1. Antenna Engineering Ch. 3.7~3.8.2 Duality & Reciprocity
Applied Electromagnetic Technology Laboratory
Department of Electronics and Computer Engineering
Hanyang University, Seoul, Korea

2. Outline Duality Theorem Reciprocity and Reaction Theorems
Reciprocity for Two Antennas
Reciprocity for Antenna Radiation Patterns

3. Duality Theorem
Two equations that describe the behavior of two different variables are of the same mathematical form.
Solutions will also be identical
The variables in the two equations that occupy identical positions are known as dual quantities.
With one set of solutions, the solutions to the other set can be formed.
Only serves as a guide to forming mathematical solutions.
No magnetic charges or current at nature

4. Reciprocity Theorem (1/4)
For physical linear network, position of an ideal voltage source and an ideal ammeter can be interchanged without affecting their readings.
Lorentz Reciprocity Theorem
Linear, isotropic medium
Two sources (J1, M1), (J2, M2) must satisfy Maxwell’s Eq.
(J1, M1)
(E1, H1)
(J2, M2)
(E2, H2)

5. Reciprocity Theorem (2/4)
With the 1st and 4th equation,
Applying vector identity
Similar application for 2nd and 3th equation,
With 2 equation results,

6. Reciprocity Theorem (3/4)
Taking volume integral of both sides,
For source-free region,

7. Reciprocity Theorem (4/4)
Another useful form of integral form for enclosed by sphere of infinite radius,
This coupling has been defined as Reaction.

8. Reciprocity for Two Antennas (1/2)
Two antennas has input impedances Z1, Z2
Linear and isotropic (not necessarily homogeneous)
One antenna works as transmitter, other one as a receiver.

9. Reciprocity for Two Antennas (2/2)
Power delivered by the generator to antenna #1,
With load admittance Y21, power delivered to the load is,
Power ratio
Under conditions of reciprocity (Y12=Y21), the power delivered in either direction is the same.

10. Reciprocity for Antenna Radiation Patterns (1/3)
The radiation pattern is a very important antenna characteristic.
Although it is usually most convenient and practical to measure the pattern in the receiving mode, it is identical, because of reciprocity, to that of the transmitting mode.

11. Reciprocity for Antenna Radiation Patterns (2/3)
For current I1 is applied at the terminal 1 and voltage V2(V20c) is measured at the open terminal of antenna #2,
Same method used for reverse situation,
If the medium between the two antennas is linear, passive, isotropic, and the waves monochromatic,
For I1 = I2,
Since the V2oc is identical to that of V1oc (reciprocity), the transmitting (V2oc) and receiving (V1oc) field patterns are also equal.

12. Reciprocity for Antenna Radiation Patterns (3/3)
The receiving mode of operation at Figure 3.5(b) for the test antenna is most widely used to measure antenna patterns because the transmitting equipment is bulky and heavy while the receiver is small and light weight in most cases.
The transmitting equipment usually consists of sources and amplifiers.
For precise measurements, it is necessary to have frequency and power stabilities.
Therefore the equipment must be placed on stable and vibration-free platforms.
This can best be accomplished by allowing the transmitting equipment to be held stationary and the receiving equipment to rotate.