1. Application of the Root-Locus Method to the Design and Sensitivity Analysis of Closed-Loop Thermoacoustic Engines
C Mark Johnson

2. Overview The Design Challenge Frequency Domain Analysis
Laplace Domain Analysis
Root Locus Approach to Instability Design
Parametric Optimisation
Addition of Heat Transfer
Parametric Output Power Response
Use of Tuning Stubs
Conclusions

3. The Design Challenge
Travelling wave loop with alternator mid-loop and 2 side branches
Simplified equivalent
Loop must oscillate: added power = losses + alternator
Steady-state frequency domain analysis
Pressure and volume velocity must match around the loop

4. Power Balance
For travelling wave loop, available acoustic power is a fraction of the loop power:
Carnot efficiency = max. possible:
Alternator absorbs acoustic power:
For steady oscillation:
Depends on ratio of volume velocities, alternator parameters & engine conditions

5. Pressure and Vol. Velocity
Simultaneous solution requires 4 independent variables (e.g. q1, q2, Ae, ra)
Power balance equality must be satisfied

6. Effect of Alternator Position
Normalised pressure, volume velocity and impedance at the alternator as a function of alternator position (q1)
Position of alternator must satisfy condition for velocity ratio from both ends of the loop
Determines the minimum thermoacoustic gain (kTA) for oscillation under varying acoustic engine conditions (Ae)

7. Conditions for Oscillation
Minimum achievable velocity ratios (determined from AHX and HHX) and corresponding onset temperature as a function of (real) pressure ratio Ae
Possible to establish boundaries to design space but…
Difficult to gain further insight into optimisation of system parameters

8. Laplace-Domain Approach
Laplace-domain analysis can be applied to study the time-domain (transient) behaviour as well as steady state
Solution of the characteristic equation for the loop gives the poles of the transfer function
Opens up new possibilities:
Parametric stability analysis of a system, for example by plotting root loci as functions of temperature, load resistance, feedback pipe parameters
Design optimisation in which geometrical and physical parameters may be varied to achieve specific targets, for example:
Maximising the real part of dominant pole pair
Minimising the temperature at which the dominant pole pair lies on the imaginary axis
Note that the pressure and velocity are not solved for explicitly: reduces the number of equations from 4 to 2

9. Two-Port Networks
Analysis based on the assumption that all thermoacoustic loop elements may be represented by a 2-port equivalent
Each element is based on a modified waveguide representation, including arbitrary shunt and series impedances
Input and output pressure and volume velocity of each element can then be represented:

10. Propagation Matrix
Feedback system representation of thermoacoustic loop
Whole loop is represented by a cascade of the S-parameter matrices for each element
The poles of the closed-loop transfer function are determined by solving the characteristic equation:
System can be solved in full to determine impulse response, frequency response etc. using Laplace solution methods e.g. poles and residues

11. Stability Analysis
In solving for the transfer function poles (two equations, two unknowns) we gain far more information about the system response than can be obtained from a simple frequency domain representation
Can explore the system stability characteristics in response to changes in geometrical and other system parameters
For a loop containing a number of undetermined parameters e.g. feedback pipe length, feedback pipe diameter, it is possible to find the region of instability and then determine an optimum operating point
Boundary of the region of instability corresponds to the condition for steady oscillations (a conjugate pair of poles placed on the imaginary axis) and thus represents the limiting case for the onset of oscillation.

12. Root Locus Method
Conditions for instability:
Temperature > 550K
Alternator load > 0.6W
Combination of pipe T=650K, RL=1W
Solve characteristic equation to find dominant pole-pair and plot as function of selected parameters
Positive real part indicates oscillatory behaviour
Amplitude determined by e.g. onset of non-linearity, heat transfer limitations etc.

13. Effect of Pipe Lengths
Real part of dominant pole for combination of pipe T=650K, RL=1W
a=0
“Strength” of instability related to positive real part of pole, a
Area inside the a=0 contour is unstable and gives an indication of the margin of error permissible in construction or the range needed for tuning purposes

14. Engine Operating Conditions
Magnitude of the impedance for the unstable region corresponds to values of engine pressure ratio Ae below 4
Phase plot shows pressure and velocity to be close to “in phase” with optimum loop parameters

15. Coupling with Heat Transfer
Heat transfer added through auxiliary equations to represent heat flow through regenerator and transfer from HHX
Lumped parameter representation of alternator and load
Frequency domain solution
Additional analyses enabled:
Engine performance (efficiency, output) under varying acoustic and heat transfer conditions
Power output (effect of changing parameters, optimisation)
Tuning (using stubs to optimise output in presence of uncertainty)

16. Example Engine 1 Stub 1 Pipe E Engine 2 Pipe 1 Stub 2 Pipe 2
Alternator
Dual series engine loop with radiant HHX (loosely based on SCORE demo 0_3)
Two tuning stubs, nominal adjustment range 0.1 to 0.6 m
Feedback pipes 100 mm, stubs 75 mm diameter
Alternator based on idealised SCORE design

17. Engine Performance
Available acoustic power strong function of regenerator temperature, radiant HHX temperature, acoustic conditions (enforced pressure ratio Ae)
Drop in power/efficiency at high temperatures due to solid conduction through regenerator

18. Stability
Real part of dominant pole for combination of pipe T=750K, RL=1W
Real part of dominant pole-pair as function of feedback pipe lengths
Significant unstable region reflecting additional gain from series engines

19. Power Response Electrical power output Hot end temperature
Simultaneously solve acoustic loop and heat transfer equations: hot end temperature varies
Peak power output at regenerator hot end temperature of ~550K corresponds to peak engine efficiency
Optimum pipe lengths do not correspond to those for greatest instability as temperature is now a variable

20. Tuning Power response Tuned response
Power response (left) determined with fixed side branch length
Optimised response (right) determined with varying side branch lengths
Possible to maintain virtually constant output power by tuning

21. Conclusions
Frequency domain techniques can be difficult to apply to design of thermoacoustic loops
Laplace domain techniques offer an attractive alternative by allowing use of standard methods such as root locus for determining feasible operating regimes
Optimisation and parametric sensitivity analysis can be performed without explicitly solving for acoustic variables
Coupling with heat transfer equations allows complete steady-state solution to be determined
Application to e.g. output power optimisation, sensitivity analysis and tuning