# Studying slotline transmission lines

PCB transmission lines are an optimal and low cost solution to make guided propagation at very high frequencies. The most popular lines are microstrip and coplanar waveguide. These transmission lines are easily realizable in a printed circuit board and whose impedance can be calculated from their dimensions. In these lines, TEM modes (transverse electromagnetic) are propagated, in which there is no component in the direction of propagation. However, there are other very popular lines that can also be used at high frequencies and are known as slotlines. In this post, we are going to study the electrical behavior of slotlines and some microwave circuits that can be done with them.

At high frequencies, lines usually behave like distributed transmission lines. Therefore, it is necessary to know its impedance so that there are no losses during propagation.

The microstrip and coplanar waveguides are very popular, since they are easily implemented on a printed circuit board, they are cheap and can be easily calculated. In both lines, the propagation mode is TEM, there are no field components in the direction of propagation, and their characteristic impedance Zc and wavelength λg depend on the line dimensions and the dielectric substrate which supports them.

There is another type of line, which is usually used at very high frequencies: the slotline. This line is one slot on the copper plane through which a transverse electric mode is propagated (specifically the TE01 mode, as shown in the following figure).

Fig. 1 –  TE01 mode on a slotline

The field is confined near the slot so that the propagation has the minimum possible losses, and as the microstrip lines, there is a discontinuity due to the dielectric substrate and air. It is used as a transmission line with substrates with a high dielectric constant (around εr≥9.2), in order to confine the fields as close as possible to the slot, although they can be used as couplings on substrates with lower dielectric constants. In this way, flat antennas can be fed with the slotlines.

In this post, we will pay attention to its use as transmission lines (with high dielectric constants), and the microwave circuits that we can make with them, studying the transitions between both technologies (slotline to microstrip).

ANALYZING THE SLOTLINE TRANSMISSION LINE

Being a transmission line and like the other lines, the slotline has a characteristic impedance Zc and a wavelength λs. But besides, using the TE01 propagation mode, the electric field component which is propagated, in cylindrical coordinates, is Eφ, as it is shown in the next figure

Fig. 2 – Eφ component

This component is calculated from the magnetic components Hr and Hz, considering the Z-axis the propagation direction, which is perpendicular to the electric field. From here, we get an expression for the propagation constant kc which is

$E_{\varphi}=\dfrac {j{\omega}{\mu_0}}{k_c^2}\dfrac {\partial H_z}{\partial r}=-{\eta} \dfrac {\lambda_s}{\lambda_0}H_r$

$k_c=\dfrac {2{\pi}}{\lambda_0} \sqrt {1- \left( \dfrac {\lambda_0}{\lambda_s} \right)^2}$

where λ0 is the wavelength of the propagated field. The first thing is deduced from the expression of kc is that we will find a cuttoff wavelength λs, from which the field propagates as mode TE01, since λ0≤λs so that kc is real and there is propagation. This means that there will be a cuttoff thickness for the substrate which will depend on the dielectric constant εr. The expression for that cuttoff thickness, where there is no propagation at TE01 mode, is

${\left( \dfrac {h}{\lambda_0} \right)}_c=\dfrac {1}{4\sqrt{{\epsilon_r}-1}}$

With these expressions, Gupta (see [1], page 283) got the expressions for the line impedance Zc and the line wavelength λs, which will allow us to typify the transmission line, making microwave circuits with slotlines.

ANALYZING A SLOTLINE

As the microstrip and coplanar waveguides, slotline can be analyzed using a FEM electromagnetic simulator. We are going to study one transmission line on an RT/Duroid 6010 substrate, which dielectric constant is εr=10,8, with 0,5mm thickness. The slot width is 5mil. According to the impedance calculations, Zc is 68,4Ω and λs, 14,6mm, at 10GHz. In a 3D view, the slotline is the next

Fig. 5 – Slotline 3D view

The next graph shows the S parameters at 50Ω impedance of generator and load.

Fig. 6 – Slotline S parameters

On the Smith chart

Fig. 7 – Slotline impedance on Smith Chart

where the impedance is 36,8-j·24,4Ω at 10GHz.

It is possible to show the propagated surface current on the line in 3D view

Fig. 8 – Slot surface current, in A/m

where it can be seen that the surface current is confined as near as possible the slot. From this current, the H-field can be derived and therefore the E-field which only has a transversal component. It can be also seen two maxima on the current magnitude, which shows that the slot distance is λs.

The FEM simulation allows us to analyze the slotline lines and build microwave circuits, knowing the characterization shown in [1].

SLOTLINE-TO-MICROSTRIP TRANSITIONS

Like the slotline is one slot made on a copper plane, transitions can be made from slotline to microstrip. One typical transition is the next

Fig. 9 – Slotline-to-microstrip transitions

Microstrip lines finish in a λm/4 open circuit stub, so the current is minimal at the open circuit and maximum at the transition location. In the same way, the slotline finishes in a λs/4 short circuit stub, with the minimum surface current at the transition location. Then, the equivalent circuit for each transition is

Fig. 10 -Equivalent circuit for a slotline-to-microstrip transition

Using the FEM simulator it is possible to study how a transition behaves. The next graph shows its S parameters. The transition has been made on RT/Duroid 6010, with 70mil thickness and 25mil slot widths. The microstrip width is 50mil and the working band is 0,7÷2,7GHz.

Fig. 11 – Transition S parameters

and showing the surface current on the transition, it ts the next

Fig. 12 – Current on the transition.

where it can be seen the coupling of the current and its distribution on the slotline.

ANOTHER MICROWAVE CIRCUITS BASED ON SLOTLINES

The slotline is a versatile line. Combined with microstrip (the microstrip ground plane can include slots), it allows us to make a series of interesting circuits, such as those shown in fig. 13

Fig. 13 – Microwave circuits with slotline and microstrip.

The 13 (a) circuit shows a balum with slotline and microstrip technology, where the microstrip is shorted to ground in the transition. The balanced part is the slotline section, since both ground planes are working like differential ports, while the unbalanced part is the microstrip, referring to the ground plane where the slots are placed. With this circuit it is possible to build frequency mixers or balanced mixers. Another interesting circuit is shown in 13 (b), a “rat-race” where the microstrip circuit is not closed, but is coupled through a slot to get the coupling. In 13 (c), a “branchline” coupler is shown, using a slotline and, finally, in 13 (d), a Ronde coupler is shown. This last circuit is ideal to equalize the odd/even mode phase velocities.

CONCLUSIONS

In this post, we have analyzed the slotline used like a microwave transmission line, compared with another technologie. Besides we have made a small behavior analysis using an FEM simulator, checking the possibilities of the line analysis (S parameters and surface current analysis) and we have shown some circuits that can be made with this technology, verifying the versatility of this transmission line.

REFERENCES

1. Gupta, K.C., et al. “Microstrip Lines and Slotlines”. 2nd. s.l. : Artech House, Inc, 1996. ISBN 0-89006-766-X.