HFE0208 GrebennikovPart3 Rev, Krótkofalarstwo
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From February 2008
High Frequency Electronics
Copyright © 2008 Summit Technical Media, LLC
High Frequency Design
COMBINERS & COUPLERS
Power Combiners, Impedance
Transformers and Directional
Couplers: Part III
By Andrei Grebennikov
Microwave hybrids
The branch-line cou-
plers or hybrids were
firstly described more
than six decades ago;
however, the problem of
their exact synthesis
remained a puzzle for a
number of years [53]. Initially, the branch-line
hybrid was analyzed as a four-arm symmetri-
cal network based on a superposition of the
results obtained in the even and odd modes
[54]. By writing the even and odd mode matri-
ces together, the characteristic impedances of
the branch lines and coupling into different
ports can be obtained. A general synthesis pro-
cedure which can be applied to any structure
of a multibranch hybrid, based on an invari-
ance of the Richard’s variable
S
=
j
tan
This multi-part article on
coupler and combiner
structures continues with
an examination of
microwave hybrids using
various topologies
Figure 30 · Microstrip branch-line quadra-
ture hybrid.
502 54
/
=
.
Ω
the microstrip branch-line hybrid shown in
Fig. 30 represents a 3-dB directional coupler,
for which power in arm 1 divides evenly
between arms 2 and 3 with the phase shift of
90°. No power is delivered to arm 4, because
the signal flowing through different paths
(lengths of
θ
to the
transformation of
S
1/
S
apart from a 180°
phase change, had become available a decade
later [55]. As a result, with highly precise com-
puter-design techniques available for branch-
line hybrids, it became possible to generate
any coupling value in the useful 0 to 15 dB
coupling range. Waveguide designs which
have been used in large complex feeds for
phase-array radars, are compact, highly pre-
dictable in amplitude and phase characteris-
tics, and handle very high power. Coaxial,
microstrip or stripline implementations of
branch-line hybrids provide simple planar
structures of moderate bandwidth capability,
up to about 2/3 of an octave.
For a fully matched case with standard 50-
→
/4) have the same ampli-
tude and opposite phases at this port. The
branch-line hybrid does not depend on the
load mismatch level for equal reflected coeffi-
cients from the outputs when all reflected
power is dissipated in the 50-
λ
/4 and 3
λ
ballast resis-
tor. However, in practice, due to the quarter-
wavelength transmission-line requirement,
the bandwidth of such a single-stage quadra-
ture branch-line hybrid is limited to 10-20%.
Figure 31 shows the calculated frequency
bandwidth characteristics of a single-section
branch-line hybrid matched at the center
bandwidth frequency with the load impedance
Z
L
=
Z
0
= 50
Ω
Ω
source and load impedances, when the char-
acteristic impedances of its transverse
branches are 50
and the characteristic
impedances of its longitudinal main lines are
Ω
, where
C
12
is the insertion loss
calculated as the ratio of powers at the input
Ω
42
High Frequency Electronics
High Frequency Design
COMBINERS & COUPLERS
Figure 32 · Microstrip branch-line quadrature
impedance-transforming hybrid.
Figure 31 · Bandwidth performance of the single-sec-
tion branch-line hybrid.
port 1 relative the output port 2,
C
13
is the coupling cal-
culated as the ratio of powers at the input port 1 relative
to the output port 3, and
C
14
is the isolation calculated as
the ratio of powers at the input port 1 relative to the iso-
lated port 4 [56]. At millimeter-wave frequencies, the
lengths of the microstrip lines can actually get shorter
than the widths and the mutual coupling between the
input lines and discontinuities at the input increases sig-
nificantly. This has a direct effect on the input/output
match, frequency bandwidth and isolation. To minimize
the effect of these problems, the branch-line hybrid can be
designed as a two-section hybrid using three-quarterwave
lines for the series main lines and quarterwave lines for
the shunt branch lines, with all inputs/outputs orthogonal
to each other [57]. As a result, the return loss is 10 dB or
better over 90% of the band, the isolation can achieve 10
dB or better over the whole band, and the difference in
the coupling can be equal or less than 1 dB over about
75% of the frequency band from 26 to 40 GHz.
If one pair of terminating resistors has different val-
ues compared to the other pair, the resulting branch-line
hybrid can operate as a directional coupler and an
impedance transformer simultaneously [58]. Design val-
ues of the branch- and main-line characteristic
impedances for a single-section branch-line hybrid shown
in Fig. 32, related to the input source impedance
Z
0S
and
output load impedance
Z
0L
, can be calculated by
Figure 33 · Broadband microwave branch-line
quadrature hybrid.
to 25-
) impedance transformation ratio can provide
approximately 20-percent frequency bandwidth with
±0.25 dB amplitude imbalance. However, for a fixed direc-
tivity, the frequency bandwidth of branch-line impedance-
transforming hybrid increases as the output-to-input
impedance ratio is reduced [58].
The operating bandwidth can be significantly
increased using multi-stage impedance-transforming
hybrids. A two-section branch-line impedance-transform-
ing quadrature hybrid is shown in Fig. 33. To design this
hybrid with the given impedance transformation ratio
r
and power-split ratio
K
2
, the branch- and main-line char-
acteristic impedances should be chosen according to
Ω
ZZr
t r
tr
2
−
−
(31)
=
1
0
S
Z
K
2
Z
=
0
S
(28)
Z
Z
2
Zr
r
t
−
⎛
⎞
⎟
1
2
=
(32)
⎜
0
S
3
ZZ
K
Z
=
00
2
SL
(29)
(
)
2
rt
2
−
r
1
+
(33)
Z
=
Z
4
0
S
t
−
1
ZZ
Z
Z
=
10
L
(30)
3
where [60],
0
S
where
K
is the voltage-split ratio between output ports 2
and 3, and
R
0
=
Z
0S
[59]. Such a hybrid with a 2-to-1 (50-
2
t
=+
41
K
44
High Frequency Electronics
High Frequency Design
COMBINERS & COUPLERS
Figure 34 · Reduced-size branch-line quadrature hybrid.
The condition of
Z
2
=
Z
3
gives maximum bandwidth when
the best performance at the center bandwidth frequency
is specified.
For an equal power division when
K
= 1, the condition
t
=
r
2 specifies a minimum value of
r
which is equal to
0.5. However, in practice, it is better to choose
r
in the
range of 0.7 to 1.3, in order to provide the physically real-
izable branch-line characteristic impedances for 50-
√
Ω
input impedance. For example, for the 50 to 35
Ω
impedance transformation using a two-stage hybrid, the
impedances are as follows:
Z
1
= 72.5
Ω
,
Z
2
=
Z
3
= 29.6
Ω
,
and
Z
4
= 191.25
. This gives the power balance between
the output ports better than 0.5 dB with the return loss
and isolation better than 20 dB over a frequency band-
width of 25% for a 2-GHz hybrid.
For monolithic microwave integrated circuit (MMIC)
applications, the overall size of the quadrature branch-
line hybrids with quarterwave transmission lines is too
large. Therefore, it is attractive to replace each quarter-
wave branch line with the combination of a short-length
transmission line and two shunt capacitors providing the
same bandwidth properties. Consider the admittance
matrix [
Y
a
] for a quarterwave transmission line shown in
Fig. 34(a) and the admittance matrix [
Y
b
] for a circuit con-
sisting of a short transmission line with two shunt capac-
itors shown in Fig. 34(b) which are given by
Ω
Figure 35 · Equivalent circuits of lumped LC-type
hybrid.
parameters can be obtained in the form of
Z
Z
=
0
sin
(36)
θ
01
10
−
1
⎡
⎢
⎤
⎥
1
[]
=
Y
(34)
C
=
(37)
a
−
jZ
ωθ
Z
cos
0
0
cos
θω
−
CZ
sin
θ
−
1
1
(35)
from which it follows that the lengths of the hybrid trans-
mission lines can be made much shorter by increasing
their characteristic impedance
Z
. For example, when
choosing the electrical length of
[]
=
Y
b
jZ
sin
θ
−
1
cos
θω
−
CZ
sin
θ
where
Z
0
is the characteristic impedance of a quarter-
wave line,
Z
and
= 45°, the characteris-
tic impedance of the transmission line increases by a fac-
tor of
θ
are the characteristic impedance and
electrical length of a shortened line, and
C
is the shunt
capacitance. By equating the corresponding Y-parameters
in Eqs. (34) and (35), the simple ratios between the circuit
θ
2.
A circuit schematic of the reduced-size branch-line
quadrature hybrid is shown in Fig. 34(c) [61]. Compared
√
46
High Frequency Electronics
to the conventional branch-line
hybrid with characteristic impe-
dances of its b
ra
nch- and main-lines
of
Z
0
and
Z
0
/
branch-line hybrid is shown in Fig.
35(a) [62]. This circuit has also some
additional advantages when each its
section can work as a separate
impedance transformer, a low-pass
filter, and a phase shifter. The circuit
can be diced into four separate sec-
tions and cascaded for the desired
transmission characteristics. The cir-
cuit analysis indicates that various
types of networks fulfill the condi-
tions required for an ideal hybrid.
Therefore, greater design flexibility
in the choice of the hybrid structure
and performance is possible. Several
possible single-section two-branch
hybrid options are shown in Fig. 35
[63]. In this case, it should be noted
that not only a low-pass section but
also a high-pass section can be
2 respectively, the cir-
cuit parameters of the reduced-size
branch-line hybrid are obtained from
√
Z
Z
⎛
⎜
⎞
⎟
−
1
θ
1
=
sin
0
(38)
Z
Z
⎛
⎜
⎞
⎟
(39)
θ
2
=
sin
−
1
0
2
2
2
Z
Z
Z
Z
=−
⎛
⎞
⎟
+−
⎛
⎞
⎟
ω
CZ
1
0
2
0
⎜
⎜
0
(eq. 40), where
θ
2
are the
electrical lengths of the shunt
branch-line and series main-line,
respectively. For a particular case of
the
θ
1
and
standard
characteristic
impedance
Z
0
= 50
, the character-
istic impedance and electrical
lengths of the transmission lines
are defined as
Ω
ZZ
=
/
2
0
θ
1
= 45°
θ
2
= 30°
as shown in Fig. 34(c).
Experimental results for a 25-GHz
reduced-size branch-line quadra-
ture hybrid show that its bandwidth
performance is slightly narrower
than that of the conventional quar-
terwave hybrid, but its overall size
is more than 80% smaller.
To further reduce the MMIC
size, the transmission lines can be
fully replaced by the lumped planar
inductors. Such an approach
becomes possible because the sym-
metrical lumped
LC
-type
- or
T
-
section is equivalent at a single fre-
quency to the transmission-line sec-
tion with the appropriate character-
istic impedance and electrical
length. The lumped-element equiva-
lent circuit of a transmission-line
π
High Frequency Design
COMBINERS & COUPLERS
respectively used. In
the latter case, the
high-pass
LC
section is
considered an equiva-
lent single-frequency
replacement for a 270-
degree transmission
line [64]. Figures 35(c)
and 35(d) illustrate the
use of both low-pass
and high-pass sections
simultaneously, while
only high-pass sections
compose the hybrid
shown in Fig. 35(e). The
performances of the
hybrids shown in Figs.
35(b) and 35(e) are very
similar to that of the
classical single-section
branch-line hybrid. The
bandwidth perfor-
mances of the hybrids
shown in Figs. 35(c)
and 35(d) are narrower because the power balance
between their output ports is much narrower. Broader
bandwidth and lower output impedances can be provided
with a two-section three-branch lumped-element hybrid.
Figure 36(a) shows the equivalent circuit of a capaci-
tively coupled lumped-element hybrid, which is used for
monolithic design of variable phase shifters [65].
However, the power and phase balance bandwidth at the
output ports of this hybrid is very narrow, in the limits of
a few percent. An alternative design of an inductively cou-
pled lumped-element hybrid is shown in Fig. 36(b) [66].
As a basic element, it includes lumped multiturn mutual-
ly coupled spiral inductors with the coupling coefficient,
which can be realized using a bifilar (a sandwich of two
multiturn spiral inductors with inner and outer wind-
ings) spiral transformer to achieve a coupling coefficient
k
= 0.707. In this case, this basic lumped-element config-
uration is completely equivalent to a transmission line in
the vicinity of the center bandwidth frequency. The induc-
tively coupled hybrid can provide a power balance within
0.2 dB and phase balance within 1° in a frequency band-
width of ±10% in 2-GHz wireless applications. However,
during the design procedure, some parasitic effects should
be taken into account. For example, the coupled inductor
itself has a significant value of internal capacitance. Also,
the finite value of the inductor quality factor results in a
modest amplitude imbalance, but leads to a significant
phase deviation from ideal quadrature 90-degree differ-
ence. In this case, to compensate for the resulting perfor-
mance degradation, the electromagnetic simulation of the
Figure 36 · Equivalent circuits
of lumped hybrid with capaci-
tive and inductive coupling.
Figure 37 · Microstrip rat-race ring hybrids.
structure and optimization of the values of the added
shunt capacitors on both sides of the circuit are required.
Two of these inductively-coupled hybrids can be cascaded
in tandem to significantly extend the frequency band-
width. As a result, the phase shift of 93 ±6°, the insertion
loss between 1 and 1.5 dB, the return loss better than 16
dB, and the isolation between the output ports better
than 18 dB were measured over the frequency range from
2 to 6 GHz [66].
A hybrid-ring directional coupler or rat-race, which is
one of the fundamental components used in microwave
circuits was described and analyzed more than six
decades ago [67]. Its operation principle is based on an
assumption that the voltage at any point along the trans-
mission line is a superposition of the forward and back-
ward propagating waves. Signal from the excitation
source spreads out in the driving point and propagates
along the line. As the forward wave reaches the far-end
termination, it reflects, propagates backward, reflects
from the near-end termination, propagates forward again,
and continues in a loop. According to this wave behavior,
a pure standing wave is set up within the ring when there
is no mechanism for dissipation other than the minor
ohmic losses associated with wave transmission. The
point of voltage minimum (zero) corresponds to the case
48
High Frequency Electronics
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From February 2008
High Frequency Electronics
Copyright © 2008 Summit Technical Media, LLC
High Frequency Design
COMBINERS & COUPLERS
Power Combiners, Impedance
Transformers and Directional
Couplers: Part III
By Andrei Grebennikov
Microwave hybrids
The branch-line cou-
plers or hybrids were
firstly described more
than six decades ago;
however, the problem of
their exact synthesis
remained a puzzle for a
number of years [53]. Initially, the branch-line
hybrid was analyzed as a four-arm symmetri-
cal network based on a superposition of the
results obtained in the even and odd modes
[54]. By writing the even and odd mode matri-
ces together, the characteristic impedances of
the branch lines and coupling into different
ports can be obtained. A general synthesis pro-
cedure which can be applied to any structure
of a multibranch hybrid, based on an invari-
ance of the Richard’s variable
S
=
j
tan
This multi-part article on
coupler and combiner
structures continues with
an examination of
microwave hybrids using
various topologies
Figure 30 · Microstrip branch-line quadra-
ture hybrid.
502 54
/
=
.
Ω
the microstrip branch-line hybrid shown in
Fig. 30 represents a 3-dB directional coupler,
for which power in arm 1 divides evenly
between arms 2 and 3 with the phase shift of
90°. No power is delivered to arm 4, because
the signal flowing through different paths
(lengths of
θ
to the
transformation of
S
1/
S
apart from a 180°
phase change, had become available a decade
later [55]. As a result, with highly precise com-
puter-design techniques available for branch-
line hybrids, it became possible to generate
any coupling value in the useful 0 to 15 dB
coupling range. Waveguide designs which
have been used in large complex feeds for
phase-array radars, are compact, highly pre-
dictable in amplitude and phase characteris-
tics, and handle very high power. Coaxial,
microstrip or stripline implementations of
branch-line hybrids provide simple planar
structures of moderate bandwidth capability,
up to about 2/3 of an octave.
For a fully matched case with standard 50-
→
/4) have the same ampli-
tude and opposite phases at this port. The
branch-line hybrid does not depend on the
load mismatch level for equal reflected coeffi-
cients from the outputs when all reflected
power is dissipated in the 50-
λ
/4 and 3
λ
ballast resis-
tor. However, in practice, due to the quarter-
wavelength transmission-line requirement,
the bandwidth of such a single-stage quadra-
ture branch-line hybrid is limited to 10-20%.
Figure 31 shows the calculated frequency
bandwidth characteristics of a single-section
branch-line hybrid matched at the center
bandwidth frequency with the load impedance
Z
L
=
Z
0
= 50
Ω
Ω
source and load impedances, when the char-
acteristic impedances of its transverse
branches are 50
and the characteristic
impedances of its longitudinal main lines are
Ω
, where
C
12
is the insertion loss
calculated as the ratio of powers at the input
Ω
42
High Frequency Electronics
High Frequency Design
COMBINERS & COUPLERS
Figure 32 · Microstrip branch-line quadrature
impedance-transforming hybrid.
Figure 31 · Bandwidth performance of the single-sec-
tion branch-line hybrid.
port 1 relative the output port 2,
C
13
is the coupling cal-
culated as the ratio of powers at the input port 1 relative
to the output port 3, and
C
14
is the isolation calculated as
the ratio of powers at the input port 1 relative to the iso-
lated port 4 [56]. At millimeter-wave frequencies, the
lengths of the microstrip lines can actually get shorter
than the widths and the mutual coupling between the
input lines and discontinuities at the input increases sig-
nificantly. This has a direct effect on the input/output
match, frequency bandwidth and isolation. To minimize
the effect of these problems, the branch-line hybrid can be
designed as a two-section hybrid using three-quarterwave
lines for the series main lines and quarterwave lines for
the shunt branch lines, with all inputs/outputs orthogonal
to each other [57]. As a result, the return loss is 10 dB or
better over 90% of the band, the isolation can achieve 10
dB or better over the whole band, and the difference in
the coupling can be equal or less than 1 dB over about
75% of the frequency band from 26 to 40 GHz.
If one pair of terminating resistors has different val-
ues compared to the other pair, the resulting branch-line
hybrid can operate as a directional coupler and an
impedance transformer simultaneously [58]. Design val-
ues of the branch- and main-line characteristic
impedances for a single-section branch-line hybrid shown
in Fig. 32, related to the input source impedance
Z
0S
and
output load impedance
Z
0L
, can be calculated by
Figure 33 · Broadband microwave branch-line
quadrature hybrid.
to 25-
) impedance transformation ratio can provide
approximately 20-percent frequency bandwidth with
±0.25 dB amplitude imbalance. However, for a fixed direc-
tivity, the frequency bandwidth of branch-line impedance-
transforming hybrid increases as the output-to-input
impedance ratio is reduced [58].
The operating bandwidth can be significantly
increased using multi-stage impedance-transforming
hybrids. A two-section branch-line impedance-transform-
ing quadrature hybrid is shown in Fig. 33. To design this
hybrid with the given impedance transformation ratio
r
and power-split ratio
K
2
, the branch- and main-line char-
acteristic impedances should be chosen according to
Ω
ZZr
t r
tr
2
−
−
(31)
=
1
0
S
Z
K
2
Z
=
0
S
(28)
Z
Z
2
Zr
r
t
−
⎛
⎞
⎟
1
2
=
(32)
⎜
0
S
3
ZZ
K
Z
=
00
2
SL
(29)
(
)
2
rt
2
−
r
1
+
(33)
Z
=
Z
4
0
S
t
−
1
ZZ
Z
Z
=
10
L
(30)
3
where [60],
0
S
where
K
is the voltage-split ratio between output ports 2
and 3, and
R
0
=
Z
0S
[59]. Such a hybrid with a 2-to-1 (50-
2
t
=+
41
K
44
High Frequency Electronics
High Frequency Design
COMBINERS & COUPLERS
Figure 34 · Reduced-size branch-line quadrature hybrid.
The condition of
Z
2
=
Z
3
gives maximum bandwidth when
the best performance at the center bandwidth frequency
is specified.
For an equal power division when
K
= 1, the condition
t
=
r
2 specifies a minimum value of
r
which is equal to
0.5. However, in practice, it is better to choose
r
in the
range of 0.7 to 1.3, in order to provide the physically real-
izable branch-line characteristic impedances for 50-
√
Ω
input impedance. For example, for the 50 to 35
Ω
impedance transformation using a two-stage hybrid, the
impedances are as follows:
Z
1
= 72.5
Ω
,
Z
2
=
Z
3
= 29.6
Ω
,
and
Z
4
= 191.25
. This gives the power balance between
the output ports better than 0.5 dB with the return loss
and isolation better than 20 dB over a frequency band-
width of 25% for a 2-GHz hybrid.
For monolithic microwave integrated circuit (MMIC)
applications, the overall size of the quadrature branch-
line hybrids with quarterwave transmission lines is too
large. Therefore, it is attractive to replace each quarter-
wave branch line with the combination of a short-length
transmission line and two shunt capacitors providing the
same bandwidth properties. Consider the admittance
matrix [
Y
a
] for a quarterwave transmission line shown in
Fig. 34(a) and the admittance matrix [
Y
b
] for a circuit con-
sisting of a short transmission line with two shunt capac-
itors shown in Fig. 34(b) which are given by
Ω
Figure 35 · Equivalent circuits of lumped LC-type
hybrid.
parameters can be obtained in the form of
Z
Z
=
0
sin
(36)
θ
01
10
−
1
⎡
⎢
⎤
⎥
1
[]
=
Y
(34)
C
=
(37)
a
−
jZ
ωθ
Z
cos
0
0
cos
θω
−
CZ
sin
θ
−
1
1
(35)
from which it follows that the lengths of the hybrid trans-
mission lines can be made much shorter by increasing
their characteristic impedance
Z
. For example, when
choosing the electrical length of
[]
=
Y
b
jZ
sin
θ
−
1
cos
θω
−
CZ
sin
θ
where
Z
0
is the characteristic impedance of a quarter-
wave line,
Z
and
= 45°, the characteris-
tic impedance of the transmission line increases by a fac-
tor of
θ
are the characteristic impedance and
electrical length of a shortened line, and
C
is the shunt
capacitance. By equating the corresponding Y-parameters
in Eqs. (34) and (35), the simple ratios between the circuit
θ
2.
A circuit schematic of the reduced-size branch-line
quadrature hybrid is shown in Fig. 34(c) [61]. Compared
√
46
High Frequency Electronics
to the conventional branch-line
hybrid with characteristic impe-
dances of its b
ra
nch- and main-lines
of
Z
0
and
Z
0
/
branch-line hybrid is shown in Fig.
35(a) [62]. This circuit has also some
additional advantages when each its
section can work as a separate
impedance transformer, a low-pass
filter, and a phase shifter. The circuit
can be diced into four separate sec-
tions and cascaded for the desired
transmission characteristics. The cir-
cuit analysis indicates that various
types of networks fulfill the condi-
tions required for an ideal hybrid.
Therefore, greater design flexibility
in the choice of the hybrid structure
and performance is possible. Several
possible single-section two-branch
hybrid options are shown in Fig. 35
[63]. In this case, it should be noted
that not only a low-pass section but
also a high-pass section can be
2 respectively, the cir-
cuit parameters of the reduced-size
branch-line hybrid are obtained from
√
Z
Z
⎛
⎜
⎞
⎟
−
1
θ
1
=
sin
0
(38)
Z
Z
⎛
⎜
⎞
⎟
(39)
θ
2
=
sin
−
1
0
2
2
2
Z
Z
Z
Z
=−
⎛
⎞
⎟
+−
⎛
⎞
⎟
ω
CZ
1
0
2
0
⎜
⎜
0
(eq. 40), where
θ
2
are the
electrical lengths of the shunt
branch-line and series main-line,
respectively. For a particular case of
the
θ
1
and
standard
characteristic
impedance
Z
0
= 50
, the character-
istic impedance and electrical
lengths of the transmission lines
are defined as
Ω
ZZ
=
/
2
0
θ
1
= 45°
θ
2
= 30°
as shown in Fig. 34(c).
Experimental results for a 25-GHz
reduced-size branch-line quadra-
ture hybrid show that its bandwidth
performance is slightly narrower
than that of the conventional quar-
terwave hybrid, but its overall size
is more than 80% smaller.
To further reduce the MMIC
size, the transmission lines can be
fully replaced by the lumped planar
inductors. Such an approach
becomes possible because the sym-
metrical lumped
LC
-type
- or
T
-
section is equivalent at a single fre-
quency to the transmission-line sec-
tion with the appropriate character-
istic impedance and electrical
length. The lumped-element equiva-
lent circuit of a transmission-line
π
High Frequency Design
COMBINERS & COUPLERS
respectively used. In
the latter case, the
high-pass
LC
section is
considered an equiva-
lent single-frequency
replacement for a 270-
degree transmission
line [64]. Figures 35(c)
and 35(d) illustrate the
use of both low-pass
and high-pass sections
simultaneously, while
only high-pass sections
compose the hybrid
shown in Fig. 35(e). The
performances of the
hybrids shown in Figs.
35(b) and 35(e) are very
similar to that of the
classical single-section
branch-line hybrid. The
bandwidth perfor-
mances of the hybrids
shown in Figs. 35(c)
and 35(d) are narrower because the power balance
between their output ports is much narrower. Broader
bandwidth and lower output impedances can be provided
with a two-section three-branch lumped-element hybrid.
Figure 36(a) shows the equivalent circuit of a capaci-
tively coupled lumped-element hybrid, which is used for
monolithic design of variable phase shifters [65].
However, the power and phase balance bandwidth at the
output ports of this hybrid is very narrow, in the limits of
a few percent. An alternative design of an inductively cou-
pled lumped-element hybrid is shown in Fig. 36(b) [66].
As a basic element, it includes lumped multiturn mutual-
ly coupled spiral inductors with the coupling coefficient,
which can be realized using a bifilar (a sandwich of two
multiturn spiral inductors with inner and outer wind-
ings) spiral transformer to achieve a coupling coefficient
k
= 0.707. In this case, this basic lumped-element config-
uration is completely equivalent to a transmission line in
the vicinity of the center bandwidth frequency. The induc-
tively coupled hybrid can provide a power balance within
0.2 dB and phase balance within 1° in a frequency band-
width of ±10% in 2-GHz wireless applications. However,
during the design procedure, some parasitic effects should
be taken into account. For example, the coupled inductor
itself has a significant value of internal capacitance. Also,
the finite value of the inductor quality factor results in a
modest amplitude imbalance, but leads to a significant
phase deviation from ideal quadrature 90-degree differ-
ence. In this case, to compensate for the resulting perfor-
mance degradation, the electromagnetic simulation of the
Figure 36 · Equivalent circuits
of lumped hybrid with capaci-
tive and inductive coupling.
Figure 37 · Microstrip rat-race ring hybrids.
structure and optimization of the values of the added
shunt capacitors on both sides of the circuit are required.
Two of these inductively-coupled hybrids can be cascaded
in tandem to significantly extend the frequency band-
width. As a result, the phase shift of 93 ±6°, the insertion
loss between 1 and 1.5 dB, the return loss better than 16
dB, and the isolation between the output ports better
than 18 dB were measured over the frequency range from
2 to 6 GHz [66].
A hybrid-ring directional coupler or rat-race, which is
one of the fundamental components used in microwave
circuits was described and analyzed more than six
decades ago [67]. Its operation principle is based on an
assumption that the voltage at any point along the trans-
mission line is a superposition of the forward and back-
ward propagating waves. Signal from the excitation
source spreads out in the driving point and propagates
along the line. As the forward wave reaches the far-end
termination, it reflects, propagates backward, reflects
from the near-end termination, propagates forward again,
and continues in a loop. According to this wave behavior,
a pure standing wave is set up within the ring when there
is no mechanism for dissipation other than the minor
ohmic losses associated with wave transmission. The
point of voltage minimum (zero) corresponds to the case
48
High Frequency Electronics
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