Connecting volume effects on dynamics of pneumatic pressure measurement systems


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 315 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 315 - 318 

 
 

CONNECTING VOLUME EFFECTS ON DYNAMICS OF PNEUMATIC 

PRESSURE MEASUREMENT SYSTEMS 
 

J. Kutin1, A. Svete2 

 
1 University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia, joze.kutin@fs.uni-lj  

2 University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia, andrej.svete@fs.uni-lj  

 

Abstract: 

The pressure sensors are often placed at a certain 

distance from the measured object. Beside the 

properties of the transmission fluid, dynamic 

characteristics of such a measurement system 

depends on the geometry and dimensions of the 

connecting elements. So far made research works 

have shown that the internal volume of the pressure 

sensor can have a large influence on the dynamic 

response. This paper is focused on theoretical 

analysis of the effects of the sensor volume on 

characteristic parameters of both the frequency and 

the time response of the system under discussion 

with a gas medium.  

Keywords: pressure measurement; pressure 

transmission; dynamic response; sensor volume; 

connecting tube 

1. INTRODUCTION 

The dynamic response of the pressure 

measurement system depends on dynamic 

characteristics of its constructional components. 

Because the pressure sensors are often placed at a 

certain distance from the measured object, an 

additional pressure transmission system is formed 

in-between. The dynamic response of the pressure 

transmission depends on geometry, dimensions and 

stiffness of the connecting elements, as well as 

thermodynamic and transport properties of the 

transmission fluid.  

The most frequently studied physical model of 

the pressure transmission system consists from a 

straight cylindrical connecting tube and an internal 

volume of the sensor (Figure 1). Following the work 

of Bergh and Tijdeman [1] the authors of this paper 

derived one of the latest, most sophisticated 

versions of the linear mathematical model, which is 

applicable for both pneumatic and hydraulic 

pressure measurement systems, and considers 

viscous, thermal and compliance effects [2]. This 

improved mathematical model was also 

experimentally validated for the case of pneumatic 

pressure measurement systems [3]. 

 
Figure 1: Physical model of the pressure transmission 

system 

In some limited frequency ranges the dynamics 

of the pressure transmission system can be well 

described by low-order lumped-parameter 

mathematical models; e.g., the second-order model 

with two parameters (natural frequency, damping 

ratio) or the first-order model with one parameter 

(time constant) [4–7]. In [2] the authors derived 

analytical approximations of these lumped 

parameters by applying an asymptotic series 

expansion to the complete mathematical model. 

The paper is organized as follows. Section 2 

presents the mathematical model for the frequency 

and the step response of the pressure transmission 

system and its analytical approximations. Section 3 

discusses effects of dimensions on characteristic 

parameters of dynamic response and exposes 

potential directions for optimal design of the 

pressure transmission systems. Section 4 shortly 

summarizes conclusions about the effects of the 

sensor volume. 

2. MATHEMATICAL MODEL 

The presented mathematical model deals with 

the pressure transmission system that consists of the 

cylindrical connecting tube with radius Rt and 

length Lt and the cylindrical sensor volume with 

radius Rv and length Lv (Figure 1). It is based on the 

following main assumptions: the internal radius of 

the tube is small enough in comparison with its 

length to neglect the end-flow effects, the 

dimensions of the sensor volume are small enough 

to neglect fluid-wave effects, and all the pressure 

variations in the system have small enough 

amplitudes for validity of the linear model. 

http://www.imeko.org/
mailto:joze.kutin@fs.uni-lj
mailto:andrej.svete@fs.uni-lj


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 316 

2.1. Complete model 
The complex frequency response function of the 

pressure measurement system under discussion can 

be written as: 

1
(i )

cosh sinh

out

v tin

t v

P
G

V nP

V n

 = =

 +  

,    (1) 

where Vt = π Rt
2Lt is the internal volume of the tube, 

Vv = πRv
2Lv is the internal volume of the sensor,  is 

the propagation wavenumber, and nt and nv are the 

effective thermodynamic indexes for the gas in the 

tube and the sensor volume, respectively (see [2] for 

the complete expressions for , nt and nv, including 

the wall compliance effects).  

The time step response is calculated from the 

frequency response function by using the Fourier 

transform, considering the rectangular pulse of the 

input pressure signal. The Fourier transform of the 

rectangular pulse with the length Tpulse is defined as: 

( )
i

i 1 e
(i ) e d

i

pulseT

t

in in
P p t t

− 

− 

−

−
 = =


 .    (2) 

The output signal in the frequency domain is 

determined as: 

(i ) (i ) (i )
out in

P G P =   ,        (3) 

and the time response is calculated by its inverse 

Fourier transform: 

i1
( ) (i )e d

2

t

out out
p t P





−

=  

 .       (4) 

With the intention to improve the efficiency of 

calculations, the procedure presented by equations 

(2–4) is implemented in calculations in this paper by 

using the discrete Fourier transform. The 

corresponding sampling frequency and the number 

of samples were adapted in a way to assure that 

errors in calculations of all presented response times 

are lower than 10-3. 

2.2. Analytical approximations 
The study in this paper is limited to the pressure 

transmission systems with the gas medium, where 

the wall compliance effects are usually not of much 

importance, so the presented analytical 

approximations taken from ref. [2] do not consider 

the compliance effects. 

For the connecting tubes with relatively small 

internal diameters (overdamped conditions) the 

pressure transmission system can be approximated 

with the first-order low-wavenumber model:  

1

1
(i )

1 i
G  =

+ 
,          (5) 

which is defined with one parameter, i.e., the time 

constant : 

2

2 2

8 1

2

t v

t t

L V

c R V

 
 = + 

  
, (6) 

where c,  and  are the speed of sound, density and 

dynamic viscosity of gas, respectively. The 

amplitude frequency response of the first order 

system is defined as: 

( )
1

2

1
(i )

1
G  =

+ 

,        (7) 

and the step response can be defined as: 

( ) /,1 1 e
t

out
p t

− 
= − .         (8) 

For the connecting tubes with relatively large 

internal diameters (underdamped conditions) the 

pressure transmission system can be approximated 

with the second-order high-wavenumber model: 

2 2 2

1
(i )

1 2 i / /
n n

G  =
+    − 

,    (9) 

which is defined with two parameters, i.e., the 

natural frequency n and the damping ratio : 

1

1

2

n
vt

t

c

VL

V

 =

+

            (10) 

( )
2

12
1

2 2

n t t v
v

t n t

L V

R c V

  +   
 = + +   

    
,   (11) 

where t and v represent the thermal effects for the 

gas in the tube and the sensor volume, respectively: 

1

Pr
t

 −
 = , 

1
1

Pr

t v
v

v v

R L

L R

   −
 = + 

 
, (12) 

where γ is the heat capacity ratio and Pr is the 

Prandtl number. For  << 1 the amplitude frequency 

response of the second order system can be 

approximated as: 

2 2 2

1
(i )

1 /
n

G  
− 

,          (13) 

and the lower envelope of the oscillating step 

response can be approximated as: 

( ),2 1 e
n t

out
p t

− 
 − .           (14) 

  

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 317 

3. RESULTS 

3.1. Frequency bandwidth 
Characteristics of the frequency response are 

analysed in terms of the frequency bandwidth, 

which is defined as the range of frequencies over 

which the relative amplitude dynamic errors do not 

exceed 10-2 or 1%. The bandwidth limit frequency 

flim is calculated as the minimum frequency, where 

the amplitude ratio reaches 0.99 or 1.01. Figure 2 

shows variations of the limit frequency for different 

dimensions of the pressure transmission system 

with dry air ( = 1.19 kg/m3, c = 343 m/s,  = 18.2 

μPa s, γ = 1.4, Pr = 0.71).  

 

(a) Sensor volume Vv = 1.77 cm3 (Lv = 2.5 mm, Rv = 15 mm) 

 

(b) Sensor volume Vv = 7.07 cm3 (Lv = 10 mm, Rv = 15 mm) 

Figure 2: Variations of the limit frequency with the tube 

radius for two different sensor volumes (tube length Lt = 

1 m, dry air) 

The dashed lines in Figure 2 represent the results 

of the first-order low-wavenumber (LW1) and 

second-order high-wavenumber (HW2) 

approximation models.  The LW1 limit frequency is 

determined using equation (7): 

( )
2

1
1

1
r

(L)

lim

e= −

+ 

,          (15) 

which results in: 

( )
2

1 1
1

2 2 1

(L)
(L) lim

lim

r

f
e


= = −

  −
. (16) 

The HW2 limit frequency is determined using 

equation (13): 

2 2

1
1

1 /
r(H)

lim n

e= +
−  

,           (17) 

which results in: 

1
1

2 2 1

(H)
(H) lim n

lim

r

f
e

 
= = −

  +
.         (18) 

The LW1 model predicts the limit frequency 

relatively well for thiner connecting tubes. The 

frequency bandwidth is inversely proportional to 

the time constant τ and, therefore, is inversely 

proportional to (1/2 + Vv/Vt). The HW2 model 

predicts the limit frequency relatively well for 

thicker connecting tubes. The frequency bandwidth 

is proportional to n and, therefore, is inversely 

proportional to (1/2 + Vv/Vt)
1/2. 

 The frequency bandwidth is the largest in the 

intermediate region of validity of the low-

wavenumber and high-wavenumber approximate 

solutions. For the presented simulation results the 

largest limit frequency is found at the tube radius of 

0.63 mm for the smaller sensor volume (Figure 2(a)) 

and at the tube radius of 0.74 mm for the larger 

sensor volume (Figure 2(b)). The optimal tube 

radius that results in the largest frequency 

bandwidth can be well predicted by the intersection 

of both approximate models. 

3.2. Settling time 
Characteristics of the step transient response are 

analysed in terms of the settling time, which is 

defined as the minimum time that the normalized 

response needs to fully settle within 0.99 and 1.01; 

i.e., the relative dynamic errors of the response do 

not exceed 10-2 or 1%. Figure 3 shows variations of 

the settling time for different dimensions of the 

pressure transmission system. 

The dashed lines in Figure 3 represent the results 

of the first-order low-wavenumber (LW1) and 

second-order high-wavenumber (HW2) 

approximation models.  The LW1 settling time is 

determined using equation (8): 

( )

1 e 1

L

set
t

r
e

−
− = − ,                    (19) 

which results in: 

ln
(L)

set r
t e= − .                                                      (20) 

 Complete model

 LW1 approx. model

 HW2 approx. model

0.1 1 10
0.1

1

10

100

 

Tube radius R
t
   /   mm

 

 

L
im

it
 f

re
q

u
e
n

c
y

 f
li

m
  

 /
  

 H
z

 Complete model

 LW1 approx. model

 HW2 approx. model

0.1 1 10
0.1

1

10

100

 

Tube radius R
t
   /   mm

 

 

L
im

it
 f

re
q

u
e
n

c
y

 f
li

m
  

 /
  

 H
z

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 318 

 

(a) Sensor volume Vv = 1.77 cm3 (Lv = 2.5 mm, Rv = 15 mm) 

 

(b) Sensor volume Vv = 7.07 cm3 (Lv = 10 mm, Rv = 15 mm) 

Figure 3: Variations of the settling time with the tube 

radius for two different sensor volumes (tube length Lt = 

1 m, dry air) 

The HW2 settling time is determined using equation 

(14): 

( )

1 e 1
H

n set
t

r
e

− 
− = − ,                (21) 

which results in: 

1
ln

(H)

set r

n

t e= −
 

.              (22) 

The LW1 model predicts the settling time relatively 

well for thiner connecting tubes. The settling time is 

proportional to the time constant τ and, therefore, is 

proportional to (1/2 + Vv/Vt). The HW2 model 

predicts the settling time relatively well for thicker 

connecting tubes. The settling time is inversely 

proportional to nζ and, therefore, is proportional 

to: 

( ) ( )

5/ 4

1

21

1
1 1

2

v

t(H)

set
vn

t v

t

V

V
t

V

V

 
+ 

 
 
 

+  + + 

;      (23) 

so if the thermal effects are neglected (v → 0, t → 

0), the settling time is found proportional to (1/2 + 

Vv/Vt)
1/4. 

Similar to the findings about the frequency 

bandwidth, the fastest response is found in the 

intermediate region of validity of the low-

wavenumber and high-wavenumber approximate 

solutions. The smallest settling time occurs at a bit 

higher tube radius in comparison with the tube 

radius that results in the largest limit frequency 

(marked in Figure 3 with the vertical dot line).  

4. SUMMARY 

As could be expected, the dynamic response of 

the pressure transmission system is deteriorated 

with the increased sensor volume. These effects 

significantly depend on dimensions of the 

connecting tube. Both the limit frequency of the 

frequency response and the settling time of the step 

response are influenced by the sensor volume in 

terms of the magnitude of (1/2 + Vv/Vt). Therefore, 

the effects of the sensor volume can be neglected if 

Vv << ½ Vt.  

5. REFERENCES 

[1]  H. Bergh, H. Tijdeman H, Theoretical and 
Experimental Results for the Dynamic Response of 

Pressure Measuring Systems. Amsterdam: Nat. 

Aero- and Astronautical Res. Inst., Report NLR-TR 

F.238, 1965. 

[2]  J. Kutin, A. Svete, “On the Theory of the Frequency 
Response of Gas and Liquid Pressure Measurement 

Systems with Connecting Tubes”, Measurement 

Science and Technology, vol. 29, pp. 125108, 

November 2018. 

[3]  A. Svete, J. Kutin, “Experimental Validation of an 
Improved Mathematical Model for Pneumatic 

Pressure Measurement Systems with Connecting 

Tubes”, Measurement Science and Technology, 

vol. 31, pp. 015101, October 2019.  

[4]  E. O. Doebelin, Measurement Systems: 
Application and Design. Boston, MA: McGraw-

Hill, 2004. 

[5]  S. A. Whitmore, B. Fox B, “Improved Accuracy, 
Second-Order Response Model for Pressure 

Sensing Systems”, Journal of Aircraft, vol. 46, pp. 

491–500, March–April 2009. 

[6]  B. F. Hall, T. Povey, “A Practical Model for 
Pressure Probe System Response Estimation (with 

Review of Existing Models)”, Measurement 

Science and Technology, vol. 29, pp. 1–12, 

February 2018. 

[7]  I. Bajsić, J. Kutin, T. Žagar, “Response Time of a 
Pressure Measurement System with a Connecting 

Tube”, Instrumentation Science and Technology, 

vol. 35, pp. 399–409, June 2007. 

 
 

 Complete model

 LW1 approx. model

 HW2 approx. model

0.1 1 10
0.01

0.1

1

10

 

Tube radius R
t
   /   mm

 

 

S
e
tt

li
n

g
 t

im
e
 t

se
t 
  

/ 
  

s

 Complete model

 LW1 approx. model

 HW2 approx. model

0.1 1 10
0.01

0.1

1

10

 

Tube radius R
t
   /   mm

 

 

S
e
tt

li
n

g
 t

im
e
 t

se
t 
  

/ 
  

s

http://www.imeko.org/