Design of a Low Noise Amplifier Using the Quarter Wave Transformers Matching Technique

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DesignofaLowNoiseAmplifierUsingthe
QuarterWaveTransformersMatching
TechniqueintheFrequencyBand[9-13]GHz
ArticleinInternationalJournalonCommunicationsAntennaandPropagation·August2015
DOI:10.15866/irecap.v5i4.7065
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International Journal on Communications Antenna and Propagation (I.Re.C.A.P.), Vol. 5, N. 4
ISSN 2039 – 5086 August 2015
Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved
248
Design of a Low Noise Amplifier Using the Quarter Wave Transformers
Matching Technique in the Frequency Band [9-13] GHz
I. Toulali1, M. Lahsaini2, L. Zenkouar3
Abstract – Low noise amplifier (LNA) constitutes one of the essential component in wireless
communication systems. It is used especially for designing different types of communication
receivers. The main function of LNA is to provide sufficient gain to overcome noise of other
blocks. Amplifier design requires a matching circuit of impedance to have a low noise, achieve
maximum power transfer and have minimum reflection. This paper presents a low noise amplifier
that operates over the frequency range [9-13] GHz and adopts quarter-wave transformers
impedance matching technique. Simulation and synthesis are made by using CAD software (ADS:
Advanced Design System) addressed to simulations of RF circuits and which is developed by
Agilent®. The proposed LNA is designed in HEMT process. As a result, the amplifier is
unconditionally stable and achieves a 20 dB gain and a good impedance matching over the
working frequency range of [9-13] GHz. Copyright © 2015 Praise Worthy Prize S.r.l. - All rights
reserved.
Keywords: Low Noise Amplifier (LNA), Matching Network, Microstrip, Quarter Wave
Transformer, Transmission Lines
I. Introduction
In the fields of terrestrial radio and radar, microwave
applications have been expanding rapidly over the last
decades. Many new application areas have emerged due
to technological advances, particularly in the domain of
mobile communication systems of the new generation
like UMTS, LTE and satellite navigation [1].
Such telecommunication systems consist of a large
assembly of circuits, which are themselves produced
using active or passive components [2]. Nowadays,
researches focuses mainly on designing competitive
microwave circuits with certain technical requirements
and offering benefits such as reduction of size and cost.
The low noise amplifier (LNA) device is a powerful
module for the RF receiver. Thus, the performance
analysis and optimization of its characteristics is the
objective of our study. Most circuits working in the
frequency range [9-13] GHz are made by planar
technology [3]. This technique consists of engraving the
circuit elements on a suitable dielectric substrate.
Therefore, several topologies and LNA design
methods have been proposed based on different
manufacturing technologies and associated to various
applications [4]-[7]. In this paper, a RF amplifier is
proposed and developed by considering two transistors in
cascade and employing the matching method based on
quarter wave transformers. The dimension of microstrip
lines are calculated and optimized to achieve the
impedance matching and optimum performance of
amplifier. Some parameters design including gain, noise
figure, and stability are used as performance indicators.
This paper is organized as follows. The first part
introduces the microstrip technology used to design the
amplifier. The second part gives an overview of the
fundamentals parameters. The third and fourth sections
present the chosen matching technique. The fifth section
discusses the simulation results, which demonstrate the
feasibility of the proposed technique. The last section
summarizes the gist of the study results.
II. The Microstrip Line
The geometry of a microstrip line is shown in Fig. 1.
A conductor of width W is printed on a thin, grounded
dielectric substrate of thickness d and relative
permittivity; a sketch of the field lines is shown in Fig.
2. The mode of propagation along a microstrip line is not
pure transverse electromagnetic (TEM) [8], [9]. The
effective dielectric constant of a microstrip line is given
approximately by [10]:
≃
+
1
2
+
−
1
2
1
1
+
(1))
The effective dielectric constant can be interpreted as
the dielectric constant of a homogeneous medium that
equivalently substitutes the air and dielectric regions of
the microstrip line, as shown in Fig. 3 [8].
The characteristic impedance can be calculated by the
formulas below according to the dimensions of the
microstrip [8]:

I. Toulali, M. Lahsaini, L. Zenkouar
Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved Int. Journal on Communications Antenna and Propagation, Vol. 5, N. 4
249
=
⎩
⎪
⎨
⎪
⎧
60
8
+
4
for
≤
1
⁄
120
[
+
1
.
393
+
0
.
667
(
⁄
+
1
.
444
)
⁄
]
for
≥
1
⁄
(2)
Fig. 1. Geometry of the microstrip transmission [8]
Fig. 2. Electric and magnetic field lines [8]
Fig. 3. Equivalent geometry, where the dielectric substrate of relative
permittivity  is replaced with a homogeneous medium of effective
relative permittivity  [8]
III. Fundamentals of the
Matching Network
A model of a single stage amplifier including input
and output matching networks is represented in Figure 4
[11].
III.1. Power Gain
After the amplifier design the evaluation of the gain is
essential. Power gains of a 2-ports circuit network, like
the one shown in Fig. 4, are defined by scattering
parameters and classified into 3 types of gain: operating
gain, transducer gain and available gain [12].
III.2. Matching Condition
Maximum transducer power gain is obtained by the
conjugate match [13].
The source and load reflection coefficients should be
chosen as:
=
∗
=
∗
(3)
For a bilateral transistor ( ≠0), the input and
output reflection coefficients are given by [11]:
=
+
1
−
=
+
1
−
(4)
where , , and  are the S-parameters of the
transistor.
III.3. Stability
The circuit design requires taking into consideration
the factor of stability. The aim objective is to avoid
amplifier oscillation when we try to optimize the gain.
The quadrupole is unconditionally stable if:
|
|
≤
1
and
|
|
≤
1
(5)
To express the conditions to the unconditional
stability, we use the Rollet factor K as defined by the
following expression:
=
1
−
|
|
−
|
|
+
|
−
|
2
|
|
(6)
and: ∆ = SS−SS. The necessary and sufficient
stability criteria are:
> 1 and |∆|< 1, the quadrupole is then
unconditionally stable. In addition, it is possible to adjust
simultaneously the input and output [14]. If this
condition is not verified |∆|> 1 where ||< 1, the
quadrupole can be stable for certain impedances. In this
way, it is said to be conditionally stable. In general,
if <−1: a simultaneous adaptation is impossible, the
quadrupole is unstable and therefore unusable as an
amplifier. Another stability coefficient is defined by the
expression (7), if > 1 the circuit is unconditionally
stable [14]:
=
1
−
|
|
|
−
∗
Δ
|
+
|
|
(7)
IV. Impedance Matching
The impedance matching plays an important role in
the microwave circuit design.

I. Toulali, M. Lahsaini, L. Zenkouar
Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved Int. Journal on Communications Antenna and Propagation, Vol. 5, N. 4
250
Fig. 4. Equivalent circuit of a single stage amplifier
Designing a microwave amplifier with an improper
impedance matching will influence the stability of the
circuit and reduce its efficiency. In fact, the use of
adaptive devices is related to several objectives.
The most important factors are a maximum power
transfer.
In other words, the output impedance of the matching
circuit must be equal to the complex conjugate value of
the load impedance. It is also important to have a
reflection coefficient as low as possible or a standing
wave ratio near unity, and a noise matching [15].
There are different types of matching network to
microwave amplifiers. The first technique is based on the
use of lumped elements [16]. However, these elements
are not very suitable to wide bands.
Therefore, several methods which are more adapted to
microwave frequency domain are used. They are based
on distributed elements such as quarter wave lines [13].
One way to adapt an antenna or more, generally a
circuit is to insert a quarter-wavelength line between the
load impedance and the input impedance line, and to
choose an adequate characteristic impedance. It is one of
the simplest technique of doing impedance matching in a
narrow band [17].
The impedance transformers generally comprise a
cascade of uniform quarter wave line sections.
Discontinuities result from different jumps of impedance,
such as a change in the width of the microstrip line.
During the selection of the matching network, some
factors should be considered such as complexity,
bandwidth, and necessary adjustment according to the
desired application.
V. Quarter Wave Transformers
V.1. General Theory
For a transmission line of the characteristic impedance
 and length l, the input impedance  can be written
as follow in [1]:
=
+
+
such as:
=
=
2
(8)
Then, if an intermediate section of transmission line
with a characteristic impedance / and a quarter
wavelength long is connected between the main line and
the load, as illustrated in Fig. 5.
Fig. 5. Quarter wave transformers [1]
The impedance presented to the main line would be
equal to [17]:
=
/
(9)
The adaptation condition is verified if: /= .
This implies that:  = , therefore the transmission
line is well suited [17]. The total reflection coefficient is
given by [17]:
=
1
1
+
(
)
(10)
Fig. 6 shows the plot of || versus , we see that a
single section quarter-wave transformer provides a
perfect match for a given frequency  [17].
Fig. 6. Characteristic bandwidth of a quarter-wave transformer
for a single section

I. Toulali, M. Lahsaini, L. Zenkouar
Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved Int. Journal on Communications Antenna and Propagation, Vol. 5, N. 4
251
It is a narrow band match. The design of multi
sections circuits increase the bandwidth [17].
 Theory of small reflections
Fig. 7 shows a quarter-wave transformer and the
different transmission and reflection coefficients.
Fig. 7. A microwave circuit with two reflecting junctions [17]
The transmission and reflection coefficients are given
by relations [17]:
=
/
−
/
+
;
=
−
/
+
/
;
=
−
−
(11)
=
1
+
=
1
+
/
−
/
+
=
2
/
/
+
(12)
=
1
+
=
1
+
−
/
+
/
=
2
+
/
(13)
So, the amplitude of the total reflected wave of the
quarter-wave transformer is given by [17]:
=
+
+ +…
=
+
(14)
 Approximate theory for multisection quarter-wave
transformers
The study of the total reflection coefficient requires
making approximations to simplify the final formula.
These approximations give the general relation which
are used to calculate the characteristic impedances of a
multi-section quarter-wave transformer as shown in Fig.
7 [17].
The first approximation consists on representing the
coefficient as follows:
(
)
=
+
+
+
…
+
=
−
+
(
=
0
,
1
,
…
,
)
(15)
The second approximation assumes that the
transformer is symmetrical, which means that the
reflection coefficients may be grouped in pairs. In this
case (15) becomes:
(
)
=
+
+
+
(
)
+
(
)
+
⋯
(16)
We have represented the relation to calculate the
impedance  for a Butterworth response. The binomial
function is defined by [17]:
(
)
=
1
+
=
2
(
)
such as:
=
2
−
+
for
:
=
0
(17)
The relation which links the characteristic impedance
 and the load impedance  to the impedances
characterizing the multi-section transformer is calculated
from the following formula [17]:
(
)
=
1
+
=
=
=+ + +⋯+
we have:
=
(18)
Using the mathematical approach:
=−
+≃1 2
the ratio of impedance is obtained by:
=
2
(19)
V.2. The Broadband Matching Technique
To achieve adaptation we will study the case of three
sections of length λ
 as shown in Fig. 8.
Fig. 8. A three sections quarter-wave transformer
The equivalent characteristic impedances obtained by
applying the Eq. (19) are given by:
=
(20)
=
(21)
=
(22)