This summary provides the key details about the document in 3 sentences: The document presents a new simplified analytical model for Chireix outphasing combiners that accounts for impedance mismatch and reflection effects. The model derives simplified expressions for instantaneous efficiency and input/output voltages as functions of the decomposition angle and combiner parameters. Validation of the model is performed through comparisons with experimental and simulation results for different Chireix combiner designs.

1. Optimal Design of Chireix Outphasing Combiners Using a New
Simplified Analytical Model
M. El-Asmar, Student Member, A. Birafane, A. B. Kouki, Senior Member IEEE and F. M. Ghannouchi, Fellow IEEE
Abstract — ew simplified analytical expressions for the To overcome the efficiency limitations of resistive
instantaneous efficiency and the input/output voltages of Chireix combiners, a class of lossless, outphasing combiners (i.e.,
outphasing combiners presented. These expressions take into
account the impedance mismatch between the amplifiers and the Chireix combiners) was considered [2], [6], [8], [9], [10], [11].
lossless combining structure and account explicitly for the all of Up until recently, all studies cited in the literature on the
the combiner’s electrical parameters. We show that the Chireix-outphasing have bee based on the model of [12], (e.g.,
maximum achievable instantaneous combining efficiency can be see [13]). However, this model does not predict, nor account
controlled in position and in value through the judicious choice of for, the level of distortion observed when the Chireix combiner
stub length and transmission line characteristic impedance. We
further show that when this choice is combined with amplified is used. In [14] a new analytical approach was introduced
signal’s distribution, the average combining efficiency can be where rigorous analytical expressions for the output signals
precisely controlled and easily maximized. Various validations of and the Chireix combiner’s efficiency were presented. This
the proposed expressions are performed by comparison to approach takes into account the unmatched and non-isolated
experimental and simulation results for five different Chireix nature of the Chireix combiner and the reflection effects
combiners. Excellent agreement between simulations and
measurements is obtained in all cases considered. associated with it in a given reference impedance system.
Index Terms — Chireix combiner, LI C-Outphasing, simplified λ/4
model, power efficiency, linearity. V 1(θ ) Vo1(θ ') Zc
G
Vo(t)
I. INTRODUCTION Zs(θ ') jB
Vin(t) Separator
Power amplifier efficiency and linearity continue to be key λ/4 Zo
V 2(θ ) Vo 2(θ ')
factors in modern wireless communication systems. There is a G Zc
trade off between these two properties and it is quite a
Zs ( −θ ')
challenge to satisfy both of them simultaneously. The LINC -jB
(LInear amplification with Non linear Components) technique,
[1], [2], [3], is a potential solution that may offer high Fig. 1. Chireix-Outphasing amplifier topology.
efficiency with good linearity. In this technique, a complex
modulated signal is split into two constant envelope, phase In the standard LINC signal decomposition, the amplitude
modulated signals that are amplified by high efficiency modulation of the input signal Vin(t) is converted into two
nonlinear amplifiers and then combined at the output, as phases modulated constant envelope signals V1(θ) and V2(θ).
shown in Fig 1. When a combiner having matched and The expressions giving the various voltages are summarized
isolated ports is used, the LINC amplifier presents good here following the same notation as in [14]:
linearity but with degraded efficiency [4]. This efficiency
reduction is more pronounced for signals with a high peak to Vin(θ ) = rmax cos(θ ) (1)
average power ratio as the out-of-phase components of the
signal cancel out in the isolation resistance of the combiner. r jθ
V 1(θ ) = max e (2)
Any distortion observed at the output signal in such a case is 2
generally caused by the imbalance between the two RF
amplifier branches. Recently, a Modified Implementation of rmax − jθ
the LINC Concept (MILC) was proposed [5] whereby a mix of
V 2(θ ) = e (3)
2
the LINC and in-phase decompositions is used to improve
efficiency. This technique was tested using resistive combiners where the carrier and the original phase modulation angle have
which still limit the overall efficiency [6], [7]. been suppressed and θ is the LINC decomposition angle
corresponding to the added phase modulation (Fig. 1). The
M. El-Asmar, A. Birafane and A. B. Kouki are with the Communications
and Microelectronics Laboratory (LACIME), department of Electrical mismatches between the two amplifiers, assumed to present an
Engineering, École de technologie supérieure, Montréal, Canada H3C 1K3. output impedance of Zo, and the inputs of the Chireix
F. M. Ghannouchi is with the Intelligent RF Radio Laboratory, Electrical combiner introduce amplitude and phase distortions in the
and Computer Engineering Department, University of Calgary, Canada.
output voltages Vo1 and Vo2 on both branches. In particular,

2. the input phase θ is changed to an output phase θ’ given the input phase θ are also presented. The proposed
implicitly by [14]: expressions are considerably simpler and easy to use than
those in [14]. Furthermore, the simplified expressions of
Chireix system proposed in this work take into account the
y2 β tan(θ ) + 1
cos(θ ') = (4) reflection and non-isolation effects associated with the Chireix
2 2 combiner. To validate the simplified model over wide range of
 y β tan(θ ) + 1 + (1 + 2 y ) tan(θ ) − y β 

2
 
2 2
 Chireix combiner parameters, ADS [15] simulations and
with y = (Zo / Zc ) being the normalized characteristic experimental measurements are carried out.
admittance of the combiner lines, i.e., zc=1/y is the normalized
characteristic impedance, see Fig. 1. The constant β in this A. Efficiency
equation is related to the stub susceptance B (see Fig. 1) by
β = B.Zo / y 2 [14]. The expression of the instantaneous The expression of the efficiency for a Chireix combiner with
stubs, taking into account the mismatch effect between the
efficiency is given in terms of θ ' by [14]:
amplifiers and the combiner inputs, is presented by equations
8 y 2 cos2 (θ ') (4) and (5). It has been shown in [14] that these expressions
η (β ,θ ') = (5) are accurate. Therefore, they are used here as a reference and
(1 + 2 y 2 cos2 (θ '))2 + y 4 (β − sin(2θ '))2 as the starting point for deriving the new simplified Chireix
While equation (5) has been proven to be accurate (see combiner efficiency model. To this end, the ADS simulator
[14]), it is far from being intuitive and it is not readily usable [15] is used to obtain the curves corresponding to the
instantaneous efficiency versus the LINC decomposition angle
for overall efficiency estimation or the design of optimal
θ for different Chireix combiner parameters. Here we denote
outphasing combiners. Alternatively, finding the explicit
the electrical length of the stubs used in the combiner by γ
expression of the instantaneous efficiency versus the input
phase θ , instead of θ ' , though mathematically feasible by (measured in degrees). This length is equal to the tangent of
combining equations (4) and (5), is quite tedious and requires the stubs’ normalized susceptance ( BZo = y 2 β ) and thus
complex mathematical manipulations. In addition, in [14] both γ = arctan ( y 2 β ) .
output voltage and impedance expressions are giving versus
the output phase θ ' as complex expressions making linearity Considering a 50 Ω load (i.e., Z = 50 Ω ) and a unit
o
assessment of the combiner quite cumbersome. In light of normalized characteristic impedance, z c = Z c / Z o = 1 , three
these considerations, finding simplified expressions for the
different combiners were simulated. These correspond to three
Chireix combiner’s efficiency and linearity characteristics stub lengths, namely γ=20o, γ=45o and γ=70o. The simulated
versus the LINC decomposition angle θ is highly desirable and values of the efficiency as a function of θ for these cases have
remains an open research question. been obtained using equations (4) and (5) in ADS and are
The rest of this paper is organized as follows. In section II shown in Fig. 2. Examining these results, we make the
the new simplified expressions for the instantaneous output following observations: (i) the maximum efficiency always
voltages and efficiency of lossless outphasing combiners as a occurs at θ = γ and (ii) the value of the maximum efficiency
function of the LINC decomposition angle θ and the
varies with γ and is not necessarily 100% in all cases. This
combiner’s parameters are presented. Section III presents
latter observation led us to consider an additional set of
theoretical and experimental validations of the newly
simulations where the value of γ was fixed at 45o while the
developed outphasing combiner model. In section IV, the
characteristic impedance of the transmission lines used was
application of the new model to the optimal design of an
varied, i.e., zc = 0.6, 1 and 1.4 .
outphasing combiner for a given modulated signal is
1.0
demonstrated with a 16QAM signal and validated γ =45 deg
experimentally. Finally, the conclusions of the present work 0.8 γ =20 deg
are presented.
Efficiency
0.6
II. THE PROPOSED SIMPLIFIED MODEL 0.4
Based on the observation of the instantaneous efficiency 0.2
zc=1
curves versus the input phase θ , this paper shows that it is γ =70 deg
possible to simplify analytically the Chireix model of [14]. 0.0
Combining judiciously the equations (4) and (5), a simplified 0 20 40 60 80 100
expression of the instantaneous efficiency as a function of the θ , degrees
input LINC decomposition angle is proposed. The simplified
expressions of both voltages and efficiency equations versus Fig. 2. Simulated efficiency of the Chireix combiner for three different stubs.

3. 8 zc 2 cos 2 (γ )
The results of this second set of simulations are presented in K ( zc , γ ) = (8)
(z )
2
Fig. 3 and show that the level of the maximum attainable c
2
+ 2 cos 2 (γ )
efficiency can be adjusted by proper choice of the
characteristic impedance of combiner lines. However this
This new simplified expression for the instantaneous
choice has no impact to the location of such a maximum, i.e.,
efficiency replaces the two complex equations (4) and (5). It
occurring at 45o in all cases.
offers a more intuitive representation and clearly shows the
1.0
zc=1 effect of the stubs on the efficiency of the Chireix system.
Indeed, knowing that the matched combiner has an
zc=1.4
instantaneous efficiency curve that follows a cos θ
2
0.8
Efficiency
zc=0.6 distribution, one can easily see that the effect of using a
0.6
Chireix combiner with stubs amounts to a shift by γ , the
electrical stub length, in the instantaneous efficiency curve.
The maximum efficiency value however can be less than 1 if
0.4
the proper choice of the line impedance zc is not made.
γ=45 deg Indeed, a graphical analysis of equation (8), whereby the K
0.2 factor is plotted versus γ for various zc values as shown in
0 10 20 30 40 50 60 70 80 90
θ , degrees Fig. 4, clearly illustrates that specific combinations of γ and
zc values lead to a maximum attainable efficiency of 100%.
Fig. 3. Simulated efficiency of the Chireix combiner for three different values
of the normalized characteristic impedance ( zc = 0.6,1,1.4 ) and a constant These curves also illustrate that the required normalized
characteristic impedance of the combiner transmission lines
stub electrical length ( γ = 45o ).
must be less than or equal to 2 , with the lower impedances
shifting the location of the maximum efficiency towards lower
Based on the above observations, and given that the resistive power levels, i.e., greater θ values. In fact, one can easily see
combiner’s efficiency is of the form of A cos (θ ) [14], we can
2 this mathematically by search for the maximum of the K
factor. Setting the derivate of equation (8) equal to 0 yields
note that the expression of the instantaneous efficiency as a
the following equation:
function of the LINC decomposition angle θ , for a general
Chireix combiner is of the form of K . cos (θ − γ ) , where K
2
zcopt = 2 cos γ (9)
represents the maximum attainable efficiency. Again, from the
above results, we can see that the value of K depends only on
where zcopt is the optimal characteristic impedance for a given
zc and γ values. Indeed, using equations (4) and (5) with
Chireix stub electric length γ . This impedance insures that
various judicious transformations we obtain the following
the instantaneous efficiency is maximum for all LINC
simplified expression of instantaneous efficiency (see details
decomposition angles θ, and that it reaches 100% when θ = γ.
in Appendix A):
1 .0
8 y 2 cos 2 (γ ) z c = 1 .4
η (γ , θ ) = cos (θ − γ )
2
(6)
(1 + 2 y )
2 z c= 1
2
cos 2 (γ ) 0 .8
z c = 2 .5
Clearly, this equation is of the form K . cos (θ − γ ) with K
2
0 .6
K
being independent of θ and constant for a given combiner. It z c = .6
is given by: 0 .4
z c = .4
8 y 2 cos 2 (γ ) 0 .2
K ( y, γ ) = (7)
(1 + 2 y )
2
2
cos 2 (γ ) 0 .0
which can be written in terms of the characteristic impedance 0 20 40 60 80 100
γ , d e g re e s
zc, y = 1 / z c , as:
Fig. 4. Variation of K factor in function of stub length γ for different values
of combiner line impedance zc.

4. Equations (6), (8) and (9) are the necessary tools to the 1
design of Chireix combiners that will give the maximum Zs ( ± β , ±θ ' ) = (14)
2
attainable average efficiency for a given modulated signal. y  2 cos 2 ( ±θ ' ) + j[± β ∓ sin( 2θ ' )]
 
Indeed, the probability density function of the modulated  
Zo
signal is first used to identify the average signal level, the level
which occurs most often, and the corresponding LINC
we obtain the new simplified expressions for Vo1 and Vo2
decomposition angle, θa. Next, from equation (6) we can
given by (see Appendix C for details):
determine directly the required stub length γ=θa, to place the
maximum of the instantaneous efficiency curve at the average
power level. Finally, using equation (9), the characteristic rmax G cos γ  cos(θ − γ )  (15)
Vo1,2 =  
impedance of the combiner lines can be determined. Clearly, 1 + 2 y 2 cos2 γ  ± j (sin(θ − γ ) + 2 y sin θ cos γ ) 
2
lower characteristic impedances are needed for higher peak to
average signals where the average corresponds to increasing θ This new expression is very simple and shows the explicit
values, θ=0o corresponding always the peak signal level. dependence of the two voltages Vo1,2 on the input phase θ .
The origin of the variation of the above voltages versus θ is
attributed to the Chireix combiner structure especially to the
B. Voltage expressions stubs value γ . The stubs effect on these voltages can be shown
Having established the new expression for the instantaneous and explained easily by the expression (15).
combining efficiency, we next consider those of the output Next, we consider the output voltage Vo, which is the result
voltages. Taking into account the impedance mismatching that of the combined voltages Vo1 and Vo2 through the Chireix
occurs at the inputs of the Chireix combiner, it was shown in combiner. The complex voltage expression Vo given in [14]
[8] that the constant envelope nature of the input signal is lost has the following form:
at the output. Indeed, the output voltages, Vo1 and Vo2 shown r
in Fig. 1, vary in magnitude and phase versus θ according to Vo ( β , θ ' ) = 2 yG max 1 + Γ ( β , θ ' ) cos(θ ' ) (16)
2
the following equation:
Equation (16) can also be simplified to an explicit form of Vo
versus the input phase θ . Indeed, using equations (6), (12),
r jθ
Vo1 = G max (1+Γ ( β ,θ ') ) e (10) (13), (14) and (16) and combining them judiciously we find
2 the following new simplified expression (see details in
Appendix B):
r − jθ
Vo 2 = G max (1+Γ ( − β ,−θ ') ) e (11)
2
2 yG cos( γ )
where Г is the reflection coefficient at the input of the Chireix Vo = r cos(θ − γ ) (17)
max
combiner. These voltages are expressed in terms of θ ' instead 1 + 2 y 2 cos 2 (γ )
of θ and equation (4) is again needed to obtain the explicit
expressions in terms of the original LINC decomposition Equation (17) shows that the output voltage of the Chireix
angle θ . Again, the presence of θ’ and the need for (4) makes system is proportional to the cos(θ − γ ) term instead of
it very difficult to gain any insight into the behavior of the cos(θ ) term. Clearly, since cos(θ ) corresponds to the linear
Chireix combiner and its impact on the linearity of the LINC
amplifier. Here we show that it is possible to simplify these behavior, a phase-only predistortion, whereby one branch is
complex expressions to simple formulas expressed explicitly phase-shifted by an angle γ relative to the other, would
as a function of θ instead of θ ' . First, we show in Appendix linearize the Chireix combiner. This intuitive result was found
A that equation (4) can be rewritten as: in [17] using complex derivations. Furthermore, we can see
that this predistortion will make the combining efficiency
tg θ ' =
( )
ta n θ 1 + 2 y 2 − ta n γ
(12) found in equation (6) to follow the form K . cos (θ ) , which is
2
1 + ta n γ ta n θ equivalent to the matched combiner efficiency when K is
maximized to get the unity value as was discussed in the
Next, combining this equation with the following expressions previous section.
for reflection coefficients Г and the input impedances Zs [14]: With the above simplified voltage expressions, we can
Zs ( ± β , ± θ ' ) − Zo analyze quantitatively the effect of the stubs on the linearity of
Γ ( ± β ,±θ ' ) = (13)
Zs ( ± β , ± θ ' ) + Zo the Chireix system. First we note that equation (17) is of the
form Vo = Vmax .cos(θ − γ ) with:
2 yG cos(γ )
Vmax = rmax (18)
1 + 2 y 2 cos2 (γ )

5. To validate the simplified expressions of the two voltages
Knowing that the input voltage Vin is equal to rmax cos(θ ) , Vo1 and Vo2 given by the equation (15) we consider the
existing model [14] given by equations (10) and (11) and
the output voltage can be written explicitly in terms of the compare it with the new equation (15). Given the known phase
input voltage as follow: relationship between Vo1 and Vo2, only their magnitudes will
2
Vmax  Vin  be shown. Fig. 6 shows the simulation results for both models
cos(γ )Vin − Vmax sin(γ ) 1 − 
Vo =  (19) for the same three stubs used above. Fig. 7 shows the
rmax  rmax  simulation results of comparison between the phase of Vo1
Equation (19) has two terms: the first term is proportional to using the model [14] and the phase of Vo1 with a simplified
Vin, hence linear, while the second is non linear versus Vin. model for same different values of zc and γ . In all cases we
The non-linearity observed in the Chireix system is thus
quantified and is presented by this second term. To avoid this see that both models are equivalent.
non-linearity the term sin(γ ) must be equal to zero.
1.0
Consequently the Chireix system becomes linear when the
stubs are withdrawn. z c = 0.6, γ = 20 o
mag(Vo1), mag(Vo2)
0.8
III. MODEL VALIDATION 0.6 z c = 1, γ = 45 o
To validate the accuracy of the proposed model of Chireix
0.4
systems, both simulations and measurements are considered.
zc = 2 , γ = 70 o
A. Theoretical validation 0.2
In this case, we consider the simulation results, using ADS,
0.0
based on the existing model [14] as described by equations (4),
0 20 40 60 80 100
(5), (10), (11) and (16) and compare them to those obtained
θ , deg rees
using the proposed model of equations (6), (15) and (17). The
models equations are compared to the circuit simulations of Fig. 6. Comparison between the existing voltages Vo1 and Vo2 of equations
the physical microstrip combiner structures. These (10) and (11) (solid lines) and the simplified model of equation (15) (symbols)
comparisons are shown in Figs. 5, 6, 7 and 8 where the for three different combiners.
existing model results are shown in solid lines while those
based on the simplified expressions of our proposed model are 100
plotted using different symbols. Fig. 5 shows the results of the 80 z c = 0 .6, γ = 2 0 o
efficiency simulation for both the existing complex and the 60
proposed simplified model for three Chireix combiners
Phase(Vo1)
40
( z c = 0.6 , γ = 20o ), ( z c = 1 , γ = 45o ), ( z c = 2 , γ = 70o ). As z c = 1, γ = 4 5 o
20
can be seen, the results are identical in all cases confirming the 0
accuracy of the proposed model.
-20
zc = 2 , γ = 70 o
1 .0 -40
z c = 1, γ = 4 5 o
-60
0 .8
0 20 40 60 80 90
θ , d eg rees
0 .6
Efficiency
z c = 0 .6 , γ = 2 0 o Fig. 7. Comparison between the output phase signals of Vo1 for the existing
0 .4 model of equation (10) (solid lines) and the simplified model of equation (14)
(symbols) for three different combiners.
0 .2
zc = 2 , γ = 70o
0 .0
0 20 40 60 80 100
θ , d e g re e s
Fig. 5. Comparison results between the existing equation (5) of the efficiency
(solid lines) and the simplified analytical model of equation (6) (symbols) for
three different combiners.

6. 0 .8
The details of the measurement procedure are as follows.
0 .7
z c = 1, γ = 4 5 o First, the generators’ output power is calibrated using the
0 .6 setup’s power meter. The instantaneous combiner efficiency is
then measured by varying the relative phase between the two
mag(Vo)
0 .5
z c = 0 .6, γ = 2 0 o generators over a 90o range. The efficiency is obtained by
0 .4 taking the ratio of the power at the output of the combiner, as
measured by the power meter, to the sum of the powers
0 .3 injected by the two generators. Two sets of measurements
zc = 2 , γ = 70o where taken in this manner to validate the results shown in
0 .2
Figs. 2 and 3. The first set corresponds to the three Chireix
0 .1 combiners with γ = 20o , 45o and 70o when zc = 1 . The
0 20 40 60 80 100 second set corresponds to three Chireix combiners with
θ , deg rees
γ = 45o and zc = 0.6, 1, 1.4 .
Fig. 8. Comparison between the output voltage Vo of the existing model of The comparison between the measured results and our
equation (16) (solid lines) and the simplified model of equation (17) (symbols)
for three different combiners.
simplified model of the combiner’s instantaneous efficiency of
equation (6) are shown in Figs. 10 and 11 where the solid lines
We have also validated the new expression of the output correspond to equation (6) while the symbols represent the
voltage Vo given by equation (17) for several values of measurements. Again, excellent agreement between the
zc and γ . Fig. 8 compares the result obtained using equation proposed instantaneous efficiency model and measurements is
(17), corresponding to our simplified model, to those obtained seen for all combiner cases considered.
by equation (16) of model [14]. Again, both models are found 1 .0
to be equivalent. γ =45
γ=2 0
0 .8
B. Experimental validation
0 .6
Efficiency
To further validate the proposed model, we fabricated three
different Chireix combiners and performed experimental
measurements on all of them. The combiners where fabricated 0 .4
γ= 70
in microstrip technology using a Duroid substrate (εr=2.33,
h=31 mils). Each combiner is made of two-quarter wavelength 0 .2
lines and two stubs; one stub having an electric length of + γ z c= 1
0 .0
and the second having an electric length of − γ , as per Fig. 1.
0 20 40 60 80 90
The experimental setup used for all measurements is made of θ
two signal generators coherently locked using external
generators, the combiner(s) and a power meter. Fig. 9 Fig. 10. Comparison between the measured efficiency (dotted) and the
simplified analytical model of equation (6) (solid lines) for variable stubs’
illustrates this setup which is operated at 2 GHz.
length ( γ = 20o , 45o and 70o ) and constant characteristic impedance
( zc = 1 ).
IV. APPLICATION TO THE DESIGN OF AN OPTIMAL
OUTPHASING COMBINER
Having established the validity of the proposed model both
theoretically and experimentally, we next focus on using the
added physical insight that the new equations bring to the
design of an optimal outphasing combiner, one that maximizes
the average combining efficiency for a given modulated signal.
The design parameters are the length of the stubs, γ, and the
characteristic impedance, zc. Here we consider a 16QAM
signal and seek the combiner that will maximize the average
combining efficiency. The signal has a symbol rate of 1MS/s
Fig. 9. Experimental test bench used to validate the simplified analytical and is filtered with a root-raised cosine filter (RRCF) having a
model. roll-off factor of 0.35, thus yielding a 1.35 MHz modulation

7. bandwidth and a peak to average ratio of 6.5 dB. We start by generators (RF and baseband), an RRCF-filtered (α=0.35)
examining the PDF of the signal as shown in Fig. 12. Next, 16QAM signal was decomposed according to the LINC
we identify from this figure the LINC decomposition angle, θ, technique using the ADS simulator. The resulting data files
corresponding to the most frequently occurring signal level, for the two signals were loaded into the signal generators and
about 60o in this case. Clearly, a combiner having a output at 2GHz. The average input and output powers of the
combiner were measured using the power meter of the setup
maximum combining efficiency at θ = 60o , as shown in Fig. and the average efficiency was determined by taking the ratio
12, would maximize the average combining efficiency. of the combiner’s output power to the sum of its input powers.
Designing such a combiner is straightforward using equations These measured results are summarized in Table I. The
(6) and (9). First, from equation (6) we determine that the stub
measured average efficiency for each combiner, η mea is
electric length must be γ = 60o while from equation (9) we
compared to the theoretically calculated average efficiency,
find the normalized characteristic impedance, zc = 0.707 , or ηth , for the same combiner. This latter efficiency is computed
Z c = 35.35 Ω in a 50 Ω system. This design procedure is not using the instantaneous combining efficiency of equation (6)
only intuitive and direct, but it is also considerably simpler and the PDF of the 16QAM signal, p (θ i ) , as follows:
than the alternative of using the original equations and 2 2
8 y cos (γ )
searching for the optimal combiner though multiple η AVG = ∑ p (θi ) 2
cos (θ i − γ ) (20)
(1 + 2 y )
simulations [16]. 2
2 2
i =1 cos (γ )
1 .0 Table 1 shows the measured and simulated average
zc =1
efficiency values for the four combiners. Good agreement,
z c = 1 .4
0 .8 within the precision of the measurement setup, between
z c = 0 .6 measured and simulated values is obtained. This table also
confirms that the designed combiner does indeed produce the
Efficiency
0 .6 maximum average efficiency for the 16QAM signal.
0 .4
γ =45 deg
0 .2
0 20 40 60 80 90
θ
Fig. 11. Comparison between the measured efficiency (symbols) and the
simplified analytical model of equation (6) (solid lines) for variable
characteristic impedance ( zc = 0.6,1,1.4 ) and constant stub length ( γ = 45o ).
To validate our design, we carried out an experimental study
consisting of comparing the average combining efficiency of
four different Chireix combiners. The first three combiners
were chosen somewhat arbitrarily and are characterized by: (i) Fig. 12. Optimization of the combiner efficiency using the PDF of 16QAM
modulated signal.
γ = 0o , zc = 70Ω , which is a stubless combiner that produces a
TABLE I
good match into a 50 Ω load, (ii) γ = 45o , zc = 50Ω , which THEORETICAL AND MEASURED AVERAGE EFFICIENCY FOR DIFFERENT TYPES OF
COMBINER WITH 16QAM SIGNAL.
gives a somewhat intuitive median solution, (iii)
γ = 60o , zc = 70Ω , which has the good stub length but the Combiner ηth ηmea
characteristic impedance suited for stubless quarter wavelength
combiners in 50 Ω. The fourth combiner is based on the above 0,28 0,25
1) γ = 0o , zc = 70Ω
design, i.e., γ = 60o , zc = 70Ω . It should be noted that
combiners 1, 2 and 4 have a maximum K value of unity (see 0,89 0,83
2) γ = 45o , zc = 50Ω
equation (9)) while K=0.648 for combiner 3 as its
characteristic impedance is not optimal. All four Chireix
0,62 0,64
combiners were fabricated using the same substrate (εr = 2.33 3) γ = 60o , zc = 70Ω
and h = 31 mils).
Using the same measurement setup described in Fig. 9, 0,947 0,87
4) γ = 60o , zc = 35Ω
where careful attention was paid to the phase coherence of the

8. V. CONCLUSION 8 y2
η=
In this paper, a simplified analytical model of the Chireix  (1 + tan γ tan θ )2 + ((1 + 2 y 2 ) tan θ − tan γ )2 
 (1 + tan 2 γ ) 
combiner efficiency and voltage expressions has been
 (1 + tan γ tan θ )2 
proposed. This model is simpler and more intuitive than the  
(1 + 2 y 2 ) tan θ − tan γ
 +4 y (1 + y ) − 4 y tan γ
2 2 2
recently published models and offers insight into the behavior 
of the Chireix combiner directly as a function of the input  (1 + tan γ tan θ ) 
signal’s amplitude. Theoretical simulations and experimental which is in form η = . We now evaluate the numerator N
measurements were carried out for various Chireix combiner D
configurations and the results obtained validated the accuracy and the denominator D separately as follows:
of the proposed model. It was also shown that the new
simplified model equations can be used for the precise and = 8 y 2 (1 + tan γ tan θ )2
easy design of a Chireix combiner for a given application.
This was demonstrated by designing an optimal combiner for a
 (1 + tan 2 γ )[(1 + tan γ tan θ ) 2 + ((1 + 2 y 2 ) tan θ − tan γ ) 2 ] 
16QAM signal. The designed combiner was demonstrated  
theoretically and experimentally to yield the maximum D =  + 4 y 2 (1 + y 2 )(1 + tan γ tan θ ) 2 
average efficiency.  − 4 y 2 tan γ (1 + tan γ tan θ )((1 + 2 y 2 ) tan θ − tan γ ) 
 
APPENDIX A
Arranging D in the form D = A tan2 θ + B tanθ + C , we obtain:
SIMPLIFICATION OF THE EFFICIENCY EXPRESSION
A = (1 + tan 2 γ )[tan 2 γ (1 + 2 y 2 ) 2 ] + 4 y 2 (1 + y 2 ) tan 2 γ − 4 y 2 tan 2 γ (1 + 2 y 2 )
Starting with equation (5):
8 y2 cos2 (θ ') (A1)
η= which can be simplified to:
(1 + 2 y cos (θ '))2 + y 4 (β − sin(2.θ '))2
2 2
A = (1 + tan 2 γ )(1 + tan 2 γ + 4 y 2 ) + 4 y 4
and developing the denominator, we botain:
8 y 2 cos2 (θ ') Similarly, we have:
η= (A2)
 (1 + tan 2 γ )[2 tan γ − 2 tan γ (1 + 2 y 2 )] 
1 + y 4 β 2 + 4 y 2 (1 + y 2 ) cos2 θ '− 4 y 4 β sin θ 'cos θ ' B=
 +8 y 2 (1 + y 2 ) tan γ − 4 y 2 tan γ (1 + 2 y 2 − tan 2 γ ) 

 
Letting tan γ = y β and simplifying by cos θ ' equation (A2)
2 2
which yields, after simplification, B = 0 . The last term is:
can be written as:
C = (1 + tan 2 γ )(1 + tan 2 γ ) + 4 y 2 (1 + y 2 ) + 4 y 2 tan 2 γ
8 y2 (A3)
η=
(1 + tan γ )(1 + tan θ ') + 4 y 2 (1 + y 2 ) − 4 y 2 tan γ tan θ '
2 2
which can be put in the form:
C = (1 + tan 2 γ )(1 + tan 2 γ + 4 y 2 ) + 4 y 4
Next, we evaluate tg (θ ') in terms of θ. Using equation (4) we
can write: We observe that: A = C . Hence, the efficiency expression
tan γ tan θ + 1 takes the following form:
cos θ ' = (A4) 8 y 2 (1 + tan γ tgθ ) 2 8 y 2
(tan γ tan θ + 1) + (tan θ + 2 y tan θ − tan γ )
2 2 2 η= = (cos θ + tan γ sin θ ) 2
A.(tan 2 θ + 1) A
1 which can be successively simplified as follows:
with cos θ ' =
2
we obtain:
1 + tan θ '
2
8 y2 1
η= (cos γ cos θ + sin γ sin θ )2
(tan γ tan θ + 1) + (tan θ + 2 y tan θ − tan γ )
2 2 2
A cos 2 γ
1 + tan θ ' =
2
8 y2
( tan γ tan θ + 1) 2
η= cos 2 (θ − γ ) = K cos 2 (θ − γ )
A cos 2 γ
Then:
(tan θ + 2 y tan θ − tan γ ) 8 y2 8 y2
2 2
where K = =
tan θ ' =
2
(A5)
A cos γ (1 + tan γ )(1 + tan γ + 4 y 2 ) + 4 y 4  cos 2 γ
2 2 2
(1 + tan γ tan θ ) 2
 
and tan θ ' =
(
tan θ 1 + 2 y
2
) − tan γ (A6)
Simplifying the expression of the K parameter yields:
1 + tan γ tan θ 8 y 2 cos 2 γ (A7)
K=
Combining equations (A3), (A5) and (A6), we obtain the (1 + 2 y 2 cos 2 γ ) 2
following equation:
Thus, the final simplified expression of the Chireix combiner’s
instantaneous efficiency is given by equation (6), namely:

9. 8 y 2 cos 2 γ 2 yG cos γ
η= cos 2 (θ − γ ) (A8) Vo = Vin (θ − γ ) (B7)
(1 + 2 y 2 cos 2 γ ) 2 1 + 2 y 2 cos 2 γ
APPENDIX C
APPENDIX B Vo1 AND Vo 2 EXPRESSIONS AT THE COMBINER'S INPUTS
SIMPLIFICATION OF THE OUTPUT VOLTAGE EXPRESSION
Taking into account of the reflection coefficient occurred by
The complex expression of Vo is as follow [14]:
the unmatched Chireix combiner, the voltage at the input of
Vo ( β , θ ') = y.G.rmax . 1 + Γ( β , θ ') .cos(θ ') (B1) the combiner could be expressed as follow [14]:
where, Γ ( β , θ ') = Z s − Z 0 . Also from [14] we know that: Vo1 = Vo1 e jθ '
Zs + Z0 Vo 2 = Vo 2 e− jθ '
Z0
Zs ( β ,θ ') = 2 (B2) where;
(
y . 2.cos 2 (θ ') + j  β − sin ( 2.θ ' ) 
  ) rmax
Vo1 = G 1 + Γ( β , θ ') (C1)
then 2
2
1 + Γ ( β ,θ ') = (B3)
Vo 2
r
= G max 1 + Γ( − β , −θ ') (C2)
1 + 2 y cos θ '+ jy 2 ( β − sin 2θ ')
2 2
2
and On the other hand, we know that:
cos θ '
Vo = 2 y.G.rmax Vo ( β ,θ ') = yGrmax 1 + Γ( β , θ ') cos(θ ')
(1 + 2 y 2 cos 2 θ ') 2 + y 4 ( β − sin 2θ ') 2
Letting tan γ = y 2 β , we have: Then, we can say that:
Vo
cos θ ' Vo1,2 = e ± jθ ' (C3)
Vo = 2 y.G.rmax 2 y cos θ '
1 + 4 y 2 cos 2 θ '+ 4 y 4 cos 4 θ '− 4 y 4 cos 4 θ '
1
+4 y 4 cos 2 θ '+ tan 2 γ − 4 y 2 tan γ sin θ 'cos θ ' Vo1,2 = Vo (1 ± j tan θ ') (C4)
2y
Evaluating Vo by equation (B2), (C4) can be written as follow:
Finally, after simplification, we can write:
cos θ ' (B4) G cos γ
Vo = 2 yGrmax Vo1,2 = rmax cos(θ − γ )(1 ± j tan θ ') (C5)
1 + tan γ + 4 y 2 (1 + y 2 ) cos 2 θ '
2 1 + 2 y 2 cos 2 γ
Using equation (A6) to replace tgθ ' , equation (C5) will be:
−4 y 2 tan γ sin θ 'cos θ '
 G cos γ 
 1 + 2 y 2 cos 2 γ rmax cos(θ − γ ) 
= 
According to equation (A2), we know that the efficiency can
Vo1,2
be written if the form:  (1 + 2 y 2 ) tan θ − tan γ  
8 y 2 cos2 (θ ')  1 ± j
 
η=  1 + tan γ tan θ 
1 + y 4 β 2 + 4 y 2 (1 + y 2 ) cos2 θ '− 4 y 4 β sin θ 'cos θ '
 G.cos γ 
 1 + 2 y 2 cos 2 γ .rmax cos(θ − 
We notice, by examining this expression and the result
Vo1,2 = 
obtained in equation (B4), that we can express Vo as a function   (1 + 2 y 2 )sin θ cos γ − cos θ sin γ 
of η as follow:  γ ) 1 ± j
 

  cos θ cos γ + sin θ sin γ 
η  G cos γ 
Vo = 2 yGrmax (B5)
8 y2  1 + 2 y 2 cos 2 γ rmax cos(θ − γ ) 
Vo1,2 = 
 sin(θ − γ ) + 2 y 2 sin θ cos γ 
Combining equations (A8) and (B5), we find the final  1 ± j
 

simplified expression of the output voltage, namely:  cos(θ − γ ) 
2 yG cos γ Finally;
Vo = rmax cos(θ − γ ) (B6)
1 + 2 y 2 cos 2 γ
rmax G cos γ  cos(θ − γ )  (C6)
which is equation (17). This equation can be expressed in Vo1,2 =  
terms of the input voltage, Vin , by: 1 + 2 y 2 cos2 γ  ± j (sin(θ − γ ) + 2 y sin θ cos γ ) 
2

10. ACKNOWLEDGEMENT
The authors wish to acknowledge Mr. Walid Hamdane for his
help with the measurement setup. This work was supported in
part by the Natural Sciences and Engineering Research
Council of Canada (NSERC).
REFERENCES
[1] D. C. Cox, "Linear amplification with nonlinear components," IEEE
Trans. Commun, vol. COM-22, pp. 1942-1945, Dec. 1974.
[2] H. Chireix, "High power outphasing modulation," Pro. IRE, vol. 23, pp.
1370-1392, Nov. 1935.
[3] X. Zhang, L. E. Larson, and P. M. Asbeck, Design of Linear RF
Outphasing Power Amplifiers. Boston, MA: Artech House, 2003.
[4] Y. Jaehyok, Y. Yang and B. Kim, "Effect of efficiency optimization on
linearity of LINC amplifiers with CDMA signal," in IEEE MTT-S Int.
Microwave Symp. Dig., vol. 2, May 2001, pp. 1359-1362.
[5] G. Poitau, and A. Kouki, “MILC: Modified Implementation of the LINC
Concept,” IEEE International Microwave Symposium, June 2006, pp.
1883-1886.
[6] G. Poitau, A. Birafane, and A. Kouki, ''Experimental characterization of
LINC outphasing combiners efficiency and linearity,'' Radio and
Wireless Conference, 2004 IEEE 19-22 Sept. 2004, pp. 87 - 90.
[7] M. Helaoui, S. Boumaiza, F. M. Ghannouchi, A. B. Kouki, and A.
Ghazel, ''A New Mode-Multiplexing LINC Architecture to Boost the
Efficiency of WiMAX Up-Link Transmitters, '' Microwave Theory and
Techniques, IEEE Transactions on Volume 55, Issue 2, Part 1, Feb.
2007, pp. 248 - 253.
[8] W. Gerhard and R. Knoechel, “Differentially coupled outphasing
WCDMA transmitter with inverse class F power amplifiers,” in IEEE
MTT-S Int. Dig., San Diego, CA, Jan. 2006, pp. 355–358.
[9] T.-P. Hung, D. K. Choi, L. E. Larson, and P. M. Asbeck, ''CMOS
Outphasing Class-D Amplifier With Chireix Combiner,'' Microwave and
Wireless Components Letters, IEEE Volume 17, Issue 8, Aug. 2007,
pp. 619 - 621.
[10] J. Grundlingh, K. Parker, and G. Rabjohn, ''A high efficiency Chireix
out-phasing power amplifier for 5GHz WLAN applications,'' Microwave
Symposium Digest, 2004 IEEE MTT-S International Volume 3, 6-11
June 2004, pp.1535 - 1538.
[11] I. Hakala, L. Gharavi, and R. Kaunisto, ''Chireix power combining with
saturated class-B power amplifiers,'' Microwave Conference, 2004. 34th
European Volume 1, 11-15 Oct. 2004. pp. 1 - 4.
[12] F. H. Raab, "Efficiency of outphasing RF power-amplifier systems,"
IEEE Trans. Commu., vol. COM-33, pp. 1094-1099, Oct. 1985.
[13] I. Hakala, D. K. Choi, L. Gharavi, N. Kajakine, J. Koskela and R.
Kaunisto, "A 2.14-GHz Chireix Outphasing Transmitter," IEEE Trans.
on Microwave Theory and Techniques, vol. 53, no. 6, pp. 2129-2138.
June. 2005.
[14] A. Birafane and A. Kouki, "On the Linearity and Efficiency of
Outphasing Microwave amplifier," IEEE Trans. on Microwave Theory
and Techniques, vol. 52, no. 7, pp. 1702-1708. Jul. 2004.
[15] Agilent Technologies "Advanced Design System,"2005C ed. Palo Alto
California, 2005.
[16] M. El-Asmar, A. Birafane, and A. B. Kouki, "Optimization of the LINC
average power efficiency using Chireix combining systems," ISIE 2006,
Montreal, Quebec, Canada, pp. 3169-3172, July 2006.
[17] A. Birafane and A. B. Kouki, "Phase-only predistortion for LINC
amplifiers with Chireix-outphasing combiners," IEEE Trans. on
Microwave Theory and Techniques, Volume 53, Issue 6, Part 2, pp.
2240 - 2250. June 2005.