Lecture #29
Small-Signal Modeling and Linear Amplification - Part 2: BJTs
Outline/Learning Objectives:
•
•
•
•
•
Describe the BJT as an amplifier.
Define and describe the small-signal-model of the BJT.
Analyze BJT amplifiers using a two-part process:
dc Analysis: (1) construct the dc equivalent circuit; (2) solve for the Q-point.
ac Analysis: (3) construct the ac equivalent circuit; (4) replace the BJT by its small-signal model; (5)
solve the ac circuit.
• Analyze the BJT common-emitter (C-E) amplifier.
• Use the electronics laboratory to investigate the electrical behavior of simple circuits and devices.
•
Transistor as an Amplifier
• Small signal parameters of bipolar junction transistor (BJT)
• Voltage gain (AV), input resistance (RIN), output resistance (ROUT)
• Maximum input and output signal amplitudes
Lecture 29
29 - 1
The Bipolar Junction Transistor Amplifier
v BE = V BE + v be and 10 = v CE + i C R (Load-Line Equation)
C
vCE(t)
10 V
6.65V
RC
t
5V
iB
3.35V
vBE(t)
8mV
8mV
t
0.7V
0.7V
+ vbe
+
- VBE
3.3kΩ
iC
vBC- +
+
vCE
+
vBE -
-8mV
BJT Common-Emitter Amplifier
i B = 15µA , i C = 1.5mA since β = 100
Q-point is i B = 15µA ; V BE = 0.7V ; i C = 1.5mA ; V CE = 5V
v CE
1.65 ∠180
A V = --------- = ------------------------ = 206 ∠180 = – 206
v BE
0.008 ∠0
Lecture 29
29 - 2
Collector Current (mA)
4
t
vBE = 0.717V
IBE = 30µA
3
vCE(t)
2
vBE(t)
Q-point
1
Lecture 29
0
2
4
6
t
vBE = 0.7V
IBE = 15µA
vBE = 0.692V
IBE = 10µA
load line
0
vBE = 0.708V
IBE = 20µA
8
10
Collector-Emitter Voltage (V)
12
29 - 3
Small-Signal Modeling
i 1 = y 11 v 1 + y 12 v 2 ; i 2 = y 21 v 1 + y 22 v 2
i b = y 11 v be + y 12 v ce ; i c = y 21 v be + y 22 v ce
v BE = V BE + v be and v CE = V CE + v ce
+
vbe
-
i B = I B + i b and i C = I C + i c .
v be = ∆v BE = v BE – V BE and
v ce = ∆v CE = v CE – V CE .
ic +
ib
vce
-
i b = ∆I B = i B – I B and i c = ∆I C = i C – I C .
ib
y 11 = -------v be
ic
y 21 = -------v be
=
v ce = 0
=
v be = 0
∂i B
∂ v BE
;
Q–P
∂i C
∂ v BE
;
Q–P
ib
y 12 = ------v ce
ic
y 22 = ------v ce
=
v ce = 0
=
v be = 0
v CE
v CE
 v BE
---------------------; β F = β F0 1 +
i C = I S exp 
 1+
V
V
V
 T
A
A
∂i B
∂ v CE
Q–P
∂i C
∂ v CE
Q–P
IS
 v BE
-------; iB =
exp  --------- .
β F0
 VT 
β F0 is the value of β F extrapolated to v CE = 0 .
Lecture 29
29 - 4
Things To Remember
Y parameters have equivalent circuit representation
The current gain is actually a function of both the BE and BC voltages
i1
+
v1
y 21 v 1
1
-----
y 11
i2
1
-------
y 22
y 12 v 2
Collector Current (mA)
v2
IB = 100 µA
4
IB = 80 µA
IB = 60 µA
2
IB = 40 µA
IB = 20 µA
0
-15
-VA
Lecture 29
+
-10
-5
0
5
10
15
Collector-Emitter Voltage (V)
29 - 5
y 12 =
∂i B
∂ v CE
= 0
Q–P
∂i C
IS
V CE
IC
 V BE
--------------------exp 
y 21 =
=
= ------- = g m
 1+
∂ v BE
VT
V
V
VT
 T
A
Q–P
∂i C
IS
IC
 V BE
1
y 22 =
= ------ exp  ---------- = ------------------------- = ----VA
∂ v CE
V A + V CE
ro
 VT 
Q–P
i C ∂β F
i C ∂β F ∂i C
1 ∂i C
1 ∂i C
----------------y 11 =
=
–
=
– 2
∂ v BE
β F ∂ v BE β 2 ∂ i C ∂ v BE
β F ∂ v BE β ∂ v BE
Q–P
F
F
∂i B
∂i C 
i C ∂β F
IC 
i C ∂β F
IC
1
1
y 11 = ----- 1 – ----- = --------------  1 – ----- = ------------- = ----- .
β F ∂ v BE 
βF ∂ iC 
βF VT 
βF ∂ iC 
β0 VT
rπ
Lecture 29
29 - 6
∂i C
Here, β 0 =
is the small-signal common-emit∂ iB
ter current gain.
The Hybrid-pi small-signal model of the BJT is as
C
B
+ ib
vbe
rπ
ro
-
shown.
ic +
vce
gmvbe
-
E
Two-Port Representation
For β 0 = 100 , V A = 75V , V T = 25mV and, V CE = 10V , we get
IC
gm
rπ
rο
µf
1 µA
4x10-5 S
2.5 MΩ
85 MΩ
3400
10 µA
4x10-4 S
250 kΩ
8.5 MΩ
3400
100 µA
0.004 S
25 kΩ
850 kΩ
3400
1 mA
0.04 S
2.5 kΩ
85 kΩ
3400
10 mA
0.4 S
250 Ω
8.5 kΩ
3400
Summary: y 11 = I C ⁄ ( β 0 V T ) = 1 ⁄ r π and y 12 = 0
y 21 = I C ⁄ V T = g m and y 22 = I C ⁄ ( V A + V CE ) = 1 ⁄ r o
Lecture 29
29 - 7
Small-Signal Current Gain
β0 = gm r
π
∂β F
∂ iC
∂β F
∂ iC
βF
> 0 for i C < I M
βo < βF
βo > βF
< 0 for i C > I M
where
βF
β 0 = -------------------------------------------------- 1 ∂β F
1 – I C  -----
 βF ∂ iC 
iC
IM
Q–P
Amplification Factor (p. 600)
V A + V CE V A
I C V A + V CE
----------------------------⋅
= ------------------------- ≈ ------µf = gm ro =
IC
VT
VT
VT
This is the maximum voltage gain an individual transistor can provide.
Lecture 29
29 - 8
Equivalent Forms of Small-Signal Model
v ce
v be = i b r π , g m v be = g m r π i b = β o i b and i c = β o i b + ------- ≈ β o i b
r
o
C
B
+ ib
vbe rπ
gmvbe
-
C
B
ic + + ib
vce vbe rπ
ro
-
-
-
E
E
Voltage-controlled current
source (VCCS) model
βoib
ic +
vce
ro
Current-controlled current
source (CCCS) model
Definition of Small-Signal for the BJT (p. 602)
 v BE
 V BE + v be
i C = I S exp  --------- = I S exp  ------------------------- .
VT
 VT 


 V BE 
 v be
 v be

I C + i c =  I S exp  -----------  exp  ------- = I C exp  ------- .
 VT  
 VT
 VT

Lecture 29
29 - 9
v be 1  v be 2 1  v be 3
Using i C = I C 1 + ------- + ---  ------- + ---  ------- + .... and i c = i C – I C , we get
VT 2  VT
6  VT
v be 1  v be 2 1  v be 3
from i c = i C – I C that i c = I C ------- + ---  ------- + ---  ------- + .... .
6  VT
V T 2  V T
2
v be
1  v be
-------Now linearity requires that   « ------- or v be « 2V T .
2  VT
VT
Using a factor of 10 to satisfy the inequality, we get, v be ≤ 5mV .
v be
Now, we can write that i C ≈ I C 1 + ------- = I C + g m v be .
V
T
ic
iC – IC
g m v be
v be 5mV
----- = ---------------- = --------------- = ------- ≤ --------------- = 0.20 .
V T 25mV
IC
IC
IC
A 5 mV change in vBE corresponds to a 20% deviation in iC from its Q-point as well as a
20% change in iE since iE ~ iC.
Lecture 29
29 - 10
Summary of BJT Small-Signal Parameters
Parameter
BJT
Transconductance gm
IC ⁄ VT
Input Resistance RIN
rπ = βo ⁄ gm = βo VT ⁄ IC
Output Resistance ROUT
( V A + V CE ) ⁄ I C ≈ V A ⁄ I C
Amplification Factor µf
g m r o = ( V A + V CE ) ⁄ V T
Small-Signal Requirement
v be ≤ 5mV
Small-Signal Model for the pnp BJT
This is identical to that of the npn transistor.
Remember that the dc currents in a pnp transistor flow in an opposite direction to that
in a npn transistor.
Lecture 29
29 - 11