux density B from which other parameters such as flux linkage, force and the induced voltage are derived.
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