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 balun
Guanella 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.
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