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Vol.98(1) March 2007 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 1

VOL 98 No 1March 2007

SAIEE Africa Research JournalSAIEE AFRICA RESEARCH JOURNAL EDITORIAL STAFF ...................... IFC

Simulation of a proton exchange membrane fuel cell stack using an electronic

equivalent circuit model

by JP Du Toit and HC vZ Pienaar .................................................................2

The development of a flexible rotor active magnetic bearing system

by EO Ranft, G van Schoor and JG Roberts .................................................8

An adaptive hybrid list decoding and chase-like algorithm for Reed-Solomon codes

by W Jin, H Xu and F Takawira .............................................................................13

Effect of characteristics of dynamic muscle contraction on crosstalk in surface

electromyography recordings

S Viljoen, T Hanekom and D Farina ...........................................................18

NOTES FOR AUTHORS ...................................................................................IBC


SIMULATION OF A PROTON EXCHANGE MEMBRANE FUEL CELL STACK USING AN ELECTRONIC EQUIVALENT CIRCUIT MODEL

J.P. Du Toit* and H.C. vZ. Pienaar**

* Dept. of Applied Electronics and Electronic Communications, Vaal University of Technology, Private Bag X021, Vanderbijlpark, 1900, South Africa. **Telkom Centre of Excellence, Vaal University of Technology, Private Bag X021, Vanderbijlpark, 1900, South Africa.

Abstract: This article describes the method to calculate the parameters of an electronic equivalent circuit model of a proton exchange membrane fuel cell stack. The model is based on that of Yu and Yuvarajan [1], but very little detail is given by them on how to calculate the component values of the circuit model based on the experimental data of a fuel cell or fuel cell stack. Here, a method will be established in which the performance data of small fuel cell stacks can be used to simulate the behaviour of much larger stacks under the same operating conditions. It is crucial to be able to simulate a fuel cell stack using an electronic circuit equivalent model when designing power converters for fuel cells, since the fuel cell stack can then be simulated together with a power converter or any other electronic circuitry. First, a mathematical model of a small, two-cell, proton exchange membrane fuel cell was calculated based on experimental data. This model was then adapted to describe the characteristics of a much larger, 100 W, fuel cell stack. Finally, the mathematical model was used to calculate the parameters for an electronic circuit model by establishing a clear relationship between the two models.

Key words: Proton exchange membrane fuel cell (PEMFC), electronic circuit model

1. INTRODUCTION

Over recent years, much emphasis has been placed on the development of fuel cells (FCs) for the replacement of internal combustion engines (ICEs) in automobiles for both economic and environmental reasons. Although there are different types of FCs currently in development, the proton exchange membrane fuel cell (PEMFC) is the most likely candidate to replace ICEs because of its lightweight and low operating temperatures. A PEMFC is an electrochemical device that converts a fuel (hydrogen) and an oxidant (oxygen or air) into electricity. Unlike a battery, a FC can supply electrical energy for as long as it is being supplied with fuel. This, together with the fact that a FC contains no moving parts, results in much longer lifetimes of FCs compared to batteries and ICEs, resulting in long-term cost savings.

A number of technical and economic issues still prevent the widespread implementation of FCs in automobiles since the threat of global warming demands that hydrogen for cars be produced from sources that do not generate greenhouse gases. However, the use of FCs in stationary applications is much more plausible [2]. This is especially true for the use of FCs in backup power systems, even more so when the hydrogen needed to fuel these devices can be produced on-site from renewable sources.

The theoretical open-circuit output voltage of a PEMFC

is 1.23 V. Due to losses in the various components of a FC, the open-circuit voltage is somewhat less, typically around 0.9 V. Furthermore, as the output current of a FC increases, its voltage drops in a non-linear fashion. The various losses in a FC, together with their influence on the cell’s output characteristics, will be discussed in the next section. If a FC stack is to be implemented in a system that would provide backup power for telecommunications equipment, for example, some form of power converter would be needed to increase the output voltage of a FC stack to a required level and that would be able to maintain a constant output voltage even under changes in load conditions.

In order to facilitate the design of a DC-DC boost converter to condition the output voltage of a FC stack, it is necessary to be able to simulate a FC stack as it operates in combination with such a converter. For this reason an electronic equivalent circuit model of a PEMFC stack has been developed [1]. Very little detail is available on the calculation of the parameters of the above-mentioned model. Furthermore, Yu and Yuvarajan [1] only calculated the parameters of the model based on already existing performance data of a PEMFC stack and did not use the model to predict the characteristics of larger stacks.

This paper will discuss the method used to calculate the parameters of an electronic circuit equivalent model of a

Vol.98(1) March 2007 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS �

PEMFC stack. This was done by obtaining a mathematical model of a small, two-cell stack, based on experimental data. The mathematical model was then adapted to model the behaviour of a larger stack and verified by comparing the model with experimental data of such a stack. Finally, the model was adapted for an even larger, 100 W FC stack, and then used to calculate the needed parameters for an electronic circuit model by establishing a relationship between the mathematical and electronic equivalent circuit model.

2. V-I CHARACTERISTICS OF A FC STACK

As the current drawn from a FC stack increases, the output voltage of the stack decreases due to various losses present in the components of the stack. The voltage-current curve of a typical FC is shown in Figure 1 and is also referred to as a polarisation curve or a V-I curve.

Figure 1: Polarisation curve of a FC stack [3].

Note that a polarisation curve shows the output voltage of a FC with regards to current density which makes it easier to compare cells with different membrane surface areas since the amount of current that can be supplied by a FC increases as its membrane surface area increases. Figure 1 shows that there are three regions of a FC polarisation curve in which the output voltage exhibits different characteristics. The following sections will discuss each of these regions as well as the mathematical equations that describe them [3].

2.1 Activation losses, fuel crossover and internal currents

Activation losses comprise the portion of the cell voltage that is lost in providing activation energy for the chemical reaction that transfers electrons between the electrodes. This voltage drop has a very non-linear form and contributes, in part, to the first region of the graph in Figure 1. Activation losses, Vact, can be described by the equation shown in (1) where, i is the current density in

mA.cm-2 and i0 is the current density at which the voltage drop begins to move away from zero. For example, if i0 is 100 mA.cm-2 there will be no voltage drop until the current density i is greater than 100 mA.cm-2. In (1), A is a constant which value depends on the electrode material as well as the type of reaction taking place.

0ln

i

iiAV n

act

0lnln iAiA (1)

Another cause of voltage drop in a FC is from fuel crossover and internal currents. Although the electrolyte membrane in a FC is designed to be impermeable to gas flow and to be only proton conductive, it is still possible for a small amount of the reactants (H2 and air) to permeate through the membrane from one side of the cell to the other. This, together with a small amount of electron flow through the membrane, causes a voltage drop in the open circuit voltage of low-temperature FCs. If a total internal current density in is caused to flow through the cell by fuel crossover and internal currents, this voltage drop can be combined with the activation losses given in the first part of (1). Since the crossover current is usually very small and only useful in explaining the initial drop in FC voltage, it can be omitted [3].

2.2 Ohmic losses

The second region of a FC polarisation curve is fairly linear and is caused by the electrical resistance of the electrodes as well as the resistance to proton flow through the membrane. This voltage drop is proportional to the current density i and can be modeled by:

irohmV (2)

where r is an area specific resistance in terms of k .cm2.

2.3 Mass transport or concentration losses

The final cause of voltage drop in a FC is shown in region 3 of Figure 1. This loss is caused by mass transportation loss or concentration losses. When the oxygen needed by the cell is supplied in the form of air, there will be a reduction of the concentration of oxygen in the air around the electrodes as the oxygen is used by the cell. On the anode side, where hydrogen is used, there will also be a reduction of hydrogen pressure as more hydrogen is consumed as a result of higher currents being drawn from the cell. The concentration loss, Vtrans is given by:

nitrans meV (3)


where m and n are constants in terms of V and cm2.mA-1

respectively.

2.4 Complete mathematical model for a FC stack

Equations (1)-(3) can be combined and subtracted from the theoretical open circuit voltage E (E = 1.23 V), to give the actual FC output voltage over its entire current range:

nimeiAiriAEV lnln 0 (4)

The first two terms in the above equation are constant, regardless of the cell current, and can be replaced by a practical open circuit voltage, EOC:

nioc meiAirEV ln (5)

In the case of a FC stack, comprised of NC identical cells, Equation (5) can be rewritten as [4]:

nicccoccSTACK meNiANirNENV ln

(6)

The constants in the above equation (r, A, m and n) were determined from the analysis of experimental data from small FC stacks of only two cells. The value of NC was then adjusted to predict the behaviour of a larger, 100 W, FC stack. The resulting equation was then used to design an equivalent electronic circuit model of a 100 W FC stack to enable the simulation of the stack together with other electronic circuitry.

In order to verify the model in Equation (6), a small two-cell stack was constructed and used to obtain an experimental polarisation curve [5]. A typical value of nwas chosen as 0.008 cm2.mA-1.

From the data in Table I, it can be seen that the open circuit voltage of the two-cell stack, NEOC is equal to 1.97 V. This means that the open circuit voltage of a single cell would be 0.985 V. By substituting values from Table I into Equation (6), the values of r, A and m were calculated. This was done by forming six equations from the data in Table I:

8

24

160

320

560

640

r

r

r

r

r

r

2.7726

4.9698

8.7641

10.1503

11.2696

11.5366

A

A

A

A

A

A

2.0650

2.2015

3.7930

7.1933

18.7867

25.8716

m

m

m

m

m

m

0.15

0.22

0.40

0.55

0.88

0.96

The above equations can be written in matrix form:

Ax = b, (7)

where:

A =

8 2.7726 2.0650

24 4.9698 2.2015

160 8.7641 3.7930

320 10.1503 7.1933

560 11.2696 18.7867

640 11.5366 25.8716

,

r

x A

m

and b =

0.15

0.22

0.40

0.55

0.88

0.96

Table I: Two-cell stack performance data.

To solve for the solution matrix, x, Equation (7) can be rewritten in the form

x = A-1b (8)

The above equation was solved and the following values were found:

Stack Voltage (V)

Current (A)

Current (mA.cm-2)

1.82 0.1 4

1.76 0.2 8

1.75 0.3 12

1.73 0.4 16

1.72 0.5 20

1.70 0.6 24

1.69 0.7 28

1.68 0.8 32

1.67 0.9 36

1.65 1.0 40

1.61 1.5 60

1.57 2.0 80

1.53 2.5 100

1.50 3.0 120

1.46 3.5 140

1.42 4.0 160

1.38 4.5 180

1.33 5.0 200

1.26 5.5 220

1.23 6.0 240

1.16 6.5 260

1.09 7.0 280

1.01 8.0 320


r = 0.0006 k .cm2

A = 0.0351 V m = 0.0147 V

Based on the above values, the mathematical model of any NC – cell stack under the same operating conditions would be:

0,008

(0,986) (0,0003)

(0,0351) ln (0,0147)STACK C C

iC C

V N N i

N i N e (9)

Note that Equation (9) only holds true when i is given in terms of current density (mA.cm-2). The equation was converted to express i in terms of Amperes (A) so that it can be used to determine the parameters of an electronic equivalent circuit model:

0,32

(0,986) (0,012)

(0,0351) ln (0,0147) (0,13)O

STACK C C O

IC O C C

V N N I

N I N e N (10)

where I0 is the stack current in terms of Amperes (A). Also note that a second constant term is introduced in Equation (10) because of the nature of the natural logarithm term in Equation (9). The next section discusses the development of the electronic circuit equivalent model of a FC stack as proposed by Yu and Yuvarajan [1] that allows a FC stack containing any number of cells to be simulated using an electronic simulation package.

3. ELECTRONIC EQUIVALENT CIRCUIT MODEL

In the previous section, the mathematical model of a FC stack consisting of NC cells was discussed. As stated earlier, an electronic circuit equivalent model was proposed by Yu and Yuvarajan [1]. However, no details were given on a method of calculating the values of the components in the equivalent circuit model based on a mathematical equation obtained from experimental data.

The electronic circuit model of a PEMFC stack is shown in Figure 2. The model is based on the non-linearity of a diode and the current control feature of bipolar junction transistors (BJTs). The diode is used to model the ohmic losses and activation losses, while the two BJTs (Q1 and Q2) are used to model concentration losses. The capacitor C and inductor L are used to measure the dynamic behaviour of the stack. Typical values of 1F and 10 mH were chosen for these two components respectively.

The relationship between the voltage across a diode (VD)and the current through it (ID) is given by the equation:

ln DD T

SD

IV nV

I, and T

kTV

q (11)

where n is the emission coefficient, ISD is the saturation current and VT is the thermal voltage in terms of the Boltzmann’s constant (k), absolute temperature (T) and electronic charge, q. This equation exactly resembles Equation (1) for the activation losses in a FC [1].

The transistors, Q1 and Q2, together with R1 and R2 form a current limiting circuit used to model the concentration losses of the FC stack. R2 acts a current sensing resistor so that when the current through it exceeds a certain limit, Q2 will start conducting, reducing the base voltage of Q1. This will cause the emitter voltage of Q1 to decrease at an exponential rate. The two transistors are assumed to be identical with current gain and base-emitter voltage VBE.The variation of the output voltage (V0) as a function of load current (I0) can be determined using the circuit in Figure 3.

The base current of Q1 (IB1) and the collector current of Q2 (IC2) can be written as

21 1

O CB

I II (12)

2

2

O

T

I R

VC SI I e (13)

where IS is the saturation current of Q1 and Q2 and VT is the thermal voltage as given by Equation (11).

The output voltage can then be written as

1 2 1 2 2R1 R2O S B C BE O C BV V I I V I I I (14)

Figure 3: Modelling the concentration losses of a FC.

Figure 2: Electronic circuit model of a PEMFC [1].


By substituting Equations (12) and (13) into Equation (14) and assuming that is large, the output voltage can be simplified to

R2

R2 R1O

T

I

VO S O S BEV V I I e V (15)

Combining Equations (11) and (15), as well as taking the ohmic resistance of the diode (RD) into account, the total FC stack output voltage can be written as:

D

R2

ln (R2 R )

ln R1O

T

O S T SD BE O

I

VT O S

V V nV I V I

nV I I e (16)

It can be seen that Equation (16) has the same form as the mathematical model of a FC stack given in Equation (10). By comparing these two equations, the following values must be calculated in order to finalise the circuit model of a FC (the thermal voltage VT is 25 mV at room temperature):

VS (Voltage of the battery) R1 and R2 The emission coefficient of the diode, nSaturation current of the diode, ISD

Saturation current of Q1 and Q2, IS

Ohmic resistance of the diode, RD

For a two-cell stack (NC = 2), the above values can be determined by comparing Equation (16) with Equation (10) and establishing the following relationships:

ln 1,972S T SDV nV I , where VS is the theoretical

output voltage of 2.46 V.

DR2 R 0,024

0,07TnV

R1. 0,0294SI

R20,32

TV

A value for R1 was chosen as 10 . The values for Equation (16) were then calculated to be:

R2 = 0.008 IS = 0.00294 A n = 2.8 3 RD = 0.016 ISD = 0.000938 A

4. SIMULATION RESULTS

The graph in Figure 4 shows the polarisation curves of a two-cell FC stack obtained from experimental data (Table I), the mathematical model (Equation (10)) as well as the results of simulating the proposed electronic circuit equivalent using the Proteus VSM simulation package.

It can be seen from Figure 4 that the three results closely match each other, proving that the two models can be used to accurately describe the behaviour of a FC stack. In order to further test the models, Equation (10) was used to calculate the performance of a four-cell stack. The resulting equation was then used to determine the parameters of the equivalent circuit model for such a stack. The simulation results, together with those obtained from calculating (10), were compared to the actual performance data obtained from a four-cell stack constructed in the laboratory. These graphs (Figure 5) show that the model can be used to predict the behaviour of larger FC stacks based in the performance data of smaller ones.

Figure 4: Simulated, calculated and measured polarisation curves of a two-cell PEMFC stack.


Figure 5: Simulated, calculated and measured polarisation curve of a four-cell stack.

5. CONCLUSION

The various losses present in a FC were discussed in terms of mathematical equations which were then used to mathematically model the behaviour of a FC stack. An already existing electronic circuit equivalent model of a FC stack was defined and the mathematical model of the FC used in order to make the circuit equivalent model

more adaptable and easier to use for a wide variety of applications. It could be seen that the data collected from testing a small FC stack could be used to model and simulate the behaviour of larger stacks.

ACKNOWLEDGEMENTS

Telkom SA Ltd., M-TEC and TFMC have graciously provided funding for this project.

6. REFERENCES

[1] D. Yu, S. Yuvarajan: “Electronic circuit model for proton exchange membrane fuel cells”, J. Power Sources, vol 142, pp. 238-242, 2005.

[2] J. Romm: The hype about hydrogen, fact and fiction in the race to save the climate, 1st edition, Washington, Island Press, 2004.

[3] J. Larmine, and A. Dicks: Fuel cell system explained, 2nd ed, West Sussex, Wiley, 2003.

[4] R. Jiang, D. Chu: “Comparative studies of polymer electrolyte membrane fuel cell stack and single cell”, J. Power Sources, vol 80, pp.226-234, 1999.

[5] J. Du Toit: Design and development of a 100 W proton exchange membrane fuel cell, Vaal University of Technology, 2006.


THE DEVELOPMENT OF A FLEXIBLE ROTOR ACTIVE MAGNETIC BEARING SYSTEM

E.O. Ranft*, G. van Schoor* and J.G. Roberts**

* School of Electrical, Electronic and Computer Engineering, North-West University, Potchefstroom Campus, Private Bag X6001, Potchefstroom, 2520, South Africa **School of Mechanical Engineering, North-West University, Potchefstroom Campus, Private Bag X6001, Potchefstroom, 2520, South Africa

Abstract: A design process comprising aspects of modelling and analysis is developed, implemented and verified for a flexible rotor active magnetic bearing system. The system is specified to experience the first three critical frequencies up to an operating speed of 10,000 rpm. Rotor stability at critical frequencies places specific constraints on the equivalent stiffness and damping parameters of the active magnetic bearing. An iterative design process is then initiated by an electromagnetic design of the radial active magnetic bearings resulting in parameters used in the detailed modelling of the system. Stiffness and damping parameters as well as system dynamic response are verified and used to design a flexible rotor. The magnetic bearing locations, displacement sensor locations and rotordynamic response are verified using finite element analysis. The design of the rotor stands central to the iterative design process since it impacts on the forces experienced by the active magnetic bearings as well as the critical frequencies of the active magnetic bearing system. Once constructed the actual active magnetic bearing system stiffness and damping parameters as well as dynamic response are compared to modelled results. The rotordynamic response is characterised by measuring the rotor displacement at pre-defined locations as the rotor traverses the critical frequencies. These results are compared with the predicted rotordynamic response.

Key words: Active magnetic bearing, design process, flexible rotor, rotordynamic analysis.

1. INTRODUCTION

Active magnetic bearings (AMBs) have a number of novel qualities rendering them invaluable machine components in the modern day industry. Their ability to suspend a rotor without mechanical contact results in a no wear and no lubrication configuration. This renders the AMB an environmentally friendly technology [1].

One application that stands to benefit from AMB technology is the Pebble Bed Modular Reactor (PBMR) currently in development in South Africa. According to Shi et al. [2] AMBs will become largely conventional in this application. Shi et al. [2] and Takizuka et al. [3] both conducted various studies on AMBs with the aim of applying them in high temperature reactor (HTR) facilities.

The need exists to establish a knowledge base on AMBs for the industry from which AMBs can be specified, implemented and maintained. Discussions with industry highlighted rotordynamic performance of an AMB suspended rotor as one of the key areas of research. An AMB suspended flexible rotor design process must be developed, implemented and verified.

2. DESIGN PROCESS

The iterative design process outlined in Figure 1 shows

the first step as a complete system specification. From the system specification a maximum bearing load capacity along with a force slew rate are estimated and used to conduct a preliminary analytical electromagnetic (EM) design. Parameters obtained from the EM design along with the calculated controller parameters are now used to simulate the complete system. The stiffness and damping parameters as well as the system’s dynamic response are verified using the simulation.

A rotor is now designed with the physical sizes obtained from the EM design. Dynamic analyses are performed on the rotor, using the verified stiffness and damping parameters, in order to obtain the rotor forces and displacements in the magnetic bearings. These results are compared to the maximum load capacity and allowable displacement due to the bearing geometry. The electromagnetic design is reviewed and the process is repeated until these parameters are within range when focus shifts to the rotordynamic performance.

The rotor design is reviewed until both the magnetic bearing locations as well as the rotor response are within range. The rotor forces and displacements are again verified and once the rotordynamic performance is within range, the design can be implemented. In the following sections the final iteration of the design process depicted in Figure 1 is discussed in detail.


2.1 System specification

A flexible rotor must be designed that has a maximum operating speed of 10,000 rpm. The rotor should pass through its first three critical frequencies of which the third is the first bending mode before reaching maximum operating speed. The AMB must be able to stably suspend the flexible rotor through these critical frequencies and allow for advanced control implementation.

2.2 Electromagnetic design

The electromagnetic design is performed using MathCAD® software which allows the designer to easily adjust variables and reduce the time it takes to obtain the optimal design. The design is based on the heteropolar radial bearing design process outlined in [4]. Figure 2 shows the standard 8 pole heteropolar bearing geometry with the following simplifying design choices:

Poles are paired which implies no flux splitting (NNSSNNSS) and simplified control Quadrant control is implemented, i.e. each pole pair (NS) is wound in series Removable coils are used

The power amplifier (PA) specifications are obtained from the maximum force slew rate needed to implement the desired control. A 3 kVA (300 V, 10 A) PA specification is obtained and since a simple proportional derivative (PD) controller is implemented, the PA must be current controlled.

Figure 2: Standard 8-pole heteropolar radial bearing [4].

A maximum load capacity per unit area constraint is placed on an AMB due to material properties such as flux saturation and maximum current density. This implies that the peak load capacity dictates the pole face area. This serves as a point of departure for the journal and stator design. The maximum load capacity for one pole pair is obtained from Equation (1):

2 20

2cosm g

s

N i AF

x

(1)

with µ0 the permeability of free space, N the number of coil turns per pole, im the coil current, Ag the pole face area and xs the air gap length. The angle is the pole angle with respect to the pole pair centre [4]. The AMB specifications obtained from the final design iteration are summarised in Table I.

Table I: AMB Specifications.

Parameter Specification Description

Fmax 500 N Maximum load capacity

dF/dt 5x106 N/s Force slew rate

g0 0.6 mm Nominal air gap

keq 500 N/mm Equivalent stiffness

beq 2.5 N.s/mm Equivalent damping

Ag 689x10-6 m2 Pole face area

2.3 Controller design

By utilising simple PD control an AMB emulates spring mass damper behaviour [5] with the equivalent stiffness and damping as given by Equations (2) and (3) respectively.

2 2eq P i sk K k k (2)

2eq D ib K k (3)

ki and ks, given by Equations (4) and (5) respectively, represent linearised system gains at the operating point of

No

No

Yes

Yes

System Specification

Electromagnetic Design

System Modelling

Rotor Design

Rotordynamic Analysis

Design Implementation

AMB Specifications within range?

Rotordynamics within range?

Figure 1: Iterative design process.


i0 and g0 and KP and KD are the respective proportional and differential controller gains.

0 0

20 0

20,

2 cosg

im m si i x g

N i AFk

i g (4)

0 0

2 20 0

30,

2 cosg

ss m si i x g

N i AFk

x g (5)

2.4 System modelling

In the process of designing and implementing a controller for a system such as an AMB an accurate model of the system becomes an invaluable tool. Figure 3 displays the nonlinear model that was used to simulate the system in MATLAB®. The simulation program contains the nonlinear force relationship and accurately simulates the switch-mode PA switching at 100 kHz.

The equivalent bearing stiffness and damping are verified using the simulation. The bearing stiffness is obtained from a steady-state condition by implementing a constant disturbance force and measuring the displacement. The bearing stiffness and damping parameters can also be determined from a step response using Equations (6) and (7) [6].

2 N/meq nk m (6)

2 N.s/meq eqb k m (7)

The damping factor ( ) is obtained from the percentage overshoot (P.O.) and used in conjunction with the settling time to determine the system’s natural frequency ( n).

2.5 Rotor design and dynamic analysis

In mathematical terms a natural frequency is an eigenvalue and a mode shape is an eigenvector. A distributed mass-elastic system has an infinite number of

Figure 4: Flexible rotor CADKEY® model.

eigenvalues and associated eigenvectors in theory, but in practice only the lowest three or four critical speeds and associated whirl modes are excited in the operating speed range of a high speed machine [7]. Mode shapes are determined by the distribution of mass and stiffness along the rotor, as well as the bearing support stiffness. Figure 4 displays the CADKEY® model of the flexible rotor. The centre mass is used to lower the third critical frequency to below the maximum operating speed.

The first three critical speeds typically vary with support stiffness, as shown by the critical speed map in Figure 5. This undamped lateral critical speed map was generated with Dyrobes® software. The insensitivity of the third critical speed to support stiffness allows a range of operating speeds that does not traverse any of the critical speeds indicated by the vertical arrow in Figure 5. This is good machine design practice from a rotordynamics standpoint. The modern trend toward higher speeds however makes it difficult to avoid approaching or traversing the third critical speed [7]. For an equivalent stiffness of 500 N/mm and damping of 2.5 N.s/mm the first, second and third critical frequencies are situated at 2,947 rpm, 4,637 rpm and 7,276 rpm respectively.

Laminations

Trantorque

Air turbine

Journal

Centre mass

KP

KDs

i0ref

21

2

1

s

mm

x

ik

m

1

s

1

s

1 xs+

-

+

+

+

+ +

-

-

22

22

s

mm

x

ik

+-

2g0+

PA

PA

iref

im1

im2

mg-xref

Figure 3: Simulation block diagram.


3. SYSTEM CHARACTERISATION

3.1 Equivalent stiffness and damping

Figure 6 shows the experimental and simulated results for a 50 µm step in the positive horizontal direction. Initially discrepancies were noted in the rise times and P.O. of the simulated and experimental responses. This was due to an unmodelled pole included in the differentiator path. With the additional pole introduced into the simulation, the simulated and experimental results correlate much closer as shown in Figure 6. From this response the following parameters are obtained: keq = 1.654x106 N/m and beq = 2.843x103 N.s/m. The corresponding critical frequencies obtained from the critical speed map (Figure 5) are estimated at 4,000 rpm, 8,200 rpm and 9,800 rpm.

0 0.05 0.1 0.15-1

0

1

2

3

4

5

6

7

8

9x 10

-5

Time (s)

Pos

ition

(m)

Experimental resultSimulated result

Figure 6: Horizontal step response.

3.2 Rotordynamic performance

The rotordynamic performance is experimentally determined by measuring rotor peak-to-peak displacement at the bearing locations as well as the centre mass during rotor acceleration. When comparing the vertical results of the right bearing shown in Figure 7 (a) to the predicted critical frequencies in Section 3.1, remarkable correlation is observed. This confirms the high equivalent stiffness value predicted by the simulation.

4. REVIEWING THE DESIGN PROCESS

The system design constitutes electromagnetic design, detailed system analysis and modelling. In the analytical analyses of the electromagnetic design no consideration is

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

20

40

60

80

100

Dis

plac

emen

t (µ

m p

k-pk

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

20

40

60

80

100

Rotational Speed (rpm)

Dis

plac

emen

t (µm

pk-

pk)

(a)

(b)

Figure 7: Right bearing (a) vertical, (b) horizontal displacement vs. rotational speed.

Figure 5: Critical speed map.


given to losses, leakage and fringing effects encountered in the electromagnetic circuit. The need for a detailed design process incorporating an analytical design, finite element method (FEM) design verification, detailed loss predictions and a detailed system simulation is apparent.

Future work is needed to include unmodelled dynamics e.g. the additional pole in the differentiator path in the analytical model of the AMB system. The analytical and simulation models may also be further refined to include actual system dynamics and nonlinearities currently not modelled e.g. magnetic material properties. A more comprehensive analytical and simulation model would have predicted the much higher equivalent stiffness of the actual system before design implementation.

5. CONCLUSIONS

The objective was to develop a design process for developing a flexible rotor double radial active magnetic bearing system. The design process was realised and verified through simulation and experimental results. This study highlights the importance of accurate modelling and the need for an integrated design tool incorporating aspects of FEM design verification, detailed loss predictions and detailed system simulation.

6. REFERENCES

[1] M.E.F. Kasarda: “An Overview of Active Magnetic Bearing Technology and Applications”, The Shock and Vibration Digest, Vol. 32 No. 2, pp. 91-99, March 2000.

[2] L. Shi, L. Zhao, G. Yang, H. Gu, X. Diao, S. Yu: “Design and experiments of the active magnetic bearing system for the HTR-10”, 2nd international topical meeting on high temperature reactor technology, Paper D04, Beijing, China, Sept. 2004.

[3] T. Takizuka, S. Takada, X. Yan, S. Kosugiyama, S. Katanishi, K. Kunitomi: “R&D on the power conversion system for gas turbine high temperature reactors”, Nuclear Engineering and Design,Vol. 233 No. 1-3, pp. 329-346, October 2004.

[4] P. Allaire, C.R. Knospe, et al: “Short course on magnetic bearings,” Alexandria Virginia United States of America: University of Virginia, 1997.

[5] G. Schweitzer, H. Bleuler, A. Traxler: Active Magnetic Bearings: Basics, Properties and Applications of Active Magnetic Bearings, Authors Reprint, Zürich, 2003.

[6] Richard C. Dorf, Robert H. Bishop: Modern Control Systems, Prentice Hall, New Jersey, Ninth Edition, pp. 36-47, pp. 227-233, 2001.

[7] John M. Vance: Rotordynamics of Turbomachinery,WILEY, New York, pp. 116-170, 1988.

Vol.98(1) March 2007 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 1�

AN ADAPTIVE HYBRID LIST DECODING AND CHASE-LIKE ALGORITHM FOR REED-SOLOMON CODES

W. Jin, H. Xu and F. Takawira

School of Electrical, Electronic and Computer Engineering, University of KwaZulu-Natal, Howard College Campus, King George V Avenue, Durban 4041, South Africa

Abstract: This paper presents an adaptive hybrid Chase-like algorithm for Reed-Solomon codes, which is based on the list decoding algorithm. The adaptive hybrid algorithm is based on the reliability threshold to exclude the more reliable bits from being processed by the list decoding algorithm and reduce the complexity of the hybrid algorithm. Simulation results show that the decoding complexities of the adaptive hybrid algorithm for both (15.7) and (31.21) Reed-Solomon codes are almost the same as those of the list decoding algorithm (without Chase algorithm) at high signal-to-noise ratios, but there is a significant improvement in FER performance.

Key words: The list decoding algorithm, Bit reliability, Adaptive scheme, Hybrid algorithm.

1. INTRODUCTION

How to approach the performance of the Maximum likelihood decoding (MLD) with less complexity is a subject which has been researched extensively, especially for Reed-Solomon (RS) codes which are powerful error-correcting codes in digital communications and digital data-storage systems. Applying the bit reliability obtained from the channel to the conventional decoding algorithm is always an efficient technique to achieve the performance of the MLD, although the exponential increase of complexity is always concomitant. In [1]-[2], the authors also use the bit reliability to improve the performance of Bose-Chaudhuri-Hocquenqhem (BCH) codes and RS codes, respectively. It is undoubted that improved performance can be achieved if we apply the bit reliability to an enhanced algebraic decoding algorithm that is more powerful than the conventional algebraic decoding algorithms.

The Guruswami-Sudan (GS) list decoding algorithm [3] that was discovered by Madhu Sudan in 1997 and developed by Guruswami and Sudan two years later [4] is one of the enhanced algebraic decoding algorithms for RS codes. In the GS list decoding algorithm, the number of errors that can be corrected increases to

1 ( 1)GSt n k n for ( , )n k RS codes, where x is

the integer of x. It is easy to show that the GS list decoding algorithm is able to correct more errors than the conventional algebraic decoding algorithms. The fundamental idea of the GS algorithm is to take advantage of an interpolation step to get an interpolation polynomial which is produced by the support symbols, the received symbols and their corresponding multiplicities. The GS algorithm then implements a factorisation step to find the roots of the interpolation

polynomial. After comparing the reliability of these codewords, which are obtained from the output of factorisation, the GS algorithm outputs the most likely one. The support set, the received set and the multiplicity set are created by the Koetter-Vardy (KV) algorithm [5] that is a practical implementation of the GS algorithm.

To further improve the performance of the GS list decoding algorithm, [6] has proposed a hybrid list decoding and Chase-like algorithm. Simulation results in [6] show that the performance of the hybrid algorithm for the (7.5) RS code can approach that of the MLD, and the performance of the (15.7) RS code can correct one more symbol error than the GS list decoding algorithm. The complexity of the hybrid algorithm in [6] depends on the number of bits which are used in the Chase-like algorithm, but the complexity is exponential with the number of bits. Actually, as signal-to-noise ratio (SNR) increases the received bits are more reliable, and it is not necessary to apply the Chase-like algorithm in the GS list decoding algorithm. To further reduce the complexity at high SNRs, we propose an adaptive hybrid algorithm which is based on the GS list decoding and the adaptive Chase-like algorithm in this paper. The adaptive hybrid algorithm is based on the reliability threshold to exclude the more reliable bits from being processed by the GS list decoding algorithm.

This paper is organised as follows. Section 2 introduces the KV soft-decision front end along with the corresponding algorithm. Section 3 gives a brief description of the adaptive Chase-Generalised Minimum Distance (Chase-GMD) algorithm. Section 4 explains how the list decoding algorithm and the Chase algorithms can be combined with further incorporation of the adaptive idea. Simulation results are given in Section 5. Section 6 draws conclusions for this paper.


2. THE KOETTER-VARDY SOFT-DECISION FRONT END

Guruswami and Sudan hinted at a possibility of a soft-decision extension to their algorithm by allowing each point on the interpolated curve to have its own multiplicity. Koetter and Vardy (KV) proposed a method to perform soft-decision decoding by assigning unequal multiplicities to different points according to their relative reliabilities. An algorithm that generates the multiplicity matrix from the reliability matrix was presented in [5]. A lower complexity algorithm for implementing the KV front-end was proposed in [7], but we still use the KV algorithm from [5] which is shown as follows.

The KV Algorithm for calculating Multiplicity Matrix Mfrom the reliability matrix subject to complexity constraint s.

Using this algorithm, we obtain the support set, the received set and the multiplicity set. The candidates of the codeword polynomial are obtained through an interpolation step and a factorisation step.

3. AN ADAPTIVE CHASE-GMD ALGORITHM

Mahran and Benasissa proposed an adaptive Chase-GMD algorithm for linear block codes in [8]. In the adaptive Chase-GMD Algorithm, an l-bit quantizer is used to classify the received bits by their reliability. A brief overview of the adaptive Chase-GMD algorithm is given as follows.

As errors are more likely to occur in the first least reliable positions of the received bits R, the Chase-GMD algorithm is firstly considered to be applied in those positions. A reliability threshold function of confidence level, r, is used in the adaptive Chase Algorithm. The higher the confidence level the higher the possibility of selecting more chase-like erasures. The threshold function T is given by:

0.5

bc

o

T S rE

RN

(1)

where cR is the rate of the RS code. 0/bE N is the bit

signal-to-noise ratio. S given in (2) is a scalar constant that depends on the number of quantisation levels.

33 2

7.0

2

45.0ll

S (2)

The threshold can be used to decide which bit should be processed by the Chase algorithm. Let the reliability of received sequence be . The bits will be used in the adaptive Chase algorithm only if their reliabilities satisfy the following condition:

min1,2, ,2j

dT T j (3)

If a bit does not satisfy the above condition then it can be ignored by the Chase algorithm even if it is the most unreliable bit in the received bits.

4. THE ADAPTIVE HYBRID ALGORITHM

The application of the Chase algorithm to the KV soft-decision front end based on the bit reliability can improve the performance of the list decoding algorithm, with an adaptive scheme reducing complexity.

Before the presentation of the adaptive hybrid algorithm, there are some definitions which should be made clear. We can obtain the support set, the received set and the multiplicity set through the KV front end. We use ‘multi-points’ to define the received symbols whose support symbols are the same, ‘low-multiplicity-points’ to define the received symbols whose multiplicities are less than the maximum multiplicity except for multi-points, and ‘high-multiplicity-points’ to define the received symbols whose multiplicities are equal to the maximum multiplicity. We refer to the high-multiplicity-points as reliable points, and other received symbols as unreliable points.

Now, the adaptive hybrid list decoding and the Chase-like algorithm contain the following steps: i. Implement the KV soft-decision front end to

obtain the support set, the received set and the multiplicity set.

ii. Use Equation (1) to obtain the threshold value with appropriate scale constant S and confidence level r.

iii. Calculate the number of multi-points in the output of the KV soft-decision front end. If more than one received symbol is found to have the same support symbols, the number of multi-points increases by

one. We denote it as multiN .

Definition: mi,j is an entry at the position ( , )i j in

multiplicity matrix M .

Algorithm:

Choose a desired value 1 1

,0 0

q n

i ji j

s m ; * ;

0M ;

While 0s do

Find the position ( , )i j of the largest entry *,i j in * ;

,*,

, 2i j

i ji jm

; , , 1i j i jm m ; 1s s ;

End while


iv. Calculate the number of low-multiplicity-points in the output of the KV soft-decision front end.

We denote it as lowN .

v. If multiN + lowN G St

use the chase-like algorithm for high-multiplicity-points. The threshold can be used to finally decide if those unreliable bits are picked up by the Chase algorithm or not.

Else use the Chase-like algorithm for both low-multiplicity-points and high-multiplicity-points. The unreliable bits selected by the Chase algorithm must also satisfy the condition mentioned above.

GSt is the number of errors that can be corrected

for ( , )n k RS codes.

vi. Output all the received sets, the corresponding support set and multiplicity set to interpolation step.

vii. Interpolation step proposed in [5]. viii. Factorisation step proposed in [5].

ix. Compare the probability of all candidates of the codeword created by different received sets and output the most likely one.

In the above proposed algorithm, we do not consider the multi-points because the list decoding algorithm has already taken them into account. In other words, the list decoding algorithm pays more attention to multi-points than other points. When the list algorithm fails, the errors coming from multi-points are not the significant source of failure for the list decoding.

In the above algorithm, we classify the output of the KV soft-decision front end into two different cases,

multiN + lowN G St and multiN + lowN G St . We will

discuss them separately.

If multiN + lowN G St , it means that the number of

unreliable points does not exceed the error-correcting ability. Even if all unreliable symbols are incorrect, the list decoding algorithm can still generate the right codeword polynomial. In this case, the errors coming from reliable symbols are the main reason for the failure of the list decoding algorithm, so we apply the Chase algorithm to reliable symbols in order to obtain more reliable received sets corresponding to the same multiplicity.

If multiN + lowN G St , it means that the number of

unreliable symbols exceeds the error-correcting ability. If all these symbols are incorrect, the list decoding algorithm can not generate the right codeword polynomial. In this case, we have to concentrate on both low-multiplicity-points and high-multiplicity-points.

Because the list decoding algorithm has already taken the multi-points into account, we do not take the multi-points into account. Before we apply the Chase algorithm to both kinds of points, we must make it clear which kind of points we should take into account first, low-multiplicity-points or high-multiplicity-points. We can extend the search scope into high-multiplicity-points by changing the unreliable bits. In order to improve the search scope, changing the unreliable bits in high-multiplicity-points is better than in low-multiplicity-points. This implies that we can obtain more candidate codeword polynomials if we choose high-multiplicity-points. It seems that we should change unreliable bits in high-multiplicity-points, but at high SNRs, we draw a different conclusion. As the SNR increases, the high-multiplicity-points (reliable points) become more and more ‘reliable’. The probability that reliable points are received correctly is very large. The performance improvement is marginal even if we invert these bits which are in the reliable points. In this paper, we only take into account low-multiplicity-points first at high SNRs.

There are several threshold values that are shown in Figure 1 and Figure 2 for 0.467.cR The scalar constant

in Figure 1 is 0.225, which is the minimum of a 4-bit quantizer. The scalar constant in Figure 2 is 0.35, which is the maximum of the same 4-bit quantizer.

It is obvious that we can change the confidence level to get different thresholds. As the confidence level increases, the number of bits that can be ignored by the Chase algorithm decreases. The confidence level can be adjusted to fit the list decoding algorithm. It is expected that the performance of the adaptive hybrid algorithm is comparable with the performance of the hybrid Chase-list algorithm in [6], but with lower complexity.

Figure1: The thresholds with S=0.225 for 0.467.cR


Figure 2: The thresholds with S=0.35 for 0.467.cR

5. SIMULATION RESULTS

All simulations are performed in an AWGN channel and BPSK transmission is assumed. For comparison purposes, we simulate the conventional decoding, the list decoding, the hybrid algorithm in [6] and the adaptive hybrid algorithm. We use the Chase-2 algorithm to improve the performance for the (15.7) RS code. We select 4 unreliable bits according to the least reliable positions based on the hybrid algorithm. A 4-bit quantizer with the confidence level 1 to 8 is used. In the simulation of the (15.7) RS code we choose S=0.26 and r=3, which are suitable for the list decoding algorithm with maximum multiplicity 2. The frame error rate (FER) performance is shown in Figure 3, and the corresponding complexity is shown in Figure 5. The simulation results of 2-bit hybrid algorithm, 3-bit hybrid algorithm and 4-bit hybrid algorithm are also shown in those figures for comparison. Based on the fact that one interpolation step and one factorisation step take almost 95% of total decoding time, we define a unit of the decoding complexity as the time taken by one interpolation step and one factorisation step in the list decoding algorithm. The hybrid algorithm can correct one more symbol error than the list decoding algorithm. Figure 5 also shows that the complexity of the adaptive hybrid algorithm decreases as the SNR increases. The complexity of the adaptive hybrid algorithm at 7dB is almost 2, which is the complexity with the 1 bit Chase-2 algorithm applied to the list decoding algorithm, but the gap between the FER performance of the adaptive hybrid algorithm and the hybrid’s is negligible.

We still use the Chase-2 algorithm to improve the performance for the (31.21) RS code. A 4-bit quantizer is also used with S=0.225 and r=3. The FER performance is shown in Figure 4, and the corresponding complexity is shown in Figure 6. The simulation results of the 1-bit hybrid algorithm and the 2-bit hybrid algorithm are also shown in those figures for comparison. The simulation

results in Figure 4 and Figure 6 show that the adaptive algorithm can reduce the complexity with small or marginal performance penalty. The complexity of the adaptive hybrid algorithm in Figure 6 can approach the list decoding algorithm without the Chase algorithm at high SNRs.

Figure 3: FER performance of the (15.7) RS code.

Figure 4: FER performance of the (31.21)RS code.

Figure 5: The complexity of the (15.7) RS code.


Figure 6: The complexity of the (31.21) RS code.

6. CONCLUSION

In this paper, an adaptive hybrid list decoding and Chase-like algorithm is presented. The adaptive hybrid algorithm is based on the reliability threshold to exclude the more reliable bits from being processed by the list decoding algorithm. In the above steps we obtain more received sets and accordingly obtain more candidate codeword polynomials. As the search scope is extended, the transmitted codeword is easily obtained. Simulation results show that the FER performance of the proposed adaptive hybrid algorithm for both (15.7) and (31.21) RS codes can be comparable with the performance of the hybrid algorithm in [6], but the complexity is much lower, especially at high SNRs.

7. REFERENCES

[1] M. P. C. Fossorier and S. Lin: “Complementary reliability-based decoding of binary block codes”, IEEE Transactions on Information Theory, Vol.43, pp. 1667~1672.

[2] T. H. Hu and S. Lin: “An efficient hybrid decoding algorithm for Reed-Solomon codes based on bit reliability”, IEEE Transactions on Communications,vol.51, no.7, pp. 1073~1081, July 2003.

[3] M. Sudan: “Decoding of Reed Solomon codes beyond the error-correcting bound”, J. Complexity,Vol. 13, pp.180~193, 1997.

[4] V.Guruswami and M. Sudan: “Improved decoding of Reed-Solomon codes and algebraic geometry codes”, IEEE Transactions on Information Theory, vol. 45 no. 6 , pp. 1757~1767, Sept. 1999.

[5] R. Koetter: “Fast generalized minimum-distance decoding of algebraic-geometry and Reed-Solomon codes”, IEEE Transactions on Information Theory,vol. 42 no. 3, pp. 721–736, May 1996.

[6] W. Jin, H. Xu and F. Takawira: “A hybrid list decoding and Chase-like algorithm of Reed-Solomon codes”, Proceedings of the 4th International Symposium on Information and Communication Technologies, pp. 87–92, Jan 3-6 2005.

[7] W. J. Gross, F. R. Kschischang, R. Koetter, and P. Gulak: “Simulation results for algebraic soft-decision decoding of Reed-Solomon codes”, Proceedings of the 21’st Biennial Symposium on Communications,pp. 356–360, June 2-5 2002.

[8] A. Mahran and M. Benaissa: “Adaptive combined Chase-GMD algorithms for block codes”, IEEECommunications Letter, Vol.8, No.4, pp.235-237, April 2004.


EFFECT OF CHARACTERISTICS OF DYNAMIC MUSCLE CONTRACTION ON CROSSTALK IN SURFACE ELECTROMYOGRAPHY RECORDINGS

S. Viljoen*, T. Hanekom* and D. Farina**

*Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria,

South Africa **Center for Sensory-Motor Interaction (SMI), Department of Health Science and Technology, Aalborg University, Aalborg, Denmark

Abstract: An investigation into the ability of different spatial filters to reduce the amount of crosstalk in a surface electromyography measurement was conducted. A simulation model was implemented to compare the performance of four spatial filters under dynamic muscle contractions. Two parameters of a dynamic muscle contraction, namely muscle shortening and varying contraction force, were evaluated separately. The normal double differential filter resulted in the best crosstalk rejection for varying contraction force simulations, while the double differential filter performed best when incorporating muscle shortening. It is furthermore suggested that crosstalk is influenced more by muscle shortening than by changes in the contraction force.

Key words: Surface electromyography, modelling, crosstalk, analytical model, spatial filters, motor unit action potential, average rectified value, mean power spectral frequency, muscle shortening, muscle contraction.

1. INTRODUCTION

In surface electromyography (sEMG) recordings, crosstalk is the unwanted signal component detected above one muscle but generated by another muscle [1, 2]. It is one of the main areas of research in sEMG measurements [3]. Crosstalk has been investigated by experimental [1, 3 - 5] and modelling [3, 4, 6 - 8] studies. It has been shown that crosstalk signals consist mainly of far-field potentials that are generated due to the extinction of the action potential at the muscle fibre endings (end-of-fibre effect) [2 – 4, 6, 9, 10, 11]. Crosstalk depends on many anatomical and physical factors of the sEMG generation system [7, 8, 12].

Crosstalk has mainly been investigated in isotonic, isometric contractions, e.g. an evaluation of methods to reduce crosstalk [6] and a comparison of the different methods employed to quantify crosstalk [4]. This eliminates some inherent sources of crosstalk (e.g. sliding of muscles under skin). Everyday tasks, however, require dynamic muscle contractions. The objective of this study was to investigate, by simulation, the effect of some parameters of dynamic muscle contraction on crosstalk.

In a dynamic contraction the muscle shortens and the force may vary over a large range. These two parameters (degree of shortening and force) were evaluated separately to examine their individual effects on crosstalk. The geometrical changes incorporated include a change in muscle length (concentric contraction) and the accompanying change in limb radius. The performance of four spatial filters with respect to

crosstalk rejection was compared for the simulated conditions.

2. METHODS

2.1 Volume conductor model

The analytical model developed by [13] was used to generate single fibre action potentials (SFAPs). A spatio-temporal function describes the generation, propagation and extinction of the SFAP. The model describes a cylindrical layered volume conductor with bone, muscle, fat, and skin tissues [13]. The layers are anisotropic and their effect on the SFAP is modelled as a two-dimensional spatial filter. A specific part of the muscle region comprises the active muscle fibres. The active muscle territory was described as an ellipse. The parameters that were fixed in the volume conductor model are reported in Table I. These include the physiological cross-sectional area (PCSA) and maximum voluntary contraction (MVC) of the muscle.

2.2 Detection system

Detection system is the general term used to refer to the electrode arrangements used to realise the spatial filters shown in Figure 2. Ten detection systems were placed circumferentially around the surface of the volume conductor with their centres in the range 0º - 45º, where 0 º is the position directly above the muscle (see Figure 1).

Four spatial filters, with inter-electrode distance (IED) 5 mm, were considered (Figure 2): monopolar (Mono),


single differential (SD), double differential (DD) and normal double differential (NDD). The detection points were halfway between the innervation zone (IZ) and the tendon region. The end-of-fibre effect is visible as a small bump at the end of each potential.

Table I: Fixed parameters in volume conductor model.

Parameter Value Reference

Bone radius 15 mm [14]

Skin thickness 1 mm [15]

Fibres / MU 50 – 1000 [16]

MU / muscle 200 [16]

Fibre density 20 fibres/mm² Computed

Conductivity of skin 1 S/m [17]

Conductivity of bone (isotropic)

0.02 S/m [17]

Conductivity of fat (isotropic)

0.05 S/m [18, 19]

Conductivity of muscle (radial + angular)

0.1 S/m [18, 19]

Conductivity of muscle (longitudinal)

0.5 S/m [18, 19]

Inter-electrode distance

5 mm Selected

PCSA of selected muscle

600 mm² [20]

% MVC where recruitment stops

85% Selected

Interpulse interval variability

15% Selected

Variation in discharge rate with force (var_dis)

15% Selected

2.3 Motor unit

The SFAPs were summed together to obtain the motor unit action potentials (MUAPs). The MU territories were circular with centres located randomly within the active muscle. Each MU was assigned a specific conduction velocity (CV). CV values were limited in the range 2-7 m/s and were assigned in agreement to the size principle, i.e., CV increased with MU size [21].

2.4 Motor unit firing patterns

The firing pattern computations are based on the model developed in [22]. The MUs were recruited on the basis of MU recruitment threshold, defined by:

ln( ) /( ) , 1,2,3...RR i nRTE i e i n (1)

where RR is the % maximum voluntary contraction (MVC) where the complete MU pool is recruited. It is

assumed that the force increases linearly from 0 to 100 % MVC in 3 s. The time at which a certain contraction force is reached, t%MVC, can thus be found from:

%%

3100MVCMVC

t (2)

The time at which a certain MU is recruited is defined by:

%( )

( )100rec MVC MVC

RTE it i t t

. (3)

MUs with negative trec values are not yet recruited at the time that the desired contraction strength is reached. Their discharge rate is thus set equal to zero. A discharge rate is computed for every MU with a positive recruitment time from Equation 4:

min( ) _ (100 ( ))ratef i f var dis RTE i (4)

with fmin = 8 pulses per second (pps) and var_dis the variation in discharge rate with force equal to 0.3 pps/%MVC. The discharge rates were limited to 35 pps. All the parameters correspond to experimental measurements [23].

2.5 Signal analysis

The average rectified value (ARV) and mean power spectral frequency (MNF) are commonly used to evaluate the characteristics of surface EMG [1, 2]. The ARV is equal to the area under the rectified signal. Its value will increase and then decrease with increasing electrode separation. The slope of the decrease provides an indication of filter selectivity with respect to crosstalk. The propagating components, which form the main part of an EMG signal, are attenuated rapidly with increasing distance. Crosstalk signals propagate with less attenuation and can thus be measured a larger distance from their origin [11]. The detection system which shows the best suppression of amplitude content with increasing distance will thus be preferential for crosstalk reduction.

The MNF value defines the frequency component that carries the largest weight (centre of gravity). It thus provides information about the frequency content of the crosstalk signal [1, 2]. The interpretation of the MNF is based on the fact that the filtering properties of a detection system are determined by its transfer function. The sEMG waveform is time and space dependent. Since the electrode separation and location are spatial parameters, they influence the spatial filtering characteristics of the electrode configuration. The temporal and spatial waveforms are linked via the CV of the muscle fibres. A change in the spatial characteristics of the electrode configuration will thus also be reflected in the temporal waveform as a filtering effect where the relative contributions of the frequency components are weighted relative to those of the original signal.


Typically, the MNF will increase with increasing electrode separation. The reason behind this is than an increase in electrode separation leads to a larger pick-up area, which in turn increases the crosstalk [4]. The effect of electrode location on the MNF is determined by the main contributors to the sEMG signal at that location, i.e. whether the MNF will increase or decrease with increasing interelectrode distance depends on the relative contributions of the propagating and non-propagating components. For example, if the electrode system is located close to the tendon, an increase in the MNF can be expected relative to the MNF observed for a detection system location midway between the tendon and IZ because the non-propagating components of the signal originating at this location contribute more to the detected waveform than the components of the true EMG signal[3].

The ARV and MNF were computed from 1 s long simulated signals at the 10 transversal locations for each spatial filter. These variables were normalized with respect to the value assumed at the first location (0º).

2.6 Simulation of muscle shortening

Three libraries of SFAPs were created using the analytical model described above. The muscle fibre length was changed between the libraries to simulate muscle shortening. A constant muscle volume was maintained by increasing the limb and active muscle radius while decreasing the fibre length. The distance between the centre of the active muscle and the centre of the bone remained constant (30 mm, see Figure 3). The detection system remained halfway between the IZ and tendon region for all the simulations.

Figure 1: The model of the limb with different layers, the active muscle, the location and territory of the MUs. The location of the four detection system centres (see Figure 2) is also shown.

Figure 2: The spatial filters implemented in the simulation and an SFAP detected with each one.

Muscle region

Skin Fat

Detection system centres Active muscle

Bone

MU territories

0 °

45 °

15 mm

50 mm

5 °

Double differential (DD)

Monopolar (Mono) Single differential (SD)

11

- 1

Laplacian (NDD)

1 1

- 2 1 - 4 1

1 1

Double differential (DD)

Monopolar (Mono) Single differential (SD)

11

- 1

Laplacian (NDD)

1 1

- 2 1 - 4 1

1 1


)2

exp()(2

2

w

zAzr

Figure 3: The muscle geometries used for incorporating muscle shortening. Increasing limb and muscle radius with decreasing muscle fibre length can be seen. The parameters describing the muscle geometry for library 1 (left), 2

(centre) and 3 (right) can be found in Table II.

Table II: Variable parameters used for simulating muscle shortening.

Library nr Limb radius (mm)

Fibre semi-length (mm)

Muscle radius (A) (mm)

Width (w) (mm)

Radial distance between muscle fibre

centres (mm)

1 60 75 15 83.29 0.29

2 63.5 55 18.5 49.58 0.31

3 68 40 23 30.99 0.34

The muscle shape was described by a Gaussian function (also called a bell-shaped curve), given in Equation 5:

(5)

where A defines the maximum radius of the active muscle, z is the longitudinal distance along the axis of the muscle and w defines the contraction width. The higher the value of w, the more contracted a muscle is. The relation between muscle volume and muscle shape is:

2( ( )) .d

d

V r z dz (6)

with d the muscle fibre semi-length. A muscle volume of 83.2 cm³ and maximum fibre semi-length of 75 mm was obtained from [20] and used for library 1 (the relaxed muscle). A fully contracted biceps brachii muscle can reduce its length to ½ of that in the relaxed state [20]. Library 3’s fibre semi-length was thus selected as 40 mm, with library 2 given a value in-between the other two, see Table II.

The limb radius was obtained by adding all the radii of the components (bone, 30 mm spacing and active muscle). A numerical process was implemented to obtain the best values of muscle radius (A) and contraction

width (w) to ensure a constant muscle volume. The complete process is described in [24]. The values obtained can be found in Table II. The fibre density decreases, because the number of fibres in the active muscle stays constant as the area increases. This is displayed by the increased radial distance between the fibre centres with increasing contraction force. CV distribution was Gaussian with mean 4 m/s and standard deviation 0.3 m/s [24].

2.7 Simulation of increased muscle force

Change in contraction force was modelled as a change in the number of recruited MUs, their discharge rate and CV. The active muscle is approximated by an ellipse with semi-axis lengths of a = 15 mm and b = 12.7 mm. These values were chosen to obtain the desired physiological cross-sectional area of 600 mm² for the biceps brachii as given by [20]. The limb radius used was 50 mm and the radial distance between muscle fibre centres 0.22 mm. The fibre semi-length was set to 60 mm.

Six simulations were performed in which the contraction force was increased from 10 to 100 % MVC. The force at which the entire MU pool was recruited was 85% MVC in all cases. For forces higher than 85% MVC, the increase in force is obtained by an increase in discharge rate only.

15 mm23 mm18.5 mm

Bone

Limb

Active Muscle

60 mm 63.5 mm 68 mm

30 mm 30 mm 30 mm

Fibre semi-length: 75 mm 55 mm 40 mm

15 mm23 mm18.5 mm

Bone

Limb

Active Muscle

60 mm 63.5 mm 68 mm

30 mm 30 mm 30 mm

Fibre semi-length: 75 mm 55 mm 40 mm


Table III: Variable parameters for simulating increasing contraction force.

Contraction force (% MVC)

Recruited Mus

Max discharge rate (pps)

Mean CV (m/s)

CV Std dev (m/s)

10 103 10.86 3.77 0.18

30 153 17.22 3.88 0.22

50 176 23.59 3.93 0.25

70 191 29.95 3.97 0.27

90 200 35 4 0.3

100 200 35 4 0.3

Since the smallest MU is recruited first, and CV is assigned increasing with MU size, the large MUs with high values of CV are not recruited at low values of contraction force. The result is a decrease in the mean CV with decreasing contraction force. This has been observed by several authors [26 - 28]. Table III shows the parameter values that were used in the simulations.

The discharge rate of each MU was computed from Equations 1 to 4. MUs with negative RTE_time values are not yet recruited at the time that the desired contraction strength is reached. Their discharge rate is thus set equal to zero.

The mean and standard deviation of CV is computed from a summation of the CV values assigned to the recruited MUs. From Table III the increase in CV with contraction force ranging from 10 to 50 % MVC is more significant than when the force increases above 50% MVC. This agrees with experimentally determined values of CV increase with contraction force [29].

When comparing the variables used for a 90 and 100 % MVC in Table III, the two simulations seem identical. The average discharge rate of the MUs will, however, be lower for a 90% contraction than for one of 100% MVC. The number of MUs discharging at the maximum rate of 35 pps will thus be lower for 90 % MVC than for 100% MVC.

3. RESULTS

All the graphs in this section are shown in normalised units (NU). Unless otherwise stated, they are normalised with respect to the value obtained at 0 º.

3.1 Effect of muscle shortening on crosstalk

Figure 4 shows the results obtained from the simulation of the three libraries. Library 1 (relaxed muscle) results in the most selective measurements, followed by library 2 and then 3. The only exception is NDD, where library 2 is more selective than library 1 at large distance.

NDD is the most selective filter at first for all three libraries, but right at the end DD increases beyond NDD in all the investigated cases.

Library 1 (relaxed muscle) results in the lowest values of MNF, followed by library 2 and then 3. At large distances library 3 has a lower MNF than library 2 for the DD and NDD detection systems.

The SD system always has the lowest MNF value. For libraries 1 and 2 NDD has the second lowest MNF values, but its values are then exceeded by those of DD. In the end the MNF of NDD is again lower than that of DD. For library 3, DD has a lower MNF than NDD right from the start, but NDD increases beyond DD's values at large distances.

NDD thus has the lowest MNF values for all three libraries at large distances. The last time its values exceed those of DD occurs at the same distance for all three libraries.

3.2 Effect of increased contraction force on crosstalk

Figure 5 shows the initial value of ARV and MNF as obtained with the different detection systems for different values of contraction force. ARV is strongly dependent on contraction force, while MNF varies only slightly when the contraction force increases. Both graphs are normalised with respect to the values obtained at 10 % MVC.

ARV increases with increasing contraction force. The increase can be explained by the fact that more MUs are recruited as the contraction force increases. Since an EMG signal is the summation of the contributions of all the active MUs, it should increase in amplitude as more MUs become active. The values measured at 25º (open markers) show a slightly larger increase than those measured at 0º (filled markers).

The MNF does not vary significantly with increasing contraction force. The variations do, however, appear to be very random in nature. This observation is independent of the circumferential position of the detection system. The same is thus seen at 0º and 25º.


NU

NU

Figure 6 shows the ARV and MNF values obtained from 10% and 100% MVC. The values for ARV and MNF obtained for the other contraction force strengths lay in-between the 10 and 100 % MVC results. The first observation is that neither the amplitude nor the frequency components of the various detection systems is largely affected by the difference in contraction force strength.

When comparing the ARV values, Mono, SD and DD show a very slight increase in selectivity with decreasing contraction force, while that of NDD remains constant.

The MNF values of Mono, SD and DD remain mostly unchanged for a 100% or 10% contraction. The decrease of NDD, DD and SD MNF with distance asymptotes at 30º for both values of contraction force, after which that of DD increases slightly. DD and NDD have slightly lower frequency components for 10% MVC as compared to 100% MVC.

(a) ARV

(b) MNF

Figure 4: The ARV (a) and MNF (b) of libraries 1 (open markers, relaxed muscle), 2 (filled markers) and 3 (open markers with line, contracted muscle) with an IED of 5 mm, fat layer = 1 mm, skin conductivity = 1 S/m.

0 5 10 15 20 25 30 35 40 45

10 -2

10 -1

10 0

D etection poin ts (degrees)

M onoS DD DN D D

0 5 10 15 20 25 30 35 40 45

0.4

0 .6

0 .8

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M onoS DD DN D D


NU

NU

(a) ARV

(b) MNF

Figure 5: The initial values of ARV (a) and MNF (b) as detected at different contraction force levels at 0º (filled markers) and 25º (unfilled markers), fibre semi-length = 60 mm, fat layer = 1 mm, skin conductivity = 1 S/m,

IED = 5 mm.

4. DISCUSSION

4.1 Effect of muscle shortening on crosstalk

The first objective of the simulation was to establish the relationship between limb geometry and amount of crosstalk. When a dynamic contraction is elicited, the muscle geometry changes along with other muscular properties (including CV, discharge rate and location of the IZ). Since these changes all occur simultaneously, it is very difficult to ascribe a change in the measured sEMG signal to a change in a single muscular property. By changing only the geometrical properties, the influence this has on crosstalk can be investigated directly.

When a muscle contracts the location of the IZ and tendon area can slide with respect to the skin and detection electrodes. It was shown by [29] that this sliding effect could result in EMG amplitude changes in excess of 200%. This may falsely be interpreted as an increase in muscle activity. A shift in biceps brachii IZ position of between 1 and 4 cm with changes in elbow angle was observed by [30], while [31] confirmed the fluctuations in EMG amplitude with changing knee angles. When an increase in amplitude is observed along with muscle shortening, it is thus not clear which part of the amplitude change is due to shifting of the IZ underneath the detection electrodes and which part (if

0 20 40 60 80 1000.96

0 .98

1

1.02

1 .04

1 .06

C ontraction force (% M V C )

M ono

S D

D D

N D D

0 20 40 60 80 1001

1 .5

2

2 .5

3

3 .5

4

4 .54 .5

C ontraction force (% M V C )

M onoS DD DN D D


NU

NU

any) is due to the muscle shortening. A simulation enables one to change selected variables and keep the rest constant, investigating the individual effect of each variable on the output.

When shortening of the muscle fibre occurs, the IZ and tendon area move closer to the centre of the muscle. This changes are relatively small, especially keeping in mind that the fibre length for the contracted muscle is reduced

by almost half relative to that of the relaxed muscle. This could imply that the propagating wave components contribute significantly to the detected crosstalk.

Library 1 simulates a relaxed muscle, with the tendon area the furthest away from the detection electrodes. From Figure 4 it is clear that library 1 always results in the best crosstalk suppression, as expected. Furthermore

be expected to increase as the muscle fibres decrease in length. This is evident from Figure 4 where crosstalk is increased with decreased muscle length. However, the

a) ARV

(b) MNF

Figure 6: The ARV (a) and MNF (b) with a contraction force of 100 % (filled markers) and 10 % (open markers), fibre semi-length = 60 mm, fat layer = 1 mm, skin conductivity = 1 S/m, IED = 5 mm.

0 5 10 15 20 25 30 35 40 45

10-2

10-1

100

D etection po in ts (degrees)

M onoS DD DN D D

0 5 10 15 20 25 30 35 40 45

0 .6

0 .8

1

D etection po in ts (degrees)

M onoS DD DN D D

When shortening of the muscle fibre occurs, the IZ and tendon area move closer to the centre of the muscle. This means that the non-propagating components origi-nate closer to the detection electrodes. The crosstalk can thus be expected to increase as the muscle fibres decrease in length. This is evident from Figure � where crosstalk is increased with decreased muscle length. However, the

changes are relatively small, especially keeping in mind that the fibre length for the contracted muscle is reduced by almost half relative to that of the relaxed muscle. This could imply that the propagating wave components con-tribute significantly to the detected crosstalk.

Library 1 simulates a relaxed muscle, with the tendon area the furthest away from the detection electrodes. From Figure � it is clear that library 1 always results in the best crosstalk suppression, as expected. Furthermore


one would expect to see more crosstalk in library 2 and the most in library 3 (increasing crosstalk as the tendon area moves closer to the detection areas). This can be seen for the Mono, SD and DD systems, and also for most of NDD. At some stage, however, library 2 results in a more selective amplitude response than library 1 (NDD detected). This might be due to the decreasing fibre density with decreasing muscle length.

Increasing muscle length was linked to a decrease in spectral frequencies by [32] and [33], while [34] confirm that the non-propagating components will have a smaller influence with increasing muscle length because of the increased distance between the source and detection electrodes. As stated by [35], MNF estimates are greatly influenced by muscle fibre shortening and are also dependent on the detection system being used. This is clear from the variability of the MNF estimates in Figure 4 (b), especially for NDD.

4.3 Effect of increased contraction force on crosstalk

A similar study in which the contraction force was increased with the same steps as in the current study, and the ARV and MNF were experimentally determined was performed by [29]. The author reported that the influence of contraction force on the initial value of the MNF in the biceps brachii is small, and possibly masked by other factors. When comparing the results obtained by other authors on MNF and contraction force, it is difficult to draw any conclusion as the results vary quite a lot from one simulation to the next. MNF variation with contraction force increase depends on the muscle being studied, the IED and the transfer function of the underlying tissue layers [26]. A simulation study was done by [23] in which they found that MNF mostly increased and then remained constant with increasing contraction force. There were, however, exceptions. They conclude that a general relationship between spectral variables and MU recruitment cannot be defined.

For the current simulation (biceps brachii muscle, IED = 5 mm, skin layer = 1 mm and fat layer = 5 mm) the MNF does not vary significantly with increasing contraction force (see Figure 6 (b)). NDD is the most selective filter for crosstalk rejection for all of the contraction force strengths studied (Figure 6 (a)). It is clear that a change in the contraction force strength did not have a significant influence on either the amplitude or frequency components of the resulting EMG signals. This could also be as a result of the purely resistive nature of the model. According to Lowery et al. [36] the components in the model which could produce significant capacitive effects are the muscle tissue, fat layer and skin. Inclusion of capacitive effects in their model resulted in a phase shift of the detected action potentials with respect to the purely resistive model and a reduction in the amplitude of the non-propagating components. The thickness of the fat layer and skin are kept constant throughout the simulations and should thus have a

relatively constant effect. However, the capacitive effects of the muscle tissue could influence the results since the thickness of the muscle tissue layer in the model changes. Limb geometry (i.e. realistic vs. cylindrical) is not expected to influence the results significantly since Lowery et al. [36] concluded that an idealized cylindrical limb model is adequate if the main parameters of interest are more qualitative features of the EMG signal, such as the approximate rate of decay of the EMG amplitude.Future research could thus focus on the inclusion of capacitive effects in the tissues.

5. CONCLUSIONS

The effects of two dynamic contraction parameters on crosstalk rejection were evaluated separately. A simulation model was used to investigate the effect of muscle shortening, and the effect of increased contraction force on crosstalk selectivity for four spatial filters. NDD resulted in marginally better crosstalk rejection in the increasing contraction force simulations. When muscle shortening was simulated, DD was the most selective for all the investigated situations, followed closely by NDD.

The MNF estimates of the EMG signal was clearly affected by muscle shortening and the accompanying geometrical changes, while increasing the contraction force did not cause a significant change in MNF. Generally, the ARV increased with muscle shortening as well as increasing muscle contraction force.

The selectivity of the detection systems was not significantly influenced by the variation in contraction force. This could be as a result of the purely resistive nature of the model. Assessment of crosstalk selectivity in dynamic contractions should thus, in future, include capacitive effects of the tissues.

ACKNOWLEDGEMENTS

This research has been supported by the National Research Foundation (South Africa) and the Italian Government under the Bilateral Research Agreement between South Africa and Italy.

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