BACKGROUND OF THE INVENTION
-
The present invention relates to a Wilkinson power divider
circuit and a corresponding Design Method, and more
particularly to a Wilkinson power divider circuit,
comprising a plurality of N transmission lines, N being an
integer equal to or greater than 2, having a respective
length of 1 = λ0/4 at a center frequency f0, where λ0 is the
wavelength at f0, and respective line impedances Zw; said
plurality of N transmission lines being connected to a
first port at a respective first end, to a respective
second port at a respective second end, and via a
respective resistor to a node at the respective second end.
-
In general, the conventional Wilkinson power divider
circuit, which is usually fabricated in stripline or
microstrip form, is a lossy multiport network that can be
made having all ports matched with isolation between the
output ports, although in a limited frequency range.
-
Wilkinson power divider circuits may be used as RF (radio
frequency) power splitters or combiners, whose main feature
is that they in theory provide perfect match to the
reference impedance, as well as perfect isolation between
the input or output ports - yet only in a limited frequency
range. For more details, see for example chapter 8 in
"Microwave Engineering" by David M. Pozar, Addison-Wesley,
1993, p. 395 ff.
-
The principle of the classical N-way equal Wilkinson power
divider, as described in E. Wilkinson, "An N-way Hybrid
Power Divider", IRE Trans. on Microwave Theory and
Techniques. Vol. MTT-8, pp. 116 - 118, January 1960, is
shown in Fig. 12, wherein Pl, P2, ..., PN+1 denote ports;
TRL1, TRL2, ..., TRLN transmission lines; 1 = λ0/4
respective line lengths at the center frequency f0; Zw =
√NxZ0 respective line impedances, Z0 being the reference
impedance (usually 50 ohms); O a node and R1 ..., RN = Z0
resistors.
-
For power splitting purposes, port P1 is the input and
ports P2, ..., PN+1 are the outputs, whereas for power
combining purposes, port Pl is the output and ports P2,
..., PN+1 are the inputs. All ports P1, P2, ..., PN+1 are
referenced to ground.
-
The term equal Wilkinson power divider means that in a
power splitting application, power into port P1 is equally
split to ports P2, ..., PN+1 and vice versa for a power
combiner. It is also possible to make Wilkinsons with
unequal power division/combining, see Pozar, section 8.3,
pp. 399 - 400. A disadvantage of doing so is that outputs
are matched to different impedances than Z0.
-
The key parameters for RF(radio frequency) power splitters/
combiners are transmission loss (|S21|, |S31|, ...),
reflection loss (|S11|, |S22|, ...), and especially
isolation (|S23|, ...). Of these, the reflection loss (|S11
|, |S22|, ...) and the isolation (|S23|, ...) are the most
frequency dependent parameters.
-
In general, Sij is the S-parameter stating the ratio (in
terms of amplitude and phase) to port i from an incoming
electromagnetic wave at port j.
-
Sij is generally complex, and may thus be written as Re{Sij}
+ j·Im{Sij} or |Sij|∠Sij. Here Re{Sij} is the real part of
Sij, Im{Sij} is the imaginary part, |Sij| is the magnitude,
and ∠Sij is the angle. Thus, the following relations hold:
|Sij| = √(Re2{Sij}+Im2{Sij})
∠Sij = arctan(Im{Sij}/Re{Sij})
-
For example, S11 is the reflection on port 1 (the
contribution from port 1 to port 1). The corresponding
return loss (in dB) is calculated from this value as -
10·log(|S11|2).
-
Similarly, the transmission gain of a 2-port device is
10·log(|S21|2) (and of course the transmission loss is
-10·log(|S21|2))..
-
In the case of the Wilkinson divider/combiner, an important
parameter is the isolation, which for a 2-port Wilkinson is
-10·log ((|S23|2) and -10·log(|S32|2) (for a symmetrical
Wilkinson, these two expressions are identical).
-
The isolation is a measure of how much energy is leaked
into port 2 when port 3 receives a certain amount of power-or
vice versa.
-
For further informations on S-parameters, reference is made
to section 5.4 of the above cited book by Pozar.
-
For example, a typical plot of the S-parameters of the
classical 2-way (N = 2) equal Wilkinson power divider
having one input and two output ports is shown in Fig. 13.
The S-parameter curves were calculated using a simple
computer design program for the analysis of microwave
circuits.
-
Example values of the parameters are f0 = 500 MHz,
Zw = √2 x Z0 = 70.7 ohms, Z0 being the reference impedance
of 50 ohms, and R1 + R2 = 2 x Z0 = 100 ohms.
-
As observed from Fig. 13, the reflection loss versus
frequency behave similarly to the isolation, whereas the
transmission loss is largely frequency independent. The
useful isolation bandwidth f2-f1 at a minimum isolation of
f.e. -30 dB of a Wilkinson is quite limited. This poses a
problem in some applications, where broadband operation is
required. It is possible to increase the bandwidth by using
stepped multiple sections, but this requires more space and
increases cost (see Pozar, sections 8.3, p. 400 - 401).
-
Thus, the technical problem to be solved is to provide an
improved Wilkinson power divider circuit having an
increased useful isolation bandwidth which may be easily
constructed as well as a method of designing such improved
Wilkinson power divider circuits having an extended
bandwidth.
SUMMARY OF THE INVENTION
-
The present invention provides a Wilkinson power divider
circuit as defined in claim 1 and a corresponding design
method as defined in claim 6.
-
Particular advantages of the Wilkinson power divider
circuit according to the invention are the increased
isolation bandwidth and the inherent DC-decoupling at the
port P1.
-
The principal idea underlying the present invention is that
the isolation is very sensitive to the match on port 1,
i.e. the reflection loss |S11|, and not nearly as sensitive
to the match on other ports (2, 3, etc.). Therefore, if a
match with wider bandwidth is achieved on port P1, the
isolation will also have a wider bandwidth. A simple series
LC-circuit (coil L + capacitor C connected in series)
having its resonance frequency fr at or near the center
frequency f0 of the isolation band is appropriate. It is in
general appropriate to adjust or detune the characteristic
impedance of the transmission lines and/or the LC-values,
in order to achieve a symmetric response around said center
frequency f0.
-
Preferred embodiments of the present invention are listed
in the respective dependent claims.
BRIEF DESCRIPTION OF THE DRAWINGS
-
The present invention will become more fully understood by
the following detailed description of preferred embodiments
thereof in conjunction with the accompanying drawings, in
which:
- Fig. 1
- shows the principle of an N-way Wilkinson power
divider circuit according to an embodiment of the
present invention (for equal power split, Z1 = Z2
= ... = ZN);
- Fig. 2
- illustrates a typical plot of the S-parameters of
the 2-way equal Wilkinson power divider according
to another embodiment of the present invention
having one input and two output ports in
comparison to the classical 2-way equal Wilkinson
power divider shown in Fig. 13;
- Fig. 3
- illustrates a typical plot of the S-parameters of
a 2-way equal Wilkinson power divider having one
input and two output ports before addition of the
LC-circuit and optimization;
- Fig. 4
- illustrates a typical plot of the S-parameters of
a 2-way equal Wilkinson power divider having one
input and two output ports after addition of the
LC-circuit and optimization;
- Fig. 5
- shows simulated results for isolation (|S23|max)
versus a termination-resistance R11 for the ideal
2-way 50 ohm equal Wilkinson power divider shown
in Fig. 3;
- Fig. 6
- shows a Smith chart (cfr. Pozar, section 3.4)
from 200 to 800 MHz, for 3 values of Zw: 60.7
ohms, 70.7 ohms (√2 x 50 ohms is optimum for the
ordinary Wilkinson) and 80.7 ohms displaying how
the input impedance of the Wilkinson behaves as a
function of frequency for different transmission
line characteristic impedances Zw;
- Fig. 7
- shows a plot of 2Δƒ/ƒ0 versus k;
- Fig. 8
- shows a plot of α versus k;
- Fig. 9
- isolation versus frequency for numerical example
of another embodiment using equation (33) for C;
- Fig. 10
- isolation versus frequency for numerical example
of another embodiment using equation (34) for C;
- FIG. 11
- isolation versus frequency for numerical example
of the classical 2-way equal Wilkinson power
divider having one input and two output ports;
- Fig. 12
- shows the principle of the classical N-way equal
Wilkinson power divider; and
- Fig. 13
- illustrates a typical plot of the S-parameters of
the classical 2-way equal Wilkinson power divider
having one input and two output ports.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
-
In the figures, identical reference signs denote identical
or equivalent elements.
-
Fig. 1 shows the principle of an N-way Wilkinson power
divider circuit according to an embodiment of the present
invention.
-
According to Fig. 1, an LC-circuit 50 including an inductor
L and a capacitor C connected in series between said first
port Pl and said first ends of said plurality of N
transmission lines TRL1, TRL2, ..., TRLN has been added to
the conventional circuit shown in Fig. 12. Z1 ..., ZN
denote line impedances which in a first approach equal the
reference impedance Z0 (of f.e. 50 ohms) times the square
root of N.
-
Fig. 2 illustrates a typical plot of the S-parameters of
the 2-way equal Wilkinson power divider according to
another embodiment of the present invention having one
input and two output ports in comparison to the classical
2-way equal Wilkinson power divider shown in Fig. 13.
-
The result of the addition of the tuned LC-circuit 50
regarding the isolation |S23'| is a considerably broadened
isolation bandwidth f2' - f1' at a minimum isolation of
f.e. -30 dB, the maximum isolation at the center frequency
f0 being somewhat decreased. Moreover, a slightly enhanced
refection loss on port P1 is observed.
-
However, in the most general case, the addition of the LC-circuit
50 has the effect that the response around said
center frequency f0 becomes asymmetric. Thus, it is
necessary to adjust or detune the characteristic impedance
of the transmission lines and the LC-values, in order to
achieve a sufficiently symmetric response around said
center frequency f0 as well as a desired minimum isolation
over the whole frequency band.
-
The latter procedure is an optimization procedure which can
be performed on a standard computer design program for the
analysis of microwave circuits. As a result, the resonance
frequency fr = 1/(2π√LC)) of the LC-circuit 50 can slightly
deviate from the center frequency f0, and the impedance of
the transmission lines from the reference value √N x Z0.
-
Fig. 3 illustrates a typical plot of the simulated S-parameters
of a 2-way equal Wilkinson power divider having
one input and two output ports before addition of the LC-circuit
and optimization.
-
The example is based on a simple 2-way equal Wilkinson
power divider. The center frequency f0 is intended to be
0.5 GHz. Figure 3 shows the simulation result. Note that
the two transmission lines have an electrical length of 90°
(i.e. physical length equal to λ/4) at 0.5 GHz, with the
characteristic impedance Zw being 70,7 ohms (= √2 x 50
ohms). The resistor between the right second ends of the
two transmission lines is 100 ohms.
-
As becomes readily apparent, the simulated isolation
bandwidth at a minimum isolation of -30 dB equals about 50
MHz.
-
Fig. 4 illustrates a typical plot of the simulated S-parameters
of a 2-way equal Wilkinson power divider having
one input and two output ports after addition of the LC-circuit
and optimization.
-
First, with 0.5 GHz, one can calculate LC from the formula
for series resonance frequency fr = 1/(2π√LC)). Choosing C
= 10 pF, one obtains L ≈ lO nH. The result after manual
optimization which is shown in Fig. 4 involves the
following parameters:
Zw = 73 ohms
L = 15 nH
C = 6.8 pF
which is slightly different from the corresponding
parameter starting values.
-
As becomes readily apparent, the simulated isolation
bandwidth at a minimum isolation of -30 dB has dramatically
increased to about 170 MHz, i.e. more than a factor of
three.
-
Note that the peak isolation is lower than in the standard
configuration, and that the isolation decreases more
rapidly at very low and very high frequencies.
-
So far, the optimization of the parameter values has been
performed in an empirical manner using a standard
simulation program. In the following, analytical
expressions will be derived to calculate the parameters L,
C of the LC-circuit 50 and the impedances Z of the
transmission lines.
-
Heretofore, the ideal 2-way 50 ohm equal Wilkinson power
divider shown in Fig. 3 has been simulated.
-
Particularly, it has been investigated how the peak
isolation changes as a function of source-impedance.
Simulated results for |S23|max at f0 = 500 MHz versus a
termination-resistance R11 is shown in Fig. 5, namely for
R11 values ranging from 40 to 60 ohms. Notice the steep
decrease in isolation when R11 deviates from the optimum 50
ohms.
-
From the plot, it is evident that the match on port 1
should be modified, if one wants to achieve isolation over
a broader bandwidth. The match on port 2 and 3 is not as
important for the isolation.
-
Next, it was investigated how the input impedance of the
Wilkinson behaves as a function of frequency for different
transmission line characteristic impedances Zw.
-
The result is shown below in Fig. 6 in a Smith chart (cfr.
Pozar, chapter 3.4) from 200 to 800 MHz, for 3 values of
Zw: 60.7 ohms, 70.7 ohms (optimum) and 80.7 ohms. One
should notice that S11 moves clockwise with increasing
frequency, as for all other passive circuits, and should
observe how the S11-circle of the Wilkinson power divider
expands and moves to the right as Zw increases.
-
It is known (f.e. see G. Gonzalez, ,,Microwave Transistor
Amplifiers - Analysis and Design", section 6.4, p. 61,
Prentice-Hall, 1984) that the input impedance Z11 (or
reflection S11) of a series combination of Z0 ohms resistive
termination (the reference impedance), a coil L and a
capacitor C will move entirely on the circle r = 1 with
frequency, or in other words on the circle given by the
normalized impedance z = 1 + jx, where z = Z/Z0 and where x
goes from -∞ to +∞ (cfr. to Fig. 6).
-
By choosing Zw > √2 x Z0 ohms for the Wilkinson power
divider, it is therefore possible to obtain a conjugate
match at two frequencies symmetrically located below and
above f0 = 500 MHz using the LC-series circuit 50, since
the input impedances of the Wilkinson power divider at
these two frequencies will be complex conjugates of each
other (as detailed below). In general, conjugate matching
is the optimum matching method (see for example section
3.6, p. 99 in Pozar).
-
From Fig. 6, it is expected that S11 is symmetrical (but
complex conjugated) around f0. This is proved as follows.
-
The input impedance at port P1 of the Wilkinson power
divider is equal to a parallel-connection of two identical
transmission lines with impedance Zw and physical length
λ/4 at f0, since the resistor R has no effect when both
ports P2 and P3 are terminated in Z0.
-
The resulting impedance of two identical impedances Z in
parallel is Z/2. Therefore, from transmission line theory,
one obtains (f.e. see Pozar, page 79)
(1) Z11 = ½·Zw(Z0+jZwtanβ1)/(Zw+jZ0tanβ1)
where β = 2π/λ and 1 is the length of the transmission
line. At 1 = λ0/4, as in this case, β1 = π/2. Therefore, Z11
at some positive amount Δ from f0 may be rewritten as
(2) Z11(Δ) = ½·Zw(Z0+jZwtan(π/2+Δ))/(Zw+jZ0tan(π/2+Δ))
and at some negative amount -Δ from f0 as
(3) Z11(-Δ) = ½·Zw(Z0+jZwtan(π/2-Δ))/( Zw+
jZ0tan(π/2-Δ))
-
Since tan(π/2-x) = -tan(π/2+x), equation (5) may be
rewritten as
(4) Z11(-Δ) = ½·Zw(Z0-jZwtan(π/2+Δ))/Zw-jZ0tan(π/2+Δ))
-
In general, for any complex numbers z1 and z2 and their
complex conjugates z1* and z2*, the relation (z1/z2)* =
(z1*/z2*) holds, so since Zw and Z0 are real,
(5) Z11(-Δ) = ½·Zw[(Z0+jZwtan(π/2+Δ))/
(Zw+jZ0tan(π/2+Δ))]*
and therefore comparing (1) and (5), one arrives at
(6) Z11(-Δ) = Z11*(Δ)
-
This means that if Z
11 = a+jb at the frequency f
1 = f
0 - Δf,
then Z
11 = a-jb at the frequency f
2 = f
0 + Δf. As S
11 is
closely related to Z
11 by S
11 = (Z
11-Z
0)/(Z
11+Z
0) then
-
So the same holds for S11, i.e. S11 is also symmetrical (but
complex conjugated) around f0.
-
The knowledge of the coordinates of the two points z = 1
+/- jα where the S11-circle of the Wilikinson intersects
with the r = f circle, makes it possible to calculate the
necessary L and C, since their combined input impedance
must be the complex conjugate of the Wilkinson splitter:
-
At ω1, ω2 we must therefore require that ZLC + Z0 = Z11*,
where ZLC is the impedance of the LC circuit and Z0 is the
respective generator or consumer impedance, or
equivalently:
(7) At ω1= 2πƒ1: 1/(jω1C) + jω1L + Z0 = Z0(1-jα)
(8) At ω2 = 2πƒ2: 1/(jω2C) + jω2L + Z0 = Z0(1+jα)
-
It is necessary that the sign of jα is negative at
f1 = f0 - Δf and positive at f2 = f0 + Δf in order to
achieve the conjugate match.
-
As mentioned, the S11-circle expands as Zw increases.
Therefore, the potential bandwidth improvement also
increases with Zw.
-
Now, from (7), we will try to find Δf as a function of Z
w,
by finding where Re{Z
11}/Z
0 = 1:
(9) Z11 = ½·Zw(Z0+jZwtan) (Zw-jZ0tan)) /
[( Zw+jZ0tan)(Zw-jZ0tan)]
where = β1, so
(10) Z11 = ½·Zw/(Zw 2+Z0 2tan2)·[Z0·Zw+Z0·Zw·tan2+
j(Zw 2·tan-Z0 2·tan)]
and therefore
defining k = Z
0/Z
w, where 0 < k < √½, as we are only
interested in Z
0/√2 < Z
w < ∞ (f.e. see this document, page
13, last paragraph).
-
Finally,
(12) Re{Z11}/Z0 = ½·(1+tan2)/(1+k2tan2)
-
Setting (12) equal to 1 in order to find 2Af means
(13) ½·(1+tan2)/(1+k2tan2) = 1 1+(2k2-1)tan2 = 0
and therefore for 0 < k < √½
(14) tan2 = 1/(1-2k2) ⇒ tan = ±1√/(1-2k2)
so
where n is any integer number, and Arctan(x) denotes the
principal value of arcus tangent of x, i.e. |Arctan(x)| ≤
2π for all x.
-
With 1 = λ0/4 = c/(4f0√εr) at f0 where εr is the relative
dielectric constant of the transmission line medium, in
general we have that
(16) 1/λ = (λ0/4)/λ = c/(f0√εr)/4(c/(f√εr)) = ƒ/4ƒ0
and therefore
(17) = β1 = 2π1/λ = πƒ/2f0
-
Now, for any solution to (15), = ± ξ + nπ, where π/4 ≤ ξ
< π/2 (i.e. 1/4 ≤ ξ/π < ½,
(18) πƒ/2ƒ0 = ±ξ+nπ ƒ = 2ƒ0(±ξ/π+n)
and since n is an integer, 1/4 ≤ ξ/π < ½, and 0 < f < 2f0,
2Δƒ is given by
(19) 2Δƒ = 2ƒ0([-ξ/π+1]-[ξ/π+0]) = 2ƒ0(1-2ξ/π)
and so
(20) 2Δƒ = 2ƒ0(1-2Arctan(1/√(1-2k2))/π)
i.e. Δf = f0(1-2Arctan(1/√(1-2k2))/π). Roughly put, 2Δƒ
equals the isolation bandwidth.
-
A plot of 2Δƒ/ƒ0 versus k is shown in Fig. 7.
-
In order to find α, we simply calculate Im{Z
11}/Z
0 at f = f
0
± Δf:
and since k
2tan
2 = k
2/(1-2k
2) at f = f0 ± Δf (from (14))
and k = Z
0/Z
w we have
-
-
A plot of α versus k is shown in Fig. 8.
-
In the following, it is derived how L and C are calculated.
-
From (1) and (2), one obtains
(24) ω1L-1/(ω1C) = -X
(25) ω2L-1/(ω2C) = X
where X = αZ0.
-
Defining Δω = ω0 - ω1 = ω2 - ω0 and ε = Δω/ω0 = Δf/f0 means
that
ω1 = ω0 - Δω = ω0 (1 - ε)
ω2 = ω0 + Δω = ω0 (1 + ε)
and therefore
(26) ω0(1-ε)L - 1/(ω0(1-ε)C) = -X
(27) ω0(1+ε)L - 1/(ω0(1+ε)C) = X
-
For Δω << ω0, i.e. ε << 1, (26) and (27) may be
approximated by
(28) ω0(1-ε)L - (1+ε)/(ω0C) = -X
(29) ω0(1+ε)L - (1-ε)/(ω0C) = X
using the approximation 1/(1+ε) ≈ 1-ε for |ε| << 1.
-
Adding (28) and (29) yields
(30) ω0L - 1/(ω0C) = 0 ω0L = 1/(ω0C)
i.e. there is series resonance of L and C at ω0 = 2πf0.
-
Similarily, subtracting (29) from (28) yields
(31) ω0Lε + ε/(ω0C) = X ω0L + 1/(ω0C) = X/ε
-
By inserting (30) into (31), one obtains
(32) L = X/2εω0 = αZ0/2εω0
(33) C = 2ε/ω0X = 2ε/ω0αZ0
-
For large values of ε (close to 1), the above two
expressions cannot be used. Instead, using a similar
approach, but without using the approximation, it can be
shown that the expression for L remains unchanged, and that
the expression for C becomes
(34) C = 2ε/(1-ε2)ω0X = 2ε/(1-ε2)ω0αZ0
-
So, with expression (32) and (34), the series connection of
L and C is not necessarily in resonance at ω0. In fact, we
obtain that the resonance frequency ωr of L and C is given
by
(35) ωr = 1/√((1-ε2)LC) = ω0/√(1-ε2)
so as ε increases, ωr also increases, away from ω0.
-
As a numerical example, with ƒ0 = 500 Mhz, Z0 = 50 Ω and Zw
= 80.7 Ω (> 70.7 Ω = √2 · 50 Ω), k becomes 0.6196, and Δƒ
becomes 285.9/2 = 142.95 MHz(eq. (20)), i.e. ƒ1 = 357.05
MHz and ƒ2 = 642.95 MHz. Furthermore, α becomes 0.3889
(eq.(23)), so that
ε =Δω/ω0 = Δƒ/ƒ0 = ½·285.9/500 = 0.2859
X = α·50 Ω = 0.3889·50 Ω = 19.445 Ω
ω0 = 2π·500 MHz
-
Inserting into (32) and (33) yields L = 10.82 nH and C =
9.36 pF. Using expression (34) instead yields C = 10.19 pF,
and ωr = 2π·521.87 MHz.
-
The resulting simulated isolation versus frequency for this
embodiment is shown in Fig. 9 and 10, respectively, and for
comparison the classical case in Fig. 11.
-
Assuming the same bandwidth for the case without LC circuit
and with LC circuit having the above derived parameters, it
can be simulated that the improvement in isolation using
the approximate formulas (33) or (34) for C equals about
6.1 dB or 6.6 db, respectively, over the frequency range
357.05 to 642.95 MHz.
-
One should realize that the method described has the effect
that if the one extends the bandwidth of the Wilkinson
power divider by a large amount using the method described,
then the maximum obtainable isolation decreases
correspondingly. Thus, in general, an appropriate tradeoff
has to be found.
-
For the general case, an N-way Wilkinson power divider,
similar results as in the above derived case for N = 2 may
be obtained. The results will be given below.
-
It should be noted, that in this general case, the
theoratical value of Zw for the ordinary Wilkinson is Zw=
√N·Z0. This means that the necessary Zw for the improved
Wilkinson is bounded by Z0/√N < Zw< ∞, i.e. 0 < k < 1/√N
(since k = Z0/Zw). In the following, it is assumed that Zw
and k are within these bounds.
-
The input impedance becomes
Z11 = (1/N)·Zw(Z0+jZwtanβ1)/(Zw+jZ0tanβ1)
so
Re{Z11}/Z0 = (1/N)·(1+tan2)/(1+k2tan2)
and
Im{Z11}/Z0 = (1/N)·Zwtan(1-k2)/(1+k2tan2)
-
To find 2Δƒ, set Re{Z11}/Z0 = 1, which results in
tan2 = (N-1)/(1-Nk2) ⇒ tan = ±√(N-1)/√(1-Nk2)
i.e.
= ±arctan[√(N-1)/√(1-Nk2)]
and so
2Δƒ = 2ƒ0(1-2Arctan[√(N-1)/√(1-Nk2)]/π)
-
As k→0, 2Δƒ becomes 2ƒ0(1-2Arctan[√(N-1)]/π), so for
increasing N, the maximum obtainable bandwidth improvement
decreases.
-
To find
α, we calculate Im{Z
11} at ƒ = ƒ
0 ± Δƒ, which
results in
so
-
Observe that for N = 2, the above results reduce to the
previously derived result for the 2-way Wilkinson. The same
equations as for the 2-way Wilkinson is used to calculate L
and C.
-
In summary, the steps for improving the bandwidth of
Wilkinson power dividers using the method described can be
summarized as follows:
- a) choosing a desired 2Δƒ;
- (b) calculating the required k (i.e. Z0/Zw) and α from
equation (20) and (23); and
- (c) calculating L and C from equations (32) and (34)
(using equation (33) instead of (34) if Δf << f0)
-
-
Although the present invention has been described with
respect to preferred embodiments thereof, it should be
understood that many modifications can be performed without
departing from the scope of the invention as defined by the
appended claims.
-
Particularly, although a series connection of a single
capacitor and inductor has been shown, a plurality of
inductors and capacitors may be used instead.
-
Moreover, the invention is not restricted to an equal
Wilkinson power divider circuit, but can be applied as well
to an unequal Wilkinson power divider circuit.
