**The
Spark Transmitter. 4. Inductive Coupling of Tuned Circuits.**

**"All is riddle, and
the key to a riddle is another riddle."**

**R.W. Emerson**

Will we ever get to the load? Well, it's getting closer . . . trust me, I'm a doctor.

**Inductive Coupling.**

Having generated power at rf, it is necessary to transfer it from the primary circuit where it originated to the secondary circuit where it will do something useful. Remembering the requirements of the maximum power theorem, it is clear that the conditions required are those of an impedance match.

In
the diagram, the primary and secondary coils are coupled inductively
by means of a mutual inductance M. This has a reactance at the
operating frequency X_{m}. For perfect matching, the
resistance coupled into the primary circuit must be equal to the
resistive load which the generator expects to see. If the load is
itself reactive, then to get a perfect match two out of the three
reactances X_{p}, X_{s}, and X_{m} must be
variable. Fortunately, if primary and secondary are tuned circuits at
resonance, things become greatly simplified and we have the
possibility of obtaining a perfect match purely by varying the mutual
inductance coupling. This we do according to the equation:

where R_{p} and R_{s}
are the resistances associated with both circuits. (The value of the
mutual inductance M can then be calculated if we know the frequency
of operation.) Well, what exactly are these resistances? For the
primary circuit, R_{p} is the resistance associated with the
generator (remember the maximum power theorem!)

But what is R_{s}? That is a
harder question to answer. It also seems likely, for a Tesla coil
secondary, that it will vary according to whether a spark is being
produced from the top electrode or not. At least with an aerial it
will be constant. With a resonant aerial in fact it's fairly easy,
being equal to the sum of the resistive losses in the aerial, the
ground and the radiation resistance. The radiation resistance is a
fictitious resistance, which if it was included in the aerial would
cause as much power loss through heating as the loss of power due to
radiation from the aerial. So in the case of a Tesla secondary, what
we need here is a "spark resistance" which would cause the
same amount of power loss as is caused by spark discharge from the
top electrode. It is by no means easy to see what value this should
be. We can, however, state with certainty that it is unlikely to be very useful trying to reduce the rf resistance of the secondary to a value much below that of the ground connection, as the two act in series.

Fortunately, we can be ignorant of the exact requirements here and adjust the mutual inductance coupling by the physical separation of the primary and secondary coils and, according to time honoured wireless practice, "tune for maximum smoke"! [You can usually tell when you've damaged an electronic component because the smoke they put in it at the factory leaks out. I don't know who it was who invented smoke as a means for indicating faulty components but all I can say is, it's a jolly clever idea and I wish I had the patents on it.]

One of the reasons (by no means the only one, nor, as it happens, anything like the best one) why Tesla secondaries give bigger sparks with larger top capacitances is because there is often insufficient mutual inductance coupling for perfect impedance matching (this may be because decreasing the separation between the coils causes sparking into the primary) and when the top capacitance of the secondary increases, the resonant frequency drops, more turns are needed on the primary (which may be too small in relation to the primary capacitance) and the effective size of the primary is increased, thereby increasing the proportion of input power converted to rf, and increasing the mutual inductance and improving the match. It may also happen that reducing the secondary reactance (corresponding to the drop in frequency) improves the impedance match and we get a better power transfer from that cause too. (We'll come to the best reason for big secondary capacitances later.) But there's more.

The mutual inductance coupling between primary and secondary can be related to their self-inductance by means of the coupling constant k:

Notice that since k is defining the relationship between magnetic flux linkages in the circuit, it can never be greater than 1. A value of 1 means that all the flux produced by the primary is linked with the secondary and vice versa. A value of k greater than 1 would mean that more than all of the flux produced by the primary is linked with the secondary and thus values of k greater than 1 (and I have seen people claim it!) means you have a problem! In fact, k = 1 is never achievable! The closest you are likely to get is in the output transformers of high quality valve amplifiers where primary and secondary are split into interleaved windings, and in specialist types of instrument transformers where the construction is similar. Power transformers used for supply distribution are also quite good, fortunately for the supply companies and the end user. Neon sign transformers and welding transformers are examples of designs where the value of k varies considerably with the load and is sometimes a lot less than 1.

Transformers of this latter type are called "magnetic leakage transformers" because the design is such that a large proportion of the flux generated by each coil can escape from the magnetic circuit associated with the other coil. Under load, the proportion which "leaks" increases. This gives intentionally poor power regulation and ensures that when a short circuit is placed across the secondary (the striking of the welder's arc, the conductive breakdown of the neon gas, or the flashing over of the primary spark gap) the output voltage is suddenly reduced until the "fault condition" is removed. Mr. Melville Clapp-Eastham in the USA can be credited with the introduction of this type of transformer for spark transmitters, and his Model E transformer has a prominent place in wireless history. Similar results can be obtained from a power transformer of the closely coupled type if there is an external inductance (choke) in series with the primary and this external choke provides the necessary "leakage inductance".

The coupling constant is independent of the number of turns in a coil. The number of turns in a coil determines the magnetic field which will be produced for a given current. The coupling constant is concerned with how the lines of magnetic force produced by one coil interact with another coil, and hence the coupling constant between two air spaced coils depends only on their physical size and disposition in space. Hence to obtain the best coupling between primary and secondary in an air-cored transformer we can only change the size and spatial relationships of the coils. With a tapped coil it may be noticed that changing the tap position changes not only the self-inductance but also the coupling constant. This is of course because when the tap is moved to a different position, the effective size and spatial relationship of that coil are changed as well as its self-inductance.

When we have the critical coupling,
which exists when the voltage output is optimised, then we have an
additional relationship between k_{c} the critical coupling
constant and the Q values of primary and secondary:

The value of coupling constant is
important in a spark transmitter because the tightness of coupling
determines the rate at which the primary loses power to the
secondary, and hence determines decrement, damping, sharpness of
tuning (loaded Q) and intensity of current at resonance (and hence
secondary voltage in a Tesla coil.) Remember those nice graphs
showing the logarithmic decrement and loaded Q? The graph of
decrement d = 0.09 and loaded Q =
34.6 corresponds to the critical coupling constant having a value of
k_{c} = 0.17, which, from the records left by the old-time
spark wireless operators, is around the maximum which can be used
with a quenched spark gap of multi-plate construction. Hence for a
critical coupling constant of k_{c} = 0.17, the product Q_{p}Q_{s}
must be 34.6. We can of course split that product between a wide
range of possible Q_{p} and Q_{s} values! If both are
equal to Q = 5.88 (the square root of 34.6) the decrement of each
circuit individually is given by the graph of d
= 0.53.

The diagram shows the effect of varying the coupling on the frequency distribution (read 'logarithmic decrement') of a spark transmitter. As the coupling is increased much beyond 20%, k = 0.2, the frequency spread increases dramatically, indicating that the logarithmic decrement has increased and that loaded Q has decreased. The square of the current, plotted on the y axis, also plummets drastically.

The next diagram shows that, in order to get the highest possible secondary current, the primary and secondary circuits have to be slightly detuned. In each case the primary circuit remains tuned to a wavelength of 650m, whilst the aerial circuit (secondary) is varied from 500 - 650m. The best result is for 585m. This was obtained in an experimental test circuit chosen to demonstrate the effect clearly, and the best detuning is here about 11%. For the average aerial and coupling k=0.17, the detuning was normally about 3%.

Curves like these are recorded by
coupling an rf ammeter into the circuit. I have seen circuit diagrams
in which the meter is placed directly in the primary circuit, but
evidently these were low voltage circuits (possible with the quenched
gap which will operate on just a few hundred volts) and it is more
usual to couple the meter to the aerial circuit and then indirectly
by means of a coupling loop. A thermocouple ammeter would be the
instrument of choice, but these curves were recorded most likely with
a hot wire meter, whose deflection is proportional to the square of
the current - hence the plot of I^{2} on the y axis.

The mutual inductance coupling ensures that everything critical to the operation of the spark transmitter (or Tesla coil) is dependent on just about everything else, and that is why trying to find the global optimisation for a Tesla coil to give the biggest spark for a given input power is so very difficult. It isn't that we have such an enormous number of variables - it's the interdependence of all of them simultaneously on each other!

All of which goes to show how very complicated the inductively coupled spark transmitter (or Tesla coil) really is. It's a nice demonstration of the fact that there is not necessarily a direct correlation between the number of components in a circuit and its complexity of operation. The spark transmitter circuit is one of the simplest - just seven components (power transformer, primary capacitor, primary inductance, primary spark gap, secondary inductance, secondary capacitance and mutual inductance) and yet a detailed description of its operation would require a lot more space than this and cartloads of higher mathematics. Any electrical circuit can be broken down into just four fundamental 'units' - inductance, capacitance, resistance and mutual inductance - and with just seven components, this circuit has the lot.

As a mere radio ham tinkering outside my sphere of professional competence I can only scratch the surface. I am left gasping with admiration at the achievements of the old timers who built and operated spark transmitters and Tesla coils often without a clue as to the frequency of operation or technical knowledge much above Ohm's law. They did it, of course, by a combination of knowing inside out what there was to know, by meticulous method and by sheer patience and dogged determination.

Oh, by the way. We have now arrived at the load.

We're not through yet though. The next section looks at transmission lines and magnifier circuits.