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Unformatted text preview: CHAPTER 12 m. (3.) Under the quasistatic assumption the carriers and hence the device under analysis are
assumed to respond to a timevaying signal as if it were a dc. bias. In the derivation of the
generalized twoport model, one speciﬁcally equates the total timevarying terminal currents
(i3, is) to the dc. currents that would exist under equivalent biasing conditions. (b) Two separate deﬁnitions are necessary because, contrary to the polarities assumed in
the development of the generalized smallsignal model, the IB and 1C currents were
previously taken to be positive ﬂowing out of the base and the collector terminals in a pnp
BJT. (As noted in Section 10.1, the direction of positive current was sochosen to avoid
unnecessary complications, serious signrelated difﬁculties, in the physical description of
current ﬂow inside the BJT when operated in the standard amplifying mode.) (c) The HybridPi model gets its name from the white arrangement of circuit elements with
“hybrid” (a combination of conductance and resistance) units. (d) Names (see the ﬁrst paragraph in Subsection 12.1.2): gm...transconductance
rc_...output resistance
ruminput resistance
r”...feedthrough resistance (e) The capacitors model the collectonbase and emitterbase pn junction capacitances which
cannot be neglected at higher frequencies. (f) The minority carrier concentration in the base continues to increase as pictured in plot (iii) of Fig. 12.4(d) until a maximum buildup consistent with the applied biases is
attained. The base current varies as Q3113 and therefore also continues to increase toward a
saturating maximum value. (In the quantitative analysis, i3 increases from ICCTt at the
start of saturation to a saturating value of [3313.) Once saturation biased, 1]: remains essentially constant at 5C 5 [CC = VCdRL (g) In words, the base transit time is the average time taken by minority carriers to diffuse
across the quasineutral base. Mathematically (see Eq. 12.22), 1.: WZIZDB. (h) the =IdIB = TB/Tt (i) An £3 < 0 aids the widthdrawal of stored charge from the quasineutral base, which in
turn reduces both the storage delay time and the fall time. (i) A Schottky diode clamp is a circuit arrangement where a Schottky diode is connected
between the collector and base of a BJT as pictured in Fig. 12.7(a). The Schottky diode
conducts at a lower forward bias than a pn junction and therefore minimizes the forward
(saturationmode) bias that is applied to the BJT under turn—on conditions. This reduces
the stored charge and speeds up the tumoff transient. (Also see Subsection 12.2.4.) 121 12.2
The BIT viewed as a twoport network and connected in the commonbase conﬁguration is pictured below. iE=IE+ie i :1 H
C E C c in 2: VEB+Ueb _ VCB+ch => out Invoking the quasistatic assumption we can write
iE(VEB+veb,VCB+vcb) E IE(VEB+veb,VCB+vcb) = IE(VEB:VCB) + ie
ic(VEB+veb.Vcn+vcb) E [C(VEBWeuVCBch) = IC(VEB.VCB) + ic
01' is = IE(VEB+veb.VCB+vcb)JEWEBNCB) ic = IC(VEB+veb.VCB+vcb) —1c(VEB.VCB) Next performing a Taylor series expansion of the first term on the righthand side of the
above equations, and keeping only first order teams, we obtain a! a;
1E(VEB+veb.VCB+vcb) = IEWEBNCB) + £4 Deb +—E‘ Deb
. . BVEB VCB BVCB VEB a: a!
[C(VEB+Ueb.VCB+Ucb) = 1C(VEB.VCB) + 4(3' Deb +~—C—l Deb
aVEB veg aVcn V53 which when substituted into the preceding equations gives . a! E 313 1e 3 veb + vcb
aVEB VCB aVCB VEB it: = __131C Deb +‘_‘31C vcb
aVEB VCB aVCB VEB 122 If the direction of positive current ﬂovi' is as deﬁned in Fig. 10.2 (+15 out and +Ic in for
an npn BIT, +1]; in and He out for a pup BIT), then introducing 311;, 315 all; 3]};
811 E = ; 812 E =
aVBEiVBC aVEBiVCB aVBCiVBE aVCBIVEB
H II 11 ﬂ npn pnp npn pnp +15 out +15 in +1}; out +1}; in _ 81C 31C _ 31C 31c 821 = = . ; 322 = =
aVEB VCB aVBE VBC aVCB VEB aVBC VBE
ﬂ 1'? 1'! fl
npn pnp npn pnp
HQ in +Ic out «Hg in +Ic out yield the emitter and collector ac. current node equations is = Enveb+glzvcb
is = gziveb +822vcb The low—frequency smallsignal equivalent circuit characterizing the ac. response of the
BIT connected in the common base conﬁguration is therefore concluded to be 12.3
From an inspection of Fig. 11.5(d), one concludes 1C E 1.1 mA at the speciﬁed operating point. Given the BIT is to be modeled using the simpliﬁed equivalent circuit of
Fig. 12.2(a), and assuming T = 300 K, one computes (referring to Eqs. 12.9), n=k_T/i= 0 259 = 5.18x 10352
13 5x106 12,4
The node equations appropriate for the B and C terminals in the HybridPi model (Fig. 12.2b) assume the form
ib = vbi‘jrtt+ ”be/”u
it: = gmvbe + “Deb/”1.1 + Ucdro But vbc =—va = vbe  use. Thus = area aa miracles A comparison of the preceding equations with text Eqs. (12.6) leads to the conclusion 1 L = _L
811 ‘rnJrru 812 m _ _L mil. .1.
821 gm ’11 8'2 ’u+’o Clearly r” = l/g12. Moreover, substituting l/rp = —glz into the other three expressions
allows us to solve for the remaining HybridPi parameters in terms of the generalized
model parameters. Speciﬁcally, II I
c:
00
s
N G: = M8]! +812) ’11
8m = 821812 r0 = I/(s’22+312) Although in a somewhat different order, the preceding are Eqs. (12.10). 124 1.2.5 Computations were ﬁrst performed to determine the VEB values required to obtain an
IC = 1 mA with and without accounting for base width modulation. These VEB voltages
were then incorporated directly into the ﬁnal program (P_12_05.m on the Instructor’s
disk). In the MATLAB program, the user is asked whether he/she wishes to input VEB and
. VEC or to use the preset values. The small incremental voltage deviations from the dc.
voltage values used in approximating the partial derivatives appearing in Eqs. (12.5) were
varied until a factor of two change in the incremental values led to no change to ﬁve
significant places in the computed gij parameters. The gi' parameters were in turn used to
compute the HybridPi parameters employing Eqs. (12. 1‘0). Sample results with and without accounting for base width modulation are tabulated
below. In both cases there is at most a thirdplace difference between the gm and r5:
computed from ﬁrst principles and the gm and r,t computed using Eqs. (12.9). As
expected, g1; and g2; are approximately zero when base width modulation is assumed to be
negligible, and therefore re and r” become inﬁnite. Finite values are obtained for r0 and r“
when base width modulation is included. Note that base width modulation has little effect
on gm but leads to a signiﬁcant increase in rm. An increase in ﬁdc s gmrﬁ is of comse
expected when base width modulation is included. No basewidth modulation gm = 3.8685x102 s VEB = 0.67416 v
r7; = 4.596OX103 .0. ‘ 1C = 1.0000 IDA
r!l z: 00 gm = 3.8612X10’2 S ...using Eq. (12.9)
r,[ = 4.6047X103 Q . With basewidth modulation included gm = 38510><102 8 V3; = 0.66961 v
r0 = 1.4932x105 Q VEC =10 v
rIt = 5.9530X103 Q [C = 1.0000 mA r” = 7.5141x107 52 gm == 3.8611X10'2 S ...using Eq. (12.9)
rn = 5.9761x103 Q ‘ 125 MATLAB program script...
%Computation of the Hybrid Pi Parameters (Problem 12.5) %Initialization clear; close format compact: format short e bw=input('Inc1ude base—width modulation? lYes, ZNo...'):
s=input('Manually input VEB and VEC? 1Yes, 2No...'): %Input Eber—Moll Parameters
BJTO %Voltages used in Calculation
VbiEnkT*log(NE*NB/ni‘2);
VbiC=kT*log(NC*NB/ni‘2); if s== VEBO=inputt'Input VEB in volts, VEB
VECO=input('Input VEC in volts, VEC
else 
VECO=10 if bw==1, VEBO=O.669606 else VEBO=0.674162 end: end '):
'); %iB and iC Calculations
VEB=VEBO: ‘
VEC=VECO:
iB=;
ic=;
for i=l:5,
if bw==1,
VCB=VEB*VEC:
BJTmod
else
end
IBO= (l— aF). *IFO+(1 aR). *IRO:
IB1= (l aF). *IFO+(laR). *IRO. *exp(—VEC/kT):
IB=(IBl. *exp(VEB/kT) —IBO);
IC:((aF. *IFO—IRO. *exp(—VEC/kT)).*(IB+IBO)./IB1+IRO“aF.*IRO);
%Reset Voltages
if i==l, VEB=VEBD 0. 0001; elSe; end
if i==2, VEB=VEBO+0.0001: else: end
if i==3, VEB=VEBO: VEC= VECO0.01; else: end
if i==4, VEC=VECC+0.01: else: end
iB=[iB,IB];
iC=[iC,IC];
end 12 6 %Compute Generalized TworPort Model Parameters
911=(iB(3)iB(2))/0.0002;
912=(iB(5)iB(4))/0.02:
921=(iC(3)—iC(2))/0.0002;
922=(iC(5)—iC(4))/0.02; fprintf('\nHybridrPi Model Parameters\n') gm:g21—glz if g22+912==0 ro=inf else ro=1/(922+g12)
end rpi=1/(g11+g12) if g12==0, rmu=inf
else,' rmu=1/g12
end fprintf('\ngm and rpi cemputed using Eqs.(12.9)\n')
gm=iC (1) /o . 0259
rpi=0.0259/iB(1) 12.6 (a) The highfrequency equivalent circuit of Fig. 12.2(c) with vce = 0 can be manipulated
into the form where
Y] = #Hmcd,
Y = —L+ 'coC
2 m J (in 127 Combining node and loop analysis we note = Ylvbe' — szcb' (I)
IDc: = gmvbeI + YZch' '1” ”Ce/1'0 (2)
icrc + 1)ce' + (fb+ic)re = 0 (3)
”be. — 19cc:r + 1)cb' = 0 (4) Eq. (4) is used to eliminate Ucb' in Eqs. (1) and (2). Eqs. (1) and (2) are then combined to
eliminate ”be" Next Eqs. (3) is used to eliminate vce'. Finally, the idib ration is formed
guns +gm—Y2
Yr—l
++Y1+Y2(26 ) 2+;n—(n+%)m+m—1 {YZ—gm)
£9: Y1+Y2
ib Yz—gm 1
Y — Y +— *1
Y1+Y2},62 (2 r0 0 Using the MA'I‘LAB program to compute lidibl versus frequency, one determines an fT = 235 MHz . Data sheets list the fr of the 2N3906 pnp BIT to be approximately 200 MHz. (It should be noted that the Electronics Workbench software program was used
to determine the dc. operating point that produced an IC = 1 mA. The series resistances
listed in the problem statement were those quoted by the EW program. Zero~bias
capacitance values employed in computing the Hybrid—Pi parameters were also extracted
from the Electronics Workbench program.) A plot of lidibl versus frequency, and the MATLAB m—ﬁle constructed to generate the
plot and determine fr, are reproduced on the next page. ~ 12—8 MATLAB program script... %Problem 12.6...fT determination %Initialization
clear; close %Parameters
gm=3.86e—2;
rpi=4.65e3:
ro=2.00e4:
rmu=3.59e6:
Ceb=23.6e12:
ch=2.32e—12:
rble:
rc=2.8: re:0: %ic/ibI vé. frequency
f=logspace(4,9,200): w=2;*pi.*f: Y1=1/rpi+j.*w.*Ceb;
Y2=1/rmu+j.*w.*ch;
R= (YZgm) ./ (Y1+Y2) ;
Den=R.*rc.*¥2 — (Y2+l/ro).*rc — 1: beta=abs(R./Den);
%Plot
loglog(f,beta);
x1abel('f (Hz)'): %beta=[ic/ibl grid
ylabel(' ic / ib l') 12,]
The Eqs. (6.68)](6.69) solution for the 11311:}: ﬂowing in a narrow base diode is ggigogtIMpqvA/ml) IDIFF = (IA
LP ND sinh(xc'/Lp) For application to a BJT we make the symbol replacements...Dp ——> DB, LP —> L3, ND —>
N13, xc‘ ——) W, and VA m) VEB. Then 2 I
[DH7F = qA D._B 1.2L C'OSth/Li) (e qVEB/kT _ 1)
LB NB $1nh(W/LB) Since W/LB << 1 in a standard transistor, the cosh/sinh factor can be expanded as noted in
the problem statement to obtain mm 3 1% , £2] .
Sinh(W/LB) "— W I + 3 (LB) lV/LB << 1 and 2 .. DB ”i N 1(W)2 V B/kT
I = A————— 1+~——— qu 1
Dm (q ME 3L3 ]( )
Introducing the substitutions cited in Subsection 7.3.2, that is,
B 133113 DBTB (I + jcorB)
and {ngEBIkT_ 1) =5 (qveb/kjjquEB/kT
yields the con'esponding a.c. relationship 2
. .. DB "i)
1 = A———
mn (q WNB Finally, by deﬁnition, 1 + 1 ——W2 + jco *WZ ) (quit) quEB/kT
3 D313 31)}; H" YD = GD +J'wCD = idifr/veb
and therefore 1210 12.3 The pictured "0n" point in Fig. 12.3(b) lies right on the [3 = Vs/Rs line. Therefore
[BB E Vs/Rs = 30pA. Inspecting the plot we ﬁnd ICC :—: VcdRL = 5.0 mA. We know Bdc = IdIB = 13/11. Although base width modulation clearly causes ﬁdc to vary someth depending on the dc. operating point, it is reasonable to employ a median value
in obtaining the desired estimate. Speciﬁcally, using the point where the load line crosses
the I]; = 15 pA characteristic, we obtain ' a_m=ttwn=w = 208
13 13 15 >(10'6 my:I : (5x103) _ = 0.80
[BBTB (30X10‘6)(208) 12— 11 ....—_....__———_..T..___.....~_..v.__. . . . 12,2 .
(a)/(b) The required plots and the generating MA’ILAB mﬁle are reproduced below. The
computational relationships used in producing the plots were A; =1 _.1__
1:3 1—x
_1. V ._
Ea: 111x) mung—o
“‘8 2 . _
l lIx) "'lféﬂ
where x = ICCTI/IBBTB 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ICC mull IBB tauB 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
ICC taut] IBB tauB MATLAB program script... 
%Rise and Storage—Delay Time plots (Prob. 12.9) %Initialization
clear: close %Rise time computation x=linspace(0.01,0.99): rise=log(l./(1—x)); %rise=tr/tauB plot(x,rise): grid x1abe1{'ICC taut / IBB tauB’); ylabel('tr / tauB')
pause %St0rageDelay Time computation delay0=log(1./x); %delay0=tsd/tauB, xi=0
delayl=log(2./(1+x}): %delay1=tsd/tauB, xi=1
plot(x,delay0,x,delayl); grid xlabel('ICC taut / IBB tauB'); ylabel(‘tsd / tauB')
text(0.08,2.8,'xi=0'); text(0.08,0.8,'xi=l') 1213 12.19 (a) Let :1 be the time when ic 2' 0.9Icc and :2 the time when ic = 0.1Icc. Making use of
Eq. (12.31b), we can then write I.CUI) = 0.9ICC = 133%[(1+§)e41/TB_§] iCUz) = 0.11cc =IBB%[(1+§)6"7JTB 5] Solving for the t’s yields 1% 1 + 5 )
m —
091CCTt/1BB’FB + .5 1 +5
‘2 3 113 w
0.lI(jcq/IBBTB +§ and per the measarements— based deﬁnition tl If = tg—tl = mln( 0.91ccw133m + 5) _ a3 I“(0.9x + 5 ) OJICCTt/IBB’IB + if 0.11: + f where x = [cc‘ﬁ/IBBTB (b) With g: 0 and 1;: 1, the part (a) relationship simpliﬁes to f_r : Iln9 ...if§=0‘
TB ‘ 0.9x+1  _
menu) "‘lfg‘rl The requested tf/t‘g versus x plot is displayed on the next page along with the script of the
MATLAB mﬁle used to generate the plot. Consistent with the analysis in Subsection 12.2.3, the plotted fall times decrease
when t: > 0. This occurs because an if; < 0 aids the withdrawal of charge from the
quasineutral base. If the x—ratio increases either due to an increase in 1 cc or a decrease in
1313, the charge storage is enhanced relative to the charge removal capability of the base 1214 current. Thus, the tf/‘IB ratio for the 5: 1 curve increases with increasing x. When E = 0,
the charge removal from the base occurs only by recombination and the falltime collector
current assumes the simple form, ic = Aexp(—t/tj3). Since If is always evaluated
employing the same relative ic values, ic(t1)/ic(t2) = constant 2 CXp(If/TB), and If/TB is
seen to be a constant independent of x. 2.5 15
a:
E
E
1
Q5
0
0 DJ 02 03 04 as as Q? as as 1
ICC taut] IBB tauB
MA'I‘LAB program script...
%Fa11 Time (Problem 12.10)
%Initialization clear; close %Fall Time computations x0=[0,1]: y0=[log(9),log(9)]; %tf/tauB when xi=0
x1=linspace(0,l): _
yl=log((0.9.*xl+1)./(0.1.*xl+1)}; %tf/tauB when xi=1
plot(x0,y0,x1,yl); grid ‘ xlabel('ICC taut / IBB tauB'): ylabel('tf / tauB')
text(0.47,2.1, 'Xi=0'): text(0.47,0.4, 'xi=1': 12—15 ...
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