Switchmode RF and Microwave Power Amplifiers

By Andrei Grebennikov, Nathan O. Sokal and Marc J. Franco

Combining solid theoretical discussions with practical design examples, this book is an essential reference on developing RF and microwave switchmode power amplifiers.

With this book you will be able to:

• Design high-efficiency RF and microwave power amplifiers on different types of bipolar and field-effect transistors using well-known and novel theoretical approaches, nonlinear simulation tools, and practical design techniques
• Design any type of high-efficiency switchmode power amplifiers operating in Class D or E at lower frequencies and in Class E or F and their subclasses at microwave frequencies, with specified output power
• Understand the theory and practical implementation of load-network design techniques based on lumped and transmission-line elements
• Combine multi-stage Doherty architecture and switchmode power amplifiers to significantly increase efficiency of the entire radio transmitter
• Learn the different types of predistortion linearization techniques required to improve the quality of signal transmission in a nonlinear amplifying system

New to this edition:

• Comprehensive overview of different Doherty architectures which are, and will be used in modern communication systems to save power consumption and reduce costs
• A new chapter on analog and digital predistortion techniques
• Coverage of broadband Class-F power amplifiers, high-power inverse Class-F power amplifiers for WCDMA systems, broadband Class-E techniques

• Unique focus on switchmode RF and microwave power amplifiers that are widely used in cellular/wireless, satellite and radar communication systems and which offer major power consumption savings
• Complete coverage of the new Doherty architecture which offers major efficiencies and savings on power consumption
• Balances theory with practical implementatation, avoiding a cookbook approach, enabling engineers to develop better designs
• Trusted content from leading figures in the field with a Foreword of endorsement by Zoya Popovic

• Language: English
• Publisher: Academic Press
• Release date: Jun 28, 2012
• ISBN: 9780124159839

Andrei Grebennikov

Dr. Andrei Grebennikov is a Senior Member of the IEEE and a Member of Editorial Board of the International Journal of RF and Microwave Computer-Aided Engineering. He received his Dipl. Ing. degree in radio electronics from the Moscow Institute of Physics and Technology and Ph.D. degree in radio engineering from the Moscow Technical University of Communications and Informatics in 1980 and 1991, respectively. He has obtained a long-term academic and industrial experience working with the Moscow Technical University of Communications and Informatics, Russia, Institute of Microelectronics, Singapore, M/A-COM, Ireland, Infineon Technologies, Germany/Austria, and Bell Labs, Alcatel-Lucent, Ireland, as an engineer, researcher, lecturer, and educator. He lectured as a Guest Professor in the University of Linz, Austria, and presented short courses and tutorials as an Invited Speaker at the International Microwave Symposium, European and Asia-Pacific Microwave Conferences, Institute of Microelectronics, Singapore, and Motorola Design Centre, Malaysia. He is an author or co-author of more than 80 technical papers, 5 books, and 15 European and US patents.

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assistance.

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 illustrated in a simple and clear form. The frequency-domain analysis is less ambiguous 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 models for various types of MOSFET, MESFET, HEMT, and BJT devices including HBTs, which are very prospective for modern microwave monolithic integrated 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–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–pull characterization and 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, 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 to its input port, which analytically can be written as

(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’s law to the circuit to be analyzed. Generally, such a system will be composed of nonlinear integro-differential equations in a nonlinear circuit. The solution to this system can be found by applying the numerical-integration methods.

The voltage v(t) in the frequency domain generally represents the multiple-frequency signal at the device input which is written as

(1.2)

where V0 is the constant voltage, Vk is the voltage amplitude, ϕk is the phase of the k-order harmonic component ωk, k, N, and N is the number of 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 necessary 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 basic behavior of a power amplifier and its operation modes. It can also serve as a good starting point for a final computer-aided design and optimization procedure.

The piecewise-linear approximation of the active device current–voltage transfer characteristic is a result of replacing the actual nonlinear dependence i=f(vin), where vin is the voltage applied to the device input, by an approximated one that 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 shown in Fig. 1.1(a).

Figure 1.1 Piecewise-linear approximation technique.

The output-current waveforms for the actual current–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 waveforms corresponding to these two dependences are practically the same for the 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 significant 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 of the linear power amplifier.

The piecewise-linear active device current–voltage characteristic is defined as

(1.3)

where gm is the device transconductance and Vp is the pinch-off voltage.

Let us assume the input signal to be in a cosine form,

(1.4)

where Vbias is the input dc bias voltage.

At the point on the plot when the voltage vin(ωt) becomes equal to a pinch-off voltage Vp and where ωt=θ, the output current i(θ) takes a zero value. At this moment,

(1.5)

and the angle θ can be calculated from

(1.6)

As a result, the output current represents a periodic pulsed waveform described by the cosinusoidal pulses with maximum amplitude Imax and width 2θ as

(1.7)

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 a zero value, one can write

(1.8)

Figure 1.2 Schematic definition of a conduction angle.

Taking into account that I=gmVin for a piecewise-linear approximation, Eq. (1.7) can be rewritten for i>0 by

(1.9)

When ωt=0, then i=Imax and

(1.10)

The Fourier-series expansion of the even function when i(ωt)=i(−ωt) contains only even components of this function and can be written as

(1.11)

where the dc, fundamental-frequency, and nth harmonic components are calculated by

(1.12)

(1.13)

(1.14)

where γn(θ) are called the coefficients of expansion of the output-current cosine waveform or the current coefficients [1, 2]. They can be analytically defined as

(1.15)

(1.16)

(1.17)

where n=2, 3, ….

The dependences of γn(θ) for the dc, fundamental-frequency, second-, and higher-order current components are shown in Fig. 1.3. The maximum value of γn(θ) is achieved when θ=180°/n. Special case is θ=90°, when odd current coefficients are equal to zero, i.e. γ3(θ)=γ5(θ)=…=0. The ratio between the fundamental-frequency and dc components γ1(θ)/γ0(θ) varies from 1 to 2 for any values of the conduction angle, with a minimum value of 1 for θ=180° and a maximum value of 2 for θ=0°, as shown in Fig. 1.3(a). Besides, it is necessary to pay attention to the fact that the current coefficient γ3(θ) becomes negative within the interval of 90°<θ<180°, 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 γn(θ)>0, achieve positive maximum values at the time moments corresponding to the middle points of the current waveform, the harmonic components, for which γn(θ)<0, can achieve negative maximum values at these time moments too. As a 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

(1.18)

Figure 1.3 Dependences of γ n ( θ ) for dc, fundamental, and higher-order current components.

In some cases, it is necessary for an active device to provide a constant value of Imax at any values of θ that require an appropriate variation of the input voltage amplitude Vin. In this case, it is more convenient to use the coefficients αn defined as a ratio of the nth current harmonic amplitude In to the maximum current waveform amplitude Imax,

(1.19)

From Eqs (1.10), (1.18), and (1.19), it follows that

(1.20)

and maximum value of αn(θ) is achieved when θ=120°/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 assumption 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 considerably in excess of 50%, potentially as high as 90%, by choosing the proper operation 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 angle of 360° or 100% 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 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 were assigned to Class-B power amplifiers [6].

To analytically determine the operation classes of the power amplifier, consider a simple resistive stage shown in Fig. 1.4, where Lch is the ideal choke inductor with zero series resistance and infinite reactance at the operating frequency, Cb is the dc-blocking capacitor with infinite value having zero reactance at the operating frequency, and RL is the load resistor. The dc supply voltage Vcc is applied to both plates of the dc-blocking capacitor, being constant during the entire signal period. The active device behaves as an ideal voltage- or current-controlled current source having zero saturation resistance.

Figure 1.4 Voltage and current waveforms in Class-A operation.

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 Vin≤Vbias−Vp, where Vp is the device pinch-off voltage. Usually, to simplify an analysis of the power-amplifier operation, the device transfer characteristic is represented by a piecewise-linear approximation. As a result, the output current is cosinusoidal,

(1.21)

with the quiescent current Iq greater or equal to the collector current amplitude I. In this case, the output collector current contains only two components – dc and cosine – and the averaged current magnitude is equal to a quiescent current Iq.

The output voltage v across the device collector represents a sum of the dc supply voltage Vcc and cosine voltage vR across the load resistor RL. Consequently, the greater the output current i, the greater the voltage vR across the load resistor RL and the smaller the output voltage v. Thus, for a purely real load impedance when ZL=RL, the collector voltage v is shifted by 180° relative to the input voltage vin and can be written as

(1.22)

where V is the output voltage amplitude.

Substituting Eq. (1.21) into Eq. (1.22) yields

(1.23)

where RL=V/I, and Eq. (1.23) can be rewritten as

(1.24)

which determines a linear dependence of the collector current versus collector voltage. Such a combination of the cosine collector voltage and current waveforms is known as a Class-A operation mode. In practice, because of the device nonlinearities, it is necessary to connect a parallel LC circuit with resonant frequency equal to the operating frequency to suppress any possible harmonic components.

Circuit theory prescribes that the collector efficiency η can be written as

(1.25)

where

(1.26)

is the dc output power,

(1.27)

is the power delivered to the load resistance RL at the fundamental frequency f0, and

(1.28)

is the collector voltage peak factor.

Then, by assuming the ideal conditions of zero saturation voltage when ξ=1 and maximum output-current amplitude when I/Iq=1, from Eq. (1.25) it follows that the maximum collector efficiency in a Class-A operation mode is equal to

(1.29)

However, as it also follows from Eq. (1.25), increasing the value of I/Iq can 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 output 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 the collector voltage represents a purely sinusoidal waveform with the voltage amplitude V≤Vcc.

Figure 1.5 Voltage and current waveforms in a Class-B operation.

Equation (1.8) for the output current can be rewritten through the ratio between the quiescent current Iq and the current amplitude I as

(1.30)

As a result, the basic definitions for nonlinear operation modes of a power amplifier through half the conduction angle θ can be introduced as

• When θ>90°, then cosθ<0 and Iq>0, corresponding to Class-AB operation.

• When θ=90°, then cosθ=0 and Iq=0, corresponding to Class-B operation.

• When θ<90°, then cosθ>0 and Iq<0, corresponding to Class-C operation.

The periodic pulsed output current i(ωt) can be represented as a Fourier-series expansion

(1.31)

where the dc and fundamental-frequency components can be obtained by

(1.32)

(1.33)

respectively, where

(1.34)

(1.35)

From Eq. (1.32), it follows that the dc current component is a function of θ in the operation modes with θ<180°, in contrast to a Class-A operation mode where θ=180° and the dc current is equal to the quiescent current during the entire period.

The collector efficiency of a power amplifier with resonant circuit, biased to operate in the nonlinear modes, can be obtained by

(1.36)

which is a function of θ only, where

(1.37)

The Class-B power amplifiers had been defined as those which operate with a negative grid bias such that the anode current is practically zero with no excitation grid voltage, and in which the output power is proportional to the square of the excitation voltage [7]. If ξ=1 and θ=90°, then from Eqs (1.36) and (1.37) it follows that the maximum collector efficiency in a Class-B operation mode is equal to

(1.38)

The fundamental-frequency power delivered to the load PL=P1 is defined as

(1.39)

showing its direct dependence on the conduction angle 2θ. This means that reduction in θ results in lower γ1, and, to increase the fundamental-frequency power P1, it is necessary to increase the current amplitude I. Since the current amplitude I is determined by the input voltage amplitude Vin, the input power Pin must be increased. The collector efficiency also increases with reduced value of θ and becomes maximum when θ=0°, where a ratio γ1/γ0 is maximal, as follows from Fig. 1.3(a). For example, the collector efficiency η increases from 78.5% to 92% when θ reduces from 90° to 60°. However, it requires increasing the input voltage amplitude Vin by 2.5 times, resulting in a lower value of the power-added efficiency (PAE), which is defined as

(1.40)

where Gp=P1/Pin is the operating power gain.

The Class-C power amplifiers had been defined as those that operate with a negative grid bias more than sufficient to reduce the anode current to zero with no excitation grid voltage, and in which the output power varies as the square of the anode voltage between limits [7]. The main distinction between Class B and Class C is in the duration of the output current pulses, which are shorter for Class C when the active device is biased beyond the cutoff point. It should be noted that, for the device transfer characteristic, which can be ideally represented by a square-law approximation, the odd-harmonic current coefficients γn(θ) are not equal to zero in this case, although there is no significant difference between the square-law and linear cases [8]. To achieve the maximum anode efficiency in Class C, the active device should be biased (negative) considerably past the cutoff (pinch-off) point to provide the sufficiently low conduction angles [9].

In order to obtain an acceptable trade-off between a high power gain and a high power-added efficiency in different situations, the conduction angle should be chosen within the range of 120°≤2θ≤190°. If it is necessary to provide high collector efficiency of the active device having a high gain capability, it is necessary to choose a Class-C operation mode with θ close to 60°. However, when the input power is limited and power gain is not sufficient, a Class-AB operation mode is recommended with small quiescent current when θ is slightly greater than 90°. In the latter case, the linearity of the power amplifier can be significantly improved. From Eq. (1.37) it follows that that the ratio of the fundamental-frequency component of the anode current to the dc current is a function of θ only, which means that, if the operating angle is maintained constant, the fundamental component of the anode current will replicate linearly to the variation of the dc current, thus providing the linear operation of the Class-C power amplifier when dc current is directly proportional to the grid voltage [10].

1.3 Load line and output impedance

The graphical method of laying down a load line on the family of the static curves representing anode current against anode voltage for various grid potentials was already well known in the 1920s [11]. If an active device is connected in a circuit in which the anode load is a pure resistance, the performance may be analyzed by drawing the load line where the lower end of the line represents the anode supply voltage and the slope of the line is established by the load resistance, i.e. the load resistance is equal to the value of the intercept on the voltage axis divided by the value of the intercept on the current axis.

In a Class A, the output voltage v across the device anode (collector or drain) represents a sum of the dc supply voltage Vcc and cosine voltage across the load resistance RL, and can be defined by Eq. (1.22). In this case, the power dissipated in the load and the power dissipated in the device is equal when Vcc=V, and the load resistance RL=V/I is equal to the device output resistance Rout[7]. In a pulsed operation mode (Class AB, B, or C), since the parallel LC circuit is tuned to the fundamental frequency, ideally the voltage across the load resistor RL represents a cosine waveform. By using Eqs (1.7), (1.22), and (1.33), the relationship between the collector current i and voltage v during a time period of −θ≤ωt<θ can be expressed by

(1.41)

where the fundamental current coefficient γ1 as a function of θ is determined by Eq. (1.35), and the load resistance is defined by RL=V/I1, where I1 is the fundamental current amplitude. Equation (1.41) determining the dependence of the collector current on the collector voltage for any values of conduction angle in the form of a straight-line function is called the load line of the active device. For a Class-A operation mode with θ=180° when γ1=1, the load-line defined by Eq. (1.41) is identical to the load-line defined by Eq. (1.24).

Figure 1.6 shows the idealized active device output I–V curves and load lines for different conduction angles according to Eq. (1.41) with the corresponding collector and current waveforms. From Fig. 1.6, it follows that the maximum collector current amplitude Imax corresponds to the minimum collector voltage Vsat when ωt=0, and is the same for any conduction angle. The slope of the load line defined by its slope angle β is different for different conduction angles and values of the load resistance, and can be obtained by

(1.42)

from which it follows that a greater slope angle β of the load line results in a smaller value of the load resistance RL for the same θ.

Figure 1.6 Collector current waveforms in Class-AB and Class-C operations.

The load resistance RL for the active device as a function of θ, which is required to terminate the device output to deliver the maximum output power to the load can be written in a general form as

(1.43)

which is equal to the device equivalent output resistance Rout at the fundamental frequency [7]. The term equivalent means that this is not a real physical device resistance as in a Class-A mode, but its equivalent output resistance, the value of which determines the optimum load, which should terminate the device output to deliver maximum fundamental-frequency output power. The equivalent output resistance is calculated as a ratio between the amplitudes of the collector cosine voltage and fundamental-frequency collector current component, which depends on the angle θ.

In a Class-B mode when θ=90° and γ=2V/Imax. Alternatively, taking into account that Vcc=V and Pout=I1V can be written in a simple idealized analytical form with zero saturation voltage, Vsat, as

(1.44)

In general, the entire load line represents a broken line PK including a horizontal part, as shown in Fig. 1.6. Figure 1.6(a) represents a load line PNK corresponding to a Class-AB mode with θ>90°, Iq>0, and I<Imax. Such a load line moves from point K corresponding to the maximum output-current amplitude Imax at ωt=0 and determining the device saturation voltage Vsat through the point N located at the horizontal axis v where i=0 and ωt=θ. For a Class-AB operation, the conduction angle for the output-current pulse between points N′ and N′′ is greater than 180°. Figure 1.6(b) represents a load line PMK corresponding to a Class-C mode with θ<90°, Iq<0, and I>Imax. For a Class-C operation, the load line intersects a horizontal axis v in a point M, and the conduction angle for the output-current pulse between points M′ and M′′ is smaller than 180°. Hence, generally the load line represents a broken line with the first section having a slope angle β and the other horizontal section with zero current i. In a Class-B mode, the collector current represents half-cosine pulses with the conduction angle of 2θ=180° and Iq=0.

Now let us consider a Class-B operation with increased amplitude of the cosine collector voltage. In this case, as shown in Fig. 1.7, an active device is operated in the saturation, active, and pinch-off regions, and the load line represents a broken line LKMP with three linear sections (LK, KM, and MP). The new section KL corresponds to the saturation region, resulting in the half-cosine output-current waveform with a depression in the top part. With further increase of the output-voltage amplitude, the output-current pulse can be split into two symmetrical pulses containing a significant level of the higher-order harmonic components. The same result can be achieved by a increasing a value of the load resistance RL when the load line is characterized by a smaller slope angle β.

Figure 1.7 Collector current waveforms for the device operating in saturation, active, and pinch-off regions.

The collector current waveform becomes asymmetrical for the complex load, the impedance of which represents the load resistance and capacitive or inductive reactances. In this case, the Fourier-series expansion of the output current given by Eq. (1.31) includes a particular phase for each harmonic component. Then, the output voltage at the device collector is written as

(1.45)

where In is the amplitude of the nth output-current harmonic component, |Zn| is the magnitude of the load-network impedance at the nth output-current harmonic component, and ϕn is the phase of the nth output current harmonic component. Assuming that Zn is zero for n=2, 3, … , which is possible for a resonant load network having negligible impedance at any harmonic component except the fundamental, Eq. (1.45) can be rewritten as

(1.46)

As a result, for the inductive load impedance, the depression in the collector current waveform reduces and moves to the left side of the waveform, whereas the capacitive load impedance causes the depression to deepen and shift to the right side of the collector current waveform [12]. This effect can simply be explained by the different phase conditions for fundamental and higher-order harmonic components composing the collector current waveform and is illustrated by the different load lines for (a) inductive and (b) capacitive load impedances shown in Fig. 1.8. Note that now the load line represents a two-dimensional curve with a complicated behavior.

Figure 1.8 Load lines for ( a ) inductive and ( b ) capacitive load impedances.

1.4 Classes of operation based upon a finite number of harmonics

Figure 1.9(a) shows the block diagram of a generic power amplifier, where the active device (which is shown as a MOSFET device but can be a bipolar transistor or any other suitable device) is controlled by its drive and bias to operate as a multiharmonic current source or switch, Vdd is the supply voltage, and I0 is the dc current flowing through the RF choke [13]. The load-network bandpass filter is assumed linear and lossless and provides the drain load impedance R1+jX1 at the fundamental frequency and pure reactances Xk at each kth-harmonic component. For analysis simplicity, the load-network filter can incorporate the reactances of the RF choke and device drain-source capacitance which is considered voltage independent. Since such a basic power amplifier is assumed to generate power at only the fundamental frequency, harmonic components can be present generally in the voltage and current waveforms depending on class of operation. In a Class-AB, -B, or -C operation, harmonics are present only in the drain current. However, in a Class-F mode, a given harmonic component is present in either drain voltage or drain current, but not both, and all or most harmonics are present in both the drain voltage and current waveforms in a Class-E mode. The required harmonics with optimum or near-optimum amplitudes can be produced by driving the power amplifier to saturation. The analysis based on a Fourier-series expansion of the drain voltage and current waveforms shows that maximum achievable efficiency depends not upon the class of operation, but upon the number of harmonics employed [13, 14]. For any set of harmonic reactances, the same maximum efficiency can be achieved by proper adjustment of the waveforms and the fundamental-frequency load reactance.

Figure 1.9 Basic power-amplifier structure and classes of amplification.

A mechanism for differentiating the various classes of power-amplifier operation implemented with small numbers of harmonic components is shown in Fig. 1.9(b) [13]. It is based on the relative magnitudes of the even (Xe) and odd (Xo) harmonic impedances relative to the fundamental-frequency load resistance R1. In this case, the classes of operation can be characterized in terms of a small number of harmonics as follows:

• Class-F: even-harmonic reactances are low and odd-harmonic reactances are high so that the drain voltage is shaped toward a square wave and drain current is shaped toward a half-sine wave.

• Inverse Class-F (Class-F−1): even-harmonic reactances are high and odd-harmonic reactances are low so that the drain voltage is shaped toward a half-sine wave and drain current is shaped toward a square wave.

• Class C: all harmonic reactances are low so that the drain current is shaped toward a narrow pulse.

• Inverse Class C (Class C−1): all harmonic reactances are high so that the drain voltage is shaped toward a narrow pulse.

• Class-E: all harmonic reactances are negative and comparable in magnitude to the fundamental-frequency load resistance.

The transition from ‘low’ to ‘comparable’ occurs in the range from R1/3 to R1/2, whereas the transition from ‘comparable’ to ‘high’ similarly occurs in the range from 2R1 to 3R1. In this case, the circular boundary is for illustration only, and the point at which an amplifier transitions from one class to another is somewhat judgmental and arbitrary, as there is not an abrupt change in the mode of operation. All power amplifiers degenerate to a Class-A operation when there is only a single (fundamental) frequency component. Class B is the special case of a pulsed operation with a conduction angle of 180°, which is represented by a half-sine current waveform based upon even harmonics. Class-D can be considered as a push–pull Class-F power amplifier, in which the two active devices provide each other with paths for the even