How to calculate single layer LC-circuit coil in Coil64?

Usually, in the description of a specific radio amateur design, for an inductor, it is immediately showed: on which former, by which wire and how many turns you need to wind, what value the LC-circuit capacitor should be. When working with Coil64, a beginner radio amateur is faced with the problem of choosing the initial data for calculating the LC-circuit coil. Foremost, the following two questions appear:

• What inductance and capacitance values to select, and based on what?
• What should be the former diameter and the wire diameter, and why?

Let's answer these questions with a specific example. Let's calculate the LC-tank coil for the 14 MHz amateur band.

In accordance with the formula for the resonant frequency of the LC-circuit, there is a fixed value of the product L*C for a determined resonant frequency f. In this case, we can choose any ratio between L and C while maintaining the value of this product. At the same time, the characteristic impedance of the circuit ρ definitely depends on the L/C ratio.

Therefore, in order to decide on the L/C ratio selecting, we need to know the value of the LC-circuit characteristic impedance. The choice of the impedance is not entirely unambiguous and often depends on the specific circuit application, however, in practical terms, you can be guided by the following simple criteria for transistor circuits, depending on the frequency range:

• On long and medium waves up to frequencies in a bit of megahertz ρ = 500...2000 ohms;
• At short waves up to frequencies of a few tens of megahertz ρ = 250...500 ohms;
• On VHF above 100 MHz ρ = 50...250 ohms;

A resonant LC-circuit with ρ less than 50 is ineffective, therefore, helical coils are not used at UHF frequencies. For lower frequencies, you can take a larger value, the higher the frequency, the less. For lamp circuits, these values can be increased by two to three times.

Open Coil64 on the “LC Circuit” tab, select the initial calculation conditions: “Impedance and frequency are known”. We set the characteristic impedance close to the average value for SW, for example 320 Ohms, and enter the frequency of 14 MHz. We press the “Calculate” button and compute: the inductance of the circuit is 3.59 μH, the capacitance is 36 pF. The LC-circuit capacitance value is calculated as the closest from the standard series E24, so we go to the initial calculation conditions: “Capacitance and inductance are known” and make a control recalculation with this capacitance value. We see that the resonant frequency of the circuit is 14 MHz, and the recalculated value of the characteristic impedance is about 316 ohms, that is within the allowable values.

Go to the “Coil” tab, select the single-layer close-winding coil. The required values of inductance and frequency have already automatically entered the input fields. We select the initial conditions for the calculation: “Former diameter and winding length”. We need to decide on the coil-former diameter. With an increase in the coil-former diameter, the coil self-capacitance increases and the self-resonance frequency decreases. It is necessary that the coil self-resonance frequency be approximately an order of magnitude higher than the operating frequency. Usually, coil-formers with a diameter of 6…12 mm are chosen for SW bandpass filters. Let's say we have formers with a diameter of 7 mm. We select the initial calculation data “Former diameter and winding length”. We enter our value for the former diameter of 7 mm. The optimal winding length is from 0.5 to 2 winding diameters. Let's take 9 mm. We press the calculate button, and we get the parameters of our coil: 26 turns of wire with a diameter of 0.22 mm. Typically, bandpass filters do not require coils with a constructive quality factor higher than 200, the calculated Q-factor of the coil is Q=129. The self-resonance frequency is 198 MHz, which is much higher than the working one, which is what we need. In this case, we immediately managed to choose the appropriate coil-former diameter. Sometimes it is necessary to sort through several values of the former diameter and the length of the winding in order to satisfy the condition for the self-resonance frequency value of the coil and obtain the desired Q-factor and the minimum self-capacitance.Further, we can correct the calculation by selecting the initial data: “Former diameter and wire diameter”. Selecting the nearest standard wire diameter of 0.21 mm will give us corrected results for our coil. The LC-circuit capacitance in the results is less than the specified one by the value of the coil's self-capacitance. It can be implemented from a constant capacitor of 15 pF and a variable trimmer of 6…25 pF.

Narrowband filters require higher Q-factor coils. For such coils, you should choose a larger former diameter, for example 20…40 mm and select the option “One-layer coil with round wire”. We set the former diameter of 20 mm, which is about three times the original value of 7 mm. We also increase the wire diameter by three times and select the standard 0.62 mm. The optimal winding pitch is approximately equal to two wire diameters, we set to 1.2 mm. For an optimal coil with a maximum Q-factor, the winding length should be close to the winding diameter. The constructive Q-factor got about 400, which is a quite high value. The actual Q-factor of the coil will be less than the calculated one, since the influence of the former material, shield and other factors is not taken into account in the calculation. Having calculated the parameters of our high-Q coil, we need to make sure that its self-resonance frequency exceeds the operating frequency by at least five times. In our case, this condition is met. The self-capacitance of the coil is also an order of magnitude smaller than the LC-circuit capacitance.

You can take the characteristic impedance close to the maximum, for example 450 ohms. And then, by selecting the diameter of the former and the wire diameter, it is possible to design the largest optimal coil with the highest possible Q-factor at this frequency, guided by the criterion for the self-resonance frequency value of the coil. It makes no sense to further increase the dimensions of the coil, since the frequency of its self-resonance will become unacceptably low. The result is in the following image:

Since the Coil64 app computes other basic parameters of the coil in addition to the inductance, we can continue the calculation in circuit simulation programs, for example, in the free LTSpice, presenting the coil as a realistic lumped element model with real loss and parasitic capacitance. Prepared initial data for the coil model in LTSpice can be found in the output field. It is enough to copy and paste them into the required fields of the coil properties.

In conclusion, it should be noted that the design of high Q-factor LC-circuits with coils that operate at frequencies close to the self-resonance is, in principle, well possible. Examples are the Tesla coil or VHF helical resonators bandpass filter. However, the exact calculation of such circuit using Coil64 based on lumped element model is not possible. It is necessary to model them as distributed-element circuit. This can be done in electromagnetic simulators or using special empirical calculation formulas.