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

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

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. 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). 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. Once obtaining the transient response of the body it is possible to compute certain parameters providing a measure of the transient current.

2 Time Domain Model of the 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

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 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: 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 E z inc is the incident electric field and the scattered electric field E z inc is expressed in terms of the vector and scalar potential: where E z inc is the incident electric field and the scattered electric field E z inc is expressed in terms of the vector and scalar potential:

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

2 Time Domain Model (cont’d) They are related with the continuity equation: They are related with the continuity equation: and R L is the resistance per unit length of the antenna length: and R L is the resistance per unit length of the antenna length:

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, -Z 0 is the wave impedance of a free space -F 0 (t); F L (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.

3 The Boundary Element Solution of the Hallen Integral Equation The Hallen integral equation can be written in the operator form: 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: 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. {I} is the vector containing unknown time dependent coefficients of the solution.

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

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

3 The Boundary Element Solution (cont’d) The time domain signals F 0 and F L can be expressed in terms of auxilliary functions K 0 (t) and K L (t): The time domain signals F 0 and F L can be expressed in terms of auxilliary functions K 0 (t) and K L (t):

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

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

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

3 The Boundary Element Solution (cont’d) The matrix system can be, for convenience, written in the form: 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. 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: The solution in time for unknown current coefficients is given by: where N T is the total number of time segments. where N T is the total number of time segments. The weighted residual approach for the time increment yields: The weighted residual approach for the time increment yields: θk denotes the set of time domain test functions.

3 The Boundary element Solution (cont’d) Choosing the Dirac impulses as test functions it follows: Choosing the Dirac impulses as test functions it follows: The space-time discretisation condition is given by: The space-time discretisation condition is given by: The resulting recurrence formula for the space-time dependent current is of the form: 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.

3 The Boundary Element Solution (cont’d) The appropriate boundary conditions at the wire ends are: 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: 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. Now, everything is ready to start the stepping procedure.

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 T 0 is the period of the waveform.

When the current along the wire at each node and time instant is known, I av (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. The distribution of average values of current is simply given by:

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

where T 0 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: The rms value of the thin wire space-time current distribution is given by:

where σ is the average conductivity and S is the cross-section of the body 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:

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:

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

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

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

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

5.1 Gaussian pulse Fig 2: Transient current induced in the human body exposed to the Gaussian pulse waveform with: E 0 =1V/m, g=2*10 9, t 0 =2ns.

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

5.1.2 Instantaneous power and absorbed energy versus time for the Gaussian pulse exposure Fig 5: Instantaneous powerFig 6: Absorbed energy

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

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

5.2.2 Instantaneous power and absorbed energy versus time for the step function exposure Fig 10: Instantaneous powerFig 11: Absorbed energy

5.3 Electromagnetic pulse (EMP) with E 0 =1.05V/m, a=4*10 6 s -1, b=4.76*10 8 s -1. Fig 12: Transient current induced in the human body exposed to the EMP waveform

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

5.3.2 Instantaneous power and absorbed energy versus time for the step function exposure Fig 15: Instantaneous powerFig 16: Absorbed energy

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