Calculation of Electric Field Induced in the Human Body for Simultaneous Exposure to Spatially Uniform ELF Electric and Magnetic Fields With a Phase Difference

International exposure guidelines such as ICNIRP guidelines and IEEE C95.1 standard are published to protect human from potential adverse health effect. These guidelines and standards establish the limit for the induced electric field, also called the basic restriction. The permissible external field strength—known as the reference level—is then conservatively derived from the basic restriction. Though the reference level is calculated assuming that the human body is exposed to electric or magnetic fields separately, in reality, simultaneous exposure to both fields may occur. Such exposures are particularly likely when a human body is positioned under overhead transmission lines. Under such circumstances, a phase difference between the electric and magnetic fields occur due to the phase difference between the power line’s voltage and current. We investigated the impact of external electric and magnetic field phase differences on the induced electric field in numerical human models. This was done under simultaneous exposure to a spatially uniform vertical electric field and horizontal magnetic fields at 50 Hz. Our computational findings revealed that the strength of the induced electric field fluctuates with the phase difference and that the variation caused by this difference varies across different body parts. The basic restrictions of the ICNIRP guidelines were met under the simultaneous exposure to electric and magnetic fields at the reference level, even when considering the phase difference.


I. INTRODUCTION
Public concerns exist regarding possible adverse health effects of exposure to electromagnetic fields. Two international guidelines addressing this issue, the International Commission on Non-Ionizing Radiation Protection (ICNIRP) [1], [2] and the Institute of Electrical and Electronics Engineers (IEEE) standard C95.1 TM -2019 [3] offer protection from environmental electromagnetic fields ranging The associate editor coordinating the review of this manuscript and approving it for publication was Su Yan . from 0 Hz to 300 GHz. These guidelines are referenced in World Health Organization (WHO) documents and establish limits for the induced (in-situ) electric fields within the human body for exposure to low-frequency environmental (external) electric and magnetic fields. Hereafter, we refer ''induced electric field'' in the human body for internal [1] or in-situ [3] electric fields. The limit for these induced electric fields is known as the ''basic restriction'' according to ICNIRP guidelines or ''dosimetric references limit (DRL)'' per the IEEE standard. The basic restriction is derived from the threshold for nervous stimulation, factoring in a safety margin [4], [5]. In the low frequency range, both external electric field and magnetic fields induce an electric field in the human body governed by distinct physical laws [6]. When exposed to an electric field, surface charges induce a capacitive current (and thus an electric field) within the body [6]. Regarding magnetic field exposure, Faraday's law indicates that eddy currents will occur within the body [6].
Permissible external field strength is determined separately for electric and magnetic field exposure, termed as the ''reference level'' in the ICNIRP guidelines or ''exposure reference level (ERL)'' according to the IEEE standard (hereafter referred as reference level in this article).
In practical scenarios, such as under high-voltage overhead power lines (50 or 60 Hz), simultaneous exposure to electric and magnetic fields may occur. In such an environment, the induced electric fields in humans for respective exposure to electric and magnetic fields should be evaluated. The induced electric fields due to the electric field and the magnetic field may either constructively or destructively interfere, depending on the location of the human body. These conditions have been evaluated using a numerical human body model in [32] and [33]. Note that the current is nearly in phase with the voltage as the power factor (cosϕ) should generally be greater than 0.9 in most cases [36], [37]. The current can be zero when the power systems are in open ended conditions. In this case the magnetic field is quite zero.
Leitgeb et al. [32] computed the induced current density in an American male model for simultaneous exposure to a uniform magnetic field of 100 µT and a uniform electric field of 5 kV/m at 50 Hz. They suggested considering superposed electromagnetic fields for conservative evaluation, because the induced electric field strength for simultaneous exposure may exceed that induced by a single component of electric and magnetic fields in some body parts. The same group evaluated induced current densities in a pregnant woman model under conditions of simultaneous exposure to a uniform magnetic field of 100 µT and a uniform electric field of 5 kV/m [33]. These studies assumed that the electric and magnetic fields are in phase. However, a phase difference exists between the current and the voltage induced by line impedance and tidal conditions [36], [37]. The impact of this phase difference between the electric and magnetic fields on the induced electric field in the body has not been quantitatively explored.
This study examined the effect of phase difference on the induced electric field for simultaneous exposure to electric and magnetic fields at reference levels according to international guidelines and standards. The focus of our discussion is the simultaneous exposure of external electric and magnetic fields, mainly occurring at 50 Hz or 60 Hz under high-voltage overhead power lines. We also assessed whether the basic restriction of exposure guidelines is still met when considering the phase difference for simultaneous exposure.

A. NUMERICAL HUMAN MODELS
The Japanese adult male detailed anatomical model ''TARO'' and female model ''HANAKO'' [38] and the 3 years old child model [39] developed at NICT (National Institute of Information and Communications Technology, Japan), as shown in Fig. 1, were used to evaluate the induced electric fields by external electric and magnetic fields. In these models, 54 types of tissue are segmented. The voxel resolutions of the model are 2 mm for the adult models and 1 mm for the child model, respectively.
The electrical conductivities of each tissue at 50 Hz are listed in Table 1. These values are based on the electrical constant reported by Gabriel [40].

B. COMPUTATIONAL METHOD
The induced electric field within the human body is separately calculated for electric and magnetic field exposures. These results are superimposed during post-processing to account for the phase difference between the electric and magnetic fields, simulating the induced electric field for simultaneous exposure.

1) COMPUTATIONAL METHOD FOR INDUCED ELECTRIC FIELD BY EXTERNAL MAGNETIC FIELD
The Scalar Potential Finite Difference (SPFD) method [7], [12], [13], [16], [41] was employed to calculate the induced electric fields by the external magnetic field within the human body. Under quasi-static conditions, the induced electric fields can be expressed as follows: Here, E, ω, A, and ϕ denote the induced electric field, angular frequency, magnetic vector potential, and electric scalar potential, respectively. A is given by A(r) = (0, −B ext x/2, B ext y/2) in the case of uniform external magnetic field B ext = (B ext , 0, 0) along with x-axis, for example. Assuming a continuity condition for the current density Eq. (1) is reduced to the differential equation and subject to the boundary condition Here, σ is the electrical conductivity of the tissue, and n is the normal component at the body surface boundary. The unknown electric scalar potentials at all the nodes are calculated by solving the simultaneous equation, discretized by integrating (3) with respect to the voxel volume and selecting the voxel node as the collocation. The biconjugate gradient stabilized (Bi-CGSTAB) method [44] was utilized to solve this simultaneous equation. The stopping criteria of the Bi-CGSTAB iterative procedure was set at 10 −6 [12] measured by the relative residual norm of the equation solution. Induced electric fields are derived from the gradient of the electric scalar potential, accounting for the difference between values at two adjacent nodes on the voxel's edge.

2) COMPUTATIONAL METHOD FOR INDUCED ELECTRIC FIELD BY EXTERNAL ELECTRIC FIELD
A two-step method combining fast multipole surface charge and SPFD methods [42], applicable when the displacement current is adequately low below MHz [43], was used to compute the induced electric field in humans for electric field exposure. The validity of this calculation method is verified by comparison with the conventional method, the quasi-static finite difference time domain (FDTD) method [42]. The first step calculates the surface charge on the body surface induced by the external electric field. The surface charge satisfies the integral equation where r i is the center location of discretized surface elements, S j is the integral area of surface elements, q i is the position vector in surface element, ϕ ext is the electric scalar potential due to external electric field and given by ϕ ext (r)= −E ext · r in the case of uniform external electric field E ext , ϕ 0 is the reference value for electric scalar potential, ρ s is the surface charge density, and N is the number of surface elements. The first and second terms in brackets ''[ ]'' within Eq. (5) represent the Coulomb potential due to the other charge elements and the contribution to the Coulomb potential due to the shadow charge elements, respectively. The latter accounts for the earth's conduction effects. Eq. (5) is formulated for all surface elements assuming that the surface charge ρ s is constant on the element. The surface charge ρ s at each element is derived by numerically solving the simultaneous equations VOLUME 11, 2023 of Eq. (5). In solving this simultaneous equation by the Bi-CGSTAB method, the Fast Multipole Method (FMM) [45] is applied to calculate the matrix-vector product to reduce the computational cost. The stopping criteria of the Bi-CGSTAB iterative procedure was set at 10 −8 in terms of the relative residual norm of the equation solution, which is of sufficient accuracy to be passed on to the later stage of analysis [42].
In the second step, the induced electric field in human is calculated using the SPFD method [41] by inputting the surface charge ρ s obtained in the first step as a source term. Due to the conservation law of current density, the boundary condition of the body surface can be expressed as following equation.
In addition, the following equation holds inside the body.
The electric scalar potential, which is an unknown quantity, is obtained by solving the simultaneous equations obtained by discretizing equations of (6) and (7) with the condition that the body average of the electric scalar potential is zero (not necessary for the ground condition), with the voxel nodes as the defining points of the electric scalar potential. The equation was solved using the Bi-CGSTAB method with the stopping criteria of the iterative procedure set to 10 −6 , similar to (1). The induced electric field in the body is obtained by taking the gradient of the acquired electric scalar potential. Our computational code's efficacy was previously validated by inter-comparison [12].

C. VECTOR ADDITION OF INDUCED ELECTRIC FIELDS BY EXTERNAL ELECTRIC AND MAGNETIC FIELDS CONSIDERING THEIR PHASE DIFFERENCE
We assume a phase difference between the voltage and current of the power line and consider the relationship between the phase difference and the induced electric field. Focusing on the electromagnetic field surrounding the power line, the power line's voltage and current generate electric and magnetic field respectively ( Fig. 2 (a)). A phase difference (−90 • to +90 • range) is known to occurs between the voltage V and the current I , depending on the power flow condition (power supply and demand), the impedance of the transmission and distribution lines, and the installation of phase modulating equipment [36], [37]. The lagging phase (−90 • to 0 • range) is introduced when the impedance or load is inductive, while the advancing phase (0 • to +90 • range) is introduced when the impedance or load is capacitive. If the phase delay of the current I with respect to the voltage V is θ, the following equation holds: where V 0 and I 0 are the amplitudes of voltage and current, respectively. Since the external electric field E ext and the external magnetic field B ext are in phase with V and I , respectively, they are expressed as follows: Here, E ext and B (0) ext are the amplitudes of the external electric and magnetic fields, respectively. Based on Eqs. (8) and (9), and considering that the induced electric field due to the electric field is advanced by 90 • compared to the external electric field because of capacitive current's nature and the induced electric field due to the magnetic field is delayed by 90 • compared to the external magnetic field by Faraday's law, the induced electric field in the body due to the electric and magnetic field can be expressed as follows: where E are the amplitudes of the induced electric and magnetic fields in the body, respectively. In other words, the electric field, induced by the magnetic field, lags behind by 180 • + θ. The vector diagram of the induced electric field components in the body caused by the electric and magnetic fields, as well as the external electromagnetic field, power line voltage and current, is shown in Fig. 2 (b).
If the current phase of the power line lags behind the voltage phase by θ, the same magnitude of phase difference also occurs between the induced electric field components due to magnetic and electric field exposures. Considering the phase difference, E (E) in and E (B) in denote the induced field due to exposure to electric and magnetic fields calculated under the assumption that the external electric and magnetic fields were in phase. Then they are added, considering the phase factor e −j θ to derive the induced electric field for simultaneous exposure to electromagnetic fields.

D. EXPOSURE SCENARIOS
We considered a scenario where a human is positioned beneath transmission lines. This would expose the human body model to a uniform vertical electric field and horizontal magnetic field. The impact of external field inhomogeneity on the induced electric field is detailed in [16]. Compared to the vertical component, the horizontal component of the electric field is negligible due to its weaker coupling with the human body. The frequency was 50 Hz. As shown in Fig. 1, the vertical electric field (E z ), left-right magnetic field (B x ), and front-back magnetic field (B y ) were considered. We examined two combinations of electric and magnetic fields for simultaneous exposure conditions: E z and B x and E z and B y , hereafter referred to as cases E z + B x and E z + B y , respectively. Single electric or magnetic field exposures (E z , B x , and B y ) were also considered as reference cases. The scenarios described above are the most likely situations, but not necessarily the most extreme conditions (e.g., [18]). For example, maintenance workers of live electrical systems can assume different positions during their work, or a generic person can be lying on the ground for any reason. In addition, due to Faraday's law, the induced field strength in magnetic field exposure strongly depends on the area of the magnetic field crossing the body part. More specifically, it strongly depends on the size and posture of the human body. Note that in the ICNIRP RF guideilnes, extremely cases are not considered. Initially, this study evaluated the fundamental characteristics of the induced electric field considering simultaneous exposure to electromagnetic fields and the phase difference between the electric and magnetic fields. Subsequently, we assessed the compliance of the induced electric field with the basic restrictions in the ICNIRP and IEEE standards.
The external field strength of the electric and magnetic field was aligned with the reference levels of the ICNIRP guidelines [1] and the IEEE C95.1 TM -2019 [3] at 50 Hz. The basic restrictions and reference levels of ICNIRP and IEEE are outlined in Tables 2 and 3, respectively. According to the ICNIRP guidelines, E z = 5 kV/m and B x = B y = 0.2 mT for the general public, and E z = 20 kV/m and B x = B y = 1 mT for occupational exposure, were assumed, respectively. Note that assuming the electromagnetic field strength equivalent to the reference level for occupational exposure may not be a realistic condition for the child model. In the IEEE standard, the reference level is E z = 5 kV/m, B x = B y = 904 µT  (head and torso), and 75.8 mT (limbs) under general public conditions.

A. FUNDAMENTAL CHARACTERISTICS OF INDUCED ELECTRIC FIELDS UNDER SIMULTANEOUS ELECTROMAGNETIC FIELD EXPOSURE CONDITIONS CONSIDERING PHASE DIFFERENCES
To evaluate the impact of superposition and phase difference of external electromagnetic fields on induced electric field strength, the induced electric field in the body was computed for the simultaneous electromagnetic field exposure cases: (i) E z + B x and (ii) E z + B y (E z = 20 kV/m and B x = B y = 1 mT). Fig. 3 illustrates the layer-averaged induced electric field on the horizontal cross-section in the male model. For the single exposure to E z only, the induced field strength peaks at the neck (around z =150 cm), arms (around z = 90−120 cm), knees (around z = 40 cm), and ankles (around z = 0−10 cm) with a small cross-sectional area, thereby resulting in a small capacitive current. Conversely, exposure to only B x or B y induces higher field strength in the head (z=150−170 cm) and torso (z= 80−140 cm), where the large magnetic flux crossing the model leads to large eddy currents.  In the case of simultaneous exposure to external electric and magnetic fields, the induced electric field strength varies for different phase differences θ at 0 • , 45 • , and 90 • .
The induced electric field for simultaneous exposure exceeds that of exposure to a single component of a magnetic or electric field, suggesting a constructive interference effect for most body parts. However, the effect is destructive at certain locations for the E z + B x case (z = 60−80 cm and z = 90−100 cm). Table 4 and Fig. 4 display the 99 th percentile values of the induced electric fields in the brain, heart, and whole body in cases E z + B x and E z + B y when the phase difference between the electric and magnetic fields ranges from −90 • to 90 • . Table 4 shows the 99 th percentile values of induced electric fields on the brain, heart, and whole body assuming simultaneous exposure with phase difference surpass those for separate electric or magnetic exposures. Specifically, in the heart, the electric fields for the cases of E z + B x (83.4−63.0 [mV/m]) were 12% to 143% larger than those for the single electric field exposure case of E z (56.4 mV/m) and B x (34.3 [mV/m]) through θ = 0 • to 90 • for the male model.The exceptions were the conditions θ = 0 • and 45 • of the case E z + B x for the male model, wherein the induced electric field strength was smaller than E z only. The increase in simultaneous exposure relative to the single electric field exposure ranged from 6% to 48%.
As demonstrated in Fig. 4, the induced electric field strength changes due to the phase change are symmetric around θ = 0 • . Additionally, the effect of phase difference on induced electric field strength varies across exposure scenarios and body parts. The electric field in the heart decreases with phase difference, whereas that of the brain slightly increases in the case of E z + B x . Variation widths in the brain for all models concerning θ = 0 • were 3%-17% for the E z + B x case and 1%-18% for the E z + B y case, both of which were smaller than those in the heart (i.e., 22%-25% for the E z + B x case and 20%-26% for the E z + B y , as listed in Table 4). For both E z + B x and E z + B y exposure, the variation of the induced electric field with phase difference for the whole body is marginal because the maximum induced electric fields inside the whole body are observed around the ankle, where the induced electric field by electric induction dominates.
The variations in the phase difference between the external electric and magnetic fields induced changes in the electric With a phase difference of 0 • , the induced electric field strength at the front of the body is high but modest at phase differences of 45 • and 90 • . At a phase difference of 0 • , the induced electric field components due to magnetic field induction at the front of the body align in the same direction as those due to electric field induction, resulting in constructive interference. Conversely, the opposite tendency is observed at the back of the body, where the electric field directions induced by the external magnetic and electric fields are in opposition.
The induced electric field components due to magnetic field induction at a phase difference of 45 • are smaller than those at 0 • and null at 90 • . The total induced electric field strength at the front of the body subsequently increases while decreasing at the back of the body as the phase difference augments from 0 • to 90 • . Consequently, as shown in Fig. 4 (a), the heart's total induced electric field strength monotonically decreases. The brain's total induced electric field strength exhibits a complex phase difference dependency because the induced electric field component induced by external electric and magnetic fields strengthen each other at the front of the head but weaken each other at the back of the head.

B. COMPLIANCE EVALUATION WITH THE BASIC RESTRICTION UNDER SIMULTANEOUS ELECTROMAGNETIC FIELD EXPOSURE CONDITIONS AT THE ICNIRP AND IEEE REFERENCE LEVELS
We assessed compliance with basic restrictions (DRLs) under simultaneous exposure to electromagnetic fields at the reference level of international guidelines such as ICNIRP guidelines [1] and the IEEE C95.1 TM -2019 [3]. Given the symmetry in induced electric field strength variations concerning θ = 0 • from Section III-A, we only considered θ > 0 • in this analysis. Table 5 presents the induced electric fields calculated under simultaneous exposure of the E z + B x and E z + B y scenarios. We assumed electromagnetic field strengths equivalent to the public and occupational exposure reference levels established in the ICNIRP guidelines. As indicated in Table 5, the induced electric fields for public and occupational exposure scenarios, as well as for all phase differences, remained significantly below the basic restrictions of 20 mV/m (CNS of the head) and 400 mV/m (all tissues of the head and body) for general VOLUME 11, 2023  public exposure. The respective values for occupational exposure were 100 mV/m (CNS of the head) and 800 mV/m (all tissues of the head and body), as detailed in Table 2. The induced electric field strength variation due to phase changes ranged from −19% to +15% for the CNS of the head and from −0.9% to 0% for the whole body across all models, as shown in Table 5. Table 6 displays the induced electric field calculated under simultaneous electromagnetic field exposure for the reference level conditions specified in IEEE C95.1 TM -2019. As the reference levels and basic restrictions under controlled environment conditions are proportional to those under general public conditions, we only evaluated conformity under general public conditions. Cases where the induced electric field exceeded the DRL listed in Table 3, are highlighted in bold in Table 6. For scenarios E z + B x and E z + B y for all models, the induced electric fields in the brain exceeded the basic restriction. The same applied to the limbs in the E z + B x scenario for the female model and in all phases of the E z + B y scenario for all models. The variation widths of induced 95462 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. electric field strength due to phase changes for the E z + B x cases ranged from −2% to +20% for the brain, −25% to −3% for the heart, −0.2% to +0.3% for the limbs, and −16% to −2% for others, respectively, across all models. Those for the E z + B y cases were −6% to −1% for the brain, −27% to −3% for the heart, −0.1% to +0.2% for the limbs, and −11% to −1% for others, respectively.

IV. DISCUSSION
This study investigated the impact of phase differences on induced electric fields under simultaneous exposure to uniform electric and magnetic fields at reference levels, as outlined by various international guidelines and standards. Our computational findings revealed that induced electric field strength varies with a phase difference, with the degree of variation differing across body parts. Specifically, we observed a decrease in the electric field within the heart as phase difference increased, while the variation within the brain was comparatively smaller. We also examined the compliance with the basic restrictions in exposure guidelines, accounting for phase differences under simultaneous exposure at reference levels. While we were able to confirm conformity under ICNIRP guidelines, there were some isolated instances of basic restriction exceedance according to the IEEE standard.
The phase difference-related variations in induced electric field strengths can be attributed to changes in phase angle between induced electric fields prompted by external VOLUME 11, 2023 electric and magnetic fields. As Fig. 1 (b) elucidates, as the phase difference between external electric and magnetic fields increases, the component of induced electric fields E (B) in parallel to E (E) in decreases. If E (B) in and E (E) in constructively interfere (i.e., they are in similar directions) at a phase difference of 0 • , the total induced electric field strength will decrease with an increasing phase difference. Conversely, the total induced electric field strength increased with the phase difference if they interfere destructively. This perspective effectively explained the observed negative dependency of induced electric field strength on phase difference in the heart, as illustrated in Fig. 4.
The varying phase difference dependency of induced electric field strength among different body parts could be explained by the body part's relative location to the center of the eddy current. In the brain, located near the center of the local eddy current in the head, the variation width was relatively small (E z + B x : 17%, E z + B y 18%, for the male model, as presented in Table 4 ). Here, E (B) in and E (E) in interfere bothconstructively and destructively. However, the heart, situated outside the center of the local eddy current in the torso, exhibited a larger variation width than the brain (E z + B x : 25%, E z + B y : 26% for the male model). This is because E (B) in and E (E) in only interfere constructively or destructively in the heart, as shown in Fig. 5. Table 4 also showed that the heart most prominently demonstrates the effects of superimposed electric and magnetic fields and phase differences. The increase in induced electric field in simultaneous exposure relative to single electric field exposure ranged from 31% to 52% for the heart, 0%-25% for the brain, and 0%-1% for the whole body across all models. The brain and the whole body were less sensitive to phase difference than the heart since these body parts are dominated by the induced electric field due to single electric field exposure. For the whole body, only single electric field exposure should be considered to derive the maximum value of the induced electric field, as it was minimally dependent on the phase difference.
We also explored conformity with international guidelines and standards. As noted in Sec. III-B, the basic restriction of the ICNIRP guideline is met even when considering simultaneous exposure to external electric and magnetic fields at the guideline's reference level and their phase difference for all models. This may be due to the fact that the head and whole body are less sensitive to phase differences, as previously discussed. The ratio of the induced electric fields to the basic restriction ranged from 16 to 69% in Table 5. This ratio takes the minimum in the case of E z + B x with the general public exposure level for the male model, θ = 90 • at the whole body and takes the maximum in the case of E z + B x with occupational exposure level for the female model, θ = 0 • at the whole body. Importantly, a past study [32] considering simultaneous exposure under in-phase conditions (American male, magnetic field 100µT + electric field 5 kV/m) also reported results below the basic restriction of the ICNIRP guideline, albeit in an earlier version of the guideline [35]. While a direct comparison cannot be made due to their reliance on a previous guideline version, our findings corroborated that the ICNIRP basic restriction is still met even when considering the effect of phase difference.
Conversely, the induced electric field for the reference level strength of IEEE standard exceeded the basic restriction. Past research [12], [13] suggested that the induced electric field may surpass the basic restriction even under separate exposures to magnetic fields at reference level strength. The basic restriction exceedance was most apparent in the limbs for the female model across all models, potentially due to the inner sides of the female model being in contact with each other, as suggested in [12]. As discussed in [46], skinto-skin contact does not result in nerve stimulation, which aligns with the IEEE standard. Of all subject parts considered for basic restrictions, the heart was the most sensitive to the phase difference, even though the induced electric fields in the heart remained below the basic restriction for all cases.

V. SUMMARY
Considering the phase difference of external electric and magnetic field, we computed the induced electric field under simultaneous exposure to spatially uniform ELF electric and magnetic fields. Our study demonstrated that induced electric field strength varied with phase difference and that this variation differs across body parts. The width of this variation depends on the relative location of the subject body part, which can be characterized by the eddy current. Of all the body parts, the heart demonstrated the most pronounced effect of superimposing electric and magnetic fields and phase differences. Despite considering the effect of phase difference, the basic restriction of ICNIRP guideline is still satisfied under the simultaneous exposure of electric and magnetic fields with reference level. Even under these conditions, the induced electric field strength was at most 69% of the basic restriction of the ICNIRP guideline. Dr. Yamazaki is a member of the Bioelectromagnetics Society and a fellow of IEE Japan.