Design Article
Comment
Peter Demchenko
Can't agree, the damping factor of a series RLC circuit NOT equal to R/(2L). ...
anonymous user
I would avoid a math error and just use an inductance bridge or RLC meter. ...
Use air-core-coil resistance to estimate inductance
Peter Demchenko, Vilnius, Lithuania; Edited by Paul Rako and Fran Granville
1/5/2012 11:00 AM EST
This Design Idea shows how to
calculate the inductance of a
multilayer air-core coil using only its
dimensions and resistance. If you know
the dimensions and the number of turns
on an air-core coil, you can easily calculate
the inductance. With the dimensions
in millimeters, the inductance, L,
in microhenries is a function of the
square of the turns, as the following
equation shows: L=0.008×D2×N2/(3D+9h+10g), where D is the average
diameter of the coil; h is the height of
the coil; and g is the depth of the coil—all in millimeters (Figure 1).
If you don’t know the number of turns, you can still calculate the inductance using the dc resistance of the coil. For this technique to be accurate requires tight and regular coil winding using enameled cylindrical wire (Figure 2). You could use the wire dimensions to give an approximate expression for the total number of turns, N: N=g×h/d2, where d is the diameter of the wire, but this Design Idea assumes that the wire diameter is unknown.
Because the length of an average
turn is equal to π×D, the total length
of the wire is (N×π×D). The square of
the cross-sectional area of the wire is
(π×d2)/4.
You can express the resistance,
R, of the coil as R=ρ×N×π×D×4/(π×d2×1000)=ρ×N×D/(250×d2)=ρ×g×h×D/(250×d2×d2), where ρ is
the wire resistivity in ohmmeters and
the resistance is expressed in ohms.
Thus, you can derive an expression
for the wire’s diameter squared: d2=
. You would
then substitute for the d2 term in
the expression for turns: N=g×h/. You can now
square both sides of the equation and
cancel terms: N2=250×g×h×R/(ρ×D).
Substituting the value of N2 into the
first equation yields L=2×D×g×h×R/(ρ×(3D+9h+10g)). Using the value of
ρ for copper wire, you get an expression
for L, which depends only on the
resistance and the physical dimensions
of the coil: L=117.7×D×g×h×R/(3D+9h+10g).
If you don’t know the number of turns, you can still calculate the inductance using the dc resistance of the coil. For this technique to be accurate requires tight and regular coil winding using enameled cylindrical wire (Figure 2). You could use the wire dimensions to give an approximate expression for the total number of turns, N: N=g×h/d2, where d is the diameter of the wire, but this Design Idea assumes that the wire diameter is unknown.
You can express the resistance,
R, of the coil as R=ρ×N×π×D×4/(π×d2×1000)=ρ×N×D/(250×d2)=ρ×g×h×D/(250×d2×d2), where ρ is
the wire resistivity in ohmmeters and
the resistance is expressed in ohms.
Thus, you can derive an expression
for the wire’s diameter squared: d2=. You would
then substitute for the d2 term in
the expression for turns: N=g×h/. You can now
square both sides of the equation and
cancel terms: N2=250×g×h×R/(ρ×D).
Substituting the value of N2 into the
first equation yields L=2×D×g×h×R/(ρ×(3D+9h+10g)). Using the value of
ρ for copper wire, you get an expression
for L, which depends only on the
resistance and the physical dimensions
of the coil: L=117.7×D×g×h×R/(3D+9h+10g).L and R are proportional to each
other yields, yielding two interesting
consequences. First, for a series RLC
circuit, the following equation defines
the damping factor, 
,
meaning that the damping factor is proportional
to the square root of R for a
given C and coil dimensions D, g, and
h. Second, the quality factor, Q, of a
coil with given values of D, g, h, and ρ
and an angular frequency of w=2πF is a
constant value: Q=wL/R=2×w×D×g×h/(ρ×(3D+9h+10g)).
,
meaning that the damping factor is proportional
to the square root of R for a
given C and coil dimensions D, g, and
h. Second, the quality factor, Q, of a
coil with given values of D, g, h, and ρ
and an angular frequency of w=2πF is a
constant value: Q=wL/R=2×w×D×g×h/(ρ×(3D+9h+10g)).|
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Marian Stofka
1/7/2012 10:02 AM EST
This comment should not be taken as a criticism, but as an attempt to increase accuracy of quite cute estimation presented in the DI.
As the wire is assumed to have circular crossection,
the winding always contains empty (non-conductive) space;
even if the wire were not covered by an enamel or other insulation.
To determine the minimum empty space theoretically;
you can imagine three parallel conductive cylinders (wires); each of which is touchimg the rest two.
In the center of the crossection of this configuration, there is an empty area of:
(dxd/4)x(sqrt(3)-(pi)/2)
In the winding with many turns two such areas can be counted per each turn.
Consequently, the filling factor of the winding drops from the value of 1, assumed in the DI, to the value of:
2x(1-(pi)/sqrt(3)) = 0.89734
The number N of turns within the crossection of the winding can therefore be no more than 90% of that, assumed in the DI.
The damping factor of a series RLC circuit equals R/(2L).
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Peter Demchenko
12/9/2012 2:11 AM EST
Can't agree, the damping factor of a series RLC circuit NOT equal to R/(2L). You omitted C minimum.
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Marian Stofka
1/7/2012 10:45 AM EST
The correct form of the second expression in previous poster is:
2x(1-(sqrt(3))/(pi))
where "sqrt" = square root
"pi" = 3.14...
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anonymous user
1/10/2012 11:40 AM EST
I would avoid a math error and just use an inductance bridge or RLC meter. However, it would be a good exercise for a lab experiment in a beginning electronics course. A little contest can motivate.
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