2017-02-12

Harmonic LC resonance in single-layer coil inductance

Much debate is on how to consider harmonic LC resonance in single-layer coil excited by an inductor. Some researchers believe that to obtain correct results it is sufficient to multiply the number of calculated harmonics to the frequency first. They come from concepts of a wave and believe that if it can multiple to share, the frequency will be inversely proportional to multiply. They forget that for different frequencies there is a speed of propagation of waves [1]. In addition, the external capacity or the capacity of the ground brings about changes in the frequency of the first harmonic, which leads to multiple error in the calculation of the operating frequency of the coil at the highest harmonics.

It is not entirely correct to call frequencies that are not multiples of the first harmonics.
On the other hand, the term "fashion" doesn't quite fit here either.
Therefore, in this work we will adhere to the term “harmonic” conditionally.

Conducted research of different coils have shown a rather unusual result, poorly explained by classical laws. For example, if the coil is not connected to the external capacitance (we call this the frequency baseline), the second harmonic is considered to be a multiplication of this frequency by two. But the "three" basic frequency is multiplied to obtain the third harmonics as expected, and the fourth. "Four" basic frequency is multiplied to obtain a sixth harmonic, and do not expect a fourth, etc. frequencies of odd harmonics having the influence of an external capacity.

Below are graphs of operating frequencies depending on the harmonic number for two different coils. On the left graph displays the results for the first coil, where: f

_{i}is the actual frequency grounded coils without external capacitance (MHz), f2_{i}is the actual frequency of the grounded coil with external capacity of 12pf, i is the number of harmonics. On the right graph displays the results for the second coil, where: f_{i}is the actual frequency of an ungrounded coils without external capacitance, f2_{i}— operating frequency grounded coils without external capacitance, i is the number of harmonics. As you can see, neither of which is simple multiplication of number of a harmonic to the basic frequency of speech can not go. It is also seen that the odd harmonics are slightly lower than expected, making the graph of the polyline. The graph shows the coils which were involved in another experiment: the left — coil

**2**, the right — coil**3**.How to count the frequency of the harmonics

To do this, we introduce some terms that can facilitate further understanding of the processes. Own base frequency, \(f_s\) — resonance frequency of the first harmonic of the coil is connected without external tanks. The capacity of the ground is considered in the calculation, if used. This frequency can be found experimentally or by calculation in the calculator, for example in this.

The total base frequency of \(f_g\) — resonance frequency of the first harmonic of the coil connected with the external tanks. Under external capacity to understand or is connected parallel to the coil condenser, or a solitary capacity connected to its hot end. This frequency also produced either experimentally or in the calculator. It is clear that without external capacitance frequency, \(f_s\) and \(f_g\) will be equal.

Then the frequency for the first harmonic will be equal to the total base and be as
\[f_1 = f_g \tag{1}\]
Frequency to even harmonics are like this:
\[f_i = \left({i+2 \over 2}\right) f_s, \quad i \in 2, 4, 6, 8 ... \tag{2}\]
the Frequencies of odd harmonics are like this:
\[f_i = \left({i+1 \over 2}\right) f_s + {f_g \over 4}, \quad i \in 3, 5, 7, 9 ... \tag{3}\]
where \(f_i\) resonance frequency of the coil at the ith harmonic, \(i\) is the number of the harmonic.

A General algorithm for calculating the resonant frequency of the coil, depending on the number of the harmonics might be:
\[f_i = \begin{cases} f_g, & \mbox{if }i=1
\\
\left({i+2 \over 2}\right) f_s, & \mbox{if }i\mbox{ even number}
\\
\left({i+1 \over 2}\right) f_s + {f_g \over 4}, & \mbox{in other cases} \end{cases} \tag{4}\]

Insights

As a result of the experiments was derived the nonlinear dependence of the resonance frequency of single layer coils from the number of a harmonic of excitatory fluctuations. Fully confirmed the hypothesis that the calculations for frequency LC resonance, and standing wave are completely different, and so are two separate and possibly completely independent of the process. Last follows different patterns described in [1].

__The materials used__

- Alan Payne. SELF-RESONANCE IN COILS, 2014