De-risking Resonances in Single Conductor Structures, Such as “Ground”

Editor’s Note: A version of this article was originally published in Safety & EMC magazine English Edition in 2019.

When fixing EMC or EMI problems, I very often find that structural resonances, which are special types of ‘accidental antennas’, see [1], are the cause.

Designers are often caught out by this because they tend to ignore the wave nature of electromagnetic (EM) propagation and incorrectly assume that everything that they call, say, ‘Ground’, must be all at the same potential at any frequency.

Conductive structures that are a single node in a schematic diagram or circuit simulation, may be called ‘ground’, or chassis, earth, 0V, reference, shield, frame, or indeed anything else. But whatever they are called, at radio frequencies (RF) these single-conductor structures can exhibit complex resonances with corresponding problems for EMC and even functionality.

In this article, I will identify two major types of single-conductor structure resonances and describe some case studies for both. I will then describe how design projects can quickly and easily be ‘de-risked’ from the possibilities of suffering either type of resonance by the use of low-cost field and circuit simulators and/or low-cost bench testing on physical ‘mock-ups’.

Structural resonances in single conductor systems occur in two ways, which in this article I call ‘plane or cavity mode’, and ‘stray LC-resonator mode’.

“Plane or Cavity Mode” Structural Resonances in Single Conductor Structures

The following figures show some slides from one of my training courses, introducing the concept of two-dimensional and three-dimensional structural resonances in metalwork.

As Figures 1-5 try to show, planes and cavities can resonate when any two dimensions at once have integer numbers of half-wavelengths. So, for example for planes we have the (1,1) (2,1) (1,2) (2,2) (3,1) (3,2) (2,3) (3,3) (4,1)….etc., etc., possible modes. And for cavities (boxes) we have the (1,1,0) (1,0,1) (2,1,0) (2,0,1) (1,2,0) (2,2,0) (3,1,0)….etc., etc., modes.

Information is available on the Internet and in textbooks about EM propagation within waveguides (i.e., open-ended metal tubes) – but these are only a small subset of the very wide range of physical single-conductor plane or cavity structures that experience structural resonances, some of which are sketched in Figure 5.

To help understand how ‘plane or cavity mode’ structural resonances can occur in single conductor structures, two case studies follow.

Example of a GNSS Product

A land-based Global Navigation Satellite System (GNSS) product had two circular PCBs, each having a bottom layer dedicated to a solid 0V plane. One PCB carried the GNSS antenna/receiver module on its top side, and the other carried the rest of the digital circuit on its top side.

These two PCBs were assembled with their bottom sides together, as shown in Figure 6, so that – in use – the GPS antenna was on the top side of the product where it had a good view of the sky and the GNSS satellites whizzing around above, and the digital circuits were on the bottom. The only connection between the two PCBs was a coaxial cable and a couple of power conductors, in the exact centre of the boards, as sketched in Figure 6.

It had been correctly felt that this sort of construction would help to minimise the amount of digital circuit noise that would be picked up by the GPS antenna and cause it to lose sensitivity. However, because the two 0V planes facing each other were at the same nominal ‘ground’ potential, it had been incorrectly assumed that the cylindrical cavity that existed between them would have no effect.

Much too much digital noise was being picked up by the GNSS antenna, and I suspected that the 100mm diameter of the PCB planes plus their single central coaxial connection was creating a cavity that would resonate strongly within the frequency range used by GNSS services: 1164 – 1610 MHz.

Connecting one port of a vector network analyser to the two PCB ‘ground’ planes measured their return loss, S11, as shown in Figure 7, and as a Smith Chart in Figure 8, clearly revealing the suspected resonance to be close to 1.43GHz.

This 1.43GHz cavity resonance was encouraging digital noises on the lower PCB to couple into the upper GNSS PCB within the frequency range that any GNSS antenna or receiver must receive. Even if this noise does not actually fall within a GNSS transmit band itself, it desensitises the low noise amplifier (LNA) in the receiver (a common problem in wireless datacomms, see [2]and [3]).

The solution was to electrically connect the two 0V planes at multiple places spread all around their perimeters. The cavity was still there, but its resonant frequency was now nearly 3GHz, sufficiently far from the GNSS range for the coupled noise to be rejected as out-of-band by the GNSS receiver.

Example of a Very Large-Area LED Video Display

Very large LED video displays are often made by stacking a number of square LED boards edge-to-edge, horizontally and vertically. Each LED board fits in a square metal frame that contains other PCBs for driving the LEDs and controlling the image.

The original design had excessive emissions from below 100MHz to over 1GHz, and after an engineer trained in good EM engineering design had modified it there was only one spectral line still not comfortably below the limit line – the video clock frequency. This one frequency was still 20dB above the limit and remarkably resistant to all efforts to reduce it further.

I realised that this problem frequency was the (1,1) resonance frequency of the 0V planes in the controller boards in each of the LED panels (allowing for the effect of the boards’ FR4 dielectrics). This was creating such a high CM noise gain exactly at the clock frequency that solving it with usual EM Engineering methods would have taken more time than the project could afford and could have added a lot of cost and weight.

The solution was to use another resonance – a series-resonant CM filter tuned to the clock frequency. Another name for such resonant filters is notch filters, because they have a very narrow and very deep stopband. Although the levels of CM emissions from the video PCBs and their cables were comfortably below the limit line for the relevant emissions standard, the chance occurrence of a plane resonance at exactly the same frequency as the clock had amplified its emissions by at least 30dB.

It is not very difficult to design a passive notch filter to give at least -30dB at its notch frequency – but tuning it to the exact frequency can be difficult using standard component values and tolerances. Sometimes it pays to accept a shallower, broader notch (i.e., a lower Q value) that is at least sufficient to solve the problem.

Conclusions on “Plane or Cavity Mode” Structural Resonances in Single Conductor Structures

I have seen very many examples of this kind of resonance problem. What they all have in common is the accidental alignment of a systematic frequency – usually a switching power converter or digital clock frequency, or one of their harmonics – with the structural resonance of a plane or a cavity (see Figures 1-8).

We imagine that such co-incidences must occur at random, but they don’t because if we are asked to create a product of a certain shape and size and put a digital processor in it, we tend to choose simple, round numbers for X, Y and Z dimensions, such as 250mm x 150mm x 90mm; and for clock frequencies, such as, say, 300MHz. We don’t choose 243 x 137 x 82 mm and a clock frequency of 316.5MHz, unless we are forced to do so by circumstances we don’t control. The result is that such ‘accidental alignments’ of noise frequencies with structural resonance frequencies happen much more often than we expect.

Stray LC Structural Resonances in Single Conductor Structures

I only have room for one case study here, and it is illustrated in Figure 9.

A flat panel computer video display had good EMC emissions when fitted flush with the large metal top of its product’s base unit but emitted excessively in a narrow range around 500MHz when placed upright and mounted on a stand. This occurred with the same display unit, it was not a variation between different units.

The cables to the display were well-shielded types with their braid shields terminated at the points where the cables exited the metal base unit and where they entered the metal display enclosure – fully in accordance with good cable shielding practices, see [4]. So, the metal of the base unit’s enclosure, the cable shields, and the metal of the display screen’s enclosure, were effectively a single conductor structure – all the same circuit node on a schematic or SPICE circuit simulator, as in the case studies above.

Shielding of flexible cables can never be perfect, and display screens can never be perfectly shielded because that would make it impossible to see their images! Consequently, there are always some CM noise voltages and currents emitted and with this display they were low enough to comply with emissions limits – when mounted flush with the metal surface of its base unit.

When mounted on the stand, the display’s enclosure had about 1pF of stray capacitance shunting it to the top of the metal base unit, and about 100nH of stray series inductance in the stand or the cables routed inside it. The simple equivalent circuit for the CM noise, shown top-right in Figure 9, reveals what is happening.

All currents – even stray CM ones – always flow in closed loops, see [5], so we see that the CM noise current must flow through the stray series inductance of the cables’ shields (and the stand, if metal) then back through the stray shunt capacitance to close its loop.

To the CM current, the equivalent circuit is series-resonant at 1/2π√(LC), which with L=100nH and C=1pF works out to be about 500MHz. What is happening is that – at this frequency – the capacitive and inductive reactances in the current loop are cancelling out because they have the same values and are in antiphase, leaving only the series resistance.

The series resistance of the cable shields in this CM loop is very small, certainly less than 1 Ohm in total, with the result that the CM currents at the resonant frequency near 500MHz were very high indeed – much higher than they were at this frequency when the panel was flush with the surface.

The simple fix for the problem was to use a plastic stand and fit several ferrite toroids around the display’s shielded cables. The ferrite material of the toroids was chosen for its dominant high resistivity around 500MHz, and several of them were placed in series.

From an equivalent circuit viewpoint, these ferrites added a resistance in series with the CM current loops, significantly damping its resonance and significantly reducing E- and H-field emissions, to below the limit line.

I must sound a note of caution here: given the large dimensions of the structure being analysed when compared with the wavelengths at frequencies as high as 500MHz – it is strictly incorrect to derive the simple stray LC-resonator equivalent CM circuit shown in Figure 9.

However, my crude and simple stray LC-resonator approach at least identified the problem that had perplexed several experienced engineers for some weeks and made a quick fix possible. So as a project de-risking method it proved very valuable.

Discussions on Single-Conductor Resonating Structures

It is easy to see that something like the single-conductor structure of Figure 9 is not a cavity resonator like those depicted in Figures 3, 4, and 5, which is why I have called it a stray LC-resonator.

When parts of the structure of a cavity resonator get rather thin or narrow their stray series inductance might become so significant that – in conjunction with the stray shunt capacitances between the larger areas of the structure – they change from being cavity resonators, to being stray LC-resonators.

For example, consider the mezzanine board in sketch c) in Figure 5.

If the two mounting pillars on the right are insulated, and if the interboard connector is a simple low-cost ‘stake’ type with only one pin connecting the 0V planes of the mezzanine and main boards together, the inductance of this connection combined with the stray capacitance between the two boards could easily make the structure act as a stray LC-resonator instead of a cavity resonator.

A worked example: if the daughter board was 100mm x 50mm and was 10mm above the motherboard, and if both boards had full-sized 0V plane layers connected by one connector pin, then we can assume the stray series inductance is 10nH (because a thin conductor on its own has about 1nH per mm) and the stray shunt capacitance is 4.4pF (because C_stray in air = 8.75A/d femtoFarads, when A is the area in square millimetres and d is the spacing in millimetres).

1/2π√(LC) with L=10nH and C=4.4pF works out to be 759MHz, whereas the lowest cavity resonant frequency for the mezzanine board, the (1,1,0) mode, see Figure 3, is 3.35GHz. This stray LC resonance at 759MHz is likely to be a problem for most modern microprocessors even if they are clocked at 100MHz or below, because their harmonics typically extend to at least 3GHz.

Using a single row of 24 pins spaced apart by (say) 2mm in a stake connector the same width as the mezzanine board and connecting them all in parallel between the two 0V planes, would reduce the stray series inductance to about 0.5nH, increasing the stray LC-resonance of the structure from 759MHz to 3.4GHz – about the same as its cavity resonance.

At 3.4GHz the noise level of the clock harmonics is probably at least 13dB lower than it is at 759MHz, so – all else being the same – we should expect to see at least 13dB lower emissions due to using a row of 24 pins to connect the 0V planes together, instead using just a single pin.

De-Risking Projects from Structural Resonators

Being able to de-risk the EMC of a project is a project manager’s dream – usually (but incorrectly – see [4]) assumed to be impossible.

Now that we have some understanding of the problems that can be caused by structural resonances in single conductor systems (such as ‘ground’ systems), we can de-risk our projects from them. All of the other possible SI, PI and EMC problems can also be de-risked in early project stages, see [6], but these are outside the scope of this article.

The situations we have to deal with are almost always much more complex than shown in Figures 1-3 which assume ideal rectangular shapes, perfect conductors, perfect metal seams, and nothing but plain metal surrounded only by air with no other conductors, PCBs, devices, etc., present. The case study in Figures 6, 7 and 8 was such a simple structure – just two circular metal plates connected together in their exact centre, with nothing but air in-between them. Likewise, the way I have calculated the stray LC-resonators above, and the case study in Figure 9, only consider very simple structures.

We need to be able to analyse the real structures of the products and systems we create, and then we need to be able to deal with any problems that we find, so that our design/development timescales are not delayed and our first physical prototypes pass their EMC tests first time.

Field and Circuit Simulators

A very powerful and valuable tool is a three-dimensional (3D) field simulator. There are many of these available for under ten thousand U.S. dollars per seat, increasing in price and functionality to several hundred thousand U.S. dollars per seat, see [7].

Even low-cost 3D simulators should be able to handle cables (treating them as metal pipes), planes, and boxes with any shapes, sizes, holes and gaps, arranged in any way.

PCBs are modelled in such simulators as thin metal plates. Imperfect conductors can be simulated, usually chosen from a materials library that contains for example mild steels: plain, tin-plated or galvanised; aluminium: plain, Alochromed or Iridited; etc. Shapes can include any angles and sections of curves, and the RF-bonding between different parts can usually be simulated as point contacts, lengths of welded seams, or lengths of conductive gaskets of various basic types.

They allow plane and cavity structural resonances to be simulated, but for the stray LC resonances it will be necessary to create equivalent circuits using SPICE or similar circuit simulators and plug in the values of the relevant stray series inductances and stray shunt capacitances extracted from the 3D simulation.

Even relatively simplistic and crude simulations such as these will highlight EMC problem areas and are best used right at the start of a project to guide the design of the general assembly to help de-risk a project’s EMC.

Workshop Bench Testing

Creating physical representations of metal structures and ‘testing’ them on an ordinary workshop laboratory bench with low-cost equipment is a quick alternative to 3D simulation, and they can also help verify (or not!) simulation results that are hard to understand. (If lab bench tests on ‘mocked-up’ assemblies give very different results from a simulation, it strongly implies that something that should have been modelled in the simulator, was not.)

Some engineers will prefer the virtual environment of the 3D simulator, while others will find they get more of a practical understanding from bench testing. Both are valuable in their own ways.

In [8] and [9] I wrote about a number of ways of using low-cost bench-testing to quickly de-risk the EMC issues associated with a project’s mechanical design. I thought that the best way to show how such bench tests could help de-risk plane, cavity, and stray LC-resonators in single-conductor structures was to actually do some experiments of my own, and these are described below.

My Workshop Bench Experiments

RF reference planes are often called ‘ground reference planes’ (GRPs), but I prefer not to use the word ‘ground’ because it often leads to a confusion with ‘safety grounding’ that causes many projects delays every year.

I constructed an RF reference plane measuring 1000mm x 500mm from two sheets of perforated zinc-plated steel, then mounted a blank single-sided PCB approximately in the middle of the reference plane, on six insulating plastic pillars: one near each corner, one in the middle of each long side, and one in the middle as shown in Figure 10.

The board’s dimensions are 250mm x 150mm and its single copper layer is on the top side, 15mm above the reference plane.

In subsequent experiments I replaced some or all of the plastic pillars with metal pillars that RF-bonded the top copper PCB plane to the reference plane, to investigate the effect on the structural resonances.

Figure 10 also shows the two hand-made probes that I used to couple noise into, and pick up noise from, the structures I created. The construction of these probes, and the reasons for their method of construction, are discussed in detail in [8].

One probe is connected to the RF output of the tracking generator accessory of the (very low-cost) Rigol DSA815 9kHz – 1.5GHz spectrum analyser. This is the noise source. The other probe is connected to the spectrum analyser’s RF input, to pick up any noise from the source and detect any structural resonances.

At first, the probes are placed at a distance from the mounted PCB and the edge of the reference plane to check their coupling over the frequency range 500MHz to 1.5GHz, shown as the yellow trace in Figure 12. The ripple is caused by the fact that neither probe is matched to the 50 Ohm impedances of the spectrum analyser and the coaxial cables used. Ignoring the ripple, we see no resonances, merely a reasonably smooth increase in ‘noise’ coupling from 500MHz to 1.5GHz.

(These probes do not excite and pick-up the resonances in the reference plane because, as described in [8], they use ferrites in their ‘handles’ to reject the effects of reference plane resonances, and to reduce the effects of hand capacitance.)

After obtaining the yellow ‘background coupling’ trace from the set-up of Figure 10, I moved the probes to being partly underneath the middle of the short edges of the mounted PCB, as shown in Figure 11, giving the results shown as the pink trace in Figure 12.

Placing the probes part-way under the middles of the short edges of the ‘floating’ PCB has generally increased the coupling between them by about 20dB across the range 500MHz to 1.5GHz.

Four resonances are visible – two of them very clearly shown at 923.3MHz and 1.245GHz. The other two resonances are out of band: one of them at/below 500MHz and one at/above 1.5GHz. The 923.3MHz resonance is 32dB above the (yellow) background coupling trace, and the 1.245GHz resonance is 26dB above it.

Calculations using the formula in Figure 3 predicts two resonances below 1.5GHz for the PCB ‘floating’ above the reference plane: the (1,1,0) mode at 1.116GHz and the (2,1,0) mode at 1.562GHz. These are rather different from the resonances measured in Figure 11, and the reason for this is the presence of the FR4 board dielectric in the cavity below the top copper plane of the single-sided board.

The effect of a 1.5mm thickness of FR4 epoxy/glass fibre material, which can be roughly assumed to have a nominal dielectric constant of 4 in this frequency range, inserted into the 15mm spacing between the top copper board plane and the reference plane, is to slow the velocity of propagation a little. This has the same effect as assuming that the space between the two planes is full of air (i.e., no board dielectric is present) but the dimensions have all increased by a little, see [10] on how this is dealt with in the concept of ‘electrical length’.

1.5mm of a dielectric constant of 4, plus 13.5mm of air, has the effect of increasing all the ‘electrical lengths’ by a factor of 30%. Plugging these 30% larger dimensions into the X, Y and Z dimensions predicts resonances at 897MHz and 1.2GHz – close enough to the values measured by Figures 11 and 12 (923.3MHz and 1.245GHz) to be considered exactly correct in crude benchtop experiments like this!

The (1,2,0) mode resonance is predicted for 1.61GHz and is probably the resonance that we can see beginning to build up at 1.5GHz – at the right-hand side of Figure 12.

(Don’t worry that the calculation method in Figure 3 assumes metal boundaries on all six sides, whereas the experiment only has metal on the top and bottom boundaries and air for the other four. Changing a pair of opposing boundaries from metal to air only affects the phases of the resonant modes – i.e., whether it is the E- or the H-fields that reach a maximum at those boundaries – it does not affect the resonant frequencies themselves. And since the crude probes I was using both emit and pick-up E-fields as well as H-fields, we don’t really know whether we are measuring one or the other, or even both at the same time! Usually, with quick low-cost lab-bench tests like this, we simply don’t care.)

Next, I replaced the two insulating pillars at the ‘noise source’ end of the PCB, with metal pillars – making sure that each one had no more than a couple of milliohms resistance between the copper PCB plane on top and the reference plane on the bottom, and applied the two probes as before as shown in Figure 11. The results, which still contain the yellow background coupling measurement from Figure 10, are shown in Figure 13.

Figure 13 shows us that the (2,1,0) resonance is hardly affected by this change (1.12GHz instead of 1.245GHz), but the (1,1,0) resonance appears to have moved from 923.3MHz to 700MHz. Could this be caused by a ‘stray LC’ resonance?

At 15mm long and 10mm diameter, [11] tells us that we should expect each metal pillar to have a stray series inductance of 4.1nH. If we simply assume they are in parallel, then because they are much further apart than their height, we can estimate the overall series inductance is 2nH. Using the C_stray in air = 8.75A/d femtoFarads equation from before and allowing for the effect of 1.5mm of FR4 in the 15mm gap, we can simply estimate the stray shunt capacitance to be 37pF. Calculating 1/2π√(LC) predicts a resonance at 585MHz.

This prediction is significantly below the 700MHz observed in Figure 13, which is hardly surprising because at this sort of frequency the dimensions concerned are significant fractions of the wavelengths, so simple calculations of parallel inductance are not accurate.

However, like the display screen example earlier, my crude and simple stray LC-resonator approach has at least identified the presence of an unwanted resonance, even if not very accurately. And – apart from the notch filter approach mentioned in the large-area display example earlier – almost all of the EMC mitigation measures we would apply to solve a resonance problem are not very frequency-specific. So, the error doesn’t actually matter.

I went on to perform a number of other experiments over a period of a day, which found the following:

1. 1. With all four corner pillars replaced by metal pillars, the resonances became much the same as for the floating PCB, and the lower-frequency stray LC-resonator around 700MHz was no longer present. This is the pink trace in Figure 14.
   2. This tells me, for example, that I should not expect RF-bonding pillars at the four corners of a mezzanine board to have much effect on the resonance frequency of the cavity beneath the board. (The general guideline for substantially eliminating a cavity resonance under a PCB, is that RF-bonding pillars should be closer together than one-tenth of the wavelength at the highest frequency of concern all over the board’s area, not just around the perimeter, see [12]).
   3. Inserting a sufficient number of samples of ferrite-loaded elastomer RF absorber into the cavity below the PCB was able to remove all the resonances, as shown by the blue trace in Figure 14.

Fixing copper tape with conductive adhesive along both long sides of the board created the ‘waveguide-below-cutoff’ behaviour (see [13]) shown by the pink trace in Figure 15. Unlike the results of placing absorber in the cavity, there is now effectively no coupling between the probes that exceeds the yellow background coupling trace below about 900MHz.

Conclusions

Conductive structures that are a single node in a schematic diagram or circuit simulation, may be called chassis, ground, earth, 0V, reference, shield, frame, or indeed anything else. Regardless of what they are called, it should never be assumed that they are always at the same potential all over.

At radio frequencies such single-conductor structures can exhibit complex resonances with corresponding problems for EMC and even functionality.

This article identified two major types of single-conductor resonant structures, and described case studies for each:

1. 1. Plane and cavity resonators; and
   2. Stray LC-resonators.

It then described how projects could quickly and easily be de-risked from the possibilities of suffering these two types of resonances by using low-cost field and circuit simulators and/or low-cost bench testing on physical ‘mock-ups’.

The author conducted some bench tests on single-conductor resonant structures, and some results were presented and the rest summarised, with the aim of showing just how easy they are to perform and how very helpful bench tests can be in reducing design and development timescales and costs.

References

1. 1. “‘Accidental antennas’ and good electromagnetic (EM) design,” Keith Armstrong’s Blog, 6^th September 2018, http://www.emcstandards.co.uk/accidental-antennas-and-good-electromagneti
   2. “De-sense,” http://www.antenna-theory.com/definitions/desense.php
   3. “Module 16: Incorporating wireless modules (transmitters and receivers) into products,” training course notes available for purchase from: https://www.emcstandards.co.uk/integrating-wireless-modules-into-products
   4. “EMC techniques in electronic design Part 2 – Cables and Connectors,” Keith Armstrong: http://www.emcstandards.co.uk/files/part_2_text_and_graphics_21_may_09.pdf
   5. “Good SI, PI and EMC require this most of all…,” Keith Armstrong’s Blog, 7^th November 2018, http://www.emcstandards.co.uk/good-si-pi-and-emc-require-this-most-of-all
   6. “Introduction to EM Engineering,” Keith Armstrong, In Compliance Magazine, July 2017, https://incompliancemag.com/DigEd/icm1707/ also available from http://www.emcstandards.co.uk/
      files/introduction_to_em_engineering_incompliance_july_2017_as_published.pdf
   7. “Simulators for SI, PI, EMC Can Minimize/Eliminate Design Iterations – and Justifying Their High Cost is Easy,” Keith Armstrong webinar free at: https://emc.live/2015/simulators-for-si-pi-emc-can-minimizeeliminate-design-iterations-justifying-their-high-cost-is-easy
   8. “Using close-field probes to reduce design risks early in a project, Part 1” by Keith Armstrong, from: http://www.emcstandards.co.uk/files/close-field_probing_to_reduce_emc_design_risks_part_1_june_2014_pdf.pdf
   9. “Using close-field probes to reduce design risks early in a project, Part 2,” by Keith Armstrong, from: http://www.emcstandards.co.uk/files/close-field_probing_to_reduce_emc_design_risks_part_2__october_2014_1.pdf
   10. “Electrical Length,” https://en.wikipedia.org/wiki/Electrical_length
   11. “Round Wire Partial Inductance Calculator,” https://chemandy.com/calculators/round-wire-inductance-calculator.htm
   12. “Advanced PCB design and layout for EMC, Part 3 – PCB-to-chassis bonding,” by Keith Armstrong, 2010, http://www.emcstandards.co.uk/files/part_3_reference_plane_to_chassis_rf-bonding.pdf. For an improved and updated version, purchase the training course notes: http://www.emcstandards.co.uk/essential-pcb-designlayout-techniques-for-cost
   13. “Waveguide Cutoff Frequency,” http://www.electronics-notes.com/articles/antennas-propagation/rf-feeders-transmission-lines/waveguide-cutoff-frequency.php