Section H4: High-Frequency Transistor Models
So far, we’ve been concentrating on the capacitors that are external to the
transistor in an amplifier circuit and provide coupling or bypass functions.
These components are treated as series capacitances and determine the
low frequency response of the amplifier, which may be found by the method
of short circuit time constants using the ac small signal model as discussed
in the previous section.
What we’re going to do now is look at the internal capacitances of the
transistor that dominate the high frequency amplifier response. All external
capacitors, which are much larger than internal capacitances, are considered
to be short circuits in the high frequency region of operation. Internal
capacitances are modeled as shunt (parallel) capacitances and involve
enhancements to the small signal model we’ve been using so far.
In Section H2, an approximation for the high frequency cutoff was given in
terms of the transfer function poles as
⎡
⎤
ω H ≅ ⎢∑ τ i ⎥
⎣
⎦
i
−1
=
1
.
∑ C i Reqi
i
Once again, we will be looking at relevant capacitances one at a time,
finding the equivalent resistance seen by the individual capacitance, and
then performing a summation to determine the cumulative effect. However,
instead of the method of short circuit time constants in which we set all
other capacitors to infinity (zero effective impedance), we will be using the
method of open circuit time constants for the high frequency cutoff. In
this method, independent sources are again set to zero (short voltage
sources and open current sources), but now all capacitors other than the one
of interest are set to zero (infinite effective impedance). Just as a heads up,
a more accurate approximation for the high frequency cutoff may be found
by including the effect of the transfer function zeros as
ωH ≅
1
1
ω P21
+
1
ω P22
+
1
ω P23
+K−
1
ω Z21
−
1
ω Z2 2
−
1
ω Z2 3
,
−K
while always keeping in mind that if we can define a dominant pole, life is
good and calculations become easy!
Miller Theorem
The Miller Theorem is very useful in the development of high frequency
equivalent transistor circuits. This theorem allows us to transform a series
capacitance into equivalent shunt (parallel) capacitances that are much
larger than the original capacitance value. Specifically, the Miller theorem
allows us to transform a circuit in the form of Figure 10.14a (below left), in
which an impedance element connects the input and output, into the
equivalent circuit of Figure 10.14b (below right) where the input and output
circuits have been separated and may be analyzed independently. By
appropriately selecting the values of the impedances Z1 and Z2, the two
circuits will behave identically and we can efficiently determine the high
frequency response of the entire amplifier circuit.
By analyzing each of the above circuits individually, and realizing that the
voltage gain, Av, and currents, I1 and I2, must remain constant regardless of
the circuit representation, your author derives expressions for the shunt
impedances Z1 and Z2 in terms of the series impedance Z as
Z1 =
Z
1 − Av
and
Z2 =
Z
1− 1
.
(Equations 10.46 & 10.47)
Av
If we can create these shunt capacitances, we can
model each side of the circuit as a simple RC network,
as shown in Figure 10.13 and reproduced to the right.
In this figure, the capacitance C exists between two of
the device terminals and the resistance R is the
equivalent resistance seen by the capacitance. Using a
voltage divider, the output voltage is equal to
v out =
(1 sC )v in
R + 1 sC
=
v in
v in
=
.
sCR + 1 1 + jωRC
The input-output relationship of this circuit is therefore given by
v out
1
1
=
=
,
v in
1 + jωRC 1 + jωτ
(Equation 10.43)
where the time constant, τ, is equal to RC.
High-Frequency BJT Model
The high frequency hybrid-π
model for the BJT is given in
Figure 10.15a of your text and is
duplicated to the right. Note an
additional terminal, B’, has been
added to the schematic and that
resistance and capacitances have
been defined between each pair
of terminals. The point denoted
as B’ represents an internal base
terminal that is not accessible to
the outside world and that is
separated by the physical base terminal, B, by a resistance denoted rbb’.
Since manufacturer’s data sheets typically specify parameter values in terms
of h-parameters, a modified version of Equation 4.18 (see Section C3) is
repeated below for reference purposes.
rπ ≈ hie = (β + 1)re ≈ βre
gm ≈
ro ≈
I
β +1
β
1
1
≈
=
≈
≈ C
hib
re
hie
hie
VT
Equation 4.18
V
1
= A
hoe
IC
Using this information, the resistances indicated in Figure 10.15a will be
defined and/or related to h-parameters as follows:
¾ Base spreading resistance, rb’b. The base spreading resistance models
the equivalent resistance of the silicon between the physical base
terminal, B, and the intrinsic base terminal B’. The value of this resistance
depends on the total current level and appears in the total input
resistance as defined below. Although this resistive component is
negligible at low frequencies since it is usually less than 100Ω, it
contributes to high frequency response.
¾ Feedback resistance, rb’c. The feedback resistance models the effect of
the collector voltage on the base current. Looking from the base, rb’c is at
least equal to βοro, where βo is the value of β at low frequencies
(sometimes called dc gain), and is typically on the order of 10βoro when
the collector-base junction is reverse biased for normal active operational
mode. Note that your author does not explicitly indicate the frequency
dependence of β in this manner, but I wanted to bring up this point
because most resources do. The feedback resistance is related to the
reverse voltage gain h-parameter, hre, through
hre =
rb'e
r
v b 'e
=
≅ b'e ,
v ce
rb'e + rb'c
rb'c
(Equation 10.54)
where the last approximation is made since rb’c >> rb’e.
¾ Input resistance. The transistor input resistance is between the base
(B) and emitter (E) terminals and is indicated in the hybrid-π model by rπ.
The input resistance is approximately equal to the h-parameter hie, where
hie is defined as the input resistance of the device with the output
shorted. If we short the output of the circuit above (i.e., the collectoremitter pair of terminals), and realize that the base-collector junction is
reverse biased so its resistance is much larger than the resistance
associated with the forward biased base-emitter junction (i.e., rb’c >>
rb’e), we get the complete expression for the transistor input resistance to
be
rπ = hie = rb'b + rb'e || rb'c ≅ rb'b + rb'e ,
(Equation 10.48)
Also included in Figure 10.15a are three capacitances, one between each of
the device terminals. Specifically
¾ Collector-junction capacitance, Cb’c. The collector-junction capacitance
is the capacitance of the collector-base junction. It is small, with typical
values in the range of 0.5 to 5pF and, although its value varies with the
width of the junction depletion region, it is usually considered to be
constant over a specific region of transistor operation. This capacitance is
indicated in spec sheets as Cob, but may also be referred to as Cµ. Using
Miller’s theorem, the series capacitance Cb’c may be transformed into two
shunt capacitances of value
C M1 = C b'c (1 − Av ) (input circuit )
.
⎛
1 ⎞
⎟⎟ (output circuit )
C M 2 = C b'c ⎜⎜1 −
Av ⎠
⎝
(Equation 10.57)
¾ Output capacitance, Cce. The output capacitance is defined between the
collector and emitter terminals. It is usually extremely small and is
generally ignored since the input capacitance is so much larger.
¾ Base-emitter capacitance, Cb’e. The base-emitter capacitance, often
called Cπ to associate it with rπ, is indicated in data sheets as Cib. This
capacitance is the sum of two parts: the emitter diffusion capacitance
(Cde) that is proportional to the dc bias current and an emitter junction
depletion-layer capacitance (Cje) that is dependent on the value of baseemitter voltage, VBE, or
C b'e = C de + C je .
Since the first term is larger than the second, Cb’e is approximately equal
to the diffusion capacitance (also called the base-charging capacitance)
and usually has a value in the range of 1 to 200pF. The value of Cb’e may
be estimated by
C b'e ≈
| I CQ |
gm
1
,
=
=
2πfT
2πfT re
2πfT VT
(Equation 10.56)
where fT is the unity gain frequency, a figure of merit defined as the
frequency at which the common emitter short circuit current gain is 0 dB
(remember that 20log(1)=0 dB). Note that, since the gain-bandwidth
product (GBP) is considered a constant figure of merit for an amplifier,
the unity gain frequency may be approximated by GBP.
Putting all this information
together, we can present
the
high-frequency
BJT
equivalent circuit in terms
of the familiar hybrid-π
model as shown to the right
and in Figure 10.15b of
your text. Note that ro is
now explicitly included in
the circuit and must be considered in the calculation of equivalent
resistances (recall that we usually neglected it for low and mid-range
frequencies because it is so large).