The Inductor as a Transmission Line.
The inductor is a time-delay and energy trap. A voltage step enters and travels back and forth through the device, with gradual trapping of energy inside. The time taken for the energy leaving the inductor to rise to equal the amount entering is dependent on the mismatch between the characteristic impedances and also the time delay involved in traversing the device. If the mismatch is great, less energy enters or leaves per round trip, so the time taken for the equality of energies is longer and therefore the inductance is greater.
The single turn inductor is a transmission line with an increase in characteristic impedance followed by a short circuit, discussed in the previous chapter. Here we discuss the two-turn inductor.
When a wavefront travelling down an infinitely long cable reaches AB, it breaks up into three parts; a reflection vRC back up the cable, and two transmitted parts. The transmitted parts are the even and odd modes of propagation discussed in the earlier chapter on crosstalk. The voltages and currents on the lines AB and PQ are
The line PQ has a short which clamps its voltage to zero.
Using basic transmission line theory, (1)-(4) can be solved, yielding
Where
.
means the transmitted voltage from the cable to even mode and
is the reflected part from the junction back into the cable. By repeating
terminal conditions for the inductor to cable direction, we get
The far end of the inductor requires special attention which eliminates some mathematics. Understanding the meaning of even- and odd-mode propagation, one can see that the odd-mode is terminated by an open circuit and the even mode by a short-circuit. Since the incident voltages are already in the correct modes and each mode is individually terminated, there will be no transfer between even and odd mode at the far end of the inductor. The odd mode is reflected positively and the even mode is simply inverted.
A computer iteration of the two-turn inductor was written by Michael S. Gibson, and the results confirmed that the two-turn inductor had the expected exponential waveform with four times the time constant for the one-turn inductor. However, the waveform only approximates to an exponential, but is made up of a series of small steps. The waveforms were published in Proc. IEEE[1]
Mike Gibson and Ivor Catt also worked on The Transformer as a Transmission Line.
The time constant, the time for the voltage in the inductor to fall to
of maximum, is
, where R is the characteristic impedance of the cable
. N is the number of iterations of the program, or traverses of the inductor,
needed to reach
. So the inductance is
, where T is the time taken to traverse the inductor once.
The quadrupling of the inductance from a one-turn inductor results from the difference in the characteristic impedance of the even and odd modes. At low mismatches of even and odd mode (when they are approximately equal), the inductance is only twice that of a one-turn inductor of the same length since the lines are uncoupled (and "crosstalk" is minimal). In this case, the two-turn inductor behaves like a single turn of twice the length. But at high mismatches between even and odd modes, that is, when the lines AB and PQ are close together, the lines are tightly coupled and the modes are trapped. This trapping of the modes increases the time taken to reach the steady state (a short circuit), and is detected as an increase in inductance.