Mathematical model of checkweigher with electromagnetic force balance system


ACTA IMEKO 
June 2014, Volume 3, Number 2, 9 – 13 
www.imeko.org 

 

Mathematical model of checkweigher with electromagnetic 
force balance system 
Yuji Yamakawa1, Takanori Yamazaki2 

1 University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, Japan 
2 Tokyo Denki University, Hatoyama, Hiki-gun, Saitama, Japan 

 

 

Section: RESEARCH PAPER  

Keywords: mass measurement; electromagnetic force balance system; dynamic behavior 

Citation: Yuji Yamakawa, Takanori Yamazaki, Mathematical model of checkweigher with electromagnetic force balance system, Acta IMEKO, vol. 3, no. 2, 
article 4, June 2014, identifier: IMEKO-ACTA-03 (2014)-02-04 

Editor: Paolo Carbone, University of Perugia  

Received March 4th, 2013; In final form August 14th, 2013; Published June 2014 

Copyright: © 2014 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits 
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited 

Funding: (none reported) 

Corresponding author: Yuji Yamakawa, e-mail: Yuji_Yamakawa@ipc.i.u-tokyo.ac.jp 

 
 

1. INTRODUCTION 
Recently, highly accurate mass measurement systems for 

packages moving along a high-speed conveyor belt —so-called 
“checkweighers”— have been increasing in importance in the 
food and manufacturing logistics industries. To achieve high-
speed (continuous) measurement, packages should be moved in 
sequence. This means that the measuring time for each package 
is very short. In the near future, continuous mass measurement 
for 300 products per minute will be required. 

In general, there are two types of checkweigher systems, 
namely the load-cell type and the electromagnetic force 
compensation (EMFC) type. In the load-cell system, the mass 
of an object can be measured by the deformation of the 
Roberval mechanism when the mass to be measured is put on a 
weighing pan. Although this type is applicable to a wide range 
of masses, it has difficulty achieving high accuracy. On the 
other hand, the EMFC system keeps a balance with the 
displacement of the lever-linked Roberval mechanism with 
electromagnetic force. Then, the mass of the object can be 
measured by the driving (or feedback) current for the 
electromagnetic force. This type has achieved high accuracy by 
the null method, but is limited in its range of measurement. 

Ono, W. G. Lee, and Kameoka have already clarified in 
detail the dynamic behavior of the load-cell system [1]-[4]. But 
the dynamic behavior of the EMFC system has not yet been 
presented. Our aim was to improve the performance of the 
checkweigher for better speed and accuracy. To do this, it was 
necessary to develop a control scheme for the EMFC. We 
introduced a dynamic model of our proposed checkweigher 
using EMFC in previous papers [5]. However, the validity of 
the model for the closed-loop system could not be confirmed. 
In this paper, the model of the measurement system is 
improved. Then, the validity of this model is confirmed by a 
comparison with actual experimental results. 

2. MEASUREMENT SYSTEM 
Figure 1 shows the overall scheme of the checkweigher. 

Only a measuring conveyor appears in the photograph; the feed 
conveyor is located to the left of the measuring conveyor and 
the sorter is located to the right. The products are moved by 
the feed conveyor, the mass of product measured by measuring 
conveyor and the product is removed by a sorter outside of the 
measurable range. 

The enlarged photograph of the mass measurement  

ABSTRACT 
Our aim was to develop a high-speed and high-accuracy mass measurement system to be used with a conveyor belt (a checkweigher). 
The objective in this paper was to present a dynamic model of our proposed measurement system. The checkweigher with an 
electromagnetic force balance system was approximated using a spring-mass-damper system as the physical model, and the equation 
of motion was derived. The model parameters could be obtained from experimental data. Finally, the validity of the proposed model 
was confirmed by comparison of the simulation results with the experimental responses. The dynamic model obtained offers practical 
and useful information to examine the control scheme and to achieve high-performance mass measurement. 

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mechanism in the checkweigher is shown in Figure 2. The mass 
measurement system consists of a weighing platform, the 
Roberval mechanism, the lever-linked Roberval mechanism, the 
counter weight, the electromagnetic force actuator and the 
displacement sensor. By applying the Roberval mechanism to 
the measurement mechanism, the mass of the product can be 
measured regardless of where the product is located on the 
weighing platform. 

The mass of the product is estimated from the current of the 
electromagnetic force actuator to control the lever displacement. 

The mass-measuring method can be explained as follows: 
1. When the product of the mass M is put on the 

measurement system, this causes the displacement of the 
Roberval mechanism. 

2. The displacement of the Roberval mechanism is 
magnified twenty times by the lever. 

3. The magnified displacement can be measured by the 
displacement sensor. 

4. The current is controlled so that the displacement of 
the lever is maintained at zero. 

5. The current is measured and the current is converted 
to the estimated mass using a linear function. 

3. MODELING 
Figure 3 and Figure 4 show the physical model of the 

measurement system. The equation of motion about mass m 
can be obtained as follows: 

, (1) 
where m is the mass of the Roberval mechanism, M is the mass 
of the product to be measured, c is the a damping coefficient, k 
is a spring constant, g is the acceleration of gravity, L (= 20 
m/m) is the lever ratio, and x is the displacement of mass m. In 
addition, mL is the mass of the lever, xL is the displacement of 
mass mL, and F (= Bli, where B is a density of magnetic flux, l a 
length of the coil, and i a current) is an electromagnetic force 
input to control the position xL of the mass mL. 

From Eq. (1), the natural frequency is 

 .     

From the experimental results, the natural frequency of the 
lever is 5 Hz. Thus, the spring constant of the measurement 
system can be given as follows: 

 . (3) 
The damper coefficient can be adjusted so as to match the 

convergence rate in the responses of the lever. 
From Eq. (1), the Roverbal displacement at steady-state for 

the open-loop system (F = 0) can be described by: 

 . (4) 

In addition, the lever displacement at steady state, which is 
the Roverbal displacement magnified by the lever, can be 
obtained by the following equation: 

 . (5) 
Simulations of the measurement system can be performed 

using Eq. (1)-(5). Figure 5 shows a block diagram of the 

 
Figure 1. Overall of checkweigher. 

 
Figure 2. Photograph of Roberval mechanism. 

 
Figure 3. Overall of measurement system. 

 
Figure 4. Physical model of measurement system. 

   

Weighing
platform

Counter
weightRoberval

mechanism

Lever

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measurement system. In simulations and experiments, the 
validity of the model can be confirmed by comparing the sensor 
outputs of the lever displacement (xL). 

4. EXPERIMENTAL AND SIMULATION RESULTS FOR STEP 
RESPONSE 

In this section, the validity of the proposed model is 
confirmed by comparing the simulation result and the 
experimental results for open-loop and closed-loop systems. 

4.1. Open-loop system 
The validity of the proposed model was first examined by 

comparing the simulation and experimental results for the 
open-loop system. 

Figure 6 depicts the experimental and simulation results for 
M = 0.01, 0.02, 0.05 and 0.1 kg. The red and blue lines indicate 
the experimental and simulation results, respectively. The start-
up operation is the time when the product of the mass M is put 
on the measurement system. When the time is 0.2 s, the 
product of the mass M is removed. At the same time, the 

 
 

Figure 5. Block diagram of measurement system. 
 

 

 
Figure 6. Comparison between experimental result and simulation result (open loop). 

  

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displacement xL is measured by the displacement sensor. In 
Figure 6, the vertical axis shows the displacement sensor output 
of the lever displacement. 

It can be seen from these figures that the responses of the 
simulation were in excellent agreement with the experimental 
results. Thus, the validity of the proposed model was confirmed 
for the open-loop system. Although the amplitude of the 
response changes depending on the mass, the convergence time 
is not changed. Thus, this measurement system can be assumed 
to be a linear system. Namely, even if the controller is added to 
the measurement system, the response does not depend on the 
mass. 

In the next section, the validity of the proposed model for 
the closed-loop system was examined. 

4.2. Closed-loop system 
This section explains the experimental and simulation results 

with EMFC. The electromagnetic force was controlled by a 
proportional-integral-differential (PID) control scheme. Taking 
the actual circuit of the D action into account, the ideal D 
action could not be implemented. So, it is assumed that the D 
action is the approximated differential action. Namely, the 
transfer function of the PID controller C(s) can be described by 

, (6) 

where kp is the proportional gain, ki is the integral gain, and kdn 
and kdd are the numerator and denominator coefficients of the 
differential gains, respectively. The control voltage Uv(s) 
(Laplace transform of uv(t)) can be adjusted as follows: 

, (7) 
where E(s) is Laplace transform of e(t), and e(t) (= xLr − xL) is 
the error between the reference xLr and the displacement sensor 
output of the lever xL. The PID control can be performed by 
FPGA (Field Programmable Gate Array) every 0.1 ms. 

Figure 7 depicts the experimental and simulation results for 
M = 0.01, 0.02, 0.05 and 0.1 kg. The red and blue lines indicate 
the experimental and simulation results, respectively. The 
simulation and experimental conditions are the same as the 
open-loop system. 

Responses such as the convergence time, the rise time and 
the settling time were almost the same for the simulation and 
experimental results. Thus, the validity of the proposed EMFC 
model was confirmed. We consider that the reasons for the 
modeling error were the frictional force in small displacements 
and the magnification mechanism of the lever. However, the 
peak value of the simulation result for M = 0.1 kg was different 
from that of the experimental result. Moreover, a high-
frequency response occurred in the experiments due to the 
system noise. 

 

 
Figure 7. Comparison between experimental result and simulation result (closed loop). 

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5. SIMULATION RESULTS OF FLOOR VIBRATION 
Here, we discuss the effect of floor vibrations in the 

simulation and the experimental results. The floor vibration 
oscillates the base of the system. The vibration input applies to 
the same position in the block diagram (Figure 5) as the mass of 
the object. In the experiment, a vibration exciter is used in 
order to oscillate the system. 

Figure 8 shows the simulation result and the experimental 
result of floor vibration. The amplitude and the frequency of 
the vibration were set at xb = 1 mm and fb = 10 Hz, respectively. 
The angular frequency ωb can be calculated by 2πfb. The mass mb 
of the base of the system was set at 0.175 kg. The acceleration 
input ab of the floor vibration was set at xb×ωb2. The red and 
blue lines indicate the experimental and simulation results, 
respectively. As can be seen from this result, exactly the same 
responses were obtained. 

In addition, Figure 9 shows the results for the simulation 
and the experiment when the amplitude and frequency of the 
floor vibration were set at xb = 1.25 mm and fb = 10 Hz, 
respectively. The red and blue lines indicate the experimental 
and simulation results, respectively. It can be seen from Figure 
9 that the same responses were obtained. The validity of the 
proposed model was thus confirmed for different amplitudes of 
vibration. 

Furthermore, Figure 10 shows the result of the simulation 
and the experiment in which the amplitude and the frequency 

of the floor vibration were set at xb = 1 mm and fb = 15 Hz. 
The red and blue lines indicate the experimental and simulation 
results, respectively. The error between the simulation and the 
experimental result was about 1 g. When the frequency of the 
floor vibration increases by 1.5 times, the acceleration increases 
2.25 (=1.52) times. Thus, the output theoretically increases 2.25 
times in the simulation. However, the output of the 
experimental result increased only about 1.75 times in this case. 
We consider that the reason was the mechanical limitation of 
the displacement in the mass measurement system in order to 
avoid the system breakdown. The simulation could not replicate 
the experimental results for this reason. 

As a result, we can evaluate the effect of the floor vibration 
by the proposed model. 

6. CONCLUSIONS 
In this paper, a dynamic model of an electromagnetic force 

compensation (EMFC) system was constructed. The model was 
approximated by a spring-mass-damper system. The model 
parameters were identified by the experimental data for an 
open-loop response. 

Then, the validation of the proposed model was examined 
for both open-loop and closed-loop systems. The simulation 
and experimental results for the open-loop system were exactly 
the same. The results for the closed-loop system were almost 
the same. As a result, the validity of the proposed EMFC model 
could be confirmed. In addition, we confirmed the validity of 
the proposed model with simulations of floor vibrations. 

In the future, an entirely new control scheme for the 
electromagnetic force compensation can be implemented using 
the proposed model. This high-speed, high-accuracy mass 
measurement system will be applicable to many industrial fields. 

REFERENCES 

[1] T. Ono, “Basic point of dynamic mass measurement”, Proc. 
SICE, pp. 43-44, 1999. 

[2] T. Ono, “Dynamic weighing of mass”, Instrumentation and 
Automation 12, No. 2, pp. 35, 1984. 

[3] W.G. Lee et al., “Development of speed and accuracy for mass 
measurements in checkweighers and conveyor belt scales”, Proc. 
ISMFM, pp. 23-28, 1994. 

[4] K. Kameoka et al., “Signal processing for checkweigher”, Proc. 
APMF, pp. 122-128, 1996. 

[5] Y. Yamakawa and T. Yamazaki, “Mathematical Model of 
Checkweigher with Electromagnetic Force Compensation”, 
Proc. XX IMEKO World Congress, TC3-O-19, 2012. 

 
Figure 8. Simulation result of floor vibration (amplitude:1 mm, frequency: 
10 Hz). 

 
Figure 10. Simulation result of floor vibration (amplitude:1 mm, frequency: 
15 Hz). 

 
Figure 9. Simulation result of floor vibration (amplitude:1.25 mm, 
frequency: 10 Hz). 

ACTA IMEKO | www.imeko.org June 2014 | Volume 3 | Number 2 | 13 


	Mathematical model of checkweigher with electromagnetic force balance system