EDM Process through Mathematical Model

ABSTRACT


INTRODUCTION
EDM is a non-traditional machining process to remove material from the workpiece using a thread of electrical discharges between the electrode and the workpiece called gap [1]. non-contact characteristics of EDM makes it a valuable technique for variety of hole manufacturing approaches [2]. However, scientific knowledge about the process is still insufficient and it is the main obstacle for its more improvements. Experimental trials are challenging due to the highly stochastic and complex nature of the process caused by some process complications such as adhesion, short-circuiting and cavitations that increase the machining time and make the process become unstable [3].
Many EDM researchers have attempted innovative ideas such as ultrasonic vibrations and flushing effect with the purpose to solve the process complications and improve the system stability. Mahardika [4] found that using ultrasonic vibrations remove adhesion and short circuit during machining procedure. Shabgard [5] revealed that EDM of Ti-6Al-4V by using ultrasonic vibrations of copper electrode enhace MRR. He also claimed that crack density and tool wear ratio (TWR) reduce at finishing regimes while recast layer, cracks density, and TWR increase at roughing regimes. Although, ultrasonic vibrations help to promote productivity and feasibility in certain conditions, however there is still a long way to go for use in industry.
Flushing transfers dielectric fluid into the pipe electrode, so debris carries away and the insulating characteristics of the dielectric are preserved, it is an easy way to improve efficiency of EDM hole drilling [6]. Pressure and suction flushing techniques have the capability of eliminating the spark eroded particles, but they can only be used in particular applications. Ebisu [7] investigated the effect of jet flushing on the debris stagnation during the cutting corner shape in 1st-cut wire EDM. He showed that applying jet flushing decreases debris stagnation in the gap only temporarily after the corner. In order to overcome the difficulty aspects related to real machining environment and its effect on system stability, modeling and simulation is an alternative way to understand the mechanism underlying the EDM procedure. Although gap spark profiles play an important role in the analysis of the EDM process, but very few mathematical models have been developed to identify the profile of EDM spark during machining procedure. Minhat et al. [8] presented the mathematical model of EDM pulses based on the initial, ignition and discharge phases. However, the model failed to determine the EDM profiles correctly, Moreover, mathematical equation did not match the profiles in all phases.
This paper proposes mathematical model to predict the dynamic behavior of the EDM spark, similar to the ideal one. The mathematical explanation of the model is located in section 2. Ignition, discharge and recovery phases have been considered properly. In section 3, MATLAB software is used to simulate machining procedure. Section 4 includes discussion about the validity of the simulated model using series of experimental data to calculate MRRs and compare with the experimental results from previous researcher. Finally, conclusion is given in Section 5.

MODEL DESCRIPTION
In this section, mathematical model during ignition, discharge and recovery phases is developed separately. The proposed model shown in Figure 1 consisted of pulse power generator and EDM spark. Pulse power generator, in turn, is made of an ac source, a transformer, rectifier and a capacitor as filter followed by a transistorized switching circuit as pulse generator. All components are supposed to be ideal in order to reach simple and clear insight into the model behavior.
Based on the model, AC source voltage vs is applied to provide required value of DC input voltage Vin through the bridge diode and transformer with the turns ratio of n1:n2 where n1 and n2 are the numbers of primary and secondary windings, respectively. Pulse generator comprised of a MOSFET switch S1 and a resistor R1. Switch S1 is controlled by low power pulses of Pw1 and used to control current from capacitor C to the gap. As can be seen in Figure 1, the EDM spark which is the gap model between electrode and workpiece consists of Rs, Rig, Rdis, Ldis and switch S2 driven by Pw2. Three defined phases of EDM pulses with related mathematical equations are explained in following.

Ignition phase
Based on the profile illustrated in Figure 2, ignition phase is occurred in the time interval from t1 to t2 which is called delay time td. Open gap voltage Voc provided a severe electric field between electrode and workpiece. An ionization path created through the dielectric and flow of the current igap is interrupted. Gap voltage Vgap in this phase is equal to Voc. Small delay time results larger spark time so more energy enters into the workpiece [9], . Equivalent model of EDM system in this phase is obtained from Figure 1 when switch S1 is close and switch S2 is open. Related equations of gap current igap and gap voltage Vgap obtained as following. As S2 is considered open so, Switch S1 is on, thus applying voltage driver between resistors Rig and Rs gives: Simplifying both sides of (5) gives, From (6) can be seen that small difference between and Vgap makes the model more close to the ideal value of igap in this phase which is zero. So by applying (7), gap voltage reaches to its maximum value called open gap voltage Voc .
Then, 0  gap i (8) Deficiency of this assumption in [8], led to a mismatch of its mathematical expression with the gap current profile in this phase.

Discharge phase
The discharge phase is occurred in the time interval from t2 to t3 which is called tdis as shown in Figure 2. During this phase, the isolating effect of the dielectric breaks down, current starts to flow while the voltage falls [10]. The spark is formed and machining continued to reach a peak gap current of Ig and a discharge voltage of Vdis. In this phase both switches of S1 and S2 from equivalent EDM model in Figure 1 Integrating both sides of the (11), By applying assumption, Vgap is obtained as follow: From (21) and (24), it is clear that, final equation of gap current and gap voltage in this phase is quite different from that of presented in [8]. Expression of gap current and gap voltage in [8] can not describe the dynamic behavior of system accurately.

Recovery phase
The recovery phase is happened during the time interval from t3 to t4 which is called trec as shown in Figure 2. The flow of current is stopped and desired insulating electric properties of the dielectric fluid are recovered [11]. The schematic circuit of the EDM model in this phase is obtained from Figure 1 when switch S1 is off and no current goes through R1 and Rs. So Vgap and igap are equal to zero. This phase is totally missed in the model presented in [8].
Since the switch 1 S is open, so:

SIMULATION
In this section, the diagram of the EDM system with ac to dc power supply and transistorized switching circuit as pulse generator designed in MATLAB and schematic diagram is shown in Figure 3. According to the estimation obtained in (7), transformer reduces the source voltage of 250V to the input voltage of Vin as near as open gap voltage Voc of 160V. Figure 4 shows simulation results of gap voltage and gap current for a selected machining process. Simulated data are chosen from previous experimental tests presented by A. Yahya. [12]. In order to optimize discharge conditions and based on the experimental test  [12], delay time td is set to 2µs which is insignificant compared with the discharge time tdis. It is clearly seen in Figure 4 where simulation results are quite correlated with desirable profile of Vgap and igap in Figure 2.

RESULT AND DISCUSSION
As shown in Figure 2, the profile of EDM spark during one cycle consists of the initial phase occurred during the time interval between t1 and t2, followed by the discharge phase from t2 to t3 and the last phase which is recovery from t3 to t4. Conforming to the equation (7), small difference between input voltage and gap voltage is selected to get the model close to the ideal profile through ignition phase. Noteworthy point to mention is that to better evaluate the procedure, proposed model did not considered noise during EDM process which results from the stochastic nature of the EDM spark. MATLAB software is used to develop the complete model of EDM system. Simulation results in Figure 4, as gap voltage and gap current, are absolutely similar to the modeling profile from Figure 2.
To verify the simulated model, predicted MRRs from series of simulations are compared with series of experimental MRRs carried out by A. Yahya [12] using steel workpiece and copper electrode. Predicted MRR is determined by inserting simulated data into equation (26) obtained by the same researcher in [13]. where  is material properties factor. For the present study, α considered to be equal to the one selected by [14] i.e.,  Table 1 presents the simulated (Predicted) MRR and the experimental (Actual) MRR during several peak gap currents Ig , discharge times tdis, recovery time trec and spark frequencies Fs. The last column of this table shows predicted error which is a comparison between the experimental and the simulated MRR under identical conditions. The average simulated error is below of 5.14 %. It seen that the simulated model has ability to predict the MRR with acceptable error.

CONCLUSION
In this paper, a time domain mathematical model of EDM system has been developed. The whole model is simulated in accordance to the EDM conditions including ignition, discharge and recovery phases. . MATLAB simulation result is quite correlated with desirable spark profiles. Validity of the simulated model is carried out by comparing MRR from the previous researcher's experimental results. It is found that, welldesigned model for EDM system can easily provide the possibility to predict dynamic behavior of pulse profiles by eliminating complications related to the stochastic nature of EDM process in experimental trial.