1. Fundamental Propagation Problem: Imagine a sphere of radius R, centered on the isotropic radiator. The radiated power uniformly illuminates the inside surface of the sphere as it passes through it (Isotropic Radiation). 1.The fraction of total transmitted power illuminating the collector is equal to the ratio of the collector area to the total area of the imaginary sphere. 2.The Power Density (Poynting Vector P, w/m 2 ) at distance R is equal to the Total Power emitted divided by the total surface area of the sphere. Isotropic Radiator 100% efficient

2. Fundamental Propagation Problem: P P (Isotropic Radiation)

3. Gain Suppose we place a reflector behind the emitter. This will double the power density at the collector. This corresponds to a Gain of 2 for the transmitting antenna. Gain is defined as the ratio of maximum far-field power density, to that of an isotropic radiator at the same dstance. Antennas can be constructed with parabolic reflectors or other field shaping structures such that the power transmitted can be concentrated into a tight cone of radiation, yielding gains greater than 1000 (30 dB i ).

4. Beamwidth If 100% of the transmitted power is concentrated into a cone with vertex angle , then the power density will be: P P Gain is the ratio of this concentrated power density to that of an isotropic radiator: Note: when gain is calculated from beamwidth, we assume 100% efficiency, and we use the term “Directive Gain” or “Directivity”, and the subscript D is included. A

5. Aperture and Beamwidth Plane Waves  null /2 Direction of first “Null” Consider Plane waves impinging on an aperture having diameter “d” Direction of maximum radiated power density.

6. Aperture and Beamwidth (cont) A plot of Power density vs. angle off of boresight in the far field would look like this: For simplicity, we will approximate the Power density as uniform within the 3 dB beamwidth  3dB, and zero outside  3dB. Note: an elliptical aperture may have different horizontal and vertical beamwidths, corresponding to the aperture’s major and minor diameters.

7. Aperture, Beamwidth, and Gain Emitting area of a circular aperture with diameter d is: The above assumes antenna efficiency of 100%. For Practical antennas with efficiency , the actual antenna gain is G =  G D., with an associated effective aperture A e =  A. We can then say: Note: if horizontal and vertical beamwidths are equal, then  =  h =  v

8. Receive Antenna Gain The principle of reciprocity states that passive antennas have exactly the same characteristics whether they are used as transmit or receive antennas. Since antennas are typically characterized by their gains (rather than effective aperture), we can now express the propagation equation in terms of the receive antenna gain rather than its aperture: “Space Attenuation”