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Mathematical Basis for k=0.6

Mark Molloy

Department of Electrical Engineering, University of South Florida

4202 E. Fowler Ave., Tampa, FL 33620

Conditions required to achieve the maximum voltage at

the secondary circuit of a Tesla transformer were first pointed

out by Drude1 and consisted of a unitary tuning ratio and a

coupling coefficient of 0.6. From that point, the search for an

optimal working point has evolved along two axes.

Targeting the maximum output voltage, Reed has observed

that an 18% increase can be obtained by using a tuning

ratio less than unity and a suitable amount of coupling.

His work has been generalized by Phung et al., providing a

set of equations in order to calculate all tuning ratio and

coupling coefficient pairs that achieve a local maximum

output.

Following an alternative track, Finkelstein has identified

the general conditions required for a complete energy

transfer from the primary to the secondary circuits: in all

cases, a unitary tuning ratio is required. The Drudes conditions

achieve complete energy transfer in the least time, but

other values of coupling coefficient can be used as well,

while the transfer completion is simply moved to a later time

instant. Finkelsteins work was continued and extended to

three coupled resonance circuits by Bieniosek and eventuallygeneralized to any number of circuits by de Queiroz.

-Marco DenicolaiHigh Voltage Institute, Helsinki University of Technology, P.O. Box 3000, FIN 02015 HUT, Finland

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This analysis follows Marco Denicolai's Master's Thesis (available online).

Marco states that his final result for the output voltage (across C2) is only possible in closed

form for the case of no damping (R1=R2=0). He ends up with the equation below for v:

and are the uncoupled resonant frequencies of the primary and the secondaryrespectively. Tis the tuning ratio, defined as the square of the ratio of the uncoupled resonance

frequencies, while V1 is the initial voltage across C1.

w1 and w2 are the resonance frequencies of the primary and secondary circuits when coupled.

v(t) reaches a maximum when the arguments of both sinusoids are equal to /2+x

where x=n,m and n and m are integers:

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We can set n=0 and solve the two equations for w2/ w1

Marco approaches his analysis from a very general case. For our case we only care about the

most common condition where the tuning ratio T=1.

For this case w2 and w1 reduce as follows:

and from above:

Solving for k:

Let's review how we got to this point:

We found an expression for the voltage across the output capacitor in terms of a

product of two sinusoids, the frequency of the first sinusoid is the sum of the coupled primary

and secondary frequencies and the frequency of the second sinusoid is the difference of the

coupled primary and secondary frequencies. Perhaps these frequencies (sum and difference)

are the two strong spikes that appear in a frequency sweep of such a circuit.

We then found the possible arguments of the sinusoids that would result in a maximum

output. These arguments are integer multiples ofadded to /2. We then took the case where

the tuning ratio, T is equal to 1 and found expressions for w2/ w1 in terms ofkand also in terms

ofm. We then set the expression in mequal to the expression in kand solved for k in terms ofm.

These are the k values associated with maximum voltage gain.

for m=1 k=0.6

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Therefore, k=0.6 is the coupling coefficient associated with the fundamental coupled frequency

where the output voltage is a maximum. QED.

I also noticed that k=0.6 is the largest k value one can get with this analysis.

It starts at k=0.6 for m=1 and then k gets smaller for larger m.

Physically a coupling coefficient of k=1 is possible. So then why does k=1 not result in themaximum voltage across the secondary?

Maybe because the mutual inductance increases the resistance of the PRIMARY

as well as increasing the voltage across the SECONDARY and current flow in the primary (not

voltage) is the vehicle by which core flux is created. Maybe k=0.6 is some happy medium

between voltage across L2 and resistance in series with L1.