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AirCore Transformers
Transformers considered hitherto have had iron or ferrite cores. A class of transformers is widely used in radiofrequency circuits without cores or with small slugs of powdered iron. In a transformer with an iron core, the exciting current required for inducing the secondary voltage is a small percentage of the load component of current. In an aircore transformer all the current is exciting current and induces a secondary voltage proportional to the mutual inductance.
Consider the circuit of Fig. 171 in which Z_{1} is complex and includes the selfinductance of the primary coil. Likewise, secondary impedance Z_{2} is complex and includes the selfinductance of the secondary coil. With a sinusoidal voltage applied, Kirchhoff's laws give the following:
where ω = 2π times operating frequency, and L_{m} is the mutual inductance between the primary and secondary coils.
where X_{M} = jωL_{m}. In the above formulas, the impedances Z_{1}, Z_{2}, and Z' are complex quantities whose real and imaginary terms depend upon the values of resistance, inductance, and capacitance in the circuit. One common practical case arises when the primary resistance is zero, or virtually zero, and the secondary coil is tuned to resonance so that Z_{2} is a pure resistance R_{2}. Under these conditions, equation 102 reduces to
where R' is the equivalent resistance in the primary. Equation 103 gives the value of mutual inductance required for coupling a resistance R_{2} so that it will appear like resistance R' with a maximum power transfer between the two coils, and states that the mutual reactance X_{M} is the geometric mean between the two values of resistance. The ratio of mutual inductance to the geometric mean of the primary and secondary selfinductances is the coupling coefficient:
The value of k is never greater than unity, even when coils are interleaved to the maximum possible extent. Values of k down to 0.01 or lower are common at high frequencies. Coupling coefficient is related to untuned transformer open and shortcircuit reactance by means of the transformer equivalent circuit shown in Fig. 107(a), p. 147. Assume that the transformer has a 1:1 ratio, and leakage inductance is equally divided between primary and secondary windings. Then if L_{1} and L_{2} are the selfinductances of primary and secondary, respectively, L_{s} is the total leakage inductance (measured in the primary with secondary shortcircuited), and L_{m }the mutual inductance,
From equation 104,
If L_{m} >> L_{s}
Equations 104 (a) and (b) are useful in estimating approximate transformer band width. A tuned aircore transformer often used in receivers is shown in Fig. 172.
Here a sinusoidal voltage E_{1} may be impressed on the primary circuit by a vacuum tube amplifier. Resistances R_{1} and R_{2} are usually the inevitable resistance of coils, but occasionally resistance is added to change the circuit response. The value of voltage E_{2} obtained from this circuit depends on the impressed frequency; in Fig. 173 it is shown for resonance at three different values of coupling. If the value of coupling is such that
we obtain a condition similar to that of equation 103, in which the maximum power or current is produced in the secondary circuit. Maximum current through condenser C_{2} gives maximum voltage E_{2}. This value of coupling is known as the critical value. Smaller coefficient of coupling gives a smaller maximum value of E_{2}. Greater coefficient of coupling results in a "double hump" as shown in Fig. 173.
The heights of resonant peaks and frequency distance between peaks depend upon circuit Q and coefficient of coupling k. The double hump curve of Fig. 173 is desirable because, with modulated waves, frequencies in adjacent channels are rejected; yet very little attenuation is offered to audio frequencies which effectively add or subtract from the carrier frequency normally corresponding to resonance. Close tuning control and high Q are essential to good response and selectivity. If the primary circuit is made to resonate at a different frequency from the secondary, audio response is much worse, and considerable distortion is likely. Moreover, the response at mean frequency is less than it would be if the circuits were properly tuned. Aircore transformers are usually made adjustable for tuning and coupling.


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