7.4.4.1 The Voltage Gain
Limitation
The development of radar brought with it the need to be able to
amplify very weak signals in the presence of strong ones, and for the first
time made the question of stability and freedom from ringing a prime
consideration in tuned amplifiers. These tuned amplifiers frequently were
required to have voltage amplifications as great as a million overall, with no
change in operating frequency permitted.
The basic criterion which must be satisfied, both for each
individual amplifier stage and for the amplifier as a whole, is that the loop
amplification of individual elements as well as of the assembled groups of
elements must be rigidly limited to assure that stability will not be
impaired. This stability problem is essentially a phasesum problem. If an
input voltage is applied to the amplifier or stage in question, then the
voltage returned through feedback to be summed into the input voltage is the
product of this voltage by the amplification "around the loop" from input back
to input

K_{L} = K_{u} •
K_{f} 
(7.3) 
where K_{u} is the forward voltage amplification to the output, and
K_{f} is the feedback "amplification" from the output back to
the input on an openloop basis.
The modified forward amplification,
K’_{u},
then takes the form:

K’_{u} =
K_{u} /(1  K_{u} K_{f} ) 
(7.4) 
and the phasor term (1 
K_{u} K_{f} ) determines both the variation of the
signal amplitude and the signal phase.
Clearly, one of the requirements of
any amplifier to which Eq. (7.3) applies is that K_{u} K_{F} 
must be small compared to unity, or a potentially unstable situation can
develop. In addition, significant phase shift in the output circuit compared
to the input can occur even with relatively small values of K_{u} K_{F} 
values as small as 0.1 or 0.2, for example. In such a situation, as much as 5
to 10 degree phase discrepancy per stage can be encountered.
Where phase stability is of prime
importance, it is evident that values of K_{u} K_{F} 
should be less than 0.01 if at all possible, as then there is reasonable
chance that the cumulative phase angle discrepancy in a system may be limited
to a fraction of a radian. The design of an amplifier meeting this limitation
can be both difficult and painstaking, and the mechanical realization of the
calculated design can be even more difficult. The design techniques described
in Reference
[35] offer possibly one of the best ways of achieving the required
results.
Early radar experience quickly
showed that the limit on per stage gain Ku
for achieving amplitude and phase
stability with minimum to modest ringing proved to be approximately 10. (It is
possible to get device gains of 100 with common grid or common base circuits,
but the required impedance transformation required to match the input circuit
for the succeeding amplifier typically reduces the overall stage gain back to
approximately 10.) This means that the maximum permitted value for
K_{f} is approximately 0.01 to 0.02, for a power isolation possibly as
much as 40 dB. Where phase stability is of primary importance, the maximum
permitted value for Kf is nearer 0.001 than 0.01.
It is very important to control and restrain the circulation of
carrier frequency currents throughout any multistage amplifier, since if five
stages overall are involved, the isolation from output back to input must be
about 0.01^{5} or 10^{10}. This is the reason that radar IF
amplifiers were designed to receive power in the vicinity of the middle stage,
and RC decoupling was used in both directions for supply voltages, and LC
decoupling for heater currents. All voltage feed points were in addition
individually bypassed, and grounds grouped within the channel in such a way as
to prevent circulation of carrier frequency currents in the channel.
Clearly, there is really nothing magic about the value of
K_{u} of
10. The magic number, if one exists, is in fact the "invariant" K_{u} • K_{f}
whose value must be sufficiently small to limit the phase and amplitude
excursions in the signal. This is the basic stability criterion. But there
definitely is an upper limit on the value of K_{u} , at least in a practical way, since there is a lower
practical limit on how small K_{f} can be made successfully in
production type equipment. The internal stage voltage gain from input to
output on control separation amplifiers can be significantly higher, since the
input admittances for these devices are sufficiently high that the return
feedback gain is severely reduced.
This limitation on voltage gain has
very interesting consequences, particularly in design for reliable operation.
The voltage gain of a bipolar transistor is given by Eq. (7.5).

K_{u} =
kLI_{C}Z_{L} 
(7.5) 
where:
K_{u} 
= 
forward voltage amplification 
I_{C} 
= 
collector current 
Z_{L} 
= 
load impedance 
k 
= 
efficiency factor @
1 
L 
= 
q/kT = 40V^{1} at
25°C 
q 
= 
electron charge 
k 
= 
Boltzmann's constant 
T 
= 
absolute
temperature 
In this equation, it is evident
that I_{C} Z_{L} is the maximum signal voltage for Class A
operation.
It is possible to relate the
voltage I_{C} Z_{L} to the minimum possible supply voltage
V_{CC}, which can be used with the ideal device in question to produce
the required operating characteristics. The minimum supply voltage may then be
defined in terms of the equation

I_{C} Z_{L} = k_{h} (V_{CC} 
V_{SAT}) 
(7.6) 
where k_{h} is a parameter
which relates the output load voltage to the supply voltage and
V_{SAT} is the maximum saturation voltage. k_{h} usually has a
value between 0.2 and 1.0. Substituting Eq. (7.6) in Eq. (7.5) gives the
result:

K_{u} =
k k_{h} L(V_{CC} 
V_{SAT}) 
(7.7) 
This equation may be solved
for the minimum supply voltage V_{CC} for a device in a circuit to
give


V_{CC}  =  K_{u}  (k
k_{h}L)^{1} +
V_{SAT} 
(7.8) 
In Eq. (7.8), the value of
ku is about 10,
typical values of k k_{h} are less than unity, and V_{SAT} is a few
tenths of a volt. As a result, with k
k_{h} = .5, for example, the
minimum value of supply voltage required for a circuit can be expected to be
roughly a twentieth of the voltage gain. This means that the range of required
supply voltage is between 0.5 and 10V, the lower voltage limit applying to the
common emitter configuration, and the higher to the common base
configuration.
The significance of this relation
cannot be overemphasized. The properties of the device and its associated
circuitry are controlled largely by the current level selected for operation,
and there is little point to
selecting a supply voltage for the output circuit which is more than
marginally greater than calculated by Eq. (7.8). Selection of a higher voltage leads either to excessive
power dissipation, excessive gain with its inherent instability, or
combinations of these conditions. In short, the selected supply voltage should
be as small as possible consistent with the demands on the circuits.
This discussion should not be construed to mean that the base
supply voltage provided for base bias current and voltage necessarily can be
as small as that for the collector. Since crude stabilization of circuits is
frequently obtained by controlling the base current in a transistor, the
supply voltage provided for this function must be sufficiently large to assure
that an adequate constancy of current level can be achieved. This and this
alone is the justification for use of a large voltage, yet the current
requirement for these circuits is sufficiently small that a substantial
decrease in power dissipation and a substantial improvement in reliability
could be achieved through the use of separate power sources for these two
functions. In comparison, then, one source of high current and low voltage is
required, and one of higher voltage but substantially smaller current also is
required. Using a common source for both clearly leads to the worst failures
of each! Also, use of two power sources allows a better matching of total
power and current to the demand resulting in a smaller, lighter, and less
expensive solution than with a single power
supply.