BJT high – frequency operation
In practical common emitter
amplifiers, capacitors C1 and C2
are used to block the DC current.
CE acts are as bypass capacitor for
shorting RE at high frequency
operation
Consider medium frequencies, which are
(i) high enough so that the coupling and bypass capacitances have
very small impedances and can be substituted by shorts
(ii) but are still much smaller than the BJT cutoff frequencies so that
the internal transistor capacitances can be considered as open circuits.
1
BJT small-signal operation
Base
current
1 µA
time
600
400
ib
2 µA
3 µA
4 µA
5 µA
-1.0
ic
200
time
0
2
4
6
Collector-emitter voltage (V)
2
BJT high-frequency equivalent circuit
We start with the DC equivalent circuit:
αnIe
αiIc
Ic
Ie
Emitter
Ideal diode
Ideal diode
Leakage diode
Collector
Leakage diode
Base
3
Additional high-frequency components of BJT terminals
(a)
(c)
(b)
The base - emitter (a),
base-collector (b),
and collector – emitter (c)
components of the BJT equivalent circuit
4
Small-signal hybrid π model
Simplified
Full
Cb'c
rbb'
Collect or
b'
rbb'
b'
rb'c
Base
Base
rb'e
Collect or
Cb'e
gmVb'e
rce
rb'e
Em it t er
gmVb'e
Em it t er
The transconductance, gm, is related to the dynamic
(differential) resistance, re, of the forward-biased emitter-base
junction:
α
∂Ic
∂Ie
Ic
gm =
=α
≈ ≈
∂Vb' e
∂Vb'e re Vth
Vth = kBT/q
5
Hybrid π model (cont.)
Cb'c
rbb'
Co llec t o r
b'
rb'c
Base
rb'e
Cb'e
gmVb'e
rce
Em it t er
The resistance rbb' is the base spreading resistance.
The resistance rb'c and the capacitance Cb'c represent the dynamic
(differential) resistance and the capacitance of the reverse-biased
collector-base junction.
6
Hybrid π model (cont.)
Cb'c
rbb'
Co llec t o r
b'
rb'c
Base
rb'e
Cb'e
gmVb'e
rce
Em it t er
ic ≈ gm vb'e
Using transconductance:
(ignoring the current through rce )
B-E differential resistance:
B-E internal voltage
(ignoring the current through Cb’e):
ic 1
β
≈
rb ' e ≈
ib g m g m
vb ' e ≈ ib rb ' e
7
Frequency dependent gain (short-circuit load)
r bb'
r b'e
ic
Cb'c
V b'e
Cb'e
gm V b'e
v b' e ≈ ib rb'e transforms into ib = vb'e [gb' e + jω(Cb' e + Cb'c )]
where, gb'e = 1/rb'e
Since ic
= gmvb'e, the short-circuit emitter current gain
ic
gm
β
βω = =
=
ib gb'e + jω (Cb'e + C b' c ) 1 + jω / ω
β
(
where
)
ωβ = 2πfβ = gb' e /(Cb'e + Cb' c )
8
The BJT cutoff frequencies
ic
gm
β
βω = =
=
ib gb'e + jω (Cb'e + C b' c ) 1 + jω / ω
β
(
)
ωβ = 2πfβ = gb' e /(Cb'e + Cb' c )
(1)Beta cutoff frequency fβ
is the frequency at which ω = ωβ
i.e. the magnitude of the common-emitter current gain
decreases by a factor of √2
f β = gb ' e / ⎡⎣ 2π ( Cb ' e + Cb ' c ) ⎤⎦
9
The BJT cutoff frequencies
r bb'
ic
Cb'c
r b'e
V b'e
Cb'e
gm V b'e
ic
gm
β
βω = =
=
ib gb'e + jω (Cb'e + C b' c ) 1 + jω / ω
β
(
)
(2) Common emitter cutoff frequency fΤ
is the frequency at which the magnitude of the commonemitter current gain equals unity, that is, |βω| = 1.
fT = f β
gm
β −1 ≈
2π ( Cb ' e + Cb ' c )
2
10
Carrier transport delays in BJTs
E
xdc
p
n
n
C
B
Electrons injected into base diffuse across base
and drift across the collector depletion region xdc
(typically xdc is much larger then the base thickness W)
At high frequencies this transit time becomes comparable with the
period of the input signal; at this point the output response will no
longer be in phase with the input signal.
11
Carrier transport delays in BJTs (cont.)
Collector depletion region delay time:
xdc
τ cT ≈
vsn
where xdc is the width of the collector-base depletion region and
vsn is the electron saturation velocity (for n-p-n transistors).
Other delays include collector and emitter charging times and
base transit time
If the total BJT delay time is τeff, the cutoff frequency:
1
fT ≈
2πτ eff
12
Two-port network BJT equivalent circuit
h-parameters equivalent circuit
i1
Transistor small
signal equivalent
Circuit
+
i2
h11
v1
h12v2
+
–
+
h21i1
h22
v2
–
–
v1 = h11i1 + h12 v 2
i2 = h21i1 + h22 v2
13
h-parameters
i1
+
i2
h11
h12v2
v1
+
–
+
h21i1
h22
v2
–
–
v1
v
i1 v = 0
2
open-circuit reverse h = 1
12 v 2 i = 0
voltage ratio
1
i2
short-circuit forward
h =
current ratio
21
i1 v = 0
2
i2
open-circuit output
h =v
admittance
22
2 i1 = 0
short-circuit input
impedance
h11 =
14
h-parameters and BJT parameters
Alternative notation:
h11 is also called the short-circuit input impedance (hi) ,
h12 is called the open-circuit reverse voltage ratio (hr),
h21 is called the short-circuit forward current ratio (hf), and
h22 is called the open-circuit output admittance (ho).
Second subscript is often used to denote the transistor configuration:
(“e” for common emitter circuit).
h-parameter
hoe
hie
hfe
Relation to parameters of hybrid π-equivalent circuit
1/ (rb' c + rb' e ) + 1/ rce + gm rb'e / (rb' c + rb' e )
rbb' + rb' e rb' c / (rb' e + rb'c )
g mrb'e rb' c /(rb' c + rb'e )
15