# «Development and Validation of Linear Alternator Models for the Advanced Stirling Convertor Jonathan F. Metscher and Edward J. Lewandowski Glenn ...»

NASA/TM—2015-218456 AIAA–2014–3858

Development and Validation of Linear Alternator

Models for the Advanced Stirling Convertor

Jonathan F. Metscher and Edward J. Lewandowski

Glenn Research Center, Cleveland, Ohio

March 2015

NASA STI Program

NASA/TM—2015-218456 AIAA–2014–3858

Development and Validation of Linear Alternator

Models for the Advanced Stirling Convertor

Jonathan F. Metscher and Edward J. Lewandowski

Glenn Research Center, Cleveland, Ohio March 2015 Acknowledgments Level of Review NASA STI Program Development and Validation of Linear Alternator Models for the Advanced Stirling Convertor Jonathan F. Metscher and Edward J. Lewandowski National Aeronautics and Space Administration Glenn Research Center Cleveland, Ohio 44135 Abstract Two models of the linear alternator of the Advanced Stirling Convertor (ASC) have been developed using the Sage (Gedeon Associates) one-dimensional modeling software package. The first model relates the piston motion to electric current by means of a motor constant. The second uses electromagnetic model components to model the magnetic circuit of the alternator. The models are tuned and validated using test data and compared against each other. Results show both models can be tuned to achieve results within 7 percent of ASC test data under normal operating conditions. Using Sage enables the creation of a complete ASC model to be developed and simulations completed quickly compared to more complex multidimensional models. These models allow for better insight into overall Stirling convertor performance, aid with Stirling power system modeling, and in the future support NASA mission planning for Stirling-based power systems.

Nomenclature ASC Advanced Stirling Convertor residual magnetic flux density (T) Br BOM beginning of mission alternator motor constant (N/A) Ki EM electromagnetic EOM end of mission Force (N) F FringeMult Sage fringe effect multiplier HR high reject temperature current (A) I JSat saturation magnetic polarization (T) Jmult Sage magnet strength multiplier alternator inductance (H) Lalt LR low reject temperature number of turns N PM permanent magnet net heat input (W) Q Ralt R1, R2 net heat input as calculated by Sage (W) Sage_Qin electromotive force (EMF) voltage (V) Vemf

Introduction Stirling technology development (Ref. 1) is continuing at the NASA Glenn Research Center as an efficient and reliable power system potentially for NASA’s deep space missions. Currently, when radioisotope power is required, NASA deep space missions use radioisotope thermoelectric generators (RTGs), which convert the heat from radioactive decay of Plutonium-238 into electric power, but they have efficiencies of 5 to 7 percent. Stirling engines are a higher-efficiency alternative that could significantly reduce the amount of material used in radioisotope power systems by a factor of 4 or more (Refs. 1 and 2).

The Advanced Stirling Convertor (ASC) (Refs. 3 and 4) developed by Sunpower, Inc., is a free-piston Stirling engine coupled with a linear alternator. The ASC is currently under extended testing at Glenn (Refs. 5 and 6). It is a reciprocating resonant system that consists of a helium-filled pressure vessel containing a piston, displacer, and linear alternator. Electrical power is extracted in the linear alternator where the reciprocating piston motion drives magnets through the alternator coil. Figure 1 is a cross section view of a generic free-piston Stirling convertor and defines the main components.

Advanced Stirling Convertor Modeling Modeling and simulation is important in the development and testing of Stirling engines as it aids in optimization of design, analysis of system performance, and understanding of physical parameters that are impractical to measure in Stirling devices. There have been both one-dimensional and multidimensional modeling and simulation efforts focusing on the ASC. One-dimensional models use nodes to directly solve the governing system equations and are advantageous due to their fast computation times and ease of setup (Ref. 7). One-dimensional models such as the System Dynamic Model (SDM) (Ref. 8) enable whole convertor simulation by linking representative elements within the Simplorer (Ansoft Corporation) commercial software package. SDM also has capability of modeling transient startup and nonlinear dynamic behavior, although this makes it more computationally intensive. SDM is limited by less sophisticated Stirling cycle thermodynamics and a simplified alternator model. Sage (Gedeon Associates) is another one-dimensional modeling package that is used to model Stirling engines. It is a steady-state modeling package that is less computationally intensive and has been continually improved over the years. Its thermodynamic computations have been shown to agree well with two-dimensional computational fluid dynamic (CFD) models (Refs. 9 and 10). Recent additions to the Sage model library allow for modeling of linear motors and alternators, enabling whole convertor modeling of the ASC.

Further detail on Sage and validating its modeling capability is discussed later in this paper.

Multidimensional simulations are typically CFD models that focus on specific regions of the Stirling engine such as the regenerator, although there has been some work toward whole engine modeling NASA/TM—2015-218456 2 (Ref. 7). Multidimensional simulations offer many advantages as outlined by Dyson (Ref. 11), such as modeling inherently three-dimensional phenomena as flow turbulence. Multidimensional simulations are computationally expensive and do not typically include linear alternator modeling to give a whole convertor simulation. The ANSYS Maxwell finite element method (FEM) software package allows multidimensional modeling of the linear alternator and has been used at Glenn to model linear alternator designs from earlier Stirling convertor efforts (Ref. 12). Maxwell has the same disadvantage of being computationally expensive and not able to model the whole convertor.

A whole convertor model would be beneficial in analyzing test data as it enables the simulation of parameters that are impractical, if not impossible, to measure and assists in system verification and validation. This paper reviews a whole convertor modeling effort using the Sage software package. As a one-dimensional model, it will allow for fast development and simulation times. Simulations are compared to test data to validate the model and determine model limitations.

Sage Overview Sage (Ref. 13) is a one-dimensional Stirling device modeling software package developed by Gedeon Associates. Sage contains a library of generic model components that can be placed and connected in the Sage graphical user interface (GUI). The model components contain the user-defined dimensions and properties and are connected to other model components through various connection interfaces (force, pressure, volume flow, heat flow, etc.). Sage components can be thought of as building blocks that are assembled to form the system of interest (Ref. 14). Figure 2 shows an example of Stirling engine components and their interconnections. Components may then have subcomponents and their own connections. This modular method facilitates quick model construction as the underlying equations are defined by the components and their interconnections. Sage allows the user to optimize parameters according to defined constraints and optimization objectives. This powerful ability enables design optimization or can assist in tuning model parameters using performance data.

The Sage library is divided into model classes (Stirling, Pulse Tube, and Low-T Cooler). The Stirling model class has been used for modeling ASC engines, but until recently was unable to model the linear alternator. The recent addition of electromagnetic (EM) components to the Sage library allows the modeling of simple circuits and linear motors and alternators, enabling whole convertor modeling of the ASC.

The Sage EM library consists of basic circuit components as well as magnetic components. It includes resistor, capacitor, and inductor model components as well as voltage and current sources.

Component properties are user defined and the components are connected through current interfaces.

These components can be used to model simple RLC circuits as shown in Figure 3, or used as part of more complex EM models and combined with magnetic model components.

The library also includes a wire coil that can be used with magnetic model components to develop linear electric actuator and generator models or similar devices such as transformers. The library contains magnetic components such as magnetic field or flux sources, airgaps between magnetic components, permanent magnet (PM) and ferromagnetic materials, and magnetic single- or two-pole components. EM Some of these high-level components have built-in subcomponents to further define the model structure. The user defines the physical dimensions of the components; however, it should be remembered that this is a one-dimensional model and the geometry is assumed axisymmetric. The solution is also time periodic and does not model transient behavior, making this unsuitable for certain system simulations or analyses.

Linear Alternator Operation A linear alternator operates on the principle of Faraday’s law in which an electromotive force (emf), or voltage, is induced along the boundary of a surface through which there is changing magnetic flux (Ref. 15). In the case of the ASC linear alternator, PMs are attached to the piston, which oscillates within the alternator coil. The magnetic field (B) from the magnets is directed across the pole gaps and through the inner and outer ferromagnetic cores, following a path of least reluctance (R) much like current through circuit follows a path of least resistance. As the piston moves through one cycle, the magnetic flux changes as its path changes. The magnetic flux passing through the alternator coil will increase and decrease in an oscillatory manner due to the changing position of the magnets within the stationary ferromagnetic cores, causing the magnetic field to change direction. This changing magnetic field passing NASA/TM—2015-218456 4 through the circular surface enclosed by the alternator coil causes a voltage to be induced (Vemf).

Equation (1 field through a surface (Eq. (2)) and the magnetic flux through each “surface” created by the turns (N) of the alternator coil are known as flux linkages (N f. 16). Vemf can be simplified as the time derivative of the flux linkages (Eq. (3)).

= (1) = (2) = (3) Vemf is in phase with piston velocity; however, the voltage at the alternator terminals (Valt) is phase shifted due to the inductance of the coil and acts to oppose changes in current. This behavior stems from Lenz’s law in which the direction of the induced current in the coil flows as to create a magnetic field opposing the change in magnetic flux through the coil. Inductance (L) is defined in Eq. (4) (Ref. 17). Sage takes a slightly different approach at calculating inductance (Eq. (5) and Ref. 14) but can be shown to be consistent by substituting the relationship between voltage and inductance shown in Equation (4).

= = (4) = (5) Linear Alternator Modeling Using Sage Sage Linear Alternator Modeling Using the Sage Transducer Component An alternator model can be created using the “transducer” component (Figure 4) in the Sage EM library. Like a physical transducer, it converts energy from one type to another. In Sage it converts mechanical energy to electrical. The component has built-in force and current connections and assumes the relationship shown in Equation (6) and energy conservation shown in Equation (7). The variable Ki is user defined to match the system characteristics. In a linear motor- or alternator-type model, Ki is the motor constant.

(6) = (7) / =

** Figure 6.—Linear alternator circuit model in Sage using the transducer component.**

Outlined are the main linear alternator model components.

Transducer Alternator Model Components Figure 5 shows a circuit diagram of a linear alternator with controlling circuit elements. Vemf represents the voltage generated by the linear alternator while Ralt and Lalt represent the resistance and inductance of the alternator, respectively. The remaining resistors R1 and R2 are the wire and lead resistance in the circuit. A tuning capacitor is used for power factor correction and an alternating current (ac) power supply controls the piston amplitude. This circuit diagram is a useful comparison to the Sage model of a linear alternator using the transducer component described earlier. Figure 6 shows a Sage model of a linear alternator (Ref. 18). The model requires three key Sage EM components to model the linear alternator. The primary component is the transducer that converts force from the piston into electric current; however, it does not account for the resistive and inductive properties of the wire coil in the alternator. A resistor and an inductor component are needed to account for these properties. The outlined components show the key linear alternator components. The remaining components model the rest of the circuit connected to the linear alternator and compare directly to the circuit diagram.

Transducer Alternator Model Tuning This method of modeling a linear alternator is simple to implement, requiring only three components, but is limited in that it ignores the underlying physical phenomena and potential losses such as eddy currents, hysteresis, and flux leakage. It also requires that the user have data to input properties such as alternator inductance and resistance as well as the motor constant Ki. For the ASC, values for alternator inductance and resistance are known. In an attempt to account for losses, an additional resistor Rloss is added in the Sage model, though this assumes the losses are proportional to current. Determining an appropriate resistive loss is not straightforward as the real losses may change with convertor operation point. The same could be true for Ki.

NASA/TM—2015-218456 6 Figure 7.—Transducer tuning parameter value as a function of rejector temperature.