Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 1390295, 19 pages
https://doi.org/10.1155/2017/1390295

Research Article

On Optimal Truncated Biharmonic Current Waveforms for Class-F and Inverse Class-F Power Amplifiers

Anamarija Juhas,¹ Stanisa Dautovic,¹ and Ladislav A. Novak²

¹Department of Power, Electronics and Communication Engineering, Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovica 6, 21000 Novi Sad, Serbia
²Computer Science Department, Indiana University Bloomington, 150 S. Woodlawn Ave., Bloomington, IN 47405, USA

Correspondence should be addressed to Ladislav A. Novak; ladnovak@indiana.edu

Received 4 October 2016; Revised 25 January 2017; Accepted 1 February 2017; Published 16 March 2017

Academic Editor: Alessandro Lo Schiavo

Copyright © 2017 Anamarija Juhas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, two-parameter families of periodic current waveforms for class-F and inverse class-F power amplifiers (PAs) are considered. These waveforms are obtained by truncating cosine waveforms composed of dc component and fundamental and either second or third harmonic. In each period, waveforms are truncated to become zero outside of a prescribed interval (so-called conduction angle). The considered families of waveforms include both discontinuous and continuous waveforms. Fourier series expansion of truncated waveform contains an infinite number of harmonics, although a number of harmonics may be missing. Taking into account common assumptions that for class-F PA the third harmonic is missing in current waveform and for inverse class-F PA the second harmonic is missing in current waveform, we consider the following four cases: (i) (ii) , (iii) and (iv) , We show that, in each of these cases, current waveform enabling maximal efficiency (optimal waveform) of class-F and inverse class-F PA is continuous for all conduction angles of practical interest. Furthermore, we provide closed-form expressions for parameters of optimal current waveforms and maximal efficiency of class-F (inverse class-F) PA in terms of conduction angle only. Two case studies of practical interest for PA design, involving suboptimal current waveforms, along with the results of nonlinear simulation of inverse class-F PA, are also presented.

1. Introduction

There is continuous interest in class-F PA and its dual inverse class-F PA (see, e.g., [1–8]). Finding optimal and suboptimal nonnegative waveforms for PAs also attracted substantial interest within the research community (see, e.g., [5, 9–16]) and can be regarded as a part of the so-called waveform engineering [15, 16].

In this paper, we consider a two-parameter model of periodic current waveform, defined within fundamental period as

where stands for is conduction angle, and are parameters, is constant, and or This family includes both continuous and discontinuous waveforms. Waveform of type is a truncated biharmonic current waveform.

The current waveform of type (1) is an even function and therefore its Fourier series expansion contains dc component and cosine terms. Coefficients of the Fourier series expansion can be expressed as

where and denotes unnormalized sinc function Coefficients of fundamental harmonic and harmonic can be also obtained from (3) by using that Notice that change of parameter and/or causes the change of the whole harmonic content of waveform of type (1).

Corresponding voltage waveform of PA is assumed to be of the form

where , and or

A number of existing models of current waveforms (both continuous and discontinuous) can be embedded in model (1), including the most used continuous model of current waveform for classical PA operation (see, e.g., [5]):

Model (5) can be obtained directly from (1) by setting and

Another widely considered continuous model of current waveform is one-parameter model of type (see, e.g., [5])

which can be also obtained from (1) by setting the value of parameter to

Typically, continuous current waveform of type (6) is paired with voltage waveform of type (4) with for biharmonic mode of PA operation [5, 17]. In [17], the value of parameter (for and ) is obtained via optimization of efficiency, subject to the constraint that harmonic coefficient is nonpositive. A special case of biharmonic mode with is also analyzed in [5, 17].

The two-parameter model of continuous current waveform for PA with has been considered in [18]. This model can be obtained from (1) by setting , and

The one-parameter model of discontinuous current waveform is used in [3]. The authors of [3] considered case in the context of class-F PA and case in the context of inverse class-F PA. Current waveform proposed in [3] for class-F can be obtained from (1) by setting , and Furthermore, parameter is determined from the condition that the third harmonic is missing in current waveform. The same model of current waveform is also used in [19] in the context of continuous class-F PA.

On the other hand, for inverse class-F, the one-parameter model of discontinuous current waveform proposed in [3] can be obtained from (1) by setting , and In this context, parameter is obtained from the condition that the second harmonic is missing in current waveform.

For class-F PA, we use the common assumption that the third harmonic in current waveform is missing, (see, e.g., [2, 3, 7]). In this case, the corresponding voltage waveform is of type (4) for Also, for inverse class-F PA, we use the common assumption that the second harmonic in current waveform is missing, (see, e.g., [2, 3]). The corresponding voltage waveform is of type (4) for . In what follows, current waveforms of type (1) satisfying condition are denoted by In Sections 2–4, we consider current waveform of type (1) for and To be more specific, we consider the following four cases:

(i),(ii) and ,(iii),(iv) and

Cases (i) and (ii) correspond to class-F mode of PA operation, whereas cases (iii) and (iv) correspond to inverse class-F mode. To the best of our knowledge, cases (ii) and (iv) are not widely explored in waveform modeling for PA design.

The efficiency of PA can be expressed via basic waveform parameters and as , provided that parameter () is equal to the quotient of fundamental harmonic amplitude and dc component of the current (voltage) waveform (see, e.g., [1]). In class-F and inverse class-F PA, a higher harmonic component could appear in at most one of the waveforms in current-voltage pair (see, e.g., [1]). This fact implies that current and voltage waveforms can be optimized independently for class-F and inverse class-F PA. Therefore, nonnegative current waveform of type that ensures maximal efficiency (optimal waveform) of class-F PA or inverse class-F PA is a waveform of type with maximal parameter

For all cases (i)–(iv), in Section 2, we prove that current waveforms of type (1) enabling maximal efficiency are continuous for all conduction angles of practical interest for class-F and inverse class-F PA. Closed-form expressions for parameters of such waveforms as functions of conduction angle are also provided in Section 2. In Section 3, an independent numerical verification of the values of parameters of optimal current waveform is described. In Section 4, we consider efficiency of class-F and inverse class-F PA, using the results obtained in Section 2. In Section 4.1, maximal efficiency of class-F PA with optimal current waveforms for or is provided. Section 4.2 is devoted to inverse class-F PA with optimal current waveforms for or In Section 5, we consider continuous current waveforms of type (6) satisfying relaxed condition for and , instead of used in the context of class-F and inverse class-F PA. Two case studies involving suboptimal current waveforms for class-F and inverse class-F are also presented in this section. As a practical validation of the proposed approach, comparisons of the results derived in this paper with the result of nonlinear simulation of inverse class-F PA with CGH40010F HEMT are provided in Section 6.

2. Optimal Current Waveform of Class-F and Inverse Class-F PA

In this section, current waveforms of type (1) that satisfy condition for and are considered. In what follows, we prove that current waveform , which in pair with voltage waveform of type (4) enables maximal efficiency, is continuous for all conduction angles of practical interest for class-F and inverse class-F PA (i.e., for those conduction angles listed in Table 1). In this section, we also provide closed-form expressions for parameters of such waveforms.

Table 1: Conduction angles for which the optimal waveform of type is continuous.

Case

Conduction angle

(i)
3
3

(ii)
3
2

(iii)
2
2

(iv)
2
3

In what follows, we first prove that optimal waveform is continuous.

As it has been mentioned earlier, Fourier coefficient of harmonic of current waveform of type (1) can be expressed in form (3). Furthermore, condition implies that parameter is equal to

From (2) and (3), it follows that basic waveform parameter of waveform reads

where is given by (8). For the prescribed conduction angle, parameter is a function of only, which implies that is also a function of only. First derivative of with respect to can be expressed in the following form:

where is a function of :

It is easy to see that the sign of depends on the sign of only. Since is negative for , and (see Figure 1), it immediately follows that decreases when increases. Consequently, the highest value of (and the highest efficiency) is attained for minimal value of

Figure 1: Functions (solid line), (dashed line), and (dotted line).

In order to be nonnegative, waveform should satisfy condition , that is, For minimal value of (see the above consideration), this condition reduces to equality. Therefore, of optimal current waveform is equal to

which further implies that optimal waveform of type is continuous; that is, it is of type (6).

After substitution of (12) into (8) and solving the resulting equation, we obtain parameter of continuous waveform of type (6) that satisfies condition , which we denote by This parameter is a function of conduction angle only:

Consequently, optimal current waveform of class-F PA (inverse class-F PA) is a continuous waveform of type (6) with parameter , and In what follows, we show that this waveform is nonnegative for the conduction angles listed in Table 1.

Let us first consider cases (ii) and (iii), that is, cases when

It is easy to show that continuous waveform of type (6) for can be expressed in factored form as

Because and for , waveform of type is nonnegative if and only if for By inspection of the first derivative of , it follows that it has only one critical point in interval Since this critical point is at , it immediately follows that is nonnegative provided that This condition can be rewritten as

where

The dotted line in Figure 2 corresponds to

Figure 2: Parameter : (dotted line), (solid line), and (dashed line).

Case (ii): and For and , expression (13) can be simplified to

The solid line in Figure 2 corresponds to It is easy to see that According to the above consideration, waveform with is nonnegative if Substituting (17) and (16) into and solving the resulting relation for lead to Consequently, waveform with is nonnegative when (third row in Table 1).

Case (iii): and Substitution of and into (13) leads to

The dashed line in Figure 2 corresponds to It is easy to show that for all According to the above discussion, waveform with is nonnegative if Substituting (18) and (16) into and solving the resulting relation for , we obtain Consequently, waveform of type with is nonnegative when (fourth row in Table 1).

A number of shapes of current waveforms of type satisfying are presented in Figure 3. Figure 3(a) corresponds to the case (case (ii)), whereas Figure 3(b) corresponds to the case (case (iii)).

(a)

(b)

Figure 3: Shapes of current waveform of type with (a) and (b)

Let us now consider cases (i) and (iv), that is, cases when

It is easy to show that continuous waveform of type (6) for can be expressed in factored form as

Let us introduce an auxiliary waveform

Because and for , waveform of type is nonnegative if and only if for First derivative of is equal to Therefore, critical points of on interval are and satisfying Notice that relations and imply , that is, Values of at critical points are and Condition can be rewritten as

where

Also, condition can be rewritten as

where

From (24), it follows that Consideration of critical points of leads to the conclusion that is nonnegative if

(a),(b) and ,(c) and

In what follows, we show that and Thus, both correspond to case (a), whereas cases (b) and (c) do not occur for waveforms of type with or Consequently, waveform with is nonnegative if The dotted line in Figure 4 corresponds to

Figure 4: Parameter : (dotted line), (solid line), and (dashed line).

Case (i): and Substitution of and into (13) leads to

The solid line in Figure 4 corresponds to It is easy to see that and According to the above discussion concerning auxiliary waveform , waveform with is nonnegative if Substituting (25) and (22) into and solving the resulting relation for , we obtain Consequently, waveform of type with is nonnegative when (second row in Table 1).

Case (iv): and For and , expression (13) can be simplified to

The dashed line in Figure 4 corresponds to From (26), it immediately follows that According to the above consideration involving auxiliary waveform , waveform with is nonnegative if Substituting (26) and (22) into and solving the resulting relation for yield Consequently, waveform of type with is nonnegative when (fifth row in Table 1).

Several examples of shapes of current waveforms of type satisfying are presented in Figure 5. Figure 5(a) corresponds to the case (case (i)), whereas Figure 5(b) corresponds to the case (case (iv)).

(a)

(b)

Figure 5: Shapes of current waveform of type with (a) and (b)

3. Verification of the Analytical Results

In this section, an independent numerical verification of the analytical results derived in Section 2 is presented. The values of the parameters of the optimal waveform of type , obtained numerically are in full agreement with the values of the parameters calculated from the closed-form expressions derived in Section 2.

Algorithm 1 provides the procedure for numerical calculation of the parameters of the optimal current waveform of type for prescribed conduction angle. Only conduction angles listed in Table 1 are considered, in accordance with the results of Section 2. The algorithm executes the brute force search for optimal current waveform through the set of nonnegative waveforms of type Notice that, for the prescribed conduction angle, there exist only one continuous and an infinite number of discontinuous nonnegative waveforms of type

Input:

         //

         // , e.g. (the value of has no influence on the output results)

        ,   // , e.g.

         // half of the conduction angle

         // (current waveform is an even function of )

Output:

         ,   // parameters of optimal waveform of type

  begin

    ;  ;

    for    to    do

            ;

            ; // expression (8)

            ;  ;

            for    to    do

                ;

                ;

                ;

            end for  //  end -loop

            ;

            if    do

                ;  ;  ;

            end if

    end for  // end -loop

    ;

    if    do

            print   “no solution found – increase the value of and try again”

    else

            ;

            find such that ; // find index of the maximum element of

            ;  ;

    end if

  end.

Algorithm 1

A short description of Algorithm 1 is as follows: in line 2, we initialize the value of counter , and we choose step size for and step size for (because current waveform is an even function); in lines 3–5, we calculate for each ; in lines 6–11, we calculate , dc component, and amplitude of fundamental harmonic of for each pair ; in line 12, we find the minimum value of for each pair ; in lines 13–16, we calculate for each pair provided that corresponding is nonnegative; in lines 17–24, we find as the maximum value of and corresponding pair Parameters of optimal waveform of type are , and

It is obvious that the numerical accuracy of the values of the parameters obtained by Algorithm 1 depends on chosen step size and step size By numerical investigation, we find that and provide numerical stability of the three most significant decimal digits.

For example, for and , numerical values of parameters and of optimal waveforms are calculated for , and On the other hand, values of parameters , and are calculated from the closed-form expressions as follows:

(a) from (17),(b),(c)substitution of and into (9) yields

The values of parameters obtained by execution of Algorithm 1 and values calculated from the closed-form expressions are presented in Figure 6. Full agreement between the numerical and analytical results can be observed for all considered conduction angles.

Figure 6: Parameters of optimal current waveform with and Numerical results: (red circle dots), (blue square dots), and (black triangle dots). Analytical results: (red dotted line), (blue dashed line), and (black solid line).

For all remaining pairs , full agreement between analytical and numerical results is also confirmed (not shown here).

4. Maximal Efficiency of Class-F and Inverse Class-F PA

Maximal efficiencies of class-F PA and inverse class-F PA with current waveforms of type (1) are provided in Sections 4.1 and 4.2, respectively. Our consideration in this section involves only conduction angles listed in Table 1.

Efficiency of PA can be expressed via basic waveform parameters and of current and voltage waveforms (see, e.g., [1]) as follows:

As it has been shown in Section 2, optimal current waveform of type (1) for class-F (inverse class-F) PA is a continuous waveform of type (6) satisfying condition ().

For continuous waveforms of type (6), parameter is given by (7). Substitution of (7) into (2)-(3) leads to the following expressions for Fourier coefficients of dc component and fundamental harmonic of waveform of type (6):

Therefore, basic waveform parameter of waveform can be expressed as

We denote basic waveform parameter of continuous waveform with by This parameter can be easily obtained from (30) by substituting ,

4.1. Maximal Efficiency of Class-F PA

In this subsection, we provide maximal efficiency of class-F PA with current waveform of type for or In both cases, we assume that voltage waveform is an optimal waveform of type (4) with which reads (see, e.g., [1])

Basic waveform parameter of is

According to the results of Section 2, optimal current waveform, which in pair with voltage waveform (32) provides maximal efficiency of class-F PA, is a continuous waveform of type (6) with parameter For , parameter is given by (17), whereas for parameter is given by (25). Solid lines in Figures 2 and 4 correspond to parameters and , respectively.

A number of shapes of optimal current waveforms of class-F PA are presented in Figures 3(a) and 5(a). Figure 5(a) corresponds to continuous waveform with (case (i): , whereas Figure 3(a) corresponds to continuous waveform with (case (ii): and ).

According to (27), efficiency of class-F PA with voltage waveform (32) and current waveform of type with for is equal to

Parameter can be obtained from (31) by substituting , and from (25). Parameter can be obtained from (31) by substituting , and from (17). In Box 1, we provide a brief overview of the calculation of maximal efficiency of class-F PA.

Class-F, case (i):

Class-F, case (ii):

Box 1: Closed-form expressions used for the calculation of maximal efficiency of class-F PA.

Maximal efficiency of class-F PA is presented in Figure 7. The solid line corresponds to (case (i)), whereas the dashed line corresponds to (case (ii)). For prescribed , maximal efficiency of class-F PA is a function of conduction angle only. The efficiency monotonically decreases with the increase of conduction angle as in classical modes of PA operation.

Figure 7: Maximal efficiency of class-F PA. Solid line corresponds to and dashed line corresponds to

For conduction angle   (so-called class-B conduction angle), from Box 1, we obtain and Thus, for , optimal waveform is equal to (see (5)). Furthermore, we obtain , which is equal to the maximum efficiency of class-F PA with voltage waveform (32) [10].

For conduction angle and , from Box 1, we obtain and , which are in agreement with the results presented in [17] (see also [5]).

These examples also confirm our analytically obtained results presented in Section 2.

4.2. Maximal Efficiency of Inverse Class-F PA

In this subsection, we provide maximal efficiency of inverse class-F PA with current waveform of type for or In both cases, we assume that voltage waveform is an optimal waveform of type (4) with which reads (see, e.g., [1])

Basic waveform parameter of is

According to the results of Section 2, optimal current waveform, which in pair with voltage waveform (34) provides maximal efficiency of inverse class-F PA, is a continuous waveform of type with parameter For , parameter is given by (18), whereas for parameter is given by (26). The dashed lines in Figures 2 and 4 correspond to parameters and , respectively.

Several examples of shapes of current waveforms that provide maximal efficiency of inverse class-F PA are presented in Figures 3(b) and 5(b). Figure 3(b) corresponds to continuous waveform with (case (iii): ), whereas Figure 5(b) corresponds to continuous waveform with (case (iv): and ).

According to (27), efficiency of inverse class-F PA with voltage waveform (34) and current waveform of type with for is equal to

Parameter can be obtained from (31) by substituting , and from (18). Parameter can be obtained from (31) by substituting , and from (26). In Box 2, we provide a brief overview of the calculation of maximal efficiency of inverse class-F PA.

Inverse class-F, case (iii),

Inverse class-F, case (iv),

Box 2: Closed-form expressions used for the calculation of maximal efficiency of inverse class-F PA.

Maximal efficiency of inverse class-F PA is presented in Figure 8. The solid line corresponds to (case (iii)), whereas the dashed line corresponds to (case (iv)). For prescribed , maximal efficiency of inverse class-F PA is a function of conduction angle only. The efficiency monotonically decreases with the increase of conduction angle as in classical modes of PA operation.

Figure 8: Maximal efficiency of inverse class-F PA. Solid line corresponds to and dashed line corresponds to

For conduction angle and , from Box 2, we obtain and , which are in agreement with the results presented in [17] (see also [5]).

Also, for conduction angle and , we obtain and , which are in agreement with the results presented in [18].

These two examples provide another verification of analytical results presented in Section 2.

5. Suboptimal Continuous Current Waveforms in PA Efficiency Analysis

In this section, we consider continuous current waveforms of type (6), , with second or third harmonic Fourier coefficient being nonpositive (in contrast to the previous more strict condition that either coefficient of second or third harmonic is equal to zero). Thus, here we extend our analysis and consider suboptimal continuous current waveforms, which are also of interest for the modeling of waveforms for PA design. Moreover, we provide two case studies involving continuous current waveforms with satisfying (case study 1) or (case study 2) in PA efficiency analysis.

5.1. Continuous Current Waveforms with Nonpositive Fourier Coefficient of the Second (Third) Harmonic

Fourier coefficient of the harmonic of current waveform of type (6) can be obtained by substituting (7) into (3):

It is easy to show that , for , and Consequently, Fourier coefficient of the harmonic is nonpositive when

The value of for which (37) becomes equality is (see (13)). Thus, (37) can be rewritten as

As it has been shown in Section 2, expression for can be reduced to:

(i)expression (25) for ; parameter is plotted with solid line in Figure 4,(ii)expression (17) for ; parameter is plotted with solid line in Figure 2,(iii)expression (18) for ; parameter is plotted with dashed line in Figure 2,(iv)expression (26) for ; parameter is plotted with dashed line in Figure 4.

Coefficients of fundamental and third harmonic of optimal voltage waveform (32) for class-F PA have opposite signs. Similarly, coefficients of fundamental and second harmonic of optimal voltage waveform (34) for inverse class-F PA have opposite signs. Therefore, when the output network of PA is passive, voltage waveform (32) can be paired with current waveform of type (6), provided that the third-harmonic Fourier coefficient of current waveform is nonpositive. Analogously, voltage waveform (34) can be paired with current waveform of type (6) provided that the second-harmonic Fourier coefficient of current waveform is nonpositive.

For example, substitution of , and (class-B conduction angle) into (37) yields According to the above discussion, Fourier coefficient of the third harmonic is positive for , equal to zero for , and negative for Therefore, small changes of parameter around the optimal value may cause sign changes of the third-harmonic current component. When output network of PA is passive, sign change of the third-harmonic current component causes sign change of the third-harmonic voltage component, which eventually may lead to a significant decrease of efficiency of class-F PA.

Similarly, substitution of and into (37) yields Fourier coefficient of the third harmonic of continuous current waveform (6) is nonpositive for all and positive for all Therefore, small change of parameter from the optimal value may also lead to a significant decrease of efficiency of class-F PA.

5.2. Case Studies

Here, we consider efficiency of PAs with suboptimal current waveforms of type (6) for We illustrate that with near-optimal current waveform almost maximal efficiency of class-F (inverse class-F) PA can be attained. In case study 1, the third-harmonic Fourier coefficient of current waveform is negative, whereas the second-harmonic Fourier coefficient is negative in case study 2. Corresponding voltages are also suboptimal waveforms: nonnegative waveform with dc component and fundamental and third harmonic in case study 1 and nonnegative waveform with dc component and fundamental and second harmonic in case study 2.

Let us consider generic PA circuit shown in Figure 9. We assume that voltage and current waveforms at the output port of transistor are

Both waveforms are normalized such that and Under common assumptions that behaves as short circuit and behaves as open circuit at fundamental and higher harmonics, voltage and current waveforms at the load are

The load impedance is equal to at fundamental harmonic, to at harmonic, and to at harmonic, and When load is passive ( and ), products and are nonpositive.

Figure 9: Generic PA circuit diagram.

Efficiency of PA with normalized waveforms (39) can be calculated as (e.g., see, [14])

Case Study 1. In this case study, we analyze the efficiency of PA with current waveform of type (6) for and negative Fourier coefficient of the third harmonic. This type of current waveform can be considered as suboptimal current waveform of class-F PA. Corresponding voltage waveform of PA is suboptimal nonnegative waveform with dc component and fundamental and third harmonic.

Let us introduce two current waveforms, one of them being very close to the optimal for class-F PA. More precisely, these waveforms are normalized waveforms of type (6) with

 subcase (1a): , and , subcase (1b): , and

According to (17), optimal value of parameter for class-F PA is In both subcases, and therefore the third-harmonic Fourier coefficients of both waveforms are negative. The value of in subcase (1b) is very close to the optimal.

In subcase (1a), substitution of , and into (28) leads to Furthermore, from (6), we obtain normalized form of current waveform

Fourier coefficients of fundamental and third harmonic of are

In subcase (1b), substitution of , and into (28) leads to From (6), we obtain normalized current waveform

Fourier coefficients of fundamental and third harmonic of are

In both subcases, we assume that voltage is nonnegative waveform of type

For passive load, along with (i.e., implies Also, along with (i.e., implies In [14], it is proved that relation holds () for all nonnegative waveforms of type (46). In subcase (1a), relation together with implies Therefore, in subcase (1a), values of impedance at the third harmonic are bounded by and Similarly, in subcase (1b), values of impedance at the third harmonic are bounded by and

In what follows, we consider maximal efficiency of PA with current waveform (42) (i.e., (44)) and voltage waveform (46) provided that and in subcase (1a) (i.e., and in subcase (1b)). The problem of finding maximal efficiency of PA with waveform pair (46) and (42) (i.e., (44)) for the prescribed third-harmonic impedance can be reduced to the problem of finding voltage waveform of type (46) with maximal absolute value of for prescribed coefficients of the third harmonic [14]. Fourier series expansion of (42) (i.e., (44)) contains only cosine terms and therefore the procedure for calculation of maximal efficiency of PA described by Algorithm in [14] can be used in both subcases. For the sake of completeness, here we provide Algorithm 2, which is analogous to Algorithm in [14]. To improve readability of the algorithm, we change the notation: in [14] is replaced with

// input: and ;

(i) choose such that and ;

(ii) calculate and ;

(iii) if then calculate and ;

else, calculate ,   and ;

(iv) calculate efficiency ;

(v) calculate   and  

Algorithm 2

Function , which appears in step (iii) of Algorithm 2, is defined as

In subcase (1a), input of Algorithm 2 (coefficients and is given by (43). Maximal efficiency of PA with waveform pair (42) and (46), as a function of normalized third-harmonic impedance , is presented in Figure 10(a). Efficiency of 0.89 is attained in the vicinity of (corresponding to

(a)

(b)

Figure 10: Maximal efficiency of PA with (a) current (42) and voltage (46) and (b) current (44) and voltage (46) as a function of normalized third-harmonic impedance.

In subcase (1b), input of Algorithm 2 (coefficients and is given by (45). Maximal efficiency of PA with waveform pair (44) and (46), as a function of normalized third-harmonic impedance, is presented in Figure 10(b). Efficiency of 0.92 is achieved in the vicinity of (corresponding to

As we mentioned earlier, the optimal value of parameter is Corresponding maximal efficiency of class-F PA (see Box 1) is Current waveform in subcase (1b) is very close to optimal current waveform for class-F PA and corresponding peak efficiency of 0.92 is very close to

Case Study 2. In this case study, we analyze the efficiency of PA with current waveforms of type (6) for and negative Fourier coefficient of the second harmonic. This type of current waveform can be considered as suboptimal current waveform of inverse class-F PA. Corresponding voltage of PA is a suboptimal nonnegative waveform with dc component and fundamental and second harmonic.

Let us introduce two current waveforms, one of them being close to the optimal for inverse class-F PA. More precisely, these waveforms are normalized waveforms of type (6) with

 subcase (2a): , and , subcase (2b): , and

According to (18), the optimal value of parameter for inverse class-F PA is In both subcases, and therefore second-harmonic Fourier coefficients of both waveforms are negative. The value of in subcase (2b) is close to the optimal.

In subcase (2a), substitution of , and into (28) leads to Furthermore, from (6), we obtain normalized form of current waveform

Fourier coefficients of fundamental and second harmonic of are

In subcase (2b), substitution of , and into (28) leads to From (6), we obtain normalized current waveform

Fourier coefficients of fundamental and second harmonic of are

In both subcases, we assume that voltage is a nonnegative waveform of type

For passive load, along with (i.e., implies Also, along with (i.e., implies In [14], it is proved that relation holds () for all nonnegative waveforms of type (52). In subcase (2a), relations and imply Therefore, in subcase (2a), values of impedance at the second harmonic are bounded by and Similarly, in subcase (2b), values of impedance at the second harmonic are bounded by and

In what follows, we consider maximal efficiency of PA with current waveform (48) (i.e., (50)) and voltage waveform (52) provided that and in subcase (2a) (i.e., and in subcase (2b)). The problem of finding maximal efficiency of PA with waveform pair (52) and (48) (i.e., (50)) for prescribed second-harmonic impedance can be reduced to the problem of finding voltage waveform of type (52) with maximal absolute value of for prescribed coefficients of the second harmonic [14]. Fourier series expansion of (48) (i.e., (50)) contains only cosine terms and therefore the procedure for the calculation of maximal efficiency of PA described by Algorithm in [14] can be used in both subcases. For the sake of completeness, here we provide Algorithm 3, which is analogous to Algorithm in [14]. We change the notation: is replaced with

// input: and ;

(i) choose such that and ;

(ii) calculate and ;

(iii) if then calculate and ;

else, calculate and ;

(iv) calculate efficiency ;

(v) calculate and

Algorithm 3

In subcase (2a), input of Algorithm 3 (coefficients and is given by (49). Maximal efficiency of PA with waveform pair (48) and (52), as a function of normalized second-harmonic impedance , is presented in Figure 11(a). Efficiency of 0.84 is attained in the vicinity of (corresponding to

(a)

(b)

Figure 11: Maximal efficiency of PA with (a) current (48) and voltage (52) and (b) current (50) and voltage (52) as a function of normalized second-harmonic impedance.

In subcase (2b), input of Algorithm 3 (coefficients and is given by (51). Maximal efficiency of PA with waveform pair (50) and (52), as a function of normalized second-harmonic impedance, is presented in Figure 11(b). Efficiency of 0.87 is achieved in the vicinity of (corresponding to

As we mentioned before, the optimal value of parameter is Corresponding maximal efficiency of inverse class-F PA (see Box 2) is Current waveform in subcase (2b) is close to the optimal for inverse class-F PA and corresponding peak efficiency of 0.87 is close to

6. Simulation

In this section, we provide results of nonlinear simulation of inverse class-F PA based on high performance CGH40010F GaN HEMT, manufactured by Cree Inc. (Section 6.1). Comparison of mathematical models of nonnegative waveforms and intrinsic waveforms of HEMT obtained in simulation is provided in Section 6.2. The example of simulated PA demonstrates that the theoretical results presented in this paper can serve as a useful tool during the design of high-efficiency PA.

6.1. Simulation Setup and Results

The circuit diagram of simulated inverse class-F PA is depicted in Figure 12. The proposed design of inverse class-F PA has been implemented in Advanced Design System (ADS). Computations were performed by using a harmonic balance simulator.

Figure 12: Circuit diagram of simulated inverse class-F PA.

The drain and gate biases are set to and , respectively. Frequency is set to , input power to , capacitance of dc blocking capacitor to , and inductance of choke to

In simulations of inverse class-F PA, we use large-signal model of CGH40010F provided by the manufacturer Cree Inc. This model allows access to the virtual ports located right at the active device [20]. Thus, it is possible to observe intrinsic waveforms, without the effect of package parasitics [20]. Furthermore, as a part of waveform engineering, to verify the class of operation, it is essential to analyze intrinsic waveforms at current generator plane of the device [16, 20]. Moreover, such sophisticated model allows designers to seek practical waveforms that approximate theoretically derived waveforms [16, 20].

Due to impedance transformation through the output parasitic network of the device, the impedance presented at the current generator plane (intrinsic plane) and load impedance often have considerably different values. To choose the initial values for load impedance at harmonic frequencies up to the third harmonic, we use the approximate equivalent scheme of CGH40010F output parasitic network proposed in [16] (Figure 13). The values of elements of parasitic network are [16] , , , , and

Figure 13: Approximate equivalent scheme of CGH40010F output parasitic network [16]. The intrinsic plane (current generator plane) and package plane are marked by a dotted line, and and denote intrinsic waveforms.

Initial values for load impedance are determined from the conditions that values of impedance presented at current generator plane (see Figure 13) at harmonic frequencies are , and By solving these equations numerically, we obtain initial values for load impedance at fundamental and second harmonic: and Solving the equation at the third harmonic yields Since this value cannot be realized with passive load (real part is negative), for the initial value, we choose , which transforms to at current generator plane.

The initial value of (see Figure 12) at fundamental frequency is estimated by using small-signal S-parameters provided in the data sheet of CGH40010 [21]. For CGH40010 terminated with , by using small-signal S-parameters (, we calculate that input impedance of CGH40010 is equal to . Initial value of is set to complex conjugate value; that is, The initial value of is set to low value, for which we choose For the initial value at the third harmonic, we choose

According to [16], theoretically predicted waveforms provide guidance to the shapes of practical waveforms of PA. During the simulation, we perform waveform engineering approach; that is, we search for target waveforms: the intrinsic current waveform as close as possible to truncated biharmonic waveform and intrinsic voltage waveform as close as possible to optimal voltage waveform with fundamental and second harmonic.

The final values of at fundamental, second, and third harmonic are , and The final values of at fundamental, second, and third harmonic are , and The results of simulation are intrinsic drain efficiency, drain efficiency, and output power at

6.2. Comparison of Waveforms

In this subsection, we provide the comparison of the theoretically predicted waveforms and intrinsic waveforms obtained in simulation. We also compare intrinsic current waveform obtained in simulation with three commonly used models of nonnegative current waveforms.

Simulated intrinsic waveforms and (see Figure 13) can be expressed as follows:

where denotes and denotes the order of harmonic balance simulation. After the change of variable these waveforms become

where and Simulated intrinsic waveforms expressed in form (54) are plotted in Figure 14. Their amplitudes and phases for are presented in Table 2. From this data, it can be concluded that voltage waveform has a significant second harmonic, whereas higher harmonics are small. Furthermore, an almost ideal difference between fundamental and second harmonic phases results in the flattening of the bottom of the voltage waveform.

Table 2: Frequency-domain data of simulated intrinsic waveforms.

0
27.733
0.000
0.398
0.000

1
38.369

0.503
0.000

2
11.761

0.015

3
0.147
31.830
0.181

4
0.349
84.385
0.059

5
0.647

0.041

Figure 14: Intrinsic waveforms obtained in simulation: (solid blue line) and (dashed red line).

Moreover, the data obtained in simulation are used to calculate impedance presented at the intrinsic plane at each harmonic () by using the following formula:

Values of , their magnitude , and arguments for are presented in Table 3. Impedance presented at the intrinsic plane at fundamental frequency has very small imaginary part; it is almost purely real. Furthermore, and These values show that the magnitude of impedance at the second harmonic is high, while magnitude of impedance at the third and fourth harmonic is low. Magnitude of impedance at the fifth harmonic is These values of impedance presented at intrinsic plane confirm inverse class-F mode of operation of simulated PA.

Table 3: Values of impedance presented at the intrinsic plane.

1
76.313 + j0.156
76.313
0.117

2
56.926 − j798.180
800.207

3
0.727 + j0.360
0.811
26.315

4
1.510 + j5.765
5.960
75.326

5
3.892 + j15.460
15.942
75.869

Intrinsic waveforms (54) in normalized form can be written as

Fourier coefficients of normalized waveforms and for are presented in Table 4.

Table 4: Fourier coefficients of normalized simulated intrinsic waveforms.

0
1.0000
0.0000
1.0000
0.0000

1

0.0028
1.2643
0.0000

2
0.4241
0.0051

0.0368

3
0.0045

0.0438

4
0.0012

0.0232

5

0.0199
0.0975
0.0230

Here, we provide an approximation of simulated intrinsic current waveform with biharmonic truncated waveform of type (6). The normalized intrinsic current waveform is approximated by normalized current waveform of type (6) for , and (conduction angle , which reads

Normalized intrinsic current waveforms obtained in simulation (red dashed line) and (solid black line) are presented in Figure 15. Good agreement between the shapes of the waveforms can be observed. Moreover, basic waveform parameter of simulated waveform (see Table 4; ) and basic waveform parameter of (calculated from (30)) are

respectively. Maximal values of current waveforms are also in good agreement.

Figure 15: Normalized intrinsic current waveform obtained in simulation (dashed red line) and (solid black line).

Next, we compare the efficiency predicted by mathematical model and intrinsic efficiency obtained in simulation. Basic waveform parameter of optimal truncated current waveform for (calculated from (31)) is equal to (corresponding to ). Basic waveform parameter of optimal voltage waveform with (see (34)) is Thus, maximal attainable efficiency of inverse class-F for is equal to

See also Box 2. By taking into account knee voltage of HEMT (from the results of simulations, we estimate that decreases to

and the corresponding maximal efficiency becomes

In simulation, we obtain that intrinsic drain efficiency is equal to

(see Table 4; and The difference between and appears because On the other hand, the efficiency predicted for inverse class-F PA with current waveform (57) and optimal voltage waveform, by taking into account knee voltage, is equal to

In what follows, we compare intrinsic current waveform obtained in simulation with the following three commonly used types of nonnegative waveforms:

(i)Rectangular waveform(ii)Maximally flat odd harmonic waveform(iii)Odd harmonic waveform with maximal amplitude of fundamental harmonic

All these waveforms are normalized in the sense that dc component is equal to 1.

(i) The ideal shape of current waveform for inverse class-F PA is square waveform (rectangular waveform with conduction angle It is easy to show that the basic waveform parameter of rectangular waveform with conduction angle is equal to

For and , corresponding values are and Although is close to , the shape of square waveform (dotted green line in Figure 16) and its maximal value are not in good agreement with the simulated intrinsic current waveform (dashed red line in the same figure).

Figure 16: Current waveforms: simulated (dashed red line), square (dotted green line), maximally flat odd harmonic for (solid black line) and odd harmonic waveform with maximal fundamental harmonic amplitude for (dash-dotted blue line).

(ii) Maximally flat odd harmonic waveform [6] denotes nonnegative waveform, which contains dc component, fundamental harmonic, and consecutive odd harmonics means only fundamental, means fundamental and third harmonic, etc.). Since it is an even function, it contains only cosine terms at fundamental and higher odd harmonics. Maximal value of maximally flat odd harmonic waveform is equal to 2 [6]. The basic waveform parameter of this waveform is equal to [6]

Corresponding values are , and so forth. Parameter slowly increases when increases and , [6]. Maximally flat odd harmonic waveform tends to square waveform when tends to infinity [6]. Maximally flat odd harmonic waveform for is plotted in Figure 16 (solid black line).

(iii) Odd harmonic waveform with maximal amplitude of fundamental harmonic contains dc component, fundamental harmonic, and consecutive odd harmonics. It is an even function and contains only cosine terms. Maximal value of this waveform is also equal to 2. Basic waveform parameter of odd harmonic waveform with maximal amplitude of fundamental harmonic is equal to [10]

Corresponding values are , and so forth. This waveform also tends to square waveform when tends to infinity. Parameter increases when increases and Odd harmonic waveform with maximal amplitude of fundamental harmonic for is plotted in Figure 16 (dash-dotted blue line).

The above consideration (illustrated in Figures 15 and 16) shows that truncated biharmonic current waveform can provide better approximation for the simulated intrinsic current waveform than other considered types. Additionally, in order to quantify similarities of normalized intrinsic current waveform obtained in simulation with four considered models of nonnegative waveforms, we calculate the following three commonly used quantitative indicators:

(a)Cross-correlation coefficient (denoted as XCORR in Table 5)(b)R2 statistics or so-called coefficient of determination (denoted as R2 in Table 5)(c)Mean squared error (denoted as MSE in Table 5)

In Table 5, abbreviation “truncated” denotes waveform (57), “square” denotes normalized square waveform, “max flat” denotes maximally flat odd harmonic waveform for , and “max ampl.” denotes odd harmonic waveform with maximal fundamental harmonic amplitude for The range of all three indicators is Better fit is indicated with XCORR closer to 1, R2 closer to 1, and MSE closer to 0. The values of all three indicators confirm that truncated biharmonic current waveform (57) provides better approximation of the simulated intrinsic current waveform in comparison to the other three commonly used types.

Table 5: Quantitative indicators of the four considered approximations for intrinsic current waveform. Abbreviations are described in the text.

Truncated
Square
Max flat
Max ampl.

XCORR
0.9955
0.9561
0.9737
0.9842

R2
0.9903
0.9071
0.9383
0.9637

MSE
0.0089
0.0858
0.0570
0.0336

7. Conclusion

In this paper, we consider two-parameter families of current waveforms of type (1) with or These families include both discontinuous and continuous waveforms. Under common assumptions that the third harmonic in current waveform for class-F and the second harmonic for inverse class-F are missing, the following four cases are analyzed: (i) , (ii) , (iii) , and (iv) It is shown that in each of these four cases current waveform enabling maximal efficiency (optimal waveform) is continuous for conduction angles listed in Table 1. Furthermore, closed-form expressions for parameters of optimal current waveforms and maximal efficiency of class-F (inverse class-F PA) in terms of conduction angle are provided in Box 1 (Box 2). Two case studies involving suboptimal continuous current waveforms in PA efficiency analysis are also presented. Finally, we provide comparison of the theoretical results derived in this paper with the results obtained in nonlinear simulation of inverse class-F PA based on CGH40010F HEMT.

Conflicts of Interest

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

The authors wish to thank Cree Inc. for providing large-signal models of GaN HEMTs. They also express their gratitude to Keysight Technologies Inc. for providing them with license for Advanced Design System (ADS) software. This work is supported by the Serbian Ministry of Education, Science and Technology Development as a part of Project TP32016.

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