Small Signal Model of a BJT
•Small signal Models are only useful for Forward active mode and thus, are
derived under this condition. (Saturation and cutoff are used for switches
which involve very large voltage/current swings from the on to off states.)
•Small signal models are used to determine amplifier characteristics
(Example: “Gain” = Increase in the magnitude of a signal at the output of a
circuit relative to it’s magnitude at the input of the circuit).
•Warning: Just like when a diode voltage exceeds a certain value, the nonlinear behavior of the diode leads to distortion of the current/voltage curves,
if the inputs/outputs exceed certain limits, the full Ebers-Moll model
must be used.
Signal notation: A transistor circuit, whether digital or analog, is typically
connected to several DC power supplies that establish the desired DC
"bias" currents and voltages throughout it. It also typically has one or more
time varying input signals that result in time varying currents and voltages
(one of which is the desired output of the circuit) being added to the DC bias
currents and voltages.
Each voltage and current in such a circuit thus has a DC bias portion and a
signal portion, which add to make the total. We use the following notation to
identify these components and the total:
DC Bias Values: To construct linear amplifiers and other linear signal processing
circuits from non-linear electronic devices we must use regions in the non-linear
characteristics that are locally linear over useful current and voltage ranges, and
operate there.
To accomplish this we must design the circuit so that the DC voltages and currents
throughout it "bias" all the devices in the circuit into their desired regions, e.g.
yield the proper bias currents and voltages:
IA , IB , IC , ID,etc. and VAG,VBG,VCG,VDG, etc.
This design is done with the signal inputs set to zero and using the large signal
static device models we have developed for the non-linear devices we studied:
diodes, BJTs, MOSFETs.
Working with these models to get the bias values, though not onerous, can be
tedious. It is not something we want to have to do to find voltages and currents
when the signal inputs are applied. Instead we use linear equivalent circuits .
Linear equivalent circuits: After biasing each non-linear devices at the
proper point the signal currents and voltages throughout the circuit will
be linearly related for small enough input signals. To calculate how they
are related, we make use of the linear equivalent circuit (LEC) of our
circuit.
The LEC of any circuit is a combination of linear circuit elements
(resistors, capacitors, inductors, and dependent sources) that correctly
models and predicts the first-order changes in the currents and voltages
throughout the circuit when the input signals change.
A circuit model that represents the proper first order linear relationships
between the signal currents and voltages in a non-linear device is call an
LEC for that device.
Our objective is to develop LECs for each of the nonlinear devices we
have studied: diodes, BJTs biased in their forward active region (FAR),
and MOSFETs biased in their sub-threshold and strong inversion FARs.
Consider the BJT as a two-port Network
BJT Hybrid Model π
Hybrid Model π
All frequencies
Better model than h parameter model
since it contains frequency sensitive
components. These are ac small signal
parameters which are determined at the
Q point
Parasitic Resistances
rb = rb’b = ohmic resistance – voltage drop inbase region caused by transverse flow
of majority carriers, 50 ≤ rb ≤ 500
rc = rce = collector emitter resistance – change in Ic due to change in Vc, 20 ≤ rc ≤
500
rex = emitter lead resistance – important if IC very large, 1 ≤ rex ≤ 3
Parasitic Capacitances
Cje0 = Base-emitter junction (depletion layer) capacitance, 0.1pF ≤ Cje0 ≤ 1pF
Cμ0 = Base-collector junction capacitance, 0.2pF ≤ Cμ0 ≤ 1pF
Ccs0 = Collector-substrate capacitance, 1pF ≤ Ccs0 ≤ 3pF
Cje = 2Cje0 (typical)
ψ0 = .55V (typical)
τF = Forward transit time of minority carriers, average of lifetime of holes and electrons,
0ps ≤ τF ≤ 530ps
Hybrid Model Pi Parameters
rπ = rb’e = dynamic emitter resistance – magnitude varies to
give correct low frequency value of Vb’e for Ib
rμ = rb’c = collector base resistance – accounts for change in
recombination component of Ib due to change in Vc which
causes a change in base storage
cπ = Cb’e = dynamic emitter capacitance – due to Vb’e stored
charge
cμ = Cb’c = collector base transistion capacitance (CTC) plus
Diffusion capacitance (Cd) due to base width modulation
gmVπ = gmVb’e = Ic – equivalent current generator
Low and Midband Frequency Hybrid Model π
At low frequencies, all Xc for hybrid model π are
very large. Since they are in parallel with a much
lower than their associated resistances, theXC
component may be ignored (replace with an open).
At frequencies over a few 100kHz, they must be
considered. At high frequencies, they shunt (short)
the parallel resistances. rμ is very large and may be
ignored (replace with an open) rb is much smaller
than rπ. Since they are in series, rb is often ignored
(replaced with a short or lumped with rπ) since the
current, IB the current through both, is also very
small. rex is very small and is often ignored
(replace with a short) unless IE is very large. ro is
very large and may be ignored at low frequencies ic
= io ≅ gm vπ (where vb’e = vπ)