Cecil got me thinking about writing exact expressions for the maximum
reactance for a given SWR, or equivalently for a given magnitude of
reflection coefficient. Barring clerical errors (and I've checked a
couple sample cases that seem OK), the following should be true.

Given r = magnitude of reflection coefficient in a system with a
real-valued reference impedance Z0, the maximum reactance will occur
when the reflection coefficient is

Rho = [2*r^2 + j*r*(1-r^2)]/(1+r^2)

Minimum reactance is obviously the complex conjugate of that.

The corresponding impedance is

Z = Z0 * (1+r^2 + j*2*r)/(1-r^2)

This may be commonly available in texts, or otherwise well known, but
I don't recall seeing it before, so thought I'd post it.

(Maximum and minimum resistance should be obvious)

Cheers,
Tom

Tom Bruhns wrote:
"Minimum reactance is obviously the complex conjugate of that (Rho)."

Fine, but from Terman`e Fig. 4-7 on page 96 of his 1955 edition it
appears reactance is maximum nearly everywhere except at 1/4-wave
intervals starting at 1/4-wave back from the antenna tip. It flips from
inductive to capacitive or vice versa abruptly at these resonant points.
Phase jumps 180-degrees at each minimum as shown in Fig.4-7b, on page
96.

Terman says on page 94 of his 1955 edition:
"When the load impedance does not equal the characteristic impedance,
the phase of the resulting line voltage (or current) then oscillates
around the phase of the voltage (or current) of the incident wave, as
illustrated in Fig. 4-5. The phase shift under these conditions tends to
be concentrated in regions where the voltage (or current) goes through
a minimum; this is increasingly the case as the reflected wave
approaches equality with the incident wave."

Terman refers behavior of antennas to this transmission line behavior.
The standing wave has loss in an antenna as in a lossy transmission
lline, but loss in an antenna includes radiation in addition to loss
from conversion to heat. At the open-circuit end of a radiating element,
there is a reflection due to the abrupt impedance change at the antenna
tip. Maximum SWR must occur at the antenna tip.

On page 97, Terman says:

S = 1+absolute Rho / 1-absolute Rho
S = SWR
Rho = reflection coefficient

or

Absolute refl;ection coefficient = S-1 / S+1

Terman says:
"This relationship is illustrated graphically in Fig. 4-9."

Fig. 4-9 shows SWR=1 when the absolute value of Rho=0 (no reflection),
and SWR=20 when the absolute value of Rho=0.88.

In the case of an open circuit antenna tip, Rho must approach 1.0 unless
the tip is located near ground on an antenna such as an inverted Vee,
for example, which has its tips near the earth. Capacitive coupling may
account for continuation of current to some extent through the gap.

Terman says on page 98:
"The expression "Transmission-line impedance" applied to a point on a
transmission line signifies the vector ratio of the line voltage to line
current at that particular point. This is the impedance that would be
obtained if the transmission line were cut at the point in question, and
the impedance looking toward the load were measured on a bridge."

The impedance is high at an antenna tip and low 1/4-wave back from the
tip. Impedance is a voltage to current ratio. Voltage is high where
impedance is high. Current is low where voltage is high, and vice versa.

Edmund A. Laport in "Radio Antenna Engineering" shows a use for the
antenna Q number to determine if the support insulators for a horizontal
dipole with a shunt (delta) center feed is likely to break down from
too-high voltage.

Current is sinusoidal, starting from the virtual (full 1/4-wave from the
feed) tip end, that is to ignore the missing 5% or so for end effects,
and to start the sine distribution at where the tip end would be had the
antenna not been discounted. Maximum current is at the center of the
dipole. Maximum current departs from sinusoidal distribution within
about plus or minus 10% from the middle of the dipole, but it`s no big
deal.

"---the voltage to ground at the end of a dipole Vm will be QVa/2, where
Va is the balanced voltage applied to the central feed point. Va=IaZa,
the the antenna current at the feed point times the feed point
impedance."

The balanced antenna driving voltage in Ed`s example on page 241 is:
Va=IaRa=7.25x95=690 volts rms unmodulated.
The potential from one end of the dipole to ground is then:
Vend=QVa/2=3950 volts
Obviously, Q in the calculation was 11.44
This 3950 volts is the working potential for strain insulators which
will support the dipole. To allow for 100% amplitude modulation, this
value must be doubled.

Q is important as the multiplication factor for the antenna feedpoint
voltage to find what voltage is to be expected at the open-circuit end
of an antenna element.

Best regards, Richard Harrison, KB5WZI