A circular waveguide is schematically depicted in Figure 5. To consider its parameters we will employ the polar coordinates with radial components ρ and angle φ.

So the corresponding electric and magnetic field components for the general case are Eρ=jkc2 (βEzρ+ωμρHzφ),Eφ=-jkc2(βρEzρ+μωHzφ,Hρ=jkc2 (ωερEzφ+βHzρ),Hρ=-jkc2 (ωεEzφ+βρHzρ)

Figure 5. Schematic of the circular transmission line.
Figure 5. Schematic of the circular transmission line.

Let’s consider the TE waves, where EZ=0, HZ0. Considering the Hz(ρ,φ,z)=Hz(ρ,φ)ejβz and applying the Laplace equation, we can get the magnetic field structure and the rest of the parameters of the waveguide, in a similar way we did before.

So        Hz(ρ,φ)=(Asinnφ+Bcosnφ)Jn(kc,ρ)Eφ(z,ρ,φ)=jωμkc(Asinnφ+Bcosnφ)Jn(kc,ρ)ejβz, here Jn(kc,ρ) and Jn‘ (kc,ρ)  are Bessel functions. The characteristics of the waveguide will be determined by the Bessel function roots pnm. So the parameters of the TE mode transmission line are:

Cut-off wave number knm=ωμεkcnm=pnma .

Propagation constant βnm=k2pnma2.
Cut-off wavelength  λcnm=2πpnm aλgnm=2πβ .
Phase velocity  vpnm=ωβ.

Dielectric attenuation constant αdnm=k2tanδ2β.
Eznm=0Hznm= (Asinnφ+Bcosnφ)Jn(kc,ρ)ejβzEρnm=-jωμkc2ρ(AcosnφBsinnφ) Jn(kc,ρ)ejβzEφnm=jωμkc(Acosnφ+Bcosnφ) Jn‘ (kc,ρ)ejβzHρnm=kc(Acosnφ+Bcosnφ) Jn‘ (kc,ρ)ejβzHφnm= –kc2ρ(AcosnφBsinnφ) Jn(kc,ρ)ejβz

Impedance  ZTEnm=β .

For the TM mode of the transmission line, applying the considerations and calculation above we can get the following results:

Cut-off wave number knm=ωμεkcnm=pnma .
Propagation constant βnm=k2Pnm2a2.
Cut-off wavelength λcnm=2πapnmλgnm=2πβ.
Phase velocity vpnm=ωβ.

Dielectric attenuation constant αdnm=k2tanδ2β.
Eznm= (Asinnφ+Bcosnφ) Jn(kc,ρ)ejβzHznm= 0Eρnm=-kc(Acosnφ+Bsinnφ) Jn‘ (kc,ρ)ejβzEφnm=jβnkc2ρ(AcosnφBcosnφ) Jn(kc,ρ)ejβzHρnm=jεωnkc2ρ(AcosnφBcosnφ) Jn(kc,ρ)ejβzHφnm= –jωεkc2ρ(Acosnφ+Bsinnφ) Jn‘ (kc,ρ)ejβz

Impedance ZTM nm=βηk.

The cut-off frequency for the waveguide can be obtained using the cut-off wavelength. The roots for Bessel functions can be found in mathematic catalogues.

Coaxial transmission line


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