Unformatted text preview: ro
o =[ gm V
∂ Ib −1
=[ Vbe =Vbeq =
∂ Vbe IC Q
VT (A.4) β
Figure A.2: HyVT iβ pi smallsignal 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∂smallsignal hybridpi 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
= shortcircuit 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 gatesource 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: Hybridpi 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 argesignal VP are usually available om d BJT a ith s
soidal Vbe A.3 Largesignal transconductance of a BJT with sin
soidal Vbe When a BJT is driven with a sinusoidal signal such that the baseemitter voltage swing approaches or exceeds a few tens of mV, the collector current waveform becomes nonsinusoidal. 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 nonzero (positive)
g m X1 X2 baseemitter voltage and collector current into quiescent and timevarying components, i.e.,
IC = IDC + iC (A.11) Vbe = VDC + vbe (A.12) where lowercase letters refer to the timevarying 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 timevarying component
of the baseemitter 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 nonsinusoidal 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 timevarying component of
the input signal is equal to zero (v1 = 0). The DC component of the collector current when
the timevarying 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
baseemitter voltage swing exceeds a...
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 Spring '08
 Staff
 Digital Signal Processing, Signal Processing, Bandwidth, message signal, Superheterodyne receiver

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