section 31d

(d) application to coaxial cable Consider an infinitely long coaxial cable driven at its left end at z = 0. A coaxial cable acts as a filter G(ω). In the frequency domain, it we drive the cable with I(ω), the output is O(ω) where O(ω, z) = G(ω, z) I(ω) G(ω) = e-γ(ω)z γ(ω) = α(ω) + iβ(ω) The transfer function G(ω,z) is like our X(ω) above, and satisfies the dispersion relations shown. In particular, it satisfies (31.9), α(ω) = α(∞) + (2/π) !Syntax Error, I dω'ω' (31.9a) β(ω) = β(∞) - (2/π) ω!Syntax Error, I dω' (31.9b) We have just argued that if α(ω) is slowly varying near ω, then K(ω) = (2/π) !Syntax Error, I dω' is roughly constant κ, and so τ(ω) = d [ ω K(ω) ] /dω ≈ κ and there is very little dispersion. One way to get α(ω) to be slowly varying is from (31.9a) in the case β(ω') ≈ 0. In this case (31.9a) says α(ω) ≈ α(∞) = constant. Thus, in a region where γ(ω) has a very small (or no) imaginary part, we expect to get small (or no) dispersion of pulses. So how does α(ω) vary with ω in a coaxial cable? We first show that γ(ω) is basically the same as the index of refraction n(ω) for the dielectric material in the cable. For a coaxial cable one has γ = where R,L,G and C are resistance, inductance, conductance (across the dielectric) and capacitance all per unit length of the cable. If we ignore R and G, we find that γ(ω) ≈ i ω => α(ω) = 0 β(ω) = ω The capacitance C of a section of coaxial cable is given by C = ε(ω)2πε0/ln(b/a) = k1 ε(ω), where ε(ω) is the dielectric "constant", which in general is not constant as a function of ω. Thus we have β(ω) = ω = ω If ε(ω) really were a constant, then we would have linear phase and constant group delay. From Maxwell's equations one knows that the index of refraction is given by n(ω) = = /k2 Then β(ω) = ω k2 n(ω) so γ(ω) is directly related to the index of refraction n(ω). One way to get α(ω) to be slowly varying is from the first equation above in the case Im(ε(ω)) ≈ 0 meaning β(ω') ≈ 0. In this case the first equation says α(ω) ≈ α(∞) = constant. Thus, in a region where the index ε(ω) has a very small (or no) imaginary part, we expect to get small (or no) dispersion of pulses. *********************************************** For an infinitely long coaxial cable of length driven at one end by some voltage Voeiωt, the voltage distance z down the cable is given by V(z,t; ω) = G(ω) V0 eiωt V(z,t) = V0e-γ(ω)z eiωt = V0 e-α(ω)z e-iβ(ω)z eiωt where α(ω) is the attenuation factor and β(ω) is the phase. The complex factor γ(ω) is given by γ = where R,L,G and C are resistance, inductance, conductance (across the dielectric) and capacitance all per unit length of the cable. The functions α(ω) and β(ω) satisfy dispersion relations of the form (31.9) α(ω) = α(∞) + (2/π) !Syntax Error, I dω'ω' (31.9a) β(ω) = β(∞) - (2/π) ω!Syntax Error, I dω' (31.9b) We have just argued that if α(ω) is slowly varying near ω, then K(ω) = (2/π) !Syntax Error, I dω' is roughly constant κ, and so τ(ω) = d [ ω K(ω) ] /dω ≈ κ and there is very little dispersion. One way to get α(ω) to be slowly varying is from the first equation above in the case Im(ε(ω)) ≈ 0 meaning β(ω') ≈ 0. In this case the first equation says α(ω) ≈ α(∞) = constant. Thus, in a region where the index ε(ω) has a very small (or no) imaginary part, we expect to get small (or no) dispersion of pulses. V = Voeiωt e then it turns out the dispersion relations This effect is also known as "dispersion". It happens that n(ω) = , where ε(ω) is called the dielectric constant of the medium, although it is sometimes not very "constant" as a function of ω. It turns out that ε(ω) has the right properties to satisfy the above pair of equations (31.9) where now ε(ω) = e-γ(ω) = e-α(ω) e-iβ(ω) : α(ω) = α(∞) + (2/π) !Syntax Error, I dω'ω' (31.9a) β(ω) = β(∞) - (2/π) ω!Syntax Error, I dω' (31.9b) We have just argued that if α(ω) is slowly varying in some region, then K(ω) = (2/π) !Syntax Error, I dω' is roughly constant κ, and so τ(ω) = d [ ω K(ω) ] /dω ≈ κ and there is very little dispersion. One way to get α(ω) to be slowly varying is from the first equation above in the case Im(ε(ω)) ≈ 0 meaning β(ω') ≈ 0. In this case the first equation says α(ω) ≈ α(∞) = constant. Thus, in a region where the dielectric constant ε(ω) has a very small (or no) imaginary part, we expect to get small (or no) dispersion of pulses. In effect, a dielectric medium is a filter, and ε(ω) is the transfer function of the filter. The input and output of this "filter" are known as the "electric field" E and the "electric displacement" D. In the frequency domain the equation of this filter is (21.2) which here reads (at any point in space) D(ω) = ε(ω) E(ω) . If ε(ω) has a small imaginary part in some range of ω, this filter is non-dispersive. For a transmission line like a coaxial cable, if we ignore conductance G through the dielectric and we ignore inductance L along the cable, if the cable is driven with voltage V0 at one end z = 0, the voltage at the other end is given by V(z,t) = V0e-γ(ω)z eiωt = V0 e-α(ω)z e-iβ(ω)z eiωt The attenuation α(ω) and phase β(ω) satisfy the dispersion relations (31.9) above and we reach the same conclusion. Away from electromagnetic resonances of the medium, ε(ω) has a very small imaginary part for a non-polar dielectric like polyethylene or teflon. Thus, if we could ignore ohmic losses in the conductors, coaxial cables using these materials as dielectrics would be non-dispersive up to infrared frequencies -- where vibrational and rotational resonances set in.