# Combiners and splitters

*Pieter-Tjerk de Boer, PA3FWM web@pa3fwm.nl*

(This is an adapted version of part of an article I wrote for the Dutch
amateur radio magazine *Electron*, August 2024.)

The previous installment [4] was about intermodulation measurements, in which a key role is played by the "combiner": a circuit that combines the signals from two signal generators. Such a circuit is also colled "coupler", or, if used the other way around, "splitter". In this installment we have a closer look at them. We'll find there is a fundamental limitation to what a coupler can do, what still is possible, how they work, and how we can build them ourselves.

## Ideal but impossible coupler

An ideal coupler would have two inputs and one output. A signal offered to one input would only go to the output, unattenuated, and not at all to the other input. If one would offer signals on both inputs, the sum of their voltages will appear at the output. Both inputs and the output are of course matched to 50 ohms, and the circuit is passive (doesn't need a power supply, doesn't contain transistors etc.) and perfectly linear.
Unfortunately, such a coupler *cannot* exit.
And that's not just because real components are never ideal,
but there is a more fundamental reason.

Suppose we would have such a coupler, and we'd connect signal generators
to both inputs with the *same* frequency and amplitude.
By the above wishlist, the output voltage would be double that on the
individual inputs.
But doubling the voltage at the same resistance (impedance) means,
according to P=U^{2}/R, quadrupling the power.
The output power would thus be twice the sum of the input powers:
that's impossible because of conservation of energy.

As long as both signals have a *different* frequency, this problem doesn't occur.
One can prove this mathematically (both signals are, in mathematical terms, orthogonal).
But one can also find it from direct reasoning.
A (small) frequency difference means that the phase difference between
both signals (slowly) changes.
At some moments both are in phase (with the above problem of the total
output power being twice the total input power);
but at other moments they are in opposite phase (resulting in zero output power);
and most of the time it's somewhere in between.
Averaged over all phases the output power is the same as the total
input power, so no problem with conservation of energy.

In a sense this is funny: even if we never intend to offer two signals
of the same frequency, the simple fact that one *could* do this is
sufficient reason that such a coupler cannot exist.
If one makes a *frequency-selective* coupler, i.e., one which also
acts as a filter, then it is possible;
an example is the so-called diplexer that is used to connect a
2-m and a 70-cm transmitter to one antenna.
The filter action makes it impossible to offer input signals with the
same frequency.

## Possible: a coupler with 3 dB loss

A circuit that*is*possible, is a coupler which takes half of the input power to the output, and dissipates the other half. It thus has a loss of 3 dB, or the output voltage is the input voltage divided by √2. If one would offer two identical signals to this circuit, the output amplitude would be 2/√2 = √2 times the input amplitude; so the total output power would then be twice each of the input powers, which is equal to the sum of the input powers: no problem with energy conservation. (In that situation, there would be no dissipation in the coupler.)

Practically such a coupler is often a so-called hybrid coupler, as depicted in figure (a), below. This coupler does not have three but four ports, which in principle are equivalent. Power offered to one port is divided equally among its "neighbour" ports, but does not arrive at the "opposite" port. One can make this into a 3 dB coupler by using e.g. ports 1 and 3 as inputs, port 2 as output, and connecting port 4 to a resitor (dummyload). Then of any power offered to port 1 or 3, half will arrive at output 2, and the other half in the dummyload.

## Practical realisation using a transformer

The usual way of building a 3-dB coupler for audio or HF frequencies is sketched in figure (b). The heart of this circuit is an autotransformer, which we presume ideal.Let's start by offering a signal to port 2, and terminate ports 1 and 3 with 50 ohms. Because of the circuit's symmetry, the current fed into port 2 will split equally between left and right, see figure (c). That's correct also from the transformer's point of view: it will enforce equal but opposite currents in the two halfs of the winding. Thus, the power is divided equally between the two resistors. Also because of symmetry, there will be no voltage (and thus no power) on port 4 (so it doesn't matter what we connect there). Furthermore, we observe that the input impedance of port 2 is not 50 but 25 ohms, as both 50 ohm loads are in parallel.

Next, assume the signal is offered to port 4, as in figure (d). Neither wire of this port is grounded, but for the principle that doesn't matter. Seen from this signal source both 50 ohm loads are in series, as demonstrated by the blue arrow. And because of symmetry there will be no voltage on port 2. We see that again the power is split equally between both resistors; and we concluded that the impedance of port 4 is not 50 but 100 ohms.

If we now offer signals to ports 2 and 4 simultaneously,
and those signals have different frequencies,
then they won't affect each other and will both be split equally over ports 1 and 3.
If they have the same frequency, then their phase difference determines where the power goes.
One should note that when using port 2 as the input (figure (c))
the currents in both resistors flow in the *same* direction,
while when port 4 is the input, they are *opposite*.
So when offering the same signal to both ports 2 and 4,
their phase difference determines on which port (1 or 3)
they are in phase or not, and thus where the total power does or does not go.

Finally, we should study what happens if a signal is offered to port 1.
This is more complicated because the symmetry is broken.
See figure (e), where I re-drew the circuit a bit, and terminated port 2 with 25 ohms and port 4 with 100 ohms,
the impedances we found above,
and I've tentatively assumed port 3 is short-circuited (dotted line).
The signal source now effectively drives two parallel circuits:
on the one hand the 100 ohm resitor to ground, and on the other hand the transformer with the 25 ohm resistor.
The transformer transforms the 25 ohm to 100 ohm (as the turns ratio is 1:2, so the impedance ratio is 1:4).
Thus, the signal source sees two times 100 ohm in parallel, resulting in 50 ohm, and the power will distribute equally between the 25 and the 100 ohm resistors.
The arrows indicate how the current flows: half the current from the generator goes to the 100 ohms,
the other half to the transformer.
The transformer enforces that in the other winding an equal but opposite current flows,
so the current through the 25 ohm resistor is twice this current:
it better be, as otherwise we couldn't put the same power into it as into the 100 ohm resistor,
because of P=I^{2}R.
But from this it also follows that the current that has flown through the 100 ohm resistor,
immediately turns around into the transformer.
Thus, *if* ports 2 and 4 are properly terminated in 25 and 100 ohm,
there is no current flowing through port 3.
So, we needn't have assumed that this port was short-circuited!
Whatever is connected to port 3, the power offered to port 1 is distributed equally between ports 2 and 4.

If we repeat the above considerations for a signal offered to port 3 instead of 1, obviously nothing changes, except that the current direction in the 100 ohm resistor (port 4) is inverted, similar to what happened when we compared offering power to ports 2 and 4.

## Practical realization

For a practically usable coupler, we'd like all ports to have a 50 ohm impedance, rather than having a mixture of 50, 25 and 100 ohms. Thus, we arrive at the schematic sketched here, with ports 1 and 3 used as inputs, port 2 used as the output via a transformer to go from 25 to 50 ohms, and port 4 terminated in 100 ohm.If you'd like to build such a combiner/splitter, I advise to have a look at IN3OTD's website [1]. Not only does he give advise for the practical construction of good transformers, but he also discusses tricks that make the circuit work over as wide a frequency ranges as possible, despite the non-idealities of the transformers. I built mine on the basis of his advice, see the photograph.

An ideal transformer has infinite inductance. If the inductance is less, performance at lower frequency suffers. IN3OTD points out that it's best if both transformers have the same inductance. Een ideale transformator heeft een oneindige zelfinductie. Als die zelfinductie niet oneindig is, gaat de goede werking bij lagere frequenties achteruit. IN3OTD wijst erop dat het dan het beste werkt als de zelfinductie van beide trafootjes gelijk is.

Also, in an ideal transformer both coils are perfectly coupled. In a practical transformer this is not quite the case. The "non-coupled" part of the inductance worsens the isolation between the inputs at higher frequencies, and IN3OTD points out that this can be partially remedied by adding a small capacitor. This "stretches" the frequency range where the isolation is "good enough", but at the cost of (even) worse isolation at even higher frequencies. By mounting a trimmer capacitor and checking its influence using a network analyzer, I found 33 pF as the best compromise in my case.

The graphs show the insertion loss and isolation as measured on my coupler between 10 kHz and 150 MHz. Above about 300 kHz the insertion loss ("doorlaatdemping") is constant within a few tenths of a dB, just slightly more than 3 dB: that's as expected, as half the power disappears into the 100 ohm resistor. Below 100 kHz the loss increases quickly, for lack of inductance in the transformers.

The isolation between both inputs is about 40 dB at 1 MHz and slowly worsens at higher frequencies; but up to about 100 MHz it's still 25 dB or more, also thanks to the 33 pF capacitor. IN3OTD even achieves 33 dB. But much more than that is not very useful anyway, as this isolation can only be enjoyed if the output is terminated nicely in 50 ohm (see the previous installment [4]).

For intermodulation measurement it's also important that the combiner doesn't produce intermodulation itself. From about 3 MHz I measure an IP3 (third-order intercept point) of +45 dBm or more, but that's about the limit of my measurement setup so it might be even better. At lower frequencies the IP3 quickly worsens, +33 dBm at 1 MHz and +23 dBm at 100 kHz. This is to be expected, because the magnetic field strength must be higher at lower frequencies for the same voltage on the transformer, so the ferrite gets closer to saturation.

For use at lower frequencies a larger ferrite core, e.g. a BN43-202, would probably work better: more inductance and less saturation.

## Using only one transformer

It is also possible to make a 3 dB coupler with only one ferrite core. Then we use port 4 instead of port 2 from figure (b) as the output. To do so, an extra winding is made on the same core, with 70% (ideally 1/√2) of the number of turns of the first winding. That serves two purposes: impedance transformation from 100 to 50 ohms, and galvanic separation, as port 4 by itself is not grounded. Of course, there's no 100 ohm resistor anymore, but now port 2 must be terminated with 25 ohms.According to [2], there's another way to use only one ferrite core, even if it feels a bit like cheating: use one core for for both transformers, by using one hole of the two-hole core for each transformer and leading the wire back along the outside of the core, rather than through the other hole. IN3OTD tried both ways, but concluded they perform less well than using two separate cores [2].

## The 6 dB coupler

Another cuopler circuit that is often used is sketched in the next figure. This is a 6 dB couplers, meaning that only a quarter of the input power reaches the output, another 3 dB less than in the previous coupler. Losing another 3 dB may not sound like a lot, but for intermodulation measurements it may be a problem; e.g., the 3rd order intermodulation products will decrease by 9 dB, so might well disappear into the noise floor. But it does have the advantage of simplicity, as it needs only one transformer.In figure (b) the schematic has been redrawn to emphasize the symmetry. There are two inputs, each connected to a generator, and four resistors of 50 ohm each. Seen from e.g. the bottom left generator, R1 and R2 are in series, R3 and R4 are also in series, and both series circuits are in parallel. Thus, each resistor gets a quarter of the power. And if all four resistors are equal, it is a bridge circuit that is in equilibrium, so nothing reaches the other generator.

In the real coupler (figure (a)), one of the four resistors (R4) is replaced by the output. The transformer is only needed to make it possible to ground also the second generator without short-circuiting a resistor. It's effectively a balun.

Because of this "lighter" task of the transformer, it is easier to make this circuit wide-band than the previously described 3dB coupler. Literature [3] even describes one usable from 300 kHz to no less than 13.5 GHz. Their "transformer" consists of a thin coaxial cable with ferrite around it; a common-mode choke.

My transformer consists of about 13 bifilar turns on again a BN43-2402 core. This was a somewhat random choice, and surely not optimal, but the measurement results are good enough for my purpose. The insertion loss (from in1 or in2 to out) is 6 dB as expected, and only increases a bit below 100 kHz. And the isolation between both inputs, when the output is terminated in 50 ohm, is over 50 dB at best, and at least 25 dB from 100 kHz to 150 MHz. This coupler also has better intermodulation performance than the previous one: the measured IP3 is better than +45 dBm above 1 MHz, and even at 100 kHz still some +30 dBm.

## Other applications

Besides the application that lead to this article, namely intermodulation measurements, such couplers are also used for other purposes. For example in reverse, as a splitter, to connect two receivers to one antenna.The 3 dB coupler is also often used to combine two power amplifiers to obtain more output power. Then one is in the situation discussed before, where both inputs (e.g. ports 1 and 3) get the same signal. If they are equally large and in phase, all power comes out at port 2. The 100 ohm resistor (port 4) only dissipates power if there is a difference between both amplifiers, e.g. due to a defect.

Every analog telephone also used to contain such a circuit: the microphone was connected to port 2, the loudspeaker to port 4, the telephone line to port 1 and a resistor to port 3. This ensured one's own microphone signal did not end up in one's own loudspeaker.

The 6 dB coupler is also used for reflection measurements with a network analyzer. For example, this circuit or a variant is used in the NanoVNA, and this application was also the motivation for [3].

## References

[1] https://www.qsl.net/in3otd/ham_radio/power_splitters/dual-core.html[2] https://www.qsl.net/in3otd/ham_radio/power_splitters/single-core.html

[3] D. Nikolay, M. Philipp: A 300kHz-13.5GHz Directional Bridge;

*Proc. 45th. European Microwave Conference*, 2015. https://www.mikrocontroller.net/attachment/482240/GHz-Directional-Bridge.pdf

[4] tn39-intermodulation-measurement.html

Text on this page is copyright 2024, P.T. de Boer, web@pa3fwm.nl .

Republication is only allowed with my explicit permission.