Chapter 1 Theoretical Foundations

Chapter 1

Theoretical Foundations

1.1 Maxwell’s Equations in the Frequency Domain

The classical theory of electromagnetic (EM) fields is ruled by the Maxwell equa-tions, comprising four partial differential equations (PDE) and a set of consti-tutive laws. The linear time-invariant (LTI) case is conveniently formulated inthe frequency domain (FD), using phasor notation. Hence the derivative withrespect to time t transforms to the algebraic factor jω, where j is the imaginaryunit and ω the angular frequency.

Let the magnetic field be denoted by H, the electric flux density by D, theelectric current density by J, the magnetic flux density by B, the electric fieldstrength by E, and the electric charge density by ρ. Then the governing PDEsread

curlH = jωD+ J (Ampere’s law), (1.1.1a)divB = 0 (Gauss’ law of magnetism), (1.1.1b)curlE = −jωB (Faraday’s law), (1.1.1c)divD = ρ (Coulomb’s law). (1.1.1d)

Herein, J and ρ are related by the continuity equation

div J = −jωρ. (1.1.2)

The proof is by taking the divergence of (1.1.1a) and the time-derivative of(1.1.1d).

The current density may be split into the conduction current density Jc, whichis related to E by a constitutive equation (Ohm’s law), and the impressed currentdensity Ji, which is a given excitation:

J = Jc + Ji. (1.1.3)

1.1.1

1.1.2 Antenna Theory 1 – R. Dyczij-Edlinger, WS 2016/17

Most often, the constitutive equations are formulated with the help of the fol-lowing material properties: the magnetic permeability μ, the electric permittivityε, and the electric conductivity σ:

B = μH, (1.1.4a)D = εE, (1.1.4b)Jc = σE (Ohm’s law). (1.1.4c)

Sometimes it is more convenient to use the inverse mappings of (1.1.4a) and(1.1.4c),

H = νB with ν = μ−1, (1.1.5a)E = ρJc with ρ = σ−1. (1.1.5b)

Herein, ν is called the magnetic reluctivity and ρ the electric resistivity, respec-tively. (Note that the symbols for charge density and resistivity are the same.)Since (1.1.4) and (1.1.5) are linear relations between vectors, the EM materialproperties take the form of second-order tensors. It is common use to express μ,ν, and ε by products of their respective free-space values μ0, ν0, ε0 and relativevalues μr, νr, εr, respectively:

ε = εrε0, with ε0 ≈ 8.854 187 818 · 10−12 F/m, (1.1.6a)μ = μrμ0, with μ0 = 4π · 10−7 H/m, (1.1.6b)

ν = νrν0, with ν0 =1

μ0

. (1.1.6c)

The relative material tensors are sometimes taken to be complex-valued, to ac-count for magnetization and dielectric losses:

εr = ε′ − jε′′, (1.1.7a)μr = μ′ − jμ′′, (1.1.7b)νr = ν ′ + jν ′′. (1.1.7c)

By plugging (1.1.7c) into Ampere’s law (1.1.1a), it is readily seen that ε′′ implies afrequency-proportional current density in phase with E, which is thus dissipative:

curlH = jωε0 (ε′ − jε′′)E+ σE

= jωε0ε′E+ ωε0ε

′′E+ σE. (1.1.8)

Antenna Theory 1 – R. Dyczij-Edlinger, WS 2016/17 1.2.1

1.2 Complex Power and Poynting VectorLet P denote the time-average power (D: Wirkleistung) and Q the reactive power(D: Blindleistung). By convention, positive Q means inductive power. Then thecomplex power S (D: komplexe Leistung) is given by

S = P + jQ. (1.2.1)

It can be shown [1, Sec. 2.4] [2, pp. 19 – 23] that the associated flux density, thetime-average, or complex, Poynting vector T (D: komplexer Poynting-Vektor),takes the form

T =1

2E×H∗. (1.2.2)

Hence the power flow through a surface Γ, with normal vector n, is obtained from

S = P + jQ =

∫Γ

T · ndΓ =1

2

∫Γ

(E×H∗) · ndΓ. (1.2.3)


1.3 Scalar and Vector PotentialsSince B is solenoidal (1.1.1b), a magnetic vector potential A may be defined by

curlA := B. (1.3.1)

Substituting (1.3.1) for B in (1.1.1b) yields

curl (E+ jωA) = 0. (1.3.2)

The irrotational nature of E+jωA implies to define an electric scalar potential ϕ:

− gradϕ : = E+ jωA, (1.3.3)E = −jωA− gradϕ. (1.3.4)

Plugging (1.3.1) and (1.3.4) into Ampere’s law (1.1.1a) and Coulomb’s law (1.1.1d),respectively, leads to a coupled pair of PDEs in terms of potentials:

curl(ν curlA) + (jωε+ σ)(jωA+ gradϕ) = Ji, (1.3.5a)− div [ε (jωA+ gradϕ)] = ρ. (1.3.5b)

Note that divA, which has no physical meaning, has not yet been defined.

1.3.1 Free-Space Solutions

For a given distribution of Ji and ρ in free space, or any domain of homogeneousand isotropic material properties without ohmic losses, (1.3.5) simplifies to

curl curlA− ω2εμA+ jωεμ gradϕ = μJi, (1.3.6a)−jωε divA− ε div gradϕ = ρ. (1.3.6b)

Substituting the Laplace operator Δ defined by

ΔA : = grad divA− curl curlA, (1.3.7a)Δϕ : = div gradϕ, (1.3.7b)

for the second-order differential operators in (1.3.6) leads to

−ΔA− ω2εμA+ grad (divA+ jωεμϕ) = μJi, (1.3.8a)−jωε divA− εΔϕ = ρ. (1.3.8b)

It is apparent that the Lorenz gauge

divA : = −jωμ0ε0ϕ (Lorenz gauge) (1.3.9)


will result in a pair of decoupled inhomogeneous Helmholtz equations,

−�A− ω2μεA = μJi, (1.3.10a)

−�ϕ− ω2μεϕ =1

ερ, (1.3.10b)

As shown in [1, Chapter 6], the solutions to (1.3.10) are given by

A(ra) =μ

4π

∫Ωq

Ji(rq)e−jβrqa

rqadΩ, (1.3.11a)

ϕ(ra) =1

4πε

∫Ωq

ρ(rq)e−jβrqa

rqadΩ, (1.3.11b)

wherein rqa denotes the distance from the source point rq to the observation pointra, and β is the propagation coefficient:

β(ω) = ω√με. (1.3.12)

Analogous results hold for electric surface current densities K and surfacecharge densities σi,

A(ra) =μ

4π

∫Γq

K(rq)e−jβrqa

rqadΓ, ϕ(ra) =

1

4πε

∫Γq

σ(rq)e−βrqa

rqadΓ. (1.3.13)

as well as for line currents ii and line charge densities τ :

A(ra) =μ

4π

∫sq

ii(rq)e−jβrqa

rqads, ϕ(ra) =

1

4πε

∫sq

τi(rq)e−jβrqa

rqads. (1.3.14)

From the potentials, the electromagnetic fields E and H are obtained by thedefinitions of A (1.3.1) and ϕ (1.3.3), respectively. For non-vanishing frequency,the calculation of ϕ may be bypassed by the following procedure:

B = curlA, H = μ−1B, jωD = curlH− Je, E = ε−1D. (1.3.15)

At great distance from the excitations, an even simpler procedure applies; seeSubsection 1.4.

Exercise 1.3.1. Eq. (1.3.15) provides a way of computing all EM fields solely fromA, i.e., from Ji, without ever referring to ϕ and ρ, respectively. How can this be?Why does (1.3.15) hold for ω �= 0 only?


(a) Ideal structure. (b) Realization. (c) Spherical coordinates.

Figure 1.3.1: Hertzian dipole centered about origin of coordinate system.

1.3.2 Example: Infinitesimal Electric or Hertzian Dipole

Definition 1 (Electric dipole and electric dipole moment p). A structure consist-ing of two electric point charges (−Q,+Q) of equal magnitude but opposite sign,which are displaced by a vector Δl, is called an electric dipole. By convention,Δl points from the negative to the positive charge. The electric dipole momentp (D: elektrisches Dipolmoment) is defined by

p = QΔl. (1.3.16)

Definition 2 (Hertzian dipole). The limit case of an electric dipole, with Δl → 0and QΔl = const ., is called a Hertzian dipole.

A Hertzian dipole may be realized by a closely spaced pair of small platesconnected to a current source; see Fig. 1.3.1b. In the time-harmonic case, theplates are periodically charged. Since size of the structure is considered to be ofinfinitely small, the charging current i flowing below the spheres along the dipoleaxis is quasi-stationary and, hence, independent of position:

i(t) =d

dtQ(t), i = iωQ, (1.3.17)

iΔl = iωp. (1.3.18)

For convenience, let the Hertzian dipole be centered about the origin of thecoordinate system and point in the z direction, as in Fig. 1.3.1a. Thanks toΔl → 0, the free-space solution (1.3.14) simplifies to

A(ra) =μ0

4π

∫sq

ii(rq)e−jβrqa

rqads −→ i|Δl|μ0

4π

e−jβra

raez for Δl → 0. (1.3.19)

Since there is no explicit dependence on source coordinates, let us drop the ob-servation point index a. As illustrated by Fig. 1.3.1c, conversion to spherical


q

qr a

r

qar

qa

cosq qa

r

0R

Figure 1.3.2: Simplifications in the far-field region.

coordinates (r, θ, ϕ) yields

A(r) = i|Δl|μ0

4π

e−jβr

r(er cos θ − eθ sin θ) . (1.3.20)

The EM fields are obtained from (1.3.20) via (1.3.15); see Exercise 1.3.2:

H(r) =curlA

μ0

= β2 i|Δl|e−jβr

4πeϕ

[j

βr+

1

(βr)2

]sinϑ, (1.3.21)

E(r) =curlH

jωε0=

β

ωε0β2 i|Δl|e

−jβr

4π

{er 2 cosϑ

[1

(βr)2− j

(βr)3

]+

eϑ sinϑ

[j

βr+

1

(βr)2− j

(βr)3

]}.

(1.3.22)

As expected from the rotational symmetry about the z axis of the structure, thefields are independent of ϕ.

Exercise 1.3.2. Derive the expressions for the EM fields (1.3.21), (1.3.22) from(1.3.20).

1.3.3 Simplifications at Great Distance from the Radiator

Let the region Ωq of non-vanishing impressed electric current density be containedwithin a ball of finite radius R0 about the origin of the coordinate system, asshown in Fig. 1.3.2. Provided that the observation point ra is located at a largedistance from the sources, |ra| � R0, the representation formula (1.3.11a) for the


magnetic vector potential A(ra) can be simplified greatly, by using asymptoticexpressions. The exact representation of rqa, derived by the law of cosines,

rqa = ra

√1− 2

(rqra

)cosψqa +

(rqra

)2

, (1.3.23)

implies that the denominator of (1.3.11a) may be replaced by ra,

A(ra) → μ

4πra

∫Ωq

Ji(rq)e−γrqadΩ for ra � R0. (1.3.24)

Due to the oscillating nature of the phase, the exponent needs more carefulconsideration. A power series expansion of (1.3.23) based on

√1 + x = 1 +

1

2x+

1

2 · 4x2 − 1 · 3

2 · 4 · 6x3 +− . . . for |x| ≤ 1 (1.3.25)

leads to

rqara

→ 1− rqra

cosψ +

(rqra

)2sin2 ψ

2+

(rqra

)3sin2 ψ cosψ

2+O

(r4qr4a

). (1.3.26)

Farfield Zone

In many applications, approximations rqa leading to phase errors no larger thanπ/8 rad (22.5◦) in the integrand are acceptable:

2π

λ|rqa − rqa| ≤ π

8. (1.3.27)

In the farfield (FF) or Fraunhofer zone, the observation point is so far awayfrom the radiator that the expansion may be truncated after the linear term:

rqa = ra − rq cosψ. (1.3.28)

As illustrated by Fig. 1.3.2, the geometrical interpretation is that all vectors rqaare parallel to ra. An elementary calculation shows that the maximum error inrqara

is given by 12( rqra)2 and occurs when cosψ = 1

2

rqra

; see Exercise 1.3.3. In termsof the diameter D = 2R0 of the antenna, the far-field zone is thus defined by

ra ≥ 2D

λD and ra � D. (1.3.29)

Thus the FF approximation to (1.3.11a) takes the form

A(ra) ≈ μ

4π

e−γra

ra

∫Ωq

Ji(rq)e+γrq cosψqadΩ for ra ≥ 2

D

λD and ra � D. (1.3.30)


Table 1.3.1: Field Regions for a Radiator of Diameter D.

Region Distance ra to observation pointMinimum Maximum

Reactive nearfield 0 min{0.62

√DλD, 2D

λD}

Radiating nearfeld∗ 0.62√

DλD 2DD

λ

Farfield 2DDλ

∞∗ For D > 1

18

√3λ (D > 0.096λ); see Exercise 1.3.5.

Radiating Nearfield Zone

In the radiating near-field zone or Fesnel zone the expansion is truncated after thequadratic term. The inner boundary of the Fresnel zone may, again, be definedby that distance for which the maximum phase error becomes π/8 rad (22.5◦).The phase error is estimated from the maximum of the cubic term, which is theleading term to be omitted from the expansion. An elementary calculation, whichis detailed in Exercise 1.3.4, leads to

ra ≥√2√3

3

√D

λD with

√2√3

3≈ 0.62. (1.3.31)

Thus the approximation to (1.3.11a) reads

A(ra) ≈ μ

4π

e−γra

ra

∫Ωq

Ji(rq)e+γrq(cosψqa− 1

2

rqra

sin2 ψqa)dΩ

for 0.62

√D

λD ≤ ra < 2

D

λD and ra � D. (1.3.32)

The immediate neighborhood of the excitations is called the reactive near-field. It is characterized by localized energy storage, as in the stationary case,and the EM fields are dominated by terms that decay faster than r−1.

The main results are summarized in Table 1.3.1.

Exercise 1.3.3. Calculate the inner boundary of the Fraunhofer zone (1.3.29).

Exercise 1.3.4. Derive the estimate (1.3.31) for the inner boundary of the Fresnelzone. Hint: Consider the cubic term in the power series expansion (1.3.26) only.

Exercise 1.3.5. It is often stated that "for electrically small antennas, the radi-ating near-field zone may not exist." More specifically, (1.3.32) appears to implythat, for D ≤ 1

18

√3, the inner radius of the Fresnel zone becomes larger than the

outer one. Please explain.


1.4 The Far-Field RegimeFor lossless media, the Fraunhofer approximation (1.3.30) simplifies to

A(ra) → μ0

4π

e−jβ|ra|

|ra|∫Ωq

JE(rq)ejβ|rq | cosψqadΩ in the FF region. (1.4.1)

Definition 3 (Radiation vector N (D: Strahlungsvektor)). The radiation vec-tor N is defined by

N(ϑa, ϕa) :=

∫Ωq

JE(rq)ejβ|rq | cosψqadΩ. (1.4.2)

The radiation vector N depends on the look angles (ϑa, ϕa) only, i.e., theangular coordinates of the observer. We thus have

A(ra, ϑa, ϕa) → μ0

4π

e−jβra

ra︸︷︷︸homogeneousspherical wave

N(ϑa, ϕa) for ra � R0. (1.4.3)

The factor in front of N describes a homogeneous spherical wave originatingfrom the center of the coordinate system, because the surfaces of equal phase arespheres (hence: spherical wave) of angle-independent magnitude (hence: homo-geneous).

1.4.1 Electromagnetic Fields

For brevity, let us drop the index a. Using spherical coordinates, the gradient ofthe spherical wave exp (−jβr)/r reads

∇e−jβr

r=

(er

∂

∂r+

eϑr

∂

∂ϑ+

eϕr sinϑ

∂

∂ϕ

)e−jβr

r= er

(−jβ

e−jβr

r− e−jβr

r2

),

(1.4.4)

and the curl of a vector field v(ϑ, ϕ) takes the form

∇× v(ϑ, ϕ) =1

r2 sinϑ

∣∣∣∣∣∣er reϑ r sinϑeϕ∂∂r

∂∂ϑ

∂∂ϕ

vr(ϑ, ϕ) rvϑ(ϑ, ϕ) r sinϑvϕ(ϑ, ϕ)

∣∣∣∣∣∣ .The respective asymptotic limits for r → ∞ are given by

∇e−jβr

r→ −jβ

e−jβr

rer for r → ∞, (1.4.5)

∇× v(ϑ, ϕ) = O(1

r) for r → ∞. (1.4.6)


Thus we obtain the following approximation for the magnetic flux density:

B = ∇×A → ∇× μ0

4π

e−jβr

rN(ϑ, ϕ) (1.4.7a)

=μ0

4π

(e−jβr

r

O( 1r)︷︸︸︷

∇×N(ϑ, ϕ)︸︷︷︸O( 1

r2)

+∇e−jβr

r︸︷︷︸→−jβ e−jβr

rer

×N(ϑ, ϕ)

), (1.4.7b)

B → −jβer × μ0

4π

e−jβr

rN(ϑ, ϕ) = −jβer ×A for r → ∞. (1.4.7c)

Accordingly, we have for H:

H =1

μ0

B → −jβ

μ0

er ×A. (1.4.8)

The displacement current density jωD is obtained from the Gauss law,

jωD = ∇×H → ∇× μ0

4π

e−jβr

r

(−jω

√ε0μ0

er ×N(ϑ, ϕ)

), (1.4.9)

jωD → ∇× μ0

4π

e−jβr

rN(ϑ, ϕ) (1.4.10)

with

N(ϑ, ϕ) = −jω

√ε0μ0

er ×N(ϑ, ϕ). (1.4.11)

By analogy to (1.4.7), we have

jωD → −jβer × μ0

4π

e−jβr

rN(ϑ, ϕ) = −jβer ×

[−jω

√ε0μ0

er × μ0

4π

e−jβr

rN(ϑ, ϕ)

]

D → jβ

η0er × (er ×A) =

jω√μ0ε0√

μ0/ε0er × (er ×A),

D → jωε0er × (er ×A), (1.4.12)E → jω er × (er ×A). (1.4.13)

By denoting transverse components by (·)T , specifically

AT := (0, Aϑ, Aϕ) in the far-field region, (1.4.14)

the Grassmann expansion theorem

A× (B× c) ≡ B(A · c)− c(A ·B) (Grassmann) (1.4.15)


leads to

E → ET , ET = −jωAT , (1.4.16a)

H → HT , HT =1

η0er × E, (1.4.16b)

wherein η0 is the characteristic impedance (D: charakteristische Impedanz) offree space, given by

η0 =

√μ0

ε0≈ 376.7 Ω. (1.4.17)

Moreover, we have

E,H ⊥ r, E ⊥ H. (1.4.18)

Eqs. (1.4.16) – (1.4.18) express the well-known property that the EM fields in thefarfield zone are described locally by planar TEM waves propagating in the radialdirection. Moreover, the FF representation of A (1.4.3) together with(1.4.16)imply that the fields decay proportionally to r−1:

|E| ∝ 1

r, |H| ∝ 1

rin the FF region. (1.4.19)

1.4.2 Total Radiated Power and Radiation Intensity

According to (1.4.16), E and H are in phase, possess transverse components only,and are perpendicular to another. In consequence, the Poynting vector becomesreal-valued and has a radial component Tr only:

T =1

2E×H∗ =

1

2E×

(1

η0er × E∗

)=

1

2η0

[er(E · E∗)− E∗

=0 (1.4.18)︷︸︸︷(E · r) ], (1.4.20)

T = erT r with Tr =1

2η0‖E‖2 er = η0

2‖H‖2 er ∈ R. (1.4.21)

Eq. (1.4.19) implies that

Tr ∝ 1

r2in the FF region. (1.4.22)

The total power Prad radiated by a bounded structure is evaluated best by en-closing it by a hull ∂Ω that is entirely located in the FF region. Using sphericalcoordinates for integrating the Poynting vector yields

Prad =

π∫θ=0

2π∫φ=0

Trr2 sin θ dθ dφ =

1

2η

π∫θ=0

2π∫φ=0

(|| E||r

)2

sin θ dθ dφ. (1.4.23)

Note that the product sin θ dθ dφ is just an infinitesimal element of solid angle Ω:

dΩ := sin θ dθ dφ. (1.4.24)


Definition 4 (Radiation intensity). The radiation intensity U is defined by thepower per solid angle Ω, in steradiant, for the limiting case of an infinitesimallysmall element dΩ:

U(θ, φ) = limΔΩ→0

ΔP

ΔΩ

∣∣∣∣(θ,φ)

=dP

dΩ

∣∣∣∣(θ,φ)

. (1.4.25)

We have

U = r2Tr, (1.4.26)

Prad =

∮Ω

U dΩ. (1.4.27)

1.4.3 Example: Farfields of Hertzian Dipole

We revisit the Hertzian dipole of Fig. 1.3.1a. Obviously, the farfields may beobtained by dropping all field components that decay faster than 1/r in (1.3.21)and (1.3.22). A direct way starts from the FF approximation of A (1.4.1). For ashort path Δlez of constant line current i, it simplifies to

A(ra) → μ0

4π

e−jβ|ra|

|ra|∫Ωq

JE(rq)ejβ|rq | cosψqadΩ → μ0

4π

e−jβ|ra|

|ra| i|Δl|ez, (1.4.28)

which happens to be the same as the exact solution. The fields follow directlyfrom (1.4.16). To identify the transverse components of A, let us employ sphericalcoordinates and, for brevity, drop the index a:

A(r) → i|Δl|μ0

4π

e−jβr

r(er cosϑ− eϑ sinϑ) , (1.4.29)

Thus,

AT → −i|Δl|μ0

4π

e−jβr

reϑ sinϑ, (1.4.30)

E → −jωAT , Eωμ0=βη0−−−−−→ η0jβi|Δl|e

−jβr

4πrsinϑ eϑ, (1.4.31)

H → 1

η0er × E, H

er×eϑ=eϕ−−−−−−→ jβi|Δl|e−jβr

4πrsinϑ eϕ. (1.4.32)

By expressing β in terms of the wavelength λ,

Eβ= 2π

λ−−−→ η0ji|Δl|λ

e−jβr

2rsinϑ eϑ, (1.4.33)

Hβ= 2π

λ−−−→ ji|Δl|λ

e−jβr

2rsinϑ eϕ, (1.4.34)


it can be seen that the EM fields are proportional to the electric size of the dipole.The corresponding Pointing vector and radiation intensity are given by

T = erTr = er1

2η0‖E‖2 = er

1

r2|i|22

η04

( |Δl|λ

)2

sin2 ϑ, (1.4.35)

U = r2Tr =|i|22

η04

( |Δl|λ

)2

sin2 ϑ, (1.4.36)

and the total radiated power becomes

Prad =

π∫ϑ=0

2π∫ϕ=0

U sinϑdϑdϕ =|i|22

η04

( |Δl|λ

)2

2π

π∫ϑ=0

sin3 ϑ︸︷︷︸(1−cos2 ϑ) sinϑ

dϑ

=|i|22

η0π

2

( |Δl|λ

)2 [− cosϑ+

1

3cos3 ϑ

]π0

=|i|22

2π

3η0

( |Δl|λ

)2

. (1.4.37)

The radiated power is seen to be proportional to the square of the dipole length.


1.5 Fictitious Magnetic Charges and CurrentsEven though magnetic charges and currents have not been observed physically,they will turn out to be very useful in the context of the Equivalence Principle.

Let ρM

denote the magnetic charge density (D: magnetische Ladungsdichte)and JM the magnetic current density (D: magnetische Stromdichte). Then thetime-harmonic Maxwell equations for electric and magnetic excitations, also in-dicated by (·)E and (·)M , respectively, are given by

curlHE − jωDE = JE, curl(−EM)− jωBM = JM , (1.5.1a)divDE = ρ

E, divBM = ρ

M, (1.5.1b)

curlEE + jωBE = 0, curlHM + jω(−DM) = 0, (1.5.1c)divBE = 0, divDM = 0. (1.5.1d)

Note that the equations for magnetic excitations have been reordered and refor-mulated such as to achieve perfect correspondence with the column on the left.Hence the two sytems are dual, with

HE ↔ −EM , DE ↔ BM , JE ↔ JM ,

BE ↔ −DM , EE ↔ HM , ρE ↔ ρM . (1.5.2a)

In consequence, the material properties need to be exchanged, too:

εE ↔ μH , μE ↔ εH . (1.5.2b)

Duality enables any result obtained for electric excitations to be transferred tothe case of magnetic excitations by applying the exchange rules (1.5.2). Whenboth electric and magnetic excitations are present, the total field is obtained asthe sum of the partial solutions, thanks to linearity:

E = EE + EM , B = BE +BM ,

D = DE +DM , H = HE +HM . (1.5.3)

1.5.1 Interface Conditions

We consider a smooth interface Γ12 with unit normal vector n12 pointing fromRegion 1 to Region 2, electric surface charge density σE, magnetic surface chargedensity σM (D: magnetische Flächenladungsdichte), electric surface current den-sity KE, and magnetic surface current density KM (D: magnetische Flächen-stromdichte). By duality, the interface conditions for the EM fields [3, Sec. 1.8]are given by

n12 · (D2 −D1) = n12 · (D2E −D1E) = σE, (1.5.4a)n12 · (B2 −B1) = n12 · (B2M −B1M) = σM , (1.5.4b)


as well as [3, Sec. 5.6]

n12 × (H2 −H1) = n12 × (H2E −H1E) = +KE, (1.5.5a)n12 × (E2 − E1) = n12 × (E2M − E1M) = −KM . (1.5.5b)

1.5.2 Scalar and Vector Potentials

Similarly to the pair of potentials (A, ϕ) in case of electric excitations, (1.5.1d)and (1.5.1c) allow to introduce a magnetic scalar potential ψ (D: magnetischesSkalarpotenzial) and an electric vector potential F (D: elektrisches Vektorpoten-zial) for the case of magnetic excitations:

BE =: curlA,(1.5.2)←→ −DM =: curlF, (1.5.6)

EE + jωA =: − gradϕ,(1.5.2)←→ HM + jωF =: − gradψ. (1.5.7)

Thus duality carries over to the potentials,

A ↔ F, ϕ ↔ ψ. (1.5.8a)

In free-space, the Lorenz gauge leads to the Helmholtz equations

−�A− ω2μ0ε0A = μ0JE,(1.5.2)←→ −�F+ ω2μ0ε0F = ε0JM , (1.5.9)

−�ϕ− ω2μ0ε0ϕ =1

ε0ρE,

(1.5.2)←→ −�ψ − ω2μ0ε0ψ =1

μ0

ρM , (1.5.10)

and the solutions

A(ra) =μ0

4π

∫Ωq

JE(rq)e−jβrqa

rqadΩ, F(ra) =

ε04π

∫Ωq

JM(rq)e−jβrqa

rqadΩ, (1.5.11)

ϕ(ra) =1

4πε0

∫Ωq

ρE(rq)e−jβrqa

rqadΩ, ψ(ra) =

1

4πμ0

∫Ωq

ρM(rq)e

−jβrqa

rqadΩ. (1.5.12)

1.5.3 Far-Field Approximations

By duality, the FF approximation for the magnetic vector potential (1.4.1) yields

F(ra) → ε04π

e−jβ|ra|

|ra|∫Ωq

JM(rq)ejβ|rq | cosψqadΩ, (1.5.13)

and the fields approximations (1.4.16) result in

E = EE + EM → −jωAT + jωη0er × F, (1.5.14)

H = HM +HE → −jωFT − jω1

η0er ×A. (1.5.15)

The fields still form a locally plane TEM wave travelling in the radial direction:

E · er = 0, H · er = 0, ηH = er × E. (1.5.16)


(a) Configuration above conducting half-space. (b) Configuration and its image in free space.

Figure 1.6.1: Method of images: Notice the polarities of the image charges andthe directions of the image currents.

1.6 The Method of ImagesEqs. (1.5.11) and (1.5.12) provide the retarded potentials for charge and currentdistributions in infinite free space. However, these equations do not apply toconfigurations above an electrically conducting half-space (z ≤ 0), as sketchedin Fig. 1.6.1a. As a remedy, the original structure may be transformed to aconfiguration embedded in free space by considering the interface (z = 0) as asymmetry plane; see Fig. 1.6.1b.

The BCs on the surface of the perfect electric conductor (PEC) are given by

ez × E = 0, vanishing tangential components, (1.6.1a)ez ·B = 0, for ω > 0, vanishing normal component. (1.6.1b)

When the conducting half-space is replaced by the images of the original materialregions and excitations, the EM fields in the upper half-space will remained thesame provided that the image charges and currents are chosen as follows:

Electric charges: ρE(−z) = −ρE(z), (1.6.2a)Electric currents: ez · JE(−z) = +ez · JE(z), (1.6.2b)

ez × JE(−z) = −ez × JE(z), (1.6.2c)Magnetic charges: ρM(−z) = +ρM(z), (1.6.2d)Magnetic currents: ez · JM(−z) = −ez · JM(z), (1.6.2e)

ez × JM(−z) = +ez × JM(z). (1.6.2f)

If the material properties are homogeneous in the entire domain, (1.5.11) and(1.5.12) will apply; else, the equivalence principle of Sec. 1.8 may help. Whenquantities like energy or radiated power are computed, one ought to bear in mindthat only one half of each value is related to the upper half space.


1.7 Reciprocity and Reactions

For the same domain and material distributions, we consider two different exci-tations, a and b.

curlHa = jωεEa + Ja, curlHb = jωεEb + Jb, (1.7.1)− curlEa = jωμHa +Ma, − curlEb = jωμHb +Mb. (1.7.2)

The following procedure is similar to the derivation of the Poynting Theorembut for the conjugate transpose:

Eb · curlHa = jωEb · εEa + Eb · Ja, (1.7.3)−Ha · curlEb = jωHa · μHb +Ha ·Mb. (1.7.4)

Thanks to

div (Eb ×Ha) = ∇ · (Eb ×Ha)

= Ha · (∇× Eb)− Eb · (∇×Ha)

= Ha · curlEb − Eb · curlHa, (1.7.5)

we have

− div (Eb ×Ha) = jωEb · εEa + jωHa · μHb + Eb · Ja +Ha ·Mb. (1.7.6)

Replacing the roles of the two configurations leads to

− div (Ea ×Hb) = jωEa · εEb + jωHb · μHa + Ea · Jb +Hb ·Ma. (1.7.7)

Subtraction yields

div (Ea ×Hb − Eb ×Ha)

= +jωEb · εEa + jωHa · μHb + Eb · Ja +Ha ·Mb

− jωEa · εEb − jωHb · μHa − Ea · Jb −Hb ·Ma. (1.7.8)

Provided that the material tensors ε and μ are symmetric,

ε = εT , (1.7.9)μ = μT , (1.7.10)

we have

Eb · εEa = Ea · εEb, (1.7.11)Hb · μHa = Ha · μHb, (1.7.12)


and (1.7.8) reduces to

div (Ea ×Hb − Eb ×Ha) = Eb · Ja − Ea · Jb +Ha ·Mb −Hb ·Ma. (1.7.13)

Applying the Gauss Theorem to (1.7.19) leads to the integral representation∮∂Ω

(Ea ×Hb − Eb ×Ha) · ndΓ =

∫Ω

(Eb · Ja − Ea · Jb +Ha ·Mb −Hb ·Ma) dΩ.

(1.7.14)

Let the boundary ∂Ω consist of three complementary components: a PEC part ΓE,a part ΓH which is a perfect magnetic conductor (PMC), and an impedanceboundary ΓZ of surface impedance Z�. Then the corresponding BCs read

n× E = 0 on ΓE, (1.7.15a)n×H = 0 on ΓH , (1.7.15b)n× E = n× Z�(H× n) on ΓZ . (1.7.15c)

Eqs. (1.7.15a) and (1.7.15b) imply that the integral on the left-hand side of(1.7.14) vanishes on ΓE and ΓH . On ΓZ we have

(Ea ×Hb) · n(1.7.15c)= {[n× Z�(Ha × n)]×Hb)} · n= (Hb × n) · Z�(Ha × n), (1.7.16)

(Ea ×Hb − Eb ×Ha) · n = (Hb × n) · Z�(Ha × n)

− (Ha × n) · Z�(Hb × n). (1.7.17)

Provided that Z� is (complex) symmetric, (1.7.17) yields zero. In consequence,the left-hand side of (1.7.14) vanishes, and we arrive at the main result∫

Ω

(Ea · Jb −Ha ·Mb) dΩ =

∫Ω

(Eb · Ja −Hb ·Ma) dΩ. (1.7.18)

The impedance relation for waves in the FF (1.4.16) shows that the BC (1.7.15c)also includes radiation into free space, using a sphere of radius R for ΓZ , withR → ∞ and Z� = η0.

1.7.1 The Lorentz Reciprocity Theorem

In the special case of domain free of excitations, (1.7.13) reduces to

div (Ea ×Hb) = div (Eb ×Ha) for J ≡ 0, M ≡ 0. (1.7.19)

Applying the Gauss Theorem leads to the integral representation∮∂Ω

(Ea ×Hb) · dΓ =

∮∂Ω

(Eb ×Ha·) dΓ. (1.7.20)


Definition 5 (Reaction). The reaction of a field (E,H)a to an excitation (J,M)b,abbreviated by 〈a, b〉, is defined by

〈a, b〉 : =∫Ω

(Ea · Jb −Ha ·Mb) dΩ. (1.7.21)

Hence the reciprocity theorem (1.7.18) may be stated in compact form as

〈a, b〉 = 〈b, a〉. (1.7.22)

1.7.2 Example: Electric Current Sheet on PEC Surface

Consider a reciprocal configuration in a domain Ω with non-vanishing electricboundary ΓE. Let the excitation a by given by a localized unit current in sometangential direction ta at some location ra ∈ ΓE, and the excitation b by a local-ized unit current in some direction eb at some interior point rb. The correspondingsurface and volume current densities Ka and Jb are given by

Ka(ra) = ta δ(r− ra), with ra ∈ ΓE, (1.7.23a)Jb(rb) = eb δ(r− rb), with rb ∈ Ω. (1.7.23b)

Herein δ is the Kronecker delta. By (1.7.23), the reactions (1.7.21) simplify to

〈a, b〉 =∫Ω

Ea(r) · eb δ(r− rb)dΩ = Ea(rb) · eb, (1.7.24)

〈b, a〉 =∫Ω

Eb(r) · ta δ(r− ra)dΩ = Eb(ra) · ta. (1.7.25)

Since Eb satisfies the BC

n× Eb = 0 ⇐⇒ Eb · ta = 0 on ΓE, (1.7.26)

the reaction 〈b, a〉 vanishes by (1.7.25),

〈b, a〉 (1.7.25)= Eb · ta

(1.7.26)= 0. (1.7.27)

Thanks to reciprocity (1.7.22), the same holds true for 〈a, b〉:

〈a, b〉 (1.7.24)= Ea(rb) · eb

(1.7.22)=

(1.7.27)0 ∀eb, ∀rb ∈ Ω. (1.7.28)

Since both the direction eb and the location rb have been arbitrary, Ea mustvanish everywhere in Ω: A sheet of electric current on a PEC surface does notproduce any field; the electric current is short-circuited by the PEC underneath[4, p. 71] [5, p. 255].Exercise 1.7.1 (Magnetic current sheet on PMC surface). Assuming a reciprocalsetting, show that a magnetic current sheet on a PMC surface produces no field.


(a) Current source. (b) Voltage source. (c) Z matrix. (d) Excitation.

Figure 1.7.1: Field models for lumped sources.

1.7.3 Reciprocal Networks

An ideal lumped current source is modeled by an impressed current i across anelectrically small gap between the terminals of two thin wires; see Fig. 1.7.1a.

Let a denote the fields due to some excitation and b denote the excitation atthe considered terminal. Then

〈a, b〉 =∫Ω

Ea · JbdΩ =

∫ 1

0

Ea · ds∫Γ

Jb · dΓ,〈a, b〉 = −ubaib. (1.7.29)

Let us describe a multi-port network with the help of the impedance matrix Z:⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

u1...ub......uN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Z11 · · · · · · · · · · · · Z1N...

...Zb1 · · · · · · Zba · · · ZbN...

......

...ZN1 · · · · · · · · · · · · Z11

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

i1......ia...iN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦. (1.7.30)

When a single current ia is excited, we have

uba = Zbaia, (1.7.31)〈a, b〉 = −Zbaiaib, (1.7.32)

Zba = −〈a, b〉iaib

. (1.7.33)

Thus the entries of the Z matrix are obtained from reactions and, for reciprocalnetworks, we have

〈a, b〉 = 〈b, a〉 =⇒ Zba = Zab, (1.7.34)

i.e., the Z matrix is complex symmetric.


1.7.4 Stationarity

In practice, the exact distributions of the fields and/or excitations a0, b0 are usu-ally not known. However, it is often possible to find approximations for whichthe error in the reaction is of higher order. Let δa and δb be some variations inconfiguration and εa, εb some scaling coefficients. Then the approximations a(ε)and b(ε) take the form

a(εa) = a0 + εa δa, (1.7.35)b(εb) = b0 + εb δb, (1.7.36)

and we have

〈a, b〉(εa, εb) = 〈a0, b0〉+ εa〈δa, b0〉+ εb〈a0, δb〉+ εaεb〈δa, δb〉. (1.7.37)

To make the errors of first order vanish, we require that the first partial derivativesof 〈a, b〉 with respect to εa and εb be zero at the exact solution.

Definition 6 (Stationarity). A bilinear form 〈a, b〉(εa, εb) is stationary (D: sta-tionär) at (εa, εb) = (0, 0), iff

∂〈a, b〉∂εa

∣∣∣∣εa=0

!= 0 and

∂〈a, b〉∂εa

∣∣∣∣εa=0

!= 0. (1.7.38)

By (1.7.37) and (1.7.36), 〈a, b〉 becomes stationary if

〈δa, b0〉 = 0 and 〈a0, δb〉 = 0, (1.7.39)

or, equivalently,

〈a, b0〉 = 〈a0, b0〉 and 〈a0, b〉 = 〈a0, b0〉. (1.7.40)

The conditions above may be interpreted as follows:

• Reactions are stationary with respect to field errors orthogonal to or awayfrom the exact excitations.

• Reactions are stationary with respect to excitation errors orthogonal to theexact fields.

By (1.7.37), we have

〈a, b〉(εa, εb) = 〈a0, b0〉+ εaεb〈δa, δb〉. (1.7.41)


(a) Original configuration. (b) Equiv. interior problem. (c) Equiv. exterior problem.

Figure 1.8.1: The equivalence principle.

1.8 The Vector Huygens PrincipleConsider a domain of inhomogeneous material properties and source distribution.Let the domain be partitioned into two complementary subdomains Ω1 and Ω2

sharing a common interface Γ12, as shown in Fig. 1.8.1a.We compare the original configuration with an equivalent interior problem

after Fig. 1.8.1b, where all excitations and fields inside the outer domain Ω2 shallvanish, whereas those inside Ω1 shall remain unchanged. The assumptions abovelead to discontinuities at the interface Γ12, specifically in the normal componentsof D and B as well as in the tangential components of E and H. They may beinterpreted as equivelent electric and magnetic surface charges

σE = −n12 ·D1, (1.8.1a)σM = −n12 ·B1, (1.8.1b)

and surface currents

KE = −n12 ×H1, (1.8.1c)KM = +n12 × E1. (1.8.1d)

Similarly, one may set up an equivalent exterior problem according to Fig. 1.8.1c,featuring unaltered excitations and fields in Ω2 and vanishing fields in Ω1. Toobtain E1 ≡ 0 and H1 ≡ 0, all excitations within Ω1 need to be replaced byequivalent surface charges

σE = n12 ·D2, (1.8.2a)σM = n12 ·B2 (1.8.2b)

and equivalent surface currents

KE = +n12 ×H2, (1.8.2c)KM = −n12 × E2. (1.8.2d)


(a) PEC wire in free space. (b) Equivalent sources in free space.

Figure 1.8.2: Equivalence principle applied to current-carrying PEC wire.

The equivalent sources in (1.8.1) and (1.8.2) are called Huygens sources, andthe underlying equivalence principle is called the Vector Huygens Principle.

Since the material properties inside the domain of vanishing fields have noeffect whatsoever, they may be modified arbitrarily. In practice, they are chosensuch as to make the equivalent configuration easier to solve:

1.8.1 Love’s Equivalence Principle

This describes the case when the material properties in the domain that remainsunchanged are homogeneous, and the material in the domain whose fields aremade vanish is chosen to be the same. Hence the characteristics of Love’s equiv-alence principle are as follows:

1. The resulting equivalent problem is for excitations in unbounded homoge-neous medium. Solutions are readily available from (1.5.11) and (1.5.12).

2. Both electric and magnetic equivalent sources are necessary to satisfy theBCs on Γ12. They are given by (1.8.1) or (1.8.2), respectively.

Thanks to Property 1, Love’s equivalence is widely applicable. It is often usedfor structures radiating into free space.

1.8.2 Example: Equivalent Sources of Wire Antennas

Fig. 1.8.2a shows a perfectly conducting wire with surface current density KE,embedded in free space. What are equivalent sources in free space such that the


EM fields outside the wire remain unchanged?Since the fields inside the PEC wire vanish, the equivalent sources are readily

determined from the jumps of Htan and Etan in the original configuration:

KequivE = n12 ×H2 = KE, (1.8.3)

KequivM = −n12 × E2 = 0. (1.8.4)

Thus, for a PEC wire of known surface current density KE, the EM fields in theexterior domain may be calculated by considering the same current distributionin infinite free space; see Fig. 1.8.2b.

1.8.3 Schelkunoff’s Equivalence Principle: PEC or PMC

It is known from Sec. 1.7.2 and Exercise 1.7.1 that

• Equivalent electric surface currents on a PEC surface of arbitrary shape donot produce any fields. They are short-circuited by the PEC.

• Equivalent magnetic surface currents on a PMC surface of arbitrary shapedo not produce any fields. They are short-circuited by the PMC.

The Schelkunoff equivalence principle [6] covers the case when the materialproperties in the domain that remains unchanged are homogeneous, and thematerial in the domain whose fields are made vanish are replaced by PEC (PMC).Its advantage is that only a single type of equivalent current, magnetic on PECor electric on PMC, needs to be considered. On the downside, the applicabilityof the Schelkunoff equivalence principle is often limited by the fact that solutionsfor domains with PEC (PMC) boundaries of general shape are not available.

1.8.4 Example: Aperture in Conducting Ground Plane

A particulaly instructive application of the Schelkunoff equivalence principle isradiation through an aperture in a conducting ground plane into free-space, asin Fig. 1.8.3a.

When the fields in Ω1 are set to zero, those in Ω2 will remain unchanged pro-vided that equivalent surface currents according to (1.8.2) are introduced on Γ12.The material in Ω1 may then be chosen to be PEC; see Fig. 1.8.3b. Applying theMethod of Images of Section 1.6 leads to an equivalent configuration in free space,which is shown in Fig. 1.8.3c. Since the image of KM is in the same direction asthe original distribution, the net magnetic surface current density will double. Incontrast, the image of KM is in the opposite direction as the original distribution,so that the total electric current vanishes; as illustrated by Fig. 1.8.3d:

KM = −2n12 × Etan,KE = 0,

}in free-space configuration. (1.8.5)


(a) Original problem. (b) Huygens sources on PEC.

(c) Equiv. free-space config.. (d) Simplified equiv. currents. (e) Simpl. currents over PEC.

Figure 1.8.3: Illustration of Schelkunoff’s equivalence principle, using a PECbody and image theory.

Applying image theory one more time, the configuration of Fig. 1.8.3d is reducedto a sheet of half the magnetic current density on the surface a PEC half-space;see Fig. 1.8.3e. The comparison of Fig. 1.8.3b and Fig. 1.8.3e illustrates thatequivalent electric surface currents on a planar, infinite PEC surface are of noeffect; they are short-circuited by the PEC.

Exercise 1.8.1. Consider a planar surface Γ in free space which is partitioned intotwo complementary parts ΓE and ΓH (ΓE ∩ ΓH = ∅,ΓE ∪ ΓH = Γ).

1. Assuming some electric surface current density KE on ΓE, show that themagnetic field H has vanishing tangential components on ΓH .

2. Assuming some magnetic surface current density KM on ΓM , show that theelectric field E has vanishing tangential components on ΓE.


(a) Free-space. (b) PEC screen with aperture. (c) Complementary PMC screen.

Figure 1.9.1: Babinet’s principle, relating to another the fields due to comple-mentary planar PEC and PMC screens.

1.9 Babinet’s Principle

It was observed by the French physician Jacques Babinet that the optical diffrac-tion patterns of complementary screens are essentially the same.

Given the following three configurations in infinite free space:

• Setting 0 comprises the excitations JE and JM in the lower half-space,z < 0; see Fig. 1.9.1a. The resulting fields, E0 and H0, are called theincident fields.

• Setting 1 comprises the same excitations, JE and JM , plus a planar PECscreen ΓE of infinite extent with an aperture, in the x-y plane; see Fig. 1.9.1b.The corresponding fields are denoted by E1 and H1.

• Setting 2 comprises the same excitations, JE and JM , plus a planar PMCscatterer ΓM which is complementary to the screen of Setting 1; see Fig. 1.9.1c.The corresponding fields are denoted by E2 and H2.

Then the incident fields in the upper half-space are obtained by summing thefields in presence of the electric and the magnetic screen, respectively [7], [2,pp. 365–371] [8, pp. 138–140]:

E0 = E1 + E2 for z > 0, (1.9.1)H0 = H1 +H2 for z > 0. (1.9.2)

To prove Babinet’s principle, let us start from a BVP that uniquely determinesthe EM fields in the upper half-space. The PDEs take the form

curlH− jωεE = 0 for z > 0, (1.9.3a)curlE+ jωμH = 0 for z > 0, (1.9.3b)


and the BCs to be imposed are

ez ×H = given on ΓH , (1.9.4a)ez × E = given on ΓE. (1.9.4b)

The goal is to express the BCs for Setting 1 and Setting 2 in terms of the incidentfields of Setting 0. For this purpose, the total fields of Settings 1 and 2 aredecomposed into the incident fields and the respective scattered fields Es,Hs:

E1 = E0 + Es1, H1 = H0 +Hs

1, (1.9.5)E2 = E0 + Es

2, H2 = H0 +Hs2. (1.9.6)

The effects of the electric screen in Setting 1 are also obtained from an equivalentelectric current distribution on ΓE in free space. They provide the sole excitationfor the scattered fields (Es

1,Hs1). Thus Hs

1 has no tangential components on ΓH ;see Exercise 1.8.1, and the EM fields on the x-y plane satisfy

ez ×Hs1 = 0

(1.9.5)⇐⇒ ez ×H1 = ez ×H0 on ΓH , (1.9.7)PEC: ez × E1 = 0 on ΓE. (1.9.8)

Hence the BVP for Setting 1 is given by the PDEs (1.9.3) subject to the BCs

ez ×H1 = ez ×H0 on ΓH , (1.9.9a)ez × E1 = 0 on ΓE. (1.9.9b)

By similar reasoning, the effects of the magnetic screen in Setting 2 lead to

PMC: ez ×H2 = 0 on ΓH , (1.9.10)

ez × Es2 = 0

(1.9.6)⇐⇒ ez × E2 = ez × E0 on ΓE, (1.9.11)

so that the BVP for Setting 2 is given by the PDEs (1.9.3) subject to the BCs

ez ×H2 = 0 on ΓH , (1.9.12a)ez × E2 = ez × E0 on ΓE. (1.9.12b)

By linearity and homogeneity of (1.9.3), the sum fields in the upper half-spacesatisfy the PDE (1.9.3). The respective BCs are

ez ×H = ez ×H0 on ΓH , (1.9.13)ez × E = ez × E0 on ΓE. (1.9.14)

The uniqueness of the solution implies the sought result (1.9.2).Exercise 1.9.1. Is Babinet’s principle, specifically (1.9.2), also valid in the lowerhalf-space? (Hint: Write down the BVP for each setting and compare.)

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