Bandspread Calculations

Part 1

When designing and building a homebrew receiver, one of the first things to be considered is the range of frequencies to be covered. Once this is decided, one must then determine the required component values for the resonant circuits. Custom coils of any inductance can be wound without too much difficulty. Variable tuning capacitors, however, are only available in certain values, and the range of choices is getting smaller and smaller, as they become obsolete. The usual design method then becomes: picking an available tuning capacitor and then winding an inductor to suit. Normally, the inductance value is determined at the lower end of the frequency range, using the tuning capacitor's maximum capacitance value (when the capacitor's plates are fully meshed) and then applying the LC resonance formula:

where the units are Farads, Henries, and Hz.

For convenience we will include a factor of 106 in the equation so that the units will be kHz, pF and µH. We then get:

or

(1)

Using this formula then, we can determine the inductance value required using the minimum frequency, and the maximum capacitance value. The upper end of the of the frequency band then just happens to be whatever results from the tuning capacitor's minimum value (plus stray circuit capacitance). For a medium wave broadcast band (MW-BCB) receiver, common variable capacitors tend to have a nearly ideal range of about 10:1 which translates well to the approximately 3:1 frequency range of the MW-BCB.

From the LC resonance formula (1), we see that frequency is inversely proportional to the square root of the capacitance (assuming the inductance is held contant). Therefore, if the tuning capacitor has a range of 10:1 then the frequency range will be √10:1 or 3.16:1. This works well for the MW-BCB, but what if we are interested in building a shortwave receiver that covers the 49 meter band? Let's say we're interested in the range: 5500 kHz to 7000 kHz. The frequency ratio is 7000/5500, or 1.2727:1, and the capacitance ratio will be the square of the frequency ratio or 1.6198:1. If the only available tuning capacitor has a range of 50 to 500 pF, then it would tune well past the desired band. The desired band would be compressed into a small part of the tuning capacitor's range. We need to spread out the band to cover the full range of the tuning capacitor, hence the term bandspreading. Ideally, we should have a tuning capacitor with a 1.6198:1 range. Fortunately, it is easy to reduce the variable range of an electrical component by adding another fixed value component either in series, or in parallel with the variable one.

Parallel Bandspreading

Parallel capacitances add together. So, if we add a 500 pF fixed capacitor in parallel with our 50-500 pF capacitor, we would now have a range of 550 to 1000, and a ratio of 1000/550, or 1.8:1 which is getting closer to the desired 1.6198. This parallel capacitor is typically called a trimmer, and we will denote it as CT. When we have a trimmer capacitor in parallel with the variable one, the ratio of capacitance is:

where

k_C  = net capacitance ratio

C_T  = parallel “trimmer” capacitor value

C_L  = variable capacitor minimum capacitance value

C_H  = variable capacitor maximum capacitance value

We can use a trial and error method to home in on the required fixed value capacitor, but that's tedious and unnecessary. We can rearrange the equation to get:

(2)

Now, we can just plug in the known values to find C_T. Let's test it out. Using the example values above: k_C = 1.6198, C_L = 50, C_H = 500, we get

We can check that the ratio is indeed 1.6198, by substituting the values back into the original formula:

We have solved the bandspreading problem by calculating the value of a trimmer capacitor that can be wired in parallel with the variable capacitor. Note, that to this point, we have not discussed the circuit inductance at all. It doesn't need to enter into the bandspread calculation, because the bandspreading is dependent only on capacitance ratio. However, to design the remainder of the tuned circuit, we now apply the resonance formula (1) to calculate the inductance. We rearrange the resonance formula to get the inductance on the left hand side:

(1a)

Using our example values at the low frequency end of the band, we get:

We can verify that this value of inductance combined with the maximum and minimum capacitance values will produce the frequency range 5500 to 7000 kHz. With the tuning capacitor at maximum capacitance,

and at minimum capacitance,

In theory, these numbers work, but we haven't discussed whether 0.712 µH is a practical inductance value. We might want to aim for something a bit higher.

Series Bandspreading

It was mentioned that the range of the variable capacitor can be reduced by adding a fixed capacitance either in parallel, or in series. We've looked at the parallel case. Now let's look at the series case. The series capacitor is typically called a padder, and we will denote it as C_P. For capacitors in series, the net capacitance C_N is governed by this relationship:

or

where C₁, C₂, C₃, etc., are all connected in series.

For the special case of just two capacitors in series this formula can be rearranged to

which should be familiar to most electronics hobbyists. However, for this discussion, the reciprocal formula works out to be more convenient.

If we add a capacitor in series with our variable capacitor, then the net capacitance, when the tuning capacitor is set to maximum, will be given by:

and when the tuning capacitor is set to minimum,

where C_Hʺ and C_Lʺ are the respective net maximum and net minimum circuit capacitances. (We will use the double-prime notation, to indicate the “padded” values of the tuning capacitor.)

The ratio of maximum net capacitance to minimum net capacitance is then given by:

or

As before, we can rearrange this equation to get the unknown value C_P by itself on the left hand side:

(3)

Using the values from the above example again, k_C = 1.6198, C_L = 50, C_H = 500, we get:

Substituting this value back into the series capacitance formula for both the high and low values, we get,

and

Taking the ratio,

which is the ratio we want.

Now if we plug the values into the LC resonance formula (1a) we can get the inductance value:

Substituting this inductance value back into the resonance formula (1) with each of the high and low padded capacitance values will verify that the frequency range is indeed 5500 to 7000 kHz. So we see that we can bandspread the tuning capacitor with either a trimmer or a padder capacitor of the correct value. However, for our example, the padder arrangement appears to result in a more practical value for the inductance.

While we have discussed what are practical inductance values, we haven’t looked at what effect these different methods of bandspreading will have on tuning linearity. If we assume that our tuning capacitor has the common “midline” characteristic, then we can see the result of parallel bandspreading in the following diagram of a dialscale:

The scale shows the tuning linearity using a 676 pF trimmer. The scale is reasonably linear until we get near the top of the band where it starts to stretch out.

Now let’s look at what we would get using 36.98 pF padder capacitor:

In this case, the scale is much more nonlinear, with the frequencies very compressed at the high end of the band. So, although the series bandspreading arrangement results in a more practical inductance value, the tuning linearity is not as good as the parallel bandspreading arrangement. This leads us to consider the possibility of using a combination of trimmer and padder to get better linearity while still having a practical inductance value.

Combined Parallel/Series Bandspreading

There is of course no reason why we can't use a combination of series and parallel capacitors to bandspread the variable capacitor. If we do this, we will end up with a required inductance value that falls somewhere between the previous two that we have calculated. Hence, 0.712 µH and 24.32 µH are the limiting inductance values for our example problem. In fact, using formulae (2) and (3) will give us the limiting values for the trimmer and padder respectively. The trimmer value given by (2) will be the maximum, while the padder value given by (3) will be the minimum. Substituting the resulting minimum padded and maximum trimmed tuning capacitor values back into (1a) will produce the the maximum and minimum inductance values. So, we already know quite a bit about our tuned circuit.

There are two possible series/parallel configurations that we can use. The trimmer can be connected across both the variable capacitor and the padder; or just across the variable capacitor, as shown in the diagram. The arrangement in the diagram is more commonly used in radio circuits. So, that is the case we examine.

Now, we will look at calculating the unknown value when the other one is known. We will start with the simplest case. Using our previous example, suppose we want to use a trimmer value of 300 pF. Since this is less than the maximum value of 676 pF given by (2), we should be able to find a  compatible padder to bandspread the circuit.

The simplest way to approach this is to define the trimmed values of the variable capacitor as:

(where the prime notation indicates that the tuning capacitor’s high and low values include the trimmer). Now, we simply substitute these values into (3) instead of C_H and C_L.

Substituting for C_H′ and C_L′

(4)

Let’s plug in the example values, C_T = 300 pF, k_C = 1.6198, C_L = 50, and C_H = 500,

Now that we know the padder and trimmer, we can calculate the net high and low circuit capacitances from these series/parallel formulae:

and

(We previously used the double-prime notation to indicate the value of the capacitance due to padding only. From here on, it will indicate capacitance values taking into account the effects of both trimming and padding.) These formulae give us 386 pF and 238 pF respectively. Substituting either of these values back into (1a) along with the corresponding frequency will give us the required inductance value of 2.17 µH. The resulting tuning characteristic is shown below:

Next, suppose we want to choose an arbitrary padder value, and then calculate the required trimmer value. Formula (4) gives us the relationship between the trimmer, padder and tuning capacitor values. It now becomes a matter of rearranging the formula so that CT is by itself on the left hand side. If we multiply all of the terms in (4) by its denominators, until there are no fractions remaining, we get

Then separating out the coefficients of C_T, we get

which is a quadratic equation in the form

where

and where

	(5a)
	(5b)
	(5c)

The solution of the quadratic is given by

The ± sign indicates that there are two possible solutions to the equation. However, only the following form (with a plus sign) will yield a positive value for the trimmer

(5)

Rather than expand the a, b, and c terms into an excessively complex formula, it will be left in the form shown, the sub-terms a, b, and c can be evaluated individually, and then those values substituted into (5)

As an example, let’s pick a padder value of 200 pF, with the remaining parameters the same as in previous examples, k_C = 1.6198, C_L = 50, and C_H = 500. Then, evaluating a, b, and c, we get

Then

Substituting the values into (1a) results in an inductance of 5.52 µH, and the resulting tuning characteristic is shown below:

Summary of Part 1

We have shown that, given a frequency range that is smaller than the uncompensated range of a variable capacitor, we can bandspread it to cover the desired range exactly, and if we pick a value for the trimmer,  then we can calculate the required value for the padder, and vice versa. And, once having both the trimmer and padder values, we can determine the required inductance value.

In Part 2 we will include the effect of stray capacitance, and we will look at the case where we pick the inductance value, and then from that, calculate the required trimmer and padder values.

Meanwhile, if you want to experiment with the effects of trimmers, padders, etc., on bandspreading, There is an on-line Bandspread Calculator here. Or, go to the downloads page and download the Tracker program. Although it is described as a superhet tracking calculator, it also performs all of the bandspreading calculations discussed here, and was used to produce the tuning characteristic illustrations.

Continue to: Part 2