A Low Cost RF Impedance Analyzer
and Rm is known precisely.
Impedance errors will occur when
either Vz or VRM is very small, corresponding to large or small load impedances in which case the accuracy will
suffer. Impedances near 50 ohms
(where the voltages are roughly the
same) and below an SWR of about 4:1
should be calculated fairly accurately.
Calculation of the cosine of the
impedance angle θ is another story.
Theoretically, Equations 4 and 7 would
produce identical results. But practically speaking, there will be a difference
depending upon the load impedance.
What must be realized is that in a
practical circuit there will be measurement errors and they will affect these
two equations differently. Below is a
numerical example to illustrate. More
general comments follow the example.
Consider, as an example, a circuit
such as shown in Figure 3, where R1 =
R2 and Z = Rm = 50 ohms (
corresponding to an SWR of 1:1 at the load
Z). Assume the measured voltages are
VRV = 0.000 V, VRM = 1.000 V, VZ =
1.000 V, and VS = 1.999 V (which has
a negative error of 1 mV) since VS
would equal 2.000 volts in a system
with SWR of 1:1. Calculating from
Equation 4 yields cos(θ) = 0.998 while
Equation 7 yields cos(θ) = 1.000 as it
should. So, we have incurred an error
of 0.2% with the first calculation and
an error in the angle of 3. 6 degrees but
there was no error with Equation 7.
Note, too, that the error can
conspire to make the cos(θ) >1, which
is theoretically impossible for the
cosine function. To illustrate, consider
the above case with Vs = 2.001 V (a
positive 1 mV error) and calculate
cos(θ) from Equation 4 as 1.002.
Now consider the same example
except that VRV = 0.005 V (a positive 5
mV error) and VS = 2.000 V (no error).
Calculation from Equation 4 yields
cos(θ) = 1.000 while Equation 7 yields
cos(θ) = 0.9999. So, we have incurred
only an error of 0.01% with the second
calculation and an error in the angle of
0.81 degrees but with an assumed,
much larger, measurement error.
Clearly then, as shown above,
Equation 7 is less sensitive to errors for
this particular case of a resistive load.
Of course, this is not rigorous proof, but
other examples at higher SWRs show a
similar advantage. The author has performed a sensitivity analysis of cos(θ)
with respect to VRV and Vs in Equations
4 and 7, respectively. Sensitivity analysis
refers to the sensitivity of a circuit
parameter or value to errors in the
component or some other value such
as voltage. The mathematical details are
too long to present here, but the results
show that Equation 7 has the least
sensitivity to errors when the load is
resistive and at low SWR. Therefore, the
circuit based on Equation 7 is chosen.
I hope that from the above
discussion you can appreciate the
importance of accurate measurements.
As a further illustration, consider using
an eight bit analog-to-digital converter
(A/D) to measure the voltages in the
above numerical example. Since we
have a maximum voltage of two volts
and 256 levels with an eight bit unit,
each level corresponds to 7. 81 mV.
Clearly, only a one level mistake may
cause a moderate angle error. On the
other hand, modern digital multimeters
have counts (levels) ranging from 2,000
to over 50,000 and 3-1/2 or more digits with 0.5% or better accuracy. This
should provide a fairly economical and
accurate means of collecting the voltage data. (More details are discussed
later on.) Now let’s take a look at how
to measure the voltages required with
an ordinary digital multimeter.
Diode Sampling Circuit
A circuit that is commonly used to
convert AC voltages to DC for measurement is the “diode voltage sampler”
(DVS) shown in Figure 5. In this case,
we want to find the voltage Vs across R.
Here’s how it works. The positive, peak
AC voltage across R is rectified by
diode D and charges capacitor C.
Once the capacitor is charged to about
the peak value of Vs, it no longer places
a load on R except for the minute
amount of current needed to keep it
charged. A high input impedance DVM
can read the voltage on C which should
be close to the peak value of VS.
Of course, for maximum accuracy,
we should add the value of the diode
voltage drop VD to the DVM reading
Vm. If VS is large, the diode drop is
often neglected or considered a small
constant of say, 0.7 volts, for a silicon
diode. And as long as the diode
voltage drop is small compared to Vs,
this works reasonably well. But if VS is
smaller, say, 1.5 volts peak, the diode
drop needs to be estimated better. And
for even smaller voltages — as we might
find in our impedance measuring
circuit — it becomes critical to know
the exact value of the diode drop to
compensate our DVM reading.
A method that I have used with
some success employs a mathematical model of the diode in the PC software. Knowing the diode parameters
enables better compensation by using
the model to estimate the diode
voltage drop. Unfortunately, it is not
perfect as it still leaves room for error
due to manufacturing tolerances and
capacitance effects. These errors can
be ameliorated somewhat by
calibrating the model with known
■ FIGURE 5. Diode
February 2008 41