RF Electronic Devices and Systems

The transmission line as parallel planes, stripes and microstripes


 Parallel planes or stripes is the most general example of transmission line. Schematically it is depicted below. The parallel stripes can carry TEM, TM and TE waves.

For analysis of TEM waves we must resolve the Laplace equation 2Φ(x,y)=0. Let’s consider the case where the bottom stripe is grounded and the top stripe carries potential V0. Geometry of the transmission lines allows us to assume that the solution will be as a linear equation for the potential Φ(x,y)=Ay+B. Applying the boundary conditions Ф(x,y)=V0yd. So the electric field is E(x,y)=-V0yd ejωμεzey where ey is the radius-vector. The magnetic field is H(x,y)=V0xdεμ ejwμεzex, where ex is a radius-vector. The potential difference between two stripes or the voltage is V=0d Eydy=V0ejwμε, the induced current is I=0W Hxdx=WdεμV0 ejωμε, the characteristic impedance is: Z0=VI=dWμε.

The same as we discussed before, TE waves are characterised as EZ=0, HZ0. Applying the reduced wave equation we can find the Ex and HZ. And the TE waves will have the following characteristics:

Cut-off wave number knm=ωμεkcnm=d .
Propagation constant βnm=k22d2 .
Cut-off wavelength λcnm=2πkcnmλgnm=2πβ.
Phase velocity   vpnm=ωβ .
Dielectric attenuation constant αdnm=k2tanσ2β.
Eznm=0Hznm=BcosnπnydejβzExnm=-jωμkcBsinπnydejβzEynm= 0Hxnm= 0Hynm= –kcBsinπndejβz Impedance  ZTEnm=kβμε .

The power flow for TE waves is PTE=Re20W0d ExHy*dxdy, the attenuation constant is α=2kc2Rskβdεμ.

TM waves are characterised by Hz=0,Ez≠0. Resolving wave equation (22y+kc2) Ez(x,y)=0, we can obtain Ez(x,y) and the remaining TM wave characteristics apply the boundary conditions.

Cut-off wave number   knm=ωμεkcnm=d .
Propagation constant   βnm=k22d2  .
Cut-off wavelength  λcnm=2πkcnmλgnm=2πβ .
Phase velocity  vpnm=ωβ .
Dielectric attenuation constant αdnm=k2tanσ2β.
Eznm=AsinπnydejβzHznm= 0Exnm=0Eynm= –kcAcosπnydejβzHxnm=jwεkcAcosπnydejβzHynm= 0 Impedance ZTMnm=βkμε.

The power flow in the transmission line can be obtained via PTE=-Re20W0d EyHx*dxdy, the power dissipated in the transmission line is Pd=Rs0W JS2dx. And the attenuation constant is α=2kRsβdμε. Figure 2 shows the field lines for TEM waves, the TEn and TMn wave field line can be calculated for certain values on n.

Figure 1. The scheme of the parallel plane transmission line.
Figure 1. The scheme of the parallel plane transmission line.
Figure 2.TEM fields in the parallel plane transmission line.
Figure 2. TEM fields in the parallel plane transmission line.

The related geometry transmission line is a stripline which can be seen in Figure 3, where the dielectric material fill the space between two conducting plates. A stripline is usually manufactured with several techniques including etching and photolithographic fabrication. The basic operation mode of a stripline is TEM waves mode.

Striplines the same as other transmission line geometries support all modes orders, but in practice they are always limited for some practical application by the voltage applied to the plates, or by the distance between them. It is not easy to apply the Laplace equation and wave equation techniques to resolve the characteristics of a stripline. Striplines can be calculated numerically. Striplines are characterised by the following formulas:

vp=cεr, β=wvp, Z0=εrcC, where c is the velocity of light in a free-space. Usually the characteristic impedance for a stripline has a more complex form and fully depends on the geometry of the stripline Z0=30πεr hWe+0.441h.

The parameter We is the effective width, and it can be calculated in the following way Weh=Wh0,                   Wh>0.35(0.35Wh),    Wh<0.35

A variety of catalogues and reference books give the Wh characteristics for a stripline. And these characteristics allow us to calculate the stripline width when designing the stripline. Wh=30πεrZ0-0.441 when εrZ0<120Ohm and Wh=0.85-1.44130πεrZ0 when εrZ0>120Ohm.

Attenuation constant of the practical stripline also depends on the stripline geometry α=2.7*10-3RsεrZ030π(hd) (1+2Whd+1πh+dhd ln2hdd), when εrZ0<120Ohm and α=0.16RsZ0h (1+h0.5W+0.7a (0.5+0.414aW+12π ln4πWa. The formulas above can be used for practical usage when designing a stripline.

Figure 3. The scheme of a stripline structure.
Figure 3. The scheme of a stripline structure.

The microstrip is the most popular transmission line in RF and microwave, depicted in Figure 4. It is fabricated with standard printed circuit techniques, can be miniaturised and integrated into any system, and is less expensive in production.

The microstrip consists of the conducting plate and dielectric layer. The outer conductor is a single flat metal ground-plane. The inner conductor is a metal plane separated from the outer plane by the dielectric layer.

The practical microstrip has a very thin dielectric layer and the resulting electromagnetic fields are quasi-TEM. So all the solutions for TEM waves are not exactly static, but quasi-static and can be described with an approximation. A microstrip line is characterised with an effective dielectric constant, that can be seen in all the approximation formulas for TEM waves for microstrips. The effective dielectric constant depends on the microstrip geometry and dielectric constant of the dielectric layer, and it is always less than the dielectric constant of the layer: εef=εr+12+εr-1211+12hW.

The characteristic impedance for the microstrip will depend on the microstrip geometry Z0=60εefln24h2+W24hW, if Wh≤1, and Z0=120πεef(Wh+1.393+0.677 ln (Wh+1.444)), where Wh≥1.

The reverse procedure of obtaining the ratio Wh from Z0 is possible as well. Attenuation of the microstrip is αd=k0εr(εef-1)tanσ2εef(εr-1)  (the dielectric losses of the transmission line), and αc=wμ02σZ0W (the conduction losses of the transmission lines).

Figure 4. a – the scheme of the microstrip transmission line, b – the real microstrip transmission line.
Figure 4. a – the scheme of the microstrip transmission line, b – the real microstrip transmission line.

(«Microwave engineering», David M.Pozar, 4th edition, Wiley; “The ARRL Handbook for radio communications 2015”, ARRL; www.researchgate.net, the optically activated switch in the transmission line.)


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