NASAlTM-20 13-216563 AIAA-2013-3812 

Development. and Integration of an Advanced 

Stirling Convertor Linear Alternator Model for 

a Tool Simulating Convertor Performance and 

Creating Phasor Diagrams 

Jonathan F. Metscher and Edward J Lewandowski 

Glenn Research Center, Cleveland, Ohio 

December 2013 
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NASAlTM-20 13-216563 AlAA-2013-3812 

Development and Integration of an Advanced 

Stirling Convertor Linear Alternator Model for 

a Tool Simulating Convertor Performance and 

Creating Phasor Diagrams 

Jonathan F. Metscher and Edward J. Lewandowski 

Glenn Research Center, Cleveland, Ohio 

Prepared for the 

11 th International Energy Conversion Engineering Conference (IECEC) 

sponsored by the American Institute of Aeronautics and Astronautics 

San Jose, California, July 14-17, 2013 

National Aeronautics and 

Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

December 2013 
Acknowledgments 

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Available electronically at http://www.sti.nasa.gov 
Development and Integration of an Advanced Stirling Convertor 

Linear Alternator Model for a Tool Simulating Convertor 

Performance and Creating Phasor Diagrams 

Jonathan F. Metscher and Edward J. Lewandowski 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

A simple model of the Advanced Stirling Convertor (ASC) linear alternator and an alternating current 

CAC) bus controller has been developed and combined with a previously developed thennodynamic 

model of the convertor for a more complete simulation and analysis of the system performance. The 

model was developed using Sage (Gedeon Associates), a one-dimensional thennodynamic modeling 

program that now includes electromagnetic components. The convertor, consisting of a free-piston 

Stirling engine combined with a linear alternator, has sufficient sinusoidal steady-state behavior to allow 

for phasor analysis of the forces and voltages acting in the system. A MA TLAB (Mathworks) graphical 

user interface (GUI) has been developed to interface with the Sage software for simplified use of the ASC 

model, calculation of forces, and automated creation of phasor diagrams. The GUI allows the user to vary 

convertor parameters while fixing different input or output parameters and observe the effect on the 

phasor diagrams or system perfonnance. The new ASC model and GUI help create a better understanding 

of the relationship between the electrical component voltages and mechanical forces. This allows better 

insight into the overall convertor dynamics and performance. 

Cf 

F 

FF 

I 

Lall 

N 

t 

Rail 

Rl, R2 

Sage_Qin 

Vem! 

Wnet 

x 

Nomenclature 

alternator motor constant (N's/m) 

force (N) 

sinusoidally varying forcing function (N) 

current (A) 

alternator inductance (H) 

number of turns 

net heat input (W) 

time (s) 

alternator resistance (n) 

resistances (n) 

net heat input as calculated by Sage (W) 

electromotive force (EMF) voltage (V) 

power (W) 

position (m) 

voltage (V) 

magnetic flux (Wb) 

NASAllM-2013-216563 
Introduction 

The Advanced Stirling Radioisotope Generator (ASRG) is a radioisotope power system being 

developed for future NASA deep space missions, where solar power is not feasible. Current radioisotope 

thermoelectric generators (RTGs) provide reliable electric power for long-duration space missions; 

however, they have low conversion efficiency (around 5 to 7 percent). The heat source used for the RTGs 

is the General Purpose Heat Source (GPHS), which generates heat by the radioactive decay of plutonium238, a limited resource that is only just starting to be produced again in the United States. The MultiMission Radioisotope Thermoelectric Generator (MMRTG) powering the Mars Science Laboratory 

Rover Curiosity uses eight GPHS modules. The ASRG is a higher-efficiency system, requiring two 

GPHS modules. The ASRG achieves this by using highly efficient Stirling engines (Ref. 1). 

The Advanced Stirling Convertor (ASC), developed by Sunpower, Inc., is a free-piston Stirling 

engine coupled with a linear alternator (Fig. I). The convertor consists of a helium-filled, hermetically 

sealed pressure vessel containing a displacer, piston, and the linear alternator. Heat is input to the system 

from a GPHS to heat the working fluid (helium). The piston is initially put into motion by the alternator 

as an alternating current (AC) voltage is applied. The working fluid is shuttled between the compression 

space and expansion space through heat exchangers and a regenerator for increased efficiency. The 

oscillating pressure of the working fluid imparts a force on the displacer. A spring located in the bounce 

space provides a restoring force for the displacer whereas the bounce space pressure acts as a gas spring 

providing a restoring force for the piston. Magnets are attached to the piston allowing piston motion to be 

converted to electric power by the linear alternator. The linear alternator provides a damping force on the 

piston as well as a spring-like restoring force. To minimize vibrations in the ASRG, two ASCs are 

mounted opposite of each other and their piston strokes are controlled electrically by adjusting the AC bus 

voltage and phase of altern,ator (Fig. 2). The ASRG requires only two GPHS modules (one per ASC), a 

factor of 4 reduction in plutonium-238 usage relative to an MMRTG, while providing comparable electric 

power output (Ref. 2). 

Figure 1.-Cutaway of the ASC. 

Figure 2.-ASRG diagram. 

NASAl1M-20 13-216563 2 
Phasor Analysis 

To accomplish control of the ASC along with maximizing efficiency requires a better understanding 

of the interaction of components and relationship of forces acting on those components. Insight into 

convertor dynamics can be gained by plotting the forces acting on components. A time-dependent plot of 

the forces acting on the piston is shown in Figure 3. 

The components of a Stirling engine oscillate at the same frequency but vary in phase (Ref. 3). The 

forces are also sufficiently sinusoidal to be represented as phasors. This is a more useful method for 

plotting the time-varying forces as it more clearly shows how forces change in response to a change in 

system parameters (Ref. 4). It is also a more intuitive method as all the force phasors added together 

should be equivalent to the inertia phasor, verifying F = rna. Figure 4 is a typical phasor diagram of 

forces acting on the piston. 

Sage Software 

The Sage software package, developed by Gedeon Associates, is a one-dimensional thermodynamic 

modeling software for Stirling machines. It contains a library of generic model components, which can be 

interconnected through the graphical user interface (GUI) to create a model. The user can set both model 

geometry as well as initial model parameters. Sage can then use an itemtive solver to find a converging 

solution to the system that balances energy flows and temperatures at interfaces, provided that the model 

is physically sOillld and input values are reasonable starting values. Sage also supports user-defined 

variables, and typically independent Sage model parameters can also be recast as dependent variables and 

defmed through algebraic expressions. Design optimization is also supported by defining constmints and 

setting optimized variables, making Sage a powerful software tool for the development and analysis of 

Stirling engines (Ref. 5). 

-Fait 

-Bounce Spring 

- Pressure Wave 

-Inertia 

Figure 3.-Time-dependent plot of piston forces. 

-All_F."" 

-Boonce Sjiace Pnos .... F_ 

-PresSUte Fo«:e 

~lnedialf...,. 

Figure 4.-Phasor diagram of piston forces. 

NASAlTM-20 13-216563 3 
21 PI'btt18~"","" 

t9 mGt _ 

-;..Pphst 15 

,.,j.,Qstdy 17 

_mGt 20 

mst ""'""£::3. ~ +riIm 28 

+1huIy 2S 

rejiocb:w 

G 

pressure source 

17 Q.tdy ..... _Qstdy 10 

~!:=.~ 

cold sink 

A>Iur .,;. 

+& 

Phsr 

di$placer driver 

Zl mGt -E- ,-~--- .. :. 

6 Pphsr-E-~ 

7 mGi -E-"f----r .. ~ ....... .. 

26 mGt. -E-~ .. _mGt 27 

25~tdy-E-~ 

acceptor 

hot sowce 

Figure 5.--8age Stirling engine model. 

To build a model; Sage model components are selected and placed into the edit space. Many 

components allow for subcomponents to be placed within the main (or parent) components. Some . 

component interfaces are automatically generated while others may be added individually. These 

interfaces are the physical inputs and outputs of components such as force, pressure, heat flow, mass flow, 

and so on. The interfaces are connected from one component to another as can be seen in Figure 5. The 

Sage model does not give a physical sense of the geometry of the system; however, it does give a sense of 

the physical interactions between components. 

It should also be noted that there are two general types of Sage components: time-ring and phasor. 

Phasor components assume purely sinusoidal motion without harmonics and input/output is given by a 

magnitude and an angle. Time-ring components do not assume purely sinusoidal motion and are solved 

along a time grid. Input/output for time-ring components are given as a Fourier series. Time-ring and 

phasor components cannot be connected unless a motion filter is applied. The motion filter forces the 

time-ring components solution grid to have sinusoidal characteristics. This can cause some loss of 

physical properties if the system has significant nonlinear properties. The ASC is modeled with phasor 

components as the steady-state motion is very nearly sinusoidal (Fig. 5). 

MATLAB Graphical User Interface 

Sage is also available as a dynamic link library (dll) file, which allows access to Sage models from 

another program running in the Windows operating system using a C++ compiler. MATLAB 

(Mathworks) is such a program that can use a e+t compiler to run the Sage dU. Model parameters can be 

accessed and changed from the dll file, and outputs can be calculated and exported. When the simulation 

is executed, MATLAB passes the input parameters to the corresponding model input in the dll file. The 

dU file calls the Sage solver to update the input parameters and run the model to fmd a solution. The 

outputs are then passed back to the dll file and are available to the MATLAB GUI. Figure 6 shows the 

interaction between MATLAB and Sage. 

Previously, a MATLAB GUI (Fig. 7) was developed to interface with the Sage dll to vary input 

parameters and plot calculated outputs in phasor diagram form (Ref. 6). The user selects the Sage model 

from a library of models and defmes the input parameters from the MA TLAB GUI. The outputs from the 

Sage model are used to determine the forces. acting on the components and plot phasor diagrams. Plots 

can be held from run to run and overlaid to show how forces vary as input parameters are changed. The 

displacer and piston phasor diagrams are drawn with the piston phase along the x-axis. 

NASAllM-20 13-216563 4 
Pass lIser Inputs 

to dll 

Pass outputs 

from Sage to 

GUI 

Call Sage 

E <tract Sa",::: 

outPUTS 

Set lise' Inputs 

as Input 

parameters 

Run mooC! 

• 

until 

a cOf1vsrasd 

sulutron IS r')ul1d 

Figure 6.-Diagram of interaction between MATLAB and Sage. 

Figure 7.-MATLAB GUI running Sage dynamic link library (dll) and creating phasor diagrams. 

The Sage model of the ASC and corresponding MATLAB phasor output shown in Figure 7 does not 

include the linear alternator dynamics. This model also requires piston amplitude to be set, whereas a 

more realistic model would allow piston amplitude to change as other operating parameters were varied. 

Another limitation concerns the heat input parameter, which is input as the hot-end temperature. To 

model an ASC powered by a GPHS, the user should be able to set the heat input in watts from the heat 

source and have the tool calculate the temperature at the hot end. 

NASAl1M-20 13-216563 5 
.-------------------------------------------~----------------------

Figure 8.-Example RLC circuit model in Sage. 

Linear Alternator Model Development 

• t1 IGI XCI...... .... I 

Gt ... ~ 

Alternators could not be modeled with older versions of Sage software. The newest version of the 

Sage software (version 9.1) now includes electromagnetic model components. The electromagnetic model 

library includes basic electronic circuit components such as voltage and current sources, resistors, 

capacitors, and inductors. Figure 8 shows a basic RLC circuit model in Sage. It also includes components 

such as wire coil, permanent magnets, ferromagnetic material, magnetic flux sources, and magnetic field 

sources. These components can be used to develop simple circuit models or combined with mechanical 

components to create more complex parts such as alternators or linear motors. The components are 

connected in the same manner as the mechanical components. The components have only single 

input/output current interfaces, but may be connected to connection blocks or voltage references, which 

can have multiple (user-defined) interface connections. It should be noted that all electromagnetic 

components are time-ring rather than phasor components. 

Another component of the library is labeled a "transducer." This part largely ignores the physics of 

the interaction of mechanical force and electric current in an alternator and assumes the relationship 

shown in Equation (1), where Cf is the measured motor constant. It also assumes ideal power transfer 

from the piston to the alternator by Equation (2). In reality there is loss associated with the alternator. 

Although not as rigorous as developing an alternator model from basic components, this method does 

provide a simplified approach of developing an alternator model to test with the existing thermodynamic 

model of the ASC. This is useful for testing the integration of the two models as well as learning some of 

the idiosyncrasies of the Sage modeling system. 

(1) 

F dx/ dt =.tJVI (2) 

Linear Alternator 

The linear alternator operates on the principle of Faraday's law. Magnets are attached to a magnet can 

on the piston of the ASC such that they pass through the alternator coil as the piston reciprocates. The 

changing magnetic field through the alternator coil induces a voltage Vemf The voltage generated is 

proportional to rate of change of magnetic flux rp through the coil (therefore proportional to the velocity 

of the piston) and proportional to the number of turns N in the coil (Eq. (3)). 

(3) 

NASAlTM-20 13-216563 6 
ACPower 

Supply 

Tuning Cap 

Load 

Rl R2 Ralt Lalt 

Figure 9.-Circuit diagram of the linear alternator and AC bus controller. 

29 FG' ~ ~ +FG' ZI 

35 IG' -<-y~IG' 31 

41 IGt.... f 

Gt 

.... 1m 45 

.... IGt 44 

31 IG' .... .... IG' 33 -"Ny.. 

Gt 

Aoit 

45 IG' .... ..,. 1m 46 

-vGt 

33 IG, .... ~ ~IG' ZI 

Gt 

- 44 1m .... ... 1m 47 

~ 

Gt 

!rod 

37 IGI .... -;- -;.IG' 39 39 IG, .... if .... IG' ~1 

ICUstarl C~Of 

4S IG' -<- 1. 471Gt .... ~ 

I 

.... IGI 35 

Figure 1 O.-Linear alternator circuit model in Sage. 

The circuit diagram in Figure 9 is a representation of the linear alternator elements and the AC bus 

controller. The coil of wire in the alternator has an inductance and resistance represented by Lall and Rail in 

the diagram. The resistors Rl and R2 represent the wire resistance of the circuit and the lead resistance, 

and are typically small values. The inductance of the alternator creates a voltage phase shift relative to the 

current, which reduces the power factor. A tuning capacitor is used to correct the phase shift caused by 

the alternator inductance. The AC power supply (or AC bus voltage) is used to control the piston 

amplitude by applying a voltage back to the alternator. The magnitude of the AC bus voltage alters the 

alternator force on the piston thereby altering the piston amplitude and phase. It is important to model and 

understand this relationship to be able to achieve control and synchronization of both pistons in an ASRG. 

The circuit can be solved simply by Kirchhoff s voltage law to find the necessary AC bus voltage; 

however, it is more useful to model this in Sage to see how circuit elements affect ASC performance. 

Figure 10 shows the linear alternator circuit in Sage using the transducer model component. A force input 

to the transducer (from a piston) is translated to a current. The amplitude and frequency of the piston 

determines the voltage generated. 

~odellntegration 

To form a complete model, the alternator needs to be combined with the ASC model, as shown in 

Figure 11. The alternator model (Fig. 10) is inserted into the ASC Sage model (Fig. 5) and a force 

interface is added to the free-piston component in order to connect to the force interface of the transducer. 

Unfortunately, the transducer component does not consider any losses due to eddy currents or hysteresis. 

To simulate some loss, a damper was added and connected to the piston. While not entirely accurate, it is 

enough for development purposes and can be calibrated based on test data at a later point. 

The ASC model assumes a constrained piston (piston with set amplitude). This is not realistic but it is 

useful in determining system characteristics with a set piston amplitude. The piston amplitude is set as an 

input and system forces are calculated. Any required force necessary to keep the piston in motion at the 

given amplitude is calculated and output as a required forcing function (Fig. l2(a». With the addition of 

the linear alternator model this can also be used as a sanity check on the model. The AC bus voltage is 

used to control piston amplitude; therefore, the required forcing function should be driven to zero with a 

reasonable voltage input as shown in Figure 12(b). 

NASAlTM-20 13-216563 7 
a) 

b) 

'::::::::0: ~mGt 19 ~~PpNt2t 

Cf'-+PpIm 31 

..... ,,--

'ejeclcr 

3 Atr;; ~ ~::::::=1 -;. mGt 30 

15 P,,",'" -E===h -..Pp,," 5 

20 mGt -E-Cf'~ritGt 11 

~sp..:e 

17 o*" ~ ?O!~ 10 

~:=~ 

""" .... 

Gt-fMrmotion fler 

fpho<>. 

~ 

Phsr 

di1p1.xerctiver 

"0 ~t +- ~::::::~ S Pphsr+~ 

7 mGt ~Cf' 

~on,pace 

""''''''', 

filF"", ... p 

Phsr 

.m.,.. 

'9 IG'''' I 

Gt 

~ IGI 53 53 IGI +- -? 1m 54 52 [Gt -<E- 71G' 55 5< IG'''' I 55 IGt'" .J,. 

OUtputs 

F 

Wnet 

FF 

OUtputs 

F 

Wnet 

FF 

-..IG. 52 -vGt 

ACe .. 

Gt voI:ogclefetenc:e 

Figure 11.-ASC constrained piston and linear alternator model. 

boundaxy foree (N. rad) 

b6unduy pcmer infl~ ('if) 

requi.red foreinq function {N, r&d) 

b6undary force (N. rad) 

boundary pcmer infJ.ow {In 

required forcing function (N • .rad) 

2.842E+02 cis! 2.929) 

8. 614!i:+Ol 

6.944E+Ol cis{-l.OtS) 

2.42912:+02 cis (;"3~ 1(2) 

-l.BOSE-OS 

1.317!-OS cis( 1.8'8) 

Figure 12.-(a) Model output with required forcing function (69.44 N). (b) Model output with required forcing 

function driven to near zero (1.809E-05 N). 

To create a more realistic model, the constrained piston component was replaced with a free-piston 

component. This allows the piston amplitude to vary based on system input parameters, including 

temperature variations of the system and AC bus voltage changes. Piston amplitude is now an output of 

the Sage solver rather than an input. 

Alternator Voltage Phasor Diagram 

Using the new model of ASC and the Sage dll, the MATLAB GUI is able to run the model and 

collect the outputs to create a voltage phasor diagram of the linear alternator. Voltage amplitudes and 

phases are taken in by the GUI and vectors created in a bead-to-tail method. Tbe pbase oftbe voltage 

through a resistor is the same as the phase of the current, hence, the voltage Rail is in phase with current. 

The phase of the current is considered to be along the x-axis in the voltage pbasor diagram. 

NASAlTM-20 13-216563 8 
-iOII~~ 

-EM' 

--;o;a -l!I! .. 

__ 

~~ *-' 

-1iKIi!;", 

_1Iiil ...... 

-,....,;,q~~ 

Figure 13.-Voltage phasor diagram of linear alternator. 

Figure 13 shows a typical voltage phasor diagram for the linear alternator. The voltage across the 

alternator resistance is in phase with the current; however, the phasor is plotted in the opposite direction 

to indicate a voltage drop. The alternator inductance and tuning capacitance phasors are -900 and 900

, 

respectively, as a result of this sign convention. The alternator voltage phasor is the voltage at the 

terminal leads and is the sum of the Vern/> alternator resistance, and alternator inductance phasors. This is 

useful to plot because the terminal lead voltage can be measured in testing while the Vern!, resistance, and 

inductance can only be calculated. 

MA TLAB GUI and Model Improvements 

A new MATLAB GUI, shown in Figure 14, was developed to interface with the new Sage ASC 

model. The GUI expands the input panel to include linear alternator parameters. A main feature of the 

interface is the ability to set either AC bus voltage and solve for the resulting piston amplitude or vice 

versa. This capability is enabled by the newly integrated Sage model. Two phasor diagrams have been 

added to the original three (piston forces, displacer forces with separated pressures, and displacer forces 

with delta pressure). The phasor diagram of alternator voltages has been added along with corresponding 

output data. A phasor diagram depicting the phase of forces relative to the phase of the piston motion has 

also been added. 

Improvements 

Other improvements have been made to the GUI such as the ability to save input and output data. The 

output data includes the force data in the output window as well as the Sage output data listing for the 

model. Previous input data can then be loaded to rerun a simulation. Phasor diagrams can be opened into 

separate figure windows for easier viewing and manipulation. 

The heat input parameter has been improved to allow the user to specify the heat input of the heat 

source in watts instead of specifying the hot-end temperature. This can be useful to simulate the heat input 

of the GPHS. Sage; however, requires hot-end temperature in Kelvin as an input parameter. The net heat 

input to the convertor is not equal to the gross heat output from the GPHS. Some heat is lost to the 

insulation, leaving a net heat input (Qin net). An iterative process is used to determine Qin net and the 

corresponding hot-end temperature. The hot-end temperature is estimated and input to Sage, which 

calculates as an output the heat input Sage J]in" This is compared to the calculated Qin_nel> which takes into 

account heat loss through insulation. The estimated hot-end temperature is adjusted based on the 

difference between Sage _Qin and Qin_net and the cycle is repeated. 

NASNTM-2013-216563 9 
Figure 14.--MATLAB GUI running Sage model and plotting phasor diagrams. 

Future Work 

The new Sage model and MATLAB GUI offer increased capability over previous models; however, 

there is still room for improvement. The linear alternator model does not accurately calculate alternator 

losses. The method currently used can change as input parameters are altered, but it is unknown if these 

changes accurately represent reality. The current model also is a very simple representation of the 

alternator. A better physical model could be developed in Sage utilizing the magnetic and coil 

components to build a more physical representation. This type of model in Sage may also be able to better 

model alternator losses. Another improvement is to combine two ASC models in Sage to develop an 

ASRG model, allowing a more complete simulation of the entire generator. Finally, the models need to be 

validated with test data to quantifY their accuracy. 

Conclusion 

The addition of the linear alternator and AC bus controller to the ASC model gives a more realistic 

representation of the system. It allows the piston amplitude to be determined based on the system input 

parameters while offering insight into how system performance is affected by changes to the AC bus 

voltage, which was not possible in the previous model. The reworked GUI complements the new model 

by simplifYing the access to the Sage (Gedeon Associates) model parameters and providing quick insight 

into model performance through phasor diagrams. This tool can be easily adapted and expanded to 

accommodate future model improvements and GUI enhancements. 

References 

1. Richardson, R.; and Chan, J.: Advanced Stirling Radioisotope Generator. Proceedings of NASA 

Science Technology Conference (NSTC 2007), 2007. 

2. Wood, J.G., et al.: Continued Development of the Advanced Stirling Convertor (ASC). American 

Institute for Aeronautics and Astronautics, 2007. 

NASAlTM-20 13-216563 10 
3. Walker, G.; and Senft, l.R.: Free Piston Stirling Engines. Springer-Verlag, 1985. 

4. Shaler, K.; and Lewandowski, E.J.: Stirling Convertor Dynamic Analysis Using Phasor Diagrams. 

Proceedings of Nuclear and Emerging Technologies for Space (NETS 2011), 2011. 

5. Gedeon, David: Sage: Object-Oriented Software for Cryocooler Design. Cryocoolers 8, R.G. Ross, 

Jr., ed., Plenum Press, New York, 1995, pp. 281-292. 

6. Saba, Dipanjan; and Lewandowski, E.J.: Development of a Phasor Diagram Creator to Visualize the 

Piston and Displacer Forces in an Advanced Stirling Convertor. Proceedings of Nuclear and 

Emerging Technologies for Space (NETS 2013), 2013. 

NASA/1M-2013-216563 11