Jeffryes W. Chapman

Vantage Partners, LLC, Brook Park, Ohio

Thomas M. Lavelle

Glenn Research Center, Cleveland, Ohio

Ryan D. May

Vantage Partners, LLC, Brook Park, Ohio

Jonathan S. Litt and Ten-Huei Guo

Glenn Research Center, Cleveland, Ohio

Propulsion System Simulation Using the Toolbox 

for the Modeling and Analysis of Thermodynamic 

Systems (T-MATS)

NASA/TM—2014-218410

November 2014
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Jeffryes W. Chapman

Vantage Partners, LLC, Brook Park, Ohio

Thomas M. Lavelle

Glenn Research Center, Cleveland, Ohio

Ryan D. May

Vantage Partners, LLC, Brook Park, Ohio

Jonathan S. Litt and Ten-Huei Guo

Glenn Research Center, Cleveland, Ohio

Propulsion System Simulation Using the Toolbox 

for the Modeling and Analysis of Thermodynamic 

Systems (T-MATS)

NASA/TM—2014-218410

November 2014

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

50th Joint Propulsion Conference

cosponsored by AIAA, ASME, SAE, and ASEE

Cleveland, Ohio, July 28–30, 2014
Acknowledgments

The authors would like to thank the NASA Aviation Safety Program’s Vehicle Systems Safety Technologies Project for funding this

work. The assistance of the OpenMDAO team was critical to enabling the authors to release this tool under an open-source license.

Available from

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Available electronically at http://www.sti.nasa.gov

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only. Their usage does not constitute an of¿ cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2014-218410 1 

Propulsion System Simulation Using the Toolbox for the Modeling 

and Analysis of Thermodynamic Systems (T-MATS) 

Jeffryes W. Chapman 

Vantage Partners, LLC 

Brook Park, Ohio 44142 

Thomas M. Lavelle 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Ryan D. May 

Vantage Partners, LLC 

Brook Park, Ohio 44142 

Jonathan S. Litt and Ten-Huei Guo 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

A simulation toolbox has been developed for the creation of both steady-state and dynamic 

thermodynamic software models. This paper describes the Toolbox for the Modeling and Analysis of 

Thermodynamic Systems (T-MATS), which combines generic thermodynamic and controls modeling 

libraries with a numerical iterative solver to create a framework for the development of thermodynamic 

system simulations, such as gas turbine engines. The objective of this paper is to present an overview of 

T-MATS, the theory used in the creation of the module sets, and a possible propulsion simulation 

architecture. A model comparison was conducted by matching steady-state performance results from a 

T-MATS developed gas turbine simulation to a well-documented steady-state simulation. Transient 

modeling capabilities are then demonstrated when the steady-state T-MATS model is updated to run 

dynamically. 

Nomenclature

Alt altitude 

Ath throat area 

BPR branch pressure ratio 

C-MAPSS40k Commercial Modular Aero-Propulsion System Simulation 40,000 lbf thrust 

Cv specific heat at constant volume 

dT delta temperature from standard day 

dt delta time 

DLL dynamic-link library 

Eff efficiency 

f(x) generic function 

FAR fuel to air ratio 
NASA/TM—2014-218410 2 

Fg gross thrust 

g acceleration of gravity

Ȗ specific heat ratio 

ht total enthalpy 

hs static enthalpy 

I shaft inertia 

J Jacobian 

k iteration 

l iteration number 

MN Mach number 

N shaft speed 

n iteration step counter 

Nc corrected shaft speed 

Ndmd demanded shaft speed into controller 

Ndot derivative of shaft speed with respect to time 

Nmech mechanical shaft speed 

NPSS Numerical Propulsion System Simulation 

NR Newton-Raphson 

PI proportional integral 

PLA power lever angle 

PR pressure ratio 

Pa ambient pressure 

Ps static pressure 

Pt total pressure 

R specific gas constant 

R-line unique line on the compressor map 

ȡ density 

S entropy 

SPR stall pressure ratio 

SS steady-state 

t time 

T-MATS Toolbox for the Modeling and Analysis of Thermodynamic Systems 

Trq torque 

Ts static temperature 

Tt total temperature 

W mass flow rate 

Wc mass flow rate corrected to local conditions 

Wf mass flow rate corrected to local conditions 

x generic variable 

X_il generic inner loop variable 

X_ol generic outer loop variable 
NASA/TM—2014-218410 3 

I. Introduction 

As computer technology has improved, advancements in propulsion system modeling have become 

synonymous with increased complexity. In many cases, this complexity stems from the integration of 

components created with different development tools. Control system design and research has been 

particularly affected by this type of integration complexity due to the need for incorporating legacy 

models into a dynamic simulation package. An effective means of reducing this model integration 

complexity is to build the complete system in a single development package, such as MATLAB/ 

Simulink (The MathWorks, Inc.). MATLAB is a design tool commonly used among control engineers 

and researchers and contains the graphical programming tool Simulink, which is designed for the 

simulation of dynamic systems. 

A number of thermodynamic simulation tools that model propulsion systems have been developed, 

which integrate with Simulink with varying levels of difficulty. These tools include Numerical Propulsion 

System Simulation (NPSS), GasTurb TurboLib, and Commercial Modular Aero-Propulsion System 

Simulation 40,000 (C-MAPSS40k). The NPSS software is a powerful and highly flexible platform in 

which propulsion system simulations can be created modularly (Ref. 1). Based on a code similar to C, 

NPSS offers modifiable gas turbine components that may be combined into larger systems. While NPSS 

is very powerful, integration of an NPSS model with other platforms, such as Simulink, can be time 

consuming (Ref. 2). GasTurb, another gas turbine modeling program, offers a graphical user interface that 

may be used to create engine simulations (Ref. 3). While providing a more visual interface than NPSS, 

this product also requires a user to work with a stand-alone software package for simulation development. 

Additionally, integration with other modeling software can be challenging, as GasTurb requires the 

creation of specialized dynamic-link library (DLL) files for cross-platform integration. Alternatively, the 

Simulink plug-in ThermoLib is available to expand the Simulink block sets to include thermodynamic 

modeling capability directly in Simulink, eliminating the need for system integration (Ref. 4). ThermoLib 

blocks, however, have limited code access; and, as a result, updating components to meet modeling 

requirements can become complicated. Another Simulink-based tool is the C-MAPSS40k software 

developed by NASA that combines a realistic turbine engine simulation with an industry-standard 

controller (Ref. 5). Not only does C-MAPSS40k solve many model integration issues by allowing a 

designer to work totally in the MATLAB/Simulink environment, it also provides the user full access to 

the development code. Despite this code access, C-MAPSS40k is not a generic tool, as it is a model of a 

particular gas turbine engine that may not represent the system a developer is interested in. 

To fill the need for a thermodynamic transient and controls design simulation package that is built in 

Simulink, has component-level flexibility, and maintains a generic simulation framework, the Toolbox for 

the Modeling and Analysis of Thermodynamic Systems (T-MATS) software package was developed by 

NASA Glenn Research Center. T-MATS is an unrestricted, open source Simulink toolbox (plug-in) that 

can be used by any industry professionals or academics interested in the development of custom 

thermodynamic and controls simulations. The T-MATS software package contains specialized modeling 

libraries that add to the Simulink block sets, which include numerical iterative solvers and advanced 

turbo-machinery component models, enabling a developer to create complex propulsion system 

simulations, such as a gas turbine engine (Ref. 6). 

 
NASA/TM—2014-218410 4 

Installation of T-MATS creates a new Simulink library with the following block sets: Effectors and 

Controls, containing controller algorithms and control system hardware simulation blocks; Numerical 

Methods, offering self-contained numerical method operations; Solvers, providing flexible iterative 

solvers; and Turbo-machinery, consisting of specialized gas turbine simulation functionality. By using 

these block sets, a user can rapidly create complex simulations by dragging blocks into a Simulink model 

and connecting them appropriately. Furthermore, T-MATS block design details are parameterized, which 

allows a developer to quickly provide the required information. In addition to block sets for 

thermodynamic modeling, T-MATS contains Model Development Tools designed to help reduce the time 

required to create a Simulink simulation. 

The remainder of this paper discusses the modeling theory, application to a gas turbine engine 

simulation, and a model matching between T-MATS and a “truth” model from NPSS. Section II presents 

an architecture overview for model development using the T-MATS software package. Simulink blocks 

and tools added to MATLAB/Simulink upon installation of T-MATS are discussed in Section III. 

Section IV details a model matching case for T-MATS blocks by comparing gas turbine data generated 

from an NPSS model, component by component, with a T-MATS model developed with the same maps 

and constants. A summary may be found in Section V. Finally, information for obtaining T-MATS is 

given in Appendix B. 

II. System Model Architecture Overview 

Simulations can be divided into two major categories: steady-state and dynamic. A steady-state 

simulation is a model that is unchanging with time, and can be used for analysis and design in the absence 

of requirements on system transients. A dynamic simulation is one that changes with time and can be 

used to model systems for controls development, a key feature of T-MATS. In thermodynamic modeling, 

both types of simulations are useful and T-MATS contains all the tools required for setting up either. 

In setting up simulations, T-MATS makes use of a generic and modifiable multi-loop architecture, 

which contains an “outer” loop iterated over time, t, and an “inner” loop set up to solve for internal 

variable imbalances over discrete iterations, k. In turbo-machinery systems, the form and solution method 

for these imbalances is often dictated by plant composition (Ref. 7). A diagram of this architecture, with a 

single “inner” loop (blue) and an “outer” loop (green), can be seen in Figure 1. The inner loop plant 

contains all components that have thermodynamic imbalances, such as turbines, compressors, and 

nozzles. Plant imbalance measurements, or dependents, f(x(k)), are routed from the inner loop plant to the 

iterative solver. The iterative solver then adjusts the plant dependents by making changes to the inner loop 

plant dependent effectors, or independents, X_il(k+1). In a typical gas turbine simulation, dependents 

often take the form of component flow errors, while each independent may specify input mass flow, 

compressor R-line, bypass pressure ratio (BPR), or turbine pressure ratio (PR). The iterative solver works 

in tandem with a “while loop” to pause the simulation at each time step, allowing the iterations to reach 

plant convergence (i.e., drive the dependents to zero) before the next time step. Components within the 

“outer” loop act to integrate the system within the time domain and may or may not be time dependent. 

These include outer loop effectors that feed signals from outside the “inner” loop, such as environmental 

conditions like altitude (Alt) or Mach number (MN), and system commands, such as power lever angle 

(PLA). Controllers, shaft dynamics, and other time-dependent portions of the simulation are located 

within the outer loop plant, which uses a feedback to the inner loop plant (y(t)) to generate inner loop 

plant inputs (X_ol(t+dt)). The architecture shown in Figure 1 is appropriate for the creation of a dynamic 

or steady-state simulation. (To create a steady-state simulation, the portion shaded in green can be 

removed since the simulation is not time dependent.) 
NASA/TM—2014-218410 5 

Figure 1.—T-MATS sample model architecture. 

III. T-MATS Components and Theoretical Development 

The T-MATS software package comes with components and tools that can be used for implementing 

models of thermodynamic systems. This section will review the components for modeling effectors, 

controls, and turbo-machinery, the iterative solvers, and the model development tools, discussing some of 

the theory behind their development, and detailing how they may be used in the simulation creation 

process. 

A. Turbo-Machinery Components 

Turbo-machinery, as it relates to T-MATS, is defined as the components that compose a gas turbine 

engine. A simple example of a turbo-machinery system is a turbojet, which contains a compressor, a 

burner, a turbine, and a nozzle in series (see the example in Section IV for further details of a turbojet 

model). The T-MATS software package contains the turbo-machinery component models listed in 

Table 1, which may be combined to create a 0-D component-level model of a gas turbine engine. A 0-D 

simulation is one in which the component-level dynamics are not modeled. It is important to note that 

T-MATS is completely open source and could be altered for higher order simulations, but the included 

turbo-machinery components are only valid for 0-D analyses. The following sub-sections describe the 

theory behind each T-MATS component used for gas turbine simulation development. 

 
NASA/TM—2014-218410 6 

TABLE 1.—DESCRIPTION OF COMPONENTS IN THE TURBO-MACHINERY BLOCK SET 

T-MATS block Description 

Ambient Converts Environmental conditions into measurable parameters (e.g., Alt to 

ambient pressure, Pa) 

Burner Transfers energy of a fuel burn to a gas stream 

Compressor Compresses gas stream based on input power 

Duct Models effects of a gas stream moving through a duct (pressure loss) 

Inlet Conditions gas stream for entry into system 

Nozzle Constricts the gas stream, modeling thrust 

Shaft Connection for rotating components 

Splitter Forks a gas stream 

Turbine Extract energy from a gas stream, creating a torque on a shaft 

Valve Split gas stream in relation to a fractional input 

1. Ambient Component 

The ambient component is used for converting the environmental variables altitude (Alt), temperature 

different from standard day (dT), and Mach number (MN) to the thermodynamic variables enthalpy, 

temperature, and pressure. Static pressure (Ps) and temperature (Ts) are determined from an empirical 

model based on Alt and MN. Total temperature (Tt) and pressure (Pt) are calculated based on the 

isentropic relations shown in Equations (1) and (2). It should be noted that the relationships in 

Equations (1) and (2) assume the specific heat ratio (J is a constant and modified forms of these 

equations should be used when applying the equations to high MN flow. Work is currently underway to 

update the ambient component to eliminate this issue. 

  ̧

1

·  ̈

©

§ J  

2

1 1 *2 MN

Ts

Tt (1)

 ̧ J /  J 1 1

·  ̈

©

§ Tt

Ts

Pt

Ps (2) 

Enthalpy is determined empirically from a T-MATS-provided table that relates enthalpy to 

temperature based on an assumed gas composition of air and Jet-A fuel. It is important to note that, in 

other components, entropy is also calculated empirically based on temperature, pressure, and the same gas 

compositions. 

2. Burner Component 

The burner component, or combustor, acts to move energy from a fuel source into the gas stream. The 

magnitude of this energy transfer is modeled by performing a weighted average of the enthalpy of the gas 

stream (htair) and the lower heating value (LHV) of the fuel source taking into account burn efficiency 

(Eff), based on fuel to air ratio (FAR) as shown in Equation (3). The total enthalpy (httotal) of the air and 

mixed flow are used to empirically calculate an exit temperature using tables, as described above for the 

ambient component. Module exit pressure and burn efficiency are based on simple scale factor multipliers 

on input pressure and enthalpy, respectively. 

ht ht *   1 FAR LHV * FAR * Eff total air   (3)
NASA/TM—2014-218410 7 

3. Compressor Component 

The compressor component represents a mechanical device that increases the pressure of a gas, which 

can take many forms, such as a centrifugal or multi-stage axial compressor, or even a fan. Whatever form 

the physical compressor takes, T-MATS assumes the compression process is isentropic and can be 

represented by a map that typically defines the relationship between corrected mass flow rate (Wc) and 

compressor PR, and contains the unique values for corrected shaft speed (Nc) and efficiency (Eff) at any 

given point. In the T-MATS compressor component, PR, Wc, and Eff are empirically calculated using Nc

and R-line, an unique line on the compressor map relating Wc and PR. Once pressure and Eff have been 

determined, temperature can be calculated based on enthalpy and entropy tables. The power required for 

the compressor to operate is a function of the change in enthalpy across the compressor (ǻht) and mass 

flow (W) as seen in Equation (4). The shaft torque required to maintain current shaft speed, which must be 

provided from a power generating component such as a turbine, is calculated as a function of power and 

shaft speed (Nmech) as seen in Equation (5). In some compressors, air is removed (or bled) and used for 

turbine cooling. This cooling flow is modeled by removing associated mass flow and power from the core 

flow as dictated by where the bleed occurs. 

Power W *   'ht (4)

Nmech

Power Torque (5)

Using compressor maps is an effective way of determining how a compressor will function when 

running in expected operating regions. Maps do not, however, give an accurate representation of 

conditions outside of these expected norms or indicate what may cause this operation. One such condition 

occurs when the PR builds too high for a given mass flow rate and the compression process fails, causing 

the potentially damaging phenomenon of compressor stall. In T-MATS, the stall line, or the line on a 

compressor map at which a stall will occur, is defined by a stall pressure ratio (SPR) that is a function of 

corrected flow into the compressor. The stall margin provides a measure of how close the compressor is 

to the stall condition through the percent difference between the current pressure ratio (PR) and the SPR, 

as shown in Equation (6). It should be noted that if the compressor component crosses the stall line, stall 

margin will become negative but no other effects will be observable in the simulation since T-MATS does 

not contain a stall model. 

$ StallMargin[%]  ̧ $*100 1

·  ̈

©

§  PR

SPR PR (6)

In general, outputs of the compressor component can be determined from the thermodynamic state of 

the flow feeding into it from upstream. In order to determine the unique operating point, however, the 

R-line must be specified. This value can be determined from a mass flow imbalance resulting from 

comparison of the input mass flow and the mass flow interpolated from the compressor maps, as 

described above. Many turbo-machinery components built in T-MATS contain similar internal 

imbalances that may be reconciled by implementing a solver external to the components. These internal 

imbalances are generally flow imbalances (or errors) through components; in the steady-state case, the 

acceleration of the engine shaft(s), Ndot, are considered imbalances. Each T-MATS component that is 

prone to flow imbalance contains an output that depends on independent turbo-machinery inputs. A list of 

component independent and dependent variables is located in Table 2. When using a solver, the number 

of independents must match the number of dependents. Changes in mass flow rate, or Ndot in steady-state 

cases, have an effect on the entire system, therefore a one-to-one relationship between independent and 

dependent variables does not typically hold. 
NASA/TM—2014-218410 8 

TABLE 2.—TURBO-MACHINERY INDEPENDENT AND DEPENDENT VARIABLES 

T-MATS component Independent variable Dependent variable 

Ambient Mass Flow Rate (W) none 

Compressor R-line Normalized Mass Flow Rate Error 

Nozzle none Normalized Mass Flow Rate Error 

Shaft Ndot (SS only) NMech (SS only) 

Splitter BPR none 

Turbine Pressure Ratio (PR) Normalized Flow Error 

Central to modeling the operation of the compressor described above is the compressor map, which 

needs to be developed accurately in order to achieve realistic compressor performance. If generic maps 

are available, the iDesign tool can be used to automatically derive the map scale factors for compressors 

and turbines, and the throat areas of the nozzles, to meet particular engine requirements. Map scale factors 

are constants used to scale compressor and turbine maps to either normalize map inputs or to change the 

size of hardware components (for example, if a compressor for a simulation needs to be half the size of 

the compressor for which the map was created). The T-MATS iDesign feature calculates these map scale 

factors, along with nozzle throat areas, by fitting each component to the overall system model at a known 

steady-state, or design, point. Though generated at the design condition, scale factors created by this tool 

may be used to run the simulation in off-design conditions. (The technique of design-point analysis is 

beyond the scope of this document; for further information, see Ref. 1.) The use of this type of map 

scaling is a prerogative of the user: some may use it often while others may prefer to develop maps that 

are tuned to the specific application. It is up to the designer to determine what is appropriate for a specific 

application. 

4. Duct Component 

The duct component governs a transition between two gas turbine components. In T-MATS this is 

modeled as a simple fractional pressure loss. 

5. Inlet Component 

An inlet acts to guide the flow to the front of the engine face and is modeled empirically as a drop in 

pressure based on total pressure relative to ambient pressure. This modeling scheme assumes changes in 

total temperature across the nozzle are negligible, which may not be true in some cases, such as at 

supersonic speed. 

6. Nozzle Component 

The nozzle component acts to generate thrust for a gas turbine by forcing the output air through a 

narrow passage. Modeling the nozzle requires a combination of iterative solving, empirical lookups, and 

physic-based equations. The first step in determining the thrust is to iteratively solve (internal to the 

nozzle block) for the velocity, static pressure, and static temperature (Ts) of the flow at the throat of the 

nozzle by assuming Mach number (MN) is equal to 1 then using the relationships shown in Equations (7) 

and (8), where total and static enthalpy (ht and hs, respectively) are determined by table lookup, assuming 

the expansion process is isentropic and specific heat ratio (J) and the specific gas constant (R) are known. 

velocity 2 *   ht  hs (7) 

R Ts

velocity MN

Ȗ * * (8) 
NASA/TM—2014-218410 9 

If the calculated static pressure is greater than the ambient pressure, the nozzle is assumed to be choked 

and the velocity, static pressure, and static temperature computed iteratively are accurate. However, if the 

calculated static pressure is less than ambient pressure, the nozzle is not choked and MN, static 

temperature, and velocity are recalculated based on ambient pressure. 

Once velocity, static pressure (Ps), and static temperature (Ts) have been determined, the exit mass 

flow and thrust can be calculated as shown in Equations (9) to (11) where R is the specific gas constant, ȡ

is density, Ath is throat area, Pa is ambient pressure, g is the acceleration of gravity, and W is mass flow. 

R Ts

Ps

* U (9)

W U* velocity * Ath (10) 

velocity   Ps Pa Ath

g

w thrust *   *  ̧

 ̧

1

·

 ̈

 ̈

©

§ (11)

7. Shaft Component 

The shaft component supplies a connection medium for turbo-machinery. Shaft acceleration (Ndot) is 

calculated as the sum of the input torques divided by the shaft inertia (I) as described in Equation (12). 

 

I

CompoentTorques Ndot

2* S*

6 (12)

8. Splitter Component 

The splitter component divides the flow into two separate streams (W_branch1 and W_branch2) 

based on a BPR, as shown in Equations (13) and (14), and is typically used to model a multi-stream 

system, such as high bypass engines. 

  BPR

W BPR W branch

 1

* _ 2 (13)

BPR

W W branch

 1 _ 1 (14)

9. Turbine Component 

The turbine component is designed to extract energy from a mass flow by directing air across a series 

of blades to spin a rotor or shaft. As with a compressor, the turbine is assumed to be isentropic and can be 

represented by a map that is typically defined as the relationship between Wc and turbine PR, and 

contains the unique values for Nc and Eff at any given point. In the T-MATS turbine component, Nc and 

turbine PR are used to empirically calculate Wc and Eff. Turbine temperature, enthalpy, torque, and power 

are calculated in the same manner as for the compressor, detailed in Equations (4) and (5), with the 

exception that power and torque are produced by the turbine then used to drive a compressor(s). In many 

cases, the mass flow entering a turbine is directly from a burner and has a temperature high enough to 

cause structural damage. For this reason, cooling flow is often added to turbine blades to create a 

protective layer of high pressure, lower temperature gas. Because the turbine is 0-D, additional gas cannot 

be added to the interior of the turbine component, so T-MATS models this cooling air as a combination of 

mass flows added either before or after the turbine. 
NASA/TM—2014-218410 10 

10. Valve Component 

The valve component is a variable bypass orifice governed by a valve position that can open and 

close to increase or decrease bypass flow. Mass flow through the valve component is governed 

empirically by looking up mass flow values based on the pressure ratio across the valve. Valve position is 

taken into account with a flow multiplier that governs the opening and closing of the orifice. The model 

assumes that the valve exit stream is piped directly to a relatively large bypass stream and that changes in 

temperature and pressure of the bypass stream are negligible. 

B. T-MATS Solving Method 

The T-MATS solvers are based around an iterative solver which operates with information defined by 

a Jacobian calculator. The iterative solver makes use of the Newton-Raphson (NR) method to step a plant 

toward a solution, a process described mathematically in Equation (15), where k is the step iteration 

number. 

       $ J

f x k

x k 1 x k  (15)

The Jacobian (J) is a linear map between the inputs (x) and outputs (f(x)) of a plant and is defined by 

perturbing each x from its initial conditions (x(0)) to find the effect on f(x). A mathematical description of 

the Jacobian is given in Equation (16). For non-linear systems, the Jacobian is only valid in a 

neighborhood local to the linearization point. 

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

w

w

w

w

w

w

w

w

 

n

m m

n

x

f

x

f

x

f

x

f

J



  



1

1

1

1

(16)

These two equations, Equations (15) and (16), represent the two main steps of operation in a 

T-MATS solver. Initially, the Jacobian calculator component creates a linear representation of the plant at 

the current operating point. This requires perturbing slightly, in turn, each input from its initial condition, 

recording the results, and numerically determining the partial derivative of each output with respect to 

each input. The Jacobian is built from these calculations, inverted, and provided to a NR solver. In the 

second step, the NR solver steps toward a solution using the Jacobian developed in the first step. Because 

this solver method makes the assumption that the plant is locally linear, there is a chance the solver will 

not converge to a solution. To reduce the chances of non-convergence, it is important to have solver 

initial conditions and perturbation sizes set to values appropriate for the system. 

In order to increase computational speed, the Jacobian calculator is not run every time step. A 

Jacobian is calculated at the beginning of the simulation and is not recalculated unless the NR solver has 

difficulty converging. The NR solver is configurable, giving users the capability of specifying the bounds 

of convergence, as well as the maximum number of iterations to try in a single time step. 

In T-MATS, the method described above is used as the basis for a steady-state and a dynamic solver. 

When running dynamic simulations, the dynamic solver block can be utilized, in conjunction with a 

“while loop” as shown in Figure 1, to ensure convergence of a simulation at each time step. During 

testing, or when determining initial conditions for the system, it is sometimes advantageous to remove the 

time iteration and simply solve the system in steady-state. The steady-state solver does not require the 

“while loop” to operate, as it treats each time step as the solver iteration, an invalid model architecture if 

the simulation contains dynamic effects. 
NASA/TM—2014-218410 11 

C. Effectors and Controls Components 

The effectors and controls components are intended for use in creating the “outer” loop hardware and 

controller algorithm models, providing a starting point for control system simulation. Included are the 

following blocks: a proportional integral (PI) controller, a fuel actuator model, and a sensor model. The 

sensor and actuator models consist of a first order lag with the ability to add Gaussian noise, faults, and 

other nonlinearities. The default PI controller is a simplified set-point regulator that is designed as a 

template that encourages controller expansion. 

D. Model Development Tools 

In addition to the block sets, T-MATS includes a set of tools, accessible from the Simulink menu bar, 

that make model creation easier and faster. Discussing these tools in detail is beyond the scope of this 

paper, however a list of the each tool and what it does has been included in Appendix B. 

IV. Turbojet Example 

Using maps, map scale factors, and other modeling constants taken from a public source NPSS model 

(Ref. 6), a turbojet plant model was created using T-MATS in order to validate the component models, 

and to serve as an example that can be distributed with the software. For this example, the entire 

simulation was created first in steady-state, to verify a model match with an NPSS “truth” model, and 

then turned into a dynamic model for time-dependent simulation. The plant model contains a compressor, 

burner, turbine, duct, and nozzle as well as an ambient component connected as shown in Figure 2, where 

the station numbers are indicated along the bottom. The compressor and turbine are connected via a 

mechanical shaft that drives the compressor using power from the turbine. 

The inputs and outputs of the turbojet plant connect with either the iterative solver (in the “inner” 

loop) or with integrators (in the “outer” loop). Solver independent and dependent variables as well as 

plant input variables, including those connected to an “outer” loop integrator, are summarized in Table 3. 

In the steady-state simulation, input fuel flow (Wf) is set to the design point dictated by the NPSS data. 

The fuel flow is allowed to vary with time in the dynamic simulation. For more information on model 

implementation with T-MATS, see Reference 6. 

Figure 2.—Turbojet example problem block diagram. 

 
NASA/TM—2014-218410 12 

TABLE 3.—PLANT INPUT AND OUTPUT CONNECTIONS 

Simulation type Independent variables 

Steady-State 

W R-line PR Ndot 

Dynamic None 

 Dependent variables 

Steady-State Compressor 

Flow Error 

Turbine Flow 

Error 

Nozzle Flow 

Error 

Nmech 

Dynamic None 

 Integration variable 

Steady-State None 

Dynamic Only Nmech from Ndot

 Other inputs 

Steady-State 

Wf Alt MN dT Dynamic

A. Steady-State Model 

Accuracy of the turbojet plant model was investigated by comparing the steady-state model data with 

the equivalent NPSS data. The percent difference between T-MATS and the NPSS “truth” model can be 

viewed in Table 4. Steady-state data were gathered for the following parameters at the stations 

immediately after each component, as well as at the compressor input: flow (W), total temperature (Tt), 

total pressure (Pt), ambient pressure (Pa), shaft torque (Trq), shaft speed (N), fuel to air ratio (FAR), and 

gross thrust (Fg). If a parameter was not calculated at a particular station, e.g., shaft torque by the burner, 

it was neglected. In all cases, the difference between the NPSS model and the T-MATS model was less 

than 0.5 percent. The largest deviations were in station 4 temperature and station 5 pressure calculations. 

The difference in thrust was 0.15 percent and engine flow differences were all negligible. Steady-state 

differences’ of these magnitudes demonstrate component-level and system-level modeling in T-MATS 

that is on par with the NPSS turbojet example model and should be acceptable in similar gas turbine 

applications. 

B. Dynamic Model 

Once the plant model accuracy had been verified, the simulation was made dynamic by placing the 

engine plant and a T-MATS dynamic solver inside a “while loop.” The “outer” loop was created by 

adding an integrator, to calculate the shaft speed, and a shaft speed control system, to produce a fuel 

demand. This setup can be represented by the block diagram in Figure 1, where the outer loop plant 

contains the shaft speed control system as well as the shaft speed integrator and the inner loop plant 

contains the engine plant model and solver. 

The plots in Figure 3 to 5 were created from simulation of the dynamic turbojet model subjected to a 

thrust “chop”, a fast reduction in shaft speed demand (Ndmd), at 15 s into the simulation, decreasing 

Ndmd from 10,000 to 8,000 rpm in 0.001 s. A PI controller is used to determine the fuel command from 

the error between Ndmd and a sensed shaft speed. Results in Figure 3 show that, as Ndmd is dropped, fuel 

flow is reduced and the shaft speed tracks the decrease in Ndmd, resulting in lower gross thrust. While the 

final drop in shaft speed was only 20 percent both the fuel and gross thrust were reduced by about 

80 percent, which is typical for gas turbine engines. Gross thrust fall time can be observed to be around 

1 s with an undershoot of about 1 percent the initial values and a settling time of about 2 s. For a small 

engine, such as the example turbofan, these types of responses are typical. 

 
NASA/TM—2014-218410 13 

TABLE 4.—TURBOJET PERFORMANCE COMPARISON 

Station Parameter T-MATS NPSS % Difference 

Station 2 Input 

W (pps) 99.97 100.0 0.03% 

Tt (qR) 518.67 518.67 0.0% 

Pt (psia) 14.40 14.40 0.0% 

Pa (psia) 14.70 14.70 0.0% 

Station 3 Compressor 

W (pps) 99.97 100.0 0.03% 

Tt (qR) 1317.0 1315.80 0.0912% 

Pt (psia) 288.60 288.0 0.2083% 

Trq (lbf*ft) 14660.0 14650.0 0.0683% 

Station 4 Burner 

W (pps) 103.0 103.0 0.0% 

Tt (qR) 3079.0 3065.0 0.4568% 

Pt (psia) 274.10 273.64 0.1681% 

FAR 0.03 0.03 0.0% 

Station 48 Turbine 

W (pps) 103.0 103.0 0.0% 

Tt (qR) 2454.0 2449.0 0.2042% 

Pt (psia) 89.94 89.60 0.3795% 

FAR 0.30 0.30 0.0% 

Trq (lbf*ft) 14660.0 14650.0 0.0683% 

Station 5 Duct 

W (pps) 103.0 103.0 0.0% 

Pt (psia) 89.04 88.70 0.3833% 

FAR 0.03 0.03 0.0% 

Station 9 Nozzle 

W (pps) 103.0 103.0 0.0% 

Fg (lbf) 10670.0 10654.0 0.1502% 

Shaft N (rpm) 9999.79 10000.0 0.0021% 

Figure 3.—Dynamic turbojet transient simulation output. 
NASA/TM—2014-218410 14 

Results in Figure 4 show temperatures and pressures at key stations during the transient. Pressures 

after the compressor and after the turbine, Pt3 and Pt48 respectively, drop about 70 percent from their 

initial values, with transient fall times, and settling times, comparable to those observed in the fuel flow 

transient (1 s and 2 s, respectively). Pressure undershoot was observed to be slightly larger after the 

compressor, 8 percent, rather than after the turbine, 5 percent. Temperatures, Tt3 and Tt48, show drops of 

70 and 50 percent, respectively, and fall times that are consistent with the pressure transients (about 1 s). 

Undershoot and settling for Tt3 (8 percent and 2 s) are significantly less than for Tt48 (30 percent and 

4 s). This difference in temperature transient response can be attributed to a delay in the energy balance 

along the shaft. 

The plot in Figure 5 shows the transient compressor operating points superimposed over the 

compressor map. The pressure ratio is initially nearly 20, and then quickly drops to 15 following the set 

point change, progressing to around 5 as the system reaches steady-state, which is a typical trend for a 

thrust chop event. Initially stall margin is low, and then grows quickly as fuel flow is cut off sharply. As 

fuel stabilizes, the stall margin moves back up to about the same value it was before the transient began. It 

can be seen that in all cases the stall line remains above the operating points, suggesting stall free 

operation. Transient performance, in this example, is dependent on PI tuning and is meant only to 

demonstrate the dynamic capability of the model. 

Figure 4.—Dynamic turbojet thermodynamic dimension outputs. 
NASA/TM—2014-218410 15 

Figure 5.—Compressor transient operating line for dynamic turbojet simulation. 

V. Summary 

The Toolbox for the Modeling and Analysis of Thermodynamic Systems (T-MATS) software 

package is an open source MATLAB/Simulink toolbox (plug-in) for the development of custom 

thermodynamic and controls models that allows for the building of an entire system simulation in a single 

computational environment. This paper presents an introduction to the T-MATS software package, 

describing possible propulsion simulation architectures, detailing the theory behind the Simulink block 

sets included in the package, and showing an example of a steady-state and a dynamic turbojet engine 

simulation created with T-MATS. A model matching was performed by comparing the steady-state 

T-MATS simulation against a well-documented NPSS steady-state engine model. Results from this model 

matching show comparable performance between the two simulations and that T-MATS is a simulation 

environment that is capable of modeling turbo-machinery. Dynamic simulation results show transients 

that are plausible and realistic for a small turbofan engine. Additional information on the capabilities and 

usage of T-MATS can be found in locations described in Appendix B and Reference 6. 

NASA/TM—2014-218410 17 

Appendix 

A. T-MATS Tool Description 

The T-MATS Model Development Tools, accessible from the Simulink menu bar, include the ability 

to toggle between Simulink GoTo and From blocks, turning the T-MATS iDesign feature on or off for all 

blocks in a model, and automatically creating GoTo and From blocks for each of the inputs and outputs of 

a subsystem, as shown in Table 5. 

B. T-MATS Download Information 

The T-MATS software can be downloaded from the Github repository at https://github.com/nasa/ 

T-MATS/releases, where lists of bugs and feature requests are also maintained. A user group has been 

created at https://groups.google.com/forum/#!forum/t-mats-user-group to enable collaboration between 

users and developers. It should be noted that although T-MATS was initially developed by NASA, 

T-MATS is open source and, as such, collaboration with the user community in finding bugs, adding 

features, and generally working toward a better product is strongly encouraged. 

TABLE 5.—DESCRIPTION OF T-MATS TOOLS 

T-MATS Tool Description 

GF_Convert Toggles between GoTo and From blocks, without changing the block name. 

It also works to toggle between Inport and Outport blocks. 

iDesign_On Turns iDesign on for every block within the model. 

iDesign_Off Turns iDesign off for every block within the model. 

Block Link Setup Creates Goto or From blocks for every input and output of a subsystem. For 

turbo-machinery blocks the tool creates linkages appropriate for T-MATS 

connections, which allow blocks to be connected quickly and easily. 
NASA/TM—2014-218410 18 

References 

1. Jones, S.M., “An Introduction to Thermodynamic Performance Analysis of Aircraft Gas Turbine 

Engine Cycles Using the Numerical Propulsion System Simulation Code,” NASA/TM—2007-

214690, March 2007. 

2. Nichols, L.D. and Chamis, C.C., “Numerical Propulsion System Simulation: An Interdisciplinary 

Approach,” AIAA-91-3554, September 1991. 

3. Gasturb, Software package, Ver. 12, GasTurb GmbH, Aachen, Germany, 2014. 

4. Thermolib, Software package, Ver. 5.2, EUTech, Aachen, Germany, 2014. 

5. May, R.D., Csank, J., Lavelle, T.M., Litt, J.S., and Guo T.H., “A High-Fidelity Simulation of a 

Generic Commercial Aircraft Engine and Controller,” AIAA-2010-6630, 46th AIAA Joint Propulsion 

Conference & Exhibit, Nashville, TN, July, 2010. 

6. Chapman, J.W., Lavelle, T.M., May, R.D., Litt, J.S., Guo, T.H., “Toolbox for the Modeling and 

Analysis of Thermodynamic Systems (T-MATS) User’s Guide,” NASA/TM—2014-216638, January 

2014. 

7. Jones, S.M., “Steady-State Modeling of Gas Turbine Engines Using The Numerical Propulsion 

System Simulation Code,” GT2010-22350, June 2010.