September 1947 QST
Table of Contents
Wax nostalgic about and learn from the history of early electronics. See articles
from ARRL's
QST, published December 1915  present. All copyrights hereby acknowledged.

Although single capacitor, single inductor resonant
tank circuits are not widely used, one place you do find them is in transistor amplifier stages in
order to peak gain in a particular frequency band. Author Jack Najork offers here a few pointers
regarding preforming quick mental calculations for how ranges of capacitance values affect the
resonant frequency. For instance the ratio of capacitance values needed to cover a certain range of
resonant frequencies is equal (ideally) to the square root of the frequency ratio. This comes into
play both when determining values during the design phase and the selecting a variable capacitor for
providing tuning over a continuous range of frequencies. The same is true for inductor selection, of
course.
Simple L and C Calculations
A Useful Formula for Bandspread and Padder Problems
By Jack Najork, W2HNH
This article discusses a wellknown but seldomappreciated simple relationship between frequency
and circuit capacitance. It has the advantage over many radio calculations in that it takes less time
to work out the formula than it does to arrive at the answer by the "cutandtry" method.
One of the most useful formulas, and at the same time one most neglected by hams who build or alter
their own gear, is the simple mathematical relationship between a change in the frequency of a tuned
circuit, such as indicated in Fig. 1, and the change in capacitance which causes it. Its most common
application is in the determination of capacitance values in bandspread or other tuningrange problems
in receivers or VFOs. It shows the frequency range that any variable condenser will cover or, conversely,
the variable capacitance necessary to cover a desired frequency range.
The relationship is expressed by
In other words, it always takes a capacitance ratio of the square of the frequency ratio to cover
the desired frequency range. If the frequency ratio, maximum to minimum, is 2 to 1, the required capacitance
range to cover it is 4 to 1; if the frequency range is 3 to 1, the capacitance range must be 9 to 1,
etc. And, of course, the converse is true  the frequency range that a variable condenser will cover
is the square root of the capacitance ratio, i.e., a capacitance range of 4 to 1 produces a frequency
change of 2 to 1. Example:
Desired frequency range  3500 kc. to 4000 kc.
Minimum circuit capacitance  30 μμfd.
Frequency ratio = 4000/3500 = 1.143
1.1432 = 1.3
Therefore the capacitance change to tune over this band must be in the ratio of 1.3 to 1. Since
the minimum circuit capacitance with the condenser set at zero is given as 30 μμfd., the maximum
must be (1.3) (30) = 39 μμfd. The difference between maximum and minimum, 9 μμfd., is the
required variation.
The formula can be twisted around, of course, to give any one of the desired values by substituting
known values for the other three factors.
In all of the foregoing, C_{min} is the minimum circuit capacitance with the variable condenser
set at zero, C_{max} the total circuit capacitance with the variable in full, f_{max}
and f_{min} respectively the highest and lowest frequencies to which the circuit tunes
and C_{var} the variation in capacitance in the tuning condenser. C_{min}, of course,
includes any fixed capacitance that may be used in the circuit either for bandspread or to obtain a
highQ circuit such as in VFOs. Of course f_{min} and f_{max}, and
C_{min}, C_{max} and C_{var}, should be in the same units, i.e., kc. or Mc.
for f and μd. or μμfd. for C.
Fig. 1  A  Circuit for f_{max}. B  Circuit for f_{min}.
C_{min} is the total minimum circuit capacitance which includes tube capacitances, minimum capacitance
of the tuning condenser and capacitance introduced by sockets and wiring as well as the distributed
capacitance of the coil. C_{var} is the variable part of the tuningcondenser capacitance (maximum
minus minimum).
In practical application it should be recognized that the variation in capacitance that may be expected
from a variable condenser is not the rated maximum capacitance shown in the catalogues; it is the difference
between the condenser's maximum and minimum capacitance. A typical small 100μμfd. variable may
have a minimum capacitance of 10 μμfd. making the variation only 90 μμfd. This minimum as
well as any capacitance introduced by mounting the condenser close to grounded metal must be included
in the minimum circuit capacitance, C_{min}. This minimum circuit capacitance usually is the
only factor that is not readily determined and since it may run as high as 50 to 75 μμfd. when
capacitance coupling is used between tetrode or pentode stages, especially when the condenser is mounted
close to the chassis, it isn't something that can be neglected entirely without introducing considerable
error in the calculations. In many cases, however, it is possible to estimate the total minimum capacitance
with an accuracy sufficient for the purpose. In the case of capacitance coupling between stages, the
output capacitance of the driving tube and the input capacitance of the driven tube are in parallel
with the tank coil and therefore form part of the minimum circuit capacitance. The values for any particular
combination of tubes may be made up from the information given in tube manuals.^{1} Variablecondenser
minimum capacitances also are usually shown in the manufacturer's catalogue.
Mounting a midgettype condenser on metal in the usual manner introduces a stray capacitance between
stator and ground equivalent to from 2 to 4 times the condenser's rated minimum capacitance. This must
be added to the circuit minimum although it does not alter the amount of capacitance variation to any
appreciable degree. Socket, wiring, and coil capacitances may add another 10 to 20 μμfd. If the
circuit minimum is to be held down, the condenser in particular must be well spaced from grounded metal
chassis and shields.
The inductance required to go with the condenser may be determined from the following formula:
where the frequency is in Mc., the capacitance in μμfd. and the inductance in μh. Coil dimensions
to give the calculated inductance may be selected from published tables or the ARRL Lightning Calculator.
1 See p. 22, February, 1947. QST, for popular .audioreceivingtube
capacitances.
Posted July 11, 2016
