We can consider the arecibo s band radar system as an

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We can consider the Arecibo S-band radar system as an extreme example. This radar operates at a frequency of 2.38 GHz or a wavelength of 12 cm. The diameter of the Arecibo reflector is 300 m. Consequently, we can estimate the directivity to be about 6 × 10 7 or about 78 dB, equating the effective area with the physical area of the reflector. In fact, this gives an overestimate, and the actual gain of the system is only about 73 dB (still enormous). We will see how to calculate the effective area of an antenna more accurately later in the semester. 1.3.2 Phasor notation Phasor notation is a convenient way of describing linear systems which are sinusoidally forced. If we excite a linear system with a single frequency, all components of the system will respond at that frequency (once the transients have decayed), and we need only be concerned with the amplitude and phase of the response. The results can be extended to multiple frequencies and generalized forcing trivially using superposition. Let us represent signals of the form A cos( ωt + φ ) as the real part of the complex quantity A exp( j ( ωt + φ )) . Here, the frequency ω is presumed known and the amplitude A and phase φ are real quantities, perhaps to be determined. The introduction of complex notation may seem like an unnecessary complication, but it simplifies things enormously. This is because differentiation and integration become algebraic operations. Henceforth, we regard the real part of the phasor quantity as being physically interesting. The last step in our calculations will usually be to evaluate the real part of some expression. It is usually convenient to combine φ and A into a complex constant representing both the amplitude and phase of the signal, i.e. C = A exp( ) . C can be viewed as a vector in the complex plane. When dealing with waves, we 13
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expect both the amplitude and phase (the magnitude and angle of C ) to vary in space. Our task is then usually to solve for C everywhere. There is no explicit time variation — that is contained entirely in the implicit exp( jωt ) term, which we often even neglect to write. Note that the components of vectors too can be expressed using phasor notation. This notation can be confusing at times but really represents no special complications. Be sure not to mistake the complex plane with the plane or volume containing the vectors. As usual, the real part of a phasor vector is what is physically interesting. The rules of arithmetic for complex numbers are similar to those for real numbers, although there are important differences. For example, the norm of a vector in phasor notation is defined as | E | 2 = E · E . Consider the norm of the electric field associated with a circularly polarized wave traveling in the ˆ z direction such that E = x ± j ˆ y ) exp( j ( ωt kz )) . The length of the electric field vector is clearly unity, and evaluating the Euclidean length of the real part of the electric field indeed produces a value of 1. The phasor norm of the electric field meanwhile gives E · E = 2. In fact, the real part of inner and outer conjugate products in phasor notation (e.g.
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