We ended part one stating that we could now calculate padders and trimmers, and from there determine what value to use for an inductor. Real tank circuits will always have some stray capacitance, and therefore we need to take it into account.

Hence, our tank circuit will now include **C**_{S}
as shown in this diagram. In Part 1, we used the notation **C**_{H}
and **C**_{L} to denote the maximum
and minimum values of the tuning capacitor **C**_{V}.
We also used “prime” notation to denote the net maximum and
minimum values after including the trimmer and padder, which we will
reiterate here:

**C**_{H}
, **C**_{L}
– The maximum and minimum capacitances of the variable capacitor CV

**C**_{H}* ′*
,

**C**_{H}* ″*
,

And, we will now add a third:

**C**_{H}* ‴*
,

Repeating the previous definitions for the above:

Hence:

(6) |

And likewise:

Hence:

(7) |

By including stray capacitance, the calculations to determine trimmer and padder values become somewhat more complex. We will work through them again, and in addition, we will include the situation where we select the tank inductance L, and then determine the required trimmer and padder values.

Starting with equation (6) and rearranging it:

Let:

Then substituting:

and rearranging

(6a) |

Similarly, we can rearrange equation (7) to get:

(7a) |

where:

Now subtract (7a) from (6a) to get:

(8) |

This allows us to get rid of
the **C**_{T}**C**_{P}
term which would otherwise make things a bit messy. And, in order to
keep things from getting too unwieldy, we will define the following
constants:

and substitute them into (8) to get:

(8a) |

Or, expressing the padder in terms of the trimmer:

(8b) |

And consequently:

(8c) |

We now have a simple expression relating **C**_{P}
to **C**_{T}.

We can use expressions (8b) and
(8c) to solve for one when the other is known as we did in Part 1 of
this discussion. Having determined both **C**_{P}
and **C**_{T}, we can then substitute
these back into the previous formulae to solve for inductance.

Now, what if we want to select an arbitrary value of inductance, and then determine the correct trimmer and padder capacitors to give the desired bandspreading? To do this, we will start with (6a) again:

(6a) |

We will define two more constants:

Substituting these into (6a):

Substituting the expression for **C**_{P}
in terms of **C**_{T} from (8b):

Multiplying out the terms:

Then grouping by terms of **C**_{T}:

(9) |

Once again we have a quadratic equation in the form

where

the coefficients are:

And the solution for **C**_{T} is:

Therefore, we can now solve equation (9) to find *C*_{T},
and then substitute the calculated value of *C*_{T}
back into (8b) to find *C*_{P}.

Again we will pick a frequency range of 5500 to 7000 kHz, and tuning
capacitor range: *C*_{L
}=50,
*C*_{H
}=500.
We will also assume a stray capacitance *C*_{S}=15
pF. For an inductance we will choose *L*=12
µH.

Referring to resonance formula (1), and remembering that the lowest frequency corresponds to the maximum net capacitance, we get:

Rearranging:

For *F*_{L}=5500, we get:

Similarly, for the maximum frequency *F*_{H}=7000,
we get:

Calculating the constants:

From these we calculate the quadratic coefficients:

Then **C**_{T} can be calculated:

And this value of *C*_{T}
can be substituted into (8b) to find *C*_{P}:

These values can be checked by substituting them back into formulae (6) and (7) to get the net circuit capacitances and then into the resonance formula (1) to verify the frequency range. That will be left as an exercise for the reader. The following dial scale shows the tuning characteristic when using a standard midline variable capacitor.

Given a desired frequency range to bandspread, a tuning capacitor with a given capacitance range, and accounting for stray circuit capacitance, we can now pick either a trimmer, a padder or an inductance value. Then, with the given component value, we can calculate the remaining two unknown component values. As mentioned at the end of Part 1 of this discussion, there is an on-line Bandspread Calculator here, which you can use to perform the calculations discussed above. The calculator determines the values of the two unknown components, when given the value of the known component. While I have not revisited the derivations of Part 1 to include stray capacitance for the cases of fixed Trimmer or fixed Padder, they are actually simpler than the fixed inductor case, and I will address them in a future update of this page. However, the calculator does perform all of these calculations while including the effect of stray capacitance.

In Part 3 we will discuss a method of bandspreading where the variable bandspread capacitor is connected to a tap on the tank inductor, and a bandset capacitor (either fixed or variable) is connected across the entire coil. This is a method which dates back to the early days of radio, and was used in both commercial and homebrew shortwave radio circuits.

Continue to: Part 3

Back to:

Bandspreading Part 1

Radio/Electronics Theory

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This page last updated: January 30, 2023

Copyright 2010, 2023 Robert Weaver