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# 329lect23 - 23 Signal transmission circular polarization...

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23 Signal transmission, circular polarization Since in perfect dielectrics the propagation velocity v p = v and the intrinsic impedance η are frequency independent (i.e., propagation is non-dispersive), d’Alembert plane wave solutions of the form E = ˆ xf ( t - z v ) and H = ˆ y f ( t - z v ) η are valid in such media. t t -4 -2 2 4 6 -6 -4 -2 2 4 6 -4 -2 2 4 6 -6 -4 -2 2 4 6 m ( t ) m ( t ) cos( ω t ) Consider a waveform f ( t ) = m ( t ) cos( ω t ) , where ω is some specific frequency having a corresponding period T = 2 π ω , m ( t ) is some arbitrary signal (e.g., a voice signal, a message) changing slowly compared to period T . In that case, f ( t ) specified above can be called narrowband AM , and ω the carrier frequency of modulating cosine of the message signal m ( t ) . 1

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The corresponding x -polarized wave fields propagating in z direction can then be represented as Field 1 E = m ( t - z v ) cos( ω t - β z x and H = m ( t - z v ) η cos( ω t - β z y where β = ω μ as usual 1 . With reference to the expressions above, we could say that the AM wave field has an x -polarized carrier . By contrast, Field 2 E = m ( t - z v ) cos( ω t - β z y represents an AM wave field with a y -polarized carrier , and so does Field 3 E = m ( t - z v ) sin( ω t - β z y but with a carrier that has been time-delayed by a quarter period. Suppose Fields 1 and 3 above were transmitted simultaneously and therefore superpose. In that case we will have a wave field with Circular polarized carrier E = m ( t - z v )[cos( ω t - β z x + sin( ω t - β z y ] 1 In dispersive media where β is a non-linear function of ω , narrowband AM can propagate as m ( t - z v g ) cos( ω t - β z x where v g ∂ω ∂β is known as group velocity — covered in detail in ECE 450. 2
which has a circular polarized carrier . Since this is just a superpo- sition of two d’Alembert waves, the accompanying H is easily found to be x y z t = 0 t > 0 CIRCULAR POLARIZATION: Field vector rotates instead of oscillating. The rotation frequency is also the wave frequency. cos( ω t - β z x + sin( ω t - β z y RIGHT CIRCULAR E H = m ( t - z v )[cos( ω t - β z y - sin( ω t - β z x ] / η .

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