antenna antenna (4.139)

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1 .6.6 The Lmts of Usable Input Levels for LNAs The sgnal voltage level delvered to the nput of an LNA from the antenna may vary n a very wde nterval, from very weak sgnals comparable to the nose level, to hgh ampltude sgnals resultng n severe nonlnear (harmonc and ntermodulaton) dstorton. For weak sgnal levels, the sgnal-to-nose rato at the output determnes the lower lmt of the nput sgnal. The acceptable mnmum value of the output S/N rato depends on the area of applcaton and the demodulaton technques appled to the ncomng carrer. As mentoned n Secton.6., the output S/N rato s determned by the nose factor (F) of the amplfer that s a measure of the nternally generated nose of the amplfer and the nput S/N rato. On the other hand, the nput S/N rato depends on the feld strength of the ncomng wave, the antenna and the couplng scheme connectng the output of the antenna to the nput of the amplfer. An antenna can be consdered as a transducer delverng a voltage to ts output port, proportonal to the strength of the electromagnetc wave llumnatng the antenna. The proportonalty factor depends on the drecton of the ncomng wave, the structure of the antenna and the frequency. It must be kept n mnd that an antenna s a resonatng system (wth very few exceptons) such that the output voltage reaches ts maxmum at a certan frequency, dependng on the structure (dmensons, shape, etc.) of the antenna. It s obvous that to maxmze the nput sgnal voltage of the LNA and correspondngly mprove the nput S/N rato, t s necessary to operate the antenna at ts resonance frequency. The open-ended maxmum output voltage of an antenna (v o antenna ) can be expressed n terms of the magntude of the electrc feld component of the ncomng wave (E) and the maxmum value of the proportonalty factor (α antenna ) that corresponds to the resonance frequency and zero angle between the drecton of the wave and the axs of the man lobe of the radaton pattern of the antenna: antenna antenna vo E (.19) Snce the current of a MOS amplfer s controlled by the nput sgnal voltage (and not sgnal power), v o antenna must be transferred to the nput of the amplfer wth possble maxmum effcency to ncrease the nput S/N rato and correspondngly, the output S/N rato. The sgnal voltage reachng the nput port of the LNA depends on the nterface between the output of the antenna and the nput of the amplfer. If we denote the

2 couplng effcency of ths nterface wth β couplng, the nput sgnal of the amplfer correspondng to a certan value of E becomes antenna couplng vn E (.10) For example; f the antenna s very close to the nput of the amplfer as mentoned n Secton.6.1 and f the nput mpedance of the amplfer s matched to the antenna at the resonance frequency, the nput voltage of the amplfer s equal to one half of v o antenna, n other words β couplng = 0.5. On the other hand, f the nput mpedance s very hgh compared to the nternal mpedance of the antenna, the nput voltage of the amplfer s equal to v o antenna and β couplng = 1. If a transmsson lne has to be used between the output of the antenna and the nput of the amplfer, the conventonal approach s to match the nput mpedance of the amplfer to the characterstc mpedance of the lne and to use the antenna at a pont where ts mpedance s real and equal to the lne mpedance. The β couplng n ths case s consderably smaller and obvously mpars the S/N rato. For hgh sgnal levels there s another problem not only for LNAs but all knds of amplfers; the nonlnear dstorton of the output sgnal whch produces harmoncs of the snusodal components of the nput sgnal and ther ntermodulaton products. The amount of nonlnearty depends on the supply voltage, the load, the poston of the operatng pont on the output characterstc curves, as mentoned n Secton.8. and obvously, on the nput sgnal level. Ths basc behavor of amplfers wll be examned on the most frequently used common source amplfer, n the followng. The nonlnearty s prmarly related to the relaton between the dran current and the gate-source voltage. The output voltage s equal to the voltage drop on the load. In case of a wde-band amplfer, the load s resstve up to the vcnty of the -db frequency. Therefore, the nonlnearty of the dran current drectly reflects on the output voltage. But n the case of a tuned amplfer, the tuned load flters out the harmoncs and the ntermodulaton products fallng outsde of ts pass band. The dran current depends not only on the gate-source voltage but also on the dransource voltage to some extent, and consequently, on the load. In Fg..55(a) the statc (DC) and the dynamc load lnes of a common source amplfer loaded wth a parallel resonance crcut are shown. The statc and the dynamc load lnes are determned by the DC resstance of the nductance (r L ) and the resonance mpedance of the load (R eff ), respectvely. The nput voltage-to-output current characterstc of ths crcut, called the dynamc transfer characterstc, can be derved from Fg..55(a) and s shown n Fg..55(b). The nonlnearty of the transfer curve s obvous; to obtan a large dynamc range for the dran current wth an acceptable dstorton, the operatng pont (Q) must be n the mddle of ths curve.

3 In case of a resstance loaded (wde-band) amplfer, the statc and dynamc characterstcs are same up to the vcnty of the upper cut-off frequency. I D I D V GS =1.1V V GS =1.0V Q V GS =0.9V Q V GS =0.8V V GS =0.7V V CEQ V CC V CE V GS =0.6V V GS (V) (a) (b) Fgure.55 (a) The DC load lne (dashed lne) and the dynamc load lne (sold lne) on the output characterstc of a transstor. (b) The dynamc transfer characterstc correspondng to the AC load. The nonlnearty of the dran current can be represented wth a Taylor seres referred to the operatng pont: di 1 d I 1 d I I I V V V 2 D D 2 D D DQ GS ( ) ( ) 2 GS GS dvgs 2! dvgs! dv Q Q GS Q Ths expresson can be converted nto g v g v g v g v 2 d 1 gs 2 gs gs gs (.11) where v gs and d are the small sgnal components of the gate-source voltage and the dran current, respectvely. The output sgnal voltage s v o d R ef f for a tuned amplfer wth a bandwdth determned by the effectve qualty factor of the

4 resonance crcut. For a wde-band amplfer, the output voltage s vo drdup to the db frequency of the crcut. The g coeffcents of the seres gven n (.11) depend on the shape of the dynamc transfer characterstc and each of them can be postve or negatve. The frst coeffcent (g 1 ) n (.11) s notng but the g m of the transstor for ths operatng pont, and others are the hgher order dervatves of g m [see references] 29. Snce the magntude of the coeffcents usually decreases wth ncreasng order, the fourth and hgher order terms wll be neglected to keep the expressons manageable. To evaluate the harmonc dstorton, a snusodal nput sgnal voltage ( vgs v cost) must be appled to the nput of the amplfer. The correspondng dran current can be solved as: 1 2 d g2 g 2 8 gv 1 gv cost gv 2 gv cos 2t gv cost (.12) To evaluate the ntermodulaton dstorton, we shall nvestgate the effects of two snusodal sgnals 0 wth dfferent frequences arrvng at the nput of the amplfer: v V cos t V cos. gs t 29 See: P. M. Jupp, D. R. Webster, Applcaton of Dervatve Superposton to low IM Dstorton IF Amplfers, Roke Manor Research Ltd, B. Razav, RF Mcroelectroncs, Prentce Hall, The assumpton of two dfferent sgnals represents the smplest case; f there are more than two sgnals the ntermodulaton occurs among all of these components.

5 For ths case, the dran current becomes: d g2( V1 V2 ) gv 1 1 g( VV 1 2 V1) cos 1t gv 1 2 g( V1V1 V2) cos2t gv 2 1cos 21t gv 2 2cos 22t gv 1cos1t gv 2cos2t gvv 2 1 2cos( 12) t gvv 2 1 2cos( 12) t gv 1V2cos(2 12) t gvv 1 2cos(2 21) t 2 2 gv 1V2cos(2 12) t gvv 1 2cos(2 21) t (.1) It s obvous that for V 1 = 0 or V 2 = 0 (.1) reduces to (.12) From (.12) t can be seen that: The dran current has acqured a DC term (the rectfcaton term) due to the even harmonc coeffcents. Ths term ndcates a shft of the operatng pont. The coeffcent of the fundamental frequency has two components; a term that ncreases lnearly wth the ampltude of the sgnal (Fg..56, lne A), and a term orgnatng from the thrd order nonlnearty and ncreases wth the thrd power of the ampltude of the sgnal (Fg..56, curve B). For an S-shaped transfer curve t can be seen that the sgn of the component orgnatng from the thrd order nonlnearty s negatve. Therefore, the actual varaton of the fundamental term s equal to (A- B) on (Fg..56).

6 d A x (A-B) B x v gs Fgure.56 The lnear term (A), and the cubc term (B) of the fundamental component of the dran current as a functon of the nput sgnal voltage. The sold lne corresponds to the actual varaton of the fundamental term of the dran current. Ths saturaton of the dran current ampltude ndcates that n the case of amplfyng a modulated sgnal, the share of the term assocated wth the cubc term must be well below than that of the lnear term n order to mantan the relatve ampltudes of the carrer and sdefrequences orgnatng from the modulaton. The usually accepted crteron s not to exceed the nput level correspondng to the -1dB fall of the output power whch corresponds to a factor of reducton of the output power, or factor of reducton of the output sgnal current or voltage. The nput voltage correspondng to ths pont can be calculated from gv V dB 1 g g gv g (.1) The coeffcent of the second harmonc s composed only of the even order parameters, and the coeffcent of the thrd harmonc s composed only of the odd order parameters. These terms are not mportant for

7 tuned amplfers, snce these harmoncs are normally fltered out by the output tuned crcut. But for wde band amplfers they may be n the pass band of the amplfer. As already mentoned, symmetrcal crcuts elmnate the even harmoncs. Another advantage of the elmnaton of the even harmoncs s the elmnaton of the DC shft of the operatng pont due to the rectfcaton term. For (.1), the above nterpretatons related to the rectfcaton term as well as the second and the thrd harmoncs are equally vald. The effects related to the ntermodulaton products can be nterpreted as follows: The frst order ntermodulaton products, ( 1 2) and ( 1 2) are far from ω 1 and ω 2 and wll be fltered out by the output resonance crcut for tuned amplfers. The second order ntermodulaton products, ( 1 2 2) and (2 1 2 ) also are prone to be fltered out. The other set of the second order ntermodulaton products, (2 1 2) and (2 2 1), (we shall name them as the close ntermodulaton products ) are crtcal. If the dfference between ω 1 and ω 2 s Δω, for example 2 1, these second order ntermodulaton products become (2 1 2) 1 (2 ) In Fg..57(a) the relatve postons of the fundamental components and these ntermodulaton products are shown for V 1 = V 2 = V. For a tuned amplfer, f one of these ntermodulaton products falls nto the pass band of the amplfer, t nterferes wth the actual sgnal (the carrer and ts sde frequences) that s ntended to be amplfed (Fg..57(b)). It s obvous that the magntude of these ntermodulaton products must be well below that of the receved sgnal. It must be noted that n case of a wde-band LNA, all these components exstng n the dran current produce correspondng output voltage components up to the db frequency of the amplfer. In ths case an nput flter s necessary to exclude all sgnals other than the carrer (and ts sde frequences) ntended to be receved.

8 I 11 I 22 I 12 I (2 ) (2 ) (2 ) (2 ) 2 1 (a) (b) Fgure.57 (a) The close ntermodularon products of two snusodal sgnals havng equal ampltude and dfferent frequences. (b) Illustraton of the nterferng effects of one of these ntermodularon product on an LNA tuned to (or to the vcnty of) ths frequency. From (.) t can be seen that the magntudes of the fundamentals and these second order ntermodulaton products are affected by g and rapdly ncrease wth the ampltudes of the ω 1 and ω 2 components, V 1 = V 2 = V, as descrbed n the followng The magntudes of the ω 1 and ω 2 components of the dran current: 9 9 I11 I22 gv 1 gv gv 1 g (.15) The magntudes of (2 1 2 ) and (2 2 1 ) components: I12 I21 g V (.16) In case of a tuned amplfer, f all these components are close to each other and are n the pass band of the amplfer, the correspondng components of the output voltage become 9 V11 V22 Reff gv 1 Reff g (.17) and V V R g V (.18) eff Therefore, the rato of the magntude of one of the fundamental components to the magntude of one of the close ntermodulaton products can be wrtten as

9 V V R 9 gv R g V 1 2 (.19) eff 1 eff 11 g1 12 R g eff g The numercal value of ths rato for the nput level correspondng to the -1 db pont can be calculated from (.1) and (.19: V g V db (.150) V g V V dB 0.15 Ths relaton provdes a hnt for a method to fnd the nput level correspondng to the -1 db pont usng a relatvely easy ntermodulaton measurement. Another metrc to evaluate the nonlnearty of an amplfer s the nput level correspondng to the so-called thrd order ntercept pont, or IP n short. IP s defned as the ntercept pont of the lne correspondng to the lnear term of one of the fundamental components of (.1) and the lne correspondng to the magntude of one of the close ntermodulaton components, shown on a log-log axes or n db scales (Fg..58). V o (dbv) V 12 1 db V 11 1dB V IP V (dbv) Fgure.58 Defnton of IP, the thrd order ntercept pont.

10 The nput levels correspondng to the IP can be calculated from (.1): g gv g V V 1.15 (.151) 1 1 IP g The nput voltage correspondng to the -1 db pont was calculated as: Therefore; V g (.1) 1dB g IP db (.152) V 1dB Ths expresson shows that the nput level correspondng to the IP s well above the level correspondng to the -1 db pont, and practcally not applcable due to the excessve nonlnearty 1. 1 Although the ntercept pont les outsde the usable lmts of the amplfers, the reason behnd the wdespead acceptance of the IP as a metrc of nonlnearty s the smplcty of ts determnaton by a relatvely easy measurement procedure.

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