Transistor Models and the Feedback Amplifier
Wes Hayward, w7zoi.
30May, 1June, 9June, 17June, 19June 2009 (converted to HTML,
27Dec09)
(Figure captions appear in blue.) (Measurement Data added at end.)
Applied science, and electronic engineering in particular depend upon models.
We never attempt to deal with a complete description of a transistor, vacuum
tube, or other device. It’s just too complicated.
Rather, we deal with models. A model is a simplified picture
of the actual device, usually steeped in the language of mathematics.
Transistor models range from very simple linear, frequency independent approximations
of physical reality to complex nonlinear frequency dependent descriptions.
The more refined and elegant models usually form the basis of computer programs
for circuit analysis. These models are potentially much more
accurate and complete than the simplified models. But there is a price
for this accuracy and completeness. A refined device model can often
get in the way when we try to understand circuits. The
more useful analysis, and ultimately the more productive design approach
treats circuits with the simplest device model that will do the job.
Only after the salient circuit behavior is determined with a simple device
model is a more refined analysis performed.
The circuit we consider here is the so called feedback amplifier used in
many RF applications. This circuit begins with a single transistor
(or FET) amplifier with two forms of feedback. When negative
feedback is applied in two ways, it allows flexibility that is not available
with just one. Eventually we will consider a special two stage
design.
Device Models
What is the simplest device model that we can use for the circuit analysis?
One traditional bipolar transistor model often regarded as the simplest is
a current driven current generator. A common emitter amplifier
is modeled as a current controlled current generator described by a parameter
β (Beta). The current flowing out of the collector is directly
proportional to the current flowing in the base. This familiar
model is shown below.
Fig 1. Simple current controlled current
generator forms a simple model for the bipolar transistor. This
is useful for both small and some large signal calculations including bias
evaluation.
This current controlled “Beta Generator” is the model we use for simple bias
calculations and sometime for amplifier analysis. But this is not the
simplest model, nor is it always the best of the simple models.
Voltage driven models are also extremely useful. The transistor
is now modeled as a current generator controlled by an input base voltage,
shown below.
Fig 2. This model for the bipolar transistor
is a current generator that is controlled by a voltage, vbe.
The “be” in the subscript for this voltage indicates that it is the base
voltage measured with respect to the emitter voltage, or vbe=vb-ve.
The voltage controlled model shown above is a small signal model.
The meaning of the term small signal is summarized in the paragraph
below.
Small Signal Modeling:
Consider the following example circuit:
Fig 3. The left schematic shows a circuit with biasing components.
The base is driven with a low impedance source, so the impedance of the biasing
divider is not important for signal flow. The input impedance
of the NPN is of little consequence with a low-Z voltage drive. The
emitter circuit contains a 510 Ω bias resistor, but the emitter is bypassed
to ground. This merely means that the signal induced variations
in emitter current flow in the bypass capacitor and not in the 510 Ω resistor.
This circuit is a grounded, or common emitter amplifier, for the emitter
is common to input and output. The output load is the parallel
combination of the 1 Meg and the 470 Ω, which is essentially just 470 Ω.
Analysis of the NPN bias shows an emitter current of 4.5 mA.
As we will find later, this establishes the transconductance, gm, for the
NPN at 0.173. The resulting small signal circuit is that at the
right, a considerable simplification over the original.
Often when performing small signal analysis,
voltage and current levels are used that seem far from small.
For example, a 1 volt signal applied to the above circuit yields a small
signal current of 0.17 amp. That output current flows in a 470
Ohm load for a small signal output voltage of, from Ohm’s Law, 81 volts.
We neglect the collision with the reality of the left circuit and calculate
a voltage gain of 81. The signal currents and voltages are both
well beyond the bias values, but this can be ignored, for the voltage gain
is the detail sought. Kilovolt or microvolt signals
both work in a small signal model.
The Essence of Emitter Degeneration
The voltage driven model presented in Fig 2 is especially useful when we
consider emitter degeneration, which is a resistance in the emitter circuit
that is not bypassed. One of the virtues of modeling with
simplified elements is that it allows the discovery of circuit behavior that
might otherwise be obscure if we tried to do the analysis with more complicated,
complete models. Consider the following general circuit.
Fig 4. The left current generator is specified
by a arbitrary transconductance, gm. A degeneration resistor
Rd is placed in series with the generator. The right model
is a simplified circuit with a new, reduced transconductance.
The signal current from the left generator of Fig 4 flows in the degeneration
impedance to create an emitter voltage other than ground. The
model at the left is analyzed to obtain a new transconductance, GM, that
describes voltage gain for the base with respect to ground.
The algebra shows that the new, upper case transconductance is given by the
simplified equation
This equation applies so long as the original transconductance, gm, is very
large. The mathematical details are in an appendix file, http://w7zoi.net/fba_with_simple_model.pdf
Another extremely useful result is that the voltage gain of such a circuit
is a simple ratio.
where the negative sign indicates an inverting amplifier.
Some Physics
The voltage drive model of Fig 2 can now be extended by evoking a little
bit of physics. This can be found in numerous texts with the
most notable (read as “my favorite”) probably being that of Gray and Meyer,
“Analysis and Design of Analog Integrated Circuits,” Second Edition, Wiley,
1984. We have learned that the transconductance of a transistor
is simply related to an emitter degeneration resistance.
Examination of a more detailed model by Ebers and Moll shows that the bipolar
transistor is a device with an exponential behavior,
This is a large signal model. Manipulation
of this equation produces the small signal approximation
where
Io is now the bias current, q is the electronic charge, k is Boltzman’s constant,
and T is absolute temperature in Kelvin. This can be reformatted
in more familiar terms as
where Ie is now the bias emitter current in mA. If we interpret
this in terms of the small signal voltage drive model of Fig 2, we conclude
that the bipolar transistor is model as the following:
Fig 5. The bipolar transistor is modeled
as a voltage controlled current source where the generator has very high
transconductance, but is then degenerated with an intrinsic emitter resistance,
re with value 26/Ie(mA).
Before applying this model to the feedback amplifier,
consider its importance. The model states that the transistor has a
gain that is proportional to the standing bias current.
If the device bias current is increased, the gain will also increase.
This is probably the most fundamental tool available to the designer.
The second is feedback.
Total Degeneration
Emitter degeneration has been presented in two forms. One was
a completely general rd. The other was the intrinsic emitter
degeneration related to bias. The two are combined below
to produce a total degeneration, which we signify with an upper case Rd.
Fig 6. The total degeneration in a common
emitter amplifier is Rd consisting of the external degeneration from rd plus
the internal or intrinsic degeneration from re.
Why would one degeneration be used over the other? The
current dependent re is part of the transistor. As such, it can
be nonlinear in the large signal equivalent model. This means
that it can generate harmonic and intermodulation distortion products in
a real world circuit. In contrast, rd is a simple resistor,
a linear element for both small and large signals. The most linear
circuits will be those with high bias current (and thus small re) in conjunction
with enough rd to constrain the gain to a modest level.
Analyzing the Feedback Amplifier
The familiar feedback amplifier is shown below in small signal form.
Bias and other DC details are omitted. This circuit uses two forms
of negative feedback: emitter degeneration and parallel collector to base
feedback.
Fig 7. Fundamental feedback amplifier.
The transistor is modeled as an ideal voltage controlled current source
with a transconductance that depends on the DC bias current, as discussed
above,
External degeneration is added, shown as rd in the above circuit.
The two degeneration resistances merge to become a single (upper case) Rd.
The simple transistor model is then used to analyze the circuit for gain
as well as input and output impedance. Details are
given in http://w7zoi.net/fba_with_simple_model.pdf .
Transducer gain is given as
Gt depends upon the feedback elements Rf and Rd as well as the terminations
RS and RL. Gain depends upon current which is described by re,
which is a part of Rd.
The input and output impedances are given by
Input impedance depends upon the output load resistance while the output
impedance depends upon the source resistance.
An interesting detail that flows from the analysis is that the amplifier
is perfectly matched with Zin=RS and Zout=RL if the total degeneration Rd
is set to Rd = RS•RL/ Rf. However, this
relationship is exact only if RS = RL.
It is otherwise just an approximation.
Consider the case of equal 50 Ohm source and load resistances and a degeneration
resistance defined by the above equation. The following
curves then describe the amplifier:
Fig 8. Gain and degeneration resistance
versus feedback R for the special case of a 50 Ohm source and 50 Ohm load
and Rf•Rd = RS•RL.
Fig 9. Input and output impedance versus
feedback for the “matched” condition, Rf•Rd = RS•RL. The calculation
was done for RL =51. If it had been exactly 50 like RS, the two curves
would have appeared on top of each other.
The following curves show an amplifier with a 50 Ohm source and a 200 Ohm
load. This is a popular configuration that offers better efficiency
and intercept when a modest power supply (12 volts) is available.
We again apply Rf•Rd=RS•RL.
Fig 10. Impedances, degeneration resistance,
and transducer gain for the feedback amplifier with 200 Ohm load, 50 Ohm
source, and Rd = RS•RL/ Rf.
An amplifier of general interest is the simple bidirectional circuit of EMRFD
Fig 6.110. This circuit is shown below, but with amplification
in only one direction.
Fig 11. A typical feedback amplifier used in
some SSB transceivers described on the web.
This circuit, in one form or another, has been used in some recent web applications.
The amplifier in this figure is biased to an emitter current of only 3.5
mA. As such, re is 7.5 Ohms. Because this amplifier uses
no external degeneration, Rd is also 7.5 Ohms.
The input impedance for any load, or the output impedance for any source
can be calculated. If the source and load are forced to
have the same value, a characteristic impedance can be evaluated for the
amplifier, shown in the figure as 87 Ohms. It is clearly not
a 50 Ohm circuit even though it is often used with 50 Ohm terminations.
It may be a poor termination for filters.
An Enhanced Transceiver Amplifier Block
The ideal amplifier for use in a transceiver is one that has input and output
impedances that are independent of the terminations. Yet
feedback amplifiers are often used, for they offer stable gain and freedom
from self oscillation, which is another form of stability.
The virtues of stable impedances and well controlled, stable gain can be
realized with a simple circuit if more than one transistor is used.
An example circuit is shown below.
Fig 12. A transceiver gain block.
Two stages are cascaded. The first stage has a 470 Ohm load resistance.
Parallel feedback and emitter degeneration force the input to 50 Ω.
The intermediate load is the source for the second stage. The
output transformer is back terminated to generate a 50 Ω output resistance.
The small signal version of this circuit is
Fig 13. The transceiver gain block in small
signal form.
The previous equations are applied to see that the input resistance of the
first stage is about 57 Ω with a net voltage gain of 16.5. The
gain can be changed by altering the usual feedback elements as well as the
load. The second stage contains no parallel feedback, but is
still gain stable because of emitter degeneration. The voltage
gain is less for this stage. A back termination guarantees a
good output match. Analysis with the circuit of Fig 13 shows
a transducer gain of 23 dB.
The circuit of Fig 12 was analyzed in LT SPICE using 2N3904 transistors biased
at 3 and 10 mA for the two stages. The 10 MHz gain was 23 dB.
Input return loss was 19 dB while the output return loss was 26 dB.
There was only a slight change in either port when the opposite port was
terminated in a 2:1 VSWR. This circuit should offer very
stable terminations for filters that are used in two directions in bidirectional
SSB transceivers.
The design presented above has not been optimized. As such, it
will probably suffer from compromised IMD as well as poor noise figure.
It still illustrates the results available when designing with simplified,
idealized models.
Measured Results
The simple calculations for the amplifier of Fig 12, followed by more detailed
simulations were just too much to ignore. On top of that, Bob
Kopski (K3NHI) beat me to the punch by building and measuring one during
an east coast rainy spell. The rains arrived here today
(June 19th) so I turned the soldering iron on and built one of my own.
The measured results are essentially what the calculations said that they
would be. The first parameter measured was gain and it came in
at 22.5 dB. (All quoted data is at either 10 or 14 MHz.
Sweeps go from 1 to 50 MHz.) Forward and reverse gains are shown
below.
Fig 14. Gain versus F.
Fig 15. Reverse Gain.
The input match is presented in the next figures.
Fig 16. Input impedance match in
rectangular form.
Fig 17. Smith Chart representation
of input impedance match from 1 to 50 MHz. Slight adjustment
would allow the input to be centered about the chart center.
The next curves show the output match.
Fig 18. Output impedance match in
rectangular form. Separate measurements indicated that
the poor wideband match at the upper frequencies resulted from the transformer,
which was 16:5 turns on a FB43-2401 toroid.
Fig 19. Output impedance match in Smith Chart form.
The “hook” corresponds to the dip in the rectangular plot.
The next figure shows the amplifier hanging on the vector network analyzer
(VNA, N2PK type) during gain testing. The circuit was built with
“ugly” methods, although I did use SMA connectors to match the VNA.
Fig 20. Photo of the amplifier board.
The circuit was tested for noise figure with a Noise Com noise diode source.
The result was a pleasant surprise with NF=5 dB. The third order
IMD was measured next for the board with results shown below.
Fig 21. Third order IMD products
are 38 dB below the top of the screen at 0 dBm per tone for OIP3=+19 dBm.
The IMD was well behaved. This amplifier is not strong enough
for use as a post-mixer stage to follow a diode ring in a serious superhet
receiver. A little more work might get it there.