HighFrequencySmallSignal bjt

High-Frequency Small-Signal BJT Model ( S.O. Kasap, 1990 - 2002: v.1.1) 1An e-Booklet

HIGH-FREQUENCY SMALL-SIGNAL BJT MODELSafa Kasap

Department of Electrical EngineeringUniversity of Saskatchewan

Canada

pn(0)

E

IB x

pn(x)

IC

IC

IE

VCE

VBE

INPUT

OUTPUT A pnp transistor operated in the activeregion in the common emitterconfiguration. The dc voltage acrossthe BE junction, VBE, controls thecurrent IE and hence IB and IC. Theinput current is the current that flowsbetween VBE and the base, which isIB. The output current is the currentflowing between VCE and thecollector, which is IC.

C

E

B

Figure 1

We consider the pnp bipolar transistor depicted in Figure 1. Under normal dc operating conditions, theminority carrier (hole) concentration profile and therefore the injected minority carrier (hole) charge in thebase will be constant, as shown in Figure 1. The base current simply replaces those majority carriers(electrons) continuously consumed in the recombination. If QB is the total amount of injected minoritycarrier charge in the base due to the dc voltage VBE, which is shown in Figure 2, then the dc base current IBis simply QB/h, as 1/h is the mean rate of recombination,

I QB Bh

=

DC base current

pn(0)

E

IB + ib x

E

B

C

VCE

VBE

RC

vbe(t) pn(x) QB

QB

p'n(0)

IC

IC + ic

IE + ie

INPUT

vce(t)

OUTPUT

A pnp transistor operated in the activeregion in the common emitter amplifierconfiguration. The applied signal vbemodulates the dc voltage across the BEjunction and hence modulates the injectedhole concentration up and down about thedc value pn(0). The solid line shows pn(x)when only the dc bias VBE is present. Thedashed lines show how pn(x) is modulatedup by a positive small signal signal vbesuperimposed on VBE.

Figure 2


Suppose that we now increase VBE by VBE, which leads to an increase in pn(0) to p'n(0), which isshown as the dashed line in Figure 2. With more minority carriers injected, there is now an additionalstored charge in the base, as indicated by the gray shaded area and labeled as QB in Figure 2. Since thestored charge, QB, in the base depends on VBE, there is a capacitive effect appearing between the BEterminals. We represent the capacitive effect across the base-emitter terminals by defining a small-signaldiffusion1 (or storage) capacitance, Cdiff, by

C QVdiff

B

BE

=

Base diffusion capacitance

The stored charge, neglecting recombination, as shown in Figure 2, is

Q eAW p eAW p eVkTB B n B no

BE= ( ) = 12 0

12

exp

Differentiating this with respect to VBE we obtain

C eQkT

e IkT

eIkT rdiff

B h B h C h

be

= = = =

where we have used

rkTeI

kTeIbe E C

=

This is the capacitance that has to be charged and discharged as the signal vbe modulates VBE. Its value isgenerally greater than the capacitance of the BE depletion region. For example, typically, for a transistorwith = 100 and a minority carrier lifetime of 1 s, when IC = 1 mA, rbe = 2.5 k and

Cr

diffh

be

= = =

102500

0 46

. nF

There is also the capacitance of the depletion region, Cdep, between the base and the emitter. Itsvalue for an abrupt junction was derived during the treatment of the pn junction and depends on the widthof the BE depletion region, WBE,

C AWdep BE

=

where WBE decreases with increasing VBE. The total small-signal capacitance across BE is thereforeCbe = Cdiff + Cdep

The small signal equivalent circuit of the BJT must therefore also include the capacitance Cbe, asshown in Figure 3 (a). As the base-collector junction is reverse biased, there is, in addition, a depletionregion capacitance between the base and the collector terminals, which is shown as Cbc in Figure 3 (a). It isapparent that Cbc provides a feedback path for the output current into the input, which, generally,deteriorates the gain of the BJT amplifier.

1 The stored charge in the base is due to the diffusion of injected minority carriers in this region.


ib

RCvberb'e

Cb'e

B

E E

C

vcev

s

Rs

S

S

vin

Cb'cB'rbb'

vb'e rce

ic = gmvb'e

ib

RCvbe rbe

Cbe

B

E E

C

vcev

s

Rs

S

S

vin

ic = gmvbe

CbcSmall signal equivalent circuitAC source Load

(a) The small-signal model including storage anddepletion region capacitances.

(b) The high frequency small-signal model includingstorage and depletion region capacitances and basespread and collector-emitter resistances.

Simplified small-signal HF model of the bipolar transistor in the CEconfiguration.

Figure 3

The structure and geometry of nearly all bipolar transistors is such that recombination invariablyoccurs over a region rather than at one specific location. Figure 4 shows a simplified schematic sketch of atypical pnp BJT fabricated by the commonly used planar technology. It is apparent that the device is notsymmetric. The surface area of the collector junction is larger than that of the emitter junction. We definethe active base region as the volume of base that contains the majority of the emitter to collector holecurrent, which is indicated (only roughly) in the figure. The base current has to supply electrons to theclosest and farthest points of recombination, which means that it flows over the whole active region of thebase, though its magnitude gets smaller farther away from the base terminal, B. Since the base material hasa finite resistivity, there is therefore a distributed voltage drop along the active region. In other words, thevoltage across the base-emitter, VBE, gets smaller as we move away from B . Thus the minority carrierinjection and hence IE gets less farther away from B . Suppose that point B ' is some mean point in theactive base region, as roughly shown in Figure 4, that represents an effective or a true base pointsuch that VBE is the effective base-emitter voltage that controls the emitter current. Thus, by definition,

I I eVkTE EOB E

= exp

Then we can calculate (or obtain by measurement) the effective resistance, say rbb, of the base region,which gives rise to the voltage drop VBE VBE:

VBE VBE = IB rbb


where rbb is called the base spread resistance. It takes into account not only the distributed voltagedrop in the active base region due to flow of IB but also some of the base material outside the active base tothe base terminal, B. Figure 4 shows that rbb is placed between the base terminal, B , and the true basepoint, B, and has the base current, IB, flowing through it.

p

E B Metallizationfor contacts

n

C

IC

ElectronflowIB

rbb'IB

Emitter tocollector holecurrent line

Nearly all the injectedminority carrier chargeis in this active region

Insulation (SiO2)p+ B'

A simplified representation of the structure of a typical pnp bipolartransistor (for example, as fabricated by diffusion processes). The basecurrent flows through a finite semiconductor region to replenish theelectrons lost by recombination. It has to supply electrons to the closestand also to the farthest point in the base. Point B' represents a meanpoint in the active base region that acts as an effective base point.

Figure 4

pn(0)

x

Base

WBCWB

W'B W'BC

pn(x)

pn(x)

VCB = 10 V

VCB = 5 V

SCLThe Early effect. When the BCreverse bias increases, the depletionwidth WBC increases to W'BC, whichreduces the base width WB to W'B.As pn(0) is constant (constant VEB),the minority carrier concentrationgradient becomes steeper and thecollector current, IC, increases.

Figure 5

As B represents the true base point, the capacitances Cbe and Cbc involving the active base are nowbetween B and E, and B and C so that they become Cbe and Cbc. The modified small signal equivalentcircuit is shown in Figure 3 (b). The model, in addition, has a resistance rce placed between the CEterminals due to the following effect. Suppose that VBE is kept constant and VCE is increased by an amountVCE. This increases the reverse bias VCB by VCB. Consequently the base width, WB, gets narrower as


shown in Figure 5 (due to the Early effect), which leads to a steeper minority carrier concentration gradientand hence to a greater collector current, say by an amount IC. Since VCE leads to IC the two outputparameters are related just as they would be in a resistance. We define a small signal collector-emitterresistance rce by

rVI

v

iceCE

C

ce

c

=

The modulation of the base width WB by VCE is not very strong and hence IC is generally verysmall. Consequently rce is quite large, typically greater than 50 k. It is represented as a resistance acrossthe CE terminals of the small signal equivalent circuit as shown in Figure 3 (b).

There are other effects, usually of secondary nature, in the small signal equivalent circuit which arebeyond the scope of this book. Their inclusion does not significantly affect the predictions of the model.Small-signal equivalent circuits, such as that in Figure 3 (b), are most useful in obtaining the frequencyresponse.

For example, suppose that an ac source, vs, is connected across the base-emitter (in series with adc bias, which is not shown in small-signal equivalent circuits). We adjust vs so that the input current, ib,is always constant. What is the current gain, , at different frequencies? The impedance between B and E,zbe, is given by rbe and Cbe in parallel. At low frequencies, we can neglect the impedance of Cbe so thatthere is only rbe. The low-frequency gain, , is then

= = = =ii

v

v

r

rc

b

m b e

b e

b e

m b e o

gg

Low frequency

where by definition o = gmrbe and represents the low-frequency current gain.At high frequencies, zbe becomes shunted by the small impedance of Cbe so that

z j Cb e b e =

1

The magnitude of the current amplification at high frequencies is

= = =

ii

v

v

z

Cc

b

m b e

b e

b e

m

b e

g g

High frequency

log scale

o

f

o

2

f

log scale

The frequency dependence of thecurrent gain shown on a log-logplot. The cutoff frequency is when= o/2.

Figure 6


We see that the current gain decreases with the frequency in the high-frequency range. At highfrequencies, Cbe shunts BE and thereby reduces vbe which results in a smaller output current, ic = gmvbe.The overall current amplification is shown in Figure 6, where the current gain exhibits a cut-off frequencyat = , where its magnitude has fallen by a factor of 2 . This occurs when the impedance of Cbe isequal to rbe

fr Cb e b e

=

12

cutoff frequency (1)

NOTATIONA areaB base terminal BJTBJT bipolar junction transistorC collector terminal of BJT; capacitanceCE common emitter configuratione electronic charge (1.602 10-19 C)E electric fieldE emitter terminal of BJTf frequencygm mutual transconductanceIB, IC, IE base, collector and emitter currentsk Boltzmanns constant (1.381 10-23 J K-1)pn(x) minority carrier concentration profile

QB total injected minority carrier charge in baseR, r resistancerbb base spread resistanceT temperatureV voltageVCC supply voltageWB base width low-frequency gain permittivity of a medium (C V-1 m-1 or F m-1) pi, 3.14159...h mean rate of recombination angular frequency

USEFUL DEFINITIONSEarly effect (Base width modulation) is the modulation of the base width by the voltage appearing across the base-

collector junction. An increase in the base to collector voltage increases the collector junction depletion layer width,which results in the narrowing of the base width, hence increasing the collector current.

Base spread resistance (rbb) is an effective resistance that represents the voltage drop from the external base terminal (B)to the actual base point (B). It is the voltage VBE that controls the collector current.

Bipolar junction transistor (BJT) is a transistor whose normal operation is based on the injection of minority carriersfrom the emitter into the base region and their diffusion to the collector, where they give rise to a collector current.The voltage between the base and the emitter controls the collector current; this is the transistor action.

Majority carriers are electrons in an n-type and holes in a p-type semiconductor.Minority carriers are electrons in a p-type and holes in an n-type semiconductor.pn junction is a contact between a p-type and an n-type semiconductor. It has rectifying properties.Recombination of an electron hole pair involves an electron in the conduction band (CB) falling in energy down into an

empty state (hole) in the valence band (VB) to occupy it. The result is the annihilation of the electron-hole pair.Recombination is direct when the electron falls directly down into an empty state in the VB as in GaAs.Recombination is indirect if the electron is first captured locally by a defect or an impurity, called a recombinationcenter, and from there it falls down into an empty state (hole) in the VB as in Si and Ge.


All material in this publication is copyrighted.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, ortransmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the author.Permission is granted to individuals for downloading this document from the authors website or his

CD-ROM for self-study only, provided a copy of the second edition of the author's McGraw-Hill textbookPrinciples of Electronic Materials and Devices, Third Edition, has been purchased or adopted as a requisite

course textbook. The permission is valid only while the book remains in adoption or it is in print.SPECIAL CUSTOM PUBLISHED e-BOOKLET

S.O. Kasap, 1990-2002The author reserves all rights

Last Updated: 27 November 2001 (v.1.1)

First published in Web-Materials(Established 1996)

http://Materials.Usask.Ca

HighFrequencySmallSignal bjt

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