Figure 1.
Specific example to illustrate Drude's prediction.
(a) Physical setup of primary and secondary inductors with dc self-inductances and
. (b) Low-frequency equivalent circuit with reciprocal mutual inductance. (c) Voltage and current spatial profiles for the first self-resonance of the secondary solenoid. (d) Equivalent circuit for frequencies near this resonance with nonreciprocal mutual inductance. This circuit is a simplification of a more complete circuit derived from a distributed-element model treating the solenoid as a transmission line, which is shown in a later figure. All parameters are defined in the text.
Figure 2.
Uniform transmission line described by the Telegrapher's equations (2).
(a) Voltage and current conventions and relation to Fig. 1. (b) Distributed-element model equivalent to (2). The equivalent circuits in all other figures are derived from this model.
Figure 3.
Equivalent circuits for the line and external system.
(a) Lumped-element model equivalent to (8) for the spatial mode. (b) Lumped-element model for a two-terminal port coupled inductively or capacitively with the line. For direct coupling,
and
. The inductance
is present only for the specific example. When the line and external system are coupled, the circuits in (a) and (b) combine according to the relationship between the lumped sources.
Figure 4.
Lengthening argument for the misrepresentation of energy.
For and no coupling, note that two constraints determine
and
: (i) the resonant frequency
, and (ii) the impedance
. As sketched for
, lengthening by one or more wavelengths does not change (i) or (ii), thus neither
or
. For fixed
and
, the energy (12) modeled by the circuit in Fig. 2(a) also does not change. However, the stored energy (11) must increase, so this circuit misrepresents energy.
Figure 5.
Exact equivalent circuit for the specific example.
The coupling with the primary inductor of Fig. 3(b) stitches together the lumped-element models of Fig. 3(a) into this single circuit. The narrowband circuit in Fig. 1(d) is an approximation of this circuit that ignores losses and the contributions of the modes
, which are nonresonant for frequencies near the fundamental
.
Figure 6.
Transmission lines with standard equivalent circuits.
These circuits are formed by external coupling stitching together the circuits of Fig. 3, as described in the text. (a) Open-circuit line from direct bottom coupling. (b) Closed-circuit line from direct top coupling. (c) Segment of line from direct top and bottom couplings, such as a coaxial cable with BNC terminals. Note that this circuit is not standard, but follows from the text. (d) Analog of loop-coupled microwave cavity from inductive coupling, or reciprocal version of Fig. 5. (e) Analog of probe-coupled microwave cavity from capacitive coupling.