NDT.net • June 2006 • Vol. 11 No.6

## An approach for the simulation of 2D pulsed eddy currents

Changqing Lee (cqlee@iastate.edu), Iowa State University USA

### Abstract

In this paper, a simple approach for the simulation of 2D pulsed eddy currents is developed by using finite element method (FEM) and realized by using FEMLAB software package. For this simulation, the impedance of a coil, the magnetic flux density around a coil, and the voltage on a coil can be calculated. Compared with the transient time stepping techniques, this approach is simple, easy to understand and easy to apply.

### Introduction

Pulsed eddy current systems use a non-sinusoidal wave to drive coils in order to measure changes in electrical and magnetic properties as single and swept frequency eddy currents do. These non-sinusoidal waveforms are usually square waves but triangle waves are also sometimes used. For the simulation discussed in this paper, I will only consider the case for square waves.

In most pulsed eddy current systems, a low pass filter is used to improve the quality of square waves. This filter simply consists of a resister and a capacitor. After filtering, the periodic driving signal is illustrated in Figure 1. Figure 1: General eddy current signal response

In the first period, the driving current is given in equation (1). (1)

Where,
I0 = current
t = time
t = time constant, determined by the filter
T = period

### Modeling

#### A. Using FEMLAB to Calculate Coil Impedance

Using FEM to calculate the impedance of a coil in eddy current testing is not uncommon. There are several approaches for computing the impedance of a coil, such as the power and energy method, and the flux method. In the power and energy method, coil impedance is found from the dissipated power and stored energy. Hence this approach yields total impedance regardless of the number of coils, their shape or location in the solution region. Therefore we apply this method to compute the impedance. When the stimulating frequency is much lower than the resonant frequency of the coil, normally the FEM calculations can be compared with Dodd and Deeds model  and good agreement can be achieved. Here, the experimental data were also obtained and they are compared with FEM calculations and Dodd and Deeds model.

Now let's briefly review the concepts for the FEM calculation of coil impedance . The coil impedance consists of two parts; one is the real part (resistance component) and the other is the imaginary part (inductance component). These two parts are computed from two field quantities, namely the dissipated power and stored energy (P and Wm). Dissipated Power, From the above discussions, we can get the impedance of the coil, For the FEM calculation, the impedance at each frequency is calculated and compared with experimental data and Dodd and Deeds model. Good agreement is achieved as shown in Figure 2. Figure 2: Comparison among the results of Dodd and Deeds model, Experimental Data and FEM Simulation at the low frequency range (40Hz-1KHz) and the high frequency range (39KHz-40KHz) for the impedance change when placed on the Al half-space and in the air.

#### B. Calculations Hall Sensor Response for Pulsed Eddy Currents

The main difficulty with modeling the pulsed eddy current phenomena is dealing with the time dependent nature of the excitation waveform. There are basically two approaches that one can adopt. One is based on the summation of time-harmonic sinusoidal solutions. The other utilizes transient time stepping techniques. For our application, since the exponential excitation is periodic, the former approach is a better choice.

The actual excitation used experimentally consists of a low-pass-filtered square wave (FSW). To calculate the response due to FSW excitation by using the summation of sinusoidal components, the coefficients for each frequency must first be calculated. Then the response of each component is calculated and summed in order to obtain the response due to FSW excitation.

Consider the axi-symmetric geometry in Figure 3, with the coil being driven by a periodic FSW signal. Assume the frequency of the exponential signal is 100 Hz and T is the period of the FSW signal, so T=1/f. The filter time constant is set to be 1/100 of the whole period. N is the number of frequencies in the Fourier analysis. Here, the first 801 frequency components are calculated. In order to save calculation time, high-frequency components are ignored on the assumption that their coefficients are small compared to the low frequency components. Figure 3: Axi-symmetric FEMLAB model used to predict the response of a Hall sensor centered within a coil above a homogeneous conducting plate.

The components an and bn (the sine and cosine magnitudes at each of the selected frequencies) are given by In the calculation, the infinite is replaced by a large number N=801.

Consider the geometry in Figure 3. If we change the conductivity of the aluminum to zero, then the coil and the Hall sensor are effectively in air. For the FEM calculation, the magnetic vector potential A is used to compute at each node. Then the other physical quantities can be derived from the magnetic vector potential A. for example, the magnetic flux density B can be calculated using the following relation Calculating the response due to the aluminum plate is a little more complicated because the response at each frequency is not in phase with the excitation. This is due to the phase rotation of the induced current with depth within the conductor. A complex number must be dealt with and two new coefficients (cn and dn) are introduced. The real part of the response is denoted by cn, the imaginary part, dn. The result of a summation yields the field due to the conductor, When the coil is in the air, the real part of the response cn is equal to zero. Thus the result of a summation yields, A graphical representation is shown in Figure 4. The actual pulse response is obtained by subtracting the response in air from the response over the conductor to yield the familiar pulsed eddy-current signal, Figure 4. Figure 4: Top Left: pulse response in air, Top Right: Pulse response over a conductor, Bottom: subtracted pulse response due to the conducting plate.

#### C. Calculation of Induced Voltage for PEC

After obtaining the impedance of the coil at each frequency component and calculating the magnitude of each frequency component (an and bn), we can calculate the induced voltage on the coil by applying the ohm law, Am is the magnitude of the imaginary part of the coil impedance (when the coil is placed in the air, it only has the imaginary part). The other two coefficients cn and dn, are the real part and imaginary part of the coil impedance when it is placed on the sample. The results are shown in Figure 5. Figure 5: Induced Voltage, Top Left: the response of the coil on the sample; Top Right: the response of the coil in the air; Bottom Left: subtracting the response in the air from the response on the sample; Bottom Right: the region of interest in the plot of bottom left.

### Conclusion

1. Pulsed eddy currents excited by FSW are introduced for numerical simulations.
2. A simple approach based on Fourier transform is realized for simulation of pulsed eddy currents.
3. It is proved that this simulated impedance of a coil agrees with experimental data and Dodd and Deeds model.
4. This approach can calculate the magnetic flux density field and induced voltage on a coil as well.

### References

1. C. V. Dodd and W. E. Deeds, J. Appl. Phys. 39, 2829 (1968).
2. S. Sharma, "Application of finite element models to eddy current probe design for aircraft inspection", ISU Ph.D. thesis, 1998.