Circuit Theory Analysis
of Acoustically Coupled Rooms
Análisis circuital de espacios acústicamente acoplados
Federico Miyara
Laboratorio de Acústica y Electroacústica, Facultad de Ciencias Exactas, Ingeniería y Agrimensura (FCEIA),
Universidad Nacional de Rosario.
Riobamba 245 bis, Rosario, Argentina
fmiyara@fceia.unr.edu.ar
Abstract— A circuit theory approach is presented for the
anal
ysis of acoustically coupled spaces. It is based on energy
flow and balance with a similar analogy to the one used for
solving dynamic thermal systems, where acoustic power is ana-
log to electric current and acoustic energy density is analog to
voltage. Room volume is equivalent to capacitor and sound
absorption is equivalent to a resistor. From these analogies and
Laplace-transformed models it is possible to get stationary or
transient response solutions.
Keywords: coupled rooms; circuit model; analogies
Resumen— Se propone un enfoque circuital basado en
flujos de energía para el análisis de espacios acústicamente
acoplados. En el mismo se establece una analogía similar a la
que se utiliza en el estudio de sistemas térmicos dinámicos en el
cual la potencia acústica es análoga a la corriente eléctrica y la
densidad de energía acústica es análoga a la tensión. Los volú-
menes son equivalentes a capacitores y la absorción sonora es
asimilable a una resistencia. A partir de estas analogías se pue-
den plantear ecuaciones diferenciales y modelos en transfor-
mada de Laplace y obtener tanto soluciones estacionarias o de
régimen permanente como transitorias.
Palabras clave: recintos acoplados; modelo circuital; analogías
I. INTRODUCTION
Physi
cal systems can be represented by different formal
frameworks. Some of them are block diagrams, graph
theory [1] and bond graphs [2]. Often the choice of one or
another depends on the nature of the process and on the kind
of the desired results. In the case of electric networks, cir-
cuit theory, based on Ohm’s and Kirchhoff’s laws, provides
an adequate approach because of the simplicity of the
graphical representation akin to the actual topology of the
physical system, and the availability of methods to write
down the equations by simple inspection. The use of the
Laplace transform allows to easily analyze circuits with
dynamic components such as capacitors and inductors.
The degree of maturity of circuit theory and the familiarity
with its techniques reached by electrical engineers and other
practitioners has prompted the extension of its principles to
other specialties such as hydraulics, mechanics and acoustics
[3]. The general approach is to select two relevant variables
that can be considered analogous to voltage and electric
current and identify physical subprocesses where these
variables are linked. In general, but not always, the variables
are chosen so that their product has dimension of power. This
allows, by means of conservative transducers, the transition
from a physical domain to another, such as is the case of
electroacoustic equivalent circuits [3].
For instance, in the case of mechanical circuits, the
variables are force and velocity, and in the case of acoustic
or hydraulic systems, pressure and flow rate (or volume
velocity). In general there is a flow type variable and a
force type variable, which are the analogs to current and
voltage respectively.
Our purpose is to introduce an equivalent-circuit approach
applicable to the analysis of acoustically coupled rooms in
both static and dynamic conditions.
II. ANALOGIES
In th
e case of acoustically coupled spaces which we shall
be dealing with, we will take the acoustic power as the analog
of current. Should we stick to the rule that the product of the
two variables has dimension of power, the second variable
should be non-dimensional, which is not convenient.
We have the precedent of thermal circuits, in which the
relevant circuit variables are power and temperature [4],
whose product has not dimension of power. They are often
used to estimate the working temperature of electronic
components (Fig. 1).
Fig. 1. An example of thermal circuit in which there is a power source (the
junct
ion), the thermal resistance between the junction and the case, R
T jc
,
the thermal resistance between the case and the environment, R
T jc
, and a
fixed ambient temperature, T
a
.
In order to select the force variable it will be useful to
analy
ze a simple case in steady state (Fig. 2).
In this example the reverberant field equation holds:
T
a
R
T jc
R
T ca
P
a
j
c
Recibido: 02/12/18; Aceptado: 12/02/19
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Fig. 2. A reverberant room in steady state with an acoustic power source W.
The power W
abs
absorbed at the surfaces and the square effective pressure
P
ef
2
of the reverberant field are shown. V is the volume, S, the surface, and
α, the absorption coefficient of the room.
W
S
cP
α
α
−
ρ=
)1(4
o
2
ef
(1)
where α is the mean sound absorption coefficient, W the
acoustic power, S the inner area of the room and ρ
o
c the
specific acoustic impedance of air [3].
While we could use P
ef
2
as the analog of voltage, it is
more convenient to use the acoustic energy density D,
which is proportional to P
ef
2
:
2
o
2
ef
c
P
D
ρ
= . (2)
This formula is valid for plane waves [3], but a diffuse field as
we are assuming is the superposition of a large number of
incoherent plane waves. Thus,
W
Sc
D
α
α
−
=
)1(4
. (3)
Given the analogy
D
↔
V
W
↔
I
we can define an analog to electric resistance, which we
shall call absorption resistance, as
Sc
W
D
R
α
α
−
==
)1(4
abs
. (4)
Now we can draw the first circuit for the reverberant steady
state (Fig.3).
Fig. 3. Circuit of the reverberant steady state in a room.
In the previous example the air absorption, which is
imp
ortant for large rooms, was not taken into account. In
this case (Fig. 4) α can be replaced by a coefficient α
tot
giv
en by
F
i
g
. 4. A reverberant room in steady state, considering the acoustic power
absorbed by the air.
S
V4
mtot
γ+α=α
. (5)
where V is the volume, S the area and γ
m
, the coefficient of
absorption in the air, in neper/m [3]. Then
Sc
R
tot
tot
)1(4
α
α−
= . (6)
It is interesting to note that this resistance can be considered
as the parallel of two resistances, one of them representing
the absorption by the surfaces, and the other, by the air.
Fig. 5. Equivalent circuit for the steady state of the reverberant field in a
room, including the absorption in the air.
Bec
ause of energy conservation, we have an equivalent of
Kirchhoff’s first law,
absair
WWW
−= . (7)
D
Sc
W
411
tot
tot
air
α−
α
−
α−
α
=
. (8)
We can look for a value α' such that
'1
'
11
tot
tot
α−
α
=
α−
α
−
α−
α
. (9)
This value represents a hypothetical surface absorption equiva-
lent to air absorption. Solving for α'
αγ+α−
γ
=α
S
V
S
V
4
)1(
4
'
m
2
m
. (10)
The
distinction between both absorption mechanisms is of
conceptual interest and also illustrates a direct application of
P
ef
2
V, S,
α
W
W
abs
R
abs
W
D
+
–
P
ef
2
V
1
, S
1
,
α
1
W
W
air
W
abs
R
abs
W
D
+
–
R
air
W
air
W
abs
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circuit theory to this model, but in practice we will work
directly with
absair
// RRR
= . (11)
III. COUPLED ROOMS IN STEADY STATE
We shall next consider two rooms coupled by a small
opening of area S
12
(Fig. 6).
Fig. 6. Coupled rooms with surface and air absorption. In room 1 there is a
sound source with acoustic power W. S
12
is the area of the opening
connecting both rooms
The power loss in each room is
1
tot1
1tot1
1loss
)1(4
D
cS
W
α−
α
= , (12)
2
tot2
2tot2
loss2
)1(4
D
cS
W
α−
α
= . (13)
On the other hand, the power arriving at each room through
the opening is given by
1
12
12
4
D
cS
W = , (14)
2
12
21
4
D
cS
W = . (15)
The following energy balance holds:
WWWW =−+
2112loss1
,
(16)
0
1221loss2
=−+ WWW
.
Replacing equations (12) through (15) in (16) we get
WD
cS
D
cS
cS
=−
+
α−
α
2
12
1
12
tot1
1tot1
44)1(4
(17)
0
4)1(44
2
12
t
ot2
2tot2
1
12
=
+
α−
α
+− D
cS
cS
D
cS
If, by analogy with equation (6), we introduce the
following circuit parameters
cS
R
1tot1
tot1
1
)1(4
α
α−
= , (18)
cS
R
2tot2
tot2
2
)1(4
α
α−
= , (19)
cS
R
12
12
4
= , (20)
the system of equations (17) can be rewritten as
WD
R
D
RR
=−
+
2
12
1
1
21
111
(
21)
0
111
2
122
1
12
=
++− D
RR
D
R
System (21) can be interpreted as the resolution of the
circuit of Fig. 7 by the node potential method.
Fig. 7. Steady-state circuit model for two acoustically coupled rooms.
This system is easily solved, getting
1221
2121
1
)(
RRR
RRR
WD
++
+
= (22)
1221
21
2
RRR
RR
WD
++
= (23)
Circuit analysis allows to get interesting conclusions.
For instance, from (20) we can see that the smaller the
opening area the larger R
12
, hence the weaker the coupling,
Indeed, from equation (22) it is seen that the energy den-
sity in room 1 practically does not depend on the char-
acteristics of room 2 (as R
12
>> R
2
), while from (23) the
energy density of the latter will be a small fraction
(≅ R
2
/R
12
) of the energy density in the former.
IV. ROOMS COUPLED BY A PARTITION
We shall consider now two rooms coupled by a partition
whose sound transmission coefficient
1
is τ
s
. In such case
equations (14) and (15) will be affected by τ
s
, i.e.,
1
12
s12
4
D
cS
W τ= , (24)
1
The sound transmission coefficient is the ratio of the
transmitted power to the incident power.
D
1
V
1
, S
1
, α
1
W
W
air1
W
abs1
V
2
, S
2
, α
2
W
air2
S
12
W
12
W
21
W
abs2
D
2
R
2
W
D
1
R
1
W
loss1
W
loss2
D
2
R
12
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2
12
s21
4
D
cS
W τ= . (25)
from which
cS
R
12s
12
4
τ
= . (26)
Besides, it will be necessary to take into account the
absorption coefficient of the partition, which will normally
be less than 1, affecting also the mean absorption of both
rooms.
An interesting example arises in the case of movable
partition walls used in large rooms to accomplish different
room distributions such as at conference and exhibition
centers. Since in general τ
s
is relatively low, coupling is
we
ak, but this is compensated because the common area is
considerable.
Another typical situation is when there is communication
through a suspended ceiling in cases in which a larger room
has been partitioned to get several smaller spaces.
V. SINGLE ROOM IN DYNAMIC CONDITIONS
Transient behavior in a room is important at least in three
situations. The first one is the onset of the sound field, the
second, the sound decay typical of reverberation, and the
third, the response to amplitude modulation (modulation
transfer function), of interest for the analysis of speech
intelligibility.
In order to analyze this problem we will start from the
theoretical energy impulse response, which consists in an
exponential decay
t
V
cS
S
V
eDtD
4
4
)1ln(
o
m
)(
γ−
α−
= . (27)
Th
is response is characterized by a time constant τ given
by
S
V
cS
V
4
)1ln(
14
m
γ+α−−
=τ . (28)
In electric circuits this kind of response is associated with
the discharge of a capacitor through a resistor, so we can
propose a circuit component similar to a capacitor connected
as shown in Fig. 8.
Fig. 8. Equivalent circuit for the reverberant state of a room.
In order that the time response be the same it is necessary
that
τ
=
RC , (29)
where R is given by (6). Solving for C,
S
V
V
C
4
)1ln(
1
m
tot
tot
γ+α−−
α−
α
= . (30)
For α << 1 we can approximate
α
≅
α
−
−
)1ln( , (31)
so that
tot
1 α−
≅
V
C . (32)
Sometimes, if α
tot
<< 1, we can approximate
VC
≅
. (33)
As can be noted, the capacity is associated with the room
volume, which is reasonable.
VI. C
OUPLED ROOMS IN DYNAMIC CONDITIONS
Consider now two coupled rooms in non stationary state.
Considering the time constant of each room as if they were
not coupled, we can draw an equivalent circuit such as
shown in Fig. 9.
Fig. 9. Dynamic condition circuit model of two acoustically coupled rooms.
In this case, the node potential equations, written by
simple inspection, are
WD
R
D
R
sC
R
=−
++
2
12
1
12
1
1
111
(34)
0
111
2
12
2
2
1
12
=
+++− D
R
sC
R
D
R
Note that the equations have been written directly using
the Laplace transform, avoiding the classic approach with
differential equations (see, for example, [5], [6]) and
taking advantage of the techniques that are commonplace
in circuit analysis.
The solution of this system of equations is completely
similar to the static case of equation (21). After some
algebraic manipulation we get
R
2
W
D
1
R
1
W
loss1
W
loss2
D
2
R
12
C
1
C
2
R
W
D
+
–
C
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( )
2
2121112121221121
2
12
122121
2122121
1
////////
////
1
)//1(//
sCRRCRRsCRRCRR
R
RRRR
sCRRRR
WD
+++−
+
= (35)
( )
2
2121112121221121
2
12
122121
122
2
121
2
////////
////
1
//
sCRRCRRsCRRCRR
R
RRRR
RR
R
RR
WD
+++−
+
= (36)
As we can see, it is a second-order system and it can be
shown that its poles are real since it is a completely dissipa-
tive system. This means that there will be a double decay
slope, which is characteristic of the reverberation of acousti-
cally coupled spaces. The effect will be more noticeable in
the receiving space, especially if the coupling is weak.
VII. C
ONCLUSION
An approach based on circuit models has been presented for
the analysis of acoustically coupled spaces. The model uses
flow and force type analogies, where the flow variable is the
acoustic power (the analog of the electric current) and the force
variable is the volume energy density (the analog of voltage).
This approach allows to represent reverberant acoustic
systems by means of a model that can be derived by simple
inspection, in a similar fashion to what is customary in
electric network analysis. The model is akin to the topology
of the system.
Furthermore, the approach allows to profit from the vast
collection of resources available for circuit resolution, a
technique that has been developed for more than a century,
including computer simulation.
Particularly interesting is the use of the Laplace trans-
form, which allows to analyze the dynamics of coupled
rooms, such as transient response and double or multiple
reverberation slopes.
It is possible, for instance, to solve for the slopes of the
energy impulse response. Of course, one should not be
tempted to get the impulse response for its use in aural-
ization since only the energy density is obtained. But it
could be possible to simulate the late response applying the
energy response to a suitably filtered random noise.
R
EFERENCES
[1]
Tutte, W.T.
Graph Theory
. Cambridge University Press, 2001.
[2]
Borutzky, Wolfgang (Editor),
Bond Graph Modelling of
Engineering Systems. Theory, Applications and Software Support
.
Springer, 2011.
[3]
Beranek, Leo.
Acoustics
. ASA, 1986.
[4]
Bergman, Theodore L.; Lavine, Adrienne S.; Incropera, Frank P.;
Dewitt, David P.
Fundamentals of Heat and Mass Transfer, 7th Ed.
,
John Wiley & Sons Inc., 2011.
[5]
Cremer, Lothar; Müller, HA.
Principles and Applications of Room
Acoustics
. Applied Science, London, 1982.
[6]
Kuttruff, Heinrich.
Room acoustics (5th edition)
. Spon Press, 2009
Revista elektron, Vol. 3, No. 1, pp. 1-5 (2019)
ISSN 2525-0159
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Enlaces de Referencia
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Copyright (c) 2019 Federico Miyara
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