BJT Base Resistance and Small Signal Modellingsmcgarry/ELEC3908/Slides/ELEC3908...BJT Base Resistance and Small Signal Modelling ELEC 3908, Physical Electronics, Lecture 19 ELEC 3908,

BJT Base Resistanceand Small Signal Modelling

ELEC 3908, Physical Electronics, Lecture 19

ELEC 3908, Physical Electronics: Base Resistance & Small Signal BJT Modeling Page 19-2

Lecture Outline

• Lecture 17 derived static (dc) injection model to predict dc currents from terminal voltages

• This lecture begins by considering the resistance associated with current flow through the base region – base resistance

• Now consider small signal operation and derive equivalent circuits for low and high frequency operation

• Will also use the high frequency equivalent circuit to define the transit frequency, a common figure of merit for bipolar transistors


Physical Origin of Base Resistance• So far BJT analysis has been 1D, i.e. no lateral effects

considered• More accurate modelling of structure requires consideration of

base current flow:– From contact, IB must flow through base material to edge of active

region– Base current flow to emitter occurs in a distributed way along path

into active device


Extrinsic Base Resistance• The extrinsic (or external) base region is that between the active area

and the base contact region, assume width is sB

• Resistance of this region can be determined from basic equation for resistance from resistivity - if ρBx is the resistivity of the extrinsic base region then

rl

As

l WBxBx B

E B= =ρ ρ


Intrinsic Base Resistance - Geometry• The intrinsic (or internal) base region is the volume of the active

device area enclosed by the neutral base width and the emitter width and length

• Modelling of this resistance is more complicated because the current flow is not 1 dimensional - base current enters the side and leaves the top (to be injected into the emitter in forward active operation)


Intrinsic Base Resistance - Modelling• Assume linear distribution of iB

• The lhs voltage due to the distributed voltage drop through the region is

• The resistance from the lhs edge to a point x is

• The linearly distributed current is

Vb

R x i x dxeffE

bE

= ∫1

0

( ) ( )

R xx

W lBi

B E( ) =

ρ

i x ix

bBE

( ) = −⎛⎝⎜

⎞⎠⎟1


Intrinsic Base Resistance - Modelling• Substituting the expressions for R(x)

and i(x) into the Veff expression gives

• Performing the integral and substituting limits gives the result

• The effective intrinsic base resistance is therefore 1/6 of the value obtained if iB was flowing through the region

Vb

xW l

ix

bdxeff

E

Bi

B E

b

BE

E

=⎛⎝⎜

⎞⎠⎟ −

⎛⎝⎜

⎞⎠⎟∫

11

0

ρ

rVi

bW lbb

eff

B

Bi E

B E′ = =

16ρ


Example 19.1: Base Resistance Calculation

• Calculate the intrinsic and extrinsic base resistances for the structure and potentials shown below. Assume the internal and external base dopings are identical, and that the base separation sB is 1.5 μm. The emitter width is 1.0 μm and the length is 10.0 μm.


Example 19.1: Solution

• Since the intrinsic and extrinsic base regions are identically doped, their resistivities will be the same, given by

• The external base resistance is therefore (note that the neutral base width was calculated in an earlier example)

• And the intrinsic base resistance is therefore

ρ ρμBx Bi

pqp= = =

× ⋅ ⋅=−

1 116 10 10 480

01319 17.. Ωcm

rs

W lBxBx B

B E= =

⋅ ×× ⋅ ×

=−

− −

ρ 013 15 1014 10 10 10

1394

4 4. .

.Ω

rb

W lBiBi E

B E= =

⋅ ×× ⋅ ×

=−

− −

16

16

013 1 1014 10 10 10

15 54

4 4ρ .

.. Ω


Low Frequency Small Signal Parameters

• The transconductance gm, the rate of change of collector current with base emitter voltage, is given by

• To account for the back injection component of base current in forward active operation, another conductance gπ, the rate of change of base current with base-emitter voltage, is used, where

gdI

dVd

dVqA D n

We

qkT

ImC

BE BE

E nB Bo

B

qV kTC

BE≡ ≈⎡

⎣⎢

⎤

⎦⎥ =

{g

dIdV

ddV

qA D pW

eq

kTI

qkT

III

gB

BE BE

E pE Eo

E

qV kTB C

g

B

C

m

F

BE

m F

π

β

β≡ ≈

⎡

⎣⎢

⎤

⎦⎥ = = =

1231


Low Frequency Small Signal Equivalent Circuit• Now assemble the base resistance and conductance parameters into an

equivalent circuit valid for small signal operation• Include a controlled current source to model the dependence of

collector current variation on base current variation, and an output resistance to model the Early effect

• Result is a low frequency hybrid-π small signal equivalent circuit


Example 19.2: Conductance Parameter Calculations

Calculate the parameters rπ and gm for the device of example 19.1.



• Using the value of IC calculated from the injection model equation

• From the value of gm found above and the forward active current gain already calculated in an earlier example

gm = ⋅ × = ×− −10 026

2 28 10 8 8 104 3

.. . mhos

ggm

Fπ β= =

×= × =

−−8 8 10

2044 3 10 233

35.

. .mhos kΩ


Depletion Capacitance in the BJT Structure• Each pn-junction in the bipolar

structure has an associated depletion capacitance

• CdepBC is the absolute depletion capacitance (F) of the base collector junction, CdepBE is the absolute capacitance (F) of the base-emitter junction

• Recall from diode discussion (lecture 13)

( ) ( )( )ˆ 0ˆ

1depSi

dep DD D bi

CC V

W V V Vε

= =−


BJT Depletion Capacitance Models

• Write models for each junction of the form of the original pn-junction model equation

• The total (absolute) capacitances are then the per unit area terms multiplied by the emitter area

( )

( )

$ ( )$ ( )

$ ( )( )

$ ( )$ ( )

$ ( )( )

C VC

V VC

W

C VC

V VC

W

depBE BEdepBE

BE biBEz depBE

Si

BE

depBC BCdepBC

BC biBCz depBC

Si

BC

BE

BC

=−

=

=−

=

0

10

0

0

10

0

ε

ε

C C A C C AdepBC depBC E depBE depBE E= =$ $


Physical Origin of Base Diffusion Capacitance• Recall that in general,

capacitance is associated with the requirement to source or sink charge when changing a terminal’s potential

• Because the stored charge in the base must be changed if VBE is changed, a capacitance will be created between the base and emitter terminals

• This is termed the base diffusion capacitance


Base Diffusion Capacitance Model

• The base diffusion capacitance Cπ (in F, not a per unit area term) can be modeled in terms of the base transit time τB, the time taken to cross the neutral base region

• The transit time is given in terms of physical parameters for a constant doped base by

C gB mπ τ=

τBB

nB

WD

=2

2


High Frequency Small Signal Equivalent Circuit• If the capacitances are added to the low frequency equivalent circuit,

the high frequency hybrid-π equivalent circuit is obtained• Note that the base-emitter capacitances appear inside the base

resistances, since they are part of the basic internal transistor structure


Example 19.3: Capacitance Calculations

Calculate the high frequency hybrid-π equivalent circuit capacitances for the device in example 19.2.



• The first step in determining the depletion capacitances is to find the zero bias depletion widths. Applying the formula for W at zero bias to each junction (using previously calculated built in potentials) gives

• The zero bias capacitances are therefore

Wq N N

V

Wq N N

V

BESi

AB DEbiBE

BCSi

AB DCbiBC

( ) .

( ) .

02 1 1

11 10

02 1 1

4 5 10

5

5

= +⎛⎝⎜

⎞⎠⎟ = ×

= +⎛⎝⎜

⎞⎠⎟ = ×

−

−

ε

ε

cm

cm

$ ( )( )

. $ ( )( )

.CW

CWdepBE

Si

BEdepBC

Si

BC0

09 4 10 0

02 3 108 2 8 2= = × = = ×− −ε ε

F cm F cm


Example 19.3: Solution (con’t)

• The capacitances at bias are therefore (note that the grading coefficient is 1/2 for uniformly doped junctions)

• The absolute depletion capacitances are therefore

( )

( )

$ ( . ).

. ..

$ ( . ).. .

.

.

.

C

C

depBE

depBC

0 89 4 10

1 0 8 0 932 5 10

0 72 3 10

1 0 7 0 7417 10

8

0 57

8

0 58

=×

−= ×

− =×

+= ×

−−

−−

F cm

F cm

2

2

( )( )

C

CdepBE

depBC

= × ⋅ × ⋅ × = ×

= × ⋅ × ⋅ × = ×

− − − −

− − − −

2 5 10 1 10 10 10 2 5 10

17 10 1 10 10 10 17 10

7 4 4 14

8 4 4 15

. .

. .

F

F


Example 19.3: Solution (con’t)

• The base transit time is found from

• The base diffusion capacitance is therefore found using the transit time and the previously calculated transconductanceas

( )τB

B

nB

WD

= =×⋅

= ×−

−2 4 2

10

214 10

2 34 92 8 10

..

. sec

C gB mπ τ= = × ⋅ × = ×− − −2 8 10 8 8 10 2 5 1010 3 12. . . F


Transit Frequency Measurement Configuration• One widely used figure of merit

in bipolar performance is the transit frequency fT, the frequency at which the current gain with the collector and emitter short circuited becomes unity


Transit Frequency Analysis• To determine fT, the high frequency small signal hybrid-p circuit can

be used• The effect of the collector-emitter short circuit is to short the output

conductance ro

• The phasor collector current will be given by (note point a is ground)

( )I V V VC m b e depBC b e b e m depBCg j C g j C= − = −′ ′ ′ω ω


Transit Frequency Analysis (con’t)• The base current phasor will be given by

• The ratio of collector current phasor to base current phasor is therefore

( )I VB b e depBCg j C j C= + +′ π πω ω

( ) ( )( )II

C

B

m depBC

depBE depBC depBE depBC m B

g j C

g j C C C j C C g=

−

+ + +≈

+ +

ω

ω ω τπ π

1


Transit Frequency Analysis (con’t)

• The transit frequency is then defined by

• And hence fT is given in terms of physical parameters as

• Faster transit time, reduced capacitance and increased transconductance all improve the transit frequency

( )( )II

C

B f f T depBE depBC m BTf C C g=

≡ =+ +

11

2π τ

( )( )fC C g

T

depBC depBE m B

=+ +

1

2π τ


Example 19.4: Transit Frequency Calculation

Calculate the transit frequency for the device of example 19.3.



• The required values have all been calculated previously, so the result is obtained by substituting the values

• This example transistor therefore has a transit frequency of 0.56 GHz. A modern commercial bipolar process would offer devices with an fT of 25-35 GHz.

( )( )fT =× + × × + ×

= ×− − − −

12 2 5 10 17 10 8 8 10 2 8 10

5 6 1014 15 3 108

π . . . .. Hz


Lecture Summary

• Current flow in bipolar structure experiences external and internal base resistance – mechanisms are very different

• Low frequency small signal equivalent circuit contains rbb’and rbx (internal/external base resistances), gπ (dependence of back injection current on VBE), gm ( transconductance) and ro the output resistance (Early effect)

• High frequency small signal equivalent circuit contains the same elements plus CdepBC, CdepBE (depletion capacitances) and Cπ (diffusion capacitance)

• The transit frequency fT incorporates many of the important high frequency parameters, and so is a useful figure of merit of high frequency performance

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BJT Base Resistance and Small Signal Modellingsmcgarry/ELEC3908/Slides/ELEC3908...BJT Base Resistance and Small Signal Modelling ELEC 3908, Physical Electronics, Lecture 19 ELEC 3908,