LinearizationofPowerAmplifierusingtheModifiedFeedForwardMethod. Linearization of Power Amplifier using the Modified Feed Forward Method

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1 lobal Joural of Researches i Egieerig: Idustrial Egieerig Volume 17 Issue 1 Versio1.0 Year 017 Type: Double Blid Peer Reviewed Iteratioal Research Joural Publisher: lobal Jourals Ic. (USA) Olie ISSN: & Prit ISSN: Liearizatio of Power Amplifier usig the Modified Feed Forward Method By Mohammad Reza Motavalli & Klaus Solbach Uiversity of Qom Abstract- A modified circuit for improvig liearizatio of power amplifier based o the model of the Feed Forward circuit amplifier is proposed. With the help of mathematical model for the sigle power amplifier, the circuit is simulated ad a demostrator is fabricated ad measured. complex Taylor series are used for modelig the power amplifier by the approximatio of the amplitude trasfer fuctio ad the leveldepedece of the trasmissio-phase of the power amplifier. This ca be uderstood as a simplified form of Volterra series. I our proof of cocept experimet, we verified the cocept but also foud that the adjustmet of the circuit is critically depedet o the drive coditios ad liearizatio is achieved oly for a arrow rage of drive power. The proposed circuit i compared with the covetioal Feed Forward amplifier i additio a sigificat icrease i efficiecy, to miimize the power of the distortio sigal 3IMDproducts at high drive levels. Keywords: volterra series, complex taylor series, 3IMD-products, pre-distortio, feed forward amplifier, power combier. JRE- Classificatio: FOR Code: LiearizatioofPowerAmplifierusigtheModifiedFeedForwardMethod Strictly as per the compliace ad regulatios of: 017. Mohammad Reza Motavalli & Klaus Solbach. This is a research/review paper, distributed uder the terms of the Creative Commos Attributio-Nocommercial 3.0 Uported Licese permittig all o commercial use, distributio, ad reproductio iay medium, provided the origial work is properly cited.

2 Liearizatio of Power Amplifier usig the Modified Feed Forward Method Mohammad Reza Motavalli α & Klaus Solbach σ Abstract- A modified circuit for improvig liearizatio of power amplifier based o the model of the Feed Forward circuit amplifier is proposed. With the help of mathematical model for the sigle power amplifier, the circuit is simulated ad a demostrator is fabricated ad measured. complex Taylor series are used for modelig the power amplifier by the approximatio of the amplitude trasfer fuctio ad the leveldepedece of the trasmissio-phase of the power amplifier. This ca be uderstood as a simplified form of Volterra series. I our proof of cocept experimet, we verified the cocept but also foud that the adjustmet of the circuit is critically depedet o the drive coditios ad liearizatio is achieved oly for a arrow rage of drive power. The proposed circuit i compared with the covetioal Feed Forward amplifier i additio a sigificat icrease i efficiecy, to miimize the power of the distortio sigal 3IMDproducts at high drive levels. Keywords: volterra series, complex taylor series, 3IMDproducts, pre-distortio, feed forward amplifier, power combier. I. Itroductio T he use of high power amplifiers with high liearity for mobile ad satellite commuicatios systems is essetial. For example, i secod ad third geeratio mobile systems, SM (lobal System Mobile), 3PP (third eeratio Partership Project), a large umber of sigals with differet frequecies are trasmitted from BS (Base Statio) by high power amplifiers at the same time [1]. To avoid iterferece of these amplified sigals, the amplifier must operate liear ad that liearizatio is ot possible by classical methods. Sice the amplitude of itermodulatio sigals at the output of the amplifiers depeds o the size of the iput sigals, iput sigals with large amplitude limit the performace of the amplifier. If the coectio betwee a amplifier iput ad output sigals display as trasformatio fuctio that is exteded i the form of a series (for example Taylor s series), we see that for the larger sigal, the role of higher degrees of expressio is more ad more importat, that is, the behavior of amplifiers is o loger liear ad amplifier operates i saturatio (o-liear) regio. The saturatio regio, due to high output power ad resultig high efficiecy i Auhtor α: Departmet of Commuicatio Techique, Faculty of Egieerig, Uiversity of Qom, Qom, Ira. motavallireza@gmail.com Auhtor σ: Departmet of Microwave Ad Rf-Techology, Faculty of Egieerig, Uiversity of Duisburg-Esse, Duisburg, ermay. hft@ui-due.de mobile commuicatios ad satellite systems play a importat role. Itermodulatio sigals with large amplitudes produce i this regio of amplifier which leads to large distortio i output. eerally, oliearity i a amplifier ca appear i two differet forms: first productio of ew frequecy compoets i the output of the amplifier ad secod depedece of gai amplitude ad phase of the amplifiers to amplitude of iput sigals. If amplifiers have bee multiple iput sigals frequecy a type of distortio sigals, that is, 3IMD-products (third order Itermodulatio Distortio) should be cosidered more tha other produced sigals i output of amplifier, because they are ear to frequecy of origial sigals (iput sigals). They are i the rage of useful badwidth amplifiers ad due to limitatios i fabricatio are ot removable i practice []. The distortio sigal of type 3IMD-products i base statios are propagated by high power amplifier i total sed badwidth ad cause distortio ad iterferece i bad of iside chael as well as the eighborig chaels. This problem occurs eve o TV chaels (by 3IMD-products ad eve IMD-products), where a large umber of chaels have placed at a close frequecy ear each other. The aim of this paper is to desig a cocept for a power amplifier with high liearity ad high efficiecy. This paper preseted the proposed circuit for improvig liearizatio of the power amplifier based o the model of the Feed Forward circuit amplifier. I sectio, the priciple of operatio of the amplifier cocept is discussed. A mathematical model of ew amplifier cocept is proposed i sectio 3. I sectio 4, a simulatio model is used to ivestigate i detail the sigals withi the circuit ad the performace ad limitatios of the amplifier. Fially i sectio 5, experimetal proof of ew amplifier cocept is preseted. Simulatio ad measuremet results are compared ad show good agreemet. II. ew amplifier desig The classical parallel power combier amplifier usig two equal liear amplifiers have bee used for may years i order to efficietly produce higher output power levels (doubles the available output power of oe sigle amplifier) ad also i order to improve the reliability ad availability of the amplifier system compoet. However, liearity of amplificatio of each Year lobal Jourals Ic. (US)

3 Year idividual amplifier is ot improved over the idividual amplifiers. Power-added efficiecy of the combier amplifier circuit ca be high whe the amplifiers are drive close to the saturatio level ad cosequetly at high itermodulatio level. Aother successful cocept i liearizatio of power amplifiers is pre-distortio, which ca yield higher power efficiecy, yet at lower cacellatio ratios of uwated products [3-6]. O the other had, amplifiers for very high liearity requiremets i mobile commuicatios successfully employ the feed forward (FF) amplifier scheme, Fig.1, which cacels the oliear itermodulatio-products (IM) of a high power primary amplifier by superpositio of sigals from a auxiliary amplifier [7-11]. However, this cocept suffers from a degradatio of the efficiecy of the amplifier which is maily due to the liearity requiremets o the auxiliary amplifier. The FF-amplifier etails a first loop to extract the itermodulatio Power divider Iput Directioal coupler While the FF-amplifier has foud wide applicatio i commuicatio systems due to its superior itermodulatio suppressio, its efficiecy problem has ispired a modified cocept which allows the auxiliary amplifier to cotribute fudametal sigal power i additio to cacellig the itermodulatio products. The ew circuit, Fig., exhibits a two-loop topology of a covetioal feed forward amplifier, however, both loops are modified ad the two amplifiers are assumed to be idetical power amplifiers. I the first loop, the iput power divider splits the iput sigal i a 1:3 ratio while at the output, the combier is 1:1. The first loop acts as a pre-distortio stage while the secod is a distortio cacellig ad power combiig loop. For the presetatio of the circuit s operatio priciple, we assume that the fudametal iput voltage sigal of the circuit is s i (t). Whe the first power splitter divides this sigal at a ratio of 1 to 3, the iput sigal to the upper power amplifier A is s(t) = 1 10 s i(t) while the sigal icidet to the lower power amplifier B is 3 s(t). The two amplifiers are assumed to be idetical with equal amplificatio υ ad distortio products s (t) ad s (t) for power amplifier A ad B, respectively, uder idetical drive coditios. distortio compoets from the power amplifier output while the secod loop iverts phase ad amplifies this sigal i a auxiliary amplifier such that it destructively superimposes ad cacels the origial itermodulatio sigal at the output coupler. Distortios from the auxiliary amplifier have to be kept very low so that this amplifier eeds to be operated far off saturatio which meas high dc power requiremet. Sice the auxiliary amplifier caot cotribute to the fudametal sigal output power, its supply power degrades the poweradded efficiecy of the FF-amplifier [1-13] ad also sice the auxiliary amplifier is drive at low iput power (low amplitude), a good ratio of cacellig of the 3IMDproducts sigals for small amplitude obtaied, i other words, cacellig of the 3IMD-products sigals for high output power is extremely low. Amplifier Fig. 1: Covetioal feed forward amplifier Delay lie Output The output voltage of power amplifier A is a superpositio of the fudametal sigal amplified by voltage gai υ ad a distortio product: υ s(t) + s (t). The upper coupler samples this combied sigal ad feeds it to the lower coupler i frot of amplifier B. Assumig both couplig coefficiets as jk (ote: complex otatio used to idicate a phase shift of 90 ), the sampled sigal offered to the iput of amplifier B is k υ s(t) k s (t), which is combied with the iput sigal 3 s(t) after its travel through the lower delay lie to give the total voltage sigal, (3 k υ) s(t) k s (t). (Note, the coupler ad delay lie isertio loss have bee eglected for simplificatio ad time delay of the both power amplifiers is compesated by two delay lie i two-loop) With our aim to drive the lower amplifier at the same fudametal sigal level as the upper amplifier, we choose k = to achieve the total voltage sigal υ s(t) υ s (t). It is see that this lower amplifier iput sigal is a pre-distorted versio of the iput sigal of the upper amplifier, with the distortio compoet as a replica of the upper amplifier distortio, but reversed i phase. Assumig the distortio compoet to be much smaller 017 lobal Jourals Ic. (US)

4 tha the fudametal compoet, the lower amplifier will amplify the fudametal sigal plus its distortio compoet by the voltage amplificatio υ. I additio, i the same way as the upper amplifier, the lower amplifier will produce a distortio compoet due to its fudametal sigal excitatio. The resultig output sigal of amplifier B is the υ s(t) s (t) + s (t). Assumig perfectly equal amplifiers A ad B, the distortio sigal origiatig from the lower amplifier is equal to the oe geerated by the upper amplifier, such that the output sigal of the amplifier B is υ s(t) s (t). I the upper sigal path, the output sigal of the power amplifier A travels through a upper delay lie by eglectig the delay lie s atteuatio, appears at the combier as: υ s(t) + s (t). By comparig sigals i the upper ad lower path, we see that the fudametal sigals are equal ad thus ca be combied to give double power while the distortio compoets icidet to the combier are equal ad i ati-phase ad thus cacel to give the total output power: υ s(t). Year Fig. : Cocept of the ew amplifier circuit with sigals idicate It has to be oted that this aalysis is oversimplified with the assumptio of perfect idetity of the two amplifiers ad it eglects the effect of the icreased drivig level of the lower amplifier due to the added predistortio sigal. With slightly uequal amplifiers ad slightly uequal drivig levels, it is still possible to achieve ear-perfect cacellatio of itermodulatio at the price of a loss i the power combiig efficiecy, as ca be show by simulatio, sectio 4. However, a major performace limitatio of the ew circuit was foud by aalysis of a experimetal amplifier system i sectio 5, to origiate i the drive power level depedece of the amplifier voltage gai (magitude ad phase) ad the isertio loss of the delay lies. III. Mathematical Model a) Characterizatio of sigle power amplifiers To describe the behavior of the etire circuit mathematically, each power amplifier should be first characterized separately, it meas that each power amplifier should be represeted as a mathematical model; all other compoets ca be described by simple mathematical models, sice their behavior ca be assumed to be liear i the regio of iterest. For oliear behavior of power amplifier, the power trasfer fuctio has bee measured, i the other words, the behavior of each amplifier is measured regardig 1 db compressio poit ad itermodulatio products of orders 3, 5, 7 ad 9 (3IMD-products, 5IMDproducts 7IMD-products ad 9IMD-products) separately. To develop the mathematical descriptio of this behavior, Amplifier output voltage to the iput voltage is expaded i a series. At first, Taylor series is used, that due to stark o-liearity, model obtaied was ot fit to actual behavior of amplifiers. Next the volterra series is used that relatio other series is more flexible ad for describtio oliearity systems is more appropriate [14-16]. The measuremets of the characteristics of the power trasfer fuctio has bee doe with two toe iput sigal. I the calculatios performed, it has bee demostrated that if both ampltudes iput sigals are equal (such as commo case i SM), volterra series becomes Taylor series with complex coefficiets [17]. Sice for creatig mathematical model of a amplifier with complex coefficiet, complex measuremets must be available, therefore, measuremets of complex voltage gai i close saturatio regio is used. For creatig mathematical model, two siusoidal voltage sigal with equal amplitudes u i ad Frequecies f 1 (ω 1 ) ad f (ω ) are expeded i Taylor series. The geerated sigals iclude mai iput frequecies ad all ew frequecies i output of amplifier ca be summarized i a geeral form as follows [17]: () m u +1 i m u +1 i u out (t) = c =m +1 cos (m + 1)ω 1 mω t + +1 m=0 m=0 =m +1 c +1 cos (m + 1)ω mω 1 t (3.1.1) 017 lobal Jourals Ic. (US)

5 Year With - N as maximum Taylor series degree - as variable idex - m as degree of the geerated itermodulatio sigals - c as taylor series coefficiets The share of the origial sigal (or mai frequecy) ad IDM-products various degrees (3 to 9) as follows: u out,f = +1 =0 +1 u out,3imd = +1 =1 u out,5imd = +1 = u i c +1 (3.1.) +1 u i c +1 (3.1.3) +1 u i c +1 (3.1.4) u out,f,1 u i,1 u out,f, u i, = u out,f,m1 9 u 3 4 i,1 9 u 3 4 i, u out,7imd = +1 =3 u out,9imd = +1 = u i c +1 (3.1.5) +1 u i c +1 (3.1.6) To determie the coefficiets series, liear equatio system of cosidered sigals usig the measured values are writte. The equatio system for mai sigals (3.1.), which is formed of the umber M 1,the correspodig measured values u i ad u out, ca be summarized as follows (Sice the measured values o the left side (3.1.) is real, the absolute terms of right side is used): 9 u u 3 i,m1 4 i,m 1 The equatio system for the 3IMD-products sigals is too similar to the maer as the mai sigals u out,3imd,1 u out,3imd, = u out,3imd,m 3 u 3 4 i,1 3 u 3 4 i, 50 u 5 16 i,1 50 u 5 16 i, N u i,1 N u i, u i,m 1 c 1 e jφ c1 c 3 e jφ c c N e jφ c N (3.1.7) equatio ad the basis of (3.1.3) for the M measured values ca be summarized as follows: N 3 N u i,m u 1 16 i,m The equatio system of the itermodulatio sigals higher order are obtaied same maer. To write equatios system related to the gai amplifier, the output voltage sigal accordig to the u out, = c 1 u i + 3 c 4 3u 3 i + 10 c 16 5u 5 i u 4 i,1 v 1 1 v 1 3 = u 4 i, v M6 1 3 u 4 i,m 8 Now magitude ad phase gai of amplifier (φ S1 = υ, υ = S 1 ) should be regarded as real values separately. Sice Taylor coefficiets should satisfy all equatios system, the idividual liear u i,1 u i, u i,m8 N 3 C 1 u i,1 u i, u i,m u i N The liear equatio system of the gai amplfier for the M 3 measured values as follows: c 1 e jφ c1 c 3 e jφ c c N e jφ c N (3.1.8) iput voltage with amplitude u i ad Frequecy f 1 (ω 1 ) is expeded i Taylor series. I mai frequecy, output voltage ca be summarized as follows: c N (3.1.9) C 3 C N ( equatio system should be solved simultaeously, that is, the followig liear equatios system for M 1, M, M 3, M 4, M 5, M 6 measurd values should be cosidered together: 017 lobal Jourals Ic. (US)

6 Liearizatio of Power Amplifier usig the Modified Feed Forward Method [U out,f ] M1 1 = [D i,f ] M1 N [C N ] N 1 [U out,3imd ] M 1 = [D i,3imd ] M N [C N ] N 1 [U out,5imd ] M3 1 = [D i,5imd ] M3 N [C N ] N 1 [U out,7imd ] M4 1 = [D i,7imd ] M4 N [C N ] N 1 [U out,9imd ] M5 1 = [D i,9imd ] M5 N [C N ] N 1 [υ out, ] M6 1 = [D i, ] M6 N [C N ] N 1 [φ υout, ] M6 1 = {[D i, ] M6 N [C N ] N 1 } With U out,f, U out,3imd, as output voltag for mai sigals, 3IDM-products, i (3.1.), (3.1.3), D i,f, D i,3imd,, as mai matrix i (3.1.), (3.1.3), D i, as mai matrix i (3.1.10) ad v 1 [υ out, ] M6 1 = v v M6 The umber of measuremets M 1, M etc. is ot the same but has bee selected depedig o the available umber of measuremet poits (legth of the curve). b) Optimizatio To determie the complex coefficiets, the trasfer characteristics of the amplifier such as a model i the form of the mathematical approximatio are preseted; i other words usig determie coefficiets model, the differece betwee the model ad measuremets are miimized. To do this, the umerical optimizatio techiques are used. For optimizatio, the complex least square method is a appropriate techique, i which the model coefficiets are determied through settig zero of the partial derivatives [18] that is: e = i=1 g i ( y m (C) y i ) (3..1) φ v1 C 1 c 1 e jφ c1 φ v C, [φυout, ] M6 1 =, [C N ] N 1 = 3 = c 3 e jφ c φ vm C 6 N c N e jφ c N Ad for phase relatioship e = i=1 g i ( y m C φ i ) (3..) With e as error fuctio, y i as measured output value y m as model (desired) output value φ i as measured output value for phase gai g i as weightig fuctio as the measured value ad C as model parameters that must be foud It is atural that all discussed equatios system must be cosidered i error fuctio. For this reaso, cost fuctio (CF) as a fuctio of the total error ad the sum of all the dividual fuctios (fuctios error) are icluded adformed. For 6 output sigals, we have: e Sum = e 1 + e + e 3 + e 4 + e 5 + e 6 = g i y 1m C y 1i i=1 + + g i y 6m C y 6i i=1 (3..3) Our research show that it is ot possible to determie all the coefficiets of the model with the same lowest error, especially the higher order ID-products ca always be modeled worse [17]. Because of that, the weightig fuctio (g 1, g, ) is added i the cost fuctio, the weightig factors for each output sigal also be used so that be able to output sigal or output sigals with chages of this weights as requires to be optimized. So (3..3) is writte i a ew form as follows: e Sum = 1 e 1 + e + 3 e e e e 6 (3..4) The accuracy of the model depeds o the series degree (). A better accuracy ca be achieved i priciple by icreasig of this value. However, it is difficult to ehace the performace for > 13. For the umerical solutio of the optimizatio problem, the fuctio misearch to fid the miimum of the cost fuctio i MATLAB has bee used. Examples of the measured results ad of the model results ca be see i Fig. 3(a), where the voltage gai magitude ad phase of amplifier A is plotted as a fuctio of iput power level for sigle-toe excitatio, ad i Fig. 3(b), where the two-toe fudametal sigal level ad the itermodulatio product levels are plotted versus iput power level. Year lobal Jourals Ic. (US)

7 Year (a) Fig. 3: Measured ad modeled trasfer characteristics of amplifier A: (a) Sigle-ad two-toe gai, (b) two-toe fudametal ad itermodulatio products of orders 3, 5, ad 7 Fig.3 shows some deviatios betwee model ad measuremets i the fudametal sigal levels as well as i the itermodulatio products; ote that i the optimizatio of the model coefficiets the largest weights were used for the fudametal ad third-order itermodulatio products. Fig. 3(a) shows a good match for the magitude ad phase of the amplifier. Error obtaied for the phase just a few degrees ad for the magitude is less tha 0.5 db. Fig. 3(b) shows the divergece of the model whe the two-toe iput power goes beyod the highest iput power level that was used as measuremet data i the calculatio of the Taylor series coefficiets ( 13.8 db). Less dramatic but otable is the characteristic behavior below the divergece regio: (b) Deviatios appear as slight oscillatios with icreasig amplitude closer to the divergece limit. Usig the mathematical model, it was possible to calculate the phase variatio of the fudametal output sigals for two-toe excitatio, Fig.4, which was ot accessible whe measurig with a spectrum aalyzer. Agai, a slight oscillatory deviatio is icluded i the calculated variatio of the phase versus iput power sice a smooth parabolic-shaped variatio should be expected. The observed oscillatory model errors, though ot large o a average over the total extet of the modeled amplifier characteristics, will be foud as major sources of error i the simulatio of the ew circuit, sectio 4. Fig. 4: Simulated phase variatio of two-toe fudametal sigals at amplifiers A ad Bas a fuctio of iput power level IV. Simulatio of the ew Cocept I order to uderstad the iterdepedecies i the itermodulatio cacellatio of the ew circuit, the experimetal circuit was simulated usig the Taylorseries models for the two idividual amplifiers ad usig scatterig matrix models for the other compoets i the circuit (perfect match for all compoets assumed. Due 017 lobal Jourals Ic. (US)

8 to the high volume of the output sigals of the amplifier A ad B, discrete umerical methods (with help DFT - Discrete Fourier Trasform) of the etire circuit i has bee simulated i MATLAB which is more flexible tha other software. Other simulatio programs like ADS 1 have bee show restrictios to model [17]. Oe obvious deviatio of the real circuit from the simplified cocept of sectio is the cosiderable isertio loss (1.6 db) of the two delay lies: With referece to the priciple of operatio ad desigatios of sigals ad test poits give i sectio II, i the first loop, this isertio loss atteuates the iput sigal before the sampled sigal from amplifier A gets subtracted, which requires a reductio of the amplitude of the sampled sigal (k < ) i order to achieve equal υ fudametal sigal amplitudes at the iput of both amplifiers. Also this reduces the amplitude of the sampled distortio sigal s (t) from amplifier A that fed ito amplifier B ad the resultig distortio sigal at the output of amplifier B will bee reduced accordigly. O the upper sigal path, the upper delay lie atteuates the fudametal ad distortio sigals i the same way before they appear at the combier. After the above-described adjustmet of the first loop couplig from amplifier A to amplifier B, the sigals at the iput of amplifier B ca be compared to the iput sigals of amplifier A. Fig. 5 presets the evelope peak voltages of the fudametal toes, of the third-order itermodulatio products ad of the total sigal, ormalized to the evelope peak voltage of the fudametal two-toe sigal at the iput of amplifier A. It is see that the fudametal toes have bee adjusted to be equal i magitude at both amplifiers for the ull iput power level of +3 dbm. At this iput power level, the itermodulatio products cotribute already more tha 10% of evelope voltage to the total iput voltage of amplifier B, which presets a serious deviatio from the assumptio made i sectio II that the drive coditios of both amplifiers are basically equal. Fig.5: Evelope peak voltage of the fudametal toes, the third-order itermodulatio products ad the total voltage at the iput of amplifier B, all ormalized to the iput voltage of amplifier A Lookig at lower drive levels, we fid for the itermodulatio products that the variatio of voltage with drive power level exhibits a oscillatory error compoet as described i sectio 3; a more realistic idicatio of this variatio is also give i the figure. The secod otable effect is that the ormalized fudametal sigal voltage at amplifier B decays with reduced drive level. The reaso becomes clear by ispectig Fig.3(a) ad Fig.4: With reducig iput power level the voltage gai of amplifier A icreases ad the isertio phase chages; thus, at the iput of amplifier B, a icreased sigal sample from amplifier A is subtracted from the 1 Advace Desig System origial iput sigal ad creates a smaller fudametal sigal compoet; the miimum aroud the drive level of -5 dbm is foud to be due to the particular costellatio of the amplificatio amplitude ad phase variatio with drive level. Lookig at the output side of both amplifiers, Fig. 6 shows the amplitudes ad phases of the fudametal ad distortio sigals at the power combier: At the +3 dbm ull drive level; we fid the fudametal sigal from amplifier B larger tha that of amplifier A, which is due to the atteuatio of upper delay lie. At the same time, the two sigals exhibit a cosiderable phase differece of 36 which i combiatio with the amplitude imbalace affects a combier loss of about 0.5 db; ote that the poweradded efficiecy of the combier circuit is further Year lobal Jourals Ic. (US)

9 reduced by about 0.8 db due to the dissipatio loss of 1.6 db from upper delay lie affectig half the geerated power of the power combier. Year (a) Fig. 6: Amplitude (a) ad phase (b) of the fudametal two-toe sigal ad the third-order itermodulatio product at the power combier as a fuctio of iput power Istead of for best combiig efficiecy, the delay lie phase ad the first loop couplig i this simulatio were optimized for itermodulatio suppressio, see by approximately equal itermodulatio product amplitudes at the ull drive level ad a close to 180 phase differece; ote the oscillatory variatio i the itermodulatio magitude plot of Fig. 6(a), which agai ca be attributed to the approximatio error i the sigle amplifiers Taylor series model. The itermodulatio sigal amplitudes are foud approximately equal because the differet effects of the isertio loss of the two delay lies i both loops early cacel, as both the itermodulatio sigals from amplifier A ad of amplifier B get reduced i amplitude. Agai, turig to lower drive power levels, we observe about equal fudametal sigal amplitudes at approx. - 5 dbm iput power, where Fig. 5 has idicated a dip i the iput sigal of amplifier B, thus compesatig the amplitude imbalace due to the upper delay lie atteuatio. At eve lower drive level, the iput sigal amplitude of amplifier B recovers from the dip ad the fudametal sigal from amplifier B becomes larger agai tha that of amplifier A. At the same time, due to the reduced fudametal sigal drive of amplifier B relative to amplifier A, this amplifier produces cosiderably less itermodulatio (s (t)) tha amplifier A, approximately a 3 db reductio for 1 db reductio i drive power. The combiatio of the two itermodulatio compoets s (t) + s (t) at the output of amplifier B thus icreases i magitude as the drive power is reduced ad the gai of amplifier A icreases. With growig differece i the two itermodulatio cotributios at the combier, the circuit loses its cacellatio effect as ca be observed i Fig. 10. (b) V. Measurmet of the Proposed Amplifier Circuit Before buildig a experimetal ew amplifier circuit, two power amplifiers were assembled ad tested: The amplifiers used 900 MHz silico FET power modules MHW916 i cascade with preamplifiers which gave a saturated output power of about 14 W at a gai of about 53 db. Measuremet of the forward trasmissio group delay was performed with sigletoe at small-sigal level usig a vector etwork aalyzer HP87C ad the result was used to specify the legth of the two delay lies i the ew amplifier circuit. Measuremets of the complex valued sigle-toe voltage gai (scatterig coefficiet s 1 ) as a fuctio of iput power level were performed usig the vector etwork aalyzer ad two toe measuremets (910 MHz ad 911 MHz) of the fudametal sigals ad up to the seveth-order itermodulatio product were performed usig a spectrum aalyzer HP8565E (Figs.7 ad 8). Both sets of measuremets were used for the modelig of the amplifiers trasfer characteristics based o a Taylor-series expasio with complex coefficiets. 017 lobal Jourals Ic. (US)

10 Network aalyzer Port 1 Port Power amplifier Atteuator (30 db) (a) Fig. 7: The experimetal sigle-toe voltage gai, (a) sketch of setup ad (b) photograph of bech equipmet ad amplifier circuit (a) Fig. 8: The measured fudametal sigals ad up to the seveth-order itermodulatio product, (a) sketch of setup ad (b) photograph of bech equipmet ad amplifier circuit The setup Sketch of the experimetal proposed amplifier circuit (called the feed forward power combier circuit) is preseted i Fig.9. The fudametal two-toe iput sigal is produced by combiig two sigal geerators ad the iput sigal split of 1:3 voltage ratio is realized by a -3 db divider with a 10 db atteuator i the upper sigal path to amplifier A. All compoets of the power combier circuit are coected by coaxial cables which itroduce some isertio loss ad phase shift. I particular, the required time delays are realized (b) (b) by about 4 m log coaxial cables R13 which itroduce about 1.6 db of extra loss. Betwee the two directioal couplers a variable atteuator ad a variable phase shifter are iserted i order to allow compesatio of amplitude ad phase offsets i the first loop. At the output side, a power atteuator is iserted betwee the power combier ad the spectrum aalyzer i order to save the istrumet from damage by high icidet power. Year (a) 017 lobal Jourals Ic. (US)

11 Year Fig. 9: Adjustmet of the circuit tured out to be difficult: I particular, the variatio of the voltage gai i magitude ad phase as a fuctio of iput power level, as see i Figs. 3 ad 4, required selectig the drive power level for optimum liearity first of all. I our proof of cocept experimet, we set the iput power level to +3 dbm which correspods to the 1 db-saturatio level of the idividual power amplifiers. With the drive sigal level fixed, the sigal at poit 1 at the iput of amplifier B was observed usig a spectrum aalyzer ad the upper delay lie was adjusted ad the variable atteuator ad phase shifter were set such that the fudametal two-toe sigals were approximately equal i level to the sigals at poit 1 at the iput to power amplifier A. This settig, at the same time gives approximately the correct pre-distortio level ecessary for itermodulatio cacellatio at the secod loop. Fial adjustmet of the variable atteuator ad phase shifter was based o the measuremet of the sigals at the combier output; either a optimum cacellatio of the third-order itermodulatio products could be achieved with the fudametal sigals from amplifiers A ad B slightly uequal i amplitude ad phase or early equal fudametal sigals could be achieved with cosiderable differece i phase ad amplitude of the itermodulatio products. Whe the amplifier circuit was adjusted for optimum itermodulatio cacellatio, phase- ad amplitude deviatios gave rise to a loss i fudametal sigal output power of about 0.4 db. I Fig. 10, the measuremet of the proposed amplifier characteristic is preseted ad is compared to the expected characteristic of the covetioal power ( b) (c) (a) The measured proposed amplifier circuit ad (b) the sketch of setup ad (c) photograph of bech equipmet ad amplifier circuit combier circuit usig two amplifiers type A i parallel if excited at the same iput power level as the amplifiers i the proposed amplifier circuit. It is see that the proposed amplifier achieves a otable extesio of the liear rage for the fudametal sigals aroud the 1 db compressio level of the idividual power amplifiers while the third-order itermodulatio products are reduced by 5 ad 45 db at the +3 dbm iput level (the two 3IMD-products are uequal, as explaied i [19]). However, the cacellatio is cofied to a limited rage of power levels aroud the ull -iput level ad the itermodulatio products level is ot improved relative to the level of the covetioal parallel power combier outside this arrow regio ad is eve worse i some parts of the iput level rage. This is a fudametal limitatio of the feed forward power combier compared to the covetioal feed forward amplifier which was ivestigated by simulatio i sectio 4. For a practical applicatio of the feed forward power combier cocept, this meas that i a operatio mode with dyamically chagig drive coditios, e.g., chagig umbers ad levels of modulated carriers as i a mobile commuicatios base statio amplifier, the loop adjustmet has to be adapted dyamically also. Otherwise, uder more static drive coditios, as, e.g., i TV-satellite power amplifiers, the feed forward cacellatio cocept could improve the liearity of preset parallel power combiig amplifiers with oly moderate adaptivity requiremets. 017 lobal Jourals Ic. (US)

12 Year Fig. 10: Measured trasfer characteristic of the experimetal ew amplifier circuit (feed forward-power combier circuit) compared to the expected characteristic of a covetioal parallel power combiig amplifier VI. Coclusio The limitatio of the feed forward power combier circuit was show to be due to the saturatio effect of the upper power amplifier, with its gai variatio offsettig the balacig of the circuit loops. However, drivig the power amplifiers ito the saturatio rage is a ecessary coditio for high power efficiecy. Poweradded efficiecy of our experimetal amplifier was about 36% which is i cotrast to aroud 10% efficiecy of covetioal FF-amplifiers. As a price, the critical drive level depedece of the combier circuit may require higher adaptivity ad cotrol of the loop adjustmets tha a covetioal feed forward amplifier, depedig o the dyamics of the sigals to amplify. By simulatio, it ca be show that at lower drive powers the limitatios get weaker as the itermodulatio cacellatio exhibits broader ull ad cacellatio is improved eve far outside the ull, similar to the characteristics of the covetioal feed forward amplifier, yet losig the advatage i power efficiecy. Refereces Référeces Referecias 1. ASIF, S. Z. "Wireless Commuicatios Evolutio to 3 ad Beyod", Artech House, 007, pp K. Solbach, Feed Forward amplifier for SM, Uiversity Duisburg-Esse, ermay, September J. Cavers, Amplifier liearizatio usig a digital predistorter with fast adaptatio ad low memory requiremets, IEEE Trasactios o Vehicular Techology, vol. 39, o. 4, pp , November, T. Ogawa, T. Iwasaki, H. Maruyama, K. Horiguchi, M. Nakayama, Y. Ikeda, H. Kurebayash, High efficiecy feed-forward amplifier usig RF predistortio liearizer ad the modified Doherty Amplifier, IEEE MTT-S Digest, SPC Electroics Corporatio. Tokyo, Japa, K. J. Cho, J. H. Kim, S. P. Stapleto, RF High Power Doherty amplifier for Improvig the efficiecy of a Feed Forward Liear amplifier, IEEE MTT-S Iteratioal Microwave Symposium Digest, Dept. of Radio Sci. & Eg., Kwagwoo Uiversity, Vol., South Korea, USA Patet, US611146, L. Muceieks, C. Robertso, B. Irvie, N. Salvador, RF Power Amplifier Liearizatio usig parallel Power Amplifier havig itermod-complemetig predisitortio paths, August, USA Patet, US,10,671, H. S. Black, Wave Traslatio System, December F. J. Tegude, Automatic characteristic of the 3- WCDMA mobile power amplifier, Uiversity Duisburg-Esse, ermay, February, N. Pothecary, Feedforward liear power amplifiers, Artech House, P.B. Keigto, High-liearity RF amplifier desig, Artech House, USA Patet, US 6,45,446 B1, J. Eiseberg, L. Altos, S. Avis, Closed loop active cacellatio techique(act)-based RF power architecture, September Kyoug-Joo Cho, Jog-Heo Kim, S P. Stapleto, RF High Power Doherty amplifier for Improvig the efficiecy of a Feed Forward Liear amplifier, IEEE MTT-S Iteratioal Microwave 017 lobal Jourals Ic. (US)

13 Year Symposium Digest, Dept. of Radio Sci. & Eg., Kwagwoo Uiv., Vol., South Korea, T. Ogawa, T. Iwasaki, H. Maruyama, K. Horiguchi, M. Nakayama, Y. Ikeda, H. Kurebayash, High efficiecy feed-forward amplifier usig RF predistortio liearizer ad the modified Doherty Amplifier,IEEE MTT-S Digest, SPC Electroics Corporatio. Tokyo, Japa, E. Chog, "The Volterra Series ad The Direct. Method of Distortio Aalysis", Uiversity of Toroto, research work J. Vuolevi, T. Rahkoe, "Distortio i RF Power Amplifier", Artech House, J. Aikio, "Frequecy Domai Model Fittig ad Volterra Aalysis Implemeted o top of Harmoic Balace Simulatio", Faculty of Techology, Departmet of Electrical ad Iformatio Egieerig, Uiversity of Oulo, research work Mohammad Reza Motavalli, Utersuchug eier Leistugs-kombiatios-Schaltug mit Feed Forward-Liearisierug, Dissertatio, Uiversity Duisburg-Esse, ermay, september, Xiagqia, K. Hogwe, C. Hogxig, The least-square method i complex umber domai, Beijig, Chia, March, N. Carvalho, J. C. Pedro", A Comprehesive Explatio of Distortio Sidebad Asymmetries, IEEE Trasactios o Microwave Theory ad Techiques, vol. 50, o. 9, September, lobal Jourals Ic. (US)