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

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 (k = 2) or third (k = 3) harmonic. In each period, waveforms are truncated to become zero outside of a prescribed interval (so-called conduction angle).The considered families ofwaveforms include both discontinuous and continuouswaveforms. Fourier series expansion of truncatedwaveform contains an infinite number of harmonics, although a number of harmonicsmay be missing. Taking into account common assumptions that for class-F PA the third (n = 3) harmonic is missing in current waveform and for inverse class-F PA the second (n = 2) harmonic is missing in current waveform, we consider the following four cases: (i) n = k = 3, (ii) n = 3, k = 2, (iii) n = k = 2, and (iv) n = 2, k = 3.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.

In this paper, we consider a two-parameter model of periodic current waveform, defined within fundamental period as  () () = {  P ( 0() + cos  −   cos ) , || ≤   , 0, where  stands for , 0 < 2  < 360 ∘ is conduction angle,  0() and   are parameters,  P > 0 is constant, and  = 2 or  = 3.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  ≥ 1 and sinc(⋅) denotes unnormalized sinc function sinc  = (sin )/.Coefficients of fundamental harmonic ( = 1) and th harmonic ( = ) can be also obtained from (3) by using that sinc(0) = 1.Notice that change of parameter   and/or  0() 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  dc > 0,  () > 0, and  = 2 or  = 3.
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  0() = − cos   and   = 0.
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  0() to  0() = − cos   +   cos   .
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  = 2 and  = 3) is obtained via optimization of efficiency, subject to the constraint that th harmonic coefficient  () is nonpositive.A special case of biharmonic mode with   =   is also analyzed in [5,17].
The one-parameter model of discontinuous current waveform is used in [3].The authors of [3] considered case  =  = 3 in the context of class-F PA and case  =  = 2 in the context of inverse class-F PA.Current waveform proposed in [3] for class-F can be obtained from (1) by setting  = 3,  0() = − cos   ,   =   / P , and  P = (1+  / max )  .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  = 2,  0() = − cos   ,   =   −1 / P , and  P = (1 +   / max )  .In this context, parameter   −1 is obtained from the condition that the second harmonic is missing in current waveform.
In what follows, current waveforms of type (1) satisfying condition  () = 0 are denoted by  (,  =0) ().In Sections 2-4, we consider current waveform  (,  =0) () of type (1) for  ∈ {2, 3} and  ∈ {2, 3}.To be more specific, we consider the following four cases: 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  =     /2, 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  (,  =0) () that ensures maximal efficiency (optimal waveform) of class-F PA or inverse class-F PA is a waveform of type  (,  =0) () 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  = 2 or  = 3 is provided.Section 4.2 is devoted to inverse class-F PA with optimal current waveforms for  = 2 or  = 3.In Section 5, we consider continuous current waveforms of type (6) satisfying relaxed condition  () ≤ 0 for  ∈ {2, 3} and  ∈ {2, 3}, instead of  () = 0 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.

Optimal Current Waveform of Class-F and
Inverse Class-F PA  1).In this section, we also provide closed-form expressions for parameters of such waveforms.
In what follows, we first prove that optimal waveform is continuous.
After substitution of ( 12) into (8) and solving the resulting equation, we obtain parameter   of continuous waveform of type (6) that satisfies condition  () = 0, which we denote by  (  =0)cont .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   =  (  =0)cont ,  ∈ {2, 3}, and  = 3 ( = 2).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  = 2.

Mathematical Problems in Engineering
It is easy to show that continuous waveform of type (6) for  = 2 can be expressed in factored form as where The dotted line in Figure 2 corresponds to  2max .Case (ii):  = 3 and  = 2.For  = 3 and  = 2, expression (13) can be simplified to The solid line in Figure 2 1).Case (iii):  = 2 and  = 2. Substitution of  = 2 and  = 2 into (13) leads to The dashed line in Figure 2 1).
Let us now consider cases (i) and (iv), that is, cases when  = 3.
It is easy to show that continuous waveform of type (6) for  = 3 can be expressed in factored form as Let us introduce an auxiliary waveform Condition (0) ≥ 0 can be rewritten as where Also, condition (  ) ≥ 0 can be rewritten as where From ( 24), it follows that  3min < −1/3.Consideration of critical points of () leads to the conclusion that In what follows, we show that  3( 3 =0)cont > −1/3 and  3( 2 =0)cont > −1/3.Thus, both correspond to case (a), whereas cases (b) and (c) do not occur for waveforms of type ).The dotted line in Figure 4 corresponds to  3max .Case (i):  = 3 and  = 3. Substitution of  = 3 and  = 3 into (13) leads to The solid line in Figure 4 corresponds to  Case (iv):  = 2 and  = 3.For  = 2 and  = 3, expression (13) can be simplified to The dashed line in Figure 4 corresponds to According to the above consideration involving auxiliary waveform (), waveform Several examples of shapes of current waveforms of type  (=3)cont () satisfying  (=3) = 0 are presented in Figure 5. Figure 5(a) corresponds to the case  = 3 (case (i)), whereas Figure 5(b) corresponds to the case  = 2 (case (iv)).

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  (,  =0) (),  ∈ {2, 3},  ∈ {2, 3}, 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  (,  =0) () 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  (,  =0) ().Notice that, for the prescribed conduction angle, there exist only one continuous and an infinite number of discontinuous nonnegative waveforms of type  (,  =0) ().

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  ()cont () of type (6) satisfying condition  3() = 0 ( 2() = 0).(19) print "no solution found -increase the value of  0, max and try again" For continuous waveforms of type (6), parameter  0() 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  ()cont =  1()cont /  dc()cont of waveform  ()cont () can be expressed as Mathematical Problems in Engineering We denote basic waveform parameter of continuous waveform  ()cont () with   =  (  =0)cont by  (,  =0)cont .This parameter can be easily obtained from (30) by substituting (31)
These examples also confirm our analytically obtained results presented in Section 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  (, 2 =0) () for  = 2 or  = 3.In Class-F, case (i): Box 1: Closed-form expressions used for the calculation of maximal efficiency of class-F PA.
These two examples provide another verification of analytical results presented in Section 2.

Suboptimal Continuous Current Waveforms in PA Efficiency Analysis
In this section, we consider continuous current waveforms of type (6),  ∈ {2,3}, with second  2()cont or third  3()cont 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

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Inverse class-F, case (iii), Box 2: Closed-form expressions used for the calculation of maximal efficiency of inverse class-F PA.
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  = 2 satisfying  3(=2)cont < 0 (case study 1) or  2(=2)cont < 0 (case study 2) in PA efficiency analysis.
Coefficients of fundamental and third harmonic ( = 3) of optimal voltage waveform (32) for class-F PA have opposite signs.Similarly, coefficients of fundamental and second harmonic ( = 2) 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  = 3,  = 2, and 2  = 180 ∘ (class-B conduction angle) into (37) yields  2 ≥ 0. According to the above discussion, Fourier coefficient of the third harmonic is positive for  2 < 0, equal to zero for  2 =  2( 3 =0)cont = 0, and negative for  2 > 0. Therefore, small changes of parameter  2 around the optimal value  2( 3 =0)cont = 0 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  =  = 3 and 2  = 180 ∘ into (37) yields  3 ≥ 0. Fourier coefficient of the third harmonic of continuous current waveform ( 6) is nonpositive for all  3 ≥ 0 and positive for all  3 < 0. Therefore, small change of parameter  3 from the optimal value  3( 3 =0)cont = 0 may also lead to a significant decrease of efficiency of class-F PA.

Case Studies.
Here, we consider efficiency of PAs with suboptimal current waveforms of type (6) for  = 2.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  dc = 1 and  dc = 1.Under common assumptions that   behaves as short circuit and  ch behaves as open circuit at fundamental and higher harmonics, voltage and current waveforms at the load are The load impedance is equal to  1 = −( 1V − j 1V )/ 1 at fundamental harmonic, to   = −( V − j V )/  at th harmonic, and to   = 0 at th harmonic,  > 1 and  ̸ = .When load is passive (Re{ 1 } ≥ 0 and Re{  } ≥ 0), products  1V  1 and  V   are nonpositive.
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  = 2 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 According to (17), optimal value of parameter  2 for class-F PA is  2( 3 =0)cont = 0.1937.In both subcases,  2 >  2( 3 =0)cont and therefore the third-harmonic Fourier coefficients of both waveforms are negative.The value of  2 in subcase (1b) is very close to the optimal.

Case Study 2.
In this case study, we analyze the efficiency of PA with current waveforms of type (6) for  = 2 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)   According to (18), the optimal value of parameter  2 for inverse class-F PA is  2( 2 =0)cont = 0.3305.In both subcases,  2 >  2( 2 =0)cont and therefore second-harmonic Fourier coefficients of both waveforms are negative.The value of  2 in subcase (2b) is close to the optimal.
In subcase (2b), input of Algorithm 3 (coefficients  1 and  2 ) 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  2 = 37.99 (corresponding to  2 = 33.18).

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 highefficiency PA.

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.The drain and gate biases are set to  dc = 28V and   = −1.95V, respectively.Frequency is set to  0 = 900 MHz ( 0 = 2 0 ), input power to  in = 18dBm, capacitance of dc blocking capacitor to   = 1F, and inductance of choke to  ch = 1 H.
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]  ds = 1.22 pF,  = 0.55 nH,   = 0.25 pF,   = 0.1 nH, and   = 0.1 Ω.
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.

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 V DiSi and  Di (see Figure 13) can be expressed as follows: where   denotes  0  and  denotes the order of harmonic balance simulation.After the change of variable   →  −  1, these waveforms become where   =  , −  1, and   =  , −  1, .Simulated intrinsic waveforms expressed in form (54) are plotted in Figure 14.Their amplitudes and phases for  ≤ 5 are presented in Table 2. From this data, it can be concluded that voltage waveform has a significant second harmonic, whereas higher harmonics ( = 3, 4, 5) are small.Furthermore, an almost ideal difference between fundamental and second harmonic phases ( 2 −  1 ≈ 179 ∘ ) results in the flattening of the bottom of the voltage waveform.Moreover, the data obtained in simulation are used to calculate impedance presented at the intrinsic plane at each harmonic ( ≥ 1) by using the following formula: Values of ( 0 ), their magnitude ( 0 ), and arguments ( 0 ) for  ≤ 5 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, (2 0 ) > 10 ⋅ ( 0 ), (3 0 ) < ( 0 )/10 and (4 0 ) < ( 0 )/10.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 (5 0 ) ≈ ( 0 )/5.These values of impedance presented at intrinsic plane confirm inverse class-F mode of operation of simulated PA.
Intrinsic waveforms (54) in normalized form can be written as Fourier coefficients of normalized waveforms V DiSi norm () and  Di norm () for  ≤ 5 are presented in Table 4.
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 In what follows, we compare intrinsic current waveform obtained in simulation with the following three commonly used types of nonnegative waveforms:  16) and its maximal value are not in good agreement with the simulated intrinsic current waveform (dashed red line in the same figure).

Figure 8 :
Figure 8: Maximal efficiency of inverse class-F PA.Solid line corresponds to  = 2 and dashed line corresponds to  = 3.

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

Figure 13 :Figure 14 :
Figure13: 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 V DiSi and  Di denote intrinsic waveforms.

Figure 15 :
Figure 15: Normalized intrinsic current waveform obtained in simulation (dashed red line) and  mod () (solid black line).
(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 2  = 180 ∘ ).It is easy to show that the basic waveform parameter of rectangular waveform with conduction angle 2  is equal to rect  (  ) = 2 sinc   .(65)For  = 100 ∘ and   = 90 ∘ , corresponding values are  rect  (100 ∘ ) = 1.1285 and  rect  (90 ∘ ) =  sq  = 4/ ≈ 1.273.Although  sq  is close to  sim  = 1.264, the shape of square waveform (dotted green line in Figure

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

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

Table 4 :
Fourier coefficients of normalized simulated intrinsic waveforms.Normalized intrinsic current waveforms obtained in simulation (red dashed line) and  mod () (solid black line) are presented in Figure15.Good agreement between the shapes of the waveforms can be observed.Moreover, basic waveform parameter of simulated waveform (see Table4;  sim  =  1 ) and basic waveform parameter of  mod () (calculated from (30)) are Maximal values of current waveforms are also in good agreement.Next, we compare the efficiency predicted by mathematical model and intrinsic efficiency obtained in simulation.Basic waveform parameter of optimal truncated current waveform for 2  = 200 ∘ (calculated from (31)) is equal to  ,max = 1.280 (corresponding to  2 = 0.4018).Basic waveform parameter of optimal voltage waveform with  = 2 (see (34)) is  ,max = √ 2. Thus, maximal attainable efficiency of inverse class-F for 2  = 200 ∘ is equal to See also Box 2. By taking into account knee voltage of HEMT (from the results of simulations, we estimate that  knee ≈ 0.5 V),  ,max decreases to ,max  ,max ⋅ 0.98 =  max ⋅ 0.98 = 0.887.(62)Insimulation, we obtain that intrinsic drain efficiency is equal to

Table 4 ;
sim  =  1 and  sim  = − 1V).The difference between  knee max and  sim intrinsic appears because  sim  ≈ 0.9875 ,max .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