Dipanjan Saha

Development of a Phasor Diagram Creator to

Visualize the Piston and Displacer Forces in

an Advanced Stirling Convertor

NASA/TM—2013-216549

December 2013

NETS–2013–6760
http://www.sti.nasa.gov

help@sti.nasa.gov
Dipanjan Saha

Development of a Phasor Diagram Creator to

Visualize the Piston and Displacer Forces in

an Advanced Stirling Convertor

NASA/TM—2013-216549

December 2013

NETS–2013–6760
Level of Review

Acknowledgments
NASA/TM—2013-216549 1

Development of a Phasor Diagram Creator to Visualize the Piston

and Displacer Forces in an Advanced Stirling Convertor

Dipanjan Saha

University of Connecticut

Storrs-Mansfield, Connecticut 06269

Edward J. Lewandowski

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

The steady-state, nearly sinusoidal behavior of the components in a free-piston Stirling engine allows

for visualization of the forces in the system using phasor diagrams. Based on Newton’s second law, F =

ma, any phasor diagrams modeling a given component in a system should close if all of the acting forces

have been considered. Since the Advanced Stirling Radioisotope Generator (ASRG), currently being

developed for future NASA deep space missions, is made up of such nearly sinusoidally oscillating

components, its phasor diagrams would also be expected to close. A graphical user interface (GUI) has

been written in MATLAB (MathWorks), which takes user input data, passes it to Sage (Gedeon

Associates), a one-dimensional thermodynamic modeling program used to model the Stirling convertor,

runs Sage, and then automatically plots the phasor diagrams. Using this software tool, the effect of

varying different Sage inputs on the phasor diagrams was determined. The parameters varied were piston

amplitude, hot-end temperature, cold-end temperature, operating frequency, and displacer spring constant.

These phasor diagrams offer useful insight into convertor operation and performance.

Nomenclature

displacer acceleration (N/m2)

piston acceleration (N/m2)

negative facing area of displacer (m2)

piston area (m2)

positive facing area of displacer (m2)

area of displacer rod (m2)

required alternator forcing function (N)

. bounce spring force on displacer (N)

. piston bounce spring force (N)

. compression force on displacer (N)

. piston pressure force (N)

displacer inertial force (N)

. expansion force on displacer (N)

piston inertial force (N)

. displacer pressure force (N)

spring force (N)

. displacer DeltaP force (N)

displacer spring constant (N/m)
NASA/TM—2013-216549 2

displacer mass (kg)

piston mass (kg)

convertor mean pressure (psig)

bounce space pressure (Pa)

compression pressure (Pa)

pressure drop across displacer (Pa)

expansion pressure (Pa)

_ net heat input into the Stirling cycle (W)

_ heat input from the GPHS (W)

heat lost through the insulation (W)

pressure vessel temperature (the APDG sets this equal to cold-end temperature plus 8 K)

(K)

acceptor temperature = hot-end temperature (K)

displacer position (m)

Introduction

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

developed for future deep space missions. One of the main components of the ASRG is the Advanced

Stirling Convertor (ASC), developed by Sunpower, Inc. The ASC (Figure 1) is a free-piston Stirling

engine that uses the heat input from a General Purpose Heat Source (GPHS) to heat the working fluid,

helium, and through the Stirling cycle convert heat energy to electrical power. During convertor startup

the piston is put into motion by the alternator, and the movement of the piston generates a pressure wave

in the working fluid. However, the displacer and piston are not oscillating in phase, and so while the

piston generates the pressure wave, the motion of the displacer and the shuttling of gas through the heat

exchangers and regenerator determines the phase shift of the pressure wave. This results in net positive

work on the piston. The piston is connected to an alternator, so that once the system is started, the

movement of the piston will generate electrical power. Both the displacer and the piston are connected to

different springs (a mechanical spring is connected to the displacer and gas springs act on the piston),

which provide restoring forces to each component. The alternator provides additional spring force to the

piston, in addition to damping. As a result, all of the components in the system oscillate at the same

frequency, but they are not all in the same phase (Walker and Senft (1985)). Therefore, it is useful to

represent the forces on the piston and displacer as phasors, which only require a magnitude and phase

angle shift. If all of the forces on the displacer and piston are accounted for, then the phasor diagram

should form a closed polygon, graphically describing Newton’s second law that F = ma.

Figure 1.—Cutaway view of ASC.
NASA/TM—2013-216549 3

Figure 2.—MATLAB GUI operation with Sage.

Software Developed

The phasor plotting tool incorporates a user friendly graphical user interface (GUI) in MATLAB that

is able to take input parameters for operating frequency (Hz), heat input from the GPHS (W) or hot-end

temperature (K), cold-end temperature (K), displacer spring constant (N/m), and either a user-defined

mean pressure (Pa) or a mean pressure can be calculated by the software as a function of hot-end

temperature. These parameters are then sent by MATLAB through a .dll file to be used as inputs for Sage,

a one-dimensional thermodynamic modeling software developed by Gedeon Associates, which runs the

model until a converging solution is found (Gedeon (1995)). MATLAB then pulls the desired outputs

from Sage and uses them to calculate the forces on the displacer and piston. This process is shown

schematically in Figure 2.

Displacer With Separated Pressure Forces

There are two ways to represent the forces acting on the displacer in the ASC. One method is to look

at the force on the displacer from the gas in the expansion space and the gas from the compression space.

Although both spaces are connected through the regenerator and are parts of the same volume, the gases

are actually represented by slightly different pressure waves (different in both magnitude and phase shift).

This is the method used by Sage to model displacer forces.

The area on the side of the displacer facing the expansion space is greater than the area on the

compression side since the compression side has less area due to a displacer rod attached to the

mechanical planar spring. Since P = F/A, a larger area and a constant pressure will result in a larger force.

The other forces acting on the displacer are a spring force and an inertial force. The inertial force is the

force required to stop the motion of the displacer and is equal to ma. The spring force is a restoring force

that should always be in phase with the inertial force and 180° to the displacer’s position phasor. There is

also a bounce spring force acting on the displacer through the displacer rod. However, both the displacer

rod area and the bounce space pressure variation are small so that the phasor’s magnitude is a small

fraction of the other forces and can often be ignored in analysis.

. = ( )( ) (1)

. = ( )( ) (2)

. = ( )( ) (3)

Start:

User Input

Pass user inputs

to .dll Call Sage Set user inputs asinput parameters

Run model until

a converged

solution is found

Extract Sage

outputs

Pass outputs

from Sage to GUI

Calculate phasor

forces and create

phasor diagrams

MATLAB

GUI

C++

Compiler .dll File SAGE
NASA/TM—2013-216549 4

= ( )( ) (4)

= ( )( ) (5)

Displacer With Pressure Drop

The other method of interpreting the forces on the displacer is to assume a uniform pressure wave in

the working space volume (both the compression and expansion spaces), and then determine the pressure

drop across the regenerator, heater, and cooler as a separate DeltaP force. The DeltaP phasor describes the

work being done by the displacer to shuttle the gas back and forth between the hot and cold ends of the

working space. This is the method used by NASA Glenn’s Stirling convertor system dynamic model

(SDM) to determine displacer forces (Lewandowski and Regan (2004)). The DeltaP phasor diagram

should have the same inertial, bounce spring, and mechanical spring force phasors as the Separated

Pressure Forces Phasor Diagram described previously.

. = ( )( ) (6)

. = ( )( ) (7)

. = ( )( ) (8)

= ( )( ) (9)

= ( )( ) (10)

Piston

The pressure wave in the compression space imparts a driving force on the piston. Since the piston is

connected to an alternator, there is also a force that the alternator applies to the piston to keep the entire

system in its oscillating state, a force which generally contains both spring and damping components. The

bounce spring also imparts a small restoring force to the piston in the same way the mechanical spring

does with the displacer. Finally, there is an inertia force on the piston, which is especially important

because of the connection between the alternator and the piston. The piston’s inertia phasor, in addition to

the alternator phasor (a direct output of Sage), are necessary inputs into the electrical components of the

ASRG system.

. = ( )( ) (11)

. = ( )( ) (12)

= ( )( ) (13)

Graphical User Interface

The GUI, shown in Figure 3, provides a user friendly method to run the relatively complex Sage

modeling software to generate phasor diagrams based on a specified set of inputs. Additionally, the user

may specify the error tolerance, the file location of the model to be run, export collected data into Excel,

and plot multiple phasor diagrams onto one plot in order to compare the effect of varying certain

parameters. The user also has a choice of whether to fix the mean pressure or whether the pressure should

be automatically calculated as a function of hot-end temperature, which was determined using an equation

that was derived theoretically and verified by a test of the ASC hardware.
NASA/TM—2013-216549 5

Figure 3.—Sample run of MATLAB GUI with varying temperature inputs.

Figure 4.—Diagram description of heat input

converger parameter relationships.

Determining Heat Input

The user also has the option of specifying heat input in watts or hot-end temperature. If the user

provides heat input, then the program will actually iterate to find the corresponding hot-end temperature

(Figure 4). Iterating is necessary because the heat input from the GPHS for a given insulation

environment and hot-end temperature is not necessarily equal to the heat supplied to the ASC as

calculated by Sage at that hot-end temperature. The amount of heat into the cycle is determined by

subtracting the heat of the GPHS from the heat lost through imperfect insulation.

Insulation

Loss

Heat input

Going Into

the Cycle

Hot-End

Temperature
NASA/TM—2013-216549 6

Figure 5.—Time-dependent curves of multiple piston forces.

Qin_net = Qin_GPHS-Qinsulation loss (14)

However, the insulation loss is actually a function of the hot-end temperature, which is a function of

the Qin_net, and as mentioned, the Qin_net is affected by the insulation loss.

Phasors

A useful way to visually represent sinusoidal functions like those shown in Figure 5 that have the

form

( ) = cos( + ) (15)

is to plot them as phasors. A phasor is a vector whose magnitude is the amplitude of the cosine function

and whose angle from the positive x axis is determined by the function’s phase shift from a reference

cosine function. If multiple functions all have the same frequency, then their phasors can be plotted on the

same set of axes. Since the ASC’s components all operate sinusoidally at the same frequency, all of the

forces acting on the piston or displacer can be plotted on the same respective set of axes. The reason

multiple sinusoidal functions would want to be plotted as phasors on the same axes is that phasors make it

easier to add and subtract time varying sinusoidal functions. For example, when trying to verify F = ma,

all of the force phasors added together should be equivalent to the inertia phasor. To add phasors together,

the head of one phasor serves as the tail of the phasor to be added to it, and the angle of both phasors to

the reference positive x axis does not change.

Sage

Developed by Gedeon Associates, Sage is a modeling software tool used to accurately model Stirling

engines using interconnected components. A screenshot from the ASC Sage model is shown in Figure 6.

Using the inputs of hot-end temperature, cold-end temperature, operating frequency, displacer spring

constant, mean pressure, and piston amplitude, Sage runs a series of nonlinear differential equations to

find a converging solution to the system. User experience with ASC operating conditions is useful to run

this model because the model requires the input parameters to be within a certain range for a converging

solution to be found. Also, Sage does not automatically change the dependent parameters in certain

components. For example, setting the hot-end temperature in Sage is not enough; that temperature must

also be set for the acceptor temperature, displacer profile, and any other temperature that is a function of

hot-end temperature.

For the .dll file to interface with Sage, it is also necessary for all subcomponents to contain different

variable names so that the .dll can pull information from the correct data location.

There are currently a number of compatibility issues to deal with when running this GUI. Presently,

Sage is a 32-bit program whose dll requires a 32-bit operating system and can only interface with a 32-bit

version of MATLAB. It was also found that one of the only compatible C++ compilers was Microsoft

Visual C/C++ version 7.1 (also known as Microsoft Visual Studio .Net 2003). Efforts to migrate to 64-bit

systems are being pursued.

0

Falt

Bounce Spring

Pressure Wave

Inertia
NASA/TM—2013-216549 7

Figure 6.—Components used in ASC Sage model.

Parameter Variations

The effect of varying five parameters was analyzed to find how they change the phasor forces for

each component, similar to the approach taken in Shaler and Lewandowski (2011). The five parameters

analyzed were piston amplitude, hot-end temperature, cold-end temperature, operating frequency, and

displacer spring constant. The piston amplitudes were varied similarly to the cases done in Lewandowski

and Schreiber (2010). The other parameters where changed at increments so that the high or low range

would not cause the Sage modeling software to not converge (although by no means are the values below

the limits for the inputs of each parameter).While each parameter was varied at an evenly divisible

increment, the other parameters were held constant at the values described in Table I.

TABLE I.—PARAMETERS USED FOR MATLAB GRAPHICAL USER INTERFACE (GUI) SIMULATION

Piston amplitude,

m

Hot-end

temperature,

K

Cold-end

temperature,

K

Operating

frequency,

Hz

Displacer spring

constant,

Values held constant 0.004473 1115 307.5 102.2 23200

Variation range 0.004173 to 0.005073 915 to 1215 307.5 to 457 102.2 to 105.2 22200 to 25200
NASA/TM—2013-216549 8

Piston Amplitude

The piston amplitude can be constrained in Sage, which is a realistic assumption since the alternating

current (AC) bus voltage can be varied in response to other parameter changes to maintain a given piston

amplitude.

Varying piston amplitude seems to have the largest effect on the expansion and compression pressure

force magnitudes and only a negligible effect on the other phasors, as shown in Figure 7. This is expected

since higher piston amplitude will compress the gas in each space further, resulting in higher pressure

wave magnitude. Piston amplitude does not appear to have a drastic effect on varying the DeltaP loss.

Only very slight angle and magnitude effects can be seen.

Increasing piston amplitude logically increases the magnitude of the inertia of the piston, but also

requires an increase in the alternator forcing phasor magnitude. As a result output power is increased.

Generally, a restoring spring should have the opposite phase angle as the mass’ displacement and the

same angle as the mass’ acceleration. This is true for the mechanical planar spring attached to the

displacer. However, there is a very minor phase difference between the bounce spring force and the

inertial force for the piston. This difference is attributable to the fact that there is a phase difference

between the piston and the displacer; and more specifically, depending on the point in time, the bounce

space gas may be compressed by the piston but is providing a restoring force to the displacer. Because the

displacer rod is small, the shift in the force phasor for the bounce spring is also small. Also,

thermodynamic losses from compressing and expanding the bounce space gas can shift the angle of the

bounce spring phasor.

Figure 7.—Effects of varying piston amplitude on displacer (top two plots)

and piston (bottom).
NASA/TM—2013-216549 9

Hot-End Temperature

Increasing hot-end temperature has minor variations in the phase angles of all of the force phasors

(Figure 8). There is a small increase of the expansion and compression pressure magnitudes, which can be

explained by the application of Boyle’s Law.

There is a negligible effect on the phasor diagram with DeltaP. The variations shown in the plot are

misleading because the axis scale is very small.

Hot-end temperature has the largest effect on the pressure wave and the alternator forcing function.

The effect of varying hot-end temperature has the most interesting results with respect to the piston.

Changing hot-end temperature has absolutely no effect on the inertial force and a very meager one on the

bounce spring force. As mentioned, the bounce spring force is dependent on the motion of the displacer as

well as the piston, explaining the slight variation. Also, since the alternator forcing phasor is dependent on

the other phasors, the real effect of hot-end temperature is on the pressure force. Interestingly, both the

pressure forces’ magnitude and angle are varied as a function of hot-end temperature and the subsequent

result is that as hot-end temperature increases, the alternator forcing angle and the alternator forcing

magnitude both increase with respect to the positive x-axis and going counterclockwise.

Cold-End Temperature

Varying cold-end temperature has the largest effect on the expansion and compression pressure

magnitudes (Figure 9). There are slight phase differences, which appear magnified by the large pressure

magnitudes.

Figure 8.—Effects of varying hot-end temperature on displacer (top two

plots) and piston (bottom).
NASA/TM—2013-216549 10

Figure 9.—Effects of varying cold-end temperature on displacer (top two

plots) and piston (bottom).

Varying cold-end temperature clearly has a large effect on the phase angles of the alternator forcing

function on the piston. Interestingly, the damping component (y component) does not seem to be

changing with variations in cold-end temperature. Since the electrical circuit of the ASRG depends on the

piston velocity for voltage and the alternator force damping component for current, and the piston is being

constrained so that its velocity is not changing, if the alternator force damping component does not

change then the convertor’s output power is not changing either. If hot-end temperature is fixed,

decreasing cold-end temperature should result in greater power because of greater convertor efficiency

(Carnot efficiency increases). Although Sage more accurately models Stirling convertor thermodynamics

than SDM, the constancy of the damping component suggests that Sage is constraining a variable like the

piston amplitude that would not be constrained in the actual system (Regan and Lewandowski (2005)).

Most of the variations in the phasor diagrams are also seen in the discussion of hot-end temperature.

These similarities are expected since a Stirling engine depends on two thermal reservoirs and changing

the temperature difference from either end will have some similar effects. The major difference however

is that changing cold-end temperature has a more noticeable impact on the piston bounce spring

magnitude than hot-end temperature did. This is also expected since the bounce space gas volume is an

order of magnitude larger than the expansion space and because the bounce space temperature should be

close to the cold-end temperature, varying that temperature will have a large effect on the effective spring

constant of the bounce spring. Changing cold-end temperature has a greater effect on the convertor’s

natural frequency than changing hot-end temperature.

To compare the phasor diagrams of the SDM model from the Sage phasor diagrams, increasing coldend temperature has a similar effect on the angles and magnitudes of each of the forces (Shaler and

Lewandowski (2011)).
NASA/TM—2013-216549 11

Operating Frequency

Varying operating frequency has the most noticeable (albeit relatively small) effect on the expansion,

compression, and inertia magnitudes and the spring/inertia phase angles, as shown in Figure 10.

A small variation in operating frequency has a noticeable effect on DeltaP’s magnitude. A 3 Hz

change brings down the magnitude by about 2 Newtons. It is unlikely that such a relationship is linear and

minimizing the DeltaP loss does not necessarily equate to the overall system being more efficient.

Varying operating frequency has a very noticeable effect on the alternator phase angle, pressure

magnitude, and inertia magnitude.

Interestingly, as operating frequency increases, both the expansion pressure angle and compression

pressure angle decrease, so both phasors are moving clockwise. As a result, the DeltaP phasor is barely

changing as a function of operating frequency. This is a trend also seen in SDM modeling of the ASC

(Shaler and Lewandowski (2011)). With all other parameter variations (except for displacer spring

constant), the expansion and compression pressure angles have always varied so that there is an effect on

DeltaP.

Figure 10.—Effects of varying operating frequency on displacer (top two

plots) and piston (bottom).
NASA/TM—2013-216549 12

Figure 11.—Effects of varying displacer spring constant on displacer (top

two plots) and piston (bottom).

Displacer Spring Constant

The direct effect of changing the displacer spring constant is to change the displacer amplitude, as can

be seen by the change in the magnitude of the displacer inertial force phasor (Figure 11). This results in

the DeltaP magnitude and phase shift increasing as the displacer spring constant increases. The change in

displacer amplitude also affects output power, as indicated by the change in alternator damping. The

alternator phase angle changes, indicating that the amount of spring force added by the alternator to

changes due the change in displacer spring constant.

Of all of the parameters varied, displacer spring constant seems to have the largest effect on the

damping component of the alternator forcing function. For all of the other parameter changes, the data

regressions have all had an R2

 value close to 1 and never less than 0.95. For the spring constant, a

quadratic regression was needed or some of the R2 values would be around 0.92.

Improvements

The software tool has certainly automated the process of generating phasor diagrams from Sage.

However, for a holistic understanding of how changing various parameters affect the ASRG, an alternator

model needs to be added. Also, different insulation models (air, vacuum, etc.) would be useful to offer

explanations to the results of different test situations. The heat input converger also needs improvement in

that its converging time is dependent upon the initial guess. Currently, the iteration method to reach a

converging solution is rather primitive and the integration of a solver would improve the software’s

performance. As mentioned previously, the ASC model used by Sage needs to be set up to allow

parameters like piston amplitude to change when other parameters like cold-end temperature are changing

to make the tool better mimic the effects seen during testing.
NASA/TM—2013-216549 13

Conclusions

Phasor diagrams are a visually useful method of interpreting steady-state sinusoidal behavior. For a

system such as the Advanced Stirling Convertor (ASC), all of whose components are oscillating nearly

sinusoidally, it is especially useful to arrange the forces acting on a component into a phasor diagram, to

find the relationship among forces as certain parameters are changed. Of the parameters changed, the

inverse relationship of the expansion pressure and compression pressures as functions of displacer spring

constant and operating frequency were most interesting. Displacer spring constant also seems to be a

useful parameter to change in that it can be used to increase the damping component of the alternator

forcing function, which could generate more power by the system.

The software used has been successful at providing a user friendly method of varying parameters in

Sage (Gedeon Associates) running the model, and plotting the phasor diagrams. In comparison to the

phasor diagrams from the system dynamic model (SDM), similar behaviors in the phasor diagrams are

seen. However, since the Sage model is a more accurate way to model the system than the SDM, better

insight can be gained by looking at its phasor diagrams. As the alternator circuit is added to the tool, a

more complete picture will be available for how and why the ASC operates the way that it does.

References

Gedeon, D. (1995): Sage: Object-Oriented Software for Cryocooler Design. Cryocoolers 8, R.G. Ross,

Jr., ed., Plenum Press, New York, NY, pp. 281 292.

Lewandowski, E.J.; and Regan, T.F. (2004): Overview of the GRC Stirling Convertor System Dynamic

Model. AIAA 2004 5671.

Lewandowski, E.J.; and Schreiber, J.G. (2010): Testing to Characterize the Advanced Stirling

Radioisotope Generator Engineering Unit. Proceedings of the Eight International Energy Conversion

Engineering Conference (IECEC 2010), Nashville, TN.

Regan, T.F.; and Lewandowski, E.J. (2005): Development of a Stirling System Dynamic Model with

Enhanced Thermodynamics. Proceedings of the Space Technology and Applications International

Forum (STAIF 2005), Albuquerque, NM.

Shaler, K.; and Lewandowski, E.J. (2011): Stirling Convertor Dynamic Analysis Using Phasor Diagrams.

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

Walker, G.; and Senft, J.R. (1985): Free Piston Stirling Engines, Springer-Verlag, Berlin.