ESE319 Introduction to Microelectronics
High Frequency BJT Model
2008 Kenneth R. Laker, update 12Oct10 KRL
1
ESE319 Introduction to Microelectronics
Gain of 10 Amplifier – Non-ideal Transistor
Gain starts dropping at about 1MHz.
Why!
Because of internal transistor
capacitances that we have ignored
in our models.
2008 Kenneth R. Laker, update 12Oct10 KRL
2
ESE319 Introduction to Microelectronics
Sketch of Typical Voltage Gain Response
for a CE Amplifier
∣Av∣dB
Low
Frequency
Band
High
Frequency
Band
Midband
ALL capacitances are neglected
Due to external blocking
and bypass
capacitors
3 dB
Due to BJT parasitic
capacitors Cπ and Cµ
20 log 10∣Av∣ dB
f
f
L
BW = f H − f L≈ f
2008 Kenneth R. Laker, update 12Oct10 KRL
H
f  Hz
H
(log scale)
GBP=∣ Av∣ BW
3
ESE319 Introduction to Microelectronics
High Frequency Small-signal Model
SPICE
CJC = Cµ0
CJE = Cje0
C
rx
C
TF = τF
RB = rx
C =
Two capacitors and a resistor added.
A base to emitter capacitor, Cπ
A base to collector capacitor, Cµ
A resistor, rx, representing the base
terminal resistance (rx << rπ)
2008 Kenneth R. Laker, update 12Oct10 KRL
C =C de
C je 0
C0
m
V CB
1

V 0c
m
V BE
1−

V 0e
≈C de 2C je0
C de = F g m
τF = forward-base transit time
4
ESE319 Introduction to Microelectronics
High Frequency Small-signal Model
The internal capacitors on the transistor have a strong effect on
circuit high frequency performance! They attenuate base signals,
decreasing vbe since their reactance approaches zero (short circuit)
as frequency increases.
As we will see later Cµ is the principal cause of this gain loss at
high frequencies. At the base Cµ looks like a capacitor of value
k Cµ connected between base and emitter, where k > 1 and may
be >> 1.
This phenomenon is called the Miller Effect.
2008 Kenneth R. Laker, update 12Oct10 KRL
5
ESE319 Introduction to Microelectronics
Frequency-dependent “beta” hfe
short-circuit
current
The relationship ic = βib does not apply at high frequencies f > fH!
Using the relationship – ic = f(Vπ ) – find the new relationship
between ib and ic. For ib (using phasor notation (Ix & Vx) for
frequency domain analysis):
1
I
=
s C  sC  V  where r x ≈0 (ignore rx)
@ node B': b
r
NOTE: s = σ + jω, in sinusoidal steady-state s = jω.

2008 Kenneth R. Laker, update 12Oct10 KRL

6
ESE319 Introduction to Microelectronics
Frequency-dependent hfe or “beta”


1
I b=
s C  sC  V 
r
@ node C: I c = g m −s C  V  (ignore ro)
Leads to a new relationship between the Ib and Ic:
g m −s C 
Ic
h fe = =
Ib 1
s C s C 
r
2008 Kenneth R. Laker, update 12Oct10 KRL
7
ESE319 Introduction to Microelectronics
Frequency Response of hfe
h fe =
g m −s C 
1
s C s C 
r
multiply N&D by rπ and set s = jω
 g m− j  C   r 
h fe =
1 j C C  r 
factor N to isolate gm

C
1− j   g m r 
gm
h fe =
1 j C C  r 
2008 Kenneth R. Laker, update 12Oct10 KRL
IC
g m=
VT
VT
r  =
IC
C
1
For small ω
=
≪1
low : low
s
gm
10
and:
low C  C   r  ≪1
1
10
C

Note: low C C   r =low C C   ≫low
gm
gm
We have:
h fe = g m r =
8
ESE319 Introduction to Microelectronics
Frequency Response of hfe cont.









C
f
1− j
1− j
1− j   g m r 
z
fz
gm
h fe =
=
g m r =

1 j C  C   r 
f

1
j
1 j
f


h fe dB
20log10 
C

f z≫ f 
C  C  r  =C  C   ≫
=>
gm gm
f
f
Hence, the lower break frequency or – 3dB frequency is fβ
gm
1
f =
=
2  C  C  r  2 C C    the upper:
gm
1
f z=
=
2 C  / g m 2 C 
where f z 10 f 
2008 Kenneth R. Laker, update 12Oct10 KRL
9
f
z
ESE319 Introduction to Microelectronics
Frequency Response of hfe cont.
Using Bode plot concepts, for the range where: f  f 
h fe = g m r =
For the range where:
f   f  f z s.t. ∣1− j f / f z∣≈1
We consider the frequency-dependent numerator term to
be 1 and focus on the response of the denominator:
h fe =

g m r
f
1 j
f
2008 Kenneth R. Laker, update 12Oct10 KRL
=


f
1 j
f

10
ESE319 Introduction to Microelectronics
Frequency Response of hfe cont.
Neglecting numerator term:
And for f / f  >>1 (but < f / f z ):
Unity gain bandwidth:
h fe =

g m r
f
1 j
f

f
1 j
f

f

=
∣h fe∣≈
f
f
f
 
f
∣h fe∣=1⇒  f ¿ ¿| f = f =1⇒ f T = f 
T
T
f T=
= f 
2
2008 Kenneth R. Laker, update 12Oct10 KRL
=

BJT unity-gain frequency or GBP
11
ESE319 Introduction to Microelectronics
Frequency Response of hfe cont.
=100
r  =2500 
−3
C  =12 pF C =2 pF g m =40⋅10 S
12
−3
1
10 ⋅10
 =
=
=28.57⋅106 rps
 C C   r  122⋅2.5
 28.57 6
f =
=
10 Hz=4.55 MHz
2 6.28
f T = f =455 MHz
g m 40⋅10−3⋅1012
z= =
Hz=20⋅10 9 rps
C
2
z
f z=
=3.18⋅109 Hz=3180 MHz
2
2008 Kenneth R. Laker, update 12Oct10 KRL
12
ESE319 Introduction to Microelectronics
Scilab fT Plot
//fT Bode Plot
Beta=100;
KdB= 20*log10(Beta);
fz=3180;
fp=4.55;
f= 1:1:10000;
term1=KdB*sign(f); //Constant array of len(f)
term2=max(0,20*log10(f/fz)); //Zero for f < fz;
term3=min(0,-20*log10(f/fp)); //Zero for f < fp;
BodePlot=term1+term2+term3;
plot(f,BodePlot);
2008 Kenneth R. Laker, update 12Oct10 KRL
13
ESE319 Introduction to Microelectronics
hfe Bode Plot
(dB)
fT
2008 Kenneth R. Laker, update 12Oct10 KRL
14
ESE319 Introduction to Microelectronics
Multisim Simulation
v-pi
Ic
Ib
v-pi
mS
Insert 1 ohm resistors – we want to measure a current ratio.
g m−s C 
Ic
h fe = =
Ib 1
s C C  
r
2008 Kenneth R. Laker, update 12Oct10 KRL
15
ESE319 Introduction to Microelectronics
Simulation Results
Theory:
Low frequency |hfe|
f T = f =455 MHz
Unity Gain frequency about 440 MHz.
2008 Kenneth R. Laker, update 12Oct10 KRL
16