# Design of an Evanescent Mode Circular Waveguide 10 GHz Filter

by Tom Apel – **K5TRA**

*(All images are at end of article) *

Ham radio operation at 10 GHz is far removed from global shortwave communication typically operating below 30 MHz. The motivation for designing the evanescent mode circular waveguide 10 GHz filter sprang from a desire to extend ham radio operation to 10 GHz. This application note describes the design of two 10 GHz filters and is a follow up to a previous article published in Microwave Product Digest^{1} describing the design of a 10 GHz low-noise amplifier (LNA). Both are key components needed for ham radio station operation at 10 GHz.

NI AWR Design Environment, specifically Microwave Office circuit design software, was used to design the filters for a range of bandwidths up to five percent. The ability of the software to optimize the design while simultaneously accounting for resonator spacing, insertion loss, and return loss, was powerful.

** Test Bench Set Up**

Before any hardware could be tested, measurement capability needed to be extended to 10 GHz. Available equipment included a vector network analyzer (VNA), spectrum analyzer, and service monitor, which provided metrology through L-band, but did not support 10 GHz measurement capability. The designer decided to build a synchronous up/down conversion test set to extend the bench equipment to 10 GHz.

To build this test set, image reject filters (**Figure 1**) were needed. **Figure 2** illustrates the block diagram of the test set, which enabled measurement of insertion gain or loss, as well as return loss with a directional coupler.

**Bandpass Filters and Immittance Inverters**

Half-inch copper tubing segments with end caps were used to realize tunable bandpass filters at 10 GHz. In order to understand how this type of filter works, some basic filter theory is useful.

**Figure 3** shows a simple two-branch ladder circuit. Many filters can be represented as cascades of series and shunt branches. In the passband frequency range, series branches are ideally short circuit and shunt branches are open, which yields 0 dB insertion loss. Alternatively, in stopband, series branches are ideally open and shunt branches are short circuit, which yields no transmission and 100 percent reflection. For example, lowpass (LP) filters are formed from series inductor and shunt capacitor cascades and bandpass (BP) ladder canonic structures are formed from ladder cascades of series resonators and shunt resonators. This can be seen in **Figure 4**.

Bandpass filters obtained from LP to BP transformation^{2,3} produce these structures with all resonators synchronously tuned to the center of the pass band.

LC ladders composed of alternating series and shunt resonators are quite far removed from waveguide filters operating at 10 GHz, requiring two conceptual steps to bridge the gap.

The first step is the introduction of impedance (K) or admittance (J) inverters. Both filter structures in **Figure 5** can be viewed as equivalent to the series-shunt LC cascade in **Figure 4**.

An impedance inverter connected to a series resonator behaves as a shunt resonator when viewed from the opposite side of the inverter. Similarly, an admittance inverter connected to a shunt resonator behaves as a series resonator when viewed from the opposite side of the inverter. Impedance or admittance inverters (sometimes called immittance inverters) are symmetric networks where shorts map to opens and opens to shorts, series capacitors map to shunt inductors, and series inductors map to shunt capacitors.

The use of immittance inverters enables reuse of the same type of resonator in an overall bandpass filter design, as illustrated in **Figure 6**. For example, combline filters can be viewed as cascades of shunt resonators and admittance inverters, where the inter-resonator coupling is set by the particular admittance of the inverter between a pair of resonators. Filters with the same input and output port impedances are usually symmetric. This is shown in **Figure 6**. This property can be used to simplify the design parameters.

J and K inverters can be realized in many ways. Quarter-wave transmission lines are a common example of an impedance (or admittance) transformer placed between similar resonators. Inverter forms commonly used in LC filter designs are shown in **Figure 7**, where the T networks are the impedance inverter representations and the π networks are the admittance forms.

Note that a T to π transformation easily shows that they are equivalent. The T inverter networks forms are better suited to interface with series resonators and the π representation is better suited to interface with shut resonators. The obvious question now arises: Where are negative valued inductors or capacitors found?

From **Figure 6**, it is clear that inverters will be placed between shunt resonators. The left side of **Figure 8** shows a pair of shunt resonators coupled with an inductive admittance inverter. The inverter negative shunt inductance can be absorbed into the resonator representation. This results in a parallel equivalent inductance larger than the original, which is represented as L” in the equivalent circuit on the right side of **Figure 8**.

It may seem like the resonant frequency has been lowered. It is important to note that when the adjacent resonator is short circuited, the resonant frequency is the same as the original resonator. When one side of an inverter is shorted, the other side is open. This is also true for interior resonators with inverters on both sides, as in the center resonator in **Figure 6**. When the adjacent resonators on both sides are short circuited, the interior resonator tunes the center frequency of the passband. In this way, all resonators are seen to be tuned to the center frequency. Filters of this type are called synchronously tuned BP filters. The relationship is a direct result of the canonic BP ladder structures from **Figure 4**.

Recognition of the resonant frequency behavior of each resonator when adjacent resonators are shorted will be important later in this note, greatly simplifying the design process. It should also be mentioned that this behavior is the basis for Dishal’s tuning method^{4}.

**Circular Waveguide**

The second step is to address the waveguide equivalent circuit. Metal pipes of various cross-section shapes have been used as low-loss transmission media at microwave frequencies for many years. Circular cross-section copper pipe is the medium of interest here. As long as the RF frequency is above a critical frequency related to the pipe radius, RF will propagate. A half-inch copper pipe is actually 0.565” inside diameter. The corresponding TE_{11} cutoff frequency is 12.2516 GHz. Below the cutoff frequency, RF field amplitude falls exponentially with distance and the transmission path becomes reactive. This is called evanescent mode propagation.

The equivalent circuit for a length of waveguide below the cutoff frequency can be seen in **Figure 9**.

The possibilities for using a below-cutoff waveguide to form admittance inverters now become apparent. **Figure 10** shows the equivalent circuit redrawn on the left side.

Since inductive admittance inverters with series branch inductance of L_{S} also have -L_{S} shunt branches, additional shunt branches of +L_{S} and -L_{S} can be added with no net change (they cancel each other). This trick enables the recognition of the segment of copper pipe that can be viewed as an inductive admittance inverter with shunt inductors on both ends. Tuning screws can be added in the E plane to introduce shunt capacitance, thereby creating the shunt resonators. For less inter-resonator coupling, the length of the pipe segment between tuning screws can be increased (L_{S} is increased).

The following equations describe the circuit behavior of a segment of half-inch diameter copper pipe. The cutoff frequency, Fc, of a circular waveguide is a function of the guide’s inside radius, a. The measured inside diameter of a half-inch copper pipe actually is 0.565”, so a=7.1755 mm.

The equivalent circuit for a length, l, of half-inch copper pipe can be calculated as follows:

These are numeric values for the circuit representations found in **Figures 9** and **10**. Clearly, by changing the length, the series inductance and thereby the coupling (admittance of inverter) can be set. The excess shunt inductance of **Figure 10 **(formula here, ), can be resonated by adding a tuning screw in the E plane. This provides a variable capacitance to form the shunt resonator.

**Design Procedure**

The preceding paragraphs provide a procedure for analyzing a length of circular waveguide operating below cutoff. To design a filter in this media, the designer could start from LP prototype tables, perform an LP to BP transformation, and map the series resonators (see **Figure 4**) into shunt resonators cascaded with inverters. This is the classic approach found in Craven^{5} and in Howard^{6} and often involves some iterative optimization.

Another approach might be to fully describe a physical filter with inter-resonator lengths and tuning capacitors as variables in a circuit analysis environment and simply use optimization to obtain a solution. This is a somewhat forceful approach and might not yield an optimum solution.

If some knowledge of desirable filter solutions is used to simplify and constrain the structure, circuit analysis with optimization can efficiently obtain a solution that is well conditioned to converge to a desired result. Recall the symmetry shown in **Figure 6**. If the physical structure and thereby the circuit description is explicitly defined to be symmetrical, then the number of variables is cut in half. Also, recall the mention of Dishal’s tuning method for synchronously tuned filters. Each resonator should be tuned to the center frequency when all adjacent resonators are short circuited. This constraint provides a closed form solution for all tuning capacitors as a function of surrounding inductors and eliminates the capacitors as variables, providing a significant advantage in preconditioning the optimizer for a desired solution. Ultimately, the designer decided to use this approach for the filter design project.

Microwave Office was used for the circuit analysis and optimization. Circuit values were all described algebraically within the software, a feature that enabled easy implementation of symmetry and synchronous tuning variable elimination. The only independent variables were the distances between tuning screws. These are l1 and l2 in the example algebraic description in **Figure 11**. All inductors and capacitors were calculated in closed form from these two values.

**Figure 12** shows a corresponding circuit description. This is a three-resonator filter with tuned port transitions.

Since it is constructed to be symmetric about the center, the details can be more easily seen in **Figure 13**. Note that loss has been introduced through the inductor unloaded Q parameter, Q_{0}. After measuring a single unloaded resonator, the unloaded Q was empirically determined to be approximately 200.

The subminiature version A (SMA) transition can be seen in **Figure 14**.

A shorted loop forms an H plane transition. This launcher is tuned with a shunt capacitance from a tuning screw. For simulation purposes, the port transition tuning was expressed was expressed as C0p. It was calculated using Dishal’s^{4} method. This can be seen algebraically in **Figure 15**. Practically, the end transition tuning will differ slightly due to the H plane transition inductance.

The optimized swept response for this example filter is illustrated in **Figure 16**. This is a five percent bandwidth solution centered on 10.4 GHz.

**Results**

A set of solutions was run over BW steps for 2, 3, 4, and 5 resonator filters. These results are tabulated in **Table 1**.

Each case is for a center frequency of 10.4 GHz. Tuning screw locations, passband loss and reflection, and -30 dB frequency points are displayed. Photos of the five resonator filters that were used in the 10 GHz test set are shown in **Figure 14** and **Figure 17**.

Brass nuts soldered to the copper tube provide solid support and grounding for resonator tuning screws. Additional lock nuts are on each tuning screw.

Resonator screws are 4-40 and SMA transition tuning screws are 2-56 thread. Mounting stand-offs are 4-40 and are attached to the filter body with “button head” screws.

The measured response of fabricated filters was very good. **Figure 18** is a screen shot of swept response from one of the image reject filters in the 10 GHz test set shown in **Figure 2**.

The measured passband is 10.18 GHz to 10.48 GHz with nominal insertion loss of 2.9 dB and ripple less than 1 dB peak. This 300 MHz BW was obtained from a three percent BW design. The 40 dB BW is 720 MHz, as shown in **Figure 19**.

For more information on waveguides and filters: http://bit.ly/2pirB3A

References

[1] T. Apel, “Design of a 10 GHz LNA for Amateur Radio Operation, Microwave Product Digest,” February 2016, p.8.

[2] G. Matthaei, L. Young and E.M.T. Jones, “Microwave Filters, Impedance-Matching Networks, and Coupling Structures,” McGraw-Hill, New York 1964, Sections 9.02 and 8.03, pp. 427-438.

[3] A. Zverev, “Handbook of Filter Synthesis,” Wiley, New York 1967, Chapter 6.

[4] M. Dishal, “Alignment and Adjustment of Synchronously Tuned Multiple Resonant-Circuit Filters,” Proc IRE-39, November 1951, pp.1448-1455.

[5] G. Craven and C.K. Mok, “The Design of Evanescent Mode Waveguide Bandpass Prescribed Insertion Loss Characteristic,” IEEE MTT-19, March 1971 pp. 295-308.

[6] J. Howard and M. Lavey, “Simplified Method Eases the Design of Bandpass Filters,” RF and Microwaves, December 2000 p. 97.

[7] S. Mandev and R. Kwok, “Evanescent Mode Filter,” SJSU EE172 Final Project, May 2013.

[8] P. Wade, “Waveguide Filters You Can Build and Tune, Part 3 Evanescent Mode Waveguide Filters,” ARRL QEX, March 2011, pp. 23-29.

[9] C. Balanis, “Circular Waveguides,” Encyclopedia of RF and Microwave, Wiley 2005, pp. 641-653.

(1971)