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Applied Electronics
A First Course in Electronics,
Electron Tubes, and Associated Circuitry.
By the Members of the Staff of the
Department of Electrical Engineering
Massachusetts Institute of Technology.
12.
USE
OF AN IDEAL OUTPUT TkANSFORllER FOR IMPEDANCE
MATCHING
Advantages of adj us tmen t of the load resistance to suit the tube are
described in t he preceding articles In practice, however, an arbitrary
choice of the load resistance to realize these advantages is usually not
feasibl e because, for example, the load may be a device already available,
or one ,,"hose design involves inherent limitations of resi s tance Hence, in
amplifiers, outpu t transformers are generally used between the tube and
the load. The characteristics of such transformers a re described in the
.
.
volume o n magnetic circuits and tr ansf orm ers. From the considerations
of Art. 11, it is apparent that a val u e of load resistance equal to the plate
resistance is not desirable when maximum power "ith a prescribed
amount of h armonic generation is wanted, and that a transformer ratio
to cause the actual load resistance to have an apparent value in the plate
circuit of about t\vice the plate resistance of the tube is-needed when the
tube is a triode and the quiescent plate voltage is specified.
The circuit diagram for an amplifier "ith an output transformer of
turns ratio a and a resistance load, is sho\vn in Fig. 28a. If the trans­
former is assumed to be ideal, the path of operation on the plate charac­
teristics is as shown in Fig. 28b. B ecause the ideal transformer has no
losses, the windings have no resistance, and the quiescent operating point
Q bas an abscissa EbO that equals Ebb- In an actual transformer, the
quiescent point lies somewhat to the left of this abscissa by the amount of
the direct voltage drop in the primary winding; that is, on a line through
point (Ebb, 0) with a slope of - (l/Rdc), where Rdc is the direct-current
resistance of the winding in series with the plate battery (see Art. 4,
ChI IX, for a somewhat similar condition in the resistance-capacitance­
coupled amplifier) . The path of the operating point on Fig. 28b is along
a load line having a slope
(1/42RL), since the apparent impedance as
viewed from the tube into the transformer is 0,2RL• Thus the resistances
-
used for the determination of the direct-current and alternating-current
conditions on the plate characteristics are different.
If the p at h of operat ion remains in the linear region of the plate charac-
Arl. /2] OUTPUT TRANSFORMERS FOR..lMPIIDANCB
MATCHING 431
teristics, the equivalent circuit for alternating components is that shown
in Fig.. 28c, which may be further simplffiedto that of Fig..2&1because
of the impedance transformation property 'of the transformer. For ma:d-
+
+
••
+
~
jl*I'I'I'~~~-4---"""""'_
[1111111(1
+
Ea:
B""
(a)
R"
0:1
(d)
(c)
Flc.. 28. Linear ClaasAt triode operation with an ideal output transformer ..
mum power output 'With a prescribed amount of harmonic generation
and a prescribed quiescent plate voltage, the considerations of Art. 11
show that the turns ratio of the transformer should be
= ~~"z.
[189]
RL
The output transformer serves a threefold purpose; namely, (a) it
makes possible the realization of the conditions for maximum power
/I
/-32
CLASS A SINGLE-STAGE AJfPUFIERS
lei. VIII
output with almost any tube and load, since its turns ratio can be selected;
(b) it eliminates J2.,RL,the direct-current component of power in the
load, because 1'bis confined to the low-resistance primary winding of the
transformer and does not exist in RL; and (e) it serves as an electrical
isolator between the tube and the load circuit, since with it the potential
of anyone point in the load circuit may be given any desired value without regard to potentials of points in the tube circuit.
13.
PAR.AI.LEL OPERATION;
CLASS Al
One method of obtaining a power output greater than that obtainable
-
-.
2i.
~.
D
--::-I"
t
+
'.
It
NI
~
R,.
)11111
£66
(a)
(b)
FIc. 29. Circuit diagram and equiva1ertt
drcuit for two identical triodes
toDDected in parallel
from a single tube is the use of two tubes connected in parallel. Another
method, which has some advantages over the parallel connection, is
discussed in Art. 14.
. When two identical tubes are connected in parallel, the total plate-
~,.t.
IJ]
13]
PARALLEL OPERATION;
At
PARALLEL
OPERATION; CLASS
CLASS At
433
433
circuit
the interelectrode
voltages is
circuit current
current for particular
particular values
values of the
interelectrode voltages
that shown
double
double that
that of either
either tube.
tube. The
The circuit
circuit diagram
diagram is that
shown in Fig. 29a.
The
voltage
The equivalent
equivalent circuit
circuit for varying
varying components
components of CWTent
current and
and voltage
the linear
the plate
plate characwhen the
the operation
operation is restricted
restricted to
to the
linear region of the
charactube that
that gives the
the
teristics
The equivalent
equivalent tube
teristics is that
that shown in Fig. 29b. The
parallel is one with
with a plate
plate resistresistsame
same performance
performance as the
the two tubes
tubes in parallel
ance
resistance of either
tube and
plate
ance equal
equal to one-half the
the plate
plate resistance
either tube
and a plate
transformer reflects
current
current double
double that
that of either
either tube.
tube. The
The ideal
ideal output
output transformer
R L into
into the
the plate
circuit of the
the load resistance
resistance RL
plate circuit
the tubes
tubes as the
the value
value
the
(N 11/N
/N 2 )2R L •
The
tubes move simultaneously
the
The operating
operating points
points for the
the two tubes
simultaneously along the
the plate
plate characteristics.
same path
path in the
the same direction
direction on the
characteristics. The
The analysis
analysis
of Art. 11 then
tube exactly
to either
then applies
applies to the
the equivalent
equivalent tube
exactly as it
it does to
either
of the
voltage eQ) the
the harmonic
harmonic
the actual
actual tubes.
tubes. For
For a sinusoidal
sinusoidal grid-signal
grid-signal voltage
generation
caused
by
nonlinearity
in
the
tubes
comprises
both
the tubes
both odd and
generation
by nonlinearity
the total
total platep]ateeven harmonics
harmonics of the
the grid-signal-voltage
grid-signal-voltage frequency
frequency in the
circuit
circuit current
current and
and the
the plate
plate voltage.
voltage.
harmonic
The
with a prescribed
prescribed amount
maximum power output
output with
amount of harmonic
The maximum
generation
plate voltage
voltage for the
the two tubes
tubes is
generation and· a prescribed
prescribed quiescent
quiescent plate
the load
resistance that
that corresponds
twice that
that for a single tube,
tube, and
and the
load resistance
corresponds to
this output
the plate
plate resistance
resistance of the
the equivalent
output is one that
that is about
about twice
twice the
equivalent
tube. Thus,
Thus, for maximum
maximum power
power output
output with a prescribed
amount of
tube.
prescribed amount
harmonic generation
generation and a prescribed
prescribed quiescent
quiescent plate
harmonic
plate voltage,
voltage,
[190]
the transformer
transformer ratio
ratio is
and the
[191.]
[191.]
14..
14
.
PUSH-PULL OPERATION;
OPERATION; CLASS
CLASS A
PUSH-PULL
All
method of connecting
connecting two tubes
tubes that
often preferable
A method
that is often
preferable to
to the
the parallel
parallel
connection discussed
discussed in the
the previous
previous article
article when increased
increased power
outtconnection
power ou
desired is shown in Fig. 30. The
The tubes
are connected
connected so that
put is desired
put
tubes are
that the
the
current in one tube
tube decreases
decreases when
other tube
increases.
~late current
when that
that in the
the other
tube increases.
This type
type of connection
connection is commonly
commonly called
called the
connection.
This
the push-pull
push-pull connection.
The
push-pull
connection
has
numerous
advantages
over
the
The push-pull connection has numerous advantages over the parallel
parallel
connection, one of the
the most
most important
important being
elimination of eventhe elimination
connection,
being the
output with a
harmonic generation.
generation. As a result,
result, the
the maximum
maximum p~wer output
harmonic
prescribed amount
amount of harmonic
harmonic generation
generation is greater
greater than
prescribed
than that
that from two
parallel, and
and the
the push-pull
push-pull circuit
circuit is extensively
extensively used not
tubes in parallel,
not only
tubes
operation but
but also for Class B and
and Class C operation.
operation.
for Class A operation
434
CLASS A SINGLE-STAGE
AMPUFIERS
[Ch. VIII
It is assumed in the following analysis of the push-pull amplifier that:
First, the operation is restricted to the negative-grid region of the tube
characteristics and consequently the grid current is considered to be
negligible; second, the transformers are ideal; third, the load is a pure
. +t~
1+
•I
+
e.
•
+
+
+
ell
•
€p
e,
N1
I
II
•
111
11(11
Ebb
Eu
e'11
...
N'l
t't
e'p
+
'(.
FIG.
•
.f
l"
+1
30. Push-pull connection of two triodes.
resistance; and, fourth, the tubes have identical characteristics. Primed
symbols are used to distinguish the quantities in one tube from those in
the other, but the arbitrary assignments of direction and polarity are
made symmetrically with respect to the common-cathode point on the
diagram. The effects of nonlinearity on the alternating-current operation
are neglected at first; later they are taken into account.
14a. Determinationoj QuiescentOperatingPoint. - The determination
of the quiescent operating point Q on the plate characteristics is made in
the same manner as for a single tube in Art. 12, Fig. 28b. Thus when
el
= 0,
e, = e~ = 0,
[192]
[193]
e, = e: = E cc •
[194J
t p = e~ = 0
[195]
and
Also
and
eb = e£ = E bO =
Ebb.
[196J
The quiescent plate currents 106 and l~o are therefore equal in the two
tubes and correspond to the point on the plate characteristics at the voltage co-ordinates given by Eqs, 194 and 196. The total quiescent plate
current through the plate-power supply or battery is thus twice the
current for one tube, and, as far as the quiescent operating conditions
are concerned, the tubes operate in parallel.
PUSH-PULL
Arl. ill
435
OPERATION; CLASS Al
The windingdirections in the output transformer with its center-tapped
primary winding are shown in Fig..32. The dots used to indicate polarity
have the significance that a late of change of flux in the core that makes
one of the dot-marked coil ends instantaneously positive with respect to
the corresponding unmarked end also makes the other two dot-marked
coil ends instantaneously positive with respect to their respective unmarked ends. For the zero-grid-signal condition
[197]
the magnetomotive force in the upper winding of N 1 turns tends to send
flux in a counterclockwise direction in the core, and that in the lower
winding of N 1 turns tends to send flux in the clockwisedirection. The net
-. .. t
p
p
•
•
+
el
---:""
If
No.
II
Nt
•
R,
•
.t
-~p
pi
ip
FIG.
31. Equivalent circuit for identical tubes with the varying components of current and
voltage restricted to the linear region of the tube characteristics.
magnetization of the core resulting from the quiescent components of the
plate currents is therefore zero. This cancellation of magnetization is one
of the principal advantages of the push-pull connection over the singletube or parallel connection. For a given power output and amount of
harmonic generation caused by nonlinearity in the transformer, the
transformer in a push-pull amplifier may be lighter in weight and less
expensive than the one in a parallel-tube amplifierJ because it is not
necessary to provide a large core with an air gap to prevent the magnetic
saturation and resulting waveform distortion caused by the average
component of plate current. The effect of a direct current superposed on
the alternating current in an iron-cored coil is discussed in the volume OP
magnetic circuits and transformers.
14b. Operationin a Linea' Region.- For a smaIl grid-signal voltage
when the path of the operating point is restrictedto thelinear regionof the
tube characteristics,the operation for varying components of current and
436
CLASS A SINGLE-STAGE
AMPLIFIERS
[en.VIII
voltage may be obtained through use of the equivalent circuit for the
tube. The total plate current and voltage are then obtainable through
superposition of the quiescent and varying components. The equivalent
circuit for varying components is that shown in Fig. 31.
If the input transformer is wound in a manner similar to the output
transformer of Fig. 32, with equal numbers of turns on the two halves
of the center-tapped secondary winding, then
e~
= -eo'
[198]
Because of the linearity and symmetry of the circuit, it follows that
i~
= -i
[199]
p'
Thus the total current through the plate-power supply or battery is
ib
+ ib =
=
loo + i p
+ 1bO -
2l so = constant
ip
[200]
[201]
and contains no varying component. For this reason, it is not important that the plate-power supply have low internal impedance when
operation is restricted to the linear region of the curves. Also, if a
self-bias resistor is used (see .Arts. 4 and 11, Ch. IX), no by-pass capacitor
is required to prevent fluctuations
in plate current from affecting the
grid voltage during strictly linear
1.1
operation.
The ideal output transformer and
•
load introduce an apparent resistance
equal to 4 (N 1/N 2)2R L between points
FIG. 32. Winding directions in the outP and p' in the equivalent circuit of
put transformer.
Fig. 31 because of the impedancetransformation property of the transformer. The equivalent circuit including this apparent resistance is
thus that of Fig. 33a, where the center connection shown dotted is
unnecessary, because no varying component of current exists in it. From
this circuit, the apparent load resistance for each tube Is seen to be
2(N 1/N 2)2R L t or half the plate-to-plateresistanceR p p , where
_.
Rl'l'
Nl)2R
= 4 (N
2
L•
~[202]
If one tube were removed from its socket in the circuit of Fig. 30, the
second tube would then have an apparent load resistance of (N 1/N 2 )2RL •
Re-insertion of the first tube would therefore double the effective load
resistance of the second tube. This reaction of one circuit on the other
would not occur if the output transformer were eliminated and a centertapped load resistor were used. Thus it may be concluded that the effect
Arl. 11]
PUSH-PUll
OPERATION,· CLASS Al
437
is associated with the autotransformer effect or coupling between the
two plate circuits by the output transformer.
This analysis indicates that for small grid-signal voltages the load
line for the path of operation of one tube on the plate characteristics in
the linear region passes through the quiescent operating point, as shown
in Fig. 28, but has a slope of
1
However, since the analysis here is restricted to operation in the linear
region of the tube characteristics,
-i,
it must not be inferred that the
maximum power output as well as
the corresponding optimum load
resistance and harmonic genera-~-----------~--~
tion can be obtained from this
load line by the methods of Arts.
8 and 11. The reaction of the second tube through the transformer
(3)
affects the path of operation in the
first, and it is shown subsequently
that over an extended region the
path of operation is nol a straight
line on the plate characteristics of
(b)
either tube, even though RL is
a pure resistance and the output
transformer is ideal.
The circuit of Fig. 33a reduces
to that of Fig. 33b .. This diagram.
shows that as far as varying com(c)
ponents are concemed the two
tubes are in series but, as was pre2i p
viously stated, they are in parallel
as far as quiescent components are
concerned.
(d)
However, the foregoing is not
the only possible point of view" FlO. 33. Equivalent circuits for the lineal
The circuit of Fig. 33c is equivaClass AJ push-pull amplifier.
lent to Figs ..33a and 33b for power
considerations alone; it is not equivalent for the voltage and current at
the load resistor. Also, Fig ..33c is the equivalent circuit including the ap ..
parent resistance offered by the transformer and load in Fig. 33d. Figure
01
/.38
/.38
CLASS
AMPliFIERS
CLASS A SINGLE-STAGE
SINGLE-STAGE AMPliFIERS
[eh. VIII
VIII
33c is iden
iden tical with
with Fig. 29b;
29b; thus
thus it may
may be stated
stated that
that the
the Rush-pull
Rush-pull CODnection
power consideranection is equivalent,
equivalent, as far as
as varW-g
varW-g components
components and
and power
consideraparallel-connected tubes
tions
tubes op'erating
op.erating into
into one-half
one-half
tions are
are concerned,
concerned, to two parallel-connected
the
the center-tapp~d winding
winding of the
the outp.ut
outp.ut transformer.
transformer. Either
Either of the forebe useful,
but they
gQing
gQing alternative
alternative concepts
concepts may' be
useful, but
they are
are app'licable
app'licable only_
to operation
the tube's
tube's characteristic
characteristic curves.
curves.
operation over
over the
the linear
linear re~on of the
Range Extending
Extending beyond the
Linear Region.Region.14c. Operation over
over a Range
the Linear
When
path of operation
beyond the
plate
When the
the path
operation extends
extends beyond
the linear
linear region
region of the
the plate
characteristics.,
plate current
the plate
current that
that are
are not
not
characteristics., harmonics
harmonics are
are generated
generated in the
present in the
the grid-signal
grid-signal voltage
voltage
present
the grid-signal
grid-signal voltage.
voltage. For
For example,
example, if the
is sinusoidal
and expressible
expressible as
sinusoidal and
eg
= ViE,
ViE, cos wt,
wt,
[203]
[203]
the
plate current
under conditions
the plate
current under
conditions of no harmonic
harmonic generation
generation is
it;
it; = IIbO
ViI pl' cos wt;
wt;
bO + ViI
[204J
[204J
but v;ith
but
with harmonic
harmonic generation
generation it
it is expressible
expressible as the
the Fourier
Fourier series,
IIbo
[po + Vi(I
Vi(I 1'1
COSwt+ Ip 22cos2wt
.. '),
cos2wt + Ip
Ip3cos3wt
), [205]
p1 cOSwt
3 cos 3wt + ...
bo + [po
as is demonstrated
push-pull
demonstrated in Art.
Art. 8. Because
Because of the
the symmetry
symmetry in the
the push-pull
the second
second tube
tube is similar
similar to Eq.
Eq, 205,
205, but
circuit, the
the plate
current in the
circuit,
plate current
but
wt replaced
replaced by
6)t + 180°. Thus
Thus
'with wt
by 6)t
ib =
ib
bO + I11'0
pO + V2[I
i~ = I bO
V2[I pipi cos (wI
(wI + 180°)
+ I11'2
p2 cos
(2wl
(2wl + 360°)
(3wt + 540°) + ..·.]
+ I p3 cos (3wt
· .]
1p o + Vi[
Vi[ -11'1
-11'1 cos wt
wt + 11'2
2wt - I p3 cos 3wt
3wt
= IIbo
Ip2 cos 2wt
bo + 1po
+ ... J.J.
[206]
[207J
[207J
feature assumed
assumed in an
an ideal
ideal transformer
transformer is that
that the
the exciting
exciting current
current
One feature
equivalent to
to the
the statement
statement that
that the
the magnetomomagnetomonegligible, which
which is equivalent
is negligible,
tive force required
required to
to magnetize
magnetize the
the core is zero. Consequently,
Consequently, the
the sum
sum
tive
the magnetomotive
magnetomotive forces caused
caused by
currents in the
the windings
windings is zero
of the
by currents
given direction
direction around
around the
the core;
core; and
and in a two-winding
two-winding ideal
ideal transtransin a given
former the
the current
current ratio
ratio is the
the inverse
inverse of the
the turns
turns ratio.
ratio. In
In the
the ideal
ideal
former
push-pull
output transformer,
transformer, the
the sum
sum of the
the magnetomotive
magnetomotive forces in a
push-pull output
the three
three windings
windings
given direction
direction around
around the
the core caused
caused by
currents in the
by currents
given
the two-winding
two-winding transformer.
transformer. Thus,
Thus, for the
the transformer
transformer
just as in the
is zero, just
Fig. 32,
in Fig.
[208J
[208J
or
[209]
PUSH-PULL OPERATION,'
Al
PUSH-PULL
OPERATION,' CLASS
CLASS Al
Art. Jj]
Jj}
439
439
A rigorous
requires a consideration
rigorous analysis
analysis of the
the push-pull
push-pull amplifier
amplifier requires
consideration 1.
of the
the leakage
reactances
the finite
finite magnetizing
magnetizing impedance
impedance and
and also of the
leakage reactances
among
transformer, but
but their
their effects
among the
the three
three windings
windings of the
the output
output transformer,
are
are neglected
neglected here.
Substitution
the current
the
Eqs. 205 and 207 in Eq. 209 gives the
current in the
Substitution of Eqs.
load as
load
i2
i2
= 2 ~~ ¢2(I
V'2(I pp11 cos wt
wt
+ I pp33 cos 3wt
3wt + ···-).
· .).
[210]
[21OJ
Thus
Thus the
the effect of the
the symmetrical
symmetrical arrangement
arrangement is to cause a cancellation
cancellation
harmonics, and
the average
average components,
components, the
the second harmonics,
and all other
other even
even
of the
harmonics
However, if the
the grid-signalvoltage
grid-signal voltage
harmonics generated
generated within
within the
the tube.
tube. However,
is nonsinusoidal, all frequencies
harmonics,
frequencies present
present in it, inclucling
including even
even harmonics,
are
the output
are amplified
amplified as usual.
usual. The
The absence
absence from the
output signal of components
components
resulting
the advantages
advantages of the
resulting from even-harmonic
even-harmonic generation
generation is one of the
push-pull
tubes.
push-pull connection
connection over
over the
the parallel
parallel connection
connection of tubes.
The
battery is the
the sum
supply or battery
sum of
The current
current through
through the
the plate-power
plate-power supply
the
From Eqs.
Eqs. 205 and
this is
and 207, this
the currents
currents through
through the
the two tubes.
tubes. From
Current
Current through
through plate-power
plate-power
i b + i~
supply = ib
supply
= 2160
2l£Jt + 211'4
211'4 cos 4wt
· .). [211]
21b{)+ 211'0
211'0+ <0.(21pp22 cos 2QJt
4wt + ···-).
The
by
The average
average value
value of this
this plate-power-supply
plate-power-supply current
current as indicated
indicated by
direct-current ammeter
ammeter increases
increases by
amount 21
a direct-current
by the
the amount
21pO when
when the
the gridgridsignal
to E g • A change
the ammeter
signal voltage
voltage is increased
increased from zero to
change in the
ammeter
indication
voltage is applied
therefore indicates
indication when
when the
the grid-signal
grid-signal voltage
applied therefore
indicates
\va
the generation
harmonics. Whereas
Whereas
wa veform distortion
distortion caused
caused by
by the
generation of harmonics.
only the
the odd
odd harmonic-generation
harmonic-generation components
components exist
the output
exist in the
output current
current
only
and
the plate-power
plate-power supply
and voltage.
voltage. the
the current
current through
through the
supply contains
contains only
only
the
the even
even harmonic-generation
harmonic-generation components.
components.
voltages, plate
plate currents,
Waveforms of the
the grid-signal
grid-signal voltages,
currents, and
and output
output
Waveforms
current,
when the
the path
path of the
the operating
point
current, illustrating
illustrating the
the operation
operation when
operating point
extends
the tube
tube characteristics,
region of the
characteristics, are shown
extends into
into the
the nonlinear
nonlinear region
in Fig. 34. The
The grid-signal
grid-signal voltages
voltages e g and
and e~ are
are sinusoidal
sinusoidal and
and 180 degrees
out
wavefonns i" and
and i~ are flattened
flattened at
at the
out of phase.
phase. The
The plate-current
plate-current waveforms
bottom
the tube
tube characteristics,
but each
bottom because
because of nonlinearity
nonlinearity of the
characteristics, but
each is a
replica
i 2 also
replica of the
the other
other displaced
displaced by
by 180 degrees. The
The output
output current
current i2
flattened near
near its crests,
the diagram
that
crests, and,
and, since the
diagram shows that
is flattened
[212)
[212J
u,
Push-Pull t\.mplliiers""
Amplifiers..'·
u, A. P-T. Sah, H Quasi Transients
Transients in Class B Audio-Frequency Push-Pull
I.RA. Proe
.., 24 (1936), 1522-154L
1.RJ!..
Proc•.,
140
CLASS A SINGLE-STAGE
AMPLIFIERS
[Cn.VIII
i 2 contains only odd harmonlcs.P as was deduced analytically in Eq. 210.
A graphical analysis for the path of operation on the plate characteristics over an extended range is not readily made for an individual
tube, because of the coupling
c
between the plate circuits of
21'l"
the two tubes through the out"
: wt
put transformer. However, as
I V
"
I
1/
'\. I
is shown subsequently, the op---1------,/1
-------" c:
/ I
eration of the circuit can be
',?,-_....... ,/ II
represented graphically by conI
t
struction of the plate characI
I
,......
teristics for a composite tube,
I
;'
"
I /
the composite tube being de1//
fined as one which, operating
into one-half the output transformer primary winding with
wt
the other half open-circuited,
gives the same current and
power in the load as the two
tubes in push-pull. The path of
operation on these composite
o
wl
characteristicsis a straight line,
and the methods of finding the
power output and harmonic
generation given in Art. 8 are
FtG. 34. Waveforms in a push-pull amplifier
when the operation extends into the nonlinear
applicable. It is assumed again
region of the tube characteristics.
in this analysis that the output
transformer is ideal, thus having no resistance, leakage reactance, exciting current, or losses, and that
the load is a pure resistance. The circuit diagram is again that of Fig.
30, and Eq. 209 applies to the
Compositetube
plate circuit.
Equation 209 shows that the
outpu t current i 2 is the same as
the output current that would
exist if an equivalent current
I
,
('(I
FIG.
35. Composite tube and circuit equivalent
existed in one-half the transto that of Fig. 30.
former primary winding. Thus
the operation of the circuit in Fig. 30 is the same as that in Fig. 35, where
Franklin, Differential Equations [or Electrical Engineers (New York: John Wiley &
Sons, 1933) 65.
15 P.
I
,4,:.141
A,:. 141
PUSH-PULL OPERATION,'
At
PUSH-PULL
OPERATION,' CLASS
CLASS At
141
141
the
the relationships
relationships among
the composite
composite tube
tube has
has a plate
plate current
current id and
and the
among
id,
id' eb,
eb, and
and et:
et:are yet
yet to be found.
found.
In
becomes
In functional
functional notation,
notation, Eq.
Eq. 213 becomes
id(e
id(ecJ
eb) = ib(e
ib(ecc ,, eh)
eh) - i'(e~, e'),
e'),
c J eb)
[214]
[214]
where
the equation
represent the
the
where the
the two terms
terms on the
the right-hand
right-hand side of the
equation represent
characteristics
tubes. These
characteristics of the
the individual
individual tubes.
These can
can be combined
combined in
in
accordance
the input
transaccordance l\ith
with the
the circuit
circuit restrictions
restrictions as follows: Since the
input transformer is ideal,
[215]
thus
thus
[216J
[216J
Also, since the
transformer acts
the ideal output
output transformer
acts as an
an autotransformer
autotransformer
plate circuits,
makes
bet\veen
between the
the t\VO
t\VOplate
circuits, it
it makes
~[217J
\Vith
the transformer
transformer and
With t\\"o
two separate
separate load resistors
resistors substituted
substituted for the
and load,
load,
Eq
jollO'Ws is
Eq ... 217 would
would not
not be correct,
correct, and
and the
the entire
entire analysis
analysis that jol1011JS
is thereEquation 199, which
which was
Vlas obonly when
when tlte
the transformer
transformer is
is used. Equation
fore true only
tained
not apply
to the
the nonlinear
nonlinear operation
the linear
linear analysis,
analysis, does not
apply to
operation
tained in the
the tube.
tube.
of the
the resistance
resistance of the
the transformer
transformer is negligible,
Since the
negligible,
[218]
E
[218]
Eoo
Ebb;
bO = E~ = Ebb;
thus
thus
e~ = Ebb
Ebb + e~ =
= Ebb
Ebb - ep
and
and
[219J
[219J
Ebb + ep ,
= Ebb
[220]
2E bb - eoeo.
e' = 2Eob
[221]
eb
eb
,vhence
whence
tubes are
are assumed
assumed to be identical,
identical, the
function ib
i b is the
same
Since the tubes
the function
the same
but they
they are
are functions
functions of different
different variables;
the function
function i~, but
variables; thus
thus
form as the
the variables
are indicated,
indicated, and
and substitution
substitution
the prime
prime may
may be dropped
dropped if the
the
variables are
Eqs. 216 and
and 221 in Eq.
Eq. 214 gives
of Eqs.
id(E cc +.e o, eb)
eb) = ib(Ecc
ib(E cc
id(E
eo, eb)
eb) + eo,
ib(Et:c-- e(J,
eg , 2E
eb).
ib(Et:c
2Ebo
bo - eo).
[222]
In this
this way,
way J once the
the power-supply
power-supply voltages
and Ect:
are selected,
selected, the
In
voltages Ebb
Ebb and
Ect: are
the
and eo,
eo, and
and the
independent variables
variables are
are reduced
reduced to
to two,
two, namely,
nam~ly, ell and
the
independent
characteristics may
may be
be graphically
graphically constructed
constructed as shown
shown in Fig.
Fig. 36, where
where
characteristics
e(Jare
are drawn.
drawn. The
The plate-current
the curves
curves for zero grid-signal
grid-signal voltage
plate-current
the
voltage e(J
obtained through
characteristic curve
curve for the
the composite
composite tube
characteristic
tube is obtained
through rotating
rotating
the plate
plate characteristic
characteristic for an
an individual
individual tube
degrees about
about
the
tube through
through 180 degrees
axis of abscissas
abscissas until
origin, then
then displacing
displacing the
the curve
curve along
along the
the axis
until its
the origin,
142
CLASS A SINGLE-STAGE
AMPLIFIERS
[eh. VII]
new origin falls at the point at which eb is equal to 2Ebb on the original
scale, and, finally, subtracting the magnitudes of the ordinates of the two
curves. Thus the length of the ordinate Xy equals .xz minus xv in Fig. 36.
Several features of the composite characteristics are at once apparent;
namely, (a) the quiescent plate current in the composite tube is zero;
i
Compositecharacteristic for ell-O;
,__
-i,,<4c,e
,J- ib(~)e,)-i.(Eu, 2E60- e,)
Quiescentoperating point for compositetube
CJt----....::..-----;~---:::.;1111""--.,.----
}I
I 1
I
I I
I
I
I
V
I
I
,
----------;
,---I-.,~2Ebb -C b .. e~~
J
.....-----r--I--Zl:;LL
I
"_____.
FIG.
36.
.,
t
/J'IJ
Plate ch~cteristic of second
individualtube for ell'" 0;
-i. (£CCI2E llO-e. )
I
Construction of a plate characteristic curve for the composite tube with zero
grid-signal voltage.
(b) the composite plate characteristic is much straighter than either of
the individual tube characteristics, although it may have more curvature
than that shown in Fig. 36; (c) the plate resistance of the composite tube
aeb/aid is one-half the plate resistance of either of the individual tubes
at the quiescent operating point; and (d) the plate resistance for the
composite tube is essentially constant over the range shown, though the
plate resistances of the individual tubes vary considerably.
The construction of the plate characteristics of the composite tube
corresponding to three particular values of grid-signal voltage eo is shown
in Fig. 37. One particular value of the grid-signal voltage is zero, another
is positive, and the third is equal in magnitude to the positive one but is
negative. The curves for zero grid-signal voltage lie between the others in
the figure and are similar to those in Fig. 36. To obtain the composite
characteristic curve for the particular positive value of grid-signal voltage, the curve for an equal negative grid-signal voltage, denoted by
i b(E cc - eo, eb) on the diagram, is rotated and displaced as previously
explained) whereupon it becomes the curve denoted by
-ib(E
cc -
ern
2E bb
-
eb).
Art. II]
PUSH-PULL
OPERATION; CLASS Al
443
The ordinates of this latter curve are then added algebraically to those 'of
the curve for the particular positive value of grid-signal voltage, denoted
by ib(Ecc + ea) eb), giving the left-hand dotted line with positive slope.
Again, on this line, the length of the ordinate xvis subtracted from
that of the ordinate Xi to give the length xy, and the resulting curve
is the characteristic of the composite tube for the particular positive
value of ego The construction of the right-hand dotted line, which is
the characteristic of the composite tube for the particular negative
i
7
I
I
I
/
Composite characteristic for eg=O;i<.t(Ecc.e
)
b
Z
/
IJ
AU
I
~mposite cha~teristic
for s~ific negative value of e.g:
tet(Ecc-~,e~)-lb(E"- eg .eh) - 1.b(Ecc+ e". 2E M- eb )
~
+--_IL...-----ol:;~--c.._...+---:II""'.....J.-...,....-.,.......~.......,.-......;.,
I
I
I
eb
Quiescent Ioperating point
I
I
I
&1
I
-~~r..r--~~--
2 Ebh-e.--
I
......
Ii
I
-i b (Ecc - eg . 2E/J6
- e. >
FIG.
37.
Construction of the plate characteristics of the composite tube for three specific
values of grid-signal voltage.
value of eg ) is done in a similar manner. Note that composite characteristics for equal positive and negative grid-signal-voltage increments are
images of each other about the quiescent operating point. This symmetry
is the reason for several important operating features of the push-pull
circuit with an output transformer.
Figure 38 shows a family of composite characteristics for two Type 45
tubes in push-pull with a plate-supply voltage of 240 volts. The characteristics therefore spread over a range of 480 volts. The grid-bias voltage
CLASS A SINGLE-STAGE
AMPLIFIERS
[eh. VIII
is -SO volts, and the heavy dotted lines with positive slopes are the two
individual tube characteristics for zero grid signal voltage. When the
ordinates of these curves are added algebraically, the result is the heavy
solid line with positive slope, which is the composite characteristic for
.100
ec .. 0 I
J
90
I
J
I
8Q
I
I
70
Type 45 tubes
Ebbte: 240 volts
Ere = - W volts
I
/
I
I
CIS
E
60
c 50
,-
.......
40
30
IblJ
20
__
10
0 1'--+
...............
...,.,:;.+-t1~lP4-~~~........~~~~--+--h...,.--+-~
-Path of
operation for
one tube
0
ell
in volts
10
Path of
operation for
composite tube;
1
slope> --
(~)'Z RL
50
~V'l
60
70
80
E
c:::
90 :-.
1...--------------..:..-------'-100
---
'-
Composite characteristics
of individual tubes
----.Cha:raeteristi~
FIG. 38.
Composite characteristics for two Type 45 tubes in a push-pull circuit at specific
values of grid-bias voltage and quiescent plate voltage.
zero grid-signal voltages. The light solid lines with positive slopes are the
composite characteristics for lO-volt increments of grid voltage constructed by the method shown in Fig. 37. The solid and dotted lines with
negative slope are discussed subsequently.
Not only the characteristic corresponding to the zero grid-signal-volt-
PUSH-PULL
Ari.N]
145
OPERATION,' CLASS Al
age condition but all the composite characteristics over a. wide range
of grid voltage are essentially straight lines. The grid-bias voltage Etc,
chosen in Fig. 38 as -50 volts, is one for Class A operation. In later discussions of Class AB and ClassB push-pull operation, it is shown that the
composite characteristics are not always straight for those operating
conditions. Note that the characteristics of the composite tube as defined
here are dependent upon the values of E ce and Ebb and thus depend on
quantities external to the tube. The composite tube differs from an ordinary tube in this respect.
The composite plate characteristics from Fig. 38 are reproduced in
Fig. 39. As far as the current, voltage, and power in the load resistor are
2Ebh
o
FIG.
39.
Plate characteristics of the composite tube with load line superposed.
concerned, the operation of the push-pull circuit is equivalent to that of
Fig. 35, where the characteristics of the composite tube are given by
Fig. 39. The quiescent operating point is on the abscissa axis at eb equals
Ebb. where id equals zero, and a load line having a slope -
(Nl)2
-
N2
RL
gives the path of operation on the characteristics, as is shown by the
solid line of negative slope in Figs. 38 and 39. The waveform distortion
that occurs because of harmonic generation in the tubes may be obtained
from this load line by the methods of Art. 8, if i b is replaced by id, and
negative values of id are recognized. Because of the symmetry of the
composite characteristics mentioned previously, no steady or evenharmonic components appear in 1,4or in the load current.
446
CLASS A SINGLE-STAGE
AMPUFIERS
SINGLE-STAGE AMPUFIERS
[Ci. VIII
VIII
Although the plate
plate currents
currents in the individual
individual tubes
tubes may
may decrease to
balf-cycle, such
zero and remain there for an appreciable
appreciable fraction
fraction of a half-cycle,
plate characteristics;
behavior is not
behavior
not apparent
apparent on the
the composite plate
characteristics; since the
operation
operation along the load line is entirely
entirely symmetrical
symmetrical about
about the
the quiescent
operating
point for positive or negative
operating point
negative values of grid-signal voltage.
paths of operation
However, the paths
operation for the
the individual
individual tubes
tubes can be found
by a process which is the reverse of the
by which the
readily by
the one by
the composite
characteristics
vertical line through
through
characteristics are constructed.
constructed. Thus, in Fig. 38, a vertical
path of operation
particular composite
the intersection at A of the path
operation and a particular
characteristic
characteristic intersects
intersects at
at B and C the
the two individual
individual tube
tube charactercharacteristics from which the composite characteristic
characteristic is constructed,
constructed, thereby
thereby
disclosing
plate currents
particular value of
disclosing the individual
individual tube
tube plate
currents for a particular
grid-signal voltage. By this method, the paths
paths of operation
operation for the individual tubes shown by the dotted
dotted curves with negative
negative slopes are constructed.
structed. They
They are curved, even though the load is purely resistive and
the output
output transformer
transformer is ideal.
For the particular
particular conditions illustrated
illustrated in Fig. 38, the individual
individual tube
plate currents
currents do not fall to zero when the range of operation
operation is limited
limited
.... o curves corresponding to zero grid voltage on the two tubes;
by the htv....
tubes;
thus the operation
operation is Class A
Ai.1 . However, if the grid-bias
grid-bias voltage is chosen
somewhat larger, the individual
individual tube currents
currents may be zero for an appreciable fraction of the cycle, and operation
operation changes to Class ABI, which is
discussed
discussed in more detail in Ch. X. Figure 40 shows a limiting
limiting example for
plate currents
just reach
individual plate
currents in it
it just
Class Al operation, since the indi'\idual
zero when the grid voltage of the opposite tube reaches zero.
zero
The considerations that
that govern the maximum power output
output '\\ith
with a
push-pull amplifier are
prescribed amount
amount of harmonic generation
generation from a push-pull
11.
quite different from those for a single-tube amplifier given in Art. 11.
Since the even harmonics generated
generated in the tubes are canceled in the outpu
outpu t
transformer,
harmonics in the amplifier is smaller
transformer, the total
total generation
generation of harmonics
in the push-pull amplifier than
than in the single-tube amplifier when the tubes
have the same operating
operating voltages and deliver the same power output
output
individually. Consequently it follows
that, for the same total
total harmonic
follows that,
generation
generation in the amplifier, the maximum power output
output from each tube
tube
The
is larger in the push-pull amplifier than
than in the
the single-tube amplifier. The
increase may be as much as 50 per cent. This
This increased power output
output is
both in the
that changes both
the operating
operating voltages
made possible by the fact that
plate circuits
effective in the plate
circuits of the individual
individual
and in the load resistance effective
tubes may
may be made under
under the specified
specified conditions. Since the
the even barpath of operation
monies are canceled, the
the path
operation may
may be extended
extended farther
farther into
into
tube characteristics
characteristics in a push-pull
push-pull amplifier than
than
the lower region of the tube
is indicated
indicated in Fig. 26, Art. 11,
11, for a single-tube
single-tube amplifier when a prescribed amount
amount of harmonic generation
generation is not to be exceeded. Accordingly,
PUSH-PULL
Art.N]
147
OPERATIONj CLASS Al
for the same amount of harmonic generation, a larger magnitude of gridbias voltage and a larger grid-signal voltage amplitude may be used in
lOO,------:-':"r--------------,
Type 45 tubes
90
Ew;'" 250volts
Eee'" 55 volts
80
70
~
6Q
.5 50
'-...
40
30
ItIJ 20
10
0
Path of
operation for
composite tube
when
50
N1 t
RL (N ) "" 1020ohms,
60
~
or R pp"= 4080ohms
70
80
E
.S
90 "":
....
'-------------,.;......L..:.:..-----..LI00
FIG. 40.
Composite characteristics for two Type 45 tubes in a. push-pull circuit for the
limiting condition of Class At operation,"
the push-pull amplifier than in the single-tube amplifier, and the power
output from each tube is therefore larger.
The change of voltages described in the preceding paragraph is one
factor that contributes to the increased value of maximum power output
in the push-pull amplifier. Another factor is the change that may be made
in the effective load resistance for each tube. Since the composite charac• This diagram is adapted from B.]. Thompson, 11 Graphical Determination of Performance
of Push-Pull Audio Amplifiers," I.R.E. Proc., 21 (1933), rig.8, p. 595, with permission.
#8
#8
CLASS
AMPliFIERS
CLASS A SINGLE-STA.GE
SINGLE-STAGE AMPliFIERS
[CA.
[CA.VIII
VIII
teristics are symmetrical
symmetrical about
about the
the curve for zero grid-signal voltage, and
parallel for Class At operation,
are practically
operation, the
the amount
amount
practically straight
straight and
and parallel
of harmonic
plate-ta-plate
independent of the
the plate-to-plate
harmonic generation
generation is essentially independent
load resistance
particular value of grid-signal-voltage
resistance for a particular
grid-signal-voltage amplitude.
amplitude.
Consequently,
Consequently, the
the considerations of Mt.
Mt. 11
11 and the
the result
result that
that the effeceffective load resistance
resistance for a tube
tube must
must be approximately
approximately equal to twice the
th~ tube
plate resistance
maximum power output
output with a prescribed
plate
resistance of the
tube for maximum
amount
prescribed quiescent plate
plate voltage
amount of harmonic
harmonic generation
generation and a prescribed
do not
Instead, the
the considerations
considerations of Art. 10
10
not apply
apply to the composite tube. Instead,
apply, and the slope of the path
path of operation
operation on the composite characteristics for maximum
maximum power output
output is equal to the
the negative
negative of the
the slope
of the
path of operation
the composite characteristics.
characteristics. The slope of the
the path
operation on the
the
composite characteristics
)2RL , which is one(Nt/N 22)2R
characteristics corresponds to (Nt!N
fourth the plate-to-plate
plate-ta-plate resistance given by 4 (Nt/N
)2R
(Nt/N 2 )2
R L • Thus
Thus the
plate resistance
plate-ta-plate resistance
plate-to-plate
resistance should equal four times the
the plate
resistance of
the composite tube. The plate
plate resistance of the
the composite tube is, however, approximately
plate resistance
resistance of the individual
individual tubes
approximately one-half the plate
at
point. The optimum
plate-to-plate resistance
at the
the quiescent operating
operating point.
optimum plate-to-plate
for the push-pull
push-pull amplifier therefore is twice the plate
plate resistance of the
individual tubes,
tubes, and the optimum
optimum value of the
the load resistance effective in
the plate
plate circuit of each tube
plate resistance of the
tube hence is equal to the plate
Eq, 177,
177, this condition
condition results in an increase of
In accordance with Eq.
tube. In
power output
output from each tube
tube over the value obtained when an effective
load resistance equal to twice the plate
plate resistance of the tube is used.
triodes used '\\ith
with an output
output
The discussion in this article applies to triodes
The
transformer for delivering power to a resistance
resistance load. The use of the
transformer
push-pull connection
connection as a balanced voltage amplifier '\\ithout
without an output
output
IX and Class AB, Class B, and Class C
transformer is discussed in Ch. IX
transformer
operation is discussed in Ch. X.
operation
15.
15.
SYMBOLS FOR VACUUM-TUBE
VACUUM-TUBE CIRCUIT
CIRCUIT ANALYSIS
ANALYSIS
SYMBOLS
In the preceding
preceding articles of this chapter
chapter a number
number of special symbols
SYmbols
introduced and defined. A large number
number of symbols
SYmbolsare
are introduced
are needed in
vacuum-tube circuits, because the
the operation
operation is complicated
analysis of vacuum-tube
by the superposition
superposition of direct
direct quantities
quantities and alternating
alternating quantities
quantities having
having
if a consistent
consistent
harmonic components. Confusion is likely to resul
resultt if
several harmonic
SYmbolsis
not defined and adhered
adhered to through
through aU
all the analysis.
set of symbols
is not
the volume on electric circuits
circuits
The methods
methods of circuit analysis given in the
The
directly applicable
applicable to vacuum-tube
vacuum-tube circuits
circuits and may be used·
used with
are directly
arbitrarily assigned positive directions of currents
currents and
and voltages..
voltages, The
The
arbitrarily
set of definitions adopted
adopted in this volume is merely one of the
consistent set
innumerable possible sets. It
It is adopted
adopted because it
it is in substantial
substantial agreeinnumerable
Art.15l
Art, 151
SYMBOLS
SYMBOLS FOR VACUUM-TUBE
VACUUM-TUBE
CIRCUIT
CIRCUIT ANALYSIS
ANALYSIS
#9
#9
ment
ment with the
the latest
latest standards
standards'"16 available;
available; thus
thus it
it is the set most
most likely
by the
publications in the
to be encountered
encountered by
the reader
reader in other
other publications
the future.
future. To
eliminate
eliminate one additional
additional source of confusion,
confusion, the
the definitions
definitions here
here are
positive direction of the
the quantities.
quantities. If
If
extended
extended to include
include the
the assigned
assigned positive
symbols
symbols defined for the
the directions
directions opposite
opposite to those chosen here are
needed, other
needed,
other symbols
symbols can be used.
Table I summarizes
summarizes the
the symbols
symbols for the
the electrical
electrical parameters
the
Table
parameters of the
tube and
and circuit
circuit and
and for some of the
the currents
currents and
and voltages
voltages that
that do not
not
tube
enter
problems. Table
enter into
into most
most of the
the problems.
Table II
II gives the
the definitions
definitions and
and symthe current
current and
and voltage
voltage components
components that
that are fundamental
fundamental to the
the
bols of the
operation
II, the
the first four rows contain
contain symoperation of triode
triode circuits.
circuits. In
In Table
Table II,
bols pertaining
pertaining to the
the total
total quantities,
quantities, and
and the
the fifth, sixth,
sixth, and
and seventh
seventh
pertain to the
rows, symbols
symbols which pertain
the varying
varying components
components and
and are useful
in
in circuit
circuit analysis
analysis when harmonic
harmonic generation
generation is neglected.
neglected. The
The last
last four
rows contain
useful in representing
contain symbols
symbols useful
representing nonsinusoidal
nonsinusoidal varying
varying comFourier series,
series. Complex
Complex quantities
quantities are indicated
indicated by
by roman
roman
ponents as a Fourier
ponents
type.
type.
TABLE
TABLE I
SYMBOLS
SYMBOLS FOR
VACUU~I\"ACUU~I-TRIODE
TRIODE
CIRCUITS
CIRCUITS
gp
plate conductance
gp = plate
conductance
"1'
"1'
=
plate resistance
plate
resistance
gil
gil = grid conductance
conductance
''/1
/1 =
grid resistance
resistance
grn
grn = grid-plate
grid-plate transconductance
transconductance (mutual
(mutual conductance)
conductance)
J.I.
JJ. =
Ecc
E
cc
I
.
f actor
deb
deb
amp l 1'fi catlOn
cation
actor = - -d
-d '.
e~
,~con~tanl
I~ con~tanl
Cg p = grid-plate
grid-plate capacitance
capacitance
C"p
Cg k = grid·cathode
grid-cathode capacitance
capacitance
C"k
Cpk
Cpk = plate-cathode
plate-cathode capacitance
capacitance
Ebb = plate-supply
plate-supply voltage
plate
Ebb
voltage rise from the cathode
cathode toward
toward the plate
or Ecd
Ecd = control-grid
control-grid supply
supply voltage
voltage rise from the
the cathode
cathode toward
toward the grid
Ecc2 = screen-grid supply
E~c2
supply voltage
voltage rise from the cathode
cathode toward
toward the screen grid
Elf
Elf = filament
filament or heater
heater supply
supply voltage
voltage (effective or direct
direct value)
value)
E, = filament or heater
E,
heater terminal
terminal voltage
voltage (effective or direct
direct value)
value)
It
It = filament or heater
heater current
current (effective or direct
direct value)
I,
electron~emission (saturation)
I, = total
total electron-emission
(saturation) current
current from the cathode
cathode
Tubes
Tubes with
with more t1"\an
than one grid
grid require
require additional
additional symbols
symbols which are
16
supplied as followS
follows'P::
supplied
"Generalized
"Generalized System
System for
for 1.l11tltigrid
1.l11tltigridTubes.
Tubes. The
The following scheme of
symbols
symbols for multigrid
multigrid tubes
tubes avoids
avoids the
the extension
extension of letter
letter subscripts
subscripts and
provides a framework
provides
framework of symbols
symbols for tubes
tubes with
with any
any number
number of grids. In
this system
system the
the grids
grids are
are numbered
numbered according
according to position,
the grid
grid immedi·
immedithis
position, the
ately adjacent
adjacent to the
the cathode
cathode or filament
filament being
being No.1,
No.1, the
the next
next grid
grid No. 2t
2t
ately
III
III Stand4rds
on Elutronics
Elutronics (New York:
York: The Institute
Institute of Radio Engineers,
Engineers) 1938),
1938)t 11-14.
TABLE II.
CompOTUml
Vollagerisefrom
cathodeto grid
Instantaneous total value
Quiescent value; steady
value when varying comnonentof grid voltaze iszero
Average value of the total
ouantitv
Instantaneous maximum of
the total Quantity
Instantaneous
value of the
. component
v
Effective value of the varyinR'comoonen t
Amplitude of the varying
component
Average value of the varyina component
Instantaneous value of the
harmonic components
Effective value of the harmonic components
Amplitude of the harmonic
components
SYMBOLS FOR TRIODE
Vollagerisefrom
cathode to plou
tb
J(
tb
eL
E co
EbO
leo
160
ELO
Ee
Eb
It:
lit
EL
Ecm
E 6ff1
lem
Ibm
ELm
eo
e1'
...
,
11'
E,
E1'
If!
11'
E.
E(1ffI
Ep m
1 (1m
[pm
s:
E po
100
11'0
E. o
eQ2 t
E til1 E gl2
E g 1m )
Vollagedrop (JQ'O$$
Currentthroughthe Currentthroughthe
theloadin the
externalcircuit
externalcircuit directionof pomive
Imvardthe grid
towardthe plate
plate current
ec
E,o
Cob
CiRCUITS
N arneand Directionof QuantiJy
Ef/2
,
• , •
...
ff1 1 •••
Cpt,
ep 2 1 •
E PIP1 E
, •
.
1 112, •••
1011 lv'll'
2 I ...
E p 1m , E p 2m l
.
J"h
•••
I glm, I ,,2m,
••
•••
t 1'l 1 t p 2 1
e•
e.1I es2, ...
•••
I ph 11'21...
I plm, I p2:m,
•••
Est, E. 2 ,
Ed
m,
Ed
•••
m , •••
en. VIII]
PROBLEMS
451
etc. In designating the voltages or currents associated with a particular
grid, the symbols given on the preceding pages will be used with the grid
number as a subscript .... Control-grid symbols are frequently used
where reference is not made to other grids. The number of the grid need
not be used in this case. It will be understood that, when no number
appears in the subscript, the reference is to the control grid."
It should be noted that one possible source of confusion lies in the fact
that some of the symbols in the last three rows of Table II are also used
with a different meaning for multigrid tubes. However, this does not
lead to difficulty in any of the problems treated in this text.
PROBLE~IS
1. A triode having the plate characteristics of Fig. 7J Ch. IV, is used with a platesupply voltage Ebb of 400 volts, a load resistance RL of 100,000 ohms, and a grid-bias
voltage E e e of - 3 volts. \Vhat is the quiescent plate current I bO?
2. A relay having a resistance of 1,000 ohms is to be operated by the plate current of a high-vacuum triode. If the available direct grid-signal voltage is 5 volts
and the relay closes at 30 rna and opens at 20 rna)
which of the triodes wbose plate characteristics
appear in some" one manufacturer's literature
100,000
should be satisfactory ? For each triode selected,
ohms
specify the plate-supply and grid-bias voltages that
+
must be used.
3. A triode has the plate characteristics given
400v
in Fig. 7) Ch, IV, except that the grid-voltage
scale is to be multiplied by ten - that is, the increment in grid voltage between adjacent curves
FIG. 41. Triode circuit for
is 10 volts instead of 1 volt. The tube is connected
Prob.3.
as shown in Fig. 41 with a 400-volt battery as
a plate-power supply and a plate-load resistance
of 100,000 ohms. The resistor Ric is so adjusted that there is a voltage of 50 volts
between the grid and the cathode.
Find the quiescent plate current I bO, the quiescent plate voltage EM, and the
required value of Rs,
4. The plate current of a particular triode is satisfactorily given by the expression
i&= 17 X 10- 6
(e+ e;)\.7
amp.
c
where €" and eb are in volts.
(a) Determine the plate current ib corresponding to a grid voltage ec of -15
volts and a plate voltage eb of 200 volts.
(b) Find the dynamic plate resistance ,~ and the mutual conductance gm of the
tube at tbe operating point specified in (a).
(c) If the tube is used as a Class Al voltage amplifier with a load resistance 01
10,000 ohms and a grid-bias voltage E co of -15 volts, what plate-supply
voltage Ebb is required to produce a quiescent plate current equal to that
determined in (a)?
(d) Determine the voltage gain of the amplifier for the conditions in (c).