Abstract
Various classes of nonnegative waveforms containing dc component, fundamental and harmonic , which proved to be of interest in waveform modelling for power amplifier (PA) design, are considered in this paper. In optimization of PA efficiency, nonnegative waveforms with maximal amplitude of fundamental harmonic and those with maximal coefficient of cosine term of fundamental harmonic (optimal waveforms) play an important role. Optimal waveforms have multiple global minima and this fact closely relates the problem of optimization of PA efficiency to the concept of conflict set. There is also keen interest in finding descriptions for various classes of suboptimal waveforms, such as nonnegative waveforms with at least one zero, nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of harmonic, nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of harmonic, and nonnegative cosine waveforms with at least one zero. Closed form descriptions for all these suboptimal types of waveforms are provided in this paper. Suboptimal waveforms may also have multiple global minima and therefore be related to the concept of conflict set. Four case studies of usage of closed form descriptions of nonnegative waveforms in PA modelling are also provided.
1. Introduction
The origin of the concept of conflict set goes back to J. C. Maxwell (Maxwell 1831–1879), who informally introduced most of features of what today is called conflict set [1]. From this reason Maxwell set or Maxwell stratum is also used as synonyms for conflict set. Roughly speaking, conflict set associated with a smooth function with parameters is the set of -tuples in parameter space for which has multiple global minima. Conflict set is also intimately related to singularity theory and catastrophe theory [1].
Although without explicit reference, many max-min/min-max engineering design problems related to nonsmooth optimizations in parameter spaces (e.g., see [2]), including problems related to the optimization of efficiency of power amplifiers (PAs) (e.g., see [3–12]), are connected to the concept of conflict set. The concept of conflict set has been also used in mathematics (e.g., see [13–16]) and physics (e.g., see [17, 18]), including subjects like black holes [19].
Nonnegative waveforms with maximal amplitude of fundamental harmonic and those with maximal coefficient of cosine term of fundamental harmonic (optimal waveforms) have multiple global minima and therefore are closely related to the concept of conflict set. The suboptimal waveforms such as(i)nonnegative waveforms with at least one zero,(ii)nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of th harmonic,(iii)nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of th harmonic,(iv)nonnegative cosine waveforms with at least one zeromay also have multiple global minima [9, 11, 12] and therefore be related to the concept of conflict set, as well. These suboptimal waveforms are clearly of interest in shaping/modelling drain (collector/plate) waveforms in PA design (e.g., see [3–12, 20, 21]).
Fejér in his seminal paper [22] provided general description of all nonnegative trigonometric polynomials with consecutive harmonics in terms of parameters satisfying one nonlinear constraint. He also derived closed form solution to the problem of finding maximum possible amplitude of the first harmonic of nonnegative cosine polynomials with consecutive harmonics.
Fuzik [3] (see also [10]) considered cosine polynomials with dc, fundamental and harmonic, for arbitrary and provided closed form solution for coefficients of optimal waveform. Rhodes in [7] provided closed form expression for maximum possible amplitude of fundamental harmonic of nonnegative waveforms containing consecutive odd harmonics. A subclass of nonnegative cosine waveforms with dc, fundamental and third harmonic, having factorized form description has been considered in [23].
High efficiency PA with arbitrary output harmonic terminations has been analysed in [9], along with maximal efficiency, fundamental output power, and load impedance.
Factorized form of nonnegative waveforms up to second harmonic with at least one zero has been suggested in [11] in the context of continuous class B/J mode of PA operation.
General description of all nonnegative waveforms up to second harmonic in terms of four independent parameters has been provided in [12]. This includes nonnegative waveforms with at least one zero, as a special case.
End point of conflict set normally corresponds to so-called maximally flat waveform, which also belongs to class of suboptimal waveforms. First comprehensive usage of maximally flat waveforms, in the context of analysis of PA, goes to Raab [20]. General description of maximally flat waveforms with arbitrary number of harmonics has been presented in [21], along with closed form expressions for efficiency of class-F and inverse class-F PA with maximally flat waveforms. Description of maximally flat cosine waveforms with consecutive harmonics has been presented in [8] in the context of finite harmonic class-C PA.
In this paper we provide general descriptions of a number of optimal and suboptimal nonnegative waveforms containing dc component, fundamental and an arbitrary th harmonic, , and show how they are related to the concept of conflict set. According to our best knowledge, this paper provides the very first usage of conflict set in the course of solving problems related to optimization of PA efficiency. Main results are stated in six propositions (Propositions 1, 6, 9, 18, 22, and 26), four corollaries (Corollaries 2–5), twenty remarks, and three algorithms. Four case studies of usage of closed form descriptions of nonnegative waveforms in PA efficiency analysis are considered in detail in Section 7.
This paper is organized in the following way. In Section 2 we introduce concepts of minimum function and gain function (Section 2.1), conflict set (Section 2.2), and parameter space (Section 2.3). In Sections 3–6 we provide general descriptions of various classes of nonnegative waveforms containing dc component, fundamental and th harmonic with at least one zero. General case of nonnegative waveforms with at least one zero is presented in Section 3.1. The case with exactly two zeros is considered in Section 3.2. An algorithm for calculation of coefficients of fundamental harmonic of nonnegative waveforms with two zeros, for prescribed coefficients of harmonic, is presented in Section 3.3. Description of nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of harmonic is provided in Section 4. Nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of harmonic are considered in Section 5.1. An illustration of results of Section 5.1 for particular case is given in Section 5.2. Section 6.1 is devoted to nonnegative cosine waveforms with at least one zero and arbitrary , whereas Section 6.2 considers cosine waveforms with at least one zero for . In Section 7 four case studies of application of descriptions of nonnegative waveforms with fundamental and harmonic in PA modelling are presented. In the Appendices, list of some finite sums of trigonometric functions, widely used throughout the paper, and brief account of the Chebyshev polynomials are provided.
2. Minimum Function, Gain Function, and Conflict Set
In this section we consider minimum function and gain function (Section 2.1), conflict set (Section 2.2), and parameter space (Section 2.3) in the context of nonnegative waveforms with fundamental and harmonic.
We start with provision of a brief account of the facts related to the concepts of minimum function and conflict set. For this purpose let us denote by a family of smooth functions of variables depending on parameters, where is -tuple of variables and is -tuple of parameters. The minimum function , associated with the function , is defined as . Therefore, the domain of the minimum function is parameter space of the function . The minimum function is continuous, but not necessarily smooth function of parameters [13, 24]. It is a smooth function if possesses unique global minimum at nondegenerate critical point [13] (critical point is degenerate if at least first two consecutive derivatives are equal to zero). In this context, the conflict set can be defined as the set of the parameters for which function has global minimum at a degenerate critical point or/and multiple global minima [13].
For a wide class of minimum functions, when the number of parameters is not greater than four, the behaviour of minimum function in a neighbourhood of any point can be described by one of “normal forms” from a finite list as stated in [24]. For example, for smooth function , the minimum function near the origin can be locally reduced to one of the following three normal forms [25]: , or . In this example, the conflict set is the set of all points for which minimum function is not differentiable because function possesses at least two global minima [25].
2.1. Minimum Function and Gain Function
In what follows we consider family of waveforms of typewhere stands for , , , , and . Waveforms of type (1) include all possible shapes which can occur, but not all possible waveforms containing fundamental and th harmonic. However, shifting of waveforms of type (1) along -axis could recover all possible waveforms with fundamental and th harmonic.
The problem of finding nonnegative waveform of type (1) having maximum amplitude of fundamental harmonic plays an important role in optimization of PA efficiency. This extremal problem can be reformulated as problem of finding nonnegative waveform from family (1) having maximum possible value of coefficient . Nonnegative waveform of family (1) with maximum possible value of coefficient is called “optimal” or “extremal” waveform.
Furthermore, let us introduce an auxiliary waveform which is smooth function of one variable and two parameters and . In terms of , the above extremal problem reduces to the problem of finding maximum possible value of coefficient that satisfiesClearly, for any prescribed pair , there is a unique maximal value of coefficient for which inequality (3) holds for all . This maximal value of associated with the pair we denote it by and call it “gain function.”
Letbe the minimum function associated with . According to (3), and satisfy the following relation: . Since is obviously nonzero it follows immediately thatA relation analogue to (5), for (fundamental and second harmonic), has been derived in [4]. According to our best knowledge, it was the first appearance of gain function expressed via associated minimum function. The consideration presented in [4] has been restricted to the particular case when . The same problem for and arbitrary has been investigated in [3] (see also [10]).
According to above consideration, the problem of finding 3-tuple with maximum possible value of for which (3) holds is equivalent to the problem of finding maximum value of gain function and corresponding pair that satisfiesThus the optimal waveform is determined by parameters , , and ; that is, Optimal waveform has two global minima (this claim will be justified in Section 4, Remark 21). Consequently, the pair , which corresponds to maximum of gain function , belongs to conflict set in parameter space.
Figure 1 shows graph of gain function for . Notice that it has sharp ridge and that maximum of gain function (point ) lies on the ridge. This maximum corresponds to the optimal waveform (solution of the considered extremal problem). The beginning of the ridge (point ) corresponds to the waveform which possesses global minimum at degenerate critical point, that is, corresponds to maximally flat waveform (e.g., see [21]). Gain function is not differentiable on the ridge and consequently is not differentiable at the point where it has global maximum. This explains why the approach based on critical points does not work and why conflict set is so important in the considered problem.
Positions of global minima of for are presented in Figure 2. According to Proposition 1, conflict set is the ray defined by and . Waveforms with parameters that belong to the conflict set have two global minima. The waveform corresponding to the end point of the ray ( and ) has global minimum at degenerate critical point (so-called maximally flat waveform [21]).
Nonnegative waveforms of type (1) with have at least one zero. To show that, it is sufficient to see that for satisfying .
The problem of finding maximum value of fundamental harmonic cosine part of nonnegative waveform of the formwhere , is also related to the problem of finding maximum of the minimum function. Optimal waveform of family (9) has two global minima (this claim will be justified in Section 5, Remark 25), and therefore corresponding 3-tuple of parameters belongs to the conflict set in parameter space of family (9).
Let us introduce an auxiliary waveformand corresponding minimum function . Inequality can be rewritten as and therefore the highest value of is attained for . It immediately follows that nonnegative waveform of type (9) with has zero for satisfying .
2.2. Conflict Set
Historically, conflict set came into being from the problems in which families of smooth functions (such as potentials, distances, and waveforms) with two competing minima occur. The situation when competing minima become equal refers to the presence of conflict set (Maxwell set, Maxwell strata) in the associated parameter space.
There are many facets of conflict set. For example, in the problem involving distances between two sets of points, the conflict set is the intersections between iso-distance lines [14]. Conflict set also arises in the situation when two wave fronts coming from different objects meet [15, 25]. In the study of black holes, conflict set is the line of crossover of the horizon formed by the merger of two black holes [19]. In the classical Euler problem, conflict set is a set of points where distinct extremal trajectories with the same value of the cost functional meet one another [18].
Conflict set is very difficult to calculate, both analytically and numerically (e.g., see [15]), because of apparent nondifferentiability in some directions. In optimization of PA efficiency, some authors already reported difficulties in finding optimum via standard analytical tools [4, 5].
In this section, we consider conflict set in the context of family of waveforms of type (2) for arbitrary . In this context, for prescribed integer , conflict set is said to be a set of all pairs for which possesses multiple global minima.
Suppose that and are the positions of global minima of . Then, the conflict set is specified by the following set of relations:Relations (12) and (13) say that has minima at and , while relations (11) and (14) imply that these minima are equal and global.
The following proposition describes the conflict set of family of waveforms of type (2).
Proposition 1. Conflict set of family of waveforms of type (2) is the set of all pairs such that and .
The proof of Proposition 1, which is provided at the end of this section, also implies that the following four corollaries hold.
Corollary 2. The conflict set has end point at . This end point corresponds to the maximally flat waveform [21].
Corollary 3. Waveforms of type (2) with parameters that belong to conflict set have two global minima at , where .
Corollary 4. Every waveform with fundamental and harmonic has either one or two global minima.
Corollary 5. Conflict set can be parameterised in terms of as follows:Notice that is monotonically increasing function on interval .
Proof of Proposition 1. Without loss of generality, we can restrict our consideration to the interval . This is an immediate consequence of the fact that is a periodic function.
Suppose that and , where , are points at which has two equal global minima. Then conflict set is specified by relations (11)–(14). From (11)–(13) it follows that relationsalso hold. Let be a pair of points associated with . ClearlyThe first and second derivatives of are equal toBy using (20)-(21), system (16) can be rewritten asFrom (19) it follows that . Multiplying (24) and (25) with and , respectively, and summing the resulting relations, we obtain . The latest relation immediately implies thatEquations (22) and (23) can be considered as a system of two linear equations in terms of and . According to (26), the determinant of this system is nonzero and therefore it has only trivial solution: According to (18), implies According to (20), impliesFurthermore, and imply that . From (29) it follows that is position of global minimum of . Clearly , which together with leads toFrom (see (19)), and (30) it follows that , which together with yields
Since is position of global minimum, it follows that . Accordingly , which together with implies that . This relation along with (19) yields
Substitution of (31) and (28) in (24) leads toNotice that is monotonically decreasing function on interval (32). Therefore parameter is monotonically increasing function on the same interval with . Consequently , which completes the proof.
2.3. Parameter Space
In parameter space of family of waveforms (2) there are two subsets playing important role in the classification of the family instances. These are conflict set and catastrophe set.
Catastrophe set is subset of parameter space of waveform . It consists of those pairs for which the corresponding waveforms have degenerate critical points at which first and second derivatives are equal to zero. Thus, for finding catastrophe set we have to consider the following system of equations:where is a degenerate critical point of waveform .
Conflict set in parameter space of waveform , as shown in Proposition 1, is the ray described by and . It is intimately connected to catastrophe set.
In what follows in this subsection we use polar coordinate system instead of Cartesian coordinate system . Examples of catastrophe set and conflict set for plotted in parameter space are presented in Figure 3. Solid line represents the catastrophe set while dotted line describes conflict set. The isolated pick points (usually called cusp) which appear in catastrophe curves correspond to maximally flat waveforms, with maximally flat minimum and/or maximally flat maximum. There are two such picks in the catastrophe curves for and and one in the catastrophe curves for and . Notice that the end point of conflict set is the cusp point.
Catastrophe set divides the parameter space into disjoint subsets. In the cases and catastrophe curve defines inner and outer part. For catastrophe curve makes partition of parameter space in several inner subsets and one outer subset (see Figure 3).
Notice also that multiplying with a positive constant and adding in turn another constant, which leads to waveform of type (see (1) and (2)), do not make impact on the character of catastrophe and conflict sets. This is because in the course of finding catastrophe set first and second derivatives of are set to zero. Clearly (34) in terms of are equivalent to the analogous equations in terms of . Analogously, in the course of finding conflict set we consider only the positions of global minima (these positions for waveforms and are the same).
3. Nonnegative Waveforms with at Least One Zero
In what follows let us consider a waveform containing dc component, fundamental and th () harmonic of the form The amplitudes of fundamental and th harmonic of waveform of type (35), respectively, are
As it is shown in Section 2.1, nonnegative waveforms with maximal amplitude of fundamental harmonic or maximal coefficient of fundamental harmonic cosine part have at least one zero. It is also shown in Section 2.2 (Corollary 4) that waveforms of type (35) with nonzero amplitude of fundamental harmonic have either one or two global minima. Consequently, if nonnegative waveform of type (35) with nonzero amplitude of fundamental harmonic has at least one zero, then it has at most two zeros.
In Section 3.1 we provide general description of nonnegative waveforms of type (35) with at least one zero. In Sections 3.2 and 3.3 we consider nonnegative waveforms of type (35) with two zeros.
3.1. General Description of Nonnegative Waveforms with at Least One Zero
The main result of this section is presented in the following proposition.
Proposition 6. Every nonnegative waveform of type (35) with at least one zero can be expressed in the following form:whereproviding that
Remark 7. Function on the right hand side of (40) is monotonically increasing function of on interval (for more details about this function see Remark 15). From (57) and (65) it follows that relation holds for every nonnegative waveform of type (35). Notice that, according to (40), implies . Substitution of and into (55) yields . Consequently, implies that amplitude of fundamental harmonic is equal to zero.
Remark 8. Conversion of (38) into additive form leads to the following expressions for coefficients of nonnegative waveforms of type (35) with at least one zero:providing that satisfy (40) and .
Three examples of nonnegative waveforms with at least one zero for are presented in Figure 4 (examples of nonnegative waveforms with at least one zero for can be found in [12]). For all three waveforms presented in Figure 4, we assume that and . From (40) it follows that . Coefficients of waveform with (dotted line) are , , , and . Coefficients of waveform with (dashed line) are , , , and . Coefficients of waveform with (solid line) are , , , and . First two waveforms have one zero, while third waveform (presented with solid line) has two zeros.
Proof of Proposition 6. Waveform of type (35), containing dc component, fundamental and th harmonic, can be also expressed in the formwhere , , , and . It is easy to see that relations between coefficient of (35) and parameters of (47) read as follows:
Let us introduce such thatUsing (50), coefficients (49) can be expressed as (45)-(46).
Let us assume that is nonnegative waveform of type (35) with at least one zero; that is, and for some . Notice that conditions and imply that . From and , by using (50), it follows thatrespectively. On the other hand, can be rewritten asSubstitution of (51) into (52) yieldsAccording to (50), it follows that ; that is,Furthermore, substitution of (54) and (53) into (47) leads toAccording to (A.2) and (A.4) (see Appendices), there is common factor for all terms in (55). Consequently, (55) can be written in the form (38), whereFrom (56), by using (A.2), (A.4), and , we obtain (39).
In what follows we are going to prove that (40) also holds. According to (38), is nonnegative if and only if
Let us first show that position of global maximum of belongs to the interval . Relation (56) can be rewritten aswhere For , relation obviously holds. From for it follows that position of global maximum of the function of type for belongs to interval . Therefore position of global maximum of the expression in the square brackets in (60) for belongs to interval . This inequality together with leads to . Since decreases with increasing , it follows that for has global maximum on interval . For , it is easy to show that . Since is constant (see (58)), it follows from previous considerations that has global maximum on interval .
To find , let us consider first derivative of with respect to . Starting from (56), first derivative of can be expressed in the following form: where Using (A.6) (see Appendices), (62) can be rewritten asFrom and , , it follows that all summands in (63) decrease with increasing providing that . Therefore for . Consequently, and imply that .
From , , and it follows that or , and therefore . Since , it follows that is attained for . Furthermore, from (60) it follows that , which together with (58)-(59) leads to Both terms on the right hand side of (64) are even functions of and decrease with increase of , . Therefore, attains its lowest value for . It is easy to show that right hand side of (64) for is equal to 1, which further implies that From (65), it follows that (57) can be rewritten as . Finally, substitution of (64) into leads to (40), which completes the proof.
3.2. Nonnegative Waveforms with Two Zeros
Nonnegative waveforms of type (35) with two zeros always possess two global minima. Such nonnegative waveforms are therefore related to the conflict set.
In this subsection we provide general description of nonnegative waveforms of type (35) for and exactly two zeros. According to Remark 7, implies and . Number of zeros of on fundamental period equals , which is greater than two for and equal to two for . In the following proposition we exclude all waveforms with (the case when and is going to be discussed in Remark 10).
Proposition 9. Every nonnegative waveform of type (35) with exactly two zeros can be expressed in the following form:where
Remark 10. For waveforms with also have exactly two zeros. These waveforms can be included in above proposition by substituting (69) with .
Remark 11. Apart from nonnegative waveforms of type (35) with two zeros, there are another two types of nonnegative waveforms which can be obtained from (66)–(68). These are (i)nonnegative waveforms with zeros (corresponding to ) and(ii)maximally flat nonnegative waveforms (corresponding to ).
Notice that nonnegative waveforms of type (35) with can be obtained from (66)–(68) by setting . Substitution of and into (66), along with execution of all multiplications and usage of (A.2) (see Appendices), leads to .
Also, maximally flat nonnegative waveforms (they have only one zero [21]) can be obtained from (66)–(68) by setting . Thus, substitution of into (66)–(68) leads to the following form of maximally flat nonnegative waveform of type (35):Maximally flat nonnegative waveforms of type (35) for can be expressed as
Remark 12. Every nonnegative waveform of type (35) with exactly one zero at nondegenerate critical point can be described as in Proposition 6 providing that symbol “” in relation (40) is replaced with “”. This is an immediate consequence of Propositions 6 and 9 and Remark 11.
Remark 13. Identity implies that (66) can be also rewritten asFurthermore, substitution of (67) into (72) leads to
Remark 14. According to (A.6) (see Appendices), it follows that coefficients (67) can be expressed asFurthermore, from (74) it follows that coefficients , , and are equal to
For example, for , (75) and (68) lead to and , respectively, which from (72) further imply thatAlso for , (75), (76), and (68) lead to , , and , respectively, which from (72) further imply that
Remark 15. According to (A.5) (see Appendices), relation (68) can be rewritten asClearly, amplitude of th harmonic of nonnegative waveform of type (35) with exactly two zeros is even function of . Since , , decreases with increase of on interval , it follows that monotonically increases with increase of . Right hand side of (68) is equal to for and to one for . Therefore, for nonnegative waveforms of type (35) with exactly two zeros, the following relation holds:The left boundary in (82) corresponds to maximally flat nonnegative waveforms (see Remark 11). The right boundary in (82) corresponds to nonnegative waveforms with zeros (also see Remark 11).
Amplitude of th harmonic of nonnegative waveform of type (35) with two zeros, as a function of parameter for , is presented in Figure 5.
Remark 16. Nonnegative waveform of type (35) with two zeros can be also expressed in the following form:where is given by (68) and . From (83) it follows that coefficients of fundamental harmonic of nonnegative waveform of type (35) with two zeros arewhere is amplitude of fundamental harmonic: Coefficients of th harmonic are given by (45)-(46).
Notice that (68) can be rewritten asBy introducing new variable, and using the Chebyshev polynomials (e.g., see Appendices), relations (85) and (86) can be rewritten aswhere and denote the Chebyshev polynomials of the first and second kind, respectively. From (89) it follows that which is polynomial equation of th degree in terms of variable . From and (87) it follows that Since is monotonically increasing function of , , it follows that is monotonically decreasing function of . This further implies that (90) has only one solution that satisfies (91). (For expression (91) reads .) This solution for (which can be obtained at least numerically), according to (88), leads to amplitude of fundamental harmonic.
For , solutions of (90) and (91) areInsertion of (92) into (88) leads to the following relations between amplitude of fundamental and amplitude of th harmonic, :
Proof of Proposition 9. As it has been shown earlier (see Proposition 6), nonnegative waveform of type (35) with at least one zero can be represented in form (38). Since we exclude nonnegative waveforms with , according to Remark 7, it follows that we exclude case . Therefore in the quest for nonnegative waveforms of type (35) having two zeros we will start with waveforms of type (38) for . It is clear that nonnegative waveforms of type (38) have two zeros if and only ifand . According to (64), implies . Therefore, it is sufficient to consider only the interval (69).
Substituting (96) into (38) we obtain Expression , according to (64) and (39), equals Comparison of (97) with (66) yields where coefficients , , are given by (67). In what follows we are going to show that right hand sides of (98) and (99) are equal.
From (67) it follows thatAlso, from (67) for it follows that the following relations hold:From (99), by using (75), (76), (100)-(101), and trigonometric identitieswe obtain (98). Consequently (98) and (99) are equal, which completes the proof.
3.3. Nonnegative Waveforms with Two Zeros and Prescribed Coefficients of th Harmonic
In this subsection we show that, for prescribed coefficients and , there are nonnegative waveforms of type (35) with exactly two zeros. According to (37) and (82), coefficients and of nonnegative waveforms of type (35) with exactly two zeros satisfy the following relation:
According to Remark 16, the value of (see (87)) that corresponds to can be determined from (90)-(91). As we mentioned earlier, (90) has only one solution that satisfies (91). This value of , according to (88), leads to the amplitude of fundamental harmonic (closed form expressions for in terms of and are given by (93)–(95)).
On the other hand, from (45)-(46) it follows that where function is defined as with the codomain . Furthermore, according to (84) and (104), the coefficients of fundamental harmonic of nonnegative waveforms with two zeros and prescribed coefficients of th harmonic are equal towhere . For chosen , according to (104) and (66), positions of zeros areFrom (106) and it follows that, for prescribed coefficients and , there are nonnegative waveforms of type (35) with exactly two zeros.
We provide here an algorithm to facilitate calculation of coefficients and of nonnegative waveforms of type (35) with two zeros and prescribed coefficients and , providing that and satisfy (103).
Algorithm 17. (i) Calculate ,
(ii) identify that satisfies both relations (90) and (91),
(iii) calculate according to (88),
(iv) choose integer , such that ,
(v) calculate and according to (106).
For , by using (93) for , (94) for , and (95) for it is possible to calculate directly from and proceed to step (iv).
For and prescribed coefficients and , there are two waveforms with two zeros, one corresponding to and the other corresponding to (see also [12]).
Let us take as an input , , and . Execution of Algorithm 17 on this input yields and (according to (94)). For we calculate and (corresponding waveform is presented by solid line in Figure 6); for we calculate and (corresponding waveform is presented by dashed line); for we calculate and (corresponding waveform is presented by dotted line).
As another example of the usage of Algorithm 17, let us consider case and assume that and . Consequently and (according to (95)). For we calculate the following four pairs of coefficients of fundamental harmonic: for , for , for , and for . Corresponding waveforms are presented in Figure 7.
4. Nonnegative Waveforms with Maximal Amplitude of Fundamental Harmonic
In this section we provide general description of nonnegative waveforms containing fundamental and th harmonic with maximal amplitude of fundamental harmonic for prescribed amplitude of th harmonic.
The main result of this section is presented in the following proposition.
Proposition 18. Every nonnegative waveform of type (35) with maximal amplitude of fundamental harmonic and prescribed amplitude of th harmonic can be expressed in the following form:if or if , providing that , , and are related to via relations (67) and (68), respectively, and .
Remark 19. Expression (108) can be obtained from (38) by setting . Furthermore, insertion of into (43)–(46) leads to the following expressions for coefficients of waveform of type (108):
On the other hand, (109) coincides with (66). Therefore, the expressions for coefficients of (109) and (66) also coincide. Thus, expressions for coefficients of fundamental harmonic of waveform (109) are given by (84), where is given by (85), while expressions for coefficients of th harmonic are given by (45)-(46).
Waveforms described by (108) have exactly one zero, while waveforms described by (109) for have exactly two zeros. As we mentioned earlier, waveforms (109) for have zeros.
Remark 20. Maximal amplitude of fundamental harmonic of nonnegative waveforms of type (35) for prescribed amplitude of harmonic can be expressed asif , or if , where is related to via (68) (or (86)) and .
From (110) it follows that (111) holds. Substitution of (86) into (85) leads to (112).
Notice that is the only common point of the intervals and . According to (111), corresponds to . It can be also obtained from (112) by setting . The waveforms corresponding to this pair of amplitudes are maximally flat nonnegative waveforms.
Maximal amplitude of fundamental harmonic of nonnegative waveform of type (35) for , as a function of amplitude of th harmonic, is presented in Figure 8.
Remark 21. Maximum value of amplitude of fundamental harmonic of nonnegative waveform of type (35) is This maximum value is attained for (see (112)). The corresponding value of amplitude of th harmonic is . Nonnegative waveforms of type (35) with have two zeros at and for , or at and for .
To prove that (113) holds, let us first show that the following relation holds for : From , it follows that , where , and therefore . By using trigonometric identity , we immediately obtain (114).
According to (111) and (112), it is clear that attains its maximum value on the interval . Since is monotonic function of on interval (see Remark 15), it follows that for . Therefore, to find critical points of as a function of it is sufficient to find critical points of as a function of , , and consider its values at the end points and . Plot of as a function of parameter for is presented in Figure 9. According to (112), first derivative of with respect to is equal to zero if and only if . On interval , this is true if and only if . According to (112), is equal to for , equal to zero for , and equal to for . From (114) it follows that and therefore maximum value of is given by (113). Moreover, maximum value of is attained for .
According to above consideration, all nonnegative waveforms of type (35) having maximum value of amplitude of fundamental harmonic can be obtained from (109) by setting . Three of them corresponding to , , and three different values of (0, , and ) are presented in Figure 10. Dotted line corresponds to (coefficients of corresponding waveform are , , , and ), solid line to (, , , and ), and dashed line to (, , , and ).
Proof of Proposition 18. As it has been shown earlier (Proposition 6), nonnegative waveform of type (35) with at least one zero can be represented in form (38). According to (43), (44), and (36), for amplitude of fundamental harmonic of waveforms of type (38) the following relation holds:where satisfy (40) and .
Because of (40), in the quest of finding maximal for prescribed , we have to consider the following two cases: (Case i) . (Case ii) .
Case i. Since implies , according to (115), it follows that . Hence impliesTherefore if (Option 1) or (Option 2) or (Option 3).
Option 1. According to (115), implies (notice that this implication shows that does not depend on and therefore we can set to zero value).
Option 2. According to (115), implies , which further leads to the conclusion that is maximal for . For , becomes .
Option 3. This option leads to contradiction. To show that, notice that and imply that . Using (A.5) (see Appendices), the latest inequality can be rewritten as . But, from , , and it follows that all summands are not positive and therefore does not hold for .
Consequently, Case i implies and . Finally, substitution of into (38) leads to (108), which proves that (108) holds for .
Case ii. Relation , according to Proposition 9 and Remark 11, implies that corresponding waveforms can be expressed via (66)–(68) for . Furthermore, and imply . This proves that (109) holds for .
Finally, let us prove that (108) holds for . According to (68) (see also Remark 11), this value of corresponds to . Furthermore, substitution of and into (109) leads to (70), which can be rewritten asWaveform (117) coincides with waveform (108) for . Consequently, (108) holds for , which completes the proof.
5. Nonnegative Waveforms with Maximal Absolute Value of the Coefficient of Cosine Term of Fundamental Harmonic
In this section we consider general description of nonnegative waveforms of type (35) with maximal absolute value of coefficient for prescribed coefficients of th harmonic. This type of waveform is of particular interest in PA efficiency analysis. In a number of cases of practical interest either current or voltage waveform is prescribed. In such cases, the problem of finding maximal efficiency of PA can be reduced to the problem of finding nonnegative waveform with maximal coefficient for prescribed coefficients of th harmonic (see also Section 7).
In Section 5.1 we provide general description of nonnegative waveforms of type (35) with maximal absolute value of coefficient for prescribed coefficients of th harmonic. In Section 5.2 we illustrate results of Section 5.1 for particular case .
5.1. Nonnegative Waveforms with Maximal Absolute Value of Coefficient for
Waveforms of type (35) with can be derived from those with by shifting by , and therefore we can assume without loss of generality that . Notice that if is even, then shifting by produces the same result as replacement of with ( remains the same). On the other hand, if is odd, then shifting by produces the same result as replacement of with and with .
According to (37), coefficients of th harmonic can be expressed aswhere Conversely, for prescribed coefficients and , can be determined aswhere definition of function is given by (105).
The main result of this section is stated in the following proposition.
Proposition 22. Every nonnegative waveform of type (35) with maximal absolute value of coefficient for prescribed coefficients and of th harmonic can be represented asif , where , orif , where , , and are related to via relations (67) and (68), respectively, and .
Remark 23. Expression (121) can be obtained from (38) by setting and and then replacing with (see (118)) and with (see also (118)). Furthermore, insertion of and into (43)–(46) leads to the following relations between fundamental and th harmonic coefficients of waveform (121):
On the other hand, expression (122) can be obtained from (66) by replacing with . Therefore, substitution of in (84) leads towhere is given by (85).
The fundamental harmonic coefficients and of waveform of type (35) with maximal absolute value of coefficient satisfy both relations (123) and (124) if and satisfy . For such waveforms, relations and also hold.
Remark 24. Amplitude of th harmonic of nonnegative waveform of type (35) with maximal absolute value of coefficient and coefficients , satisfying , isTo show that, it is sufficient to substitute (see (118)) into .
Introducing new variable, and using the Chebyshev polynomials (e.g., see Appendices), relations and (125) can be rewritten aswhere and denote the Chebyshev polynomials of the first and second kind, respectively. Substitution of (128) into (127) leads to which is polynomial equation of th degree in terms of variable . From and (126) it follows that
In what follows we show that is monotonically increasing function of on the interval (130). From (see Remark 23) and (81) it follows that and therefore can be rewritten asObviously is even function of and all cosines in (131) are monotonically decreasing functions of on the interval . It is easy to show that , , decreases slower than when increases. This implies that denominator of the right hand side of (131) decreases slower than numerator. Since denominator is positive for it further implies that is decreasing function of on interval . Consequently, is monotonically increasing function of on the interval (130).
Thus we have shown that is monotonically increasing function of on the interval (130) and therefore (129) has only one solution that satisfies (130). According to (128), the value of obtained from (129) and (130), either analytically or numerically, leads to amplitude of th harmonic.
By solving (129) and (130) for we obtain
Insertion of (132) into (128) leads to the following explicit expressions for the amplitude , :Relations (133)–(135) define closed lines (see Figure 11) which separate points representing waveforms of type (121) from points representing waveforms of type (122). For given , points inside the corresponding curve refer to nonnegative waveforms of type (121), whereas points outside curve (and ) correspond to nonnegative waveforms of type (122). Points on the respective curve correspond to the waveforms which can be expressed in both forms (121) and (122).
Remark 25. The maximum absolute value of coefficient of nonnegative waveform of type (35) is This maximum value is attained for and (see (124)). Notice that is equal to the maximum value of amplitude of fundamental harmonic (see (113)). Coefficients of waveform with maximum absolute value of coefficient , are Waveform described by (137) is cosine waveform having zeros at and .
In the course of proving (136), notice first that holds. According to (123) and (124), maximum of occurs for . From (124) it immediately follows that maximum value of is attained if and only if and , which because of further implies . Since maximum value of is attained for , it follows that corresponding waveform has zeros at and .
Proof of Proposition 22. As it was mentioned earlier in this section, we can assume without loss of generality that . We consider waveforms of type (35) such that and for some . From assumption that nonnegative waveform of type (35) has at least one zero, it follows that it can be expressed in form (38).
Let us also assume that is position of nondegenerate critical point. Therefore implies and . According to (55), second derivative of at can be expressed as . Since it follows immediately that
Let us further assume that has exactly one zero. The problem of finding maximum absolute value of is connected to the problem of finding maximum of the minimum function (see Section 2.1). If waveforms possess unique global minimum at nondegenerate critical point then corresponding minimum function is a smooth function of parameters [13]. Consequently, assumption that has exactly one zero at nondegenerate critical point leads to the conclusion that coefficient is differentiable function of . First derivative of (see (43)) with respect to , taking into account that (see (50)), can be expressed in the following factorized form:From (138) and (139), it is clear that if and only if . According to Remark 12, assumption that has exactly one zero implies . From (51), (48), and it follows that , which, together with , implies that . Assumption , together with relations and , further implies andInsertion of into (38) leads toSubstitution of into (45) and (46) yields and , respectively. Replacing with and with in (141) immediately leads to (121).
Furthermore, , , and (118) imply that According to (38)–(40) and (142), it follows that (141) is nonnegative if and only ifNotice that implies that the following relation holds:Finally, substitution of (144) into (143) leads to , which proves that (121) holds when .
Apart from nonnegative waveforms with exactly one zero at nondegenerate critical point, in what follows we will also consider other types of nonnegative waveforms with at least one zero. According to Proposition 9 and Remark 11, these waveforms can be described by (66)–(68) providing that .
According to (35), implies . Consequently, implies that . On the other hand, according to (123), holds for waveforms of type (121). The converse is also true; and imply , which further from (35) implies . Therefore, in what follows it is enough to consider only nonnegative waveforms which can be described by (66)–(68) and , with coefficients and satisfying .
For prescribed coefficients and , the amplitude of th harmonic is also prescribed. According to Remark 15 (see also Remark 16), is monotonically decreasing function of . The value of can be obtained by solving (90) subject to the constraint . Then can be determined from (88). From (106) it immediately follows that maximal absolute value of corresponds to , which from (104) and (120) further implies that Furthermore , according to (107), implies that waveform zeros are Substitution of into (66) yields (122), which proves that (122) holds when .
In what follows we prove that (121) also holds when . Substitution of into leads to As we mentioned earlier, relation (142) holds for all waveforms of type (121). Substituting (142) into (147) we obtainThis expression can be rearranged asOn the other hand, for waveforms of type (122), according to (68), relations (148) and (149) also hold. Substitution of (see (145)) and (67) into (122) leads toFurthermore, substitution of (142) into (145) implies that . Finally, substitution of and (149) into (150) leads to (141). Therefore (141) holds when , which in turn shows that (121) holds when . This completes the proof.
5.2. Nonnegative Waveforms with Maximal Absolute Value of Coefficient for
Nonnegative waveform of type (35) for is widely used in PA design (e.g., see [10]). In this subsection we illustrate results of Section 5.1 for this particular case. The case is presented in detail in [12].
Coefficients of fundamental harmonic of nonnegative waveform of type (35) with and maximal absolute value of coefficient for prescribed coefficients and (), according to (123), (124), (134), (94), and (120), are equal toif ,where and , if . The line (see case in Figure 11) separates points representing waveforms with coefficients satisfying (151) from points representing waveforms with coefficients satisfying (152). Waveforms described by (151) for have exactly one zero at . Waveforms described by (151) and (152) for also have zero at . These waveforms as a rule have exactly two zeros. However there are two exceptions: one related to the maximally flat nonnegative waveform with coefficients , , and , which has only one zero, and the other related to the waveform with coefficients , , and , which has three zeros. Waveforms described by (152) for have two zeros. Waveforms with have only third harmonic (fundamental harmonic is zero).
Plot of contours of maximal absolute value of coefficient , , for prescribed coefficients and is presented in Figure 12. According to Remark 25, the waveform with maximum absolute value of is fully described with the following coefficients: , and . This waveform has two zeros at .
Two examples of nonnegative waveforms for and maximal absolute value of coefficient , , with prescribed coefficients and are presented in Figure 13. One waveform corresponds to the case (solid line) and the other to the case (dashed line). The waveform represented by solid line has one zero and its coefficients are , , , and . Dashed line corresponds to the waveform having two zeros with coefficients , , , and (case ).
6. Nonnegative Cosine Waveforms with at Least One Zero
Nonnegative cosine waveforms have proved to be of importance for waveform modelling in PA design (e.g., see [10]). In this section we consider nonnegative cosine waveforms containing fundamental and th harmonic with at least one zero.
Cosine waveform with dc component, fundamental and th harmonic, can be obtained from (35) by setting ; that is,
In Section 6.1 we provide general description of nonnegative cosine waveforms of type (153) with at least one zero. We show that nonnegative cosine waveforms with at least one zero coincide with nonnegative cosine waveforms with maximal absolute value of coefficient for prescribed coefficient . In Section 6.2 we illustrate results of Section 6.1 for particular case .
6.1. Nonnegative Cosine Waveforms with at Least One Zero for
Amplitudes of fundamental and th harmonic of cosine waveform of type (153) are and , respectively. According to (42), for nonnegative cosine waveforms of type (153) the following relation holds: This explains why th harmonic coefficient in Proposition 26 goes through interval .
Waveforms (153) with can be obtained from waveforms with by shifting by , and therefore, without loss of generality, we can assume that .
Proposition 26. Each nonnegative cosine waveform of type (153) with and at least one zero can be represented asif , orwhere if .
Remark 27. Identity implies that (156) can be rewritten asFurthermore, substitution of (157) into (160) leads to
Remark 28. All nonnegative cosine waveforms of type (153) with at least one zero and , except one of them, can be represented either in form (155) or form (156). This exception is maximally flat cosine waveform with which can be obtained from (155) for or from (156) for . Maximally flat cosine waveform with can also be obtained from (70) by setting . Furthermore, setting in (71) leads to maximally flat cosine waveforms for and .
Remark 29. Nonnegative cosine waveform of type (155) with and has exactly one zero at . Nonnegative cosine waveform described by (156) with and has two zeros at , where . For , nonnegative cosine waveform of type (153) reduces to (clearly, these two waveforms both have zeros).
Remark 30. Transformation of (155) into an additive form leads to the following relation: where . Similarly, transformation of (156) leads to the following relation:where is given by (158), , and . Notice that coefficients of maximally flat cosine waveform, namely, and , satisfy relation (162). They also satisfy relation (163) for .
Remark 31. Nonnegative cosine waveforms of type (153) with at least one zero coincide with nonnegative cosine waveforms with maximal absolute value of coefficient for prescribed coefficient .
In proving that Remark 31 holds, notice that expression (155) can be obtained from (121) by setting . Furthermore, if , then , which together with and (118) implies . In this case becomes . On the other hand, if , then , which together with and (118) implies . In this case becomes . Therefore, every nonnegative cosine waveform of type (155) has maximal absolute value of coefficient for prescribed coefficient , when .
Let us now show that expression (156) can be obtained from (122) by setting and . For waveforms of type (122), according to (118), and imply and . Substitution of and into leads to . Furthermore, substitution of into (145) yields . Insertion of , , and into (122) leads to (156). Therefore, every nonnegative cosine waveform of type (156) has maximal absolute value of coefficient for prescribed coefficient , when .
Proof of Proposition 26. Let us start with nonnegative cosine waveform of type (153) with . According to Remark 7, implies that . Substitution of into (155) and using (A.2) (see Appendices) lead to . Consequently, (155) holds for . On the other hand, substitution of into (158) yields . Furthermore, substitution of and (or ) into (156), along with performing all multiplications and using (A.2), leads to . Consequently, (156)–(158) hold for and .
It is easy to see that and for some imply . Therefore in what follows we assume that and .
Cosine waveforms are even functions of . Therefore, if nonnegative cosine waveform has exactly one zero it has to be either at 0 or at . On the other hand, if nonnegative cosine waveform with has exactly two zeros then these zeros are placed at , such that is neither 0 nor .
In order to prove that (155) holds for , let us start by referring to the description (38) of nonnegative waveforms with at least one zero. As we mentioned earlier, for nonnegative cosine waveform with exactly one zero (denoted by ) it is either or . Therefore in both cases . Substitution of into (43), together with and , leads to Clearly , , and , according to (44) and (46), imply . Since it follows that also holds, which further implies or . In the case when , from (164) and (43) we obtain , which further implies that . Consequently (155) holds for . In the case when , from (164) and (45) we obtain if , or if . Relations and , according to (40), imply that . Substitution of , , and (164) into (38) leads to (155), which proves that (155) holds for . On the other hand, relations and , according to (40), imply that . Substitution of , , and (164) into (38) also leads to (155), which proves that (155) also holds for . Consequently (155) holds for .
In what follows we first prove that (156)-(157) hold for . For this purpose let us start with nonnegative waveforms with two zeros described by (66). As we mentioned before, nonnegative cosine waveforms with two zeros have zeros at and , such that and . Relations and , according to (84), imply and therefore From and it follows that . Insertion of into (45) yields . Relations and (82) imply that . Substitution of and into (66)–(68) leads to (156)–(158), which proves that (156)–(158) hold for and .
Finally, substitution of and into (161) leads to Waveform (166) coincides with waveform (155) for , which in turn proves that (156) holds for and . This completes the proof.
6.2. Nonnegative Cosine Waveforms with at Least One Zero for
In this subsection we consider nonnegative cosine waveforms with at least one zero for (for case see [12]).
Cosine waveform with fundamental and third harmonic reads
For and , according to (155), nonnegative cosine waveform of type (167) with at least one zero can be expressed asFrom , it immediately follows that, for and , can be expressed asFor and , from (158) it follows that . This relation, along with (160) and (157), further implies that can be expressed asproviding that . From , it follows that (170) also holds for and , providing that .
Maximally flat nonnegative cosine waveform of type (167) with (minimum at ) reads . Dually, maximally flat nonnegative cosine waveform with (minimum at ) reads .
In what follows we provide relations between coefficients and of nonnegative cosine waveforms of type (167) with at least one zero.
For , conversion of (168) into an additive form immediately leads to the following relation:Conversion of (170) into an additive form leads to , which can be also expressed as . For , relations , , and lead toSimilarly for , conversion of (169) into an additive form leads to the following relation: For waveform of type (170) with , relations , , and lead to
Every cosine waveform of type (167) corresponds to a pair of real numbers and vice versa. Points in grey area in Figure 14 correspond to nonnegative cosine waveforms for . The points at the boundary of grey area correspond to nonnegative cosine waveforms with at least one zero. A number of shapes of nonnegative cosine waveforms with and at least one zero, plotted on interval , are also presented in Figure 14. The boundary of grey area in Figure 14 consists of four line segments described by relations (171)–(174). The common point of line segments (172) and (173) is cusp point with coordinates and . Another cusp point, with coordinates and , is the common point of line segments (171) and (174). The common point of line segments (171)-(172) has coordinates and common point of line segments (173)-(174) has coordinates . These points are represented by white circle dots and they correspond to maximally flat cosine waveforms (e.g., see [21]). White triangle dots with coordinates and refer to the nonnegative cosine waveforms with maximum value of amplitude of fundamental harmonic.
7. Four Case Studies of Usage of Nonnegative Waveforms in PA Efficiency Analysis
In this section we provide four case studies of usage of description of nonnegative waveforms with fundamental and th harmonic in PA efficiency analysis. In first two case studies, to be presented in Section 7.1, voltage is nonnegative waveform with fundamental and second harmonic with at least one zero. In remaining two case studies, to be considered in Section 7.2, voltage waveform contains fundamental and third harmonic.
Let us consider generic PA circuit diagram, as shown in Figure 15. We assume here that voltage and current waveforms at the transistor output arewhere stands for . Both waveforms are normalized in the sense that dc components of voltage and current are and , respectively. Under assumption that blocking capacitor behaves as short-circuit at the fundamental and higher harmonics, current and voltage waveforms at the load areIn terms of coefficients of voltage and current waveforms, the load impedance at fundamental harmonic is , whereas load impedance at th harmonic is . All other harmonics are short-circuited ( for and ). Time average output power of PA (e.g., see [10]) with waveform pair (175) at fundamental frequency can be expressed asFor normalized waveforms (175) with and , dc power is . Consequently, PA efficiency (e.g., see, [10, 26]) is equal toThus, time average output power of PA with pair of normalized waveform (175) is equal to efficiency (178).
Power utilization factor (PUF) is defined [26] as “the ratio of power delivered in a given situation to the power delivered by the same device with the same supply voltage in Class A mode.” Since the output power in class-A mode is (e.g., see [9]), it follows that power utilization factor for PA with pair of normalized waveforms (175) can be expressed as
7.1. Nonnegative Waveforms for in PA Efficiency Analysis
In this subsection we provide two case studies of usage of description of nonnegative waveforms with fundamental and second harmonic () in PA efficiency analysis. For more examples of usage of descriptions of nonnegative waveforms with fundamental and second harmonic in PA efficiency analysis see [12].
Case Study 7.1. In this case study we consider efficiency of PA for given second harmonic impedance, providing that voltage is nonnegative waveform with fundamental and second harmonic and current is “half-sine” waveform frequently used in efficiency analysis of classical PA operation (e.g., see [10]).
Standard model of current waveform for classical PA operation has the form (e.g., see [10, 26])where is conduction angle and . Since is even function, it immediately follows that its Fourier series contains only dc component and cosine terms:The component of the waveform (180) iswhere . The coefficient of the fundamental harmonic component readsand the coefficient of harmonic component can be written in the formFor “half-sine” current waveform, conduction angle is equal to (class-B conduction angle). According to (182), this further implies that . To obtain normalized form of waveform (180), we set which implies that . Furthermore, substitution of and in (180) leads toSimilarly, substitution of and into (183) and (184) leads to the coefficients of waveform (185). Coefficients of fundamental and second harmonic, respectively, are
On the other hand, voltage waveform of type (35) for reads This waveform contains only fundamental and second harmonic, and therefore all harmonics of order higher than two are short-circuited ( for ). For current voltage pair (185) and (187), load impedance at fundamental harmonic is , whereas load impedance at second harmonic is . According to our assumption, the load is passive and therefore and , which further imply and , respectively.
It is easy to see that problem of finding maximal efficiency of PA with current-voltage pair (185) and (187) for prescribed second harmonic impedance can be reduced to the problem of finding voltage waveform of type (187) with maximal coefficient , for prescribed coefficients of second harmonic (see Section 5).
The following algorithm (analogous to Algorithm 22 presented in [12]) provides the procedure for calculation of maximal efficiency with current-voltage pair (185) and (187) for prescribed second harmonic impedance. The definition of function , which appears in the step (iii) of the following algorithm, is given by (105).
Algorithm 32. (i) Choose such that ,
(ii) calculate and ,
(iii) if then calculate and ; else, calculate , , , and ,
(iv) calculate efficiency ,
(v) calculate and .
In this case study, coefficients of fundamental and second harmonic of current waveform are given by (186). Maximal efficiency of PA associated with the waveform pair (185) and (187), as a function of normalized second harmonic impedance , is presented in Figure 16(a). As can be seen from Figure 16(a), efficiency of 0.78 is achieved at the edge of Smith chart, where second harmonic impedance has small resistive part. Corresponding PUF calculated according to (179) is presented in Figure 16(b). Peak efficiency and peak value of are attained when second harmonic is short-circuited (which corresponds to ideal class-B operation [10, 26]).
(a)
(b)
For example, for second harmonic impedance and current waveform (185), from Algorithm 32 it follows that . Furthermore, according to step (iii) of above algorithm, maximal efficiency of PA is attained with voltage waveform of type (187) with coefficients , , , and (see Figure 17). Corresponding efficiency, PUF, and normalized second harmonic impedance are , , and , respectively.
On the other hand, for second harmonic impedance and current waveform (185), from Algorithm 32 it follows that . Then, according to step (iii) of above algorithm, maximal efficiency is attained with voltage waveform of type (187) with coefficients , , , and (see Figure 18). Efficiency, PUF, and normalized second harmonic impedance are , , and , respectively.
Case Study 7.2. As another case study, let us consider the efficiency of PA, providing that current waveform is nonnegative cosine waveform up to third harmonic with maximum value of amplitude of fundamental harmonic [22] (see also [8]):and voltage waveform is nonnegative waveform of type (187). Load impedances at fundamental, second, and third harmonic are , , and , respectively. According to our assumption, the load is passive and therefore and , which further imply and , respectively.
Because current waveform (188) contains only cosine terms and voltage waveform is the same as in previous case study, the procedure for calculation of maximal efficiency of PA with waveform pair (187)-(188) is the same as presented in Algorithm 32. In this case study the coefficients of fundamental and second harmonic of current waveform are and , respectively.
Maximal efficiency of PA associated with the waveform pair (187)-(188), as a function of normalized second harmonic impedance , is presented in Figure 19(a). Efficiency of 0.8 is achieved at the edge of Smith chart, where second harmonic impedance has small resistive part. The theoretical upper bound is attained when second harmonic is short-circuited. When this upper bound is reached, both second and third harmonic are short-circuited which implies that we are dealing with finite harmonic class-C [6, 8], or dually, when current and voltage interchange their roles, with finite harmonic inverse class-C [6, 9]. Corresponding PUF, calculated according to (179), is presented in Figure 19(b). Peak value of is attained when second harmonic is short-circuited.
(a)
(b)
For example, for second harmonic impedance and current waveform (188), from Algorithm 32 it follows that . Furthermore, according to step (iii) of Algorithm 32, maximal efficiency of PA is attained with voltage waveform of type (187) with coefficients , , , and (see Figure 20). Corresponding efficiency, PUF, and normalized second harmonic impedance are , , and , respectively.
On the other hand, for and current waveform (187) it follows that . Then, according to step (iii) of Algorithm 32, the maximal efficiency is attained with voltage waveform of type (187) with coefficients , , , and (see Figure 21). Efficiency, PUF, and normalized second harmonic impedance are , , and , respectively.
7.2. Nonnegative Waveforms for in PA Efficiency Analysis
In this subsection we provide another two case studies of usage of description of nonnegative waveforms in PA efficiency analysis, this time with fundamental and third harmonic ().
Case Study 7.3. Let us consider current-voltage pair such that voltage is nonnegative waveform with fundamental and third harmonic:and current is nonnegative cosine waveform given by (188). Load impedances at fundamental, second, and third harmonic are , , and , respectively. According to our assumption, the load is passive and therefore and , which further imply and .
In this subsection we consider the problem of finding maximal efficiency of PA with waveform pair (188)-(189) for given third harmonic impedance. As we mentioned earlier, problem of finding maximal efficiency of PA with current-voltage pair (188)-(189) for prescribed third harmonic impedance can be reduced to the problem of finding voltage waveform of type (189) with maximal coefficient , for prescribed coefficients of third harmonic (see Section 5.2).
The following algorithm provides the procedure for calculation of maximal efficiency with current-voltage pair (188)-(189). The definition of function , which appears in step (iii) of the following algorithm, is given by (105).
Algorithm 33. (i) Choose such that ,
(ii) calculate and ,
(iii) if then calculate and ; else, calculate , , , and ,
(iv) calculate efficiency ,
(v) calculate and .
In this case study coefficients of fundamental and third harmonic of current waveform are and , respectively. For the waveform pair (188)-(189), maximal efficiency of PA as a function of normalized third harmonic impedance is presented in Figure 22. Efficiency of 0.8 is reached when third harmonic impedance has small resistive part. Peak efficiency is achieved when third harmonic is short-circuited.
For the present case study, in what follows we show that power utilization factor is proportional to efficiency. For voltage waveform of type (189) it is easy to see that holds. This relation along with the fact that waveform that provides maximal efficiency has at least one zero implies that . On the other hand, current waveform (188) is cosine waveform with positive coefficients and therefore . Consequently, according to (179), the following relation holds: Clearly, the ratio is constant and therefore in this case study PUF can be easily calculated from the corresponding efficiency. Accordingly, peak efficiency and peak value of are attained for the same voltage waveform (when third harmonic is short-circuited).
In the first example, current waveform (188) and imply that . Then, according to Algorithm 33, the voltage waveform of type (189) that provides maximal efficiency has the following coefficients: , , , and (see Figure 23). Efficiency, PUF, and normalized third harmonic impedance are , , and , respectively.
In the second example, current waveform (188) and imply that . Then, according to Algorithm 33, the voltage waveform of type (189) that provides maximal efficiency has the following coefficients: , , , and (see Figure 24). Efficiency, PUF, and normalized third harmonic impedance are , , and , respectively.
Case Study 7.4. In this case study let us consider current-voltage pair, where current is normalized waveform of type (180) with conduction angle (207°) and voltage is nonnegative waveform of type (189). Substitution of and into (182) leads to . Furthermore, substitution of and into (180) leads toSimilarly, substitution of and into (183) and (184) for yields coefficients of fundamental and third harmonic of waveform (191):
Because current waveform (191) contains only cosine terms and voltage waveform is the same as in previous case study, the procedure for calculation of maximal efficiency of PA with waveform pair (189)–(191) is the same as that presented in Algorithm 33. In this case study the coefficients of fundamental and third harmonic of current waveform are given by (192).
For the waveform pair (189) and (191), maximal efficiency of PA as a function of normalized third harmonic impedance is presented in Figure 25. Efficiency of 0.84 is obtained in vicinity of (corresponding to ). Peak efficiency is achieved for voltage waveform of type (189) with coefficients , , and .
In the course of finding power utilization factor, notice that current waveform of type (191) attains its maximum value for . Insertion of and for voltage waveform of type (189) into (179) leads to Again, the ratio is constant and PUF can be easily calculated from the corresponding efficiency. Accordingly, peak value of and peak efficiency are attained for the same voltage waveform.
In the first example, current waveform (191) and imply that . Then, according to Algorithm 33, voltage waveform of type (189) which provides maximal efficiency has coefficients: , , , and (see Figure 26). Efficiency, PUF, and normalized third harmonic impedance are , , and , respectively.
In second example, current waveform (191) and imply that . Then, according to Algorithm 33, voltage waveform of type (189) which provides maximal efficiency has coefficients: , , , and (see Figure 27). Efficiency, PUF, and normalized third harmonic impedance are , , and , respectively.
8. Conclusion
In this paper we consider a problem of finding general descriptions of various classes of nonnegative waveforms with fundamental and th harmonic. These classes include nonnegative waveforms with at least one zero, nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of th harmonic, nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of th harmonic, and nonnegative cosine waveforms with at least one zero. Main results are stated in six propositions (Propositions 1, 6, 9, 18, 22, and 26), four corollaries (Corollaries 2–5), twenty remarks, and three algorithms. Four case studies of usage of closed form descriptions of nonnegative waveforms in PA efficiency analysis are considered in detail in Section 7.
Appendices
Here we provide a list of finite sums of trigonometric functions used in this paper (Appendix A) and brief account of the Chebyshev polynomials (Appendix B).
A. List of Some Finite Sums of Trigonometric Functions
Dirichlet kernel (e.g., see [27]) is as follows:
Fejér kernel (e.g., see [27]) can be expressed in the following equivalent forms:
Lagrange’s trigonometric identity (e.g., see [28]) is as follows:
In what follows we show that the following three trigonometric identities also hold:
Denote , , and .
Notice that , which immediately leads to (A.4).
Identity (A.5) can be obtained as follows:
From it follows that , which leads to (A.6).
B. The Chebyshev Polynomials
The Chebyshev polynomials of the first kind can be defined by the following relation (e.g., see [29]):The Chebyshev polynomials of the second kind can be defined by the following relation (e.g., see [29]):
The Chebyshev polynomials satisfy the following recurrence relations (e.g., see [29]):The first few Chebyshev polynomials of the first and second kind are , , , , , and .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the Serbian Ministry of Education, Science and Technology Development, as a part of Project TP32016.