Lecture 3B
MOSFET High Frequency Model and Amplifier Frequency Response
Objectives
„ To review the small signal BJT models at low frequencies
„ To study the high frequency BJT models
„ To estimate the BJT unity-gain frequency
Introduction
In the last two lectures we examined the time and frequency response of the STC circuits.
As we said the importance of studying the STC circuits is that the analysis of a complex
amplifier circuit can be usually reduced to the analysis of one or more simple STC
circuits. The frequency response of the amplifier circuits will be explored starting next
lecture. Before starting this study it will be constructive to review in this lecture the BJT
small-signal models, and examining the high-frequency models. Also, the transistor cutoff frequency which is considered a figure of merit at high frequency operation will be
estimated for both transistors.
The BJT small-signal model
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The small signal model of the BJT amplifier is shown in figure 5. Figures 5-a,b
are for the π-model, where Figures 5-c,d are for the T-model.
These models are valid for both NPN and PNP transistors.
For the same operating point, the BJT has higher transconductance and higher
output resistance that the MOSFET.
The small-signal parameters are controlled by the Q-point (operating point).
gm Vπ
Figure 1 small signal-models of the BJT
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The BJT small-signal parameters may be summarized in Table 3
Table 1 BJT small signal parameters
Symbol
Parameter
Value
I
gm = C
VT
Transconductance
gm
rπ
re
VT is the thermal voltage = kT/q, which
equals 25mV at room temperature.
k is Boltzman's constant
T is the absolute temperature in
Kelvins
q is the electron charge
Base input resistance
⎛V ⎞ β
VT
=β⎜ T ⎟=
IB
⎝ IC ⎠ g m
β is the common-emitter current gain
Emitter input resistance
⎛V ⎞ α
VT
=α ⎜ T ⎟ =
IE
⎝ IC ⎠ g m
α is the common-base current gain
rπ =
re =
Symbol
ro
Parameter
Output resistance
Value
ro =
V A +V CE
IC
VA
IC
VA is the early voltage.
The BJT high-frequency model:
Figure 2 The high-frequency hybrid- π model of the BJT
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The high frequency hybrid-π model for the BJT is shown in Figure 6.
This model is useful for signal frequencies up to a several tens of megahertz, after
which a more detailed model becomes necessary.
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Typically, the base-emitter junction capacitance Cπ is in the range of few pF to
few tens of pF, while the collector-base junction capacitance Cμ is in the range of
fraction of pF to few pF
The base resistor rx is added partly to account for the comparatively long internal
connection from the base external connection and the actual internal base
connection.
A representative resistance value for this lumped resistor is in the range of 50Ω to
perhaps 200Ω.
This resistor ordinarily can be neglected for hand estimates.
Note that rx becomes the dominant input resistance for frequencies so high that Cπ
effectively short-circuits rπ.
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A second base-width modulation effect, characterized by a resistor connected
between the base and collector is omitted; its influence is dominated by the
collector junction reverse-bias capacitance Cμ.
The emitter junction (diffusion) capacitance Cπ represents the charge store to
support the current flow across the base.
The BJT Cutoff frequency:
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As defined earlier, it is the frequency at which the current gain of the transistor
becomes one. (i.e. no more active element). It is calculated by finding the short
circuit collector current in terms of the base current.
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Using the high frequency model of BJT we can draw the circuit to estimate the
cut-off frequency of the BJT as shown in Figure 7.
sC μ V π
I c = ( g m − s C μ )V π
Figure 3 Circuit used to estimate the BJT cutoff frequency
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Applying nodal analysis at the input and output nodes as we did earlier. We can
estimate the cut-off frequency as follows:
Vπ
+ s C πV π + s C μ V π
rπ
V
I b = π + s (C π + C μ ) V π
rπ
g m − sC μ
I
hfe ≡ c =
1
Ib
+ s (C π + C μ )
rπ
Ib =
Assuming
g m >> sC μ ⇒
Assuming
g m >> sC μ ⇒
hfe ≅
βo
1+
s
where ω p =
ωp
g m rπ
1 + s (C π + C μ )rπ
g m rπ
hfe ≅
1 + s (C π + C μ )rπ
hfe ≅
1
(C π + C μ )rπ
∴ Unity gain bandwidth
(ωT ) = βo ω p =
hfe ≡
hfe ≅
hfe ≅
gm
(C π + C μ )
g m − sC μ
Ic
=
1
Ib
+ s (C π + C μ )
rπ
g m rπ
1 + s (C π + C μ )rπ
βo
1+
s
ωp
assuming g m >> sC μ
where ω p =
1
(C π + C μ )rπ
∴ Unity gain bandwidth (ωT ) = βo ω p =
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gm
(C π + C μ )
We can observe from the last analysis that the common-emitter current gain (hfe)
frequency response is similar to a simple pole with ωp as the pole frequency. This
may be drawn as shown in Figure 8
|hfe| [dB]
3 dB
βo
-6 dB/octave
[log scale]
0 dB
β
T
Figure 4 Bode plot of |hfe|
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As we can see from the last equation. Higher ωT means higher gm and lower
internal BJT capacitances which means better amplifier operation.
Typically, fT is ranging from about 100MHz to Tens of GHz.