1. Estimation of the critical temperature ratio
for thermoacoustic engines based on
adaptive control which maintains
closed-loop system at stability limit
Kazuaki Sakurai, Yasuhide Kobayashi and Noboru Yamada
Nagaoka University of Technology (Japan)

2. Background TC TH
Stack
(Regenerator)
Resonator
Sound wave
Thermoacoustic engines utilizing the thermoacoustic phenomena increase the efficiency on heat recovery
It is important to know the critical temperature ratio (CTR) in thermoacoustic systems
It is difficult to know it at the design stage when heat exchangers and stacks have complex structures

3. Power consumption of loudspeaker becomes minimum at CTR
Background
In order to estimate CTR, we have proposed an experimental method
The tube pressure amplitude is maintained to be a constant value by an open-loop control
The open-loop control has a manual tuning of driving signal
for loudspeaker
Power consumption of loudspeaker becomes minimum at CTR

4. Background
In order to estimate CTR, we have proposed an experimental method
To carry out the open-loop control, it is necessary
to previously measure the frequency responses
at various temperature ratios
The resonance frequency is calculated
in order to set the driving frequency
Labor saving of the measurement is desired

5. It will be shown that a relationship between the
Objective
Propose an adaptive control system
Increases the tube pressure amplitude till a reference value by destabilizing the closed-loop system in order to hold the critical point
By using this method, there is no need to carry out the
frequency responses, since the resonance frequencies are automatically determined by proposed control system
It will be shown that a relationship between the
adaptive gain and temperature ratio can be used an alternative method to estimate the CTR

6. Experimental apparatus
Standing-wave thermoacoustic engine
D/A
A/D PC PA
loudspeaker TH TC
Wspk
sensor1
sensor2
core
1312 mm
565
164
502 p2 p1 u vi vs
360
・ A loudspeaker and
two pressure sensors
are located
・ The thermoacoustic core
is composed of a stack
and two heat exchangers
・ The cold-side heat
exchanger’s temperature TC
is kept at room temperature

7. Experimental apparatus
Standing-wave thermoacoustic engine
D/A
A/D PC PA
loudspeaker TH TC
Wspk
sensor1
sensor2
core
1312 mm
565
164
502 p2 p1 u vi vs
360
A preliminary experiment has been carried out to obtain a rough estimation of CTR
It has been found that the CTR of the engine is approximately 1.3

8. Experiment method The driving signal of the loudspeaker (τ =11.5 ms )
The objective of the control system is to drive the loudspeaker with as small amplitude as possible in driving signal so that
the amplitude of p2 matches to the reference value
・The reference value of p2 is set to 200 Pa.
・The heater temperature is adjusted so that TH / TC
　varies from 1.00 to 1.35, 8 points with increment 0.05
The driving signal of the loudspeaker
(τ =11.5 ms )
This is known as a phase-delay controller which has been conventionally used to suppress the thermoacoustic instabilities

9. ＾ ＾ ＾ Experiment method |・| Block diagram of the control systemu
e-sτ
G(t)
LPF PI
π/2
|・|
P2* - P2 ＾
・The absolute value signal of p2 goes through
to a low-pass filter and estimated pressure
amplitude P2 is obtained ＾
・The difference between P2 and the
reference value P2*are sent to the
PI controller ＾
G(t)
・The output of PI controller
itself is used as the
feedback gain G(t)
At each setting of TH / TC, the steady-state feedback gain G is measured

10. ＾ Result Time response G ex. TH / TC= 1.0 p2
As shown in blue line, the absolute value of the gain is increased so as to oscillate the engine p2 ＾ G
Time(s)
Pressure (Pa)
Pressure (Pa),Gain
It is automatically adjusted to achieve a constant pressure amplitude at the reference value

11. Result Time response ex. TH /TC= 1.0
Enlarged view of the time response of p2 in steady state
It is observed that the pressure amplitude converges to 400 Pa
which implies the steady-state oscillation
400 Pa
Pressure (Pa)
Time(s)

12. Result Feedback gain G Linear relationship is observed
【TH / TC < 1.3】
・ Adjusted gain shows negative value
・ Loudspeaker acts to assist the
sound wave generation
【TH / TC = 1.3】
・ Gain is almost zero
【TH / TC = 1.35】
・ Positive gain is observed
・ Loudspeaker acts to suppress
the sound wave
If we assume the linear relationship, it is possible to
predict the CTR by performing measurement of gains
for several points in the region below the CTR

13. Result Oscillation frequency
The oscillation frequency has been adjusted automatically
It is confirmed that the oscillation frequency by adaptive control
has the same tendency with
frequency response experiment
The frequency response experiment as in the conventional method can be omitted with this proposed method

14. Conclusion ・We proposed an adaptive control system which maintains
the tube pressure amplitude to be a reference value.
The open-loop gain is automatically adjusted by destabilizing
the system around the critical point.
・The results show that the pressure amplitude is successfully
converged to the reference one.
This implies the closed-loop stability of the proposed
adaptive control system.
・The steady-state frequencies for various temperature ratios
are automatically obtained in similar values by manual control.
・A linear relation between the adaptive gain and temperature
ratio is also obtained, which provides an alternative method
to estimate the CTR.　

15. Future work ・ Theoretical guarantee for the closed-loop
stability of the proposed adaptive
　 control system
・ Automatic adjustment of time delay
in controller