Switchmode RF and Microwave Power Amplifiers

686 Pages·2012·12.51 MB·English

by  Grebennikov

CHAPTER
1
Power Amplifier Design
Principles
INTRODUCTION
This introductory chapter presents the basic principles for understanding the
power amplifiers design procedure in principle. Based on the spectral-domain
analysis, the concept of a conduction angle is introduced, by which the basic
Classes A, AB, B, and C of the power-amplifier operation are analyzed and illus-
trated in a simple and clear form. The frequency-domain analysis is less ambigu-
ous because a relatively complex circuit often can be reduced to one or more sets
of immittances at each harmonic component. Classes of operation based upon a
finite number of harmonics are discussed and described. The different nonlinear
modelsfor various typesofMOSFET,MESFET,HEMT,andBJTdevices includ-
ing HBTs, which are very prospective for modern microwave monolithic inte-
grated circuits of power amplifiers, are given. The effect of the input device
parameters on the conduction angle at high frequencies is explained. The design
and concept of push(cid:1)pull amplifiers using balanced transistors are presented. The
possibility of the maximum power gain for a stable power amplifier is discussed
and analytically derived. The device bias conditions and required bias circuits
depend on the classes of operations and type of the active device. The parasitic
parametric effect due to the nonlinear collector capacitance and measures for its
cancellation in practical power amplifiers are discussed. In addition, the basics of
the load(cid:1)pull characterizationand distortion fundamentals are presented.
1.1 Spectral-domain analysis
The best way to understand the electrical behavior of a power amplifier and the
fastest way to calculate its basic electrical characteristics such as output power,
SwitchmodeRFandMicrowavePowerAmplifiers. 1
©2012ElsevierInc.Allrightsreserved.
2 CHAPTER 1 Power Amplifier Design Principles
power gain, efficiency, stability, or harmonic suppression is to use a spectral-
domain analysis. Generally, such an analysis is based on the determination of the
output response of the nonlinear active device when applying the multiharmonic
signal toits input port, which analyticallycan bewritten as
iðtÞ5f½vðtÞ(cid:3) (1.1)
where i(t) is the output current, v(t) is the input voltage, and f(v) is the nonlinear
transfer function of the device. Unlike the spectral-domain analysis, time-domain
analysis establishes the relationships between voltage and current in each circuit
element in the time domain when a system of equations is obtained applying
Kirchhoff’slawtothecircuittobeanalyzed.Generally,suchasystemwillbecom-
posedofnonlinearintegro-differentialequationsinanonlinearcircuit.Thesolution
tothissystemcanbefoundbyapplyingthenumerical-integrationmethods.
The voltage v(t) in the frequency domain generally represents the multiple-
frequency signal at the deviceinput which is written as
XN
vðtÞ5V 1 V cosðω t1φ Þ (1.2)
0 k k k
k51
where V is the constant voltage, V is the voltage amplitude, φ is the phase of
0 k k
the k-order harmonic component ω , k51, 2, ..., N, and N is the number of
k
harmonics.
The spectral-domain analysis, based on substituting Eq. (1.2) into Eq. (1.1) for
a particular nonlinear transfer function of the active device, determines the output
spectrum as a sum of the fundamental-frequency and higher-order harmonic
components, the amplitudes and phases of which will determine the output signal
spectrum. Generally, it is a complicated procedure that requires a harmonic-
balance technique to numerically calculate an accurate nonlinear circuit response.
However, the solution can be found analytically in a simple way when it is neces-
sary to only estimate the basic performance of a power amplifier in terms of the
output power and efficiency. In this case, a technique based on a piecewise-linear
approximation of the device transfer function can provide a clear insight to the
basicbehaviorofapower amplifieranditsoperation modes.Itcan alsoserveasa
goodstartingpointforafinalcomputer-aideddesignandoptimizationprocedure.
Thepiecewise-linearapproximationoftheactivedevicecurrent(cid:1)voltagetrans-
fer characteristic is a result of replacing the actual nonlineardependence i5f(v ),
in
where v is the voltage applied to the device input, by an approximated one that
in
consists of the straight lines tangent to the actual dependence at the specified
points. Such a piecewise-linear approximation for the case of two straight lines is
showninFig.1.1(a).
The output-current waveforms for the actual current(cid:1)voltage dependence
(dashed curve) and its piecewise-linear approximation by two straight lines (solid
curve) are plotted in Fig. 1.1(b). Under large-signal operation mode, the wave-
forms corresponding to these two dependences are practically the same for the
1.1 Spectral-domain analysis 3
i i
0 Vp νin 0 ωt
Vbias (b)
V
in
ωt
(a)
FIGURE1.1
Piecewise-linearapproximationtechnique.
most part, with negligible deviation for small values of the output current close to
the pinch-off region of the device operation and significant deviation close to the
saturation region of the device operation. However, the latter case results in a sig-
nificant nonlinear distortion and is used only for high-efficiency operation modes
when the active period of the device operation is minimized. Hence, at least two
first output-current components, dc and fundamental, can be calculated through a
Fourier-series expansion with a sufficient accuracy. Therefore, such a piecewise-
linear approximation with two straight lines can be effective for a quick estimate
of the output power and efficiency ofthe linear power amplifier.
The piecewise-linear active devicecurrent(cid:1)voltage characteristic is defined as
(cid:2)
0 v #V
i5 in p (1.3)
g ðv 2V Þ v $V
m in p in p
where g is the device transconductance andV is the pinch-off voltage.
m p
Let us assume the input signal to bein acosine form,
v 5V 1V cosωt (1.4)
in bias in
where V is the input dcbiasvoltage.
bias
4 CHAPTER 1 Power Amplifier Design Principles
i i
I
I max
V π 2π ωt
p
0
0 vin
2θ
θ
I
Vbias Vin
ωt
FIGURE1.2
Schematicdefinitionofaconductionangle.
At the point on the plot when the voltage v (ωt) becomes equal to a pinch-off
in
voltage V and where ωt5θ, the output current i(θ) takes a zero value. At this
p
moment,
V 5V 1V cosθ (1.5)
p bias in
and the angleθ can be calculatedfrom
V 2V
cosθ52 bias p: (1.6)
V
in
As a result, the output current represents a periodic pulsed waveform described
bythecosinusoidalpulseswithmaximumamplitudeI andwidth2θ as
max
(cid:2)
I 1Icosωt 2θ#ωt,θ
i5 q (1.7)
0 θ#ωt,2π2θ
where the conduction angle 2θ indicates the part of the RF current cycle, during
which a device conduction occurs, as shown in Fig. 1.2. When the output current
i(ωt)takes azero value, onecan write
i5I 1Icosθ50: (1.8)
q
1.1 Spectral-domain analysis 5
Taking into account that I5g V for a piecewise-linear approximation,
m in
Eq. (1.7) can berewrittenfor i.0 by
i5g V ðcosωt2cosθÞ: (1.9)
m in
When ωt50, then i5I and
max
I 5Ið12cosθÞ: (1.10)
max
The Fourier-series expansion of the even function when i(ωt)5i(2ωt) con-
tains only even components ofthisfunction andcan bewritten as
iðωtÞ5I 1I cosωt1I cos2ωt1 ... 1I cosnωt (1.11)
0 1 2 n
where the dc, fundamental-frequency, and nth harmonic components are calcu-
lated by
ðθ
1
I 5 g V ðcosωt2cosθÞdωt5Iγ ðθÞ (1.12)
0 2π m in 0
2θ
ðθ
1
I 5 g V ðcosωt2cosθÞcosωtdωt5Iγ ðθÞ (1.13)
1 π m in 1
2θ
ðθ
1
I 5 g V ðcosωt2cosθÞcosðnωtÞdωt5Iγ ðθÞ (1.14)
n π m in n
2θ
where γ (θ) are called the coefficients of expansion of the output-current cosine
n
waveform or the current coefficients [1,2]. They can be analytically defined as
1
γ ðθÞ5 ðsinθ2θcosθÞ (1.15)
0 π
(cid:3) (cid:4)
1 sin2θ
γ ðθÞ5 θ2 (1.16)
1 π 2
(cid:5) (cid:6)
1 sinðn21Þθ sinðn11Þθ
γ ðθÞ5 2 (1.17)
n π nðn21Þ nðn11Þ
where n52, 3, ....
The dependences of γ (θ) for the dc, fundamental-frequency, second-, and
n
higher-order current components are shown in Fig. 1.3. The maximum value of
γ (θ) is achieved when θ5180(cid:4)/n. Special case is θ590(cid:4), when odd current
n
coefficients are equal to zero, i.e. γ (θ)5γ (θ)5...50. The ratio between the
3 5
fundamental-frequency and dc components γ (θ)/γ (θ) varies from 1 to 2 for any
1 0
values of the conduction angle, with a minimum value of 1 for θ5180(cid:4) and a
6 CHAPTER 1 Power Amplifier Design Principles
γn(θ) γ1(θ)/γ0(θ)
0.9 1.9
γ (θ)/γ (θ)
1 0
0.8 1.8
γ(θ)
1
0.7 1.7
γ(θ)
0
0.6 1.6
0.5 1.5
0.4 1.4
0.3 1.3
0.2 1.2
γ(θ)
2
0.1 1.1
0 30 60 90 120 150 θ, grad
(a)
γ(θ)
n
0.08
γ(θ)
3
0.06
0.04
γ(θ) γ(θ)
5 4
0.02
0
−0.02
−0.04
−0.06
−0.08
0 30 60 90 120 150 θ, grad
(b)
FIGURE1.3
Dependencesofγ (θ)fordc,fundamental,andhigher-ordercurrentcomponents.
n
maximum value of 2 for θ50(cid:4), as shown in Fig. 1.3(a). Besides, it is necessary
to pay attention to the fact that the current coefficient γ (θ) becomes negative
3
within the interval of 90(cid:4),θ,180(cid:4), as shown in Fig. 1.3(b). This implies the
proper phase changes of the third current harmonic component when its values
are negative. Consequently, if the harmonic components, for which γ (θ).0,
n
1.2 Basic classes of operation: A, AB, B, and C 7
achieve positive maximum values at the time moments corresponding to the
middle points of the current waveform, the harmonic components, for which
γ (θ),0, can achieve negative maximum values at these time moments too. As a
n
result, a combination of different harmonic components with proper loading will
result in flattening of the current or voltage waveforms, thus improving efficiency
of the power amplifier. The amplitude of the corresponding current harmonic
component can be obtained by
I 5γ ðθÞg V 5γ ðθÞI: (1.18)
n n m in n
In some cases, it is necessary for an active device to provide a constant value
of I at any valuesof θthat require an appropriate variation ofthe input voltage
max
amplitudeV .Inthiscase,itismoreconvenienttousethecoefficientsα defined
in n
as a ratio of the nth current harmonic amplitude I to the maximum current wave-
n
form amplitudeI ,
max
I
α 5 n : (1.19)
n I
max
From Eqs(1.10),(1.18), and(1.19), itfollows that
γ ðθÞ
α 5 n (1.20)
n 12cosθ
and maximum value ofα (θ) is achieved when θ5120(cid:4)/n.
n
1.2 Basic classes of operation: A, AB, B, and C
As established at the end of 1910s, the amplifier efficiency may reach quite high
values when suitable adjustments of the grid and anode voltages are made [3].
With resistive load, the anode current is in phase with the grid voltage, whereas it
leads with the capacitive load and it lags with the inductive load. On the assump-
tion that the anode current and anode voltage both have sinusoidal variations, the
maximum possible output of the amplifying device would be just a half the dc
supply power, resulting in an anode efficiency of 50%. However, by using a
pulsed-shaped anode current, it is possible to achieve anode efficiency consider-
ably in excess of 50%, potentially as high as 90%, by choosing the proper opera-
tion conditions. By applying the proper negative bias voltage to the grid terminal
to provide the pulsed anode current of different width with the angle θ, the anode
current becomes equal to zero, where the double angle 2θ represents a conduction
angle of the amplifying device [4]. In this case, a theoretical anode efficiency
approaches 100% when the conduction angle, within which the anode current
flows, reduces to zero starting from 50%, which corresponds to the conduction
angleof 360(cid:4) or100% duty ratio.
Generally, power amplifiers can be classified in three classes according to
their mode of operation: linear mode when its operation is confined to the
8 CHAPTER 1 Power Amplifier Design Principles
substantially linear portion of the active device characteristic curve; critical mode
when the anode current ceases to flow, but operation extends beyond the linear
portion up to the saturation and cutoff regions; and nonlinear mode when the
anode current ceases to flow during a portion of each cycle, with a duration that
depends on the grid bias [5]. When high efficiency is required, power amplifiers
of the third class are employed since the presence of harmonics contributes to the
attainment of high efficiencies. In order to suppress harmonics of the fundamental
frequency to deliver a sinusoidal signal to the load, a parallel resonant circuit can
be used in the load network which bypasses harmonics through a low-impedance
path and, by virtue of its resonance to the fundamental, receives energy at that
frequency. At the very beginning of 1930s, power amplifiers operating in the first
two classes with 100% duty ratio were called the Class-A power amplifiers,
whereas the power amplifiers operating in the third class with 50% duty ratio
wereassigned toClass-Bpoweramplifiers [6].
To analytically determine the operation classes of the power amplifier,
consider a simple resistive stage shown in Fig. 1.4, where L is the ideal choke
ch
inductor with zero series resistance and infinite reactance at the operating
V v
i cc
2V
cc
C
b
V
L
v ch vR RL Vcc
v
in Vcc
ωt
0 π 2π
i
i
I
I
q
V
b v ωt
0 V in 0 π 2π
p
V
in
ωt
FIGURE1.4
VoltageandcurrentwaveformsinClass-Aoperation.
1.2 Basic classes of operation: A, AB, B, and C 9
frequency, C is the dc-blocking capacitor with infinite value having zero reac-
b
tance at the operating frequency, and R is the load resistor. The dc supply volt-
L
age V is applied to both plates of the dc-blocking capacitor, being constant
cc
during the entire signal period. The active device behaves as an ideal voltage- or
current-controlledcurrent source having zero saturation resistance.
For an input cosine voltage given by Eq. (1.4), the operating point must be
fixed at the middle point of the linear part of the device transfer characteristic
with V #V 2V , where V is the device pinch-off voltage. Usually, to sim-
in bias p p
plify an analysis of the power-amplifier operation, the device transfer characteris-
tic is represented by a piecewise-linear approximation. As a result, the output
current is cosinusoidal,
i5I 1Icosωt (1.21)
q
with the quiescent current I greater or equal to the collector current amplitude I.
q
In this case, the output collector current contains only two components (cid:1) dc and
cosine (cid:1)and the averaged current magnitude is equal toa quiescentcurrent I .
q
The output voltage v across the device collector represents a sum of the
dc supply voltage V and cosine voltage v across the load resistor R .
cc R L
Consequently, the greater the output current i, the greater the voltage v across
R
the load resistor R and the smaller the output voltage v. Thus, for a purely real
L
load impedance when Z 5R , the collector voltage v is shifted by 180(cid:4) relative
L L
tothe inputvoltage v andcan bewritten as
in
v5V 1Vcosðωt1180(cid:4)Þ5V 2Vcosωt (1.22)
cc cc
where V is the outputvoltage amplitude.
Substituting Eq. (1.21) into Eq. (1.22) yields
v5V 2ði2I ÞR (1.23)
cc q L
where R 5V/I, andEq.(1.23) can be rewritten as
L
(cid:3) (cid:4)
V v
i5 I 1 cc 2 (1.24)
q R R
L L
which determines a linear dependence of the collector current versus collector
voltage.Suchacombinationofthecosinecollectorvoltageandcurrentwaveforms
is known as a Class-A operation mode. In practice, because of the device nonli-
nearities, it is necessary to connect a parallel LC circuit with resonant frequency
equaltotheoperatingfrequencytosuppressanypossibleharmoniccomponents.
Circuittheoryprescribes that the collector efficiencyη can bewritten as
P 1 I V 1 I
η5 5 5 ξ (1.25)
P 2I V 2I
0 q cc q
where
P 5I V (1.26)
0 q cc
10 CHAPTER 1 Power Amplifier Design Principles
is the dcoutputpower,
IV
P5 (1.27)
2
is the power delivered to the load resistance R at the fundamental frequency f ,
L 0
and
V
ξ5 (1.28)
V
cc
is the collector voltage peak factor.
Then, by assuming the ideal conditions of zero saturation voltage when ξ51
and maximum output-current amplitude when I/I 51, from Eq. (1.25) it follows
q
that the maximum collector efficiencyin aClass-A operation mode is equal to
η550%: (1.29)
However, as it also follows from Eq. (1.25), increasing the value of I/I can
q
further increase the collector efficiency. This leads to a step-by-step nonlinear
transformation of the current cosine waveform to its pulsed waveform when the
amplitude of the collector current exceeds zero value during only a part of the
entire signal period. In this case, an active device is operated in the active region
followed by the operation in the pinch-off region when the collector current is
zero, as shown in Fig. 1.5. As a result, the frequency spectrum at the device out-
put will generally contain the second-, third-, and higher-order harmonics of the
fundamental frequency. However, due to the high quality factor of the parallel
resonant LC circuit, only the fundamental-frequency signal is flowing into the
load, while the short-circuit conditions are fulfilled for higher-order harmonic
components. Therefore,ideally thecollectorvoltage representsapurely sinusoidal
waveform withthe voltage amplitude V#V .
cc
Equation (1.8) for the output current can be rewritten through the ratio
between the quiescentcurrent I and the current amplitude Ias
q
I
cosθ52 q: (1.30)
I
As a result, the basic definitions for nonlinear operation modes of a power
amplifier through halfthe conduction angleθcan be introduced as
(cid:129) When θ.90(cid:4),then cosθ,0 and I .0,correspondingtoClass-AB operation.
q
(cid:129) When θ590(cid:4),then cosθ50 and I 50,correspondingtoClass-Boperation.
q
(cid:129) When θ,90(cid:4),then cosθ.0 and I ,0,correspondingtoClass-Coperation.
q
The periodic pulsed output current i(ωt) can be represented as a Fourier-series
expansion
iðωtÞ5I 1I cosωt1I cos2ωt1I cos3ωt1 ... (1.31)
0 1 2 3