to go to our page on Doherty combiners
here to go to our page on balanced amplifiers
here to go to our main page on couplers and splitters
New for August 2012! Learn about the effect of isolation load impedance match on the balanced amplifier output match.
New for July 2012! Our section on graceful degradation has been upgraded.
New for May 2011! Click
here to learn about a special consideration for solid state
here to learn about the load-pull effects on quadrature and
here to check out a presentation that the Unknown Editor gave
at a IEEE chapter meeting in December 2008 on the topic of power
combiners (and much more!)
We've posted a wide variety of
couplers and splitters. It's time we
explored one of the main applications of splitters, which is combining
amplifiers. This will take some time, and some class participation.
For December 2008, there are some new examples using the Lim Eom
network which are linked from the bottom of this page.
Combining can be done with reactive
splitters, in phase matched splitters (Wilkinsons), 90 degree hybrids
and 180 degree hybrids. It's important to understand the trade space
of why you'd pick one over another.
The two signals must be coherent,
that is, they must be at the exact same frequency, and close in
phase and amplitude. Being at the same frequency is not an issue
when you are combining power amplifiers, because the signals started
out from the same source before you split it. Stop here if you don't
get that point, and consider your brother-in-law's offer of going
50% on a Quiznos
franchise... at least you'll always have toasted snacks coming
Why power combine?
All power amplifier technologies
have limitations. Power combining allows you to overcome them. By
far the most active area of power combining is in solid state power
amplifiers, often abbreviated SSPAs.
Over the years the holy grail
of power combining is to replace a microwave tube with a solid state
power amp. At first, tube engineers laughed at the puny efforts
of solid state engineers, it takes a ridiculous amount of one watt
amps to replace a hundred or even kilowatt amp. But over the years,
as GaAs became pHEMT, and silicon carbide and GaN technology developed,
suddenly they (we) don't look so stupid. Once you get to a magic
number of eight devices needed to form the tube replacement, it
becomes doable. How did we pick eight as the magic number? It's
In a binary combiner, when the
loss per combined section is -1 dB (not a very great splitter),
the max power output would occur when eight amplifiers are combined.
Sixteen amplifiers combined would have less power than eight in
this case. Also, consider that the yield of an assembly with more
than eight power amplifiers can be very low, and rework sometimes
creates more problems than it solves.
Solid state power amps are not
the only reason for power combining. High-power transmitters that
need to be highly reliable (like television transmitters) can employ
redundant tube amplifiers that are power-combined to make up to
50,000 watts of power. Because the power combiner allows the amps
to be isolated from each other, it is possible to turn on one amplifier,
and leave the other one (or two!) as spares for when the first one
burns out. These are very special systems, the isolation loads take
a pounding when an amplifier is turned off. Often the loads are
liquid cooled, that's the best way to remove 10 kilowatts of unwanted
dissipation! Perhaps one of our readers can send
us more information on this interesting topic...
The bandwidth of many power combining
schemes is very narrow, but in some cases that won't matter. For
ultra-wideband applications, stay away from spatial combiners and
coupled designs. A corporate structure using wideband Wilkinson
power splitters can provide as much bandwidth as anyone really needs.
Update August 2011! Wait,
we have to eat our words about bandwidth, thanks to Mark! Here's
a spatial combiner that offers 2-20 GHz:
If you see those CAP guys, tell
them they owe Microwaves101 for that free plug...
Maintaining low loss in a combiner
is very important, especially in a corporate combiner. If you lose
1 dB in each stage, by the time you get to a three stage combiner
you're throwing away half of your output power! Often overlooked
are the routing losses associated with distributing the splits out
to all those amplifiers... who cares if your Wilkinson has 0.2 dB
loss if you lose 0.5 dB routing to the next split? For "N-way"
efficiency, nothing beats a radial combiner.
There is a famous paper about
binary versus spatial combining by Robert York, we found an on-line
copy here. This paper is referred to simply as "York's
Paper" around the industry.
The size of will be a function
of the technique, as well as the media. For example, a radial combiner
using stripline will be much smaller than a radial combiner using
Isolation is one of the most
important qualities of a combiner network. Ideally you don't want
any of the amplifiers to "see" each other. In practice
it is often difficult to achieve 20 dB isolation between all branches
of the network, but that might be enough. The most common problem
of poor isolation in a combiner is that spurious oscillations can
This section was (mostly) re-written for July 2012. Enjoy!
If you are to be involved in designing solid state power amps, it is important that you understand this entire section on graceful degradation, especially the part about failure isolation.
Below is a simple Wilkinson power divider, with one leg terminated in a short circuit. We have added a transmission line that is controlled by a swept variable to move from 0 to 360 degrees in 45 degree steps. Click the images to zoom in. This is our way of simulating what the power splitter will do when an amplifier fails; the failure will likely be a short or an open, but you won't know the phase, so here we will examine the full range of phases possible, thanks to Microwave Office. The insertion loss (purple) is 3.01 dB at center frequency (left Y axis), just as it would be if that reactive termination was 50 ohms. When driven as a combiner, the loss of an ideal Wilkinson is zero dB, as two coherent signals add up. When there is only one signal, you get 3 dB loss in an isolated, 3 dB splitter.
The match seen at port 1 (red, which will be at the output of a two-way power amp) is 6.02 dB at center frequency (a 3:1 mismatch, see right Y axis). Your transmitter system is not going to like this, possible failure is just one great reason to put an isolator on the output of an SSPA (the other great reason is to prevent load pull). Perhaps most importantly, the match seen at Port 2 (brown) is what the remaining power amp will see. It is still perfect at the center frequency, or at least as perfect as what the SSPA will see in the system, because you put a circulator between the antenna and the SSPA. You are going to add a circulator, unless your spouse is correct and you never listen, and he/she is probably fooling around with the yoga instructor who is a great listener... sorry you had to hear about that here of all places.) This is failure isolation at its best, the blown amplifier has not compromised the remaining amplifier's impedance environment.
What about the assumption that a failed amp presents an open or short circuit? When an amplifier is barbecued through mishandling, this is likely the case. But don't believe everything you read on a free site on the internet. An amplifier could fail if its input RF bond comes loose. In this case the input would be an open circuit, while the output might present a 3:1 mismatch to 50 ohms. Does this change the analysis? Think about that for homework and send us the answer and we'll post it.
Where does the power go, when you lose that 3 dB? It depends on the direction you are passing energy. When you apply power to Port 1, 25% of the power of that power is reflected (3:1 mismatch) and 25% is dissipated in the isolation resistor. This is the case of the power divider on the input of a two-way SSPA. When you apply power to Port 2, half the power is dissipated in the isolation resistor, and none is reflected (at least under ideal conditions at center frequency). This is the case of the combiner on the output of a two-way SSPA. If you want to design for fault tolerance, that isolation resistor should be sized to handle half of the power of one of the amplifiers that drives past it. Two-watt power amps? The isolation load load should handle one watt. Unless you are using a Gysel combiner, which has two loads (in that case they each need to handle 1/2 watt).
The return loss at Port 1 is worth thinking about. Why is it exactly a 3:1 mismatch? In the easiest case to analyze, TL1 is 0 degrees and the lower arm sees a short circuit. Looking back from Port 2, the 100 ohm resistor is in parallel with the system impedance (50 ohms) and so behaves as 33.33 ohms. The 70.7 ohm quarterwave TL2 makes that into 150 ohms (a 3:1 mismatch). The shorted stub TL3 has no effect as it is a quarterwave. As for analyzing other cases where TL1 is not a short circuit, what do you think this is, an IEEE article? You can verify that the length of TL1 has no effect on the mismatch, using linear simulator like Microwave Office.
A Microwaves101 Theorem.... (drum roll...)
Let's propose a theorem, and offer no analysis (yet) to back it up. For any two-way, ideal, resistor-isolated coupler, if you disconnect one of the arms (open or short), the common port will see a 3:1 mismatch. Thus a two-way SSPA is doomed to have just 6 dB return loss (or worse) at either port when one amplifier fails.
Now let's look at a simple reactive splitter (a Wilkinson without the resistor). If one leg is connected to a short circuited transmission line, what happens? The insertion loss (purple, left Y axis, not the scale is -40 dB to 0 dB) can be anywhere between 0.511 dB and tens of dB (infinite in the ideal case). The 0.511 dB case results when the arm is shorted, such that if forms a quarter wave stub (open circuit), and the circuit looks like a 70.7 ohm transformer, so both S11 and S22 look like 100 ohms, or a 2:1 VSWR, or 9.546 dB (best-case) return loss. Although the "loss" is quite low in this one case compared to what happens in a Wilkinson, your power amplifier won't be happy, as you are load pulling it considerably. Note that because this is a passive reciprocal network, the magnitude of S11 is equal to the magnitude of S22 and the brown and red lines lie on top of each other. In the worst case, the return loss is zero. Why would anyone design an SSPA using a reactive splitter? Simple answer: they don't.
Reactive splitter isolation test
Now let's look at what happens in various two-way SSPAs when one amplifier fails. We chose 20 dB gain for our amplifier, and we put in sections of transmission lines to rotate the phase of the failed amp (U2, which is grey because it is discombobulated).
Let's look at the loss of gain in four cases, using a Wilkinson, a branchline, a rat race, and a Gysel. The brown line represents the gain of the SSPA when both amplifiers are functioning. You will notice that the Wilkinson has the most tolerance to failure, in that its gain deteriorates less across the bandwidth. We didn't plot S11 and S22, and they are both -6.02 dB at center frequency for any phase in the failed amp. Note that at center frequency, the gain of 20 dB is reduced by -6.02 dB (75%) and you have to pass through two isolated splitters that are no longer combining the coherent amplifier outputs.
Note that six dB of gain does NOT imply that you have lost 6 dB of available power. You lost three dB on the input side, so if you can drive the SSPA sufficiently, you have only lost 3 dB of available power. Note that the available power was decreased by 3 dB, as you lost one of the two amplifiers, so compared to the output power of two amplifiers, you are down 6 dB. Thanks to Austin for asking us to clarify that!
What happens in the case of a coupled-line coupler used in an SSPA (such as a Lange)? In this example we used an ideal coupled transmission line which has 3 dB split at center frequency. It is not intuitive, but coupled-line couplers used in power combining are immune to the effects of the phase angle of the reflection coefficient of the failed amp. What does this say about our forefathers at companies such as Avantek using thousands of Lange couplers to build power amplifiers back in the day? They were pretty smart!
How lets look at a four-way amplifier. In the case below, we have a four-way Wilkinson combiner which was used to create an SSPA. Gain drops by 2.5 dB when one amp fails, which is -20xlog(3/4). We also plotted the reflection coefficient, it is now 12 dB instead of -6 dB when we had a two-way amp with one failure.
How about an eight-way? This time we used a corporate structure. We disabled only the bottom amplifier and ran the circuit through all phases. If you want to know what happens when two amps fail in an eight way, maybe you should consider a TWT if your MMIC amps are so unreliable...
The result is that the gain drops by just 1.16 dB (now 18.84 instead of 20 dB), which is -20xlog(7/8). The reflection coefficient is now almost -18 dB. Do you sense a pattern (or two) going on here?
When you combine power sources
(power amplifiers driven at the same phase angle), if you have good
isolation, an amplifier can fail in the network and output power
of the network will degrade "gracefully", as opposed to a single source that fails catastrophically.
However, it isn't as simple as
calculating the fraction of power amplifiers that are left operating
(thanks to Shane for the correction). Some of the "leftover"
power gets dissipated in the isolation resistor network. You have to consider the available power, which is degraded by some dissipation in the output network, which gives rise to a squared term.
Time for a Microwaves101
rule of thumb!
In an N-way combiner, if one or more amplifiers fail, the gain and delivered power will both (ideally) be reduced to the square of the fraction of
working amplifiers provided that your system can enough drive power to keep the amplifiers saturated. For example, one failure out of eight results
in 76.6% power [equal to (7/8)^2] or -1.16 dB reduction. One failure out of four results
in 56.25% power (-2.5 dB). If one amp fails in a two-way combiner, you only
have 25% of the original combined power (-6.02 dB). Yikes! The available power drops exactly as you think it would, as (N-X)/N where N is the number of amps and X is the number of failures, while the delivered power goes as the square of fraction of available power. Something to consider if you are designing for the possibility of failure.
Below we have plotted available power, delivered power and dissipated power as amplifiers are lost in an eight way power amp with one watt cells. When one amp is lost, the available power is 7 watts, but only 6.25 watts is delivered as 0.875 watts is dissipated in the isolation network. Note that the isolation network dissipation is highest when half the amplifiers are off, as two watts are delivered from four watts available. By the time you get to just one amplifier on, only 1/8 watt is delivered from 1 watt available (7/8 watt dissipated).
There is a special case where you might want to operate just one amplifier in an N-way combiner: you might want to check the phase errors between the amplifiers by measuring S21 angle of each individually (see below). If you want to drive them to saturation, you need to increase the drive power significantly. In the case of one amp in an eight way, the drive must be increased 9 dB. Note that you should consider the ability of your isolation network to dissipate 7/8 of the power of one of the cells in this case. Exactly which resistors get pounded in an eight-way isolation network? We'll save that analysis for another day, but you should start thinking about it if you were to try this type of test. One more thing: the impedance match of an amplifier that is turned off (instead of failed) will likely have a real component, our analysis above assumed that the amp failed to an open or short circuit. If you are in the SSPA game, you should get S-parameters of your amplifiers in the off state so you can model this event properly. These are often called "cold S-parameters". If you plan to operate the amplifiers at a non-zero gate bias when they are inactive, you should take S-parameters of them under this condition.
Phase efficiency in an SSPA
Even your lunch is telling you to beware of phase errors...
Ideally, all of your power amplifier signals will add up coherently, and two plus two will equal four. Now wake up to reality: your power-combined amplifier
will have transmission phase errors, and these will cause you to lose power.
It's your job to minimize this problem! Phase errors can occur within
the power splitter, the individual amplifiers, or the power combiner itself. At Microwaves101 we call this term the "phase efficiency" of a power-combined power amplifier, but you can call it anything you want.
Below is a plot that we generated
using Excel, to compute loss in power versus RMS phase errors, in
this case for a four-way combiner. In computing degradation due
to phase errors, be sure to convert power to voltage first. If anyone
wants, we can supply the spreadsheet that did this calculation. Here are some data points that are "off the chart" for your contemplation: if RMS phase error is 45 degrees (like two amps out by 90 degrees) you lose 3 dB power, and half of your combined power goes to the isolation load. At this point, you added two amplifiers together and got the same result as a single amplifier. Pretty stupid on your part. Also, if the RMS error was 90 degrees (two amplifiers fighting each other at 180 degrees out of phase, which is the absolute worst you can achieve if you tried), all of the power goes to the load and it goes up in smoke.
Update August 2012: that curve above was generated by painstaking means of generating four signal voltages in Excel, varying the phase and totaling up the wave. Doesn't it look familiar? It should, because it is approximated by a cosine-squared function.
Let's take a look at an example where we calculate the RMS error of four amplifiers as we shift the phases apart. Below are the data on how we did this, A1, A2, A3 and A4 are the amplifiers, which all start out at 0 degrees phase. Then we gradually shifted first A1, then A2 toward 90 degrees, then A3 and A4 toward -90 degrees. RMS phase error is calculated in the far right column.
Now let's look at calculated phase efficiency versus RMS phase error, calculated both ways. It doesn't fit exactly, but it's really close. The voltage model curve will change depending on how the phase errors occur. If they were "more random", the two curves might lie on top of each other, don't you think? And who really cares if the fit starts to go south after 30 degrees RMS? Engineers need to focus on what is important, not what is unimportant, if you want to be a valuable to your company or customers.
Warning to plagiarists: let's pause here a moment and stake out the following claim: there is no other source of information that we can find that shows how SSPA phase efficiency fits a cosine-squared relationship with RMS phase error. So when you go off and publish a paper that includes an important fact that you learned here, be sure to remember and cite us as your reference. We'll track you down and tell your boss, this has happened before....
Now let's put out some Microwaves101 rule
#110: The phase efficiency is very nearly calculated as the cosine squared of the RMS phase error. If you want to have less than
0.5 dB combiner loss due to phase errors, make sure that your RMS
phase error is less than 20 degrees (equivalent to two amplifiers out of phase by 40 degrees). In the case of a two-way combiner, the phase efficient is [cos(phi/2)]2 where phi is the phase angle between them (RMS value is 1/2 of the two-way phase spread).
Here is the proof of how two equal amplitude signals combine to a voltage of 2Rcos(X/2). Power is the square of voltage. We'll clean it up someday when we are bored. Thanks for the help, Parrish!
What happens in an SSPA where many amplifiers are power combined, with uniform phase distribution? Another Rule of Thumb is appropriate:
#111: Following rule 110, suppose you pick amplifiers from the "Waffle Pak of Uniformly Distributed Phase", or WPUDP, spanning X degrees (more specifically, WPUDP-X). The expected RMS phase error is X divided by the square-root of twelve. You should recognize that term as the square root of the variance (σ, not σ2 which is the variance) of a uniform distribution, something that should be covered in Six Sigma training but is usually omitted to make the course so easy that an imbecile can become a "black belt." So, if your amplifiers are uniformly spread over 90 degrees, your RMS error is expected to be 26.0 degrees, and you should see 80.8% phase efficiency (-0.93 dB below what is possible). Of course, basing this calculation on just the waffle pak distribution ignores phase tolerance contributions of wirebonds and combiners, you are on your own to estimate these and recalculate the phase distribution. If you don't understand (and memorize) rules 110 and 111, you might want to pick a different career other than SSPAs....
Variations in amplitude also
cause a loss in power in an SSPA. But it is not as epic as phase
errors. Typically you will have all of your amplifiers in gain compression
and they are likely to be within 1 dB of each other. The loss in
efficiency at this point is just a percent or two (much less than
Yes, we need to follow up that
statement with some math some day...
What style you pick has a big
effect on second and third harmonics. This topic is covered in Practical
RF System Design, by William F. Egan, this is a good book
to buy if you want to move up the RF food chain. We'll summarize
the outcomes according to Egan but skip the math. We also had some
great help on this topic from Jack K. of Matrix
90 degree combiners reduce the
amplitudes of second order distortion products by 3 dB compared
to the individual amplifiers because the level at each amplifier
is 3 dB below a single stage amplifier and the slope of the second
order distortion versus signal level is 2:1 so reducing the signal
by 3 dB reduces the distortion by 6 dB and the signal to distortion
changes by 3 dB.
Third order distortions products
which include 3A, 2A+B, A+B+C are canceled while 2A-B and A+B-C
do not cancel. The level of the 2A-B and A+BC are lower by 6 dB
because the level at each amplifier is 3 dB below a single stage
amplifier and the slope of the third order distortion versus signal
level is 3:1 so reducing the signal by 3 dB reduces the distortion
by 9 dB and the signal to distortion changes by 6 dB.
Unfortunately it is the 2A-B
and the A+BC products that fall in band and cause problems.
180 degree combiners ideally
eliminate even order harmonics and intermods. They have no effect
on odd-order intermods and harmonics.
Where do the "missing"
signal products go? into the isolation load!
Coupled, corporate, radial
If you need to combine more than
two amplifiers, you need to make this decision. The following figures
were taken from U.S. Patent 4,933,651, Multichannel Combiner
Divider, from a discussion under "prior art", to illustrate
the first three cases. Thanks to Myron for correcting the patent
number! Google now has the best
patent search available on the web, in case you were wondering...
The coupled combiner seems simple
enough, but each coupler needs a different coupling coefficient
to be successful. The bandwidth of the approach is limited, and
often the loss is high, even when waveguide is used. If you opt
to use this, we wish you luck! The coupling coefficients (ignoring
accumulating "real" losses) need to follow the sequence:
1/k, 1/(k-1), 1/(k-2), 1/(k-3)...
for an eight-way combiner, the
coupling coefficients in dB are:
A corporate combiner is shown
below. This is a "third order" binary combiner, which
combined eight sources (coherent amplifiers). This is a very straightforward
combiner to develop, however, the loss can pile up, with each additional
split the final combiner needs to span an ever-increasing distance.
Also, the isolation between each amplifier is not equal.
Below is a radial combiner. This
type of design is also not for the faint of heart, however, it offers
the most bandwidth. Wilkinson's
original concept was a radial combiner.
often suffer from poor isolation, and they can have tricky line
impedance requirements. But they provide the least loss, and the
This image came from US patent
5,214,394, High-efficiency, bi-directional spatial power combiner
amplifier. In this combiner, a two-dimensional array of amplifiers
is illuminated from a feed horn. Each amplifier has two antennas
at opposite polarization, vertical for input and horizontal for
output for example. An ortho-mode transducer or circulator can be
used to separate the input and output signals. By locating the input
and output an the same side of the array, the inventors have solve
the problem of heat sinking.
For the spatial
combiner, it is difficult to achieve uniform power split to each
amplifier, and there is "spillover loss" associated with
the illumination going outside the array. Gain is low, in the case
of the reflecting array above, you can't exceed the isolation of
the two polarizations; you might need a second, smaller array to
drive the first one. The bandwidth will be narrow, and it is hard
to imagine that you could make such a combiner cheaply. Other
than that, Mrs. Lincoln, how was the play?
Reactive, quadrature, 180 degree
More to come..
The term "reactive"
in this case means that no resistors are used to terminate the errors
of an out-of-phase condition (like they would in a Wilkinson structure).
Reactive combiners can sometimes suffer from oscillations which
are the result of the "odd mode" where adjacent amplifiers
become 180 degrees out of phase. Also, if one amplifier fails, you
are in a world of hurt as it will surely pull down the other amplifiers.
Quadrature (90 degree) combiners
This is by far the most popular
amp combining method, when you only have two amplifiers to combine.
Examples include the Lange, branchline, and overlay couplers.
The input and output return losses
of amplifiers are vastly improved when 90 degree combiner is used.
The reflected power from the amps is dissipated in the load on the
"isolated" port. For an explanation of this, go to our
page on quadrature couplers.
Using quadrature combiners can
provide an appreciable degree of immunity to load-pull effects,
for example, when a solid-state power amplifier is connected to
an antenna that has less than perfect impedance match. Learn
about this phenomenon here!
180 degree combiners
180 degree splitters include
the rat-race, and
the waveguide magic tee.
The VSWR of the amplifiers is
NOT reduced in a 180 combiner.
In-phase isolated combiners
These provide isolation between
the amplifiers, the classic example is the Wilkinson. The Wilkinson
splitter also has a fairly wideband response and by adding more
sections the bandwidth can be increased.
The Gysel is also a popular in-phase
combiner. It offers a distinct advantage over the Wilkinson, in
that the isolation loads are one-ports and can be pulled away from
the splitter, such that much more power can be dissipated. If you
broke into the transmitter floors inside the Empire State Building
you might find an entire
"herd" of Gysels combining redundant tubes that power
up a score of commercial
broadcasting systems! By the way, you'd think that having a
megawatt of RF power from one site might cause a little concern
about health issues, compared to all the hype
Impedance match of isolation load: effect on balanced amplifier output match
What is the effect of a mismatched isolation load on the output impedance of an SSPA? The answer is "it depends". For now, let's consider the effect in a two-way combiner that uses ideal quadrature couplers.
Let's also assume that the output port sees a perfect match, while we allow the other ports to have reflection coefficients as numbered below, which can be assumed to be vector quantities. This figure represents the initial condition when a signal of "Vin" is present at the output port.
At the first split, each amplifier sees 1/SQRT(2) of the input signal. The phases through a coupled line are as follows: the through path gets -90 degree phase shift, while the coupled path gets 0 degrees. Got that?
Now let's look at the returning voltages after the first bounce. The signal that returns to the output port is 1/2 of the input signal multi pled by the difference in reflection coefficient as the amplifiers. This is part of the magic of a balanced amplifier. No matter how crappy the amplifiers are impedance matched, they cancel out (at least on the first bounce). The signal at the output port leaves the system, as we are assuming that it the one port that is actually matched to fifty ohms. If the amplifiers are imperfectly matched, there can be substantial signal at the isolation load, for example, of rho2 and rho3 are 0.5, fully 1/2 of the input voltage arrives here. By "substantial" we mean relative to the signal that entered. This is likely a small signal, so we are not talking about any real power. There are other considerations for sizing that load for power which we will address somewhere else.
Now what happens after the second bounce? This is the case if the isolation load is mismatched. The reflected signal lands on the amplifiers, reduced by the isolation load's reflection coefficient rho4.
On the third bounce, another signal exits the amplifier, but it is greatly reduced compared to the input signal. This reflected signal is present even if the two amplifiers have the exact same reflection coefficient.
One the fourth bounce, some more of the signal reflects off of the isolation load, but it is much smaller, and if those amplifiers were identical, this signal would be zero.
At the fifth bounce some more signal exits, and some more is returned to the isolation load. This is making us dizzy!
So far we have collected three terms of signal leaving the output port. Let's string them all together, and note that the composite reflection coefficient is the signal exiting divided by the signal incident (all those "Vin" terms go away):
You can see how an infinite series can be constructed, but the terms keep getting smaller and smaller.
Now let's put in place one of the conditions that is a primary property of a balanced amplifier: The two amplifier's reflection coefficients won't be well matched to fifty ohms, but they should be equal in magnitude and phase. In this case, all but the second term go to zero and we are left with:
You can see that unless the two amplifiers present short circuits, the reflection coefficient due to the termination is reduced in effect. For example, if each power amplifier had a 3:1 VSWR at equal phase, the amp reflection coefficients are 0.5. If the load was also a 3:1 mismatch, the net reflection coefficient is 0.125 (it is reduced by a factor of four. Sweet!
Next: is it possible that a poorly-matched isolation load could load-pull your power amplifier? The short answer: normally, no, but if something comes out of balance, yes.
Oops, we forgot to mention, the above analysis was only at center frequency. Let's use some expensive EDA software from a nameless company that should sponsor Microwaves101.
Here is an ideal balanced amp. We installed the "Sniffer" at the outputs of each amp so that we can look at what reflection coefficients they see. We added some phase lengths to each output that we will sweep such that any possible phase angle is presented when we mismatch the amps. We also put in another transmission line on the input of the upper amp which we can use to drive the amps out of phase. For now it is set to zero.
Below is the output reflection coefficient which is zero at center frequency but increases at the band edges. Nothing in this world is perfect, you have to live with it.
Now we will mismatch the amps to 3:1 VSWR (0.5 reflection coefficient) as well as the isolation load:
Lo and behold, the math that we did upstairs is valid, the net output reflection coefficient is 0.125 for any reflection phase angle at the amps. But only at center frequency.
Now it's time for some examples...
Jack's three-way splitter used
as a three-way combiner
Moved to this
Lim-Eom splitter used as a combiner
combiner on this page
combiner ion this page