“Propagation Delay” is defined as the length of time it takes for a pure traveling wave to travel from one end of a wire, transmission line, or inductor to the other end. If reflected energy is present, propagation delay is NOT equal to the phase delay. In fact, in the presence of reflections, phase delay measurements often yield faster-than-light signal velocities which are, of course, impossible. It is very difficult to measure a valid propagation delay in the presence of reflections.
Phase Propagation Factor  (Axial Propagation Factor), β, is the velocity of propagation of an EM wave through a medium such as a transmission line or an inductor (loading coil), in radians per meter. The phase propagation factor through free space divided by the phase propagation factor in a medium is the velocity factor, VF, in that medium. Both the propagation factor, β, and the attenuation factor, α, appear in the exponential equations for RF waves.
The formulas for calculating transmission line characteristics from impedance measurements are taken from Chipman . If a transmission line stub is 1/8λ (45 degrees) long, the measured short-circuit impedance, Zsc, will be (approximately) equal to the measured open-circuit impedance, Zoc.
The resistive portion of Zoc and Zsc is often very small compared to the reactive portion and thus can be considered negligible such that:
Thus one of the rules of thumb for 1/8λ pieces of transmission line is: The characteristic impedance, ZØ, of a 1/8λ stub is equal to the absolute value of the reactance. This applies to low-loss stubs where |R| is much smaller than |X| which is the usual case. Additionally, for any length of stub:
Note that the results of measuring the very high impedances associated with stub self-resonance (around lengths that are multiples of 1/4λ) are suspect because of large deviations in measurement accuracy and precision. None of the following measurements are near a multiple of 1/4λ.
The AIM4170 impedance measurements were performed on a simple "300" ohm piece of twinlead. The following graphic is the result of a 10 MHz to 24 MHz frequency scan to determine the open-circuit impedance (Zoc) and short-circuit impedance (Zsc) of a 69 inch piece of product #52 from The Wireman (advertised as 300 ohm window line). Note: A calibration procedure was performed to calibrate out the effects of the balun between the AIM4170 coax connector and the balanced feedline.
Where the Zoc line intersects the Zsc line is the frequency at which the 69" length of twinlead is 1/8λ (45 degrees) long and that happens at 17.67 MHz. Where Zoc=Zsc is also the characteristic impedance, ZØ, of the twinlead. So we know ZØ=274 ohms and the stub is 45 degrees in length at 17.67 MHz. From that data, we can calculate the velocity factor, VF, of the feedline to be 0.826=69"/83.5" which is 1/8λ of the twinlead divided by 1/8λ in free space at 17.67 MHz.
The characteristics of the stub were also measured at 7 MHz. The open-circuit impedance is 0−j873.8 ohms and the short-circuit impedance is 2.4+j86.87 ohms. So ZØ=SQRT(Zoc*Zsc)=275.6 ohms. The length of the stub is ARCTAN(Zsc/ZØ)=17.5 degrees. If we calculate the length at 7 MHz based on the 45 degree length at 17.67 MHz, we get 17.8 degrees so the results of all measurements are very close to the expected values.
Like a transmission line, an air-core inductor is associated with a characteristic impedance, ZØ, and a propagation factor, β, whose dimensions are radians per meter, from which a velocity factor, VF, may be calculated. Unlike a transmission line, these inductor parameters are not constant with changing frequency.
According to the Corum paper , Figure 3, If phase is important, any coil electrically longer than approximately 15 degrees should be analyzed using the distributed network model, not the lumped inductor model, because the current past 15 degrees is not uniform and the voltage rise is non-linear. Given that fact of physics, the question of the number of electrical degrees occupied by an inductor (loading coil) in an antenna system is of utmost importance and cannot simply be ignored as it has been in the past through the zero-length presumptions of the lumped inductor model. (Using the lumped inductor model to argue that a loading coil is a lumped inductor is a Catch-22 type of argument.)
The Hamwaves Inductance Calculator  bases its outputs on the principles of physics presented in the Corum paper . Like transmission line formula development, the length of the coil is considered to be infinite such that reflections never occur. In addition to the inductance results, the calculator also yields the characteristic impedance, ZØ, in ohms and the propagation factor, β, in radians per meter (from which the velocity factor, VF, of the inductor can be calculated).
Turns Zoc, ohms Zsc, ohms ZØ ohms (calc) Degrees(calc)
7T 0+j153.9 867−j7236 1059 8.3
22T 0+j650.3 470−j3405 1495 23.5
49T 0+j1866 214−j1316 1577 49.8
The graph of Zoc and Zsc vs Frequency for the 80m band turned up an interesting fact. Zoc was equal to Zsc at 3.82 MHz indicating that the 75m Texas Bugcatcher loading coil is 1/8λ long, i.e. 45 deg long, at 3.82 MHz. An electrical length of 1/8λ is certainly long enough to exhibit transmission line characteristics. Contrary to numerous assertions and publications, the loading coil is obviously NOT acting as a lumped inductor at that frequency. In fact, this author is willing to assert that: All 75m mobile air-core loading coils exhibit transmission line characteristics which fall outside the boundary conditions presumed by the lumped inductor model. Here is a graphic of the AIM4170 scan for the Texas Bugcatcher coil.
Note the similarities to Fig. 1, the waveforms for the twinlead stub. For this author, the conclusion seems inescapable: When the 75M Texas Bugcatcher loading coil is used on 3.82 MHz, it occupies close to 45 degrees of a resonant (90 deg) 75M mobile antenna. Of course, the mobile environment surrounding the Texas Bugcatcher loading coil can cause a change in the characteristics of the coil (but not enough to satisfy the lumped inductor model).
The 6.625 inch Texas Bugcatcher coil is 45 degrees long at 3.82 MHz which, in free space, is 386.4 inches. Therefore, the velocity factor, VF, for this loading coil is 0.017 which is characteristic of Slow Wave Structures, described by Ramo, Whinnery, & Van Duzer , an earlier version/edition of which was the author’s EE fields and waves textbook at Texas A&M back in the 1950’s.
The classical current "droop" at the top of a loading coil is primarily the result of interference between the forward current and the reflected current at that point (see Fig. 5 below). The alleged "droop" actually has little to do with current attenuation and more to do with interference between the forward and reflected current. If the antenna was one half wavelength long and the loading coil was positioned differently, the current at the bottom of the coil could be greater than the current at the top of the coil. In fact, it is possible to design a system in which the current is flowing into the bottom and into the top of the coil at the same time. Such is the nature of distributed networks and does not violate any of the laws of physics.
Assuming zero losses and equal forward and reflected currents, here is what happens to the total (superposed) current through the base-loaded Bugcatcher Coil used in a mobile antenna on 75m.
Such is easy to observe using EZNEC. VERT1.EZ is a file shipped with EZNEC. It is a 33 ft 1/4 wavelength monopole on 40m, i.e. it is electrically 90 degrees long. The author chose 90 segments for the vertical so each segment is electrically one degree long. With a phase of zero degrees for the source current, the phase in segment 30 is -1.79 degrees, in segment 60 is -2.94 degrees, and in segment 90 is -3.82 degrees. From segment 30 to segment 60 is the middle 30 degrees of the antenna yet the current phase shift is only 1.15 degrees. How can an RF signal travel 30 degrees while only experiencing a 1.15 degree phase shift? The current is not "traveling". Almost all of it is "standing" in the standing waves on the standing wave antenna.
The phase shift of the forward traveling wave is indeed 30 degrees through 30 degrees of wire and so is the phase shift of the reflected traveling wave. But those two phasors are rotating in opposite directions so their phases are opposite in sign. When one adds the two phasors together, he resulting phase is very close to a fixed zero degrees. That’s how the phase of the current can be -3.82 degrees at the end of a 90 degree antenna and is perfectly consistent with standing wave environments.
If we replace the center 30 degrees of the 90 degree antenna with a loading coil, why is a surprise that a loading coil indicates approximately the same amount of current phase shift as in the wire that it replaces, i.e. very few degrees? If RF exhibits a 1.15 degree phase shift in 30 degrees of wire, why should it be very much different in 30 degrees of loading coil when both are in standing wave environments?
2. Another well known amateur radio operator apparently measured the s21 group delay of a large air-core loading coil suitable for use in 75m mobile operation. His conclusions suffer from the same misconceptions as the earlier phase measurements at each end of a loading coil during transmit. If the phase difference at each end of a loading coil in a standing wave environment cannot be used to determine the propagation delay through the coil, then neither can the slope of the phase delay vs frequency (i.e. group delay) be used in a valid manner to determine the electrical length of a loading coil.
The fallacy in such thinking can easily be illustrated by trying to apply the same phase delay and group delay measurements techniques to determine the electrical length of a transmission line stub. That there is virtually no phase delay between one end of an open or shorted stub and the other end is easily proven by bench measurements. The only case where the phase delay in a transmission line is equal to the propagation delay is when the line is terminated in a load equal to its characteristic impedance, ZØ, i.e. when reflections do not exist. That same principle applies to loading coils. The lumped inductor mindset of the above two amateur radio operators seems to be responsible for some technically invalid concepts.
Ramo and Whinnery  discuss phase velocity and group velocity in their classic, well respected series of books the first edition of which was published in 1946. Here are some quotes concerning phase velocity and group velocity. The bold/italic emphasis is this author’s doing.
Page 262: "Group velocity is often referred to as the 'velocity of energy travel'. This concept has validity for many important cases, but it is not universally true." One of the cases where it is not true is for standing wave antennas.
Concerning phase velocities faster than the speed of light page 303: "There is no violation of relativistic principles by this result, since no material object moves at this velocity." i.e. EM (photonic) energy cannot be transferred through a large air-core 75m loading coil in 3 ns.
Page 304: "The concept of a phase velocity, and the understanding of why it may be greater than the velocity of light, is essential to the discussion of guided waves in later chapters …" It is also essential in understanding the velocity of propagation of energy in a standing wave antenna. EM energy simply cannot travel faster than the speed of light.
Concerning the cutoff frequency in a waveguide page 406: "At cutoff (Fig. 8.4b), the waves simply bounce laterally back and forth between planes without any forward progression so that group velocity is zero and phase velocity infinite." That infinite phase velocity statement is also true for a pure standing wave. When reflections are large, as they are on a standing wave mobile antenna, the phase velocity and group velocity measurements cannot be used to determine the velocity of propagation and therefore cannot be used to determine the electrical length of a loading coil.
The previous measurements indicate that the inductive reactance, Zsc, of the Texas Bugcatcher loading coil equals +j2489 ohms at 3.82 MHz, so for resonance we need a whip with a capacitive reactance of −j2489 ohms. (That much of the lumped inductor model is valid.) EZNEC tells us that a whip 82 inches long will have a capacitive reactance of −j2489 ohms thus matching the +j2489 ohm inductive reactance of the loading coil. With only a small error, one can assume that the velocity factor of the 82 inch whip is 1.0 resulting in a calculated electrical length of 9.55 degrees.
45 degrees plus 9.55 degrees equals 54.55 degrees - so where is the missing 35.45 degrees of the 90 degree antenna? Of course, 35.45 degrees is the phase shift at the impedance discontinuity between the loading coil and the whip which is easy to calculate once we know the characteristic impedance of the whip at the top of the loading coil. Note that the phase shift at the impedance discontinuity depends on the ratio of the two characteristic impedances involved, i.e. the phase shift is proportional to ZØcoil/ZØwhip = Z01/Z02 = 2489/419 = 5.94 The 419 ohm characteristic impedance, ZØwhip, at the top of the loading coil is the absolute value of the capacitive reactance of the whip (Xc=−j2489 ohms) divided by the cotangent of the length of the whip in degrees (9.55 degrees).
Knowing the value of ZØcoil and ZØwhip allows us to plot the loading coil inductive reactance and the whip capacitive reactance on a Smith Chart. The upper half of the following Smith Chart is normalized to ZØcoil=Z01=2489 ohms, the characteristic impedance of the coil and the lower half is normalized to ZØwhip=Z02=419 ohms, the characteristic impedance of the whip. Starting at Z=Ø, the 45 degree loading coil traces a curve on the Smith Chart along the outer inductive reactance circle to the value of 1.0. The whip is unterminated, so we plot its length back from infinity at the tip for a length of 9.55 degrees along the outer capacitive reactance circle to a value of 5.94. The "missing" degrees, i.e. the phase shift, θ, at the impedance discontinuity between the coil and the whip (between 1.0 and 5.94 on the Smith Chart) can be calculated from the following formula where ArcCot(|Xwhip|/ZØwhip) is the number of degrees occupied by the whip and ArcTan(Xcoil/ZØcoil) is the number of degrees occupied by the base loading coil.:
Note that 360 physical degrees around a Smith Chart is equal to 180 electrical degrees, i.e. 1/2 wavelength, so the 35.45 degree phase shift angle above is physically double that value, i.e. 70.9 degrees on paper. The "angle of the reflection coefficient" scale on the Smith Chart can be used for finding the phase shift if one remembers to divide that scalar value by two.
We have ignored the resistance values for the Smith Chart graph because the feedpoint resistance at the bottom of the coil is around 25 ohms and when normalized to ZØcoil=2489 ohms, is only 0.01 on the Smith Chart. Likewise, the resistive value looking into the whip is 0.35 ohms and when normalized to ZØwhip=419 ohms, is only 0.0008 on the Smith Chart. So the actual normalized impedance values are 0.01+j1.0 ohms for the coil and 0.0008−j5.94 for the whip.
Note that the phase shift at the ZØcoil to ZØwhip impedance discontinuity depends on the ratio of ZØcoil/ZØwhip. This same Smith Chart analysis can be used on dual-ZØ stubs.  If we take a 45 degree section of ZØ(1)=300 ohm twinlead and connect 9.55 degrees of ZØ(2)=50 ohm coax, we will have created a 1/4 wavelength short dual-ZØ open-circuit stub very similar to the base-loaded 75m mobile antenna above. The phase shift at the 300 ohm to 50 ohm junction will be 35.45 degrees, the same phase shift that occurs at the 2489 ohm to 419 ohm junction in the 75m mobile antenna.
We can conclude from the above analysis that the primary reason that a center-loaded mobile antenna requires a larger inductor than a base-loaded mobile antenna of the same total length, is that the whip on the center-loaded antenna is shorter and therefore has more capacitive reactance to be neutralized by the loading coil. By keeping the same whip length as one changes from a base-loaded to center-loaded configuration at the same resonant frequency, one needs slightly less inductance, not more, for the same resonant frequency.
The author has avoided any discussion of ferrite or iron core toroid-based loading coils which indeed may come closer to satisfying the boundary conditions of the lumped inductor model than large air-core loading coils.
 Corum K. L. and Corum J. F., "RF coils, helical resonators and voltage magnification by coherent spatial modes," Microwave Review, IEEE, Vol. 7, No. 2, Sep. 2001, pp. 36-45
 Inductance Calculator - Below the inductance calculator, the same web page also contains valuable technical and historical information including a link to the above Corum article.
 The ARRL Antenna Book, 20th edition, page 16-7, Fig. 10: "The loading coil acts as the lumped constant." Here’s what the loaded antenna current
actually looks like according to EZNEC/AutoEZ modeling:
 The equation for a pure traveling wave is of the form: I(x,t) = Imax[cos(kx+ωt)]
Note: (given that k and ω are constants) unlike a pure standing wave, the phase of a pure traveling wave is a function of position, x, and time, t, i.e. Phase = f(x,t)
 Ramo, Whinnery, & Van Duzer, Fields and Waves in Communications Electronics, 3rd edition
 Dual-Z0 Shortened Stubs