Statistical distributions are derived for power dissipation inside linear or nonlinear circuit elements driven by a circuit source and illuminated by an external random electromagnetic field. The probability density functions of the normalized power dissipated inside rectilinear, planar, and volumetric resistive elements are derived by splitting the total internal field in components with appropriate bias fields. These results are extended and generalized to several canonical configurations, for the combined internal and external deterministic or random excitation of single or multiple circuit elements. The results are fundamental to the analysis of the operation and malfunction of electronic circuits and their elements when placed inside complex electromagnetic environments, such as reverberation chambers, or on printed circuit board, as well as to their electromagnetic compatibility, emissions, and susceptibility.
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 The increasing spatial and spectral density of the electromagnetic (EM) environment in which modern electronic equipment is operating calls for accurate, efficient and reliable EM characterization methods. Precise and quantified statements regarding the confidence limits and expected value for EM quantities of interest are essential and critical in estimating safety, integrity, reliability and compatibility of interacting electronic systems and their elements or subsystems. With regard to modeling of such systems, a range of full wave computational EM methods is now available and routinely used for their numerical simulation. However, these methods carry large data processing efforts and computational costs with them when applied to complex, intricately detailed or electrically large objects. Moreover, the discretization (meshing) performed when numerically implementing these methods introduces an inevitable staircasing effect. This poses an inherent limitation on their accuracy, especially for those electrically large or resonant systems that exhibit extreme, that is, nonperturbative sensitivity of the field to relatively small changes in the configuration or EM properties of boundary surfaces, as well as to their inherent mechanical or constitutive uncertainties (tolerances).
 Statistical electromagnetism offers an efficient alternative method of analysis with experimentally demonstrated validity in a variety of application areas. Crucially, unlike deterministic numerical methods, its accuracy and simplicity increase with increasing complexity of the configuration. Information on uncertainty and sensitivity of the fields and related quantities is inherent to such a formulation, making it ideally suited for benchmarking, validation and standardization. Some of the main challenges of statistical EM are the extension of ideal first- and second-order statistical characterizations to realistic conditions of operation (e.g., extension to relatively low frequency, finite regions, finite conductivity of boundaries, partial coherence, etc.), by introducing an appropriate number of additional distribution parameters and degrees of freedom, and the independent calculation of the statistics of physical quantities based on an adequate EM model (i.e., expressing the mean, standard deviation, etc., in terms of geometrical, topological and EM quantities, independently, without recourse to statistical estimation methods using collected numerical data). The latter is needed for proper validation of the model, because a simple comparison between the statistical distributions of theoretical and experimental data is only based on variates that have been normalized or standardized with the aid of their own estimated sample statistics (typically the experimental sample mean or sample standard deviation), thus cancelling quantitative information on the dimensioned quantities of interest. In other words, such a comparison only confirms the validity of the type of distribution and “noise color,” not the accuracy of the EM parameters characterizing this distribution in absolute as opposed to relative terms.
2. Statistical Modeling of Electric and Electronic Circuits
 Statistical electromagnetism has been successfully used in modeling the dynamics of complex EM fields and systems, for example, inside mode-tuned or mode-stirred reverberation chambers [see, e.g., Kostas and Boverie, 1991; Deus et al., 1995; Arnaut, 2003b]. Dynamical EM environments are, however, not restricted to enclosures with time-varying boundaries or geometries (spatial dynamics). For example, an electronic component on a semiconductor chip is subjected at any one time to EM interference from a particular set of instantaneously active elements in its vicinity (temporal dynamics). The details of the interfering states depend on the switching state and associated current and voltage of each electronic element, as well as possible transient effects [Arnaut, 2003a]. As time elapses, the configuration of the ambient interference changes, often quasi-randomly (depending on clock frequencies, mark space ratio of pulse trains, switching delays, instantaneous carrier modulation, etc.), even though the overall system remains mechanically static. Such temporal dynamics can be likened to the twinkling of lights in a Christmas tree, in which the intensity of the light received by a particular Christmas ball changes in time with the temporal variations of the spatial pattern of currently active lights, their relative magnitudes and phases, and synchronization between them in a quasi-unpredictable manner.
 In this paper, we study the effect of an external illuminating random field on the field magnitude and energy density inside circuit elements. The analysis focuses on resistive elements, yielding an immediate interpretation in terms of dissipated power. However, the results can be extended to the characterization of apparent and reactive power with little extra effort, because of their simple mathematical relationship to dissipated power. All fluctuations considered here are assumed to operate at the macroscopic level. The complexity of EM environments can be further extended to include the complexity of the circuit itself. In such cases, modeling as a random network proves to be effective. In this paper, however, we shall assume the circuit elements and the network topology to be deterministic.
3. Ideal Distributions of the External Field
 Theoretical analysis and experimental work has demonstrated that inside an idealized mechanically mode-tuned or mode-stirred reverberation chamber, the complex field generated by a narrowband continuous wave input signal exhibits a circular Gauss normal distribution [Kostas and Boverie, 1991]. Therefore it can serve as a canonical source for a random EM field [Michielsen and Fiachetti, 2004]. More general limit distributions beyond circular Gauss normality have also been deduced on the basis of a facet model [Arnaut, 2003a]. As an extension, dynamic facets can be introduced to model the effect of switched sources surrounding a particular test location. The results are transferrable to generic “low”-dimensional systems with finite and randomly fluctuating numbers of degrees of freedom, also in other areas of applied physics or engineering, for example, for conduction, scattering and diffraction processes at microscales and nanoscales.
 With respect to the susceptibility of electronic systems, two quantities of primary interest are field strength (amplitude) and power density [Stratonovich, 1981; Arnaut, 2003a]. For each Cartesian electric field component Eα = E′α − jE″α (α = x, y, z) of the total complex electric field Et, the E′α and E″α are assumed to be independent and identically distributed Gauss normal variates with zero mean value and identical standard deviation σα = /. To avoid confusion between field components and circuit components, we shall refer to them as “components” and “(circuit) elements,” respectively. Similar characteristics exist for the magnetic induction Bα inside a deterministic isotropic medium, whose fluctuations are statistically independent of those of Eα, through scaling of Eα by . Thus the respective probability density functions (pdfs) are
Extensions to anisotropic and non-Gauss normal fields are possible [Arnaut, 2002, 2003b]. The standard deviation is directly associated with the complexity, that is, the equivalent number of degrees of freedom (independent sources) of the random field. Although no dc components can be transferred via radiation, nonzero mean (bias) fields can exist in principle, when direct illumination by a deterministic or relatively slowly varying field takes place. The analysis in this paper can be extended without difficulty to incorporate such radiated biasing fields. In the present paper, the analysis is limited to coupling via electric fields and nonsolenoidal currents only, so that only E will be of concern. The usual notational convention is followed in which random variables (variates) are denoted by uppercase letters, and their values or deterministic quantities by corresponding lowercase symbols.
 The local power or energy density S associated with E, normalized with respect to the local impedance z and associated with m Cartesian field components evaluated at ℓ locations, thus exhibits a chi-square distribution with 2p = 2mℓ (m = 1,2 or 3) degrees of freedom (χ2p2), provided that all p components are statistically independent and identically distributed. The explicit pdf of S = (E′12 + E″12) + … + (E′p2 + E″p2) is then [e.g., Arnaut, 2001a]
that is, a one-parameter distribution characterized by its standard deviation = (2) σα2. In particular, the power density of a quasi-monochromatic ideal random field exhibits a χ62 pdf for its first-order statistics, whereas the power density associated with each Cartesian component of the field exhibits a χ22 pdf [Kostas and Boverie, 1991].
 The distributions derived in this paper apply to the density in a single point location inside the circuit element. In order to obtain the total dissipated power throughout the complete element, an appropriate volume integration must be carried out. However, since the magnitude (envelope) of a random field is now itself a random function of spatial coordinates—unlike the constant magnitude of a deterministic purely time-harmonic plane wave field—care must be exercised in this process. Specifically, if the physical origin of the dissipation involves nonlocal electronic current flow, rather than being of a local molecular dispersive nature (e.g., dielectric relaxation), then spatial integration must be consistent with a local averaging procedure for the field. The pertinent quantities are the variance function, scale of fluctuation and point interval correlation function of the underlying electric and magnetic field [Arnaut, 2001b]. If this field is ideal Gauss normal, then the locally averaged field remains Gauss normal but with an appropriately scaled, namely, reduced standard deviation governed by the size of the integration domain (physical size of the element). In this case, all “point” distributions presented in this paper remain applicable upon spatial integration as well. On the other hand, if the underlying field deviates from Gauss normality (quasi-random field), then upon local averaging this field tends closer to Gauss normality, on account of the central limit theorem which is valid under rather general but specific conditions, for example [Beckmann, 1967]. In this case, the distributions obtained in this paper are merely asymptotic [Arnaut, 2003b] and approximately valid upon increased local spatial averaging. The required extent of such averaging depends mainly on the correlation properties of the field, and the size of the circuit element relative to the wavelength and the scale of fluctuation. For example, an ideal incoherent field exhibits a spatial correlation function (kd) = sinc(kd) across a characteristic distance d. It is emphasized that the analysis in this paper only considers first-order (probability) distributions; no assumptions regarding second-order (coherence) properties are required nor indeed made.
4. Linear Circuit Elements
 We consider a single circuit element with linear voltage-current characteristic. We assume a deterministic internal voltage source with constant or quasi-stationary amplitude producing a voltage v0 between the point contacts of the circuit element under consideration (Figure 1). In addition, the circuit is subjected to external excitation by a random (i.e., highly irregular, rapidly and unpredictably fluctuating) time-harmonic vector electric field E, at a given frequency. The latter may be an unwanted disturbance or intentional test field, or be the resultant phasor associated with the field generated by nearby sources of scattering or radiation, at the frequency of interest. A constant or quasi-static deterministic bias circuit field (dc field) in conjunction with a three-dimensional (3-D) random radiation field serves as a stochastic representation of this scenario. Before analyzing this case for arbitrary excitation, we first investigate a few particular but fundamental cases.
4.1. Rectilinear Dissipative Circuit Element (1-D Dissipation)
 In “thin” “long” circuit elements (e.g., lumped resistors, resistive wires, cables, printed circuit boards (PCB) tracks, etc.) with characteristic length d and vanishingly small surface area and volume, the energy is locally dissipated along a single spatial direction 1α (Figure 1 with h = w = 0). This direction may in general be a function of position x inside the element. For a deterministic circuit, an internal bias field u0 = v0/d is generated inside the element by the circuit voltage v0. The local incident field E has a projected Cartesian complex component Eα = Eα · 1α associated with this dissipation process. Since Eα exhibits a circular symmetric pdf in the complex plane defined by E′α and E″α, u0 can always serve as a phase reference for Eα. Since u0 is deterministic, u0 + Eα is still Gauss normal with respect to both its in-phase and quadrature components, but now centered around nonzero and zero mean values for the in-phase and quadrature components, respectively. The overall field acting on the element can therefore be represented by an anisotropic 1-D biased phasor in the complex plane (top diagram in Figure 2). The pdf of the corresponding normalized power density S ≡ Wα is therefore an anisotropically biased χ22 distribution, that is,
with normalization constant
in which I0(·) denotes a modified Bessel function of the first kind and order zero, and M−η,ζ(·) is a Whittaker function [Gradshteyn and Ryzhik, 1994]. The distribution (3) follows upon variate transformation of the corresponding field magnitude ∣Uα∣ = ∣u0 + Eα∣ = , namely, (wα) ∝ (uα = )/(2), where ∣Uα∣ exhibits a Nakagami-Rice n distribution, that is,
As is well known [Beckmann, 1967], the latter pdf characterizes the distance from the origin for a biased random walk in the plane. As such, it represents a deterministic rectilinear field component (bias) with a superimposed noise field acting along the same spatial direction as the bias field.
 The graph of (3) for u0 = σα = 2 V/m is shown in Figure 3a as the curve labeled “p = 1.” Its value at s = 0 decreases away from unity when u0/σα increases from zero. The negative exponential (χ22) pdf is retrieved when u0 = 0, as is verified from (3), and reaches its maximum at s = 0.
4.2. Planar Circuit Element (2-D Dissipation)
 In this case, the circuit element is quasi-two-dimensional, for example, a resistive patch with vanishingly small thickness (Figure 1 with either h = 0 or w = 0). The internally generated voltage v0 acts between two opposite point or edge contacts across the element.
 First, for an idle state of the element (u0 = 0), its total excitation is generated purely externally, and hence it is a randomly polarized field parallel to the plane of this element as defined by two mutually orthogonal spatial directions 1α and 1β. In phasor space, since u0 = 0 this excitation defines a phasor which extends equally, on average, across two mutually orthogonal complex planes associated with Eα and Eβ (top and middle diagrams in Figure 2, with u0 = 0). Since Eα and Eβ are independent and identically distributed components, the associated power density exhibits a χ42 distribution, as given by (2) with p = 2.
 For an activated circuit element (u0 ≠ 0), the local bias field is still locally one dimensional, even though its local direction may vary within the plane with position x. Its effect on the associated phasor is therefore across only one of the two complex planes (top and middle diagrams in Figure 2, with u0 ≠ 0). The resultant normalized power density S is therefore the incoherent sum of a biased χ22-distributed variate Wα and a standard χ22-distributed variate Wβ, that is, S = Wα + Wβ with Wα ↔ χ22 (μ + u02, σ), Wβ ↔ χ22 (μ, σ), whence its pdf is the convolution of (wα) with (wβ):
where I1(·) represents the modified Bessel function of the first kind and order one, and
In (6), the limits of the domain Ω follow from the requirements that 0 ≤ s − wβ < +∞ and 0 ≤ wβ < +∞, which are simultaneously satisfied provided 0 ≤ wβ ≤ s and 0 ≤ s < +∞. The graph of (7) for u0 = σα = 2 V/m is the curve labeled “p = 2” in Figure 3a. Compared to the case “p = 1,” the extra spatial dimension for dissipation is seen to give rise, in particular, to a smaller skewness of the pdf.
4.3. Volumetric Circuit Element (3-D Dissipation)
 In this general case, the dissipation is through the internal flow of energy in all three spatial directions, which applies to “boxy” resistive elements having appreciable dimensions in all three directions (Figure 1, with w, h, d ≠ 0). The excitation field u0 generated by v0 acts between the contact points or planes and is still locally 1-D deterministic, whereas the externally generated internal field is now fully 3-D random and statistically isotropic. Hence the field phasor is distributed across three mutually orthogonal planes, with an anisotropic deterministic bias in one plane. Hence S = Wα + Wβ + Wγ with Wα ↔ χ22 (μ + u02, σ), Wβ ↔ χ22 (μ, σ) and Wγ ↔ χ22 (μ, σ) (Figure 2). The pdf of the total dissipated power is obtained from the incoherent sum of a biased χ22 distributed and an unbiased χ42 distributed random variable that are mutually independent, yielding
The graph of (9) for u0 = σα = 2 V/m is the curve labeled “p = 3” in Figure 3a.
5.1. Single Circuit Element Driven by Internal Deterministic Source and Immersed in Multisource External Random Field
 In an actual complex EM environment, a driven single circuit element of spatial dimension m (m = 1, 2 or 3) is often irradiated by multiple (q) external random sources. We assume that these sources generate statistically independent and isotropic 3-D Gauss normal random electric fields E(j) (j = 1, …, q) that are identically distributed. Generalizing the results in section 4 for a single source, the local S is now the incoherent sum of q biased χ22 contributions and q unbiased χ2(m−1)2 contributions (if m = 1, then the latter contributions vanish). In view of the statistical independence of the E(j), the associated pdf fS(s) would therefore be obtained as the convolution of the pdf of a q-D biased χ2q2 variate and that of an unbiased χ2(m−1)q2 variate, resulting generally in an elliptical (i.e., spherically asymmetric) fS(s) with 2(m − 1)q + 2q = 2mq degrees of freedom. However, the bias is the internally generated deterministic field u0, which for a given circuit element is identical for all external sources. Therefore the former contribution collapses to a merely 1-D biased χ2q2 variate, resulting in a reduction of the number of degrees of freedom of fS(s) to just 2(m − 1)q + 2. Hence, for the normalized energy density in this case, from (2) and (3),
up to a normalization constant. In (11), Sp′ denotes the S characterized by (2) with p′ = (m − 1)q. It is verified that for m = 1, 2 and 3 with a single source (q = 1), the pdf (12) reduces to (3), (7) and (9), respectively.
Figure 3a shows (12) for selected values of p ≡ (m − 1)q + 1 with u0 = σα = 2 V/m. In general, the aforementioned reduction of the skewness of the pdf with increasing p is confirmed. The corresponding mean μS and standard deviation σS associated with (12) are shown in Figure 3b as a function of p, after normalization with respect to their values μS(1) and σS(1) for p = 1, as given by
Both μS and σS are seen to exhibit a quasi-linear increase with p, whereas the relative spread (normalized standard deviation, coefficient of variation) σS/μS decreases in accordance with 1/. The decrease demonstrates that statistical characterization becomes increasingly more accurate (in the sense of yielding an increasingly sharper value for S) for increasing complexity of the ambient EM environment.
5.2. Single Circuit Element Driven by Internal Random Voltage Source and Immersed in Single-Source External Random Field
 This constitutes another canonical example in which a circuit source generates a signal-plus-noise voltage v0 + V0 and results in random fluctuation U0 = V0/d for the field inside the circuit element. The fluctuation U0 is assumed to be independent from the one generated by the external random field E. The internal field now exhibits a mean value u0 and standard deviation = /d, for which we assume ≤ σα for definiteness but without loss of generality. The physical origin of the fluctuations of U0 may be fluctuations of the driving source (circuit) voltage, thermal agitation inside a resistive element (fluctuation-dissipation) [see, e.g., van Zon et al., 2004]. The nature and scale of this origin is of no concern in the current analysis, as long as the effect is an electromagnetic one.
 First, we consider again a 1-D element directed along 1α. If U0 fluctuates much more rapidly or slowly than Eα then S is again a biased χ22 variate whose bias is now time-varying (nonstationary process), that is, with ∣eα(t)∣ or u0(t) replacing u0 in (3), where the overbar signifies temporal averaging with respect to the scale of fluctuation of Eα or U0, respectively [Arnaut, 2001b]. By contrast, if the rates of fluctuation of U0 and Eα have comparable order of magnitude, then S is generally differently distributed across two mutually orthogonal complex planes, because U0 and Eα are mutually random and statistically independent with in general different bias (local average) and standard deviation, and remain incoherent. Therefore, in this case, with U0 ↔ N (u0, ) and Eα ↔ N (0, σα), we obtain S = W0 + Wα = U02 + ∣Eα∣2 as the sum of a χ22 variate and a biased χ22 variate, but which are now characterized by mutually different parameters for their underlying Gaussian pdfs. Hence fS(s) follows as
with normalization constant C′1, where
If = σα then, since U0 and Eα are statistically independent, S reduces to a biased χ42 variate whose pdf is given by (12) with m = 2 and q = 1, that is, (7). For planar (m = 2) and volumetric (m = 3) dissipation, as well as for higher values of m in case of multiple sources, (14) generalizes to
where Cm is a normalization constant.
Figure 4 shows (16) for m = 1,2 and 3, with u0 = σα = 2 V/m and = 0.5 V/m. Associated mean and standard deviation are obtained as μS = 8.52, 16.51, 24.49 V/m and σS = 8.01, 11.31, 13.77 V/m, respectively.
 If the internal voltage source is a pure noise source, that is, with zero mean (u0 = 0), then the expressions for fS(s) simplify considerably. For example, for volumetric dissipation (m = 3),
Corresponding pdfs for rectilinear and planar circuit elements follow in a similar manner. In the further degenerate case when = σα in addition (spherically symmetric U0 + Eα), (17) reduces to a χ82 distribution, as expected.
5.3. Multiple Noninteracting Circuit Elements Driven by Internal Deterministic Source and Immersed in Single-Source External Random Field
 The previous analyses apply to the local energy density inside a single circuit element. Consider now the generalization in which the normalized S is formed as the incoherent sum of individual Si generated inside ℓ circuit elements (i = 1,…,ℓ) (the statistical average 〈S〉 follows as a special case, through scaling of iSi by ℓ−1). These individual elements are assumed to be located sufficiently far apart with respect to the ambient wavelength, so that the E(xi) are spatially statistically homogeneous and independent among the locations xi. Furthermore, it is assumed for now that the fluctuations in one element do not affect those in any other element, or at least are negligibly small compared to the fluctuations induced by E. The lifting of the latter restriction will be discussed in section 5.4. For such a sum, the associated pdf fS(s) is then the ℓ-fold convolution of (3), (7) or (9), or a combination thereof:
up to a normalization constant. If all circuit elements are identical and connected either in series or in parallel, then all Ei are being biased by the same amount u0, whence all Si exhibit the same fSi(si). In this case, (19) reduces to a spherically biased χ2p2 pdf where p = mℓ. In other network topologies, the individual bias fields = /di may be mutually different, with their respective values being governed by the usual theorems of network analysis, obtainable from voltage or current dividers for the and = /ri, where ri is a deterministic resistance at location xi. On the other hand, on account of the spherical symmetry of fE(e), the pdf (19) remains independent of the physical orientations of the circuit elements. In other words, the bias field for the ith circuit element depends on the internally generated voltage across this element but, provided E is statistically isotropic, does not depend on the specific spatial orientation of the ith element in the circuit. In the more general case of anisotropic random E, however, the standard deviations of Si may be different for different circuit elements, because of their dependence on , whence (19) still applies in this case but with different in the expressions for the (si).
 As a first example, consider two 1-D resistors r1 and r2 (ℓ = 2, m = 1) immersed in a spherically symmetric ideal random field E, for which we obtain with the aid of (3) and (19)
up to a normalization constant. Since the number of degrees of freedom increases to 2mi when more circuit elements are added, S as characterized by (20) becomes more localized (i.e., has a smaller normalized standard deviation) compared to (3). As a second example, if r1 is 1-D but r2 is planar, and if r1 and r2 are coplanar and immersed in a statistically anisotropic field E ( ≠ ) and connected to leads extending in different directions and of that plane, then from (3), (7) and (19),
again up to a normalization constant. Figure 5 shows (20) for u01 = 2 V/m, u02 = 1 V/m, σα = 5 V/m, and (21) for u01 = 2 V/m, u02 = 1 V/m, = 5 V/m, = 3 V/m.
5.4. Interconnected Circuit Elements
 So far, the fluctuations induced by the external E in the various circuit elements have been assumed to be statistically independent. However, because of mutual coupling (via interconnections, radiative EM interaction, or both), a variation in the current or voltage for element ri located at xi and induced by a change of Eα(xi), will propagate toward other elements rj(xj) with j ≠ i. As a result, a nonlocal fluctuation is being added to each local one. Since both fluctuations originate from the same external source, the local and nonlocal fluctuations are mutually coherent.
 For conducted coupling, the effect of coupling as a result of nonlocal fluctuations induced by E in neighboring elements rj can be taken into account by attaching an additional circuit noise source (e.m.f.) to each rj in series. For radiated coupling, the effect of interaction can be accounted for by calculating the mutual impedances between r0 and rj, from which the input impedance follows and replaces rj. Thus, in general, the total field experienced by r0 is then
where U0 = E and the UjU0 [rj≠0(xj)] are the fluctuations originating from E at xj that propagate toward r0 and contribute with respective weights κj that are determined by the value of rj, its relative distance to r0, etc. Note that propagation across thin wire conductors is necessarily one-dimensional. Consequently, for conducted coupling, U0 and Uj exhibit mutually different dimensionality when m > 1, irrespective of whether the overall network and its elements have been interconnected into a rectilinear, planar or fully 3-D topology.
 Since both U0 and Uj are Gauss normal variates, their combined effect in linear networks remains Gauss normal, but with modified standard deviation. For example, for serially connected 1-D resistors with resistivities ρj that are subjected to an ideal homogeneous isotropic incoherent random field E, the induced current I = I0 + j≠0Ij = Eα/ρ0 + j≠0 (Eα/ρj) has standard deviation σI = σα. Therefore the standard deviation for the dissipated power then increases by a factor
Generally, the incorporation of interconnections and interaction between circuit elements into the analysis leaves all preceding distributions to remain applicable, but with modified parameters (namely, increased standard deviations but reduced normalized deviations) compared to those for isolated elements.
 When the impedance of the circuit elements contains a reactive component, the rate of fluctuation (i.e., the degree of fluctuation within a specified time interval) may be significantly different from that for the external field, even though the standard deviation (scale of fluctuation) itself may remain unchanged. Inductive or capacitive effects may alter the fluctuations that are induced by a given external field and coupled to the circuit element of interest. When the circuit excitation remains quasi-stationary, the calculation of distributions of the overall field, as outlined above, remains applicable provided this nonuniform averaging is accounted for using the associated variance function [Arnaut, 2001b].
6. Nonlinear Circuit Elements
 If the conduction is non-ohmic such as, for example, in circuits containing diodes, transistors, nonlinear resistors, etc., then the externally induced current and the total power density exhibit different pdfs from the expressions given above. For example, for a randomly fluctuating source voltage driving a circuit element with a current-voltage characteristic of the form i = kun (k constant) in the absence of external illumination, the pdf of the dissipated power follows in the usual manner as
up to a normalization constant. Figure 6 shows the normalized pdf for selected values of n with k = 1 A/Vn and u0 = 0.1 V/m, normalized with respect to the mean value . For linear elements (n = 1) with u0 = 0, the negative exponential distribution is of course retrieved, as is readily verified.
 In this paper, statistical distributions were derived for power dissipation in electric or electronic circuits that are subjected to an external ideal random field, for a number of canonical configurations. Practically important statistics and sample statistics, such as η% confidence intervals, expected value and uncertainty of the maximum value of the dissipated power, etc., all follow from the presented probability density functions using standard statistical methods. The above analysis shows that the probability density functions of dissipated power in low-dimensional systems (i.e., with relatively few degrees of freedom) are markedly asymmetric and may show significant deviations from the power density for an ideal unbiased random field. Nevertheless, as the effective dimension mℓ or the equivalent number of independent excitation sources q increases, fS(s) approaches Gauss normality in accordance with the central limit theorem. When the number of degrees of freedom increases, the relative uncertainty as quantified by the normalized standard deviation σS/μS decreases, suggesting an increasing accuracy of statistical characterization with increasing EM complexity of the system.
 This work was sponsored in part by the 2003–2006 Electrical Programme of the UK Department of Trade and Industry National Measurement System Policy Unit (project E03E54).