Index
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Ferrites in HF-applications
Transmission-line
transformers Examples
(published in
Electron # 1, 2002) General This chapter is the fifth 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.
The fourth article is an introductory
to transmission-line 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.
Guanella and Ruthroff 1 : 4 unun
Guanella A schematic diagram of a Guanella 1 : 4 unun
may be found in figure 17. Two transmission-lines are connected in parallel
at the input, while the outputs are connected in series and in series with
the load Rb.
Transformer analysis The voltages and currents have been drawn in
figure 17 according to the principles 31 - 33. Each following analysis will
start off with this 'exercise' as this first step already completes half of
the analysis. At the generator side, transmission-lines are
connected in parallel, so each is connected to a voltage 'u'. At the load
side, the center conductor of the bottom transmission line is connected to
the outer conductor of the upper line, effectively putting the outputs of the
transformer in series, so the output voltages will add. With the inputs
connected in parallel and each transmission-line is carrying a voltage 'u',
the load is connected across a voltage '2.u', making this construction a 1 : 2
voltage transformer. At the input the transmission-lines each are
carrying a current 'i'; the generator therefore carries a current '2 . i'. With voltages and currents in position in
figure 17, all impedances will be referenced to the load in the next step. A voltage of '2.u' is found across the load
with a current of 'i' flowing. Load resistance is characterized by: Rb = 2.u / i. The generator carriers a voltage 'u', with a
current '2.i' flowing. This will make the generator 'see' an impedance: (u / 2i) Zin =
--------- * Rb = Rb
/ 4
(2u / i) This indeed is a 1 : 4 impedance transformer.
Each of the transmission-lines is carrying a
current 'i' with a voltage 'u' across. Optimal characteristic impedance for
these lines follows from:
(u / i) ZC =
-------- * Rb = Rb
/ 2
(2u / i) When this transformer is to operate in a 50
Ohm environment (Zin), optimal load resistance will be 4 x 50 = 200
Ohm while transmission-lines should have a characteristic impedance of 100
Ohm, for which RG 62 comes close at 93 Ohm. This type of cable still might be
found in older data networks. Sleeve impedance No voltage is found across the outside of the
bottom transmission-line, so no (parasitic, return) current will flow. At the upper transmission-line, the outside
is at ground potential at the generator and at a potential of 'u' at the
load. The outside of this line therefore is effective connected across the
generator (same potential). This parallel reactance we like to be of
'negligible' influence, for which we derived this to be at least 2,5 x the
system impedance, so we calculate: Zsleeve = 2,5 x Zin =
2,5 x Rb / 4 = 0,625 Rb In our 50 Ohm example, Zsleeve will
be 125 Ohm at the lowest design frequency, for this example 1,5 MHz. When
selecting a n = √(125 / ω . mC . F) = 3,3 (3) Note that we calculated the upper
transmission-line (length!); the bottom line will have the identical length
(equal delay Guanella principle). Maximum core load We calculate maximum voltage across the
sleeve reactance for this core (A = 1,18 .10- UL (induction) <
0,89 .Bsat .f .n .A =
0,89 .0,33 .1,5 .106 .3 .1,18 .10-4 =
156 V. As we expect core dissipation to set the
application limit, we calculate maximum voltage across the sleeve reactance
for maximum 4 Watts of core loss: UL(dissipation) = ÖPmax ∙ (Q/6 + 1/Q) ∙ XL = 52 V. One or
two cores? This Guanella 1 : 4 transformer
may be constructed on one ferrite toroide, when winding in the same
direction. In the above example, the transformer could be made by placing six
turns on the core and cutting half way. At this position lines are to be
connected in parallel (effectively creating a parallel tap on the line) while
at the other side the lines are connected in series. Different characteristic impedance? The characteristic impedance of the
transmission-line is especially important at the high-frequency limit of the
transformer. We calculated transmission-line impedance to be 100 Ohm in our
50 to 200 Ohm example. Since 50 Ohm line is more common we may want to
construct the transformer with this type of line. As an example a transformer
with this line impedance was exhibiting SWR = 2 at 100 MHz. Measuring the
same transformer when terminated into 100 Ohm, as required for a 25 to 100
ratio, SWR = 1,1 at 200 MHz. In this last measurement set-up two effects
are active at the same time. One effect points at the importance of the characteristic
impedance of the transmission-line in the transformer, the other points at
the effect of the relative line length as related to mismatching
impedance. To further investigate the effect of
mismatching we calculated behavior over frequency of a Guanella 1 : 4
impedance transformer with m = Rb / Z0 (or reversed!)
as a parameter. The transformer has been constructed on a
In figure 18 high
frequency cut-off is going down rapidly with rising mismatch. At a
mismatching of 2 (red line) high frequency cut-off is down to 25 MHz. at the
SWR = 1,5 definition and to 45 MHz. at SWR = 2. When total line
length was Ruthroff Since the line output is completely decoupled
from the input (basic rule 33), we may connect any of the output terminals to any other point in the circuit
or to ground, as depicted by the dashed ground symbol in figure 17. Note the
lower terminal of the upper transmission-line to be at a potential 'u', which
is equal to the generator potential. Therefore we may directly connect this
terminal to the generator and leave out the bottom transmission-line. This
situation is depicted in figure 19, the Ruthroff 1 : 4 unun.
After applying basic rules 31 - 33, voltages
and currents may be placed in the figure. As with the Guanella, the generator
with a voltage 'u' is supplying a current '2.i', the transmission-line is
carrying a voltage 'u' with a current 'i' while the load is carrying a
current 'i' with a voltage '2.u' across. The Ruthroff will therefore exhibit
the same characteristics as a 1 : 4 Guanella impedance transformer, including
the requirements as to the transmission-line. For a 50 - 200 Ohm application, the
transmission-line should have a characteristic impedance of 100 Ohm. When
constructing this transformer on a Also with
this transformer a
few tests have
been performed to determine
sensitivity to the transmission-line characteristic impedance. For these
tests the transformer has been constructed using three turns 93 Ohm
transmission - line on a With our SWR = 1,5 definition,
frequency range is 0,5 - 120 MHz. The same transformer has been constructed
on the same toroide, this time using 50 Ohm transmission-line. This time the
frequency range was 0,5 - 100 MHz. We noticed before that the Ruthroff concepts
adds direct voltages to delayed ones. This makes this transformer more
sensitive to the transmission-line length. For this reason designers
sometimes prefer the somewhat 'larger' Guanella to the Ruthroff principle.
Guanella and Ruthroff 1:4 balunGuanella The Guanella balun in figure 20 is very much
like the figure 17 design. Since all outputs are decoupled from the input any
of the output terminals may be connected to ground, also the center
connection. In this latter situation the unun of figure 17 is transformed
into a the balun of figure 20.
After we placed all currents and voltages into
the figure according to the basic rules 31 - 33, an almost identical
situation arises and all impedances are the same as derived before. One of
the differences is the voltage at the lower output terminal of the bottom
transmission-line, that is now at a voltage of '-u', while the voltage on the
upper terminal of the top line is at 'u' voltage. Therefore it is now the
outside of the lower terminal that is carrying the voltage 'u', where we
should ensure a low parasitic current in stead of the outside of the upper
line. When constructing this transformer for an
impedance ratio of 50 - 200 Ohm, we again should be using a Ruthroff In figure 20 we find a voltage of 'u' at the
top of the load, that is equal to the generator voltage. Therefore we may
connect this side of the load directly to the generator and leave out the
upper transmission-line. The circuit of figure 20 then resolves into figure
21, the Ruthroff 1 : 4 balun.
Identical to the Ruthroff unun, this circuit will
function for short length of the transmission-line (low delay). The pattern
of voltages and currents according to the basic rules 31 - 33 is identical to
the Guanella, so all impedance calculations apply including the
characteristic impedance of the transmission-line and the minimum impedance
of the sleeve impedance (toroide size, ferrite type, number of turns) at the
lowest frequency of operation. Guanella and Ruthroff 1 : 9 unun Guanella The Guanella 1 : 9 is build-up in an
identical way as the Guanella 1 : 4 unun, using the same basic elements.
Inputs are in parallel and all outputs in series, putting a voltage of three
times the input voltage across the load. This makes for an impedance ratio of
1 : 9 as may be appreciated from figure 22. Note the un-even number of lines
will exclude a balancing output terminal to ground.
Analysis Applying the basic principles
of 31 - 33 will complete the picture
of voltages and currents in the figure. From this it follows that a voltage of
'3.u' is across the load with a current 'i' flowing. At the generator is a
voltage 'u' with a current of '3.i' flowing. The generator therefore 'sees'
an impedance of: (u / 3i) Zin =
--------- * Rb = Rb
/ 9 (3u / i) Each of the transmission-lines is
carrying a voltage 'u' with a current 'i' flowing, requiring a characteristic
impedance of:
(u / i) Zc =
-------- * Rb = Rb
/ 3 (3u / i) Transmission-lines When we
like to design a transformer for an impedance range 50 - 450 Ohm, the
characteristic impedance of the transmission-lines should be 450 / 3 = 150
Ohm. This is a value normally not available in coaxial lines. Twisting a pair
of plastic isolated, mounting wires ( Sleeve impedance As we derived before, total parallel
reactance at the input should be higher than 2,5 x Zin = 2,5 x Rb
/9 = 0,28 Rb. In figure 22 we notice no voltage across the
length of the lower transmission-line, a voltage of 'u' across the middle
line and a voltage of '2.u' across the upper line, translating into two times
the impedance for the upper line to the middle line for equal currents. These
impedance are found in parallel at the input, so we may write: 2 x Zsleeve in parallel to Zsleeve
= 0,28 Rb, from which we derive Zsleeve = 0,42 Rb
en 2 Zsleeve = 0,84 Rb. When designing a transformer for an impedance
range 50 - 450 Ohm, the upper sleeve reactance at the lowest operating
frequency therefore should be 0,84 x 450 = 378 Ohm. When constructing this on
a n = √(378 / ω . mC . F) = 5,7 (6) Although the sleeve impedance of the middle
line could be halved, the line should be of equal length according to the
Guanella principle of equal delay. This will improve the lower operating
frequency. When looking somewhat more in detail to
figure 22, it appears the lower two transmission-line to represent a copy of
the Guanella 1 : 4 unun transformer. We where allowed to construct this at a
single toroide as described above. This also applies to Guanella 1 : 9,
provided we apply the number of turns as dictated by the upper line (on a
separate core). Since the number of turns is doubled for this application, we
better select the small diameter Teflon RG188 in stead of RG58 when
constructing on a The transformer construction now consists of
two cores, one with 12 turns cut halfway with one side in series and the
other in parallel plus a second core with 6 turns with one side connected in
parallel tot the parallel side of the
first transformer, the other side in series with the series-side. Maximum load As usual we calculate maximum load for this
transformer. The upper core is strained most since it is carrying a voltage
of '2.u'. Calculating maximum voltage across this inductor for linear
application with 6 turns on a UL(inductie) = 0,89 .Bsat
.f .n .A = 0,89 .0,33 .1,5 106 . 6.1,18 .10-4 =
312 V. Since this represents a voltage of '2.u',
total input voltage is halved at 166 V. Next we calculate maximum voltage for maximum
core dissipation, which we found to be 4 Watt for this core size __________________ UL(dissipation) = ÖPmax ∙ (Q/6 + 1/Q) ∙ XL = 103 V. Again half this voltage is at the input, allowing
maximum 52 Volt at the generator which translates to 54 Watt of continuous
power in a 50 Ohm system. Higher power will be allowed at different
modulation schemes as may be calculated form the 'enhancement' table, even up
to a couple of hundred Watt. In the latter situation, the maximum voltage for
linear use should be checked again. Application area. The above type of transfer ratio usually will be applied at low impedance ranges, e.g. to transform to and from the in- and output stages of power amplifiers. The transformer will then operate in a 5,6 - 50 Ohm environment, where the requirements to the sleeve impedance will be lowered by the same amount.
Ruthroff
Identical to earlier designs, also the
Guanella 1 : 9 transformer may be transformed into a Ruthroff variation.
Since the outside of the middle transmission-line is at a voltage of 'u',
this may be directly connected to the generator to yield figure 23, the
Ruthroff 1 : 9 unun.
Analysis As usual we start off by putting all voltages
and currents into the figure according to basic principles of 31 - 33. This
time the exercise is somewhat more complicated, depending on the starting
position. For this analysis it is best to start form the (single) current
through the load and the upper transmission-line. With all currents and
voltages in position we calculates impedances as related to the load: (u / 3i) Zin =
--------- * Rb = Rb
/ 9 (3u / i) For the upper transmission-line
we find the familiar relation: (u / i) Zboven =
-------- * Rb = Rb
/ 3. For a 50 op 450 W transformer, the characteristic
impedance is 150 W (3u / i) The impedance of the lower
transmission-line however has been changed: (u / 2i) Zonder =
-------- * Rb = Rb
/ (3u / i) Note this transformer is requiring different transmission-lines for optimal performance. Sleeve impedance In figure 23 we notice the outer conductors
of both transmission-lines to be in parallel. Since both are carrying the
same voltage the transformer may be constructed on the same core. Across this
common line we find a voltage 'u', which makes the inductance appear directly
across the generator. Since we derived this parallel reactance to be minimum
2,5 x the parallel impedance at the lowest operational frequency, we may
calculate: Zsleeve = 2,5 x Zin =
2,5 x Rb/9 = 0,28 Rb
When designing the transformer for an
impedance range of 50 - 450 Ohm, sleeve impedance will be 125 Ohm. We next
calculating the number of turns for this transformer when constructed around
a n = √(125 / ω . mC . F) = 3,3 (3) Maximum load Calculating maximum voltage across the sleeve
inductor for linear application: UL(induction) = 0,89 .Bsat
. f .n .A = 0,89 .0,33 .1,5 106 . 3.1,18 .10-4 =
156 V. Next maximum voltage for maximum allowable
core dissipation: __________________ UL(dissipatie) = ÖPmax ∙ (Q/6 + 1/Q) ∙ XL = 52 V. This is equal to the generator voltage,
making this transformer applicable to a maximum and continuous system power
of 54 Watt in a 50 Ohm environment. For non-continuous loads, higher power
may be possible, but voltages should be checked against maximum voltage for
linear use limits.
Different transformer ratio In the above Ruthroff 1 : 9 transformer, a terminal may be found carrying a voltage of '2.u'. This will allow an impedance ratio of 1 : 4. Since both the 1 : 9 and 1 : 4 ratio's appear in the same design, it is possible to apply this transformer in the odd ratio of 9 : 4 ( 1 : 2,25) as well. Ruthroff 1:2,25 unun In figure 24 the Ruthroff 1 : 2,25 transformer may be found, as derived from the Ruthroff 1 : 9 design. The generator is now at the '2.u' position, with the original generator terminal grounded. This transformer has been applied to the Multiband trap antenna.
Analysis Again we apply the basic rules as in 31 - 33,
and apply voltages and current to the design, which is again more complicated
than before. The load 'sees' a voltage of '3.u' across with a current '2.i'
flowing. The generator is at a voltage of '2.u' with a current of '3.i'
flowing. This makes the input impedance equal to:
(2u / 3i) Zin =
----------- * Rb = 4
Rb / 9 = 0.44 Rb (which is a ratio of 2.25!)
(3u / 2i) Across the lower transmission-line is a
voltage 'u' with a current 'i' flowing. Optimal characteristic impedance
therefore will be: (u / i) ZC lower =
---------- * Rb = 2
Rb / 3 = 0.66 Rb (3u / 2i) In an analogue way we calculate the upper
transmission-line carrying a voltage 'u' with a current '2.i' flowing: (u / 2i) Zboven =
----------- * Rb
= Rb / 3 =
0.33 Rb (2i / 3u) Again we discover different transmission
lines for optimal performance. For the multiband trap antenna as mentioned the transformer was designed for a 50 - 112,5 Ohm range and had been constructed using two transmission-lines RG58 (50 Ohm). Performance of this transformer was within SWR < 1,5 over more than all HF bands (see further on). For this chapter we constructed the transformer again, this time using the calculated characteristic impedances of 37,5 and 75 Ohm. Bandwidth indeed is larger as may be expected. From this and earlier experiments it is shown characteristic impedance to be an important design parameter but sensitivity to this parameter is low enough to allow some constructional freedom. Sleeve impedance In figure 24 we find the outer conductors of
both transmission-lines to be in parallel at the output and at the input.
Since both lines are carrying the same voltage, the transmission-lines may be
constructed on the same core. Across this common line we find a voltage 'u'
, which is half the generator voltage. We derived earlier that the parallel
reactance should be 2,5 x the parallel impedance (0,44 Rb) at the
lowest operational frequency. We may therefore write: Zsleeve = 0,5 x 2,5 x 0,44 Rb
= 0,55 Rb When designing this transformer for an
impedance range 50 - 112,5 Ohm, the sleeve inductance should be at least 0,55
x 112,5 Ohm = 61,9 Ohm. When constructing this transformer around a n = √(61,9 / ω . mC . F) = 2,3 (3) Maximum load We calculate the maximum allowable voltage
across the core inductor for linear application: UL(induction) = 0,89 .Bsat
. f .n .A = 0,89 .0,33 .1,5 106 . 3.1,18 .10-4 =
156 V. which is half the voltage across the
generator. The maximum voltage for maximum core loss of
4 Watt for this size of core is: umantel(dissipatie) < Ö Pmax. (Q/6 + 1/Q) . XL = 56 V. again at half the generator voltage, allowing
this transformer to be applied up to 112 volt or 250 Watt of system power in
a 50 Ohm system under continuous load. Depending on the modulation scheme,
maximum load may be much higher but maximum voltage for linear use should be
checked again. Measurements to a 1:2,25 Ruthroff transformer
Input impedance
A practical 1 : 2,25 transmission-line
transformer has been measured in a 50 - 112,5 Ohm environment. For this measurement,
the transformer was constructed using a
Table 8: Measurements to a Ruthroff 1 : 2,25
impedance transformer At the lower frequencies (up to about 5 MHz) ,
input impedance mainly consists of the sleeve inductance Xp,
roughly equal to 2 x Zin as calculated. Up to 10 MHz. we first see
permeability (m’) drop with frequency, up to 30
MHz. followed by the increasing value of core loss (m”). Above 40 MHz. the influence of
parasitic parallel capacitance is becoming visible, further enhanced by the
effects of transmission-line delay that also become noticeable above 50 MHz. In the last column, the input SWR figures
have been calculated, showing this transformer to be applicable from below
0,5 MHz. to over 50 MHz. Measuring harmonics
From our calculations we found this
transformer to be applicable to a maximum of about 250 Watt of continuous
loading. The calculations for maximum allowable induction permitted even higher
voltages so we do not expect linearity problems to occur with this device. To check, two identical Ruthroff 1 : 2,25
transformer have been connected back to back to a generator delivering 350
Watt of continuous power at 3,6 MHz., 14,2 MHz. and 21,3 MHz., with the
second transformer terminating into a high-power, 50 Ohm 'dummy-load'
resistor with a - 46 dB output. Measurements have been performed using a HP
Spectrum analyzer with an over 60 dBc
measurement range. No (additional) harmonics could be measured which
again ensures confidence to the calculations. The last article in this series is on measurements to
ferrite materials. This may be useful to determine type and quality of materials
already in your possession or to determine peculiarities in components in
your system. Bob J. van
Donselaar, |
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