1

ECE 255:

BJT Small Signal Model (Sedra and Smith, 7th Ed., Sec. 7.2.2)

Mark Lundstrom School of ECE

Purdue University West Lafayette, IN USA

Lundstrom: Fall 2019

ECE 255: Fall 2019 Purdue University

2) AC analysis

2

+υo

−

RC = 5 kΩic

υi

We need an ac small signal model for the BJT

Lundstrom: Fall 2019

Outline

3

1)  Small signal model for BJT 2)  Small signal amplifier analysis 3)  NPN vs. PNP 4)  T-model

Lundstrom: Fall 2019

ac collector current

4

N P N

IC + ic

IB + ib

IC = ISeVBE VT → iC = ISe

VBE+υbe( ) VT

iC = ISeVBE VT eυbe VT

IC + ic = ISeVBE VT 1+υbe VT( )

ic = ISeVBE VT υbe VT( ) = IC υbe VT( )

ic =ICVT

υbe = gmυbe Ohm’s Law

Lundstrom: Fall 2019

Summary: s.s. collector current

5

N P N

ic

ib ic = gmυbe

gm = ICVT

+υbe

−

“transconductance”

Note that the ac model parameter, gm, depends on the dc bias current, IC.

Lundstrom: Fall 2019

s.s. base current

6

N P N

ic

ib

ic = gmυbe

gm = ICVT

ib =icβ= gm

βυbe =

υbe

β gm= υbe

rπ

ib =υbe

rπ

gmrπ = β

+υbe

−

Lundstrom: Fall 2019

s.s. eqv. circuit model: gm form

7

icib

B

E

C

gm = IC VT gmrπ = β

+υbe

−E

C B

ib ic

rπ gmυbe

+υbe

−

“hybrid pi model”

s.s. eqv. circuit model: beta form

8

gm = ICVT

gmrπ = βE

C B ib ic

rπ gmυbe

+υbe

−

gm = β rπ

ic =βrπυbe = βib

βib

ic = gmυbe

VCCS

CCCS

Lundstrom: Fall 2019

s.s. eqv. circuit model: vce dependence

9

E

C B ib ic

rπ gmυbe

+υbe

−

This model says that the collector current does not depend on the collector-emitter voltage.

Lundstrom: Fall 2019

Output resistance

10 E

VCE

IC

IC = ISeqVBE kBT 1+

υCE

VA

⎛

⎝⎜⎞

⎠⎟

dIC

dVCE

= ISeqVBE kBT 1VA

⎛

⎝⎜⎞

⎠⎟=

′IC

VA

ic =

υce

r0 ro ≈

VA

IC

VA = “Early Voltage”

dIC

dVCE

≈δ IC

δVCE

=icυceVBE1, IB1

ic =

dIC

dVCE

⎛

⎝⎜⎞

⎠⎟υce

Hybrid pi model

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gm = ICVT

gmrπ = β

ro =VA ICE

C B

ib ic

rπ gmυbe

+υbe

−ro

+υce

−

Lundstrom: Fall 2019

ic = gmυbe +υce r0

ib =υbe rπ

Doing it with math

12 Lundstrom: Fall 2019

f x, y( ) f x +δ x, y +δ y( ) = ?

f x +δ x, y +δ y( ) ≈ ∂ f∂x y

δ x + ...∂ f∂y x

δ y + ...

IC VBE ,VCE( ) = ISeqVBE kBT 1+

υCE

VA

⎛

⎝⎜⎞

⎠⎟

IB VBE ,VCE( ) = IS

βeqVBE kBT

Outline

13

1)  Small signal model for BJT 2)  Small signal amplifier analysis 3)  NPN vs. PNP 4)  T-model

Lundstrom: Fall 2019

Small signal circuit

14

+υo

−

RC = 5 kΩiC = IC + ic

υi

We now have an ac small signal model for the BJT

Lundstrom: Fall 2019

1) Use simple s.s. model

15

gm = ICVT

gmrπ = β

E

C B ib ic

rπ+υbe

−gmυbe

Lundstrom: Fall 2019

Draw the s.s. equivalent circuit

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υOUT =VOUT+υo

VCC

RCiC = IC + ic

VBB+−

υi

Lundstrom: Fall 2019

Small signal circuit

17

+υo

−

RC

icυi

C B ib

rπ+υbe

−

gmυbe

E

Same small signal circuit

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+υo

− RC

ic

υi

C B ib

rπ+υbe

−

gmυbe

E

Lundstrom: Fall 2019

ac analysis

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υo = −gmυbeRC

β = 100

RC = 5 kΩIC = 1mA

gm = ICVT

= 1mA0.026 V

= 38.5 mS

gmrπ = β

rπ = β gm = 100 0.039 = 2.6 kΩAυs

= −192

υo

υbe

= −gmRC

2) Use Hybrid pi model with output resistance

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gm = ICVT

gmrπ = βE

C B

ib ic

rbe gmυbe

+υbe

−ro

+υce

−

ro =VA ICLundstrom: Fall 2019

Draw the s.s. equivalent circuit

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υOUT =VO+υo

VCC

RCiC = IC + ic

RBB

VBE+−

υi

Lundstrom: Fall 2019

Small signal circuit

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+υo

− RC

ic

υi

C B ib

rπ+υbe

−

gmυbe

E

Lundstrom: Fall 2019

ro

ac analysis

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υo = −gmυbeRC || r0β = 100

RC = 5 kΩIC = 1mA

gm = ICVT

= 1mA0.026 V

= 38.5 mS

gmrπ = β

rπ = β gm = 100 0.039 = 2.6 kΩAυs= −183

υo

υi

= −gmRC || r0

VA = 100 V

ro =VAIC

= 100 kΩ

υi =υbe

Outline

24

1)  Small signal model for BJT 2)  Small signal amplifier analysis 3)  NPN vs. PNP 4)  T-model

Lundstrom: Fall 2019

Hybrid pi model of BJT

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gm = ICVT

gmrπ = βE

C B

ib ic

rπ gmυbe

+υbe

−ro

+υce

−

ro =VA ICLundstrom: Fall 2019

Question: What does the small signal model for a PNP transistor like?

NPN DC to AC

26 Lundstrom: Fall 2019

N P N

IC = β IB

IB = IC β

IE =β +1β

IC

+VBE ≈ 0.7 V

-

ic = gmυbe

ib =υbe rπ

ie = gm +1rπ

⎛⎝⎜

⎞⎠⎟υbe

N P N

+υbe

-

+VCE > 0-

PNP DC to AC

27 Lundstrom: Fall 2019

P N P

IC = β IB

IB = IC β

IE =β +1β

IC

+VBE ≈ −0.7 V

-

ic = gmυbe

ib =υbe rπ

ie = gm +1rπ

⎛⎝⎜

⎞⎠⎟υbe

P N P

+υbe

-

+VCE < 0-

Small signal hybrid pi model for NPN or PNP

28 Lundstrom: Fall 2019

gm = IC VT

gmrπ = β

E

ro =VA IC

C B

ib ic

rπ gmυbe

+υbe

−ro

+υce

−

Outline

29

1)  Small signal model for BJT 2)  Small signal amplifier analysis 3)  NPN vs. PNP 4)  T-model

Lundstrom: Fall 2019

Hybrid pi model

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E

C B

ib ic

rπ gmυbe

+υbe

−

+υce

−βib

Lundstrom: Fall 2019

ic = gmυbe

ib =υbe rπ

BJT T-Model

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E

C

B

ib

ic

ic = gmυbe

+υbe = iere

−

re =rπ

β +1

Lundstrom: Fall 2019

ic = gmυbe

ib =υbe rπ ?

ie

Question: Where does ro go?

Summary

32 Lundstrom: Fall 2019

The small signal model of a BJT consists of two resistors and one voltage-controlled current course. The values of the ac model parameters are determined by the dc bias current.

Circuit analysis consists of two steps: 1) dc analysis to determine the OP, and 2) ac small signal analysis using the ac circuit model.

We will stick with the hybrid pi model for MOSFETs and BJTs, but the book also uses the T model from time to time.

BJT s.s. model

Lundstrom: Fall 2019 33

1)  Small signal model for BJT 2)  Small signal amplifier analysis 3)  NPN vs. PNP 4)  T-model

Practice: Draw the s.s. equivalent circuit

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υOUT =VO+υo

VCC

RCiC = IC + ic

VBB+−

υi

RE

1)  Hybrid pi w/o ro 2)  Hybrid pi w ro 3)  T-model w/o ro 4)  T-model w ro