Index
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Ferrites in HF-applications
Transmission-line transformers Introductory
(published in Electron # 12, 2001) General This chapter is the fourth in a series of articles on ferrite materials in HF-applications. The first article is an introduction to this field with an overview of some widely applied materials and most important properties. The second article is on materials back-ground and most important HF-application formulas. The third article is on HF inductors and transformers. It is advisable to read the articles in the above order especially since each next chapter is building on information and formula's already explained earlier and referencing to this. Introduction Transmission-lines have been designed to transport energy from one position to the next under low-loss conditions. These line are applied when other types of energy transport no longer fulfill this requirement because of loss or radiation mechanisms or bandwidth reasons. Since transmission-lines especially are applied at higher frequencies and wide bandwidth they may also be successfully applied in impedance transformers when smartly combining inputs and outputs of these lines. Currents and / or voltages may be added up to fulfill required impedance conditions which may be different from the input to the output of these transformers. The upcoming paragraphs will discuss the design and construction parameters of transmission-line transformers, the application area's and frequency limits and the requirements to the ferrite cores of these devices. Current to transmission-line transformer A transmission-line
transformer usually consists of a (number of) basic element(s), build very
much like the current transformer in the previous chapter and is repeated in
figure 9a.
A transmission-line transformer consists of two, closely coupled conductors to keep the electro-magnetic field 'inside'. This condition applies to coaxial transmission lines and nearly so to symmetrical lines. The basic transformer element therefore is represented in figure 9a as a current transformer with equal number of turns. Since coupling is 'perfect', i1 is equal to i2, but of opposite phase. Since this 'ideal transformer' is perfect (lossless), u2 is equal to u1. Transmission-line
transformers may be designed for various transformer ratio's. When connecting
output terminal D to input terminal A and the load resistor Rb between terminal C and ground, the output
voltage u2 is added to the input voltage u1, doubling
the voltage across Rb. This will turn
the 1 : 1 transformer of figure 9a into a 1 : 2 (auto, voltage-) transformer
of figure 9b, making up to a 1 : 4 impedance transformer.
Since 'winding' BD in figure
9b is across the load, the generator will have to supply an additional
current to the circuit, that is not shared by the load. This additional
current we like to make as small as practical. To the differential
electro-magnetic field between the conductors it makes no difference whether
these are lay-out in a straight line or being curved or even rolled up into a
coil. The (transmission-) line is still determining the transmission
characteristics. The coiled-up transmission-line however is acting as an
inductor, presenting a high(er) impedance with
frequency for all currents common to both inductors (common-mode). This is
particularly the case for all currents in the BD winding, which we now are
free to suppress by making the common mode impedance high by adding a
(ferrite) core to this common-mode inductor. The common-mode impedance
is represented in figure 10 as the
parallel impedance Zsleeve.
Let's take a closer look at this sleeve impedance. When making the transmission-line very long, sleeve impedance will be high since impedance is scaling at about 1 μH / meter with line length in a first order approximation. Signals at the output of a transmission-line will be delayed with respect to the input, depending on the velocity factor and line length (RG58; vf = 0,66). Therefore, a long transmission-line will exhibit a high sleeve impedance (which is nice) and also a much delayed output signal, which is may be not so nice when adding input to output signals as in figure 9b. This effect would make transmission-line transformer frequency-dependent, which we like to avoid in wide-band applications. We may avoid signal delay by keeping the transmission-lines as short as practical. High sleeve impedance may be obtained by making the line into a coil (inductance) and even shorter lines will become practical when constructing this inductor on a ferrite core. Depending on type of ferrite and number of turns we may now make sleeve impedance high enough to choke (outside) parasitic currents while at the same time keeping the lines as short as practical. Since the signal is 'locked-up' inside the transmission-line, outside constructions do not influence this signal, whether this is a lossless inductor or a lossy choke; the transmission-line is still fully in charge of transporting the signal from input to output. So the
transmission-lines of the transformer should be short. But just how short is
reasonable? If we are to design the transformer of figure 9b on a For some transmission-line transformer types it is possible to utilize one ferrite core for more than one transmission line. We will signal at each design when this will be allowed without creating opposing magnetizing fields in the core that would negatively influence choke action. Jerry Sevick, W2FMI, made a collection of his professionally designed transmission-line transformers available to the public in his book: 'Transmission Line Transformers', available from Amazon and ARRL. Next to many transformers Jerry explains how to measure and construct these as well as how to obtain the odd transmission-line of non-standard characteristic impedance. A few basic transmission-line transformers
Transmission-line transformers have arrived not long after 'commercial' transmission-lines became available. Basic type for matching un-balanced to balanced type of circuits may be the oldest of these transformers and is depicted in figure 11.
The high sleeve impedance effectively decouples the transformer output from the input, making the output 'free-floating' and disconnects the load from ground reference at the input: an effective un-balance to balance transformer. In 1944 Guanella extended this basic transmission-line transformer in an article called 'Novel Matching Systems for High Frequencies'. The Guanella type consists of one or more basic elements, connected in series or in parallel at the input and / or output of the transformer of which an example may be seen in figure 12.
At the input
the two transmission-lines are in parallel to the Some time after Guanella, Ruthroff
analyzed in
Since the
output of the transmission-line is 'free-floating', it may be connected to
any other point of the circuit, also to ground or to the input terminal. By
connecting to the input, the output voltage is effectively put in series with
the input voltage, again adding up to two times the input voltage across the
load Rb. This simplified construction
yields a 1 : 4 impedance transformer, analogue to the double
transmission-line Guanella-type. Because of the difference in circuit
lay-out, differences will also be found in behavior as to frequency,
transformer loading, sensitivity to sleeve impedance and termination
resistance etc. This will be discussed in the upcoming chapters.
Transformer bandwidth
The practical bandwidth of a transmission-line transformer is limited at the low-frequency side by the sleeve impedance, since this impedance is situated in parallel tot the signal path either directly (Ruthroff) or indirectly depending on load conditions (Guanella in an unbalanced situation). High frequency limit is determined by more than one parameter e.g. transmission-line length and / or line-length differences, (characteristic) termination and /or parasitics in series (inductive) or in parallel (capacitive) to the transformer. Well designed transmission-line transformers however may be applied for very wide bandwidth, depending on the specific type of transformer. In general this type of transformers easily outperform flux-transformers, as described in an earlier paragraph. Bandwidth definition Transformer 'bandwidth' is not strictly defined. For this discussion, bandwidth will be defined as the range of frequencies of a terminated transformer that will exhibit at the input of the transformer SWR < 1,5. Different definitions are very well possible as may be found in the earlier mentioned book on transmission-line transformers by Sevick, who prefers SWR < 2 frequencies. Low frequency side Low frequency side is determined by the sleeve impedance, Zsleeve, that is found in parallel to the signal path or may be found here at 'adverse' loading conditions. At the low frequency side, Zsleeve is a more or less 'pure' inductor, so Zsleeve = Xp. When placed in parallel to the system impedance Zin, this reactance will make a non-characteristic input impedance, we may calculate as:
Many systems will be designed around 50 Ohm. When requiring SWR< 1,5, we may calculate the lowest sleeve impedance as:
At a lower operating
frequency of 1,8 MHz. this translates to an
inductance of 10,8 mH. When
we decide to construct this transformer around a If we would like to design a 'perfect' transformer, we may decide to use more turns to lower SWR. If it were possible to place three times as much turns at this core, creating a nine times as high a parallel impedance, we still would find SWR = 1,11 at the transformer input. As stated earlier, we will apply SWR < 1,5 as the limiting condition of our transformer designs, so for all further transformer designs we will calculate sleeve impedance at the lowest operating frequency through: Xp = 2,5 ∙Zin
We noticed
in the chapter on ferrite chokes that when parasitic parallel capacitance is
becoming the dominant factor, choke impedance will drop sharply with
frequency. Transmission-line transformers rely on input to output isolation
for operation, so like chokes a transformer will be at its limiting frequency
when the parasitic capacitance across the sleeve impedance no longer isolates
the output from the input of the transformer. A good design ensures low
parasitic parallel capacitance by avoiding input winding to be too close to
the output winding and / or in- between windings to overlap. Conclusions on bandwidth In conclusion
we may decide that a high band-width transmission-line transformer should be
made with a low number of turns. To still obtain high enough input to output
isolation, a relatively high permeability ferrite should be selected, so
rather 4A11 (43) than 4C65 (61) type of materials. A better low frequency
response (more turns, better SWR) should be checked against high frequency
behavior with will require less turns for less delay (difference), and better
tolerance against non characteristic termination. Transformer
load Transmission-line transformers often are applied in wide-band high(er) power application. Power requirements therefore usually are an important factor and it pays to have a clear view on the power determining factors. Since transmission-lines in these application usually are very short, transmission-line loss usually is insignificant to other parameters. We will look at these other factors in the next paragraphs. Various
parameters determine the maximum power load on a transmission line with
maximum voltage as one of the more important. As it happens, maximum allowed
line voltage will not easily be exceeded in amateur transformers. For RG58
material, a maximum of 1000 volt is specified, which in theory will be
reached at 10 kW. in a 50 Ohm system. Next parameter is maximum power, which
is to be de-rated with frequency and is specified for RG58 as 350 Watt at 30 MHz. It is therefore usually maximum current that will
set the power limit, at 30 MHz. not to exceed For many transmission-line constructions it is more convenient to apply small diameter miniature Teflon transmission-line, since loss is not an important factor at short line length. Since center conductor is even smaller, maximum power (current) should be checked even at medium power applications. Maximum core induction load In earlier chapters we found that for linear applications, maximum induction, Bmax = 0,2 Bsat should not be exceeded. With a certain selection of core type and size, maximum voltage for linear application is further determined by the number of turns according to: UL(induction) < 0,89 ∙ Bsat ∙ f ∙ n ∙ A Maximum core power load Usually the most important factor to determine maximum signal power in a transmission-line transformer is core temperature. As stated before, it is advisable to keep maximum core temperature rise below 30 K, to remain below Curie temperature for some ferrites but also to stay out of the temperature regime when transmission-line support materials start losing mechanical integrity. In formula 19 we found a direct way to determine maximum voltage across an inductor as related to internal power dissipation, to stay below this maximum core load: __________________ UL(dissipation) = ÖPmax ∙ (Q/6 + 1/Q) ∙ XL For transmission-line transformers, this maximum voltage is found across the sleeve choke, that ensures input to output isolation. From this formula it may be decided that higher number of turns directly increases the maximum inductor voltage, as will higher ferrite volume (related to Pmax and A) and lower ferrite loss (m"). In general it
is advisable to always check maximum voltage for core dissipation as well as
for induction when designing these transformers, especially when applying the
enhancement factors of table Guanella and Ruthroff transformers
Starting with this chapter a number of transmission-line transformers will be designed and analyzed, all adhering to a few very simple and basic rules. These rules may seem all too obvious but show to be very powerful when analyzing complicated designs. ° Currents in the coupled conductors of a transmission-line are equal in magnitude but opposite in phase (total coupling rule). (31) ° For every transmission-line the output voltage will be identical (magnitude and phase) to the input voltage (no loss rule). (32) ° There is no other connection between the
transformer input and output than through the transmission-line
(total decoupling rule).
(33)
Transmission-line
transformer 1 : 1
The most basic transmission-line transformer is no more than the basic element, usually applied as a balance to unbalance transformer, a balun, as in figure 14.
This basis
element is consisting of a short stretch of transmission-line, usually coiled
up around a (ferrite) core.
The generator
voltage at the input is also present at the output (basic rule 32). The
generator current is going through the transmission-line, through the load
and returns through the transmission-line in equal magnitude and opposite
phase (basic rule 31) to the generator. This return current will flow through
the inner side of the transmission-line, since the outside of the line is
coiled-up around a high permeability core and will present a high impedance;
the output is well isolated from the input (basic rule 33).
Because of this
high isolation, the output is 'free-floating' with equal (infinite) impedance
to ground for both output terminals, a balanced situation. This effect turns
this basic element into an effective balance to unbalance transformer, a
balun.
The analysis of this basic element 'transformer' is trivial, but will be explicitly spelled out since this 'recipe' will be followed throughout all further analysis and will turn out to be a very practical tool. Since we find a current, 'i' through the load and a voltage 'u' across, we may write: Rb = u / i Every other impedance in this circuit will be expressed in this ratio of voltage and current and will therefore be referenced to the output impedance. In figure 14 the generator is delivering a current 'i', at a voltage 'u', so will 'see' an input impedance: Zin = u / i
which is equal to Rb The transformer (impedance) ratio indeed is 1 : 1. The transmission-line is carrying a current 'i' and 'sees' a voltage 'u' across. The characteristic impedance of this transmission-line therefore should be: Z0 = u / i = Rb The transmission-line also should have a characteristic impedance which is equal to the load resistor. This is obvious for the above situation but will be different for other types of transmission-line transformers. A
1 : 1 balun application example The balun as
analyzed above will be applied quite often when connecting a symmetrical
antenna to an
a-symmetrical transmission-line. Without a balun, the antenna is
connected to the generator through the transmission line. At the same time,
one side of the symmetrical antenna is connected to the outside of the
transmission line. This outside is also acting as an additional, vertical
antenna, that not only is sensitive to all vertically polarized
'electro-smog' around the house (computer, TV, especially your neighbor's),
but will also radiate in this direction causing unwanted EMC in your shack
and again at your neighbor's. Furthermore, the antenna will exhibit a
different radiation pattern than you catered for. This antenna situation is
depicted in figure 15.
The amount of energy the out-side feed-line
will radiate is depending amongst other parameters on the line length. Some length may represent a low impedance
into which a major amount of current will flow, turning this line into the
more important radiator. To avoid this situation, the antenna should be
decoupled from the (outside) line, for instance by means of a sleeve choke,
current balun or 1 : 1 transmission-line balun. For this analysis we will divide the antenna
radiation resistance evenly over each dipole half and model the complete
antenna plus feed-line system as in figure 16.
In figure 16 we find the balun as part of the feed-line, directly connected to the antenna. For our model calculation, we presume the feed-line to the transceiver to be short and the transceiver to be connected to ground, as a worst-case situation. The antenna will exhibit a radiation resistance of 50 Ohm at an operating frequency of 1,5 MHz. To render the sleeve current insignificant as compared to the antenna current, sleeve impedance (reactance plus loss) should be at least four times as high (making power sixteen times as low). At the design frequency, sleeve impedance should therefore be: 4 x 50 = 200 Ohm. When selecting
a F = m0 ∙ A / l = AL / mi = 940 .10–7 / 700 = 1,34 .10-9 Using formula 8, and the complex permeability from table 2, the winding factor at our operating frequency is: AZ(1,5MHz) = F ∙ mC(1,5 MHz) = 1,34 ∙ 10-9 ∙ 916 = 1227 ∙10-9 With 5 turns of RG58 transmission-line around this toroide we calculate sleeve impedance at: Zsleeve = w ∙ n² ∙ AZ(1,5MHz) = 289 W which is more than eleven times as high as the impedance in parallel (half radiation resistance). This effectively stops any significant current from flowing in the (outside) sleeve, and back to the shack to cause any type of interference. Induction and power We calculate maximum allowable field strength for linear operation inside the balun core using formula 5: UL < 0,89 ∙ Bsat ∙ f ∙ n ∙ A = 0,89 ∙ 0,33 ∙ 1,5 ∙ 106 ∙ 5 ∙ 1,18 ∙ 10-4 = 257 V. The maximum voltage is across half the radiation resistance in the model of figure 16, effectively doubling the allowable transmitter voltage to 514 V. The second
limiting factor is the maximum allowable power-loss in the core, which we
determined at 4 Watt for a temperature rise of 30 K for this core size. Using
formula 19, we may now calculate maximum voltage across this __________________ UL(dissipation) = ÖPmax ∙ (Q/6 + 1/Q) ∙ XL = 86 V. Since this again is halved at the antenna, the maximum allowable voltage at the antenna is 172 V., which translates into 592 Watt of system power in a 50 Ohm system. When this will not satisfy our requirements, we find in formula 19 that we may improve by enlarging the number of turns, 'n' or the core size (factors 'A' and Pmax). Remember this calculation to be made for a continuous power situation, so enhancement factors may be applied according to table 4 depending on the transmitting mode. When applying these factors, one should always re-check maximum voltage for linear use with formula 5. Alternatives to the 1 : 1 balun For this
particular functionality we may have decided to construct an 'air balun'
type, by coiling the transmission-line into an inductor without a ferrite
core. If we would like the same type of performance at the design frequency,
an air coil with a diameter of about A variation to
winding the transmission-line around a ferrite core is to 'wind' a ferrite
core around a transmission-line. In this variation, a number of ferrite rings
is slipped onto the line as beads on a string. Since these rings in effect
are one-turn inductors, a larger number is needed to obtain the desired
impedance especially since these (usually small) ferrite rings exhibit a much
lower winding factor each. As an example for a comparable performance choke
and using RG58 type of coax, we would need 56 toroides
of 4A11 type (43) ferrite, TN 10/6/4 size, adding up to a balun length of The parasitic
parallel capacitor however is much smaller than in the More types of 1 : 1 balun are possible and practical and will be discussed in a dedicated article on these components in Balun. For more information on transmission-line transformers, please look at the next chapter. Bob J. van Donselaar, on9cvd@veron.nl |
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