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

Authors: Anamarija Juhas; Stanisa Dautovic; Ladislav A. Novak
Journal: Mathematical Problems in Engineering
 (2017)

---

## 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(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 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 (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.


---

## Body

## 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(1)ikτ=IPA0k+cos⁡τ-Akcos⁡kτ,τ≤τp,0,τp<τ≤180°,where τ stands for ωt,0<2τp<360° is conduction angle, A0(k) and Ak are parameters, IP>0 is constant, and k=2 or k=3. This family includes both continuous and discontinuous waveforms. Waveform of type i(k)(τ) 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(2)Idck=IPτpπA0k+sinc⁡τp-Aksinc⁡kτp,(3)Ink=IPτpπ2A0ksinc⁡nτp+sinc⁡n-1τp+sinc⁡n+1τp-Aksinc⁡k-nτp+sinc⁡k+nτp,where n≥1 and sinc⁡(·) denotes unnormalized sinc function sinc⁡x=(sin⁡x)/x. Coefficients of fundamental harmonic (n=1) and kth harmonic (n=k) can be also obtained from (3) by using that sinc⁡(0)=1. Notice that change of parameter Ak and/or A0(k) 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(4)vnτ=Vdc1-γVncos⁡τ-ancos⁡nτ,where Vdc>0,γVn>0, and n=2 or n=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]):(5)iclassicτ=IPcos⁡τ-cos⁡τp,τ≤τp,0,τp≤τ≤180°.Model (5) can be obtained directly from (1) by setting A0(k)=-cos⁡τp and Ak=0.Another widely considered continuous model of current waveform is one-parameter model of type (see, e.g., [5])(6)ikcontτ=IPcos⁡τ-cos⁡τp-Akcos⁡kτ-cos⁡kτp,τ≤τp,0,τp≤τ≤180°,which can be also obtained from (1) by setting the value of parameter A0(k) to(7)A0k=-cos⁡τp+Akcos⁡kτp.Typically, continuous current waveform of type (6) is paired with voltage waveform of type (4) with n=k for biharmonic mode of PA operation [5, 17]. In [17], the value of parameter Ak (for k=2 and k=3) is obtained via optimization of efficiency, subject to the constraint that kth harmonic coefficient Ik(k) is nonpositive. A special case of biharmonic mode with Ak=ak is also analyzed in [5, 17].The two-parameter model of continuous current waveform for PA withn=k=2 has been considered in [18]. This model can be obtained from (1) by setting k=2,IP=Imaxf(h2,c),A0(k)=c/f(h2,c), and Ak=-h2.The one-parameter model of discontinuous current waveform is used in [3]. The authors of [3] considered case n=k=3 in the context of class-F PA and case n=k=2 in the context of inverse class-F PA. Current waveform proposed in [3] for class-F can be obtained from (1) by setting k=3,A0(k)=-cos⁡τp,Ak=βF/IP, and IP=(1+βF/Imax)ID. Furthermore, parameter βF 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 k=2,A0(k)=-cos⁡τp,Ak=βF-1/IP, and IP=(1+βF/Imax)ID. In this context, parameter βF-1 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,I3(k)=0 (see, e.g., [2, 3, 7]). In this case, the corresponding voltage waveform is of type (4) for n=3. Also, for inverse class-F PA, we use the common assumption that the second harmonic in current waveform is missing, I2(k)=0 (see, e.g., [2, 3]). The corresponding voltage waveform is of type (4) for n=2. In what follows, current waveforms of type (1) satisfying condition In(k)=0 are denoted by i(k,In=0)(τ). In Sections 2–4, we consider current waveform i(k,In=0)(τ) of type (1) for k∈{2,3} and n∈{2,3}. To be more specific, we consider the following four cases:(i)
n=k=3,(ii)
n=3 and k=2,(iii)
n=k=2,(iv)
n=2 and k=3.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 efficiencyη of PA can be expressed via basic waveform parameters γI and γV as η=γIγV/2, provided that parameter γI (γV) 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 i(k,In=0)(τ) that ensures maximal efficiency (optimal waveform) of class-F PA or inverse class-F PA is a waveform of type i(k,In=0)(τ) with maximal parameter γI.For all cases (i)–(iv), in Section2, 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 k=2 or k=3 is provided. Section 4.2 is devoted to inverse class-F PA with optimal current waveforms for k=2 or k=3. In Section 5, we consider continuous current waveforms of type (6) satisfying relaxed condition In(k)≤0 for k∈{2,3} and n∈{2,3}, instead of In(k)=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.


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

In this section, current waveformsi(k,In=0)(τ) of type (1) that satisfy condition In(k)=0 for n∈{2,3} and k∈{2,3} are considered. In what follows, we prove that current waveform i(k,In=0)(τ), 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 typei(k,In=0)(τ) is continuous.
Case
n
k
Conduction angle2τp
(i)
3
3
2
τ
p
≥
97.4
°
(ii)
3
2
2
τ
p
≥
96.4
°
(iii)
2
2
2
τ
p
≥
146.4
°
(iv)
2
3
2
τ
p
≥
151.1
°In what follows, we first prove that optimal waveform is continuous.As it has been mentioned earlier, Fourier coefficient ofnth harmonic of current waveform of type (1) can be expressed in form (3). Furthermore, condition In(k)=0 implies that parameter Ak is equal to(8)AkIn=0=2A0ksinc⁡nτp+sinc⁡n-1τp+sinc⁡n+1τpsinc⁡k-nτp+sinc⁡k+nτp.From (2) and (3), it follows that basic waveform parameter γI(k,In=0)=I1(k,In=0)/Idc(k,In=0) of waveform i(k,In=0)(τ) reads(9)γIk,In=0=2A0ksinc⁡τp+1+sinc⁡2τp-AkIn=0sinc⁡k-1τp+sinc⁡k+1τpA0k+sinc⁡τp-AkIn=0sinc⁡kτp,where Ak(In=0) is given by (8). For the prescribed conduction angle, parameter Ak(In=0) is a function of A0(k) only, which implies that γI(k,In=0) is also a function of A0(k) only. First derivative of γI(k,In=0) with respect to A0(k) can be expressed in the following form:(10)dγIk,In=0dA0k=IPτpπIdck,In=02fn,kτp,where f(n,k)(τp) is a function of τp:(11)fn,kτp=sinc⁡k-nτp+sinc⁡k+nτp-1×sinc⁡k-1τp+sinc⁡k+1τp-2sinc⁡τpsinc⁡kτp×sinc⁡n-1τp+sinc⁡n+1τp-2sinc⁡τpsinc⁡nτp-sinc⁡k-nτp+sinc⁡k+nτp-2sinc⁡kτpsinc⁡nτp×1+sinc⁡2τp-2sinc2⁡τp.It is easy to see that the sign of dγI(k,In=0)/dA0(k) depends on the sign of f(n,k)(τp) only. Since f(n,k)(τp) is negative for n∈{2,3},k∈{2,3}, and 2τp∈(0,360°) (see Figure 1), it immediately follows that γI(k,In=0) decreases when A0(k) increases. Consequently, the highest value of γI(k,In=0) (and the highest efficiency) is attained for minimal value of A0(k).Figure 1
Functionsf(2,2)(τp) (solid line), f(3,3)(τp) (dashed line), and f(2,3)(τp)=f(3,2)(τp) (dotted line).In order to be nonnegative, waveformi(k,In=0)(τ) should satisfy condition i(k,In=0)(±τp)≥0, that is, A0(k)≥-cos⁡τp+AkIn=0cos⁡kτp. For minimal value of A0(k) (see the above consideration), this condition reduces to equality. Therefore, A0(k) of optimal current waveform is equal to(12)A0k=-cos⁡τp+AkIn=0cos⁡kτp,which further implies that optimal waveform of type i(k,In=0)(τ) is continuous; that is, it is of type (6).After substitution of (12) into (8) and solving the resulting equation, we obtain parameter Ak of continuous waveform of type (6) that satisfies condition In(k)=0, which we denote by Ak(In=0)cont. This parameter is a function of conduction angle only:(13)AkIn=0cont=sinc⁡n-1τp-sinc⁡n+1τpksinc⁡k-nτp-sinc⁡k+nτp.Consequently, optimal current waveform of class-F PA (inverse class-F PA) is a continuous waveform of type (6) with parameter Ak=Ak(In=0)cont,k∈{2,3}, and n=3(n=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 whenk=2.It is easy to show that continuous waveform of type (6) for k=2 can be expressed in factored form as(14)ik=2contτ=IPcos⁡τ-cos⁡τp1-2A2cos⁡τ+cos⁡τp,τ≤τp,0,τp≤τ≤180°.Because IP>0 and cos⁡τ-cos⁡τp>0 for |τ|<τp, waveform of type ik=2contτ is nonnegative if and only if 1-2A2(cos⁡τ+cos⁡τp)≥0 for τ<τp. By inspection of the first derivative of 1-2A2(cos⁡τ+cos⁡τp), it follows that it has only one critical point in interval τ<τp. Since this critical point is at τ=0, it immediately follows that ik=2contτ is nonnegative provided that 1-2A2(1+cos⁡τp)≥0. This condition can be rewritten as (15)A2≤A2max,where(16)A2max=121+cos⁡τp.The dotted line in Figure 2 corresponds to A2max.Figure 2
ParameterA2: A2max (dotted line), A2(I3=0)cont (solid line), and A2(I2=0)cont (dashed line).Case (ii):n=3 and k=2. For n=3 and k=2, expression (13) can be simplified to(17)A2I3=0cont=5cos⁡τp41+4cos2⁡τp.The solid line in Figure 2 corresponds to A2(I3=0)cont. It is easy to see that sgnA2(I3=0)cont=sgncos⁡τp. According to the above consideration, waveform i(k=2)cont(τ) with A2=A2(I3=0)cont is nonnegative if A2(I3=0)cont≤A2max. Substituting (17) and (16) into A2(I3=0)cont≤A2max and solving the resulting relation for τp lead to τp≥arccos(2/3)≈48.2°. Consequently, waveform i(k=2)cont(τ) with A2=A2(I3=0)cont is nonnegative when 2τp≥96.4° (third row in Table 1).Case (iii):n=2 and k=2. Substitution of n=2 and k=2 into (13) leads to(18)A2I2=0cont=8sin2⁡τp34τp-sin⁡4τp.The dashed line in Figure 2 corresponds to A2(I2=0)cont. It is easy to show that A2(I2=0)cont>0 for all τp(0<τp<180°). According to the above discussion, waveform i(k=2)cont(τ) with A2=A2(I2=0)cont is nonnegative if A2(I2=0)cont≤A2max. Substituting (18) and (16) into A2(I2=0)cont≤A2max and solving the resulting relation for τp, we obtain 2τp≥146.4°. Consequently, waveform of type i(k=2)cont(τ) with A2=A2(I2=0)cont is nonnegative when 2τp≥146.4° (fourth row in Table 1).A number of shapes of current waveforms of typei(k=2)cont(τ) satisfying In(k=2)=0 are presented in Figure 3. Figure 3(a) corresponds to the case n=3 (case (ii)), whereas Figure 3(b) corresponds to the case n=2 (case (iii)).Figure 3
Shapes of current waveform of typei(k=2)cont(τ) with (a) A2=A2(I3=0)cont and (b) A2=A2(I2=0)cont.
(a)
(b)Let us now consider cases (i) and (iv), that is, cases whenk=3.It is easy to show that continuous waveform of type (6) for k=3 can be expressed in factored form as(19)ik=3contτ=IPcos⁡τ-cos⁡τp1+3A31-cos2⁡τp-A32cos⁡τ+cos⁡τp2,τ≤τp,0,τp≤τ≤180°.Let us introduce an auxiliary waveform(20)gτ=1+3A31-cos2⁡τp-A32cos⁡τ+cos⁡τp2.Because IP>0 and cos⁡τ-cos⁡τp>0 for |τ|<τp, waveform of type i(k=3)cont(τ) is nonnegative if and only if g(τ)≥0 for τ<τp. First derivative of g(τ) is equal to g′(τ)=2A32cos⁡τ+cos⁡τpsin⁡τ. Therefore, critical points of g(τ) on interval |τ|<τp are τ=0 and τ=τc satisfying 2cos⁡τc+cos⁡τp=0. Notice that relations 2cos⁡τc+cos⁡τp=0 and |τc|<τp imply cos⁡τp<0, that is, τp>90°. Values of g(τ) at critical points are g(0)=1-A3(1+2cos⁡τp)2 and g(τc)=1+3A3(1-cos2⁡τp). Condition g(0)≥0 can be rewritten as(21)A3≤A3max,where(22)A3max=11+2cos⁡τp2.Also, condition g(τc)≥0 can be rewritten as(23)A3min≤A3,where(24)A3min=-131-cos2⁡τp.From (24), it follows that A3min<-1/3. Consideration of critical points of g(τ) leads to the conclusion that i(k=3)cont(τ) is nonnegative if (a)
-1/3≤A3≤A3max,(b)
A3<-1/3 and τp≤90°,(c)
A3min≤A3<-1/3 and τp>90°.In what follows, we show that A3(I3=0)cont>-1/3 and A3(I2=0)cont>-1/3. Thus, both correspond to case (a), whereas cases (b) and (c) do not occur for waveforms of type i(k=3)cont(τ) with A3=A3(I3=0)cont or A3=A3(I2=0)cont. Consequently, waveform i(k=3)cont(τ) with A3=A3(I3=0)cont(A3=A3(I2=0)cont) is nonnegative if A3(I3=0)cont<A3max(A3(I2=0)cont<A3max). The dotted line in Figure 4 corresponds to A3max.Figure 4
ParameterA3: A3max (dotted line), A3(I3=0)cont (solid line), and A3(I2=0)cont (dashed line).Case (i):n=3 and k=3. Substitution of n=3 and k=3 into (13) leads to(25)A3I3=0cont=4sin3⁡τpcos⁡τp6τp-sin⁡6τp.The solid line in Figure 4 corresponds to A3(I3=0)cont. It is easy to see that sgnA3(I3=0)=sgncos⁡τp and A3(I3=0)cont>-1/3. According to the above discussion concerning auxiliary waveform g(τ), waveform i(k=3)cont(τ) with A3=A3(I3=0)cont is nonnegative if A3(I3=0)cont≤A3max. Substituting (25) and (22) into A3(I3=0)cont≤A3max and solving the resulting relation for τp, we obtain 2τp≥97.4°. Consequently, waveform of type i(k=3)cont(τ) with A3=A3(I3=0)cont is nonnegative when 2τp≥97.4° (second row in Table 1).Case (iv):n=2 and k=3. For n=2 and k=3, expression (13) can be simplified to(26)A3I2=0cont=591+4cos2⁡τp.The dashed line in Figure 4 corresponds to A3(I2=0)cont. From (26), it immediately follows that A3(I2=0)cont>1/9. According to the above consideration involving auxiliary waveform g(τ), waveform i(k=3)cont(τ) with A3=A3(I2=0)cont is nonnegative if A3(I2=0)cont≤A3max. Substituting (26) and (22) into A3(I2=0)cont≤A3max and solving the resulting relation for τp yield τp≥arccos(1/4)≈75.52°. Consequently, waveform of type i(k=3)cont(τ) with A3=A3(I2=0)cont is nonnegative when 2τp≥151.1° (fifth row in Table 1).Several examples of shapes of current waveforms of typei(k=3)cont(τ) satisfying In(k=3)=0 are presented in Figure 5. Figure 5(a) corresponds to the case n=3 (case (i)), whereas Figure 5(b) corresponds to the case n=2 (case (iv)).Figure 5
Shapes of current waveform of typei(k=3)cont(τ) with (a) A3=A3(I3=0)cont and (b) A3=A3(I2=0)cont.
(a)
(b)


## 3. Verification of the Analytical Results

In this section, an independent numerical verification of the analytical results derived in Section2 is presented. The values of the parameters of the optimal waveform of type ik,In=0τ,n∈2,3,k∈2,3, obtained numerically are in full agreement with the values of the parameters calculated from the closed-form expressions derived in Section 2.Algorithm1 provides the procedure for numerical calculation of the parameters of the optimal current waveform of type i(k,In=0)(τ) 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 i(k,In=0)(τ). Notice that, for the prescribed conduction angle, there exist only one continuous and an infinite number of discontinuous nonnegative waveforms of type i(k,In=0)(τ).Algorithm 1
Input:
n,k // (n,k)∈{(3,3),(3,2),(2,2),(2,3)}
IP // IP>0, e.g. IP=1 (the value of IP has no influence on the output results)
A0,min,  A0,max,qmax // A0k1⋯qmax∈A0,min,A0,max, e.g. A0,min=-1,A0,max=1.5
τp // half of the conduction angle
rmax // τ(1⋯rmax)∈[0,τp] (current waveform is an even function of τ)
Output:
γI(k,In=0)∗,  A0(k)∗,Ak(In=0)∗ // parameters of optimal waveform of type i(k,In=0)(τ)
(
1
)  begin
(
2
) s←0;A0,step←(A0,max-A0,min)/(qmax-1);  τstep←τp/(rmax-1);
(
3
) for  q=1  to  qmax  do
(
4
)   A0(k)←A0,min+(q-1)A0,step;
(
5
)   AkIn=0←2A0ksinc⁡nτp+sinc⁡n-1τp+sinc⁡n+1τp/sinc⁡k-nτp+sinc⁡k+nτp; // expression (8)
(
6
)   Idc←0;  I1←0;
(
7
)   for  r=1  to  rmax  do
(
8
)    τ←(r-1)τstep;
(
9
)    ik,In=0(r)←IP(A0k+cos⁡τ-AkIn=0cos⁡kτ);
(
10
)    Idc←Idc+(τstep/π)ik,In=0(r);I1←I1+2τstep/πik,In=0rcos⁡τ;
(
11
)   end for  //  end r-loop
(
12
)   imin←minr⁡ik,In=01⋯rmax;
(
13
)   if  imin≥0  do
(
14
)    s←s+1;γI(k,In=0)(s)←I1/Idc;  A0(k)_temp(s)←A0(k);  Ak(In=0)_temp(s)←Ak(In=0);
(
15
)   end if
(
16
) end for  // end q-loop
(
17
) smax←s;
(
18
) if  smax=0  do
(
19
)   print   “no solution found – increase the value of A0,max and try again”
(
20
) else
(
21
)   γI(k,In=0)∗←maxs⁡(γI(k,In=0)(1⋯smax));
(
22
)   find s∗ such that γI(k,In=0)(s∗)=γI(k,In=0)∗; // find index s∗ of the maximum element of γI(k,In=0)
(
23
)   A0(k)∗←A0(k)_temp(s∗);  Ak(In=0)∗←Ak(In=0)_temp(s∗);
(
24
) end if
(
25
)  end.A short description of Algorithm1 is as follows: in line 2, we initialize the value of counter s, and we choose step size A0,step for A0k∈A0,min,A0,max and step size τstep for τ∈0,τp (because current waveform is an even function); in lines 3–5, we calculate Ak(In=0) for each A0k∈A0,min,A0,max; in lines 6–11, we calculate i(k,In=0)(τ), dc component, and amplitude of fundamental harmonic of i(k,In=0)(τ) for each pair (A0(k),Ak(In=0)); in line 12, we find the minimum value of i(k,In=0)(τ) for each pair (A0(k),Ak(In=0)); in lines 13–16, we calculate γI(k,In=0) for each pair (A0(k),Ak(In=0)) provided that corresponding i(k,In=0)(τ) is nonnegative; in lines 17–24, we find γI(k,In=0)∗ as the maximum value of γI(k,In=0) and corresponding pair (A0(k)∗,Ak(In=0)∗). Parameters of optimal waveform of type i(k,In=0)(τ) are A0(k)∗,Ak(In=0)∗, and γI(k,In=0)∗.It is obvious that the numerical accuracy of the values of the parameters obtained by Algorithm1 depends on chosen step size τstep and step size A0,step. By numerical investigation, we find that τstep≤τp/1000 and A0,step≤0.001 provide numerical stability of the three most significant decimal digits.For example, forn=3 and k=2, numerical values of parameters A2(I3=0)∗,A0(k=2)∗ and γI(k=2,I3=0)∗ of optimal waveforms are calculated for A0(k)∈[-1,1.5],A0,step=0.001,50°≤τp≤180°, and τstep=τp/1000. On the other hand, values of parameters A2(I3=0)cont,A0(k=2)cont, and γI(k=2,I3=0)cont are calculated from the closed-form expressions as follows:(a)
A2(I3=0)cont from (17),(b)
A0(k=2)cont=-cos⁡τp+A2I3=0contcos⁡2τp,(c)
substitution ofA0(k)=A0(k=2)cont and Ak(In=0)=A2(I3=0)cont into (9) yields γI(k=2,I3=0)cont.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 withn=3 and k=2. Numerical results: A0(k=2)∗ (red circle dots), A2(I3=0)∗ (blue square dots), and γI(k=2,I3=0)∗ (black triangle dots). Analytical results: A0(k=2)cont (red dotted line), A2(I3=0)cont (blue dashed line), and γI(k=2,I3=0)cont (black solid line).For all remaining pairsn,k∈3,3,2,2,2,3, 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.Efficiencyη of PA can be expressed via basic waveform parameters γI and γV of current and voltage waveforms (see, e.g., [1]) as follows:(27)η=γIγV2.As it has been shown in Section2, optimal current waveform of type (1) for class-F (inverse class-F) PA is a continuous waveform i(k)cont(τ) of type (6) satisfying condition I3(k)=0 (I2(k)=0).For continuous waveforms of type (6), parameter A0(k) 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):(28)Idckcont=IPτpπsinc⁡τp-cos⁡τp-Aksinc⁡kτp-cos⁡kτp,(29)I1kcont=IPτpπ1-sinc⁡2τp-kAksinc⁡k-1τp-sinc⁡k+1τp.Therefore, basic waveform parameter γI(k)cont=I1(k)cont/Idc(k)cont of waveform i(k)cont(τ) can be expressed as(30)γIkcont=1-sinc⁡2τp-kAksinc⁡k-1τp-sinc⁡k+1τpsinc⁡τp-cos⁡τp-Aksinc⁡kτp-cos⁡kτp.We denote basic waveform parameter of continuous waveform i(k)cont(τ) with Ak=Ak(In=0)cont by γI(k,In=0)cont. This parameter can be easily obtained from (30) by substituting Ak=AkIn=0cont,(31)γIk,In=0cont=1-sinc⁡2τp-kAkIn=0contsinc⁡k-1τp-sinc⁡k+1τpsinc⁡τp-cos⁡τp-AkIn=0contsinc⁡kτp-cos⁡kτp.


### 4.1. Maximal Efficiency of Class-F PA

In this subsection, we provide maximal efficiency of class-F PA with current waveform of typei(k,I3=0)(τ) for k=2 or k=3. In both cases, we assume that voltage waveform is an optimal waveform of type (4) with n=3 which reads (see, e.g., [1])(32)vn=3τ=Vdc1-23cos⁡τ+39cos⁡3τ.Basic waveform parameter γV of v(n=3)(τ) is γV(n=3)=2/3.According to the results of Section2, 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 Ak=Ak(I3=0)cont. For k=2, parameter A2(I3=0)cont is given by (17), whereas for k=3 parameter A3(I3=0)cont is given by (25). Solid lines in Figures 2 and 4 correspond to parameters A2(I3=0)cont and A3(I3=0)cont, respectively.A number of shapes of optimal current waveforms of class-F PA are presented in Figures3(a) and 5(a). Figure 5(a) corresponds to continuous waveform i(k=3)cont(τ) with A3=A3(I3=0)cont (case (i): n=k=3), whereas Figure 3(a) corresponds to continuous waveform i(k=2)cont(τ) with A2=A2(I3=0)cont (case (ii): n=3 and k=2).According to (27), efficiency of class-F PA with voltage waveform (32) and current waveform of type i(k)cont(τ) with Ak=Ak(I3=0)cont for k∈{2,3} is equal to(33)ηFk=13γIk,I3=0cont.Parameter γI(k=2,I3=0)cont can be obtained from (31) by substituting n=3,k=2, and A2(I3=0)cont from (25). Parameter γI(k=3,I3=0)cont can be obtained from (31) by substituting n=3,k=3, and A3(I3=0)cont from (17). In Box 1, we provide a brief overview of the calculation of maximal efficiency of class-F PA.Box 1:Closed-form expressions used for the calculation of maximal efficiency of class-F PA.
Class-F, case (i):n=k=3,2τp≥97.4°
A
3
(
I
3
=
0
)
c
o
n
t
=
4
si
n
3
⁡
τ
p
cos
⁡
τ
p
6
τ
p
-
sin
⁡
6
τ
p
η
F
k
=
3
=
1
3
⋅
1
-
sinc
⁡
2
τ
p
-
3
A
3
I
3
=
0
c
o
n
t
sinc
⁡
2
τ
p
-
sinc
⁡
4
τ
p
sinc
⁡
τ
p
-
cos
⁡
τ
p
-
A
3
I
3
=
0
c
o
n
t
sinc
⁡
3
τ
p
-
cos
⁡
3
τ
p
Class-F, case (ii):n=3,k=2,2τp≥96.4°
A
2
I
3
=
0
c
o
n
t
=
5
cos
⁡
τ
p
4
1
+
4
cos
2
⁡
τ
p
η
F
k
=
2
=
1
3
⋅
1
-
sinc
⁡
2
τ
p
-
2
A
2
I
3
=
0
c
o
n
t
sinc
⁡
τ
p
-
sinc
⁡
3
τ
p
sinc
⁡
τ
p
-
cos
⁡
τ
p
-
A
2
I
3
=
0
c
o
n
t
sinc
⁡
2
τ
p
-
cos
⁡
2
τ
pMaximal efficiency of class-F PA is presented in Figure7. The solid line corresponds to k=3 (case (i)), whereas the dashed line corresponds to k=2 (case (ii)). For prescribed k, 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 tok=3 and dashed line corresponds to k=2.For conduction angle2τp=180°  (so-called class-B conduction angle), from Box 1, we obtain A2(I3=0)cont=0 and A3(I3=0)cont=0. Thus, for 2τp=180°, optimal waveform is equal to iclassic(τ) (see (5)). Furthermore, we obtain ηF=3π/6, which is equal to the maximum efficiency η=(π/6)cot(π/6) of class-F PA with voltage waveform (32) [10].For conduction angle2τp=126° and n=k=3, from Box 1, we obtain A3(I3=0)cont=0.2043 and ηF(k=3)=0.9636, which are in agreement with the results presented in [17] (see also [5]).These examples also confirm our analytically obtained results presented in Section2.


### 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 typei(k,I2=0)(τ) for k=2 or k=3. In both cases, we assume that voltage waveform is an optimal waveform of type (4) with n=2 which reads (see, e.g., [1])(34)vn=2τ=Vdc1-2cos⁡τ+12cos⁡2τ.Basic waveform parameter of v(n=2)(τ) is γV(n=2)=2.According to the results of Section2, optimal current waveform, which in pair with voltage waveform (34) provides maximal efficiency of inverse class-F PA, is a continuous waveform of type i(k)cont(τ) with parameter Ak=Ak(I2=0)cont. For k=2, parameter A2(I2=0)cont is given by (18), whereas for k=3 parameter A3(I2=0)cont is given by (26). The dashed lines in Figures 2 and 4 correspond to parameters A2(I2=0)cont and A3(I2=0)cont, respectively.Several examples of shapes of current waveforms that provide maximal efficiency of inverse class-F PA are presented in Figures3(b) and 5(b). Figure 3(b) corresponds to continuous waveform i(k=2)cont(τ) with A2=A2(I2=0)cont (case (iii): n=k=2), whereas Figure 5(b) corresponds to continuous waveform i(k=3)cont(τ) with A3=A3(I2=0)cont (case (iv): n=2 and k=3).According to (27), efficiency of inverse class-F PA with voltage waveform (34) and current waveform of type i(k)cont(τ) with Ak=Ak(I2=0)cont for k∈{2,3} is equal to (35)ηIFk=12γIk,I2=0cont.Parameter γI(k=2,I2=0)cont can be obtained from (31) by substituting n=2,k=2, and A2(I2=0)cont from (18). Parameter γI(k=3,I2=0)cont can be obtained from (31) by substituting n=2,k=3, and A3(I2=0)cont from (26). In Box 2, we provide a brief overview of the calculation of maximal efficiency of inverse class-F PA.Box 2:Closed-form expressions used for the calculation of maximal efficiency of inverse class-F PA.
Inverse class-F, case (iii),n=k=2,2τp≥146.4°
A
2
I
2
=
0
c
o
n
t
=
8
sin
3
⁡
τ
p
3
4
τ
p
-
sin
⁡
4
τ
p
η
I
F
k
=
2
=
1
2
⋅
1
-
sinc
⁡
2
τ
p
-
2
A
2
I
2
=
0
c
o
n
t
sinc
⁡
τ
p
-
sinc
⁡
3
τ
p
sinc
⁡
τ
p
-
cos
⁡
τ
p
-
A
2
I
2
=
0
c
o
n
t
sinc
⁡
2
τ
p
-
cos
⁡
2
τ
p
Inverse class-F, case (iv),n=2,k=3,2τp≥151.1°
A
3
I
2
=
0
c
o
n
t
=
5
9
1
+
4
cos
2
⁡
τ
p
η
I
F
k
=
3
=
1
2
⋅
1
-
sinc
⁡
2
τ
p
-
3
A
3
I
2
=
0
c
o
n
t
sinc
⁡
2
τ
p
-
sinc
⁡
4
τ
p
sinc
⁡
τ
p
-
cos
⁡
τ
p
-
A
3
I
2
=
0
c
o
n
t
sinc
⁡
3
τ
p
-
cos
⁡
3
τ
pMaximal efficiency of inverse class-F PA is presented in Figure8. The solid line corresponds to k=2 (case (iii)), whereas the dashed line corresponds to k=3 (case (iv)). For prescribed k, 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 tok=2 and dashed line corresponds to k=3.For conduction angle2τp=160° and k=n=2, from Box 2, we obtain A2(I2=0)=0.4090 and ηIF(k=2)=0.9560, which are in agreement with the results presented in [17] (see also [5]).Also, for conduction angle2τp=246° and k=n=2, we obtain A2(I2=0)=0.2006 and ηIF(k=2)=0.8407, which are in agreement with the results presented in [18].These two examples provide another verification of analytical results presented in Section2.


## 4.1. Maximal Efficiency of Class-F PA

In this subsection, we provide maximal efficiency of class-F PA with current waveform of typei(k,I3=0)(τ) for k=2 or k=3. In both cases, we assume that voltage waveform is an optimal waveform of type (4) with n=3 which reads (see, e.g., [1])(32)vn=3τ=Vdc1-23cos⁡τ+39cos⁡3τ.Basic waveform parameter γV of v(n=3)(τ) is γV(n=3)=2/3.According to the results of Section2, 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 Ak=Ak(I3=0)cont. For k=2, parameter A2(I3=0)cont is given by (17), whereas for k=3 parameter A3(I3=0)cont is given by (25). Solid lines in Figures 2 and 4 correspond to parameters A2(I3=0)cont and A3(I3=0)cont, respectively.A number of shapes of optimal current waveforms of class-F PA are presented in Figures3(a) and 5(a). Figure 5(a) corresponds to continuous waveform i(k=3)cont(τ) with A3=A3(I3=0)cont (case (i): n=k=3), whereas Figure 3(a) corresponds to continuous waveform i(k=2)cont(τ) with A2=A2(I3=0)cont (case (ii): n=3 and k=2).According to (27), efficiency of class-F PA with voltage waveform (32) and current waveform of type i(k)cont(τ) with Ak=Ak(I3=0)cont for k∈{2,3} is equal to(33)ηFk=13γIk,I3=0cont.Parameter γI(k=2,I3=0)cont can be obtained from (31) by substituting n=3,k=2, and A2(I3=0)cont from (25). Parameter γI(k=3,I3=0)cont can be obtained from (31) by substituting n=3,k=3, and A3(I3=0)cont from (17). In Box 1, we provide a brief overview of the calculation of maximal efficiency of class-F PA.Box 1:Closed-form expressions used for the calculation of maximal efficiency of class-F PA.
Class-F, case (i):n=k=3,2τp≥97.4°
A
3
(
I
3
=
0
)
c
o
n
t
=
4
si
n
3
⁡
τ
p
cos
⁡
τ
p
6
τ
p
-
sin
⁡
6
τ
p
η
F
k
=
3
=
1
3
⋅
1
-
sinc
⁡
2
τ
p
-
3
A
3
I
3
=
0
c
o
n
t
sinc
⁡
2
τ
p
-
sinc
⁡
4
τ
p
sinc
⁡
τ
p
-
cos
⁡
τ
p
-
A
3
I
3
=
0
c
o
n
t
sinc
⁡
3
τ
p
-
cos
⁡
3
τ
p
Class-F, case (ii):n=3,k=2,2τp≥96.4°
A
2
I
3
=
0
c
o
n
t
=
5
cos
⁡
τ
p
4
1
+
4
cos
2
⁡
τ
p
η
F
k
=
2
=
1
3
⋅
1
-
sinc
⁡
2
τ
p
-
2
A
2
I
3
=
0
c
o
n
t
sinc
⁡
τ
p
-
sinc
⁡
3
τ
p
sinc
⁡
τ
p
-
cos
⁡
τ
p
-
A
2
I
3
=
0
c
o
n
t
sinc
⁡
2
τ
p
-
cos
⁡
2
τ
pMaximal efficiency of class-F PA is presented in Figure7. The solid line corresponds to k=3 (case (i)), whereas the dashed line corresponds to k=2 (case (ii)). For prescribed k, 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 tok=3 and dashed line corresponds to k=2.For conduction angle2τp=180°  (so-called class-B conduction angle), from Box 1, we obtain A2(I3=0)cont=0 and A3(I3=0)cont=0. Thus, for 2τp=180°, optimal waveform is equal to iclassic(τ) (see (5)). Furthermore, we obtain ηF=3π/6, which is equal to the maximum efficiency η=(π/6)cot(π/6) of class-F PA with voltage waveform (32) [10].For conduction angle2τp=126° and n=k=3, from Box 1, we obtain A3(I3=0)cont=0.2043 and ηF(k=3)=0.9636, which are in agreement with the results presented in [17] (see also [5]).These examples also confirm our analytically obtained results presented in Section2.


## 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 typei(k,I2=0)(τ) for k=2 or k=3. In both cases, we assume that voltage waveform is an optimal waveform of type (4) with n=2 which reads (see, e.g., [1])(34)vn=2τ=Vdc1-2cos⁡τ+12cos⁡2τ.Basic waveform parameter of v(n=2)(τ) is γV(n=2)=2.According to the results of Section2, optimal current waveform, which in pair with voltage waveform (34) provides maximal efficiency of inverse class-F PA, is a continuous waveform of type i(k)cont(τ) with parameter Ak=Ak(I2=0)cont. For k=2, parameter A2(I2=0)cont is given by (18), whereas for k=3 parameter A3(I2=0)cont is given by (26). The dashed lines in Figures 2 and 4 correspond to parameters A2(I2=0)cont and A3(I2=0)cont, respectively.Several examples of shapes of current waveforms that provide maximal efficiency of inverse class-F PA are presented in Figures3(b) and 5(b). Figure 3(b) corresponds to continuous waveform i(k=2)cont(τ) with A2=A2(I2=0)cont (case (iii): n=k=2), whereas Figure 5(b) corresponds to continuous waveform i(k=3)cont(τ) with A3=A3(I2=0)cont (case (iv): n=2 and k=3).According to (27), efficiency of inverse class-F PA with voltage waveform (34) and current waveform of type i(k)cont(τ) with Ak=Ak(I2=0)cont for k∈{2,3} is equal to (35)ηIFk=12γIk,I2=0cont.Parameter γI(k=2,I2=0)cont can be obtained from (31) by substituting n=2,k=2, and A2(I2=0)cont from (18). Parameter γI(k=3,I2=0)cont can be obtained from (31) by substituting n=2,k=3, and A3(I2=0)cont from (26). In Box 2, we provide a brief overview of the calculation of maximal efficiency of inverse class-F PA.Box 2:Closed-form expressions used for the calculation of maximal efficiency of inverse class-F PA.
Inverse class-F, case (iii),n=k=2,2τp≥146.4°
A
2
I
2
=
0
c
o
n
t
=
8
sin
3
⁡
τ
p
3
4
τ
p
-
sin
⁡
4
τ
p
η
I
F
k
=
2
=
1
2
⋅
1
-
sinc
⁡
2
τ
p
-
2
A
2
I
2
=
0
c
o
n
t
sinc
⁡
τ
p
-
sinc
⁡
3
τ
p
sinc
⁡
τ
p
-
cos
⁡
τ
p
-
A
2
I
2
=
0
c
o
n
t
sinc
⁡
2
τ
p
-
cos
⁡
2
τ
p
Inverse class-F, case (iv),n=2,k=3,2τp≥151.1°
A
3
I
2
=
0
c
o
n
t
=
5
9
1
+
4
cos
2
⁡
τ
p
η
I
F
k
=
3
=
1
2
⋅
1
-
sinc
⁡
2
τ
p
-
3
A
3
I
2
=
0
c
o
n
t
sinc
⁡
2
τ
p
-
sinc
⁡
4
τ
p
sinc
⁡
τ
p
-
cos
⁡
τ
p
-
A
3
I
2
=
0
c
o
n
t
sinc
⁡
3
τ
p
-
cos
⁡
3
τ
pMaximal efficiency of inverse class-F PA is presented in Figure8. The solid line corresponds to k=2 (case (iii)), whereas the dashed line corresponds to k=3 (case (iv)). For prescribed k, 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 tok=2 and dashed line corresponds to k=3.For conduction angle2τp=160° and k=n=2, from Box 2, we obtain A2(I2=0)=0.4090 and ηIF(k=2)=0.9560, which are in agreement with the results presented in [17] (see also [5]).Also, for conduction angle2τp=246° and k=n=2, we obtain A2(I2=0)=0.2006 and ηIF(k=2)=0.8407, which are in agreement with the results presented in [18].These two examples provide another verification of analytical results presented in Section2.


## 5. Suboptimal Continuous Current Waveforms in PA Efficiency Analysis

In this section, we consider continuous current waveforms of type (6), k∈{2,3}, with second I2(k)cont or third I3(k)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 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 k=2 satisfying I3(k=2)cont<0 (case study 1) or I2(k=2)cont<0 (case study 2) in PA efficiency analysis.


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

Fourier coefficient of thenth harmonic of current waveform of type (6) can be obtained by substituting (7) into (3):(36)Inkcont=IPτpnπsinc⁡n-1τp-sinc⁡n+1τp-kAksinc⁡k-nτp-sinc⁡k+nτp.It is easy to show that sinc⁡(k-n)τp-sinc⁡(k+n)τp>0, for 2τp∈0,360°,k∈2,3, and n∈2,3. Consequently, Fourier coefficient of the nth harmonic is nonpositive when(37)Ak≥sinc⁡n-1τp-sinc⁡n+1τpksinc⁡k-nτp-sinc⁡k+nτp.The value of Ak for which (37) becomes equality is Ak(In=0)cont (see (13)). Thus, (37) can be rewritten as(38)Ak≥AkIn=0cont.As it has been shown in Section 2, expression for Ak(In=0)cont can be reduced to: (i)
expression (25) for (n,k)=(3,3); parameter A3(I3=0)cont is plotted with solid line in Figure 4,(ii)
expression (17) for (n,k)=(3,2); parameter A2(I3=0)cont is plotted with solid line in Figure 2,(iii)
expression (18) for (n,k)=(2,2); parameter A2(I2=0)cont is plotted with dashed line in Figure 2,(iv)
expression (26) for (n,k)=(2,3); parameter A3(I2=0)cont is plotted with dashed line in Figure 4.Coefficients of fundamental and third harmonic(n=3) of optimal voltage waveform (32) for class-F PA have opposite signs. Similarly, coefficients of fundamental and second harmonic (n=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 ofn=3,k=2, and 2τp=180° (class-B conduction angle) into (37) yields A2≥0. According to the above discussion, Fourier coefficient of the third harmonic is positive for A2<0, equal to zero for A2=A2(I3=0)cont=0, and negative for A2>0. Therefore, small changes of parameter A2 around the optimal value A2(I3=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 ofn=k=3 and 2τp=180° into (37) yields A3≥0. Fourier coefficient of the third harmonic of continuous current waveform (6) is nonpositive for all A3≥0 and positive for all A3<0. Therefore, small change of parameter A3 from the optimal value A3(I3=0)cont=0 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 k=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 Figure9. We assume that voltage and current waveforms at the output port of transistor are(39)vτ=1+a1vcos⁡τ+b1vsin⁡τ+anvcos⁡nτ+bnvsin⁡nτ,iτ=1+a1icos⁡τ+∑m=2∞amicos⁡mτ.Both waveforms are normalized such that Vdc=1 and Idc=1. Under common assumptions that Cb behaves as short circuit and Lch behaves as open circuit at fundamental and higher harmonics, voltage and current waveforms at the load are(40)vLτ=a1vcos⁡τ+b1vsin⁡τ+anvcos⁡nτ+bnvsin⁡nτ,iLτ=-a1icos⁡τ-∑m=2∞amicos⁡mτ.The load impedance is equal to z_1=-(a1v-jb1v)/a1i at fundamental harmonic, to z_n=-(anv-jbnv)/ani at nth harmonic, and to z_m=0 at mth harmonic, m>1 and m≠n. When load is passive (Re⁡z_1≥0 and Re⁡z_n≥0), products a1va1i and anvani are nonpositive.Figure 9
Generic PA circuit diagram.Efficiency of PA with normalized waveforms (39) can be calculated as (e.g., see, [14])(41)η=-a1ia1v2.Case Study 1. In this case study, we analyze the efficiency of PA with current waveform of type (6) for k=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) withsubcase (1a):Idc(k=2)cont=1,k=2,2τp=160°, and A2=0.3,subcase (1b):Idc(k=2)cont=1,k=2,2τp=160°, and A2=0.2.According to (17), optimal value of parameter A2 for class-F PA is A2(I3=0)cont=0.1937. In both subcases, A2>A2(I3=0)cont and therefore the third-harmonic Fourier coefficients of both waveforms are negative. The value of A2 in subcase (1b) is very close to the optimal.In subcase (1a), substitution ofIdc(k=2)cont=1,2τp=160°,k=2, and A2=0.3 into (28) leads to IP=10.56. Furthermore, from (6), we obtain normalized form of current waveform(42)i1aτ=10.56cos⁡τ-cos⁡80°-0.3cos⁡2τ-cos⁡160°,τ≤80°,0,80°≤τ≤180°.Fourier coefficients of fundamental and third harmonic of i(1a)(τ) are(43)a1i1a=1.5505,a3i1a=-0.2040.In subcase (1b), substitution ofIdc(k=2)cont=1,2τp=160°,k=2, and A2=0.2 into (28) leads to IP=7.048. From (6), we obtain normalized current waveform(44)i1bτ=7.048cos⁡τ-cos⁡80°-0.2cos⁡2τ-cos⁡160°,τ≤80°,0,80°≤τ≤180°.Fourier coefficients of fundamental and third harmonic of i(1b)(τ) are(45)a1i1b=1.6060,a3i1b=-0.0081.In both subcases, we assume that voltage is nonnegative waveform of type(46)vτ=1+a1vcos⁡τ+b1vsin⁡τ+a3vcos⁡3τ+b3vsin⁡3τ.For passive load, Re⁡z_1≥0 along with a1i(1a)>0 (i.e., a1i(1b)>0) implies a1v≤0. Also, Re⁡z_3≥0 along with a3i(1a)<0 (i.e., a3i1b<0) implies a3v≥0. In [14], it is proved that relation λ3v≤1 holds (λ3v=a3v2+b3v2) for all nonnegative waveforms of type (46). In subcase (1a), relation λ3v≤1 together with λ3v=z_3a3i1a implies z_3≤1/a3i1a. Therefore, in subcase (1a), values of impedance at the third harmonic are bounded by z_3≤1/a3i1a and Re⁡z_3≥0. Similarly, in subcase (1b), values of impedance at the third harmonic are bounded by z_3≤1/a3i1b and Re⁡z_3≥0.In what follows, we consider maximal efficiency of PA with current waveform (42) (i.e., (44)) and voltage waveform (46) provided that z_3≤1/a3i1a and Re⁡z_3≥0 in subcase (1a) (i.e., z_3≤1/a3i1b and Re⁡z_3≥0 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 a1v 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 33 in [14] can be used in both subcases. For the sake of completeness, here we provide Algorithm 2, which is analogous to Algorithm 33 in [14]. To improve readability of the algorithm, we change the notation: θ0v-ξv/3 in [14] is replaced with φ1v.Algorithm 2
// input:a1i and a3i;
(i) choosez_3=r3+jx3 such that z_3≤1/a3i and r3≥0;
(ii) calculatea3v-jb3v=-z_3a3i and λ3v=a3v2+b3v2;
(iii) if27λ3v2≤1-2a3v3 then calculate a1v=-1-a3v and b1v=-3b3v;
else, calculateλ1v=3λ3v3-λ3v,φ1v=1/3atan2⁡b3v,a3v,  a1v=-λ1vcos⁡φ1v and b1v=-λ1vsin⁡φ1v;
(iv) calculate efficiencyη=-a1ia1v/2;
(v) calculatez_1=-a1v-jb1v/a1i  and  z_3n=z_3/Rez_1.Functionatan⁡2(y,x), which appears in step (iii) of Algorithm 2, is defined as(47)atan⁡2y,x=arctan⁡yxifx≥0,arctan⁡yx+πifx<0,y≥0,arctan⁡yx-πifx<0,y<0.In subcase (1a), input of Algorithm2 (coefficients a1i and a3i) is given by (43). Maximal efficiency of PA with waveform pair (42) and (46), as a function of normalized third-harmonic impedance z_3n=z_3/Re⁡z_1, is presented in Figure 10(a). Efficiency of 0.89 is attained in the vicinity of z_3n=1.26 (corresponding to z_3=0.94).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.
(a)
(b)In subcase (1b), input of Algorithm2 (coefficients a1i and a3i) 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 z_3n=33.16 (corresponding to z_3=23.84).As we mentioned earlier, the optimal value of parameterA2 is A2(I3=0)cont=0.1937. Corresponding maximal efficiency of class-F PA (see Box 1) is ηF(k=2)=0.9286. Current waveform i(1b)(τ) 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 ηF(k=2)=0.9286.Case Study 2. In this case study, we analyze the efficiency of PA with current waveforms of type (6) for k=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) withsubcase (2a):Idc(k=2)cont=1,k=2,2τp=220°, and A2=0.4,subcase (2b):Idc(k=2)cont=1,k=2,2τp=220°, and A2=0.34.According to (18), the optimal value of parameter A2 for inverse class-F PA is A2(I2=0)cont=0.3305. In both subcases, A2>A2(I2=0)cont and therefore second-harmonic Fourier coefficients of both waveforms are negative. The value of A2 in subcase (2b) is close to the optimal.In subcase (2a), substitution ofIdc(k=2)cont=1,2τp=220°,k=2, and A2=0.4 into (28) leads to IP=2.764. Furthermore, from (6), we obtain normalized form of current waveform(48)i2aτ=2.764cos⁡τ-cos⁡110°-0.4cos⁡2τ-cos⁡220°,τ≤110°,0,110°≤τ≤180°.Fourier coefficients of fundamental and second harmonic of i(2a)(τ) are(49)a1i2a=1.1932,a2i2a=-0.1023.In subcase (2b), substitution ofIdc(k=2)cont=1,2τp=220°,k=2, and A2=0.34 into (28) leads to IP=2.606. From (6), we obtain normalized current waveform(50)i2bτ=2.606cos⁡τ-cos⁡110°-0.34cos⁡2τ-cos⁡220°,τ≤110°,0,110°≤τ≤180°.Fourier coefficients of fundamental and second harmonic of i(2b)(τ) are(51)a1i2b=1.2351,a2i2b=-0.0132.In both subcases, we assume that voltage is a nonnegative waveform of type(52)vτ=1+a1vcos⁡τ+b1vsin⁡τ+a2vcos⁡2τ+b2vsin⁡2τ.For passive load, Re⁡{z_1}≥0 along with a1i(2a)>0 (i.e., a1i(2b)>0) implies a1v≤0. Also, Re⁡{z_2}≥0 along with a2i(2a)<0 (i.e., a2i(2b)<0) implies a2v≥0. In [14], it is proved that relation λ2v≤1 holds (λ2v=a2v2+b2v2) for all nonnegative waveforms of type (52). In subcase (2a), relations λ2v≤1 and λ2v=z_2a2i2a imply z_2≤1/a2i2a. Therefore, in subcase (2a), values of impedance at the second harmonic are bounded by z_2≤1/a2i2a and Re⁡{z_2}≥0. Similarly, in subcase (2b), values of impedance at the second harmonic are bounded by z_2≤1/a2i2b and Re⁡{z_2}≥0.In what follows, we consider maximal efficiency of PA with current waveform (48) (i.e., (50)) and voltage waveform (52) provided that |z_2|≤1/|a2i(2a)| and Re⁡{z_2}≥0 in subcase (2a) (i.e., z_2≤1/a2i2b and Re⁡{z_2}≥0 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 a1v 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 32 in [14] can be used in both subcases. For the sake of completeness, here we provide Algorithm 3, which is analogous to Algorithm 32 in [14]. We change the notation: θ0v-ξv/2 is replaced with φ1v.Algorithm 3
// input:a1i and a2i;
(i) choosez_2=r2+jx2 such that z_2≤1/a2i and r2≥0;
(ii) calculatea2v-jb2v=-z_2a2i and λ2v=a2v2+b2v2;
(iii) if2λ2v≤1-a2v then calculate a1v=-1-a2v and b1v=-2b2v;
else, calculateλ1v=8λ2v1-λ2v,φ1v=1/2atan2⁡b2v,a2v,a1v=-λ1vcos⁡φ1v and b1v=-λ1vsin⁡φ1v;
(iv) calculate efficiencyη=-a1ia1v/2;
(v) calculatez_1=-a1v-jb1v/a1i and z_2n=z_2/Re⁡z_1.In subcase (2a), input of Algorithm3 (coefficients a1i and a2i) is given by (49). Maximal efficiency of PA with waveform pair (48) and (52), as a function of normalized second-harmonic impedance z_2n=z_2/Re⁡z_1, is presented in Figure 11(a). Efficiency of 0.84 is attained in the vicinity of z_2n=4.89 (corresponding to z_2=4.12).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.
(a)
(b)In subcase (2b), input of Algorithm3 (coefficients a1i and a2i) 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 z_2n=37.99 (corresponding to z_2=33.18).As we mentioned before, the optimal value of parameterA2 is A2(I2=0)cont=0.3305. Corresponding maximal efficiency of inverse class-F PA (see Box 2) is ηIF(k=2)=0.8777. Current waveform i(2b)(τ) in subcase (2b) is close to the optimal for inverse class-F PA and corresponding peak efficiency of 0.87 is close to ηIF(k=2)=0.8777.


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

Fourier coefficient of thenth harmonic of current waveform of type (6) can be obtained by substituting (7) into (3):(36)Inkcont=IPτpnπsinc⁡n-1τp-sinc⁡n+1τp-kAksinc⁡k-nτp-sinc⁡k+nτp.It is easy to show that sinc⁡(k-n)τp-sinc⁡(k+n)τp>0, for 2τp∈0,360°,k∈2,3, and n∈2,3. Consequently, Fourier coefficient of the nth harmonic is nonpositive when(37)Ak≥sinc⁡n-1τp-sinc⁡n+1τpksinc⁡k-nτp-sinc⁡k+nτp.The value of Ak for which (37) becomes equality is Ak(In=0)cont (see (13)). Thus, (37) can be rewritten as(38)Ak≥AkIn=0cont.As it has been shown in Section 2, expression for Ak(In=0)cont can be reduced to: (i)
expression (25) for (n,k)=(3,3); parameter A3(I3=0)cont is plotted with solid line in Figure 4,(ii)
expression (17) for (n,k)=(3,2); parameter A2(I3=0)cont is plotted with solid line in Figure 2,(iii)
expression (18) for (n,k)=(2,2); parameter A2(I2=0)cont is plotted with dashed line in Figure 2,(iv)
expression (26) for (n,k)=(2,3); parameter A3(I2=0)cont is plotted with dashed line in Figure 4.Coefficients of fundamental and third harmonic(n=3) of optimal voltage waveform (32) for class-F PA have opposite signs. Similarly, coefficients of fundamental and second harmonic (n=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 ofn=3,k=2, and 2τp=180° (class-B conduction angle) into (37) yields A2≥0. According to the above discussion, Fourier coefficient of the third harmonic is positive for A2<0, equal to zero for A2=A2(I3=0)cont=0, and negative for A2>0. Therefore, small changes of parameter A2 around the optimal value A2(I3=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 ofn=k=3 and 2τp=180° into (37) yields A3≥0. Fourier coefficient of the third harmonic of continuous current waveform (6) is nonpositive for all A3≥0 and positive for all A3<0. Therefore, small change of parameter A3 from the optimal value A3(I3=0)cont=0 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 k=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 Figure9. We assume that voltage and current waveforms at the output port of transistor are(39)vτ=1+a1vcos⁡τ+b1vsin⁡τ+anvcos⁡nτ+bnvsin⁡nτ,iτ=1+a1icos⁡τ+∑m=2∞amicos⁡mτ.Both waveforms are normalized such that Vdc=1 and Idc=1. Under common assumptions that Cb behaves as short circuit and Lch behaves as open circuit at fundamental and higher harmonics, voltage and current waveforms at the load are(40)vLτ=a1vcos⁡τ+b1vsin⁡τ+anvcos⁡nτ+bnvsin⁡nτ,iLτ=-a1icos⁡τ-∑m=2∞amicos⁡mτ.The load impedance is equal to z_1=-(a1v-jb1v)/a1i at fundamental harmonic, to z_n=-(anv-jbnv)/ani at nth harmonic, and to z_m=0 at mth harmonic, m>1 and m≠n. When load is passive (Re⁡z_1≥0 and Re⁡z_n≥0), products a1va1i and anvani are nonpositive.Figure 9
Generic PA circuit diagram.Efficiency of PA with normalized waveforms (39) can be calculated as (e.g., see, [14])(41)η=-a1ia1v2.Case Study 1. In this case study, we analyze the efficiency of PA with current waveform of type (6) for k=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) withsubcase (1a):Idc(k=2)cont=1,k=2,2τp=160°, and A2=0.3,subcase (1b):Idc(k=2)cont=1,k=2,2τp=160°, and A2=0.2.According to (17), optimal value of parameter A2 for class-F PA is A2(I3=0)cont=0.1937. In both subcases, A2>A2(I3=0)cont and therefore the third-harmonic Fourier coefficients of both waveforms are negative. The value of A2 in subcase (1b) is very close to the optimal.In subcase (1a), substitution ofIdc(k=2)cont=1,2τp=160°,k=2, and A2=0.3 into (28) leads to IP=10.56. Furthermore, from (6), we obtain normalized form of current waveform(42)i1aτ=10.56cos⁡τ-cos⁡80°-0.3cos⁡2τ-cos⁡160°,τ≤80°,0,80°≤τ≤180°.Fourier coefficients of fundamental and third harmonic of i(1a)(τ) are(43)a1i1a=1.5505,a3i1a=-0.2040.In subcase (1b), substitution ofIdc(k=2)cont=1,2τp=160°,k=2, and A2=0.2 into (28) leads to IP=7.048. From (6), we obtain normalized current waveform(44)i1bτ=7.048cos⁡τ-cos⁡80°-0.2cos⁡2τ-cos⁡160°,τ≤80°,0,80°≤τ≤180°.Fourier coefficients of fundamental and third harmonic of i(1b)(τ) are(45)a1i1b=1.6060,a3i1b=-0.0081.In both subcases, we assume that voltage is nonnegative waveform of type(46)vτ=1+a1vcos⁡τ+b1vsin⁡τ+a3vcos⁡3τ+b3vsin⁡3τ.For passive load, Re⁡z_1≥0 along with a1i(1a)>0 (i.e., a1i(1b)>0) implies a1v≤0. Also, Re⁡z_3≥0 along with a3i(1a)<0 (i.e., a3i1b<0) implies a3v≥0. In [14], it is proved that relation λ3v≤1 holds (λ3v=a3v2+b3v2) for all nonnegative waveforms of type (46). In subcase (1a), relation λ3v≤1 together with λ3v=z_3a3i1a implies z_3≤1/a3i1a. Therefore, in subcase (1a), values of impedance at the third harmonic are bounded by z_3≤1/a3i1a and Re⁡z_3≥0. Similarly, in subcase (1b), values of impedance at the third harmonic are bounded by z_3≤1/a3i1b and Re⁡z_3≥0.In what follows, we consider maximal efficiency of PA with current waveform (42) (i.e., (44)) and voltage waveform (46) provided that z_3≤1/a3i1a and Re⁡z_3≥0 in subcase (1a) (i.e., z_3≤1/a3i1b and Re⁡z_3≥0 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 a1v 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 33 in [14] can be used in both subcases. For the sake of completeness, here we provide Algorithm 2, which is analogous to Algorithm 33 in [14]. To improve readability of the algorithm, we change the notation: θ0v-ξv/3 in [14] is replaced with φ1v.Algorithm 2
// input:a1i and a3i;
(i) choosez_3=r3+jx3 such that z_3≤1/a3i and r3≥0;
(ii) calculatea3v-jb3v=-z_3a3i and λ3v=a3v2+b3v2;
(iii) if27λ3v2≤1-2a3v3 then calculate a1v=-1-a3v and b1v=-3b3v;
else, calculateλ1v=3λ3v3-λ3v,φ1v=1/3atan2⁡b3v,a3v,  a1v=-λ1vcos⁡φ1v and b1v=-λ1vsin⁡φ1v;
(iv) calculate efficiencyη=-a1ia1v/2;
(v) calculatez_1=-a1v-jb1v/a1i  and  z_3n=z_3/Rez_1.Functionatan⁡2(y,x), which appears in step (iii) of Algorithm 2, is defined as(47)atan⁡2y,x=arctan⁡yxifx≥0,arctan⁡yx+πifx<0,y≥0,arctan⁡yx-πifx<0,y<0.In subcase (1a), input of Algorithm2 (coefficients a1i and a3i) is given by (43). Maximal efficiency of PA with waveform pair (42) and (46), as a function of normalized third-harmonic impedance z_3n=z_3/Re⁡z_1, is presented in Figure 10(a). Efficiency of 0.89 is attained in the vicinity of z_3n=1.26 (corresponding to z_3=0.94).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.
(a)
(b)In subcase (1b), input of Algorithm2 (coefficients a1i and a3i) 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 z_3n=33.16 (corresponding to z_3=23.84).As we mentioned earlier, the optimal value of parameterA2 is A2(I3=0)cont=0.1937. Corresponding maximal efficiency of class-F PA (see Box 1) is ηF(k=2)=0.9286. Current waveform i(1b)(τ) 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 ηF(k=2)=0.9286.Case Study 2. In this case study, we analyze the efficiency of PA with current waveforms of type (6) for k=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) withsubcase (2a):Idc(k=2)cont=1,k=2,2τp=220°, and A2=0.4,subcase (2b):Idc(k=2)cont=1,k=2,2τp=220°, and A2=0.34.According to (18), the optimal value of parameter A2 for inverse class-F PA is A2(I2=0)cont=0.3305. In both subcases, A2>A2(I2=0)cont and therefore second-harmonic Fourier coefficients of both waveforms are negative. The value of A2 in subcase (2b) is close to the optimal.In subcase (2a), substitution ofIdc(k=2)cont=1,2τp=220°,k=2, and A2=0.4 into (28) leads to IP=2.764. Furthermore, from (6), we obtain normalized form of current waveform(48)i2aτ=2.764cos⁡τ-cos⁡110°-0.4cos⁡2τ-cos⁡220°,τ≤110°,0,110°≤τ≤180°.Fourier coefficients of fundamental and second harmonic of i(2a)(τ) are(49)a1i2a=1.1932,a2i2a=-0.1023.In subcase (2b), substitution ofIdc(k=2)cont=1,2τp=220°,k=2, and A2=0.34 into (28) leads to IP=2.606. From (6), we obtain normalized current waveform(50)i2bτ=2.606cos⁡τ-cos⁡110°-0.34cos⁡2τ-cos⁡220°,τ≤110°,0,110°≤τ≤180°.Fourier coefficients of fundamental and second harmonic of i(2b)(τ) are(51)a1i2b=1.2351,a2i2b=-0.0132.In both subcases, we assume that voltage is a nonnegative waveform of type(52)vτ=1+a1vcos⁡τ+b1vsin⁡τ+a2vcos⁡2τ+b2vsin⁡2τ.For passive load, Re⁡{z_1}≥0 along with a1i(2a)>0 (i.e., a1i(2b)>0) implies a1v≤0. Also, Re⁡{z_2}≥0 along with a2i(2a)<0 (i.e., a2i(2b)<0) implies a2v≥0. In [14], it is proved that relation λ2v≤1 holds (λ2v=a2v2+b2v2) for all nonnegative waveforms of type (52). In subcase (2a), relations λ2v≤1 and λ2v=z_2a2i2a imply z_2≤1/a2i2a. Therefore, in subcase (2a), values of impedance at the second harmonic are bounded by z_2≤1/a2i2a and Re⁡{z_2}≥0. Similarly, in subcase (2b), values of impedance at the second harmonic are bounded by z_2≤1/a2i2b and Re⁡{z_2}≥0.In what follows, we consider maximal efficiency of PA with current waveform (48) (i.e., (50)) and voltage waveform (52) provided that |z_2|≤1/|a2i(2a)| and Re⁡{z_2}≥0 in subcase (2a) (i.e., z_2≤1/a2i2b and Re⁡{z_2}≥0 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 a1v 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 32 in [14] can be used in both subcases. For the sake of completeness, here we provide Algorithm 3, which is analogous to Algorithm 32 in [14]. We change the notation: θ0v-ξv/2 is replaced with φ1v.Algorithm 3
// input:a1i and a2i;
(i) choosez_2=r2+jx2 such that z_2≤1/a2i and r2≥0;
(ii) calculatea2v-jb2v=-z_2a2i and λ2v=a2v2+b2v2;
(iii) if2λ2v≤1-a2v then calculate a1v=-1-a2v and b1v=-2b2v;
else, calculateλ1v=8λ2v1-λ2v,φ1v=1/2atan2⁡b2v,a2v,a1v=-λ1vcos⁡φ1v and b1v=-λ1vsin⁡φ1v;
(iv) calculate efficiencyη=-a1ia1v/2;
(v) calculatez_1=-a1v-jb1v/a1i and z_2n=z_2/Re⁡z_1.In subcase (2a), input of Algorithm3 (coefficients a1i and a2i) is given by (49). Maximal efficiency of PA with waveform pair (48) and (52), as a function of normalized second-harmonic impedance z_2n=z_2/Re⁡z_1, is presented in Figure 11(a). Efficiency of 0.84 is attained in the vicinity of z_2n=4.89 (corresponding to z_2=4.12).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.
(a)
(b)In subcase (2b), input of Algorithm3 (coefficients a1i and a2i) 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 z_2n=37.99 (corresponding to z_2=33.18).As we mentioned before, the optimal value of parameterA2 is A2(I2=0)cont=0.3305. Corresponding maximal efficiency of inverse class-F PA (see Box 2) is ηIF(k=2)=0.8777. Current waveform i(2b)(τ) in subcase (2b) is close to the optimal for inverse class-F PA and corresponding peak efficiency of 0.87 is close to ηIF(k=2)=0.8777.


## 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. (Section6.1). 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 Figure12. 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 toVdc=28V and Vg=-1.95V, respectively. Frequency is set to f0=900MHz(ω0=2πf0), input power to Pin=18dBm, capacitance of dc blocking capacitor to Cb=1μF, and inductance of choke to Lch=1μH.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] Cds=1.22pF, L=0.55nH, Cp=0.25pF, Lp=0.1nH, and Rp=0.1Ω.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 vDiSi and iDi denote intrinsic waveforms.Initial values for load impedance are determined from the conditions that values of impedance presented at current generator plane (see Figure13) at harmonic frequencies are Z_(ω0)=(70+j0)Ω,Z_(2ω0)=(700+j0)Ω, and Z_(3ω0)=(0+j0)Ω. By solving these equations numerically, we obtain initial values for load impedance at fundamental and second harmonic: Z_L(ω0)=(48.5+j29)Ω and Z_L(2ω0)=(3.8+j45.9)Ω. Solving the equation at the third harmonic yields Z_L(3ω0)=(-0.2-j13.7)Ω. Since this value cannot be realized with passive load (real part is negative), for the initial value, we choose Z_L(3ω0)=(0-j13.7)Ω, which transforms to Z_(3ω0)=(0.176-j0.001)Ω at current generator plane.The initial value ofZin (see Figure 12) at fundamental frequency is estimated by using small-signalS-parameters provided in the data sheet of CGH40010 [21]. For CGH40010 terminated with Z_L(ω0)=(48.5+j29)Ω, by using small-signalS-parameters (900MHz,IDQ=200mA), we calculate that input impedance of CGH40010 is equal to (1.1-j8.7)Ω. Initial value of Z_in(ω0) is set to complex conjugate value; that is, Z_in(ω0)=(1.1+j8.7)Ω. The initial value of Z_in(2ω0) is set to low value, for which we choose Z_in(2ω0)=(1+j0)Ω. For the initial value at the third harmonic, we choose Z_in3ω0=50+j0Ω.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 ofZL at fundamental, second, and third harmonic are Z_L(ω0)=(51.4+j31.8)Ω,Z_L(2ω0)=(0+j65)Ω, and Z_L(3ω0)=(0-j15)Ω. The final values of Zin at fundamental, second, and third harmonic are Z_in(ω0)=(1.9+j9.2)Ω,Z_in(2ω0)=(0.5-j0.5)Ω, and Z_in3ω0=50+j0Ω. The results of simulation are ηintrinsicsim=0.8746 intrinsic drain efficiency, ηsim=0.8568 drain efficiency, and Pout=39.80dBm output power at 900MHz.


### 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 waveformsvDiSi and iDi (see Figure 13) can be expressed as follows:(53)vDiSiτs=V0,DiSi+∑m=1MVm,DiSicos⁡mτs+θm,s,iDiτs=I0,Di+∑m=1MIm,Dicos⁡mτs+ψm,s,where τs denotes ω0t and M denotes the order of harmonic balance simulation. After the change of variable τs→τ-ψ1,s these waveforms become(54)vDiSiτ=V0,DiSi+∑m=1MVm,DiSicos⁡mτ+θm,iDiτ=I0,Di+∑m=1MIm,Dicos⁡mτ+ψm,where θm=θm,s-mψ1,s and ψm=ψm,s-mψ1,s. Simulated intrinsic waveforms expressed in form (54) are plotted in Figure 14. Their amplitudes and phases for m≤5 are presented in Table 2. From this data, it can be concluded that voltage waveform has a significant second harmonic, whereas higher harmonics (m=3,4,5) are small. Furthermore, an almost ideal difference between fundamental and second harmonic phases (θ2-θ1≈179°) results in the flattening of the bottom of the voltage waveform.Table 2
Frequency-domain data of simulated intrinsic waveforms.
m
V
m
,
D
i
S
i
[
V
]
θ
m
[
°
]
I
m
,
D
i
[
A
]
ψ
m
[
°
]
0
27.733
0.000
0.398
0.000
1
38.369
-
179.883
0.503
0.000
2
11.761
-
0.687
0.015
-
94.766
3
0.147
31.830
0.181
-
174.485
4
0.349
84.385
0.059
-
170.941
5
0.647
-
121.226
0.041
-
17.095Figure 14
Intrinsic waveforms obtained in simulation:vDiSi (solid blue line) and iDi (dashed red line).Moreover, the data obtained in simulation are used to calculate impedance presented at the intrinsic plane at each harmonic (m≥1) by using the following formula:(55)Z_mω0=-Vm,DiSiejθmIm,Diejψm.Values of Z_(mω0), their magnitude Z(mω0), and arguments φ(mω0) for m≤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, Z(2ω0)>10·Z(ω0),Z(3ω0)<Z(ω0)/10 and Z(4ω0)<Z(ω0)/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 Z(5ω0)≈Z(ω0)/5. 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.
m
Z
_
(
m
ω
0
)
[
Ω
]
Z
(
m
ω
0
)
[
Ω
]
φ
(
m
ω
0
)
°
1
76.313 + j0.156
76.313
0.117
2
56.926 − j798.180
800.207
-
85.921
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
75.869Intrinsic waveforms (54) in normalized form can be written as(56)vDiSi_normτ=vDiSiτV0,DiSi=1+∑m=1Mamvcos⁡mτ+bmvsin⁡mτ,iDi_normτ=iDiτI0,Di=1+∑m=1Mamicos⁡mτ+bmisin⁡mτ.Fourier coefficients of normalized waveforms vDiSi_norm(τ) and iDi_norm(τ) for m≤5 are presented in Table 4.Table 4
Fourier coefficients of normalized simulated intrinsic waveforms.
m
a
m
v
b
m
v
a
m
i
b
m
i
0
1.0000
0.0000
1.0000
0.0000
1
-
1.3835
0.0028
1.2643
0.0000
2
0.4241
0.0051
-
0.0031
0.0368
3
0.0045
-
0.0028
-
0.4542
0.0438
4
0.0012
-
0.0125
-
0.1456
0.0232
5
-
0.0121
0.0199
0.0975
0.0230Here, 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 k=2,A2=0.415, and τp=100° (conduction angle 2τp=200°), which reads(57)imodτ=4.632cos⁡τ-cos⁡100°-0.415cos⁡2τ-cos⁡200°,τ≤100°,0,100°≤τ≤180°.Normalized intrinsic current waveforms obtained in simulation (red dashed line) and imod(τ) (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; γIsim=a1i) and basic waveform parameter of imod(τ) (calculated from (30)) are(58)γIsim=1.264,(59)γImod=1.267,respectively. Maximal values of current waveforms are also in good agreement.Figure 15
Normalized intrinsic current waveform obtained in simulation (dashed red line) andimod(τ) (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 for2τp=200° (calculated from (31)) is equal to γI,max=1.280 (corresponding to A2=0.4018). Basic waveform parameter of optimal voltage waveform with n=2 (see (34)) is γV,max=2. Thus, maximal attainable efficiency of inverse class-F for 2τp=200° is equal to(60)ηmax=12γI,maxγV,max=12·1.280·2=0.905.See also Box 2. By taking into account knee voltage of HEMT (from the results of simulations, we estimate that Vknee≈0.5V),γV,max decreases to(61)γVknee=γV,maxV0,DiSi-VkneeV0,DiSi=γV,max·0.98=2·0.98=1.386,and the corresponding maximal efficiency becomes(62)ηmaxknee=12γI,maxγV,max·0.98=ηmax·0.98=0.887.In simulation, we obtain that intrinsic drain efficiency is equal to(63)ηintrinsicsim=12γIsimγVsim=12·1.264·1.384=0.875(see Table 4; γIsim=a1i and γVsim=-a1v). The difference between ηmaxknee and ηintrinsicsim appears because γIsim≈0.9875γI,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(64)ηmodknee=12γImodγV,max·0.98=12·1.267·2·0.98=0.878≈ηintrinsicsim.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 harmonicAll 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 angle2τp=180°). It is easy to show that the basic waveform parameter of rectangular waveform with conduction angle 2τp is equal to(65)γIrectτp=2sinc⁡τp.For τp=100° and τp=90°, corresponding values are γIrect(100°)=1.1285 and γIrect(90°)=γIsq=4/π≈1.273. Although γIsq is close to γIsim=1.264, 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 forq=2 (solid black line) and odd harmonic waveform with maximal fundamental harmonic amplitude for q=2 (dash-dotted blue line).(ii) Maximally flat odd harmonic waveform [6] denotes nonnegative waveform, which contains dc component, fundamental harmonic, and q consecutive odd harmonics (q=0 means only fundamental, q=1 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](66)γImaxflatq=22q+1‼22q‼2q+2‼.Corresponding values are γImaxflat(0)=1,γImaxflat(1)=9/8=1.125,γImaxflat(2)≈1.172, and so forth. Parameter γImaxflat(q) slowly increases when q increases and γImaxflat(q)<limq→∞γImaxflat(q)=4/π≈1.273, [6]. Maximally flat odd harmonic waveform tends to square waveform when q tends to infinity [6]. Maximally flat odd harmonic waveform for q=2 is plotted in Figure 16 (solid black line).(iii) Odd harmonic waveform with maximal amplitude of fundamental harmonic contains dc component, fundamental harmonic, andq 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](67)γImaxamplq=2q+2cotπ2q+2.Corresponding values are γImaxampl(0)=1,γImaxampl(1)=2/3≈1.155,γImaxampl(2)=(1+2)/2≈1.207, and so forth. This waveform also tends to square waveform when q tends to infinity. Parameter γImaxampl(q) increases when q increases and γImaxampl(q)<limq→∞γImaxampl(q)=4/π≈1.273. Odd harmonic waveform with maximal amplitude of fundamental harmonic for q=2 is plotted in Figure 16 (dash-dotted blue line).The above consideration (illustrated in Figures15 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 Table5)(b)
R2 statistics or so-called coefficient of determination (denoted as R2 in Table5)(c)
Mean squared error (denoted as MSE in Table5)In Table 5, abbreviation “truncated” denotes waveform (57), “square” denotes normalized square waveform, “max flat” denotes maximally flat odd harmonic waveform for q=2, and “max ampl.” denotes odd harmonic waveform with maximal fundamental harmonic amplitude for q=2. The range of all three indicators is [0,1]. 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


## 6.1. Simulation Setup and Results

The circuit diagram of simulated inverse class-F PA is depicted in Figure12. 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 toVdc=28V and Vg=-1.95V, respectively. Frequency is set to f0=900MHz(ω0=2πf0), input power to Pin=18dBm, capacitance of dc blocking capacitor to Cb=1μF, and inductance of choke to Lch=1μH.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] Cds=1.22pF, L=0.55nH, Cp=0.25pF, Lp=0.1nH, and Rp=0.1Ω.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 vDiSi and iDi denote intrinsic waveforms.Initial values for load impedance are determined from the conditions that values of impedance presented at current generator plane (see Figure13) at harmonic frequencies are Z_(ω0)=(70+j0)Ω,Z_(2ω0)=(700+j0)Ω, and Z_(3ω0)=(0+j0)Ω. By solving these equations numerically, we obtain initial values for load impedance at fundamental and second harmonic: Z_L(ω0)=(48.5+j29)Ω and Z_L(2ω0)=(3.8+j45.9)Ω. Solving the equation at the third harmonic yields Z_L(3ω0)=(-0.2-j13.7)Ω. Since this value cannot be realized with passive load (real part is negative), for the initial value, we choose Z_L(3ω0)=(0-j13.7)Ω, which transforms to Z_(3ω0)=(0.176-j0.001)Ω at current generator plane.The initial value ofZin (see Figure 12) at fundamental frequency is estimated by using small-signalS-parameters provided in the data sheet of CGH40010 [21]. For CGH40010 terminated with Z_L(ω0)=(48.5+j29)Ω, by using small-signalS-parameters (900MHz,IDQ=200mA), we calculate that input impedance of CGH40010 is equal to (1.1-j8.7)Ω. Initial value of Z_in(ω0) is set to complex conjugate value; that is, Z_in(ω0)=(1.1+j8.7)Ω. The initial value of Z_in(2ω0) is set to low value, for which we choose Z_in(2ω0)=(1+j0)Ω. For the initial value at the third harmonic, we choose Z_in3ω0=50+j0Ω.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 ofZL at fundamental, second, and third harmonic are Z_L(ω0)=(51.4+j31.8)Ω,Z_L(2ω0)=(0+j65)Ω, and Z_L(3ω0)=(0-j15)Ω. The final values of Zin at fundamental, second, and third harmonic are Z_in(ω0)=(1.9+j9.2)Ω,Z_in(2ω0)=(0.5-j0.5)Ω, and Z_in3ω0=50+j0Ω. The results of simulation are ηintrinsicsim=0.8746 intrinsic drain efficiency, ηsim=0.8568 drain efficiency, and Pout=39.80dBm output power at 900MHz.


## 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 waveformsvDiSi and iDi (see Figure 13) can be expressed as follows:(53)vDiSiτs=V0,DiSi+∑m=1MVm,DiSicos⁡mτs+θm,s,iDiτs=I0,Di+∑m=1MIm,Dicos⁡mτs+ψm,s,where τs denotes ω0t and M denotes the order of harmonic balance simulation. After the change of variable τs→τ-ψ1,s these waveforms become(54)vDiSiτ=V0,DiSi+∑m=1MVm,DiSicos⁡mτ+θm,iDiτ=I0,Di+∑m=1MIm,Dicos⁡mτ+ψm,where θm=θm,s-mψ1,s and ψm=ψm,s-mψ1,s. Simulated intrinsic waveforms expressed in form (54) are plotted in Figure 14. Their amplitudes and phases for m≤5 are presented in Table 2. From this data, it can be concluded that voltage waveform has a significant second harmonic, whereas higher harmonics (m=3,4,5) are small. Furthermore, an almost ideal difference between fundamental and second harmonic phases (θ2-θ1≈179°) results in the flattening of the bottom of the voltage waveform.Table 2
Frequency-domain data of simulated intrinsic waveforms.
m
V
m
,
D
i
S
i
[
V
]
θ
m
[
°
]
I
m
,
D
i
[
A
]
ψ
m
[
°
]
0
27.733
0.000
0.398
0.000
1
38.369
-
179.883
0.503
0.000
2
11.761
-
0.687
0.015
-
94.766
3
0.147
31.830
0.181
-
174.485
4
0.349
84.385
0.059
-
170.941
5
0.647
-
121.226
0.041
-
17.095Figure 14
Intrinsic waveforms obtained in simulation:vDiSi (solid blue line) and iDi (dashed red line).Moreover, the data obtained in simulation are used to calculate impedance presented at the intrinsic plane at each harmonic (m≥1) by using the following formula:(55)Z_mω0=-Vm,DiSiejθmIm,Diejψm.Values of Z_(mω0), their magnitude Z(mω0), and arguments φ(mω0) for m≤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, Z(2ω0)>10·Z(ω0),Z(3ω0)<Z(ω0)/10 and Z(4ω0)<Z(ω0)/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 Z(5ω0)≈Z(ω0)/5. 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.
m
Z
_
(
m
ω
0
)
[
Ω
]
Z
(
m
ω
0
)
[
Ω
]
φ
(
m
ω
0
)
°
1
76.313 + j0.156
76.313
0.117
2
56.926 − j798.180
800.207
-
85.921
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
75.869Intrinsic waveforms (54) in normalized form can be written as(56)vDiSi_normτ=vDiSiτV0,DiSi=1+∑m=1Mamvcos⁡mτ+bmvsin⁡mτ,iDi_normτ=iDiτI0,Di=1+∑m=1Mamicos⁡mτ+bmisin⁡mτ.Fourier coefficients of normalized waveforms vDiSi_norm(τ) and iDi_norm(τ) for m≤5 are presented in Table 4.Table 4
Fourier coefficients of normalized simulated intrinsic waveforms.
m
a
m
v
b
m
v
a
m
i
b
m
i
0
1.0000
0.0000
1.0000
0.0000
1
-
1.3835
0.0028
1.2643
0.0000
2
0.4241
0.0051
-
0.0031
0.0368
3
0.0045
-
0.0028
-
0.4542
0.0438
4
0.0012
-
0.0125
-
0.1456
0.0232
5
-
0.0121
0.0199
0.0975
0.0230Here, 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 k=2,A2=0.415, and τp=100° (conduction angle 2τp=200°), which reads(57)imodτ=4.632cos⁡τ-cos⁡100°-0.415cos⁡2τ-cos⁡200°,τ≤100°,0,100°≤τ≤180°.Normalized intrinsic current waveforms obtained in simulation (red dashed line) and imod(τ) (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; γIsim=a1i) and basic waveform parameter of imod(τ) (calculated from (30)) are(58)γIsim=1.264,(59)γImod=1.267,respectively. Maximal values of current waveforms are also in good agreement.Figure 15
Normalized intrinsic current waveform obtained in simulation (dashed red line) andimod(τ) (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 for2τp=200° (calculated from (31)) is equal to γI,max=1.280 (corresponding to A2=0.4018). Basic waveform parameter of optimal voltage waveform with n=2 (see (34)) is γV,max=2. Thus, maximal attainable efficiency of inverse class-F for 2τp=200° is equal to(60)ηmax=12γI,maxγV,max=12·1.280·2=0.905.See also Box 2. By taking into account knee voltage of HEMT (from the results of simulations, we estimate that Vknee≈0.5V),γV,max decreases to(61)γVknee=γV,maxV0,DiSi-VkneeV0,DiSi=γV,max·0.98=2·0.98=1.386,and the corresponding maximal efficiency becomes(62)ηmaxknee=12γI,maxγV,max·0.98=ηmax·0.98=0.887.In simulation, we obtain that intrinsic drain efficiency is equal to(63)ηintrinsicsim=12γIsimγVsim=12·1.264·1.384=0.875(see Table 4; γIsim=a1i and γVsim=-a1v). The difference between ηmaxknee and ηintrinsicsim appears because γIsim≈0.9875γI,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(64)ηmodknee=12γImodγV,max·0.98=12·1.267·2·0.98=0.878≈ηintrinsicsim.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 harmonicAll 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 angle2τp=180°). It is easy to show that the basic waveform parameter of rectangular waveform with conduction angle 2τp is equal to(65)γIrectτp=2sinc⁡τp.For τp=100° and τp=90°, corresponding values are γIrect(100°)=1.1285 and γIrect(90°)=γIsq=4/π≈1.273. Although γIsq is close to γIsim=1.264, 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 forq=2 (solid black line) and odd harmonic waveform with maximal fundamental harmonic amplitude for q=2 (dash-dotted blue line).(ii) Maximally flat odd harmonic waveform [6] denotes nonnegative waveform, which contains dc component, fundamental harmonic, and q consecutive odd harmonics (q=0 means only fundamental, q=1 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](66)γImaxflatq=22q+1‼22q‼2q+2‼.Corresponding values are γImaxflat(0)=1,γImaxflat(1)=9/8=1.125,γImaxflat(2)≈1.172, and so forth. Parameter γImaxflat(q) slowly increases when q increases and γImaxflat(q)<limq→∞γImaxflat(q)=4/π≈1.273, [6]. Maximally flat odd harmonic waveform tends to square waveform when q tends to infinity [6]. Maximally flat odd harmonic waveform for q=2 is plotted in Figure 16 (solid black line).(iii) Odd harmonic waveform with maximal amplitude of fundamental harmonic contains dc component, fundamental harmonic, andq 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](67)γImaxamplq=2q+2cotπ2q+2.Corresponding values are γImaxampl(0)=1,γImaxampl(1)=2/3≈1.155,γImaxampl(2)=(1+2)/2≈1.207, and so forth. This waveform also tends to square waveform when q tends to infinity. Parameter γImaxampl(q) increases when q increases and γImaxampl(q)<limq→∞γImaxampl(q)=4/π≈1.273. Odd harmonic waveform with maximal amplitude of fundamental harmonic for q=2 is plotted in Figure 16 (dash-dotted blue line).The above consideration (illustrated in Figures15 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 Table5)(b)
R2 statistics or so-called coefficient of determination (denoted as R2 in Table5)(c)
Mean squared error (denoted as MSE in Table5)In Table 5, abbreviation “truncated” denotes waveform (57), “square” denotes normalized square waveform, “max flat” denotes maximally flat odd harmonic waveform for q=2, and “max ampl.” denotes odd harmonic waveform with maximal fundamental harmonic amplitude for q=2. The range of all three indicators is [0,1]. 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 k=2 or k=3. These families include both discontinuous and continuous waveforms. Under common assumptions that the third (n=3) harmonic in current waveform for class-F and the second (n=2) harmonic for inverse class-F are missing, the following four cases are analyzed: (i) n=k=3, (ii) n=3,k=2, (iii) n=k=2, and (iv) n=2,k=3. 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.



---

*Source: 1390295-2017-03-16.xml*