PIERS Proceedings, Xi’an, China, March 22–26, 2010
950
Behavioral Models for Power Amplifier Using a Difference-frequency
Dual-signal Injection Method
Hui Wang and Peiguo Liu
School of Electronic Science and Engineering, National University of Defense Technology, Changsha, China
Abstract— In this paper a difference-frequency dual-signal injection method is developed to
modeling of RF power amplifiers (PAs) in nonlinear microwave systems. In this method, the
dual-signal is with different frequency spacing. By varying the frequency spacing of dual-signal
and the power level, behavioral models for PAs can be extracted without memory effects and
with memory effects. For verification, the difference-frequency dual-signal is implemented with a
10 W class-AB PA and tested using a dual sinusoidal signal with 10 MHz frequency spacing
from 0.9 GHz ∼ 1 GHz. From the experiment results, it is found that the model improves
adjacent channel power ratio (ACPR) prediction accuracy by over 8 dB and 2 dB, compared
to the conventional memory polynomial model and with conventional Volterra-based behavioral
model respectively.
1. INTRODUCTION
Radio frequency Power amplifiers (PAs) are indispensable components in a microwave system and
are inherently nonlinear [1]. It is well know that there is an approximate inverse relationship
between the PAs efficiency and their linearity. To increase their efficiency, PAs are sometimes
driven into their nonlinear operating range. For such reasons, behavioral modeling for RF PAs has
grown to become a topic of great interest for all those involved in microwave engineering [3–9].
In this paper, we propose a difference-frequency dual-signal injection method to modeling of RF
power amplifiers (PAs) in nonlinear microwave systems. This paper is organized as follows. In Section 2, dual-signal injection method is described. The IM distortion (IMD) and AM/AM, AM/PM
are extracted, and the models without memory effects and with memory effects are described in
Section 4. The experiments are given in Section 5, and with a conclusion in Section 6.
2. DUAL-SIGNAL INJECTION METHOD
Figure 1 shows the block diagram for IMD and AM/AM, AM/PM measurement and model extraction. A power synthesizer is used to synthesize a dual sinusoidal signal into the power amplifier
under test.
The dual-signal can be described as
x (t) = x1 (t) + x2 (t) = A1 cos (ω1 t + θ1 ) + A2 cos (ω2 t + θ2 )
where A1 = A2 = A, and θ1 and θ2 are two independent stochastic variables.
Using the trigonometric identity
·
¸
·
¸
(ω1 + ω2 ) t (θ1 + θ2 )
(ω1 − ω2 ) t (θ1 − θ2 )
x (t) = 2A cos
+
· cos
+
2
2
2
2
= 2A cos [ωc t + θc ] · cos [ωm t + θm ]
Signal Source
Power
Synthesizer
Signal Source
DUT
Coupler
Power Amplifier
difference-frequency
dual-signal
Figure 1: Block diagram for model extraction.
Vector
Signal
Analyzer
(1)
(2)
Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010
951
where
(ω1 + ω2 )
2
(ω1 − ω2 )
=
2
ωc =
ωm
(3)
(4)
The first cosine term in (2) corresponds to the suppressed carrier at the frequency between the
two input signals, while the second cosine term modulates the carrier at the baseband frequency
ωm .
3. IM DISTORTION AND AM/AM, AM/PM
This paper is devoted to providing a comprehensive study of the PAs’ in-band and out-of-band
distortion: intermodulation distortion and AM/AM and AM/PM characteristics.
3.1. IM Distortion
In [11], it is said “When an amplifier is excited by multiple frequencies and driven into its nonlinear
operating range, it generates numerous mixing products.”
The input signal x (t) can be described as (1), and the output would be given by
y (t) =
N
X
i=1
i
ai [x (t)] =
N
X
ai [A1 cos (ω1 t + θ1 ) + A2 cos (ω2 t + θ2 )]i
(5)
i=1
which shows that the output would be composed of a very large number of mixing terms involving
all possible combinations. The new frequency components could be described as
ωm,n = mω1 + nω2
(6)
In (6), m and n are integers; the sum of the magnitudes of these integers is defined as the order
of IMD. Typically only third order IMD (the frequency components at 2ω1 − ω2 and 2ω2 − ω1 ) is
studied because it is the greatest magnitude and most likely to fall into an adjacent receive band.
3.2. AM/AM, AM/PM
AM/AM and AM/PM measurements are consisted in acquiring the input-output relations of the
PAs in both amplitude and phase. We describe (2) as
x (t) = r (t) cos (ωc t + ϕ (t))
where r (t) and ϕ (t) are the amplitude and phase of x (t).
When
y (t) = g (r (t)) cos (ωc t + ϕ (t) + f (r (t)))
(7)
(8)
where g (·) and f (·) represent the AM/AM and AM/PM distortions, respectively, and r (t) and
ϕ (t) are the envelope and phase of the input signal, respectively.
4. BEHAVIORAL MODELS FOR POWER AMPLIFIER
A memoryless PA can be modeled with AM/AM and AM/PM [9]. The input signal is rewritten as
xRF (t) = r (t) cos (ωc t + ϕ (t))
(9)
4.1. Memoryless Models
The memoryless model has been used for many years because of its easier computational implementation, its relative efficiency and acceptable level of accuracy can be achieved in narrow band
systems.
Then the output signal can be written as
yRF (t) = g (r (t)) cos (ωc t + ϕ (t) + f (r (t)))
(10)
yRF (t) = g (r (t)) cos (f (r (t))) cos (ωc t + ϕ (t)) − g (r (t)) sin (f (r (t))) sin (ωc t + ϕ (t)) (11)
PIERS Proceedings, Xi’an, China, March 22–26, 2010
952
4.2. Memory Polynomial Model
The memory polynomial model is widely used to describe nonlinear effects in PAs with memory
effects. The general form of this model can be written as
y (n) =
Q
X
h1 (q) x (n − q) +
q=0
y (n) =
Q
X
h3 (q1 , q2 , q3 ) x (n − q1 )x (n − q2 ) x∗ (n − q3 )
(12)
q1 ,q2 ,q3 =0
Q
K X
X
akq x (n − q) |x (n − q)|k−1
(13)
k=1 q=0
where x (n) and y (n) are the input and output, respectively. akq are the model coefficients. K is
the polynomial function order, and Q is the memory depth.
4.3. Volterra-based Behavioral Model
The Volterra-based behavioral model assumes that the response of a nonlinear system with memory,
having input x (n) and output y (n), can be expressed as
y (n) =
Q
K X
X
...
k=1 q1 =0
Q
X
hm (n − q)
qm =0
m
Y
x (n − qj )
(14)
j=1
A Volterra-based behavioral model is combination of linear convolution and a nonlinear power
series; it provides a general way to model a nonlinear system with memory. However, high computional complexity and great number of coefficients makes this method impractical in some real
applications, because of the degree of nonlinearity and the length of the system.
Recently, to overcome these problems, pured Volterra series [4], modified Volterra series [5],
dynamic deviation reduced Volterra series [6–8] have been used for reducing the complexity and
simplifying calssical Volterra series based behavioral models for RF power amplifiers.
4.4. The Proposed Model
In the conventional memory polynomial model, the baseband output (13) is given as a function of
the baseband complex input samples [x (n) , x (n − 1) , · · · , x (n − Q)]. Here, we use the envelope
of the input to instead the q samples, then the output y (n) is written as
y (n) =
Q
K X
X
k−1
akq x (n) |x (n − q)|
= x (n)
k=1 q=0
Q
K X
X
akq |x (n − q)|k−1
(15)
k=1 q=0
5. MODEL VALIDATION
To analyze the performances of different behavioral models, a 450 ∼ 1500 MHz, 10 W LDMOS PA is
simulated, which is the commercial amplifier MW6S010N manufactured by Freescale Semiconductor
Inc. The input is a dual sinusoidal signal with 10 MHz frequency spacing from 0.9 GHz ∼ 1 GHz.
-40
dBm (Output Power)
-60
-80
-100
-120
-140
-160
-180
-800
-600
-400
-200
0
200
400
600
800
Frequency (MHz)
Figure 2: Comparison of the power spectrums of the behavioral models of the PA.
Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010
953
This was calculated using harmonic-balance simulator in Agilent’s Advanced Design System (ADS).
The model is identified by minimizing the mean square error (MSE), and method given by M.
Isaksson [2] is used for the synchronization of input and output signals.
Figure 2 shows the power spectrum density (PSD) of both models of the PA. It is clear that
the spectrums of the models track the output spectrum of the PA well, both in-band and in the
adjacent channels.
6. CONCLUSION
We have proposed a difference-frequency dual-signal injection method to modeling of RF power
amplifiers (PAs) in nonlinear microwave systems. The proposed model can be used to simulate the
IMD, and AM/AM and AM/PM that depend on the history of the input signal.
To validate the proposed modeling technique, a 450 ∼ 1500 MHz, 10 W LDMOS PA is simulated.
It is found that the model improves ACPR prediction accuracy by over 8 dB and 2 dB, compared
to the conventional memory polynomial model and with conventional Volterra-based behavioral
model respectively.
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