BY GEORGE R. STEBER WB9LVI
■ FIGURE 2. Series and parallel
equivalent circuit models at the
same measuring frequency.
alone and is typically expressed as a
complex quantity such as R + jX where
R is the resistive part (real) and X is the
reactive part (imaginary). The reactive
part is usually frequency dependent
and is preceded by the math operator
“j” to indicate it is the imaginary part.
For many resistors, the reactive part is
very small, except for wire wound
types. And so, it is typically ignored.
On the other hand, many inductors and capacitors are lossy (as seen
in Figure 1) and have both a resistive
and reactive part with the reactive part
being positive for inductors and
negative for capacitors. Measurements
made on transmission lines and
antennas also show both a resistive
and reactive part which can be either
inductive or capacitive, depending on
the measuring frequency.
The models of Figure 1 show the
resistive part modeled in series with
the reactance. Another model is
sometimes used with parallel resistors
as shown in Figure 2. The series and the
parallel models are equivalent at the
same frequency of measurement and
can be converted from one to the other
using the equations shown on Figure 2.
The impedance analyzer software
(discussed later) basically calculates
the impedance in polar form from
voltage measurements provided to it.
This can be converted to rectangular
form as shown in Equation 1:
Z= Z∠è= R+ jX
where Z is impedance in ohms, Z is
the magnitude of Z, θ is the angle of
Z, R is the resistive part of Z, and X is
the reactive part of Z. The two forms
in Equation 1 are related by:
Z = R2 +X2 and
è = tan−1(X / R)
The impedance analyzer program
will display the results in both polar
and rectangular form on the computer
screen. From here, it is relatively easy
(knowing the measurement frequency) to calculate the values of the series
and parallel models and to display
them on the screen. Other parameters
such as the Q (quality factor), or
D (dissipation factor), SWR, and
reflection coefficient are also
calculated and displayed. Now let’s
see how to measure impedance in
polar form using just a voltmeter.
In recent articles [1, 2], I described
a powerful method of measuring
impedance at audio frequencies using
a simple circuit and a computer with a
sound card. Briefly mentioned was the
“three voltmeter method” (TVM) as
one technique that was looked at
while working on the “least mean
squares” (LMS) method which — by
the way — was found to be superior.
The LMS method would be preferred
here too, but unfortunately it would be
difficult and expensive to realize at
higher frequencies. So the simpler
TVM will be implemented here.
According to the grapevine, some
antenna analyzers on the market today
use an approach similar to the TVM
described here and use a microprocessor for data collection. So, this article
may give you some insight on how
those units work, as well. The TVM is,
■ FIGURE 3.
February 2008 39