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K Quiz Answer
Subject: K Quiz Answer
Date: Fri, 30 May 1997 08:24:54 -0400 (EDT)
From: msr7-at-po.cwru.edu (Mark S. Rzeszotarski, Ph.D.)
To: tesla-at-pupman-dot-com
Hello Coilers,
Last week I posted a short quiz on the coupling coefficient K.
Three different secondaries were considered, and two different primaries
were employed. In each case, the primary inside and outside dimensions
remain constant, only the number of turns changed. Similarly, the
secondary
diameter and height was held constant, and again only the number of
turns
changed (by varying the wire gauge).
Recall that the coupling coefficient K equals the mutual
inductance
M divided by the square root of the product of the primary and secondary
inductances. The mutual inductance is the degree of inductive coupling
between the primary and secondaries. Typical conventional tesla coils
employ coupling between .05 and 0.20 or so, (perhaps a bit higher for
you
folks with fast rotary spark gaps).
Since K=M/sqrt(Lp x Ls), and only Np and Ns (the number of
turns)
were varied, what I was really asking was how does M depend on Np and
Ns?
It turns out that Lp is proportional to Np^2 for a flat pancake coil.
The
same is true for a solenoid so Ls is proportional to Ns^2. What is
perhaps
not obvious is the fact that M is proportional to Np times Ns for most
coil
geometries we use in a tesla coil (solenoid, flat pancake, inverted
cone,
for five turns or more in the primary). As a result, by keeping the
same
geometry and only varying the number of turns, K remains constant since
as
Lp and Ls vary, so too does M. In this case K=0.23 for all coil pairs.
Several folks arrived at this conclusion fairly quickly. John
Couture pointed out that there is an approximation formula for M in
Terman's
"Radio Engineers' Handbook" for a solenoidal primary with solenoidal
secondary (not quite applicable here). I warn the readers that this
formula, as well as a series approximation published by Dwight, are
prone to
severe roundoff errors because several large numbers are subtracted from
each other to arrive at a small value numerical solution. These
formulas
work best if you break up the coil into a series of small coils
(mathematically) and then sum up the individual contributions.
Neumann's
formula, consisting of a double line integral, is the more appropriate
formula, but requires numerical integration for arbitrary wire
geometries.
This was used in posing the quiz, along with experimental verification.
For the newbie, M is the inductive coupling between the primary
and
secondary coil. The voltage we induce in the secondary coil is directly
proportional to M, so we may want to make it large. However, if the
coupling is too tight we start to introduce a splitting of the
frequencies
observed in the secondary, due to the complex phase shift of the induced
current. (The secondary current is 90 degrees out of phase with the
primary
current.) The degree of coupling at which this occurs is called
critical
coupling, and it depends on the losses in the primary and secondary.
We
usually try to operate coils near critical coupling. Once this
frequency
splitting starts, the secondary coil may start to break out with spark
at
locations along the secondary coil away from the top, which can cause
the
coil to self destruct. You can also get significant kickback into the
primary circuit, potentially causing damage. Initially, use very loose
coupling (secondary coil raised up 4-8 inches above the bottom turn of a
flat or conical primary coil), and lower it in stages after you get your
coil working well.
Several people also asked how to measure K. There are a variety
of
methods:
Method 1:
Set up your primary and secondary coils in the geometry you
intend
to use. Connect the two coils (primary and secondary) in series with
each
other and measure the inductance with an L meter. Call it La. Now
reconnect the two coils in series, reversing the leads on the secondary
coil. Measure it again to get Lb. M is then determined from the
formula: M
= absolute value of [La-Lb]/4
Method 2:
If you have an inductance meter, this is the method of choice.
Set
up your coil in the geometry you intend to use. Connect up your
inductance
meter to the primary with the secondary in its proper position for
normal
operation. Disconnect all leads to the secondary (including ground).
Measure the primary coil inductance with the secondary coil open
circuited,
and call it Lpso. Next, short circuit the secondary windings using a
small
piece of wire extending from the top of the secondary to the bottom.
Again,
measure the primary coil inductance, and call it Lpss. One can now
compute
the coupling coefficient k using the formula:
k = SQRT [ 1.0 - (Lpss / Lpso) ]
where SQRT means take the square root.
Method 3:
This method measures M instead of K. Place a small 1% precision
resistor (50-100 ohms) in series with your primary coil, and apply 60
cycle
A.C. Measure the A.C. voltage across the resistor, and calculate the
primary current Ip from the expression Ip=V/R, where V is the applied
A.C.
voltage in the circuit. Measure the induced voltage developed across
the
secondary coil (Vs). Now compute M from the relationship Vs=w x M x Ip,
where w is 2 times pi times the operating frequency (60 Hz). This
measurement should be done at a low frequency, well below the resonance
of
the coil. Tesla used this method at Colorado Springs to measure M.
Method 4:
If you are using tight coupling (K > 0.2) you can use an
oscilloscope to find the two resonance peaks due to frequency splitting.
Tune the primary and secondary so their resonances match. Then drive
the
primary with a signal generator and monitor the signal of the secondary
with
the scope (using a probe with 1/2 meter of wire dangling several feet
away
from the coil). Vary the frequency around resonance (Fres), and you
will
see two peaks, one at frequency F1 below Fres, and another at frequency
F2
above Fres. If Fres is the natural resonant frequency for both coils,
then:
Fres=sqrt[2 / (1 / F1^2 + 1 / F2^2) ] (reality check)
and
K = 1 - 2 / [1 + (F2^2/F1^2) ]
Flame away!
Mark S. Rzeszotarski, Ph.D.