application of the root-locus method to the design and sensitivity analysis of closed-loop...
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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