ECE 3040 - Microelectronic Circuits
Lecture 13
Modern BJT, HBT, and
Small-Signal Models for BJT
Instructor: Dr. Shyh-Chiang Shen
Study:
Pierret 11.3
Jaeger 5.12,5.13,13.5
Lecture Outline
• Advanced bipolar transistor technologies
– BiCMOS
– HBT
• BJT SPICE model
• BJT biasing circuits
• BJT small signal model
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Modern BJT Fabrication Processing
•
•
Bipolar transistors found their niche application
mostly in analog circuit as well as power
amplification applications
Monolithically integrated Bipolar transistors with
CMOS transistors (BiCMOS) were developed for
mixed-signal integrated circuit applications
– Low power consumption for CMOS in digital
circuits
– High speed and power output of BJT for analog
circuits
– A/D, D/A in DSP ASICs
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Semiconductor Heterojunctions
•
•
•
Semiconductor stacks with different
bandgap but identical lattice constant
form “hetero-junction”
First hetero-junction semiconductor
concept was proposed by Dr. William
Shockley in 1948 (US patent:
2569,347)
Hetero-junction semiconductors
– Enable the realization of quantum
mechanical concepts such as super
lattice, quantum wells, quantum
wire, and quantum dots
– Were implemented in semiconductor
devices such as semiconductor laser
diodes (LD), LED, pHEMT, and
HBT, etc
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Heterojunction Bipolar Transistors (HBT)
•
Advantages
npn HBT:
– High emitter efficiency
– Short base transit time with thin
base
– High base doping concentration
•
Compound semiconductors are
the only choices for realization of
HBT
1
γ≈
1+
∆E
DE N BW
exp(− V )
DB N E LE
kT
– SiGe HBT
– AlGaAs/GaAs HBT
– InGaP/GaAs HBT
– InP/InGaAs HBT
! High current gain and high speed
transistors
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Review of Ebers-Moll Model (npn BJT)
-- With symbols adapted from Jaeger
Complete Ebers-Moll equations (npn transistor) are given by combining
forward and reverse characteristics:
⎛
⎛v
⎜
v BE ⎞⎟ ⎤⎥
i E = I exp
− 1⎥ − α R I CS ⎜⎜ exp ⎜ BC
⎟
⎜V
V T ⎠ ⎥⎦
⎜
⎝ T
⎝
⎛
⎛v
⎡
⎤
⎛
⎞
⎜
v BE ⎟ ⎥
⎢
⎜
iC = α F I ES ⎢exp ⎜
− 1⎥ − I CS ⎜⎜ exp ⎜ BC
⎟
⎜V
⎢⎣
⎥⎦
⎝ VT ⎠
⎜
⎝ T
⎝
⎡
⎢
ES ⎢
⎢⎣
⎛
⎜
⎜
⎝
⎞ ⎟⎞
⎟⎟ − 1⎟⎟
⎠ ⎟⎠
⎞ ⎞⎟
⎟⎟ − 1⎟⎟
⎠ ⎟⎠
α F I ES = α R I CS
iB = i E − iC = ⎛⎜⎝1 − α
⎡
⎞
⎢
⎟
F ⎠ ES ⎢
⎢⎣
I
exp
⎛
⎜
⎜
⎝
⎞
BE ⎟
⎟
T ⎠
v
V
⎤
⎥
⎥
⎥⎦
− 1 + ⎛⎜⎝1 − α
Center for Compound Semiconductors
⎛
⎜
⎞
⎜
⎟
R ⎠ CS ⎜
⎜
⎝
I
⎛v
exp ⎜ BC
⎜V
⎝ T
⎞ ⎟⎞
⎟⎟ − 1⎟⎟
⎠ ⎟⎠
Dr. S.-C. Shen, ECE3040B
Review of Ebers-Moll Model (pnp BJT)
-- With symbols adapted from Jaeger
Complete Ebers-Moll equations (pnp transistor) are given by:
iE = I
⎡
⎢
ES ⎢
⎢⎣
exp
iE = α F I
⎡
⎢
ES ⎢
⎢⎣
⎛
⎜
⎜
⎝
⎞
EB ⎟
⎟
T ⎠
v
V
⎤
⎥
⎥
⎥⎦
⎛
⎜
⎜
R CS ⎜
⎜
⎝
⎛
⎤
⎜
⎥
⎜
⎥
CS ⎜
⎥⎦
⎜
⎝
−1 − α I
⎛
⎜
⎜
⎝
v EB ⎞⎟
exp
−1 − I
V T ⎟⎠
⎛v
exp ⎜ CB
⎜V
⎝ T
⎛v
exp ⎜ CB
⎜V
⎝ T
⎞ ⎟⎞
⎟⎟ − 1⎟⎟
⎠ ⎟⎠
⎞ ⎞⎟
⎟⎟ − 1⎟⎟
⎠ ⎟⎠
α F I ES = α R I CS
iB = ⎛⎜⎝1 − α
⎡
⎞
⎢
⎟
F ⎠ ES ⎢
⎢⎣
I
⎛
⎛v
⎜
v EB ⎞⎟ ⎤⎥ ⎛
⎞
⎜
exp
− 1⎥ + ⎜⎝1 − α R ⎟⎠ I CS ⎜ exp ⎜ CB
⎟
⎜V
V T ⎠ ⎥⎦
⎜
⎝ T
⎝
⎛
⎜
⎜
⎝
Center for Compound Semiconductors
⎞ ⎟⎞
⎟⎟ − 1⎟⎟
⎠ ⎟⎠
Dr. S.-C. Shen, ECE3040B
Review of CE Output Characteristics in BJT
•
•
For iB = 0, transistor is cutoff.
If iB > 0, iC also increases.
– B-E junction is forced into forward
biased region to allow the establishment
of iB (vCE = vBE-vBC >0)
– For vBC < 0, npn transistor is in
forward-active region, iC = βF iB ~
independent of vCE.
– For vBC >0 transistor is in saturation.
•
For vCE < 0, roles of collector and
emitter reverse
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Review of CB Output Characteristics in BJT
• npn transistor
– For vCB > 0, is in forward-active
region, iC ≅ iE ~ independent of vCB.
– For vCB < 0, base-collector diode
becomes forward-biased and iC grows
exponentially (in negative direction)
as base-collector diode begins to
conduct.
Center for Compound Semiconductors
npn
pnp
Dr. S.-C. Shen, ECE3040B
Example 1: Simplified Analysis
in Forward-Active Region
• Problem: Estimate terminal currents and base-emitter voltage
• Given data: IS =10-16 A, αF = 0.95, VBC = VB - VC = -5 V, IE = 100 µA
• Assumptions: Simplified transport model assumptions, room
temperature operation, VT = 25.0 mV
• Analysis: Current source forward-biases base-emitter diode, VBE > 0,
VBC < 0, we know that transistor is in forward-active operation region.
I C = α F I E = 0 .95 ×100 µ A = 95 µ A
αF
0.95
βF =
=
= 19
1− α F 1− 0 .95
IE
100 µ A
IB =
=
= 5µ A
β F +1
20
V BE = V T ln
Center for Compound Semiconductors
⎛
⎜
⎜
⎜⎜
⎝
αF IE
IS
⎞
⎟
⎟
⎟⎟
⎠
= 0.69 V
Dr. S.-C. Shen, ECE3040B
Example 2: Simplified Analysis
in Forward-Active Region
• Problem: Estimate terminal currents, base-emitter and base-collector
voltages.
• Given data: IS = 10-16 A, αF = 0.95, VC = +5 V, IB = 100 µA
• Assumptions: Simplified transport model assumptions, room
temperature operation, VT = 25.0 mV
• Analysis: Current source causes base current to forward-bias baseemitter diode, VBE > 0, VBC <0, we know that transistor is in forward-active
operation region.
I C = β F I B = 19 ×100 µ A = 1.90mA
I E = (β F + 1)I B = 20 ×100 µ A = 2.00mA
⎛
⎜
⎜
⎜
⎜
⎝
I
⎞
⎟
⎟
⎟
⎟
⎠
V BE = V T ln C = 0.764 V
I
S
V BC = V B − V C = V BE − V C = − 4 .24 V
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Example 3: Simplified Analysis
in Forward-Active Region
•
•
•
•
Problem: Find Q-point
Given data: βF = 50, βR = 1 VBC = VB - VC = -9 V
Assumptions: Forward-active region of operation, VBE = 0.7 V
Analysis:
V BE + 8200 I E + V EE = 0
8.3V
∴ IE =
= 1.01 mA
8200 Ω
IE
1 .02 mA
=
= 19 .8 µ A
IB =
β F +1
51
I C = β F I B = 0 .990 mA
V CE = V CC − (−V BE ) = 9 + 0.7 = 9.7 V
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
BJT SPICE Model
•
Lumped elements in a BJT SPICE
Model:
– Capacitances associated with the
physical structure
– Diode current source iS
– Substrate capacitance CJS related to
the large area pn junction that
isolates the collector from the
substrate and one transistor from the
next.
– RB is resistance between external
base contact and intrinsic base
region.
– Collector current must pass through
RC on its way to active region of
collector-base junction.
– RE models any extrinsic emitter
resistance in device
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
BJT SPICE Model Typical Values
Saturation Current IS = 3x10-17 A
Forward current gain BF = 100
Reverse current gain BR = 0.5
Forward Early voltage VAF = 75 V
Base resistance RB = 250 Ω
Collector Resistance RC = 50 Ω
Emitter Resistance RE = 1 Ω
Forward transit time TT = 0.15 ns
Reverse transit time TR = 15 ns
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Biasing for BJT
• Goal of biasing is to establish known Q-point which in turn
establishes initial operating region of the transistor.
• For a BJT, the Q-point is represented by (IC, VCE) for an npn
transistor or (IC, VEC) for a pnp transistor.
• The Q-point controls values of diffusion capacitance,
transconductance, input and output resistances.
• In general, during circuit analysis, we use simplified
mathematical relationships derived for a specified operation
region, and the Early voltage is assumed to be infinite.
• Two practical biasing circuits used for a BJT are:
– Four-Resistor Bias network
– Two-Resistor Bias network
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Example: Four-Resistor Bias Network for BJT
V EQ = V CC
R1
R EQ = R 1 R 2 =
R1 R 2
R1 + R 2
R1 + R 2
V EQ = R EQ I B + V BE + R E I E
4 = 12 ,000 I B + 0.7 + 16 ,000 (β F + 1)I B
V EQ − V BE
4 V - 0 .7V
∴ IB =
=
= 2.68 µ A
6
R + (β + 1)R
1.23 ×10 Ω
EQ
β F = 75
F
I C = β F I B = 201 µ A
E
IE = (βF +1)IB = 204 µA
VCE =VCC − RC IC − RE IE
⎛
⎜
⎜
⎜⎜
⎝
VCE =VCC − RC +
⎞
RF ⎟
α
I = 4.32 V
⎟
⎟⎟ C
F⎠
Double-check: F. A. region correct ? Yes.
Q-point is (201 µA, 4.32 V)
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Four-Resistor Bias Network for BJT (cont.)
• All calculated currents > 0, VBC = VBE - VCE = 0.7 - 4.32 = - 3.62 V
• Hence, base-collector junction is reverse-biased, and assumption of
forward-active region operation is correct.
• Load-line for the circuit is:
⎛
⎞
⎜
⎜
⎜⎜
⎝
V CE = V CC − R C +
RF
α
⎟
⎟
⎟⎟ C
F ⎠
I = 12 − 38 ,200 I C
The two points needed to plot the load
line are (0, 12 V) and (314 µA, 0).
Resulting load line is plotted on
common-emitter output characteristics:
IB = 2.7 µA, intersection of
corresponding characteristic with load
line gives Q-point.
4.32 V
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Four-Resistor Bias Network for BJT:
Design Objectives
• We know that
IE =
V EQ − V BE − R EQ I B
RE
≅
V EQ − V BE
RE
for
R EQ I B << (V EQ − V BE )
• This implies that IB << I2, so that I1 = I2. So base current doesn’t disturb
voltage divider action. Thus, Q-point is independent of base current as
well as current gain.
• Also, VEQ is designed to be large enough that small variations in the
assumed value of VBE won’t affect IE.
• Current in base voltage divider network is limited by choosing I2 ≤ IC/5.
This ensures that power dissipation in bias resistors is < 17 % of total
quiescent power consumed by circuit and I2 >> IB for β > 50.
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Four-Resistor Bias Network for BJT: Design
Guidelines
• Choose Thévenin equivalent base voltage V
CC
4
≤ V EQ ≤
V CC
2
• Select R1 to set I1 = 9IB. R = V EQ
1
9IB
• Select R2 to set I2 = 10IB.
R2 =
V CC − V EQ
10 I B
• RE is determined by VEQ and desired IC.
RE ≅
V EQ − V BE
IC
• RC is determined by desired VCE.
RC ≅
V CC − V CE
IC
Center for Compound Semiconductors
− RE
Dr. S.-C. Shen, ECE3040B
Four-Resistor Bias Network for BJT: Example
•
•
•
•
Problem: Design 4-resistor bias circuit with given parameters.
Given data: IC = 750 µA, βF = 100, VCC = 15 V, VCE = 5 V
Assumptions: Forward-active operation region, VBE = 0.7 V
Analysis: Divide (VCC - VCE) equally between RE and RC. Thus, VE = 5 V
and VC = 10 V
RC =
RE =
V CC − V C
IC
VE
= 6 .67 k Ω
= 6 .60 k Ω
IE
V B = V E + V BE = 5 .7 V
I
I B = C = 7 .5 µ A
βF
Center for Compound Semiconductors
I2 =10IB = 75.0 µA
I1 = 9IB = 67.5 µA
R1 =
R2 =
VB
= 84.4 kΩ
9I B
VCC −VB
10I B
=124 kΩ
Dr. S.-C. Shen, ECE3040B
Two-Resistor Bias Network for BJT: Example
• Problem: Find Q-point for pnp transistor in 2-resistor bias circuit with
given parameters.
• Given data: βF = 50, VCC = 9 V
• Assumptions: Forward-active operation region, VEB = 0.7 V
• Analysis:
9 = V EB + 18 ,000 I B + 1000 (I C + I B )
∴ 9 = V EB + 18 ,000 I B + 1000 (51 )I B
9V − 0.7V
= 120 µ A
69 ,000 Ω
I C = 50 I B = 6.01 mA
∴ IB =
V EC = 9 − 1000 (I C + I B ) = 2.88 V
V BC = 2 .18 V
Forward-active region operation is correct ? Yes
Q-point is : (6.01 mA, 2.88 V)
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Introduction to BJT Amplifiers
• BJT is used as an amplifier when biased in the forwardactive region
• In these regions, transistors can provide high power gain
• Bias is provided to stabilize the operating Q-point in a
desired operation region
• Q-point also determines
–
–
–
–
Small-signal parameters of transistor
Voltage gain, input resistance, output resistance
Maximum input and output signal amplitudes
Power consumption
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
BJT Amplifier -- Common Emitter
"BJT is biased in active region by dc voltage source VBE. Q-point is set at
(IC, VCE) = (1.5 mA, 5 V) with IB = 15 µA.
"Total base-emitter voltage is:
vBE =VBE + vbe
"Collector-emitter voltage is:
equation.
vCE =10−iC RC
Center for Compound Semiconductors
This is the load line
Dr. S.-C. Shen, ECE3040B
BJT CE Amplifier
$If the change in operating current
and voltage are small enough, iC and
vCE waveforms are undistorted
replicas of the input signal.
$ Small vBE change causes large vCE
change. Voltage gain Av is given by:
#How does the amplification work?
a) 8 mV peak change in vBE gives 5 µA
change in iB and 0.5 mA change in iC.
b) 0.5 mA change in iC produces a 1.65
V change in vCE .
Center for Compound Semiconductors
Vce 1.65∠180
Av =
=
= 206∠180 =−206
V
0.008∠0
be
Minus sign indicates 1800 phase shift
between input and output signals.
Dr. S.-C. Shen, ECE3040B
DC and AC Analysis
• DC analysis:
– Find d.c. equivalent circuit by substituting all capacitors with open
circuits and inductors with short circuits.
– Find Q-point from d.c. equivalent circuit by using appropriate largesignal transistor model.
• AC analysis:
– Find a.c. equivalent circuit by substituting
•
•
•
•
•
all capacitors with short circuits,
inductors with open circuits,
d.c. voltage sources with a ground connection, and
d.c. current sources with open circuits.
transistor using small-signal model
– Combine end results of dc and ac analysis to yield total voltages and
currents in the network. (a.c. components + d.c. components)
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
DC Equivalent for BJT Amplifier
No d.c. current flow
through capacitors!!
• All capacitors in original amplifier circuits are replaced
with open circuits, disconnecting vI, RI , and R3 from
circuit.
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
AC Equivalent for BJT Amplifier
RB = R1 R2 =10kΩ 30kΩ
R = RC R 3 = 4.3kΩ100 kΩ
Assume capacitors are all shortckted @ freq. of interest!!
Simplify ckt.
Center for Compound Semiconductors
Simplify ckt.
Dr. S.-C. Shen, ECE3040B
How to Replace the BJT with Lumped
Circuit Elements for DC and AC Analysis?
We Need a Small Signal Model..
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Small-Signal Operation of Diode (I)
• The slope of the diode characteristic at
the Q-point is called the diode
conductance and is given by:
gd =
gd ≅
∂iD
∂v D
ID
VT
Q − point
≅
⎛
IS
VD ⎞⎟ ID + IS
⎜
= exp⎜ ⎟ =
VT
VT
⎝ VT ⎠
ID
0.025V
= 40ID for ID >> IS
• gd is small but non-zero for ID = 0
because slope of diode equation is
nonzero at origin.
1
r
=
• Diode resistance is given by: d
gd
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Small-Signal Operation of Diode (II)
⎡
⎛ v ⎞ ⎥⎤
⎛ V + v ⎞ ⎥⎤
⎢
iD = I exp ⎜ D ⎟ −1⎥
∴I D + id = I S ⎢exp ⎜ D d ⎟ −1⎥
⎢
⎜V ⎟ ⎥
⎜ V
⎟ ⎥
⎢⎣
⎝ T ⎠ ⎥⎦
⎝
⎠ ⎥⎦
T
⎡
⎤
⎛ V ⎞ ⎤⎥
⎛
⎞⎡
⎛
⎞2
⎛
⎞3
⎢
v
v
v
v
⎥
⎢
1
1
I D + id = I S ⎢exp ⎜ D ⎟ −1⎥ + I S exp ⎜⎜ D ⎟⎟⎢ d + ⎜⎜ d ⎟⎟ + ⎜⎜ d ⎟⎟ + ...⎥
⎢
⎜V ⎟ ⎥
2 ⎝ VT ⎠ 6 ⎝ VT ⎠
⎥
⎝ V T ⎠⎢⎣V T
⎢⎣
⎦
⎝ T ⎠ ⎥⎦
⎡
⎢
⎢
S⎢
⎢⎣
Subtracting ID from both sides of the equation,
⎡
⎢
d
S ⎢
⎢ T
⎣
id = (ID + I )
⎞2
⎛
⎜ d⎟
⎟
⎜
⎝ T⎠
v 1 v
+
V 2V
Maclaurin’s Series
⎞3
⎛
⎜ d⎟
⎟
⎜
⎝ T⎠
v
+1
6V
⎤
⎥
⎥
⎥
⎦
+...
For id to be a linear function of signal voltage vd , vd <<2VT = 0.05V or vd ≤ 5 mV
This represents the requirement for small-signal operation of the diode.
iD = ID + id
⎛
⎞
⎜ d⎟
⎟
S ⎜
⎝ T⎠
v
∴id = (ID + I )
= gdvd ⇒ iD = ID + gdvd
V
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Small-Signal Operation of BJT
⎛v
iC = I exp ⎜ BE
⎜V
⎝ T
⎡
⎢
⎢
S⎢
⎢⎣
⎞⎥⎤
⎟⎟⎥⎥
⎠⎥⎦
⎡
⎢
C⎢
⎢
⎣
⎛
V BE ⎞⎟
vbe ⎟⎞
⎜
∴iC = IC + ic = I S exp
exp ⎜ ⎟
VT ⎟⎠
⎝ VT ⎠
⎛
⎜
⎜
⎝
2
⎤
3
vbe 1 ⎛⎜ vbe ⎞⎟ 1 ⎛⎜ vbe ⎞⎟
⎥
IC + ic = I 1+ + ⎜ ⎟ + ⎜ ⎟ + ...⎥
VT 2 ⎝ VT ⎠ 6 ⎝ VT ⎠
⎥
⎦
⎡
⎢
be
C⎢
⎢ T
⎣
2
3
⎤
⎛
v
vbe ⎞⎟ 1 ⎛⎜ vbe ⎞⎟
⎥
1
⎜
∴ic = iC − IC = I
+ ⎜ ⎟ + ⎜ ⎟ + ...⎥
V 2 ⎝ VT ⎠ 6 ⎝ VT ⎠
⎥
⎦
For linearity, ic should be proportional to vbe with vbe <<2VT or vbe ≤ 0.005V
⎛
⎜
C⎜
⎝
IC
vbe ⎟⎞
∴ic = I 1+ ⎟ = IC + vbe = IC + gmvbe
VT ⎠
VT
Change in ic that corresponds to small-signal operation is:
vbe 0.005
= vbe = ≤
= 0.200
IC IC
VT 0.025
ic
gm
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Two-Port Models (I)
• The general two port model can be arranged in a variety of ways.
• Currents can be written in terms of voltages
– Admittance or Y-parameters
i1 = y11v1+y12v2
i1
i2
i2 = y21v1+y22v2
+
+
Two
Port
• Voltages can be written in terms of
V2
V1
Network
currents
– Impedance or Z-parameters
v1 = z11i1+z12i2
v2 = z21i1+z22i2
• Note: the results i1 and i2 for e.g. are not the “outputs” v2 and i2
• There are also the s-parameters and the abcd-parameters which are
most common in high speed RF circuit descriptions
– The abcd parameters “map” inputs to outputs are therefore readily
cascaded
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Two-Port Models (II)
• Currents and voltages can be written in
terms of currents and voltages
i
– g-parameters (very common usage)
i1 = g11v1+g12i2
v2 = g21v1+g22i2
i2
1
+
V1
-
Two Port
Network
+
V2
-
• Currents and voltages can be written in
terms of currents and voltages
– Hybrid or h-parameters
v1 = h11i1+h12v2
i2 = h21i1+h22v2
• Note: the results i1 and i2 for e.g. are not
the “outputs” v2 and i2
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Two-Port Network for BJT Using Y-Parameters
General “Y-parameter” Network
BJT “Y-parameter” Network
i1=y11v1 + y12v2
ib=y11vbe + y12vce
i2=y21v1 + y22v2
ic=y21vbe + y22vce
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Admittance Parameters and BJTs
ib=y11vbe + y12vce
ic=y21vbe + y22vce
Referred to as the short-circuit parameters
= short circuit input conductance
= reverse short circuit transconductance
= forward short circuit transconductance
= short circuit output conductance
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Consider the BJT as a Two-port Network
Under Common-Emitter Configuration
• npn
βo is most often
taken as a
constant, βF
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Small-Signal Model of BJT: Summary
y12 =
y 21 =
Using 2-port y-parameter network,
ib = y11vbe + y12vce
ic = y vbe + y22vce
y 22 =
21
ib
v ce
=
v
be
= 0
ic
v be
=
v
ce
= 0
ic
v ce
ib
=
v
be
= 0
∂i B
∂v CE
=0
Q − po int
∂i C
∂v BE
∂i B
IC
VT
=
IC
V A + VCE
Q − po int
∂i C
∂v CE
=
Q − po int
IC
=
=
The port variables can represent either y11 = v
∂v BE
β o VT
be v = 0
Q − po int
time-varying part of total voltages and
ce
currents or small changes in them away
βo is the small-signal commonfrom Q-point values.
emitter current gain of the BJT.
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Hybrid-Pi Model of BJT
Transconductance:
gm = y21 =
IC
VT
≅ 40 IC
Input resistance:
• The hybrid-pi small-signal model is
the intrinsic representation of the
BJT.
• Small-signal parameters are
controlled by the Q-point and are
independent of geometry of the
BJT
Center for Compound Semiconductors
1 β oVT β o
rπ =
=
=
y21
IC
gm
Output resistance:
1 V A +VCE V A
ro =
=
≅
y22
IC
IC
Dr. S.-C. Shen, ECE3040B
Small-Signal Model for pnp BJT
• For pnp transistor
iB = IB -ib
iC = IC -ic = βF IB − βF ib
• Signal current injected into
base causes decrease in
total collector current which
is equivalent to increase in
signal current entering
collector.
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Hybrid-Pi Model of BJT
Transconductance:
IC
gm = y21 = ≅ 40IC
VT
Input resistance:
• The hybrid-pi small-signal
model is the intrinsic
representation of the BJT.
• Small-signal parameters are
controlled by the Q-point
and are independent of
geometry of the BJT
Center for Compound Semiconductors
rπ =
1 β oVT β o
=
=
y21 I C
gm
Output resistance:
1 VA +VCE VA
=
≅
ro =
y22
IC
IC
Dr. S.-C. Shen, ECE3040B
Equivalent Forms of Small-Signal Model
for BJT
•
•
Voltage-controlled current source gmvbe can be transformed into currentcontrolled current source,
Basic relationship ic = βib is useful in both dc and ac analysis when BJT is
in forward-active region.
vbe = ibrπ
∴gmvbe = gmibrπ = βoib
vce
ic = βoib + ≅ βoib
ro
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B
Small-Signal Current Gain and Amplification
Factor of BJT
Amplification factor is given by:
⎛
⎜
C ⎜
⎜
⎜
T⎝
⎞
I VA +VCE ⎟⎟ VA +VCE
µF = g ro =
=
⎟
V
VT
I
⎟
⎠
C
m
βF
βo = gmrπ = ⎡
⎢
⎢
⎢
⎢
⎣
1− I
⎛
⎜
⎜
C⎜
⎜
⎝
For VCE << VA,
⎤
⎞
⎥
⎟
⎥
F⎟
⎥
⎟⎟
⎥
C ⎠ Q − point ⎦
1 ∂β
β F ∂i
βo > βF for iC < IM , and βo < βF
for iC > IM , however, βF and βo
are assumed to be equal.
µF ≅
VA
VT
≅ 40VA
µF represents maximum voltage gain
individual BJT can provide and
doesn’t change with operating point.
Center for Compound Semiconductors
Dr. S.-C. Shen, ECE3040B