On Lower Bound Antenna Efficiency Measurements
in a Reverberation Chamber
Jason B. Coder #1 , John M. Ladbury #2 , Mark Golkowski ∗3
#
National Institute of Standards and Technology
325 Broadway, Mail Stop 687.02, Boulder CO 80305
1
2
∗
jason.coder@nist.gov
john.ladbury@nist.gov
Department of Electrical Engineering, University of Colorado Denver
1205 5th St, Denver CO 80204
3
mark.golkowski@ucdenver.edu
Abstract—This paper addresses a few specific aspects of measuring the lower bound of antenna efficiency in a reverberation
chamber. While the initial method for measuring the lower bound
of efficiency has been presented, three key revisions are discussed
here: (1) an updated notation, (2) a revised method for calculating
the lower bound of efficiency, and (3) a new method for combining
stirring techniques. The updated antenna model notation is
designed to be more general and applicable to situations with n
antennas. The revised efficiency calculation targets an issue of the
original method where the minimum bounding circle exceeded
the unit circle. Introducing a new method of combining stirring
techniques addresses a weakness of the original model. For the
model to work well, it needs a very large number of paddle
positions that generate a good statistical approximation of the
environment (in this case, a reverberation chamber). As a possible
remedy to this weakness, we propose a different way of combining
stirring techniques.
I. I NTRODUCTION
A new antenna model was initially presented in [1]. This
model is different from others in that it models the antenna
and it’s imperfections as an unknown two-port network. An
extension of this model was presented in [2] where it was
shown that the model can be used to calculate the lower
bound of antenna efficiency. The initial model and application
to antenna efficiency have been demonstrated, and a few
improvements can be made to the underlying model and
efficiency calculation to improve it’s overall usability.
First, the notation originally developed with the model
revolved around the S-parameter nomenclature. This proved
useful because we typically describe unknown two-port networks using S-parameters. The downside of the S-parameter
notation is that it becomes very cumbersome when the network
expands to multiple antennas in an arbitrary environment.
Here, we outline a new notation that attempts to overcome
this issue. This change in notation does not modify any of the
mathematical formulae or functions of the two-port antenna
model.
The second issue addressed is the calculations performed
on an antenna with a significant impedance mismatch. Using
the efficiency calculations presented in [2], the lower bound
978-1-4673-2060-3/12/$31.00 ©2012 IEEE
of transmitting efficiency can not be calculated because the
minimum radius bounding circle will exceed the unit circle.
Since antennas are not well-matched at all frequencies, this is
an unavoidable issue. In section III we detail a method to help
resolve this issue.
In addition to these two minor revisions, we also examine
the use of multiple stirring techniques. In other reverberation
chamber applications, stirring techniques are combined by
averaging data together [3], [4]. However, this is not a good
approach when using the two-port antenna model. Some
applications of the two-port antenna model rely heavily on
the statistical distribution of measured data points. Any averaging could have a profound impact on the calculated results.
This would imply that a combination of stirring techniques
can’t be used when the two-port antenna model is employed.
We propose a new method of combining stirring techniques
that ”groups” data points together instead of averaging them
together. This grouping preserves the original statistical distribution of the data. While this method also works to reduce
the measurement uncertainty (via an increase in the number
of samples), it may not decrease the measurement noise.
II. A R EVISED N OTATION
The notation presented in [1] and [2] focused on being
similar to something readers were already familiar with: twoport networks (and their notation). While the previous notation
may make it easier to understand the basic concepts of the
model, it does not easily allow the model to be expanded to
multiple antennas in an environment. The equations presented
in [1] and [2] remain valid, and the new notation can be
directly substituted in place of the old notation.
Figure 1 shows a signal flow graph of the unknown two-port
network with the revised notation. In previous publications,
the notation has revolved around S-Parameters. If the antenna
were placed in an environment (i.e. a reverberation chamber),
it too could be modeled as a two-port network. The two-port
network that represents the environment would simply be cascaded onto the end of the antenna’s model two-port network.
In this case, the only part of the environment that needs to be
216
Fig. 1. A signal flow graph of the two-port antenna model showing the
revised notation.
represented is the S11 of the reverberation chamber, denoted
RC
as S11
in Figure 2.
A simple example of moving from the old notation in [1]
and [2] to the new notation, we can look at the equation for
the transmitting efficiency. In the previous notation, this was
given as:
ηT =
|S21 |2
.
1 − |S11 |2
(1)
In the new notation, the equation for transmitting efficiency
remains the same, except the appropriate terms replace the
S-parameters:
ηT =
|T1 |2
.
1 − |Γ1 |2
Fig. 2. An example of S11 data that causes the minimum radius bounding
circle to exceed the unit circle.
(2)
The remaining equations in [1] and [2] can be translated in a
similar manner. All of the remaining equations and derivations
shown here will utilize the new notation.
III. A R EVISED E FFICIENCY C ALCULATION
An algebraic method for calculating the lower bound of
transmitting efficiency using the two-port model was first
presented in [2]. It was based on using a set of measured data
to calculate the parameters necessary to solve (2). This method
proves suitable and useful for antennas that are well-matched
at the frequencies of interest. Unfortunately, if the lower bound
of transmitting efficiency for a poorly matched antenna is of
interest, that method can have difficulty providing a result.
This problem stems from the use of a minimum radius
bounding circle. Such a circle is used around set of measured
complex S11 data. If a significant amount of mismatch exists,
it can cause the cluster of data to be near the unit circle. When
a minimum bounding circle is then put around that data, the
bounding circle may exceed the unit circle, as in Figure 2.
This creates a situation where the efficiency calculation can
not be completed.
By manipulating the equations, it is possible to redistribute
the data in such a way that the minimum bounding circle does
not exceed the unit circle. This isn’t to say that this improved
method solves all of the cases. If the reflections are very high,
there may be a some S11 values that exceed the unit circle by
a very small fraction (i.e. 1.001). These cases are rare, but can
be caused by a combination of measurement noise and very
high reflections. Regardless of the cause, they aren’t solvable
using this method.
This new iterative method begins the same way as the
previous method; with a set of measured S11 data from a
vector network analyzer (VNA). This data is acquired while
the antenna under test (AUT) is inside a reverberation chamber
(in this case).
Once the data are acquired, we will start processing by
assuming that the antenna is 100% efficient. This assumption,
albeit incorrect will allow us to calculate all four of the
parameters in the two-port network. These four parameters
will be used as a starting point for the iterative process. As
the iterations progress, these four parameters will change to
more reasonable values. The following three equations can be
used to calculate the network parameters:
< S11 >= Γ1 ,
(3a)
S1 = −Γ1 ,
(3b)
T1 = R1 =
p
1 − |Γ1 |2 ,
(3c)
where the bar over Γ1 represents the complex conjugate
operation. Once the four network parameters are calculated
assuming the antenna has 100% efficiency, the data is transRC
plane (referred to as ΓL in previous
formed to the S11
publications) using this equation:
RC
S11
=
S11 − Γ1
.
T1 R1 + (S11 − Γ1 )S1
(4)
After this transform is complete, the minimum radius
bounding circle is found. Figure 3 shows the original S11 data
RC
transformed to the S11
plane.
217
plane. From this point, the center and radius of the circle in
the S11 plane can be used to calculate a new S1 :
S1,new =
1
(S1,center − Γ1 ),
S1,radius
(6)
where S1,new is the new S1 parameter calculated from the
points of the circle, S1,radius and S1,center are the radius and
center of the circle used to calculate the new S1 , respectively.
This new S1 can then be used to find the transmitting efficiency:
ηT X =
S1,radius (1 − |S1,new |2 )
.
1 − |Γ1 |2
(7)
Unless the antenna being used truly is 100% efficient, this
process will have to be repeated at least one more time.
Fig. 3. The original measured S11 data after it has been transformed to
RC . The solid red circle is the minimum radius bounding circle. The unit
S11
circle is shown in solid green.
Now that the minimum radius circle that bounds all of the
RC
S11
points has been found, the circle must be transformed
back to S11 using a reworked version of (4):
"
!
#
1
S11 =
T1 R1 + Γ1 .
(5)
1
− S1
S RC
11
Note that (5) contains several operations: reciprocal, scale,
and offset. Each of these operations must be done on the
circle as a whole. Rather than defining a circle by its discrete
points, using only the radius and center of the circle should
be sufficient to complete the required operations.
Figure 4 shows the circle transformed back into the S11
RC plane after it
Fig. 4. The minimum radius bounding circle from the S11
is transformed back to the S11 plane. The blue dot represents the center of
the circle, and the solid green line is the unit outline of the unit circle.
RC plane.
Fig. 5. Data and minimum radius bounding circle (red) in the S11
The unit circle is shown in solid green.
RC plane after it
Fig. 6. The minimum radius bounding circle from the S11
is transformed back to the S11 plane. The blue dot represents the center of
the circle, and the solid green line is the unit outline of the unit circle.
218
This means calculating a set of new network parameters and
RC
repeating the S11 and S11
transformations again, as described
above. To repeat the process, a new T1 needs to be calculated:
q
(8)
T1,new = S1,radius (1 − |S1,new |2 )
As the process is repeated a second time, Figure 5 shows the
RC
original measured S11 data transformed into the S11
plane.
To compare the results from iteration to iteration, compare
Figure 3 to Figure 5.
An interesting characteristic of Figure 5 is that the minimum
radius bounding circle exceeds the unit circle. While an exact
mathematical explanation for this has not yet been determined,
it appears to be an artifact of the processing that only occurs
in intermediate iterations, and only for a small fraction of the
data points. Further investigation is required to determine the
cause of this anomaly. We would be more concerned if any of
the redistributed points or center of the circle exceed the unit
circle. Similarly, the radius of the bounding circle exceeding
1 would be a significant issue.
Similar to Figure 4, Figure 6 shows the second iteration
of data after the minimum radius bounding circle has been
transformed back to the S11 plane.
The calculation ends when the lower bound of transmitting efficiency stops significantly changing from iteration-toiteration. In practice, this is usually three iterations.
It is very important to note that just as in [2], this calculation
of transmitting efficiency remains a lower bound. Even with
this revised method, there is still variability in the exact
value of S1 . This variation in S1 is due to the fact that we
do not have enough paddle positions to properly represent
the probability distribution function (PDF) of the fields in
the reverberation chamber. In the next section we will show
how the grouping method can further improve this lower
bound calculation by adding more independent samples and
improving our representation of the PDF.
IV. T HE G ROUPING M ETHOD
For the purposes of comparing the proposed grouping
method to common averaging techniques, consider the case
where a single antenna is placed inside the working volume
of a reverberation chamber. We then connect the antenna to
a VNA and measure S11 at 10 different frequencies over
100 stepped paddle positions. It is generally accepted that
averaging the complex S11 data over all 100 paddle positions
will lead to an approximation of the free-space S11 value
of that antenna. Following the common averaging technique,
one could then average all 100 paddle positions at all 10
frequencies together (assuming all are statistically independent
of each other) to get a better approximation of the antenna’s
free-space S11 [5].
While there is nothing wrong with averaging the data to
calculate the free-space S11 , some information is lost during
the process. One of the key pieces of information that is lost
(or changed) is the original PDF of the measured S11 data. As
an example, Figure 7(a) shows a scatter plot of measured S11
(a)
(b)
Fig. 7. The top (a) shows a scatter plot of 100 S11 values. A histogram of
|S11 | is shown on the bottom (b).
data at 100 paddle positions. Figure 7(b) shows a histogram
of |S11 |. Figure 8 shows the same data averaged over 10
frequencies (approximately 4 MHz span). Note the significant
differences in the histograms between the two cases.
Now consider the case where the parameter of interest
isn’t the free-space value of S11 , but the lower bound of
transmitting efficiency. Since the transmitting efficiency is
largely dependent on the size of the ”cluster” of data and it’s
PDF, different answers would be obtained if the user chose to
use the data from Figure 7 or 8.
To avoid the loss of information caused by averaging points
together, group them together. Now, instead of 100 points
(each an average of 10 frequencies), we have 1,000 points by
simply grouping the 100 points at each of the 10 frequencies
together. This is accomplished by simply appending arrays
219
(a)
(a)
(b)
(b)
Fig. 8. The top (a) shows a scatter plot of 100 S11 values averaged over 10
frequencies. A histogram of |S11 is shown on the bottom (b).
of data together; no mathematical operations are performed.
Figure 9 shows the scatter plot of the grouped S11 data. This
is the same data shown in Figure 1, but this data is grouped
instead of averaged. Figure 9(b) shows the histogram of the
grouped |S11 | data. Because the grouped data set contains
1,000 points (instead of 100), the PDF of the magnitude can
be better approximated.
Here, the grouping method is demonstrated with frequency
stirring, it also applies to any other combination of stirring
techniques. Using this method, any number of stirring methods
could be combine to create one, large data set. However, when
frequency stirring is used, caution must be exercised for two
reasons. First, the bandwidth of frequency stirring must not
be so large that characteristics of the chamber or antenna
under test change significantly. In the case of calculating
the transmitting efficiency of a narrow-band antenna, the
Fig. 9. The top (a) shows a scatter plot of 1,000 S11 that have been grouped
together. A histogram of |S11 | is shown on the bottom (b).
use of frequency stirring may be restricted to a very small
bandwidth to avoid averaging over important characteristics
of the antenna.
Second, the frequency points must be an adequate distance
apart so they are statistically independent from one another.
The general rule is that frequency points must be at least one
mode bandwidth apart, defined as f /Q [5]. For the data used
in Figures 8 and 9, the mode bandwidth was approximately
85 KHz. The points averaged/grouped together are 400 KHz
apart.
Once the data are grouped together, they can be processed
using existing equations and procedures designed for whatever
application the user desires. The grouping method is meant to
be used before any other data processing has been done.
To show the use of the proposed grouping method, the lower
bound of transmitting efficiency of a single antenna placed
220
inside a reverberation chamber will be calculated.
V. A NALYSIS
As indicated in section IV, the calculation of transmitting
efficiency relies heavily on the distribution of the data. Too few
paddle positions or too few independent samples will produce
a small cluster, and thus a lower lower bound of efficiency.
One approach to prevent this from happening is to increase the
number of paddle positions. But, at a certain point, an increase
in paddle positions doesn’t increase the number of independent
samples in the data. This is why use of the proposed grouping
method is appealing - it provides another way to increase the
number of independent samples without increasing the number
of paddle positions.
Figure 10 shows a comparison of transmitting efficiency
data. All of this data was calculated with the method discussed
in section III, and represents a lower bound of transmitting
efficiency. The difference between the curves is the number
of paddle positions and stirring methods. The red curve (on
the bottom) is data that was calculated using only 100 paddle
positions. The two overlapping curves in the middle are data
taken from 1,000 paddle positions and 100 paddle positions
grouped together with 10 frequencies (to create 1,000 total
points). The grouped data shows more variation than the purely
stepped data. To give an idea of what’s possible, the top,
black curve shows the lower bound of transmitting efficiency
when 500 paddle steps are grouped with 20 frequencies (to
create 10,000 total points). The data for 100 and 1,000 paddle
positions has been smoothed with a 10-point moving average,
after the transmitting efficiency has been calculated. The
curves that feature grouped data do not have any smoothing
applied.
When the data was grouped together before calculating the
transmitting efficiency, it was grouped together in discrete
blocks. In other words, when 10 frequencies’ worth of points
were grouped together the grouping was not done as a ”moving
window.” Bins 4 MHz in size (in this case) were setup so that
each of the grouped data sets did not share any points with
the other bins of data.
This figure demonstrates two important principles. First, the
grouping method produces results that are very similar to the
data sets where only discrete paddle positions were employed.
Second, the grouping technique could produce better results
for only a small increase in measurement time. The amount
of extra measurement time required to sample at additional
frequencies is far less than taking additional paddle positions.
With reference to the data displayed in Figure 10, the data
that produced the ”100 Steps, 10 Freq.” curve took only a few
minutes longer than the ”100 Steps” data.
Uncertainties for the lower bound transmitting efficiency
shown in Figure 10 are based on the measurement uncertainties. In this case, the reflection coefficient S11 , is the only
measured parameter. The uncertainty in the S11 measurement
is 0.1 dB (with a coverage factor of 2). This does not mean
that the true transmitting efficiency of the antenna is within
Fig. 10. A comparison between using only paddle positions and using the
proposed grouping method.
the uncertainty bound, and this should not be confused with a
definite estimate.
VI. C ONCLUSION
A few updates to the two-port antenna model and lower
bound efficiency calculation have been presented. The first
update revised the model notation to be more general and
user-friendly for applications involving multiple antennas in an
arbitrary environment. The second update revised the way in
which the lower bound of transmitting efficiency is calculated.
The revised calculation employs an iterative approach in an
effort to successfully calculate the efficiency of antennas with
a significant mismatch. Finally, a new way of processing
reverberation chamber data was presented. The proposed
method groups points together instead of averaging them,
thus providing a better estimate of the probability density
function in the reverberation chamber. The grouping method
was demonstrated with conventional stepped paddle stirring
and frequency stirring.
R EFERENCES
[1] J. M. Ladbury and D. Hill, “An improved model for antennas in reverberation chambers,” in Proc. IEEE International Symposium on EMC, Ft.
Lauderdale, FL, July 2010.
[2] J. B. Coder, J. M. Ladbury, and M. Golkowski, “A novel method for
finding the lower bound of antenna efficiency.” Long Beach, CA: IEEE
Electromagnetic Compatability, July 2011.
[3] K. Rosengren, P. Kildal, C. Carlsson, and J. Carlsson, “Characterization
of antennas for mobile and wireless terminals by using reverberation
chambers: improved accuracy by platform stirring,” in Proceedings of
the 2001 IEEE International Symposium on Antennas and Propagation,
vol. 3. IEEE Antenna and Propagation, July 2001, pp. 350–353.
[4] P. Kildal and C. Carlsson, “Study of polarization stirring in reverberation
chambers used for measuring antenna efficiencies,” in Proceedings of
the IEEE International Symposium on Antennas and Propagation, vol. 2.
IEEE Antennas and Propagation, August 2001, pp. 486–489.
[5] J. Ladbury, G. Koepke, and D. Camell, “Evaluation of the nasa langley
research center mode-stirred chamber facility,” National Institute of
Standards and Technology, Boulder, CO, Tech. Rep. 1508, January 1999.
221