Arabi, E., Morris, K., & Beach, M. (2017). Analytical Formulas for the
Coverage of Tunable Matching Networks for Reconfigurable
Applications.
IEEE Transactions on Microwave Theory and
Techniques
,
65
(9), 3211-3220.
https://doi.org/10.1109/TMTT.2017.2687902
Peer reviewed version
Link to published version (if available):
10.1109/TMTT.2017.2687902
Link to publication record in Explore Bristol Research
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1
Analytical Formulas for the Coverage of Tunable
Matching Networks for Reconfigurable Applications
Eyad Arabi, Member, IEEE, Kevin A. Morris, Member, IEEE, and Mark A. Beach, Member, IEEE,
Abstract—Tunable matching networks are essential compo-
nents for agile radio frequency systems. To optimally design such
networks, the total area they cover on the Smith chart needs to be
determined. In this work, the coverage areas of typical matching
networks have been determined analytically for the first time. It
has been found that the coverage area is encompassed by up to
five arcs. Analytical expressions for the centers and radii for these
arcs have been derived. The theoretical analysis is provided for
four typical matching networks and verified by circuit simulation
and measured data. Moreover, a dynamically load-modulated
power amplifier has been designed using the presented theoretical
techniques, which demonstrates a measured improvement in the
power added efficiency of up to 5% in the frequency range of
(0.8 - 0.9) GHz.
Index Terms—Smith chart, tunable matching networks, power
amplifiers, re-configurable, dynamic load modulation
I. INTRODUCTION
R
ECONFIGURABLE wireless transceivers are becoming
crucial for future systems such as long-term evolution
(LTE) and LTE-advanced. Such systems are required to be
frequency agile to enable optimal utilization of the congested
frequency spectrum. Therefore, these systems require tunable
components like antennas, filters, and matching networks
(MN).
Beside frequency agile systems, tunable MN are being used
heavily for applications such as tunable/wideband antennas
[1], efficiency-enhanced and load-sensitive power amplifiers
(PAs) [2], [3], and range-adaptive wireless power transfer [4].
One of the main parameters the RF designer needs to know
about a tunable MN, is the set of all complex impedances that
can be matched to a specific load (typically 50 Ω). This set
defines what is known as the coverage of the MN.
The Smith chart continues to serve as an indispensable tool
for the analysis and visualization of complex loads and reflec-
tion coefficients [5]. Compared to the complex impedance-
plane, the Smith chart is a superior tool because multiple
quantities can be read directly, such as the complex impedance
and admittance and the reflection coefficient. Therefore, the
coverage of MNs defined in the complex plane of the Smith
chart is of great benefit. This is particularly useful for PAs
because the coverage can be plotted along with other criteria
such as load-pull contours, noise circles, gain circles, etc.
In [6]–[11] the boundaries have been presented using sim-
ulations, which are less legible, do not indicate the limits of
Manuscript received December 1st, 2016; revised January 21st, 2017;
accepted March 21st, 2017.
The authors are with the department of Electrical and Electronic Engi-
neering, University of Bristol, BS8 1UB, Bristol, UK. E-mail: {eyad.arabi,
kevin.morris}@bristol.ac.uk
the tuning elements, and do not provide any physical insight
about the networks. In [12], theoretical formulas have been
presented; however, they provide the coverage for discrete
impedance points (states) and do not give the continuous
coverage, and they are not directly related to the tunable
capacitors commonly used in matching networks. Even though
the formulas in [13]–[15] can produce the coverage for the full
dynamic range, they are presented for the complex rectangular
plane, not the Smith chart and for Π networks only. In this
work, the coverage areas of four of the commonly used net-
works have been thoroughly investigated. Theoretical formulas
have been derived for the loci of the closed boundaries of the
coverage areas. These formulas are derived for the complex
space of the reflection coefficient rather than the impedance
and can, therefore, be plotted directly on the Smith chart. It
is assumed that the MN is connected to a resistive load at
one end, and the reflection coefficient seen looking towards
the other end defines the coverage area as illustrated in Fig.
1 (a). This configuration is different from the one in [6]–[10],
[12] where a resistive load is connected at one end with the
other end conjugately matched to a known impedance. The
configuration used here is particularly suitable for PAs where
conjugate matching is not necessarily required. The theoretical
analysis has been verified by circuit simulation as well as
measured data, which agrees well with the theory.
The theoretical formulas presented here are compact; there-
fore, very practical for use in CAD tools and provide a useful
instrument for the analysis of tunable MN.
1
1
C
min
C
max
min
min
max
min
max
max
min
max
Matching NW.
Resistive Load
Z
in
Γ
in
(a) (b) (c)
Fig. 1. Illustration of the boundary of the coverage area. (a) Problem
description. (b) Networks with only one tunable element. (c) Networks with
two tunable elements.
II. THEORY
The coverage of a tunable MN can be defined as the set
of all complex impedances that can be matched to a specified
load at a particular frequency. If the MN has only one tunable
2
component, its coverage will be a linear path as shown in Fig.
1(a). If the MN has more than one tunable component, then
the coverage is typically an area on the Smith chart as shown
in Fig. 1(b). The boundary of such area will be referred to
here as the coverage boundary and is always a closed path.
In this work, MNs with two tunable elements (C
1
and C
2
)
are analyzed. Nevertheless, the theoretical methods introduced
here can be applied to networks with a higher number of
tunable elements, but the formulas become large and less
practical. The two elements are not necessarily identical and
can take any values between C
min
and C
max
. The load is
assumed to be pure resistive taking a value of Y
0
Ω
−1
.
The first step of the analysis is to sweep both capacitors
within their limits and observe the area covered by the
matching network. It has been found that the coverage area
is bounded by four arcs. The first two arcs are plotted by
letting C
1
∈ {C
min
, C
max
} and C
2
takes all the real numbers
between C
min
and C
max
. The remaining two arcs are plotted
by letting C
2
∈ {C
min
, C
max
} and C
1
takes all the real
numbers between C
min
and C
max
. These four arcs are parts
of circles which can be completely plotted by extending the
limits of the capacitors to -∞ and ∞. For lossless networks,
all of these circles are tangent to the |Γ| = 1 circle. The
value of the capacitor at the tangent point can be calculated
by solving for Y
in
that falls on the |Γ| = 1 circle according
to:
|Γ| =
Y
0
− Y
in
Y
0
+ Y
in
= 1 (1a)
|Y
0
− Y
in
| = |Y
0
+ Y
in
|, (1b)
which is satisfied only when the real part of Y
in
is either
zero or ∞ corresponding to either an open or a short circuit,
respectively. A short circuit can be achieved by a shunt branch
with zero impedance (if the branch has only a capacitor the
capacitance can be set to ∞). An open circuit, on the other
hand, can be achieved by a series branch with infinitely large
impedance (if the branch has only a capacitor, its value can
be set to zero).
The centers and radii of the four circles can be calculated
from the tangent point and any other point. If the tangent
point is denoted A(x
A
, y
A
) and the other point is denoted
B(x
B
, y
B
), the center (x
c
, y
c
) and radius (R
c
) of the circle
can be calculated, as derived in Appendix B, by:
x
c
=
x
A
x
2
B
− x
2
A
+ y
2
B
− y
2
A
2 (−x
2
A
− y
2
A
+ x
A
x
B
+ y
A
y
B
)
, (2a)
y
c
=
y
A
x
2
B
− x
2
A
+ y
2
B
− y
2
A
2 (−x
2
A
− y
2
A
+ x
A
x
B
+ y
A
y
B
)
and (2b)
R
c
=
p
(x
c
− x
A
)
2
+ (y
c
− y
B
)
2
, (2c)
respectively.
The four circles for a hybrid network are plotted in Fig. 2
together with the coverage area using the theoretical formulas
presented here and verified by a commercial simulator. It can
be observed that a fifth circle (referred to in the figure as
C
0
2
) is also needed to complete the boundary. This circle is
a function of the inductors and transmission lines as well as
the frequency and is referred to here as the auxiliary circle.
For given values of these parameters, impedances inside the
auxiliary circle can not be matched even if the values of the
tunable capacitors extend from -∞ to ∞ [16]. As illustrated
in Fig. 2, the auxiliary circle is traced when C
1
is swept while
C
2
is assigned a critical value (C
0
2
), which resonates with the
transmission line when C
2
= Y
0
/ω tan(θ). Formulas for C
0
2
are derived for various network topologies in the following
section. The auxiliary circle becomes part of the boundary only
if C
0
2
falls within the tuning range of C
2
. Since the auxiliary
circle defines a forbidden region, the constant circles of C
1,min
and C
1,max
can not intersect with this circle, but only shares a
single point with it and thus the three circles must be tangent.
In the following sections, four different matching networks
are analyzed and the derived formulas to calculate the five
circles are presented.
Circuit simulation Theory
+ +
1
C
2max
C
1max
C
2min
C
0
2
C
1min
Fig. 2. Illustration of the four main circles and the auxiliary circle that
form the boundary of the matching network. The coverage is verified by
a commercial simulator.
A. T-Matching Network
A lumped T-type MN is illustrated in Fig. 3(a). To ana-
lyze this network as mentioned in the previous section, the
coordinates of the points of the C
1
and C
2
circles need to be
calculated.
1) C
1
variable and C
2
constant: For the first case, C
2
∈
{C
min
, C
max
} while C
1
can take any real number between
these two limits. To plot the complete circle, however, we will
let C
1
∈ R
≥0
. The first point to be calculated is the point at
which this circle is to the outer circle (|Γ| = 1). This point
is referred to as point A and corresponds to C
1
= 0. Using
this value, the input admittance (Y
in
) becomes infinite, and
the real and imaginary parts of the input reflection coefficient
(Γ
in
) are
x
A1
= 1 and (3a)
y
A1
= 0. (3b)
For the second point (point B), C
1
is assumed to take an
infinitely large value, which corresponds to an RF short circuit.
Using this assumption, the real and imaginary values of Y
in
can be calculated as:
<{Y
in
} =
Y
0
ω
2
LC
2
2
(ω
2
LC
2
)
2
+ (Y
0
ωL)
2
(4a)
3
and
={Y
in
} =
ωL
−ω
2
C
2
2
− Y
2
0
+ Y
2
0
ω
2
LC
2
(ω
2
LC
2
)
2
+ (Y
0
ωL)
2
, (4b)
from which the real and imaginary values of Γ
in
are:
x
B1
=
− (<{Y
in
})
2
+ Y
2
0
− (={Y
in
})
2
(<{Y
in
} + Y
0
)
2
+ (={Y
in
})
2
and (5a)
y
B1
=
−2Y
0
={Y
in
}
(<{Y
in
} + Y
0
)
2
+ (={Y
in
})
2
, (5b)
respectively.
The coordinates of points A and B can be directly used in
equation (2) to calculate the centers and radii of the circles.
Two circles are obtained: one for C
2
= C
min
and another for
C
2
= C
max
.
C
1
C
2
L
1
L
2
Z
0
, θ
L
L
C
1
C
2
C
1
C
2
C
1
C
2
(a) (b)
(c) (d)
Y
2
, Γ
2
Y
in
Γ
in
Y
in
Γ
in
Y
in
Γ
in
Y
in
Γ
in
Fig. 3. Schematics of the four topologies analyzed in this work. (a) T-type. (b)
Π-type. (c) Ladder-type. (d) Hybrid Π. Γ
in
= x + jy is the input reflection
coefficient and Y
in
is the input admittance.
2) C
1
constant and C
2
variable: For this case C
1
∈
{C
min
, C
max
} while C
2
∈ R
≥0
. The first point in this case
corresponds to C
2
= 0, which gives the following values of
the real and imaginary of Γ
in
x
A2
=
Y
2
0
1 − ω
2
LC
1
2
− (ωC
1
)
2
Y
2
0
(1 − ω
2
LC
1
)
2
+ ω
2
C
2
1
(6a)
y
A2
=
−2ωY
0
C
1
1 − ω
2
LC
1
Y
2
0
(1 − ω
2
LC
1
)
2
+ ω
2
C
2
1
. (6b)
This point has a unity magnitude regardless of the value of C
1
and, therefore, always lies on the outer circle. For the second
point, the assignment: C
2
= ∞ is used and the real and
imaginary values of Γ
in
are:
x
B2
=
Y
2
0
1 − 2ω
2
LC
1
+ ω
2
Y
2
0
L
2
− C
2
1
Y
2
0
(1 − 2ω
2
LC
1
)
2
+ ω
2
(Y
2
0
L + C
1
)
2
(7a)
and
y
B2
=
ωY
0
Y
2
0
L − C
1
1 − 2ω
2
LC
1
− ωY
0
Y
2
0
L + C
1
Y
2
0
(1 − 2ω
2
LC
1
)
2
+ ω
2
(Y
2
0
L + C
1
)
2
,
(7b)
respectively. These two points can be used in equation (2) to
calculate the centers and radii of the circles of C
1
= C
min
and C
1
= C
max
.
3) Auxiliary Circle: The T-type network can match
impedances up to a maximum resistance. The first derivative
of the real part of the input impedance with respect to C
2
can be used to determine the value of C
2
that provides this
resistance as derived in Appendix B-B, and given by:
C
0
2
=
1
ω
2
L
. (8)
The necessary condition for the auxiliary circle to be part of
the boundary is for this value to fall within the limits of C
2
:
C
min
< C
0
2
< C
max
. (9)
The auxiliary circle itself can be plotted by using the value
of C
2
defined in (8) with the formulas of section II-A1
In Fig. 4-(a), an illustration of the boundary of a T-type MN
is illustrated for L = 10nH and frequencies of 0.5 GHz, 1.2
GHz, and 2.5 GHz. The complete circles are also included for
the case of 1.2 GHz.
1
1
1
1
(a) (b)
(c) (d)
0.5 GHz 1.2 GHz 2.5 GHz
C
1,max
C
1,min
C
0
2
C
1,min
C
2,min
C
2,min
C
2,max
C
1,max
C
2,max
C
0
2
C
1,max
C
0
2
C
1,max
C
1,min
C
2,min
C
1,min
C
2,max
C
1,min
Fig. 4. Illustration of the boundary at three different frequencies with the
values: C
min
=0.5 pF, C
max
=15 pF, θ=50, Z
0
=50 Ω, L
T
=10 nH, L
Π
=6.2
nH and L
1
=L
2
= 13 nH. (a) T-Network. (b) Π-Network. (c) Ladder Network.
(d) Hybrid-Π Network.
4
B. Π-Type Matching Network
A typical Π-Type MN is illustrated in Fig. 3(b). It consists
of an inductor between two shunt capacitors. The analysis of
this network can be performed by the same method used in
the previous section.
1) C
1
variable and C
2
constant: For this case, the coordi-
nates of the first point are calculated by putting C
1
= ∞. At
this case the input of the MN appears to have a zero impedance
and, therefore, the real and imaginary values of Γ
in
are
x
A1
= −1 and (10a)
y
A1
= 0 (10b)
respectively.
For the second point, C
1
is assigned a value of zero and
the real and imaginary parts of the input admittance (Y
in
) are
<{Y
in
} =
Y
0
1 − ω
2
LC
2
+ ω
2
Y
0
LC
2
(1 − ω
2
LC
2
)
2
+ (ωY
0
L)
2
(11a)
and
={Y
in
} =
ωC
2
1 − ω
2
LC
2
− ωY
2
0
L
(1 − ω
2
LC
2
)
2
+ (ωY
0
L)
2
(11b)
respectively. The real and imaginary parts of the reflection
coefficient (x
B1
and y
B1
) can be calculated from Y
in
using
(5).
2) C
1
constant and C
2
variable: For the first point of this
case, C
2
is assigned a value of ∞ (RF short circuit). The real
and imaginary parts of Γ
in
are calculated to be
x
A2
=
(ωY
0
L)
2
−
1 − ω
2
LC
1
2
(ωY
0
L)
2
+ (1 − ω
2
LC
1
)
2
and (12a)
y
A2
=
2ωY
0
L
1 − ω
2
LC
1
(ωY
0
L)
2
+ (1 − ω
2
LC
1
)
2
(12b)
respectively. This point has a unity magnitude regardless of
the value of C
1
. For the second point, C
2
is assigned a value
of zero. The real and imaginary parts of Y
in
are:
<{Y
in
} =
Y
0
1 + (ωY
0
L)
2
, and (13a)
={Y
in
} =
ω
C
1
− Y
2
0
L + C
1
(ωY
0
L)
2
1 + (ωY
0
L)
2
, (13b)
respectively. These values can be used to calculate the coor-
dinates of Γ
in
using (5).
3) The Auxiliary Circle: This network can match
impedances up to a maximum conductance. The first derivative
of the real part of the input admittance with respect to C
2
can be used to calculate the value of C
2
that produces this
conductance as shown in Appendix B-A and given by the
following formula:
C
0
2
=
1
ω
2
L
. (14)
The condition for the auxiliary circle to be part of the
boundary is the same as the one defined for the T-network in
equation (9), and the auxiliary circle can be plotted by using
the value of C
0
2
calculated in (14) with the formulas of section
II-B1.
In Fig. 4-(b), an illustration of the boundary of a Π-type
MN is illustrated for L = 6.2 nH and at frequencies of 0.5
GHz, 1.2 GHz, and 1.5 GHz. The complete boundary circles
are also plotted.
C. Ladder Matching Network
This type of MN is illustrated in Fig. 3(c). It consists of two
L-sections connected in series. The analysis of this network is
presented in the following sections.
1) C
1
variable and C
2
constant: For this case, C
1
is
assigned the values of zero and ∞. When C
1
= ∞ the real
and imaginary parts of Γ
in
are clearly given by:
x
A1
= −1 and (15a)
y
A1
= 0 (15b)
respectively. When C
1
= 0, on the other hand, the real and
imaginary parts of Y
in
are calculated as:
<{Y
in
} =
<{Y
2
} (1 − ω={Y
2
}L
1
) + ω<{Y
2
}={Y
2
}L
1
(1 − ω={Y
2
}L
1
)
2
+ (ω<{Y
2
}L
1
)
2
(16a)
and
={Y
in
} =
={Y
2
} (1 − ω={Y
2
}L
1
) − ω (<{Y
2
})
2
L
1
(1 − ω={Y
2
}L
1
)
2
+ (ω<{Y
2
}L
1
)
2
,
(16b)
respectively, where
<{Y
2
} =
Y
0
1 + (ωY
0
L
2
)
2
, and (16c)
={Y
2
} =
ω
C
2
1 + (ωY
0
L
2
)
2
− Y
2
0
L
2
1 + (ωY
0
L
2
)
2
. (16d)
<{Y
in
} and ={Y
in
} can be used to calculate the real and
imaginary parts of Γ
in
using equation (5).
2) C
1
constant and C
2
variable: For this case C
2
is as-
signed zero and ∞. For C
2
= ∞, which is an RF short circuit,
the real and imaginary parts of Γ
in
are given respectively by
x
A2
=
(ωY
0
L
1
)
2
−
1 − ω
2
L
1
C
1
2
(1 − ω
2
L
1
C
1
)
2
+ (ωY
0
L
1
)
2
, and (17a)
y
A2
=
2ωY
0
L
1
1 − ω
2
L
1
C
1
(1 − ω
2
L
1
C
1
)
2
+ (ωY
0
L
1
)
2
. (17b)
This point is on the outer circle of the Smith chart because it
has a magnitude of one regardless of the value of C
1
. For the
second point, C
2
= 0 and the real and imaginary parts of Y
in
are given respectively by:
