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Estentccollector cor rcnt ange from a few tenths of a

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Unformatted text preview: ro o =[ gm V ∂ Ib −1 =[ Vbe =Vbeq = ∂ Vbe IC Q VT (A.4) β Figure A.2: HyVT iβ -pi small-signal mo del for BJT br d = = IC Q rπ (tens to hundreds of k Ω), rx ∼ (a few tens of Ω), and rµ >β ro . Since ro and rµ are relatively .025β large resistances, and each rs shunted by a capacim nce, ro40IC Q µ can usually be ignored at iπ and r g ta is high frequencies. On data sheets CµIC Qoften given as Cob , which is the output capacitance w t er s o Q i t r e q in sm l - mmon- r ae scc ndug t hat e n c ypi al values fu e µ r d the theaclosignal hbaneIConfiscuancioui.esTentccollector cor rCnt. ange from a few tenths of a pF to a few pF . Data sheets mneanot o dee Cπ ehplbetla, ioutowismaldihage theqvalncyof ifT alsr sa p A useful li ay r m giv l for txe ici h yv b r f ll in l c i th fr ue ue s gn fo u particular bias currCnb,iasherent is the∂small-signal hybrid-pi mo del shown in Figure A.2. T the D e t w poi = [ IC ] (A.5) gm V =Vb m ∂ Vbe 1 be geq fT = , (A.6) 2π Cπ + Cµ rµ IC Q = short-circuit current gain has a magnitude of unity; it and fT is the frequency where the V T is often called the maximum frequency of oscil lation. The value of fT depends on gm and, rx Cµ B C therefore, on how the transistor is biaβed. s (FFET) ( ET) Figure A.3 lo oks very similar to the BJT mo del (Figure A.2). The gate-source resistance, Fi , i r g A. r lo k r v ry si ar la t t th i B e T mo o e gs ure A. hi . The i at -olds ( e . rgsgus e ene3allyolas geecompmied rto o he empJdance df lC(F.igWhen t2)s conditgon ehsourci.eresistan r su is ge t e high arge comp rgs c to th o imp d a o ce he mod . atgs , fficiennlyrally lfrequencies) ared an be e mitteedfrnm tof Cgs .elWhen this condition holds ( at sufficiently high frequencies) rgs can be omitted from the mo del. C dg G + vg s C dg G -+ vg s S- D D rgs Cg s Cg s rds rgs gm vgs C ds rds gm vgs C ds S Figure A.3: Hybrid-pi equivalent circuit for the FET For a junction FEFigure A)3:the ybanscpnductivale ns pircuirtiforal ho FETquare ro ot T (JFET ., H tr rid- o i equ anc e i t cropo t on t t e the s of the drain current, ID . The proportionality constant depends on the saturation drain currentr IDSunctid nhFEiTchoff Eoltagthe Pt,ransconductance is proportional to the square ro Fo , a j S , an o t e p n (JF v T), e, V i.e. of the drain current, ID . The propo2 tionality constant depends on the saturation dr r gmo= age, V IDS S ID . (A.9) current, IDS S , and the pinchoff v lt |VP | P , i.e. The parameters IDS S and VP are usually availa2 from device data sheets. ble gm = |VP | IDS S ID . (A A.3 parLmeters IDS S and transconductance frof a evice datwsheets. inuThe a arge-signal VP are usually available om d BJT a ith s soidal Vbe A.3 Large-signal transconductance of a BJT with sin soidal Vbe When a BJT is driven with a sinusoidal signal such that the base-emitter voltage swing approaches or exceeds a few tens of mV, the collector current waveform becomes non-sinusoidal. mple, suppose that the lo op is broken to the right of Z3 in Figure 5.3b. of Z3 normally lo oks into Z1 when the lo op is closed, we terminate the lo o n in Figure 5.5. Then the lo op gain is computed by exciting the circuit at Z3 + VI N - Z1 gmVI N Z2 + Z1 VOU T - Figure 5.5: Feedback lo op terminated with Z1 pened lo op and computing the output across Z1 : VOU T − g m Z1 Z2 A= − O Alo = VVIU T = −lom Z2 ||NZ1 + 1 +)Z21Z1Z33 = Z1gm Z1 Z23 g ( = ZZ3 Z + Z VI + +Z2 +Z N seful insights can be gained if we assume for the moment that Z1 and Z2 , i.e., Z1 = j X1 , Z2 = j X2 . We allow Z3 to have a non-zero (positive) g m X1 X2 base-emitter voltage and collector current into quiescent and time-varying components, i.e., IC = IDC + iC (A.11) Vbe = VDC + vbe (A.12) where lower-case letters refer to the time-varying component of the quantity. Later, we will make the assumption that the transistor bias network acts to keep the DC component of the collector current at a nearly constant value. Suppose that the time-varying component of the base-emitter voltage is sinusoidal, i.e., Vbe = VDC + v1 cosω t (A.13) and let x = v1 q/kT = v1 /25mV (at room temperature). Then IC = IS exp[VDC q/kT ] exp[ x cosω t] (A.14) The term exp[ x cos ω t] is a non-sinusoidal periodic function of time and can be expanded in a Fourier series. The series is exp[ x cos ω t] = Io (x) + 2 ∞ ￿ In (x) cos(nω t) (A.15) n=1 where the coefficients In (x) are values of the modified Bessel function of the first kind. Using this relationship, the collector current waveform can be written as ￿ IC = IDCo [Io (x) + 2 In (x) cos(nω t)] (A.16) n=1 Here IDCo is the DC component of collector current when the time-varying component of the input signal is equal to zero (v1 = 0). The DC component of the collector current when the time-varying component of the input signals is not zero is given by IDC = IDCo Io (x). (A.17) This function is plotted in Figure A.4, which shows that Io (x) grows very rapidly when the base-emitter voltage swing exceeds a...
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