CHAPTER 6 

LOSSES

This is the most important chapter of the book. It contains most of the hidden secrets. Wrapped up in the subject of cavity losses are the major compromises that make performance with economy possible. In this chapter we will talk about the concepts of energy transfer, skin effect, cavity proportions, and most important of all, bandwidth vs. insertion loss.

Energy Transfer

Let's begin by considering how energy, our receiver or transmitter signal, gets into and out of a cavity. We'll use a bandpass cavity, as an example, it's simpler. The basic principle is the same for all cavity types, however. Notice figure xx

Figure xx Energy Transfer in a Bandpass Cavity

Energy enters the cavity from the input transmission line and appears at the input loop. The loop, for all practical purposes is a simple inductor. It is obviously more complicated then that, but this is an adequate approximation.

Then the magnetic field around the loop excites the cavity into oscillation. In so doing, the cavity absorbs the energy delivered to the loop by the input transmission line. At the output loop the complement takes place. The magnetic field in the cavity excites the loop and the energy becomes RF current in the output transmission line. In the process the energy changed from current in the line to an E-H field in the volume of the cavity then back to current in the line.

If everything were as we would like it, this energy transfer process would be 100% efficient. Nothing would be lost and the cavity would be invisible to the line, at the resonant frequency. At other frequencies, of course, the losses are high. That's what we want. But unfortunately, even at resonance, duplexer cavities are not perfect -- there are losses. We must understand them if we are going to gain a practical understanding of duplexers.

Skin Effect

The single biggest loss that takes place in cavity resonators is caused by RF skin effect. In most devices, skin effect is not a serious concern, but in duplexer cavities it is a major problem. Energy as we have said is transferred in the cavity by the E-H field, but the field induces currents in the walls. If the walls were perfect conductors, this would be no problem. But the walls have resistance. 

As we have all know, the effective resistance of conductors at radio frequencies is higher than at DC or low frequency. As frequency increases, electric currents in a conductor concentrate in certain parts of the conductor. The rest of the conductor may as well not be there. Effectively the conductor becomes smaller and the resistance increases. At VHF and UHF the effect is significant.

Let's take the example of a round wire. Current in the wire creates a magnetic field around the wire as concentric lines of force. The magnetic field inside the wire, links only with current inside the lines of force but not with current outside the lines of force, nearer the surface.

This causes a progressive increase of inductance toward the center. The added inductance impedes the flow of current and the effective cross section of the wire is reduced. We can say that current tries to redistribute itself to be encircled by the smallest number of magnetic flux lines.

As we move away from round conductors the effect gets even worse. I mentioned earlier that wire is better than straps for making loops in cavities. This is the reason. Round conductors experience the smallest amount of skin effect loss.

How bad is skin effect? As I said, it is quite severe at VHF and UHF frequencies. Let's look at some numbers. The complete equation for calculating skin effect is very complex, but if we make a few reasonable assumptions we can cut it down to size.

Skin depth (inches) = .0026 / [root-2] Frequency (MHz)

Here is a table of this equation solved for common RF frequencies through UHF.
 
Frequency
Skin Depth
1 MHz 
.0026 in.
3 MHz 
.0015 in
10 MHz 
.00082 in.
30 MHz 
.00047 in.
100 MHz 
.00026 in.
300 MHz 
.00015 in
1 GHz 
.000082 in.

Skin Depth vs. Frequency

At 450 MHz, the skin depth is only .00012 inches. Current still flows below this, but decreases to 36.8% for every skin depth. By three skin depths the current is very small. Therefore, the effective thickness of the conducting surface of the inside of the cavity is roughly a mere .00036 in. Copper may be a good conductor, but as a skin this thin it has appreciable resistance.

Surface Treatment

The only practical way of minimizing skin effect losses is to electroplate the inside of the cavity with a more conductive metal. Since most of the current flows on the surface, this technique can be effective for some frequencies. Commercial cavities are often plated.

Silver, however, is really the only choice for plating copper cavities. All other metals used for electroplating have lower conductivity, even gold. Contrary to popular opinion, gold is not used on connectors for conductivity but for corrosion resistance. Copper is a better conductor than gold. Silver is only marginally better than copper.

Below roughly 1 GHz, however, the cost of silver plating is very high for the small benefit obtained. The skin depth is thick enough to require a thick plate. Therefore silver plating is only practical for the higher UHF frequencies and above. At 450 MHz and below, especially for home construction, it is a waste of time in an attempt to lower skin effect losses. Bare copper is perfectly adequate.

Well then, what about corrosion, you may be thinking. I had the same question when I first started making cavities. So I did an experiment. I took one of my cavities that was extremely dirty, and measured its performance. I hadn't cleaned it after soldering the center conductor. It was crusty black with corrosion.

Then I took it all apart and polished the pieces to mirror cleanness and carefully reassembled it, making sure that everything was in the same position. I was dumbfounded to discover that the performance was the same. Apparently the corrosion or patina, as it is called, that collects on the surface of copper is conductive. It is also not very thick. Here skin effect works for us. Since that time, I only give the inside of my cavities modest cleaning for aesthetic reasons.

One place that electroplating is effective, however, is copper on steel. Copper as compared to silver is relatively inexpensive, even in a thick plate. Also, steel is a much poorer conductor than copper. Several commercial manufacturers make excellent cavities out of copper plated steel. I have not attempted it in my home workshop, though I suspect that it would work well.

Cavity Impedance

The second major loss issue in duplexer filters is the characteristic impedance of the cavity. Remember, a typical duplexer cavity is a short length of large diameter, open air transmission line. Like any transmission line it has a characteristic impedance. In open air, the characteristic impedance is determined mainly by the relative diameters of the inner and outer conductors.

One of my first question was, does the characteristic impedance of the cavity have anything to do its performance as a filter? The answer is yes, quite a bit. In my early studies, I was surprised to find out that there is a "magic ratio" for the inner to the outer diameter of the conductors. Notice figure xx. It can be found in most books on transmission line theory.


Figure xx Total losses in a transmission line vs inner to outer 
conductor diameter ratio and characteristic impedance.

You will notice that losses become very low when the inside of the outer conductor is 3.6 times as large as outside of the inner conductor. This ratio, using athe commonly available formula found in many books, translates to a characteristic impedance of 77 ohms. 

As a sidelight, we also see why 75 ohm coax is normally used for receiving purposes, notably for TV cable. At this characteristic impedance, coax has its lowest losses? Why then, you may ask, do we use 50 ohm coax for transmitters? Don't we want transmitting coax to be low loss too? Yes, but 75 ohm coax can't handle nearly as much power as lower impedance cable. For example, RG59 can only handle half as much power as RG58. 

The best power handling capacity in coaxial line is achieved at roughly 30 ohms characteristic impedance, but cable this low in impedance is almost impossible to manufacture. There would be almost no space betwen the inner and the outer conductor. 50 ohms, therefoe, is a good compromise both for carrying RF power and for low loss. 

In a cavity, however, losses are of paramount important. The power is easily handled by the large diameter of the cavity.So we use the optimum impedance of 77 Ohms. As we've seen 77 Ohms translates to an outer to inner diameter ratio of 3.6 to 1. Small variations are not too critical. For home construction, common tubing sizes are close enough. For example, you can use 1/2 inch copper water pipe with 2" pipe. 3/4 inch and 3 inch pipe work well together. Both are sufficiently close to the magic 3.6 to 1 ratio.

When I learned about this magic ratio, I wondered, "Doesn't that create a mismatch between the 50 ohm coax and the cavity?" Yes it does, but it does not matter. The mismatch is actually much greater than this. I'll have more to say on this later when we talk about lines, but for the moment, let me state a Golden Rule. 

Any mismatch that takes place at the input of a cavity is reversed at the output if the loops are symmetrical.

Bandwidth Verses Insertion Loss

Now we come to the most important duplexer concept of all, bandwidth vs. insertion loss. If everything were as we wished, a duplexer filter would pass only the frequency we wanted and would totally reject all other frequencies. What's more, there would be no loss at the resonant frequency. In other words the bandwidth would be extremely narrow and the insertion loss would be zero. As we know, this does not happen in practice.

It is possible to have narrow bandwidth. It is directly proportional to the Q of the cavity. The Q, as we have seen, is determined mostly by skin effect losses. If we Assume that we have the ideal conductor diameter ratio of 3.6 to 1, a reasonable approximation for the Q of a copper cavities without a load is,

Q = 107 x Diameter(in.) x [root-2 Frequency(MHz)]

From this we can determine the bandwidth by,

Bandwidth = Frequency / Q

Here is a table calculated from these equations for common sizes of outer conductor, at 450 MHz.
 
Diameter 
Q (unloaded)
Bandwidth(MHz)
1 in. 
2300 
195 kHz
2 in. 
4500 
98 kHz
3 in. 
6800 
65 kHz
4 in. 
9100 
49 kHz
5 in. 
11300 
39 kHz
6 in. 
13600 
33 kHz

Q and Bandwidth of Copper Cavities, with 3.6:1 diameter ratio, at 450 MHz.

Notice the bandwidth figures. You may be surprised that they are so narrow. If you've had practical experience with duplexers, you probably were expecting Megahertz not Kilohertz, and you would be right.

The reason for the difference is quite simple. The values are for unloaded cavities. In use, duplexer cavities exhibit far poorer bandwidth because they are loaded by the external equipment connected to them. In a real duplexer, there is a 50 ohm load on both the input and the output. One is the 50 ohm load of the antenna at the tee junction. At the other ports, the duplexer sees the 50 ohm load of either the receiver or the transmitter. Each individual cavity also sees the same loads as its loads, since in a well designed duplexer the cavities downstream are more or less transparent.

Therefore, every cavity is doubly loaded by 50 ohm loads. What this does is to establishes a new effective working Q for each cavity called the loaded Q. It is far, far less than the unloaded Q given in figure xxx. That's why the working bandwidth of duplexer cavities is much broader.

The Effect of Coupling

Before we can get an idea of how much the external loads reduce the unloaded Q of the cavity, however, we must look at another process that happens inside a cavity, that is, coupling. The 50 ohm load impedance created by an external device is fixed, but the amount it actually loads the cavities is not.

The amount of loading depends on the arrangement of the loops in the cavities. You will remember that their size and orientation to the magnetic field determined how much they couple to the cavity. If a loop is made large enough and is correctly positioned, it will place the entire external load on the cavity. If, however, it is made smaller or is rotated away from the magnetic field it will not place the entire load on the cavity. This is the first of the two most important cavity concepts of all. The tighter the coupling, the wider the bandwidth.

Well, you might be saying, then let's never use tight coupling. The bandwidth will always be good. That's a valid idea, except for one thing. It is the second most important rule of all. The lighter the coupling the worse the insertion loss. Notice figure xx. Here I show the same cavity with the loops progressively adjusted for light, medium and tight coupling. Notice how the bandwidth decreases as the insertion loss increases. We can't have both -- it's simple physics.

If we do not care about how much loss is present in a duplexer than small cavities can provide all the Q we need. But if we don't want to give up receiver sensitivity and transmitter power to insertion loss, then we must use bigger cavities and couple more tightly to them. The loaded Q of a tightly coupled larger cavity may be the same as the loaded Q of a lightly coupled smaller cavity, but the insertion loss will be much less.

A good illustration of this takes place in the helical resonators that are used in good receivers. They achieve extremely narrow bandwidths by allowing large insertion losses. The coupling is made very light and the working Q is excellent, but the signal loss is high. It's made up for by amplification in the front end of the receiver.

A Basic Compromise

It should be obvious then, that a balance between bandwidth and insertion loss must be established in all practical duplexers. If we insist on too little insertion loss, the bandwidth will be too great. If we strive for too narrow a bandwidth, then insertion loss will be excessive.

So what kind of a compromise do we make? The answer to that question is, "It depends on the situation." There is no fixed answer. Each individual installation has a different answer, but there is a starting point. I call it critical coupling.

Notice figure xx again. If you begin with light coupling, curve A, and then begin to tighten the coupling, the bandwidth will only slowly begin to increase, but the insertion loss will rapidly decrease. If you continue, suddenly you will reach a point, curve B, where the conditions will reverse. Bandwidth will begin to increase dramatically and the improvement in insertion loss will slow down. The transition point I call critical coupling. It is graphically illustrated in figure xx.

The importance of the point of critical coupling is that it represents a maximum in duplexer design. Adjusting the coupling tighter than critical is counter productive. The additional small improvement that you will obtain in insertion loss is not worth the price of the great increase in bandwidth.

Therefore, if one is designing a duplex for general purpose applications, then you should set the loops for critical coupling. Most commercial duplexers are built this way. I also begin with critical couping in my home-brew designs. I make the loops just large enough to achieve critical coupling when they are placed exactly perpendicular to the magnetic field. Actually, I make the loops just a little larger and twist them slightly to achieve critical coupling. As we learned earlier, the results will be the same.

A very significant point to make here is to ask the question, how much loss will we have at critical coupling? This is where the uninformed duplexer user often makes a mistake. It is a common misconception to expect too little insertion loss and for it to be the same for all cavities. By now you should realize the falacy in both of these opinions. As we have learned, insertion loss is determined almost entirely by skin effect, which is a function of cavity diameter. The informed user needs to realize that insertion loss is a fact of life and to have some idea of what they will be. Figure xx gives my own rough guidelines, based on experience, for for cavities of common diameter at 450 MHz. We'll discuss other frequencies in a later chapter.
 
Diameter  Insertion loss
1 in. 1.5 dB
1.5 in 1 dB
2 in. .7 dB
3 in. .5 dB
4 in. .4 dB
5 in. .3 dB
6 in. .25 dB

Figure xx Insertion loss in unplated copper cavities, at critical coupling, 
at 450 MHz, with an i/o diameter ratio of 3.6:1.

The figures I have given are for cavities installed in a completed duplexer. Therefore, a small amount of the loss is due to other factors. With a little care you may be able to achieve slightly better results. I prefer to work with worth case expectations.

The Next Step

Now that we have a starting point, critical coupling, we are going to take another step that most duplexer users do not take. We are going to knowledgeable reduce the coupling to acheive a better balance between economy and performance. An oversized, overprices critically coupled duplexer can always be made to work well, but it is not the best economy.

But before we can hope to achieve an effective balance, we must get some things straight about how much loss is acceptable in a duplexer. Because, if we are going to back the duplexer off from critical coupling we are going to get more loss. It will be to our benefit, but it brings up an attitude problem. Duplexer uses do not like to accept losses

First let's consider the transmitter side. You probably won't like this, but the two-way industry is generally willing to accept up to a 3dB loss in transmitter power to achieve adequate bandwidth. That's a 50% loss in power. It's not what we're shootinf for, it is just our upper limit for acceptable insertion loss. Even so, knowing the amateur radio community as I do, I expect a lot of screaming and shouting over this point. Amateurs traditionally want every microwatt of power they can get. Commercial operators usually know better.

Unless you have all the money in the world and have all the space you need in a repeater cabinet, losses as high as 3dB must be accepted in many practical repeater installations. Often we can do better, but in adverse situations, 3 dB loss is entirely acceptable.

Besides, losing half your power in a worse case situation. is really not that bad in terms of repeater performance. Especially at UHF, but even at VHF, the service area of a repeater with half the power is not that much smaller. The terrain is a much bigger factor than power. That's why most repeaters are less than a hundred watts. A few watts is more than adequate. A kilowatt is not necessary, as we all know. So there is no need to cry over losing half of your power in a duplexer if it will give you large gains in economy with adequate performance. A couple extra watts is not worth the price.

The same is true for the receiver side of a duplexer, except that the range of acceptable losses is even greater. Again we would like as little loss as possible, but once again it is not always that important. I've seen several cases where intentionally increasing the insertion losses on the receive side in excess of 3dB has improved the repeater's performance. It is an option always worth investigating.

As we saw in an earlier chapter, the noise floor at your particular repeater site is the biggest determining factor of how much sensitivity you need. Review figure xx of chapter xx. If you have, let's say, a consistent noise floor of 100 dBm at your site, you only need an overall sensitivity of roughly xx microvolts in your receiver. Then if your receiver has a sensitivity of xx microvolts ( -1xx dBm), simple math says that you can afford to take a xx dB loss in the receive side of your duplexer without any loss in performance.

If you do, you will benefit greatly in bandwidth. Recall the bandwidth improvement of figure xx. Striving for excess gain and minimum insertion loss is not always the best choice. You will gain much more by using only enough gain and insertion loss to let your receiver hear efficiently, and no more. I can recall the first repeater site on which I saw this principle applied. I had adjusted the duplexer for minimum insertion loss on the bench. When I put it on the hill top it worked okay, but I was experiencing a little interference from a nearby paging transmitter. So reluctantly, I twisted the loops in the receiver bandpass cavities for increased insertion loss and narrow bandwidth. The interference immediately went away, and to my great surprise the repeater could hear far weaker signals. Case in point.