Lecture 30

Reciprocity Theorem

30.1 Reciprocity Theorem

Figure 30.1: The geometry for proving reciprocity theorem. We perform two experiments:

With sources J1 and M1 turned on, generating fields E1 and H1, and J2 and M2 turned off,

and vice versa.

Reciprocity theorem is one of the most important theorems in electromagnetics. With it we

can develop physical intuition to ascertain if a certain design or experiment is wrong. It

also tells us what is possible and what is impossible in design of many systems. Reciprocity

theorem is like “tit-for-tat” relationship in humans: Good-will is reciprocated with good will

while ill-will is reciprocated with ill-will. Not exactly: In in electromagnetics, this relationship

can be expressed precisely and succinctly using mathematics. We shall see how this is done.

Consider a general anisotropic inhomogeneous medium where both μ(r) and ε(r) are

described by permeability tensor and permittivity tensor over a finite part of space as shown in

Figure 30.1. This representation of the medium is quite general, and it can include conductive

299

300 Electromagnetic Field Theory

media as well. It can represent complex terrain as well as complicated electronic circuit

structures in circuit boards or microchips, and complicated antenna structures.

When only J1 and M1 are turned on, they generate fields E1 and H1 in this medium.

On the other hand, when only J2 and M2 are turned on, they generate E2 and H2 in this

medium. Therefore, the pertinent equations, for linear time-invariant systems, for these two

cases are1

∇ × E1 = −jωμ · H1 − M1 (30.1.1)

∇ × H1 = jωε · E1 + J1 (30.1.2)

∇ × E2 = −jωμ · H2 − M2 (30.1.3)

∇ × H2 = jωε · E2 + J2 (30.1.4)

From the above, we can show that (after left dot-multiply (30.1.1) with H2 and (30.1.4) with

E1),

H2 · ∇ × E1 = −jωH2 · μ · H1 − H2 · M1 (30.1.5)

E1 · ∇ × H2 = jωE1 · ε · E2 + E1 · J2 (30.1.6)

Then, using the above, and the following identity, we get the second equality in the following

expression:

∇ · (E1 × H2) = H2 · ∇ × E1 − E1 · ∇ · H2

= −jωH2 · μ · H1 − jωE1 · ε · E2 − H2 · M1 − E1 · J2 (30.1.7)

By the same token,

∇ · (E2 × H1) = −jωH1 · μ · H2 − jωE2 · ε · E1 − H1 · M2 − E2 · J1 (30.1.8)

If one assumes that

μ = μt, ε = εt (30.1.9)

or when the tensors are symmetric, then H1 · μ · H2 = H2 · μ · H1 and E1 · ε · E2 = E2 · ε · E1.2

Upon subtracting (30.1.7) and (30.1.8), one gets

∇ · (E1 × H2 − E2 × H1) = −H2 · M1 − E1 · J2 + H1 · M2 + E2 · J1 (30.1.10)

1The current sources are impressed currents so that they are immutable, and not changed by the environ-

ment they are immersed in [47].

2It is to be noted that in matrix algebra, the dot product between two vectors are often written as at · b,

but in the physics literature, the transpose on a is implied. Therefore, the dot product between two vectors

is just written as a · b.

Reciprocity Theorem 301

Figure 30.2: The geometry for proving reciprocity theorem when the surface S does not

enclose the sources.

Figure 30.3: The geometry for proving reciprocity theorem when the surface S encloses the

sources.

Now, integrating (30.1.10) over a volume V bounded by a surface S, and invoking Gauss’

divergence theorem, we have the reciprocity theorem that



S

dS · (E1 × H2 − E2 × H1)

= −



V

dV [H2 · M1 + E1 · J2 − H1 · M2 − E2 · J1] (30.1.11)

When the volume V contains no sources (see Figure 30.2), the reciprocity theorem reduces to



S

dS · (E1 × H2 − E2 × H1) = 0 (30.1.12)

The above is also called Lorentz reciprocity theorem by some authors.3

Next, when the surface S contains all the sources (see Figure 30.3), then the right-hand

side of (30.1.11) will not be zero. On the other hand, when the surface S → ∞, E1 and

3Harrington, Time-Harmonic Electric Field [47].

302 Electromagnetic Field Theory

H2 becomes spherical waves which can be approximated by plane waves sharing the same β

vector. Moreover, under the plane-wave approximation, ωμ0H2 = β × E2, ωμ0H1 = β × E1,

then

E1 × H2 ∼ E1 × (β × E2) = E1(β · E2) − β(E1 · E2) (30.1.13)

E2 × H1 ∼ E2 × (β × E1) = E2(β · E1) − β(E2 · E1) (30.1.14)

But β · E2 = β · E1 = 0 in the far field and the β vectors are parallel to each other. Therefore,

the two terms on the left-hand side of (30.1.11) cancel each other, and it vanishes when

S → ∞. (Furthermore, they cancel each other so that the remnant field vanishes faster than

1/r2. This is necessary as the surface area S is growing larger and proportional to r2.)

As a result, (30.1.11) can be rewritten simply as



V

dV [E2 · J1 − H2 · M1] =



V

dV [E1 · J2 − H1 · M2] (30.1.15)

The inner product symbol is often used to rewrite the above as

〈E2, J1〉 − 〈H2, M1〉 = 〈E1, J2〉 − 〈H1, M2〉 (30.1.16)

where the inner product 〈A, B〉 =

V dV A(r) · B(r).

The above inner product is also called reaction, a concept introduced by Rumsey. There-

fore, the above is rewritten more succinctly as

〈2, 1〉 = 〈1, 2〉 (30.1.17)

where

〈2, 1〉 = 〈E2, J1〉 − 〈H2, M1〉 (30.1.18)

The concept of inner product or reaction can be thought of as a kind of “measurement”. The

reciprocity theorem can be stated as that the fields generated by sources 2 as “measured”

by sources 1 is equal to fields generated by sources 1 as “measured” by sources 2. This

measurement concept is more lucid if we think of these sources as Hertzian dipoles.

30.1.1 Conditions for Reciprocity

It is seen that the above proof hinges on (30.1.9). In other words, the anisotropic medium has

to be described by symmetric tensors. They include conductive media, but not gyrotropic

media. A ferrite biased by a magnetic field is often used in electronic circuits, and it cor-

responds to a gyrotropic, non-reciprocal medium.4 Also, our starting equations (30.1.1) to

(30.1.4) assume that the medium and the equations are linear time invariant so that Maxwell’s

equations can be written down in the frequency domain easily.

4Non-reciprocal media are important for making isolators in microwave. Microwave signals can travel from

Port 1 to Port 2, but not vice versa.

Reciprocity Theorem 303

30.1.2 Application to a Two-Port Network

Figure 30.4: A geometry for proving the circuit relationship between two antennas using

reciprocity theorem. Circuit relationship is possible when the ports of the antennas are small

compared to wavelength.

The reciprocity theorem can be used to distill and condense the interaction between two

antennas over a complex terrain as long as the terrain comprises reciprocal media. In Figure

30.4, we assume that antenna 1 is driven by current J1 while antenna 2 is driven by current

J2. Since the system is linear time invariant, it can be written as the interaction between

two ports as in circuit theory as shown in Figure 30.5. Assuming that these two ports are

small compared to wavelengths, then we can apply circuit concepts like potential theory at

the ports.

Figure 30.5: The interaction between two antennas in the far field of each other can be

reduced to a circuit theory description since the input and output ports of the antennas are

small compared to wavelength.

Focusing on a two-port network as shown in Figure 30.5, we have

[V1

V2

]

=

[Z11 Z12

Z21 Z22

] [I1

I2

]

(30.1.19)

Then assuming that the port 2 is turned on with J2 6 = 0, while port 1 is turned off with

J1 = 0. In other words, port 1 is open circuit, and the source J2 will produce an electric field

304 Electromagnetic Field Theory

E2 at port 1. Consequently,

〈E2, J1〉 =



V

dV (E2 · J1) = I1



Port 1

E2 · dl = −I1V oc

1 (30.1.20)

Even though port 1 is assumed to be off, the J1 to be used above is the J1 when port 1

is turned on. Given that the port is in the circuit physics regime, then the current J1 is a

constant current at the port when it is turned on. The current J1 = ˆlI1/A where A is the

cross-sectional area of the wire, and ˆl is a unit vector aligned with the axis of the wire. The

volume integral dV = Adl, and hence the second equality follows above, where dl = ˆldl. Since

Port 1 E2 · dl = −V oc

1 , we have the last equality above.

We can repeat the derivation with port 2 to arrive at

〈E1, J2〉 = I2



Port 2

E1 · dl = −I2V oc

2 (30.1.21)

But from (30.1.19), we can set the pertinent currents to zero to find these open circuit

voltages. Therefore, V oc

1 = Z12I2, V oc

2 = Z21I1. Since I1V oc

1 = I2V oc

2 by the reaction concept

or by reciprocity, then Z12 = Z21. The above analysis can be easily generalized to an N -port

network.

The simplicity of the above belies its importance. The above shows that the reciprocity

concept in circuit theory is a special case of reciprocity theorem for electromagnetic theory.

The terrain can also be replaced by complex circuits as in a circuit board, as long as the

materials are reciprocal, linear and time invariant. The complex terrain can also be replaced

by complex antenna structures.

30.1.3 Voltage Sources in Electromagnetics

Figure 30.6: Two ways to model voltage sources: (i) A current source Ja driving a very

short antenna, and (ii) A magnetic frill source (loop source) Ma driving a very short antenna

(courtesy of Kong, ELectromagnetic Wave Theory [31]).

Reciprocity Theorem 305

In the above discussions, we have used current sources in reciprocity theorem to derive certain

circuit concepts. Before we end this section, it is prudent to mention how voltage sources are

modeled in electromagnetic theory. The use of the impressed currents so that circuit concepts

can be applied is shown in Figure 30.6. The antenna in (a) is driven by a current source.

But a magnetic current can be used as a voltage source in circuit theory as shown by Figure

30.6b. By using duality concept, an electric field has to curl around a magnetic current just

in Ampere’s law where magnetic field curls around an electric current. This electric field will

cause a voltage drop between the metal above and below the magnetic current loop making

it behave like a voltage source.5

30.1.4 Hind Sight

The proof of reciprocity theorem for Maxwell’s equations is very deeply related to the sym-

metry of the operator involved. We can see this from linear algebra. Given a matrix equation

driven by two different sources, they can be written succinctly as

A · x1 = b1 (30.1.22)

A · x2 = b2 (30.1.23)

We can left dot multiply the first equation with x2 and do the same with the second equation

with x1 to arrive at

xt

2 · A · x1 = xt

2 · b1 (30.1.24)

xt

1 · A · x2 = xt

1 · b2 (30.1.25)

If A is symmetric, the left-hand side of both equations are equal to each other. Subtracting

the two equations, we arrive at

xt

2 · b1 = xt

1 · b2 (30.1.26)

The above is analogous to the statement of the reciprocity theorem. The above inner product

is that of dot product in matrix theory, but the inner product for reciprocity is that for

infinite dimensional space. So if the operators in Maxwell’s equations are symmetrical, then

reciprocity theorem applies.

5More can be found in Jordain and Balmain, Electromagnetic Waves and Radiation Systems [48].

306 Electromagnetic Field Theory

30.1.5 Transmit and Receive Patterns of an Antennna

Figure 30.7: The schematic diagram for studying the transmit and receive properties of

antennas.

Reciprocity also implies that the transmit and receive properties of an antenna is similar to

each other. Consider an antenna in the transmit mode. Then the radiation power density

that it will yield around the antenna is6

Srad = Pt

4πr2 G(θ, φ) (30.1.27)

where Pt is the total power radiated by the transmit antenna, and G(θ, φ) is its directive gain

function. It is to be noted that in the above

4π dΩG(θ, φ) = 4π.

Effective Gain versus Directive Gain

At this juncture, it is important to introduce the concept of effective gain versus directive

gain. The effective gain, also called the power gain, is

Ge(θ, φ) = feG(θ, φ) (30.1.28)

where fe is the efficiency of the antenna, a factor less than 1. It accounts for the fact that

not all power pumped into the antenna is delivered as radiated power. For instance, power

can be lost in the circuits and mismatch of the antenna. Therefore, the correct formula the

6The author is indebted to inspiration from E. Kudeki of UIUC for this part of the lecture notes [126].

Reciprocity Theorem 307

radiated power density is

Srad = Pt

4πr2 Ge(θ, φ) (30.1.29)

If this power density is intercepted by a receive antenna, then the receive antenna will see

an incident power density as

Sinc = Srad = Pt

4πr2 Ge(θ, φ) (30.1.30)

The effective area or aperture of a receive antenna is used to characterize its receive

property. The power received by such an antenna is then

Pr = SincAe(θ′, φ′) (30.1.31)

where (θ′, φ′) are the angles at which the plane wave is incident upon the receiving antenna

(see Figure 30.7). Combining the above formulas, we have

Pr = Pt

4πr2 Ge(θ, φ)Ae(θ′, φ′) (30.1.32)

Now assuming that the transmit and receive antennas are identical. We swap their roles

of transmit and receive, and also the circuitries involved in driving the transmit and receive

antennas. Then,

Pr = Pt

4πr2 Ge(θ′, φ′)Ae(θ, φ) (30.1.33)

We also assume that the receive antenna, that now acts as the transmit antenna is transmitting

in the (θ′, φ′) direction. Moreover, the transmit antenna, that now acts as the receive antenna

is receiving in the (θ, φ) direction (see Figure 30.7).

By reciprocity, these two powers are the same, because Z12 = Z21. Furthermore, since

these two antennas are identical, Z11 = Z22. So by swapping the transmit and receive

electronics, the power transmitted and received will not change.

Consequently, we conclude that

Ge(θ, φ)Ae(θ′, φ′) = Ge(θ′, φ′)Ae(θ, φ) (30.1.34)

The above implies that

Ae(θ, φ)

Ge(θ, φ) = Ae(θ′, φ′)

Ge(θ′, φ′) = constant (30.1.35)

The above Gedanken experiment is carried out for arbitrary angles. Therefore, the constant is

independent of angles. Moreover, this constant is independent of the size, shape, and efficiency

of the antenna, as we have not used their shape, size, and efficienty in the above discussion.

One can repeat the above for a Hertzian dipole, wherein the mathematics of calculating Pr

308 Electromagnetic Field Theory

and Pt is a lot simpler. This constant is found to be λ2/(4π).7 Therefore, an interesting

relationship between the effective aperture (or area) and the directive gain function is that

Ae(θ, φ) = λ2

4π Ge(θ, φ) (30.1.36)

One amusing point about the above formula is that the effective aperture, say of a Hertzian

dipole, becomes very large when the frequency is low, or the wavelength is very long. Of

course, this cannot be physically true, and I will let you meditate on this paradox and muse

over this point.

7See Kong [31][p. 700]. The derivation is for 100% efficient antenna. A thermal equilibrium argument is

used in [126] and Wikipedia [127] as well.

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