# Self-Resonant Frequency of Single-Layer Solenoid Coils

Author: R.J.Edwards G4FGQ © 18th November 2004Coils have a distributed self-capacitance equivalent to a lumped capacitance connected between its ends. Consequently, all coils have a parallel resonant frequency. At resonance there's a very high impedance between the ends of the coil. A common application is use of a coil as a resonant RF choke.

For a single-layer solenoid the self-capacitance can be calculated from its dimensions. From the inductance, loss resistance and Q, the resonant frequency and impedance-vs-frequency response can be found.

A choke can be made from a length of close-wound, neatly-wound coaxial cable inserted in a coaxial line of the same type to minimise undesirable RF currents flowing on the outer surface of the coaxial line's outer conductor. It is necessary to calculate the approximate self-resonant frequency of the coil and the range of frequencies over which the choke impedance is high enough to be effective in a particular wideband application.

The impedance of a coaxial choke can be as high as several megohms at resonance, but in practice, an off-resonance impedance of 1000 ohms may be sufficient. Much depends on length of line in wavelengths and its terminating resistance.

Some Crude but Small Approximations

When a coil is connected between its connecting leads the small stray
capacitance between the leads is placed across the coil and the coils'
resonant frequency decreases. In the case of a coil in a long antenna
wire or feedline the question arises over what lengths of lead is the
capacitance between leads effectively in parallel with the coil? So what
is the coil's resonant frequency?

As the length of the coil and the spacing between the leads increases the stray lead-to-lead capacitance decreases. On the other other hand it increases as the lead length increases but in a decreasing manner until it remains constant. Because capacitances of coil and leads are both distributed their resultant is not the arithmetic sum of the two - it is mathematically indeterminate.

The capacitance of a very short open-circuited lead IS calculable and increases in direct proportion to its length. But when leads are long only the length of lead immediately adjacent to the coil affects capacitance across the coil and it is necessary to guess the short length of the leads involved. At least, the guesswork allows the uncertainty to be estimated. Fortunately all is very uncritical. In the absence of any better guesses it is suggested lead length be set equal to coil length and results will probably be in the right ball-park.

- Number of Turns = Coil Length divided by Winding Pitch.
- Winding pitch = Coil length divided by number of turns.
- Winding Pitch = Wire Diameter + Twice Insulation Thickness + Twice air space.
- For a coax choke, conductor diameter is diameter of coaxial outer conductor.

For a coax choke inserted in a long line, lead length is the initial short length of the line which has an appreciable affect on capacitance across the coil. The calculated resonant frequency with leads connected to the coil occurs when the line is cut and the resulting short leads are open circuited at their outer ends.

If instead of a neatly wound coil a 'jumble-wound' coil of the same mean diameter and number of turns is used then the self-resonant frequency and Q may fall by crudely 30 or 40 percent. More important, impedance vs frequency above resonance instead of smoothly decreasing is likely to vary erratically and the choke may not be effective at a few unpredictable frequencies above resonance. Ineffective may mean only that the choke impedance is less than 1000 ohms.

A minor advantage of a solenoid-wound coaxial choke over a ferrite-ring type is that the line impedance Zo remains constant throughout the system.

Random and Spurious Effects

At frequencies well above the fundamental resonant frequency there
are further resonances between the coil and the coax line. Theoretically,
at 3, 5 and 7 times the fundamental resonance, there will be other very
high coil impedances which cause no problems. But at the even harmonics
the impedance may fall to very low values which may render the choking
action ineffective.

The actual frequencies at which even harmonics fall into an amateur band depend on line length and on coil location and are unpredictable. To be fairly sure of avoiding spurious resonances operating frequencies should be limited to twice the fundamental resonant frequency. On the other hand the chance of a spurious response falling in an amateur band is not very high and even if it does it may not have a serious effect anyway.

Choke Performance See
Test Circuit

Performance is in terms of the insertion loss in decibels of RF power
flowing along the wire, or coaxial outer conductor, due to insertion of
the choke. The load impedance immediately on the choke is also indicated
since this has a bearing on performance. Both figures are necessarily very
approximate. They depend on line length in waves at the test frequency.
Source impedance is 50 ohms.

Other Thoughts on the Subject

The only accurate numerical results of this program are the self-resonant
frequency of the coil in free-space and the resonant frequency of the
coil with open circuit leads connected, neither of which exists in practice
because the leads, whatever their length, must be connected to the remainder
of a more complicated circuit. Whatever is done affects distributed stray
capacitances and mutual impedances and so also affect the coil's resonant
frequency - the frequency at which the coil's impedance Z is a maximum.

The actual resonant impedance Z is very high but is seldom important. It is the bandwidth over which Z is more than some relatively low value which matters. It depends entirely on the choke's application, narrowband or wideband.

In the case of a choke made with coaxial line, the dimensions of which are very non-critical, it is the impedance at band edges which matters most. The effectiveness of which depends on the coil's location along the line, the line's length, and the standing waves which always exist on it.

A ferrite choke has sensibly zero transmission loss. But a large coaxial solenoid can have appreciable transmission loss due to its long coaxial length. Also a ferrite choke can have a fairly flat response and a wider usable bandwidth.

Circuit for Performance Test on Coaxial Choke

50 ohms L R Unbalanced Feedline o--------o---QQQQQQ------o-----------------------------o Length in Wavelengths CHOKE o Z Source o-------------------o Adjustable Z at Test Freq. C Termination Z Z o Ground Ground o

Coil Q is calculated from R. Capacitor Q is assumed to be 2000. Line termination represents an uncertain low resistance ground connection.

Principal Computed Results

- Self-capacitance of coil
- Resonant frequency of isolated choke
- Resonant frequency of choke when inserted in line
- Magnitude of choke impedance vs test frequency
- Insertion loss of choke in dB, vs line length in wavelengths, at test frequency

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