Small Signal BJT Models (5/21/00)
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1.4 - SMALL SIGNAL MODEL OF THE BJT
INTRODUCTION
Objective
The objective of this presentation is:
1.) Concept of the small signal model
2.) The small signal model for the BJT
Outline
• Transconductance small signal model
• Input resistance, output resistance of the common emitter model
• Extensions of the small signal BJT model
• BJT frequency response
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
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TRANSCONDUCTANCE SMALL SIGNAL MODEL
Categorization of Electrical Models
Time Dependence
Linear
Linearity
Time Independent
Time Dependent
Small-signal, midband
Small-signal frequency
response - poles and
zeros
(.AC)
R in, A v, R out
(.TF)
Nonlinear DC operating point
iD = f(v D ,v G ,v S ,v B )
(.OP)
Large-signal transient
response - Slew rate
(.TRAN)
Based on the simulation capabilities of SPICE.
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
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What is a Small Signal Model?
• A small signal model is a linear model which is independent of amplitude. It may or may not have time
dependence (i.e. capacitors).
• The small signal model for a nonlinear component such as a BJT is a linear model about some nominal
operating point. The deviations from the operating point are small enough that it approximates the
nonlinear component over a limited range of amplitudes.
Illustration of the pn diode:
iD
iD
Small Signal
Model
id
+
vD
iD
id
ID
vd
ID
vD
Time
VD
vD
vd
Time
ECE 4430 - Analog Integrated Circuits and Systems
Fig.1.4-1
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 4
BJT, Common-Emitter, Forward-Active Region
Effect of a small-signal input voltage applied to a BJT.
Emitter
Depletion
Region
iC = IC + ic
iB = IB + ib
vi
VCC
VBE
np(0) = npo exp VBE+vbe
Vt
np(0) = npo exp VBE
Vt
Emitter
vi ⇒ ib
⇒
,,,,,,,
,,,
,,,,,,,
,,,,,,
,,,
,,,,,,,
,,,
,
,,,
Carrier
Concentration
∆Qh
Collector
Depletion
Region
∆Qe
IC+ic
IC
WB
Base
x
Collector
Fig.1.4-2
ic
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 5
Transconductance of the Small Signal BJT Model
The small signal transconductance is defined as
diC |
ic ic
∆ iC
=v =v
⇒
gm ≡ dv Q = ∆ v
BE
BE
be
i
ic = g m v i
The large signal model for iC is
vBE
iC = IS exp
Vt
∴
⇒
 d
v B E
IS
V BE IC
I S exp V  = V exp V = V
t Q
t
t
t
 dvBE
gm = 
|
IC
gm = V
t
Another way to develop the small signal transconductance
vi
 V BE +vi
 V BE
 vi 
 vi 


1  vi  2 1  vi  3
iC = IS exp V  = IS exp V  expV  = IC expV  ≈ IC1 + V + 2 V  + 6 V  + · · ·
t 
t

 t 
 t
 t
 t
 t


But
iC = IC + ic
∴
v i IC  v i  2 IC  v i  3
IC
ic ≈ IC V + 2  V  + 6  V  + ··· ≈ V vi = gm vi
t
t
 t
 t
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 6
INPUT AND OUTPUT RESISTANCE SMALL SIGNAL MODEL
Input Resistance of the Small Signal BJT Model
In the forward-active region, we can write that
iC
iB = β
F
Small changes in iB and iC can be related as
d  iC 
∆ iB = di β ∆iC
C  F
The small signal current gain, βo, can be written as
∆ iC
βo = ∆ i =
B
ic
=i
d  iC  b
diC βF
1
Therefore, we define the small signal input resistance as
v i βovi
βo
=g
rπ ≡ i = i
b
c
m
βo
rπ = g
m
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 7
Output Resistance of the Small Signal BJT Model
In the forward-active region, we can write that the small signal output conductance, go (ro = 1/go) is
diC |
ic
∆ iC
=v
⇒
ic = govce
go ≡ dv Q = ∆v
CE
CE
ce
The large signal model for iC , including the influence of vCE, is
v C E
vBE

iC = IS  1 +
 exp
VA 
Vt

diC |
VBE
IC
 1 
go ≡
dvCE Q = IS V A  exp V t ≈ V A
∴
VA
ro =
IC
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 8
Simple Small Signal BJT Model
Implementing the above relationships, ic = gmvi, ic = govce, and vi = rπib, into a schematic model gives,
C
B
E
B
+
vi
E
ib
rπ
ic
gmvi
ro
C
C
+
vce
E
B
E
Fig. 1.4-3
Note that the small signal model is the same for either a npn or a pnp BJT.
Example:
Find the small signal input resistance, Rin, the output resistance, Rout, and the voltage gain of the
common emitter BJT if the BJT is unloaded (RL = ∞), vout/vin, the dc collector current is 1mA, the Early
voltage is 100V, and βο at room temperature.
gm =
IC 1mA
1
=
=
V t 26mV 26 mhos or Siemans
V A 100V
R out = ro = I = 1mA = 100kΩ
C
ECE 4430 - Analog Integrated Circuits and Systems
βo
R in = rπ = g = 100·26 = 2.6kΩ
m
vout
vin = -gm ro = - 26mS·100kΩ = -2600V/V
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 9
EXTENSIONS OF THE SMALL SIGNAL BJT MODEL
Collector-Base Resistance of the Small Signal BJT Model
Recall the influence of V on the base width:
,,
,,
,,
,,
Carrier
Concentration
np(0) = npo exp vBE
Vt
Emitter
,,,
,,,
,,,
,,,
Collector depletion
region widens due to a
change in vCE, ∆VCE
iC
iC+∆iC
WB
Base
Initial
Depletion
Region
x
Collector
∆WB
Fig.1.3-6
We noted that an increase in vCE causes and increase in the depletion width and a decrease in the total
minority-carrier charge stored in the base and therefore a decrease in the base recombination current, iB1.
This influence is modeled by a collector-base resistor, rµ, defined as
∆vCE ∆ vCE ∆ i C
∆ iC
= ∆i ∆i = r o ∆i
≈ βoro (lower limit if base current is all recombination current)
r µ = ∆i
Β1
C
Β1
Β1
In general, rµ ≥ 10 βoro for the npn BJT and about 2-5 βoro for the lateral pnp BJT.
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 10
Base-Charging Capacitance of the Small Signal BJT Model
Consider changes in base-carrier concentrations once again.
Emitter
Depletion
Region
iC = IC + ic
iB = IB + ib
vi
VCC
VBE
,,,,,,,
,,,
,,,,,,,
,,,,,,
,,,
,,,,,,,
,,,
,
,,,
Carrier
Concentration
np(0) = npo exp VBE+vbe
Vt
np(0) = npo exp VBE
Vt
Emitter
∆Qh
Collector
Depletion
Region
∆Qe
IC+ic
IC
WB
Base
x
Collector
Fig.1.4-2
The ∆vBE change causes a change in the minority carriers, ∆Qe = qe, which must be equal to the change in
majority carriers, ∆Qh = qh. This charge can be related to the voltage across the base, vi, as
qh = C bvi
where Cb is the base-charging capacitor and is given as
qh τ F i c
IC
Cb = v = v = τF g m = τF V
i
i
t
W B2
The base transit time τF is defined as
2Dn
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 11
Parasitic Elements of the BJT Small Signal Model
Typical cross-section of the npn BJT:
Collector
Base
Emitter
n+ emitter
p+
n+
p-
isolation
rex
rb Cje
rc3 Cµ p base
Ccs
n collector
Cje
p- isolation
Cµ
Ccs
rc1
rc2
n+ buried layer
p- substrate
p+
p
p-
ni
n-
n
n+
Metal
Fig.1.4-4
Cje = base-emitter depletion capacitance (forward biased)
Cµ0
= collector-base depletion capacitance (reverse biased)
Cµ =
v C Bm

1 ψ0 

Resistances are all bulk ohmic resistances. Of importance are rb, rc, and rex. Also, rb is a function of IC.
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 12
Complete Small Signal BJT Model
rµ
B
rb
Cπ
Cµ
B'
rπ
+
v1
-
gmv1
rc
ro
C
Ccs
rex
E
E
Fig. 1.4-5
The capacitance, Cπ, consists of the sum of Cje and Cb.
Cπ = Cje +Cb
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
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Example
Derive the complete small signal equivalent circuit for a BJT at IC = 1mA, V CB = 3V, and V CS = 5V.
The device parameters are Cje0 = 10fF, ne = 0.5, ψ0e = 0.9V, Cµ0 = 10fF, nc = 0.3, ψ0c = 0.5V, Ccs0 =
20fF, ns = 0.3, ψ0s = 0.65V, βo = 100, τF = 10ps, V A = 20V, rb = 300Ω, rc = 50Ω, rex = 5Ω, and rµ =
10βoro.
Solution
Because Cje is difficult to determine and usually an insignificant part of Cπ, let us approximate it as 2Cje0.
∴ Cje = 20fF
Cµ0
10fF
=
= 5.6fF
Cµ =
V C Bne 
3 0.3

1 +

 1+

0.5


ψ
0c 

gm =
IC 1mA
V t = 26mV = 38mA/V
and
Ccs0
20F
Ccs =
=
= 10.5fF
V C Sns 
5 0.3

1 +

 1+

0.65


ψ
0s 

Cb = τF gm = (10ps)(38mA/V) = 0.38pF
∴ Cπ = Cb + Cje = 0.38pF+0.02pF = 0.4pF
βo
rπ =
gm = 100·26Ω = 2.6kΩ
V A 20V
ro = I = 1mA = 20kΩ
C
ECE 4430 - Analog Integrated Circuits and Systems
and
rµ = 10βοro = 10·100·20kΩ = 20MΩ
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 14
FREQUENCY RESPONSE OF THE BJT
Transition Frequency, f T
fT is the frequency where the magnitude of the short-circuit, common-emitter current equal unity.
Circuit and model:
io
Cµ
rb
ii
Cπ
+
v1
-
rπ
ii
rc
gmv1
ro
Ccs
io
Fig.1.4-06
Assume that rc ≈ 0. As a result, ro and Ccs have no effect.
rπ
∴ V1 ≈
1+ rπ(C π+C b)s Ii
Now,
Io (jω )
β (jω ) = I (jω ) =
i
and
Io ≈ g m V 1 ⇒
Io (jω )
=
Ii(jω )
gm rπ
βo
=
(C π+C b)s
(C π+C b)s
1+ gm rπ
1+
β
o
gm
gm
βo
(C π+C b)jω
1+ β o
gm
At high frequencies,
gm
β (jω ) ≈ jω (C +C )
π
b
⇒
ECE 4430 - Analog Integrated Circuits and Systems
gm
When | β(jω )| =1 then ω T = +C or
Cπ b
1 gm
f T = 2π
Cπ+Cb
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 15
Illustration of the BJT Transition Frequency
β as a function of frequency:
|β(jω)|
1000
βo
-6dB/octave
100
10
1
ωβ
0.1ωT
ωT
ω (log scale)
Fig.1.4-7
Note that the product of the magnitude and frequency at any point on the -6dB/octave curve is equal to ωT.
For example,
0.1 ωT x10 = ω T
In measuring ωT, the value of |β(jω)| is measured at some frequency less than ωT (say ωx) and ωT is
calculated by taking the product of |β(jωx)| and ωx to get ωT.
ECE 4430 - Analog Integrated Circuits and Systems
 P.E. Allen, 2000
Small Signal BJT Models (5/21/00)
Page 16
Current Dependence of f T
C π Cµ Cb Cje Cµ
Cje Cµ
1
Note that τT = ω = g + g = g + g + g = τF + g + g
Τ
m
m
m
m
m
m
m
At low currents, the Cje and Cµ terms dominate causing τT to rise and ωT to fall.
At high currents, τT approaches τF which is the maximum value of ωT.
For further increases in collector current, ωT decreases because of high-level injection effects and the
Kirk effect.
Typical frequency dependence of fT:
fT (GHz)
10
8
6
4
2
0
10µA
100µA
ECE 4430 - Analog Integrated Circuits and Systems
1mA
IC
10mA Fig.1.4-8
 P.E. Allen, 2000