Application of the Root-Locus Method to the Design and Sensitivity Analysis of Closed-Loop Thermoacoustic Engines

C Mark Johnson

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

The Design Challenge

1. Loop must oscillate: added power = losses + alternator2. Steady-state frequency domain analysis3. Pressure and volume velocity must match around the loop

HHX

Side branch 1

Side branch 2

AHX Transmission line 1

Transmission line 2

Back volume

Front volume

Alternator Regenerator

Ui

YA

re/2 re/2

kTA Ui YH

pe

Ue

Ysb1 Ysb2 Yb Yf

TL1 TL2 ZA Ua

pa p´a

Ue

Travelling wave loop with alternator mid-loop and 2 side branches

Simplified equivalent

Power Balance

For travelling wave loop, available acoustic power is a fraction of the loop power:

2,

1

21

Re1 TA

TA

ee

e

TA

TAlac

loop

aca

k

k

AZ

r

k

k

P

P

Carnot efficiency = max. possible:

22

2

1

211Re

TAa

TAeTATAee

e

a

kr

krkkZA

U

U

1C

HTA T

Tk

Alternator absorbs acoustic power: For steady oscillation:

eeee UZAp

gase

eee S

cZ

,

2aaUr

L

mC

a RR

BlR

Sr

0

2

2

1

Depends on ratio of volume velocities, alternator parameters & engine conditions

Pressure and Vol. Velocity

Simultaneous solution requires 4 independent variables (e.g. 1, 2, Ae, ra)

Power balance equality must be satisfied

12

11

1

12

11

11

sin1cos

sin1cos1

bHee

bHee

bHee

bbHeee

a

yyZZ

ZayyZaj

yyZZ

ZbZyyyZb

U

U

12

11

12

11

111

12

11

21

1111

sin11cos1

sin11cos1

bHee

babHee

aHeebHeeabHeeaee

bHee

abHee

baHeebHeeabHeeaee

aaaaa

yyZZ

ZbZyxyyZ

Z

ZarZZyZbyyZbxyyZarZbj

yyZZ

ZaxyyZ

Z

ZbZyrZyZayyZaxyyZbrZa

Ujxrpp

222

22

222

2

22

sin11

1cos

1

sin11

cos1

1

fATA

eeefA

TA

eee

fAee

TA

ffAee

TAe

a

yyZk

rZa

Zyy

k

rZaj

yyZZ

Zb

k

ZyyyZb

kU

U

222

2

222

sin1

cos

sin1

cos1

ZyZbk

ZZbj

k

rZaZy

k

rZa

U

p

AeeTA

ee

TA

eeeA

TA

eee

e

a

Effect of Alternator Position

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)

Normalised pressure, volume velocity and impedance at the alternator as a function of alternator position (1)

0

0.5

1

1.5

2

2.5

3

3.5

0 30 60 90 120 150 180 210 240 270 300 330 360

-200

-150

-100

-50

0

50

100

150

200

|Ua|

|Z|

|pa|

arg(Z)

Conditions for Oscillation

Possible to establish boundaries to design space but… Difficult to gain further insight into optimisation of system

parameters

Minimum achievable velocity ratios (determined from AHX and HHX) and corresponding onset temperature as a function of (real) pressure ratio Ae

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8

Engine pressure ratio (Ae)

Vo

lum

e ve

loci

ty r

atio

Ua/

Ue

400

425

450

475

500

525

550

On

set

tem

per

atu

re (

K)

|Ua|/|Ue| min reverse

|Ua|/|Ue| min forward

T onset

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

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:

1

1

2221

1211

2

2

U

p

SS

SS

U

p

U1

Y p2

Z/2 Z/2

gTA U1

p1

Ub Ua

ps U2

Propagation Matrix

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:

(p1,U1) (pn,Un) (p,U)

+

+

Pn

n

jnn SP

1

01 nP

Feedback system representation of thermoacoustic loop

System can be solved in full to determine impulse response, frequency response etc. using Laplace solution methods e.g. poles and residues

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.

Root Locus Method

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.

500

505

510

515

520

525

530

535

540

545

550

-4 -3 -2 -1 0 1 2 3

Temperature

Pipe lengths

Alternator load

HHX

Side branch 1

Side branch 2

AHX Transmission line 1

Transmission line 2

Back volume

Front volume

Alternator Regenerator

Conditions for instability:Temperature > 550KAlternator load > 0.6Combination of pipe lengths @ T=650K, RL=1

Effect of Pipe Lengths

“Strength” of instability related to positive real part of pole, Area inside the =0 contour is unstable and gives an

indication of the margin of error permissible in construction or the range needed for tuning purposes

2.2

5

2.2

7

2.2

9

2.3

1

2.3

3

2.3

5

2.3

7

2.3

9

2.4

1

2.4

3

2.4

5

2.4

7

2.4

9

2.5

1

2.5

3

2.5

5

2.5

7

2.5

9

2.6

1

2.6

3

2.6

5

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

alpha

Length pipe 1

Le

ng

th p

ipe

2

1.5-2

1-1.5

0.5-1

0-0.5

-0.5-0

-1--0.5

-1.5--1

-2--1.5

-2.5--2

-3--2.5

-3.5--3

-4--3.5

-4.5--4

-5--4.5

-5.5--5

-6--5.5

=0

Real part of dominant pole for combination of pipe lengths @ T=650K, RL=1

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

2.25 2.27 2.29 2.31 2.33 2.35 2.37 2.39 2.41 2.43 2.45 2.47 2.49 2.51 2.53 2.55 2.57 2.59 2.61 2.63 2.650.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Length pipe 1

Len

gth

pip

e 2

90000-100000

80000-90000

70000-80000

60000-70000

50000-60000

40000-50000

30000-40000

20000-30000

10000-20000

0-10000

2.25 2.27 2.29 2.31 2.33 2.35 2.37 2.39 2.41 2.43 2.45 2.47 2.49 2.51 2.53 2.55 2.57 2.59 2.61 2.63 2.650.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Length pipe 1

Len

gth

pip

e 2

100-120

80-100

60-80

40-60

20-40

0-20

-20-0

-40--20

-60--40

-80--60

-100--80

-120--100

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)

dx

dTkAkA

dx

dT

T

Ufff

fsA

p

f

ffpUH m

ssggm

m

m

2*

2*

**

2 1Re1

111111

inQHH 2122

Example

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

Pipe E

Pipe 1

Pipe 2

Engine 1

Engine 2

Stub 1

Stub 2

Alternator

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

0

20

40

60

80

100

120

140

350 400 450 500 550 600 650 700

Regen hot end temp (K)

Aco

ust

ic p

ow

er (

W)

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

Eff

icie

ncy

(P

a/Q

in)

Pa Ae=6 Pa Ae=3 Pa Ae=1.8

Efficiency Ae=6 Efficiency Ae=3 Efficiency Ae=1.8

0

20

40

60

80

100

120

140

350 400 450 500 550 600 650 700

Regen hot end temp (K)

Aco

ust

ic p

ow

er (

W)

850K 950K 1050K

Stability

Real part of dominant pole-pair as function of feedback pipe lengths

Significant unstable region reflecting additional gain from series engines

0.7

0.7

4

0.7

8

0.8

2

0.8

6

0.9

0.9

4

0.9

8

1.0

2

1.0

6

1.1

1.1

4

1.1

8

1.2

2

1.2

6

1.3

1.3

4

1.3

8

1.4

2

1.4

6

1.5

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

alpha

Length pipe 1

Le

ng

th p

ipe

2

14-16

12-14

10-12

8-10

6-8

4-6

2-4

0-2

-2-0

-4--2

-6--4

-8--6

-10--8

Real part of dominant pole for combination of pipe lengths @ T=750K, RL=1

Power Response

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

0.7

0.78

0.86

0.94

1.02 1.1

1.18

1.26

1.34

1.42 1.5

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

Power output (We)

Length pipe 1

Len

gth

pip

e 2

200-220

180-200

160-180

140-160

120-140

100-120

80-100

60-80

40-60

20-40

0-20

0.7

0.78

0.86

0.94

1.02 1.1

1.18

1.26

1.34

1.42 1.5

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

Hot end temperature (K)

Length pipe 1

Len

gth

pip

e 2

700-750

650-700

600-650

550-600

500-550

450-500

Electrical power output Hot end temperature

Tuning

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

Power response Tuned response

0

50

100

150

200

250

0.86 0.91 0.96 1.01 1.06 1.11

Length pipe E

Po

wer

(W

e)

0

100

200

300

400

500

600

700

800

Reg

en h

ot

end

tem

p (

K)

Pe Thot

0

50

100

150

200

250

0.86 0.91 0.96 1.01 1.06 1.11

Length pipe E

Po

wer

(W

e)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Stu

b le

ng

th (

m)

Pe Lsb1 Lsb2

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