MEASURES OF POST-PROCESSING THE HUMAN BODY RESPONSE TO TRANSIENT FIELDS

1. MEASURES OF POST-PROCESSING THE HUMAN BODY RESPONSE TO TRANSIENT FIELDS Dragan Poljak Department of Electronics, University of Split R.Boskovica bb, 21000 Split, Croatia Email: Dragan.Poljak@fesb.hr The scope: The calculation procedures for some meausures to evaluate human response to transient electromagnetic radiation.

2. CONTENTS • Introduction • Time Domain Representation of the Body • Some Measures for the Body Transient Response • Computational Examples • Conclusion

3. 1 Introduction • The transient current induced inside the body is the key parameter in the analysis of the interaction of human beings with transient fields. • Transient exposures can be analyzed using the human equivalent antenna model (Poljak, Tham, Sarolic, Gandhi IEEE Trans EMC Feb 2003). • Once obtaining the transient response of the body it is possible to compute certain parameters providing a measure of the transient current.

4. 2 Time Domain Model ofthe Human Body The time domain study of the EMP coupling to the body is based on a cylindrical body representation(L=1.8m, a=5cm) Fig 1 The human body exposed to transient radiation

5. 2 Time Domain Model (cont’d) • The dimensions of the human equaivalent antenna (L=1.8m, a=5cm) stay within the thin wire approximation and the effective frequency spectrum of the EMP (5MHz). • The integral equation for the transient current along the body is obtained by enforcing the condition for the total tangential electric field component at the wire surface: • where Ezincis the incident electric field and the scattered electric field Ezinc is expressed in terms of the vector and scalar potential:

6. 2 Time Domain Model (cont’d) • where the vector potential is defined by: and the scalar potential is given by: • where ρsand J are space-time dependent surface charge and surface current density.

7. 2 Time Domain Model (cont’d) • They are related with the continuity equation: • and RLis the resistance per unit length of the antenna length:

8. The integral equation for the transient current along the body is given by: Integrating the Pocklington equation yields the Hallen integral equation: • I(z,, t-R/c) is the unknown current to be determined, • c is the velocity of light, • Z0 is the wave impedance of a free space • F0(t); FL(t) are related with the current reflections from wire ends • -R is the resistance per unit length of the antenna length The time domain Hallen equation is solved via the Galerkin-Bubnov indirect boundary element approach.

9. 3 The Boundary Element Solution of the Hallen Integral Equation • The Hallen integral equation can be written in the operator form: - L is linear integral operator, - I is the unknown function to be determined for a given excitation Y • The unknown solution for current is given in the form of linear combination of the basis functions: {f} is the vector containing the basis functions {I} is the vector containing unknown time dependent coefficients of the solution.

10. 3 The Boundary Element Solution(cont’d) • The request for minimization of the interpolation error yields: • Applying the boundary element algorithm the local matrix system for i-th source element interacting with j-th observation element is given as follows:

11. 3 The Boundary Element Solution(cont’d) • The calculation procedure is more efficient if the known excitation is also interpolated over wire segment: where {E} is the time dependent vector containing known values of the transient excitation. • Hence, the matrix equation becomes:

12. 3 The Boundary Element Solution(cont’d) • The time domain signals F0and FL can be expressed in terms of auxilliary functions K0(t) and KL(t):

13. 3 The Boundary Element Solution(cont’d) • defined by relations: and

14. 3 The Boundary Element Solution(cont’d) • The rearranging of the matrix equation yields:

15. 3 The Boundary Element Solution(cont’d) • The interaction matrices [A], [C] and [D] are of the form: where G(z,z,) is the corresponding Green function. • [B] matrix is given by expression: • and the resistance matrix is of the form: • The matrix system (19) can be, for convenience, written in the form: • (23) • where g is the space-time dependent vector representing the entire right hand side of the matrix equation (20). • The solution in time for unknown current coefficients is given by: • (24) • where NT is the total number of time segments. • The weighted residual approach for the time increment yields: • (25) • Nt is the total number of time samples, • θk denotes the set of time domain test functions. • Choosing the Dirac impulses as test functions it follows: • (26) • The space-time discretisation condition is given by: • (27) • The resulting recurrence formula for the space-time dependent current is of the form: • (28) • Ng is the total number of space nodes, • the horizontal line over matrix [A] denotes the absence of diagonal terms. • The appropriate boundary conditions at the wire ends are: • (29) • The initial condition to be satisfied over the entire length of the wire at t=0 is: • (30) • Now, everything is ready to start the stepping procedure.

16. 3 The Boundary Element Solution(cont’d) • The matrix system can be, for convenience, written in the form: where g is the space-time dependent vector representing the entire right hand side of the matrix equation. • The solution in time for unknown current coefficients is given by: where NT is the total number of time segments. • The weighted residual approach for the time increment yields: θk denotes the set of time domain test functions.

17. 3 The Boundary element Solution(cont’d) • Choosing the Dirac impulses as test functions it follows: The space-time discretisation condition is given by: The resulting recurrence formula for the space-time dependent current is of the form: Ng is the total number of space nodes, the horizontal line over matrix [A] denotes the absence of diagonal terms.

18. 3 The Boundary Element Solution(cont’d) • The appropriate boundary conditions at the wire ends are: • The initial condition to be satisfied over the entire length of the wire at t=0 is: Now, everything is ready to start the stepping procedure.

19. 4 Measures of a transient response ° average value of the transient current ° root-mean-square value of the transient current ° instantaneous power ° average power ° total absorbed energy ° specific absorption 4.1 Average value of a transient current The average value of a time varying current i(t) is defined as: where T0 is the period of the waveform.

20. The distribution of average values of current is simply given by: When the current along the wire at each node and time instant is known, Iav(x) is simply given by: and the performing of a straight-forward integration yields: where {T} is the vector containing the time domain linear shape functions.

21. 4.2 Root-Mean-Square measure for a transient response Instantaneous power delivered to a resistance RL by a transient current i(t) is: while the corresponding average power Pav is determined by the integral relation: from which the rms current is then:

22. The rms value of the thin wire space-time current distribution is given by: where T0 is the time interval of interest. When the current along the wire at each node and time instant is known, the rms value of the wire current can be computed from the following relation:

23. and the performing of a straight-forward integration yields: where {T} is the vector containing the time domain linear shape functions. 4.3 Instantaneous power Instantaneous power is defined by integral: where σ is the average conductivity and S is the cross-section of the body

24. The BEM solution for instantaneous power is given by: where I is the current at i-th node and k-th instant. 4.4 Average power The average absorbed power is defined as:

25. And can be written in the form: where Irms is the effective value of the current. The BEM solution can be written as follows:

26. 4.5 Absorbed power The total absorbed energy can be obtained integrating the instantaneous power: The BEM solution is given by:

27. 4.6 Specific Absorption The specific absorption rate can be defined as: The BEM solution is given as follows:

28. 5 Computational examples • Gaussian pulse • unit step function • EMP waveform Types of excitation considered:

29. 5.1 Gaussian pulse with: E0=1V/m, g=2*109, t0=2ns. Fig 2: Transient current induced in the human body exposed to the Gaussian pulse waveform

30. 5.1.1 Spatial distribution of the average and rms values along the body for the Gaussian pulse exposure Fig 3: Average value of current Fig 4: RMS value of current

31. 5.1.2 Instantaneous power and absorbed energy versus timefor the Gaussian pulse exposure Fig 5: Instantaneous power Fig 6: Absorbed energy

32. 5.2 Step function where u(t) denotes the unit step Fig 7: Transient current induced in the human body exposed to the step function waveform

33. 5.2.1 Spatial distribution of the average and rms values along the body for the step function exposure Fig 8: Average value of current Fig 9: RMS value of current

34. 5.2.2 Instantaneous power and absorbed energy versus timefor the step function exposure Fig 10: Instantaneous power Fig 11: Absorbed energy

35. 5.3 Electromagnetic pulse (EMP) with E0=1.05V/m, a=4*106s-1, b=4.76*108 s-1. Fig 12: Transient current induced in the human body exposed to the EMP waveform

36. 5.3.1 Spatial distribution of the average and rms values along the body for the step function exposure Fig 13: Average value of current Fig 14: RMS value of current

37. 5.3.2 Instantaneous power and absorbed energy versus timefor the step function exposure Fig 15: Instantaneous power Fig 16: Absorbed energy

38. 6 Conclusion • The exposure of human body to transient electromagnetic fields is analysed in this work. • Time domain formulation is based on the human equivalent antenna representation of the body • Some useful measures for the analysis of the body transient response are proposed: ° average value of the transient current ° root-mean-square value of the transient current ° instantaneous power ° average power ° total absorbed energy ° specific absorption • The related numerical results are presented.