Authors: Darioush Agahi, William Domino

Conexant Systems Inc.

Newport Beach, CA

Note: This article originally appeared in the June 2000
edition of "Applied Microwave & Wireless"
(now out of print) *Both authors now work for Skyworks Solutions Inc.*
In the design of wireless portable devices, antenna efficiency is a variable that can have a great effect on
overall system performance, and yet may not always receive the attention it deserves. As an example, RF engineers
must frequently make critical tradeoffs in receiver design in order to improve sensitivity by mere fractions of a
dB, but a poor antenna efficiency can easily cause a degradation of several dB. This pitfall can occur in systems
such as GSM, where many tests are performed using a cable connection to the antenna port; a handset may easily
pass such tests, only to be later hampered by its antenna in the field. This paper is targeted at the very
important parameter of antenna efficiency, and a measurement technique that can be used to quantify it.

Antenna “efficiency” must be distinguished from antenna “gain”. Antenna gain is a directional quantity that
refers to the signal strength that can be derived from an antenna relative to a reference dipole. Efficiency, on
the other hand, quantifies the resistive loss of the antenna, in terms of the proportion of power that is actually
radiated versus the power that is first delivered to it. It is not a directional quantity.

**The Model**
We model the antenna’s loss as a resistor placed in series with the radiation resistance, as shown in Figure
1. Since the model includes no reactances, there is an implicit assumption that the measurements must be taken at
resonance. The equations to be derived later require this assumption.

Figure1 Model of Antenna Loss

The antenna efficiency (see appendix) is

h =
(1)

Note that it is immaterial whether the antenna is matched to the source resistance R

_{S}. While it is
certainly desirable and necessary to match the antenna in actual use, the match is not part of the problem of
finding the above resistance ratio. Therefore we need only relate the radiated power to that which is transferred
forward at the point shown in Figure 1. What is needed, then, is a way to effectively separate the resistances R

_{LOSS}
and R

_{RAD} by way of measurement, so that the efficiency can be calculated.

**The Wheeler Cap**
Wheeler [1] sets forth just such a method, where a hollow conductive sphere is placed over the antenna at the
radius of transition between the antenna’s energy-storing near-field and its radiating far-field. This transition
radius occurs at a distance of

l/2

p, and thus the sphere is
referred to as the “radiansphere”. The role of the conductive sphere is to reflect all of the antenna’s radiation
while causing minimal disturbance to the near-field. In theory, a complete sphere is appropriate for reflecting
the radiation of a small dipole, which is an approximation of an isotropic antenna, while in practice a monopole
with a ground plane can be capped with a half-sphere. The half-spherical “Wheeler cap” is shown in Figure 2. For
900MHz, the cap’s radius is 5.3cm.

Figure 2 Half-Spherical Wheeler Cap

If all of the power radiated by the antenna is reflected back by the cap and not allowed to escape, then in
our model this is the equivalent of setting R

_{RAD} to zero. By making separate S

_{11}
measurements with the cap in place and with the cap removed, we gather enough information to find the resistances
and the antenna’s efficiency.

The spherical or half-spherical cap is intended for physically small antennas; simple dipole or monopole
antennas must therefore also be electrically short. Given the cap’s radius, it is not possible to fit a monopole
of length

l/4 under it. To test such an antenna we replace the
half-spherical cap with a cylindrical cap, keeping the radius at

l/2

p.
Such a cylindrical cap is shown in Figure 3.

Figure 3 Cylindrical Wheeler Cap

**Efficiency of Short Antennas: The Constant-Power-Loss Method**
For an electrically short antenna (<

l/10), the radiation resistance
is typically small in comparison to the 50

W
source resistance of the measuring system. The radiation resistance of an ideal short monopole [2] is

R_{RAD MONO} =
(2)

So, for example, a 1/20-wavelength monopole, which fits comfortably under the half-spherical cap, exhibits
about a 4

W radiation resistance. With such a small value of R

_{RAD},
the power lost in the resistance R

_{LOSS} is about the same whether the cap is in place or removed, that
is, with zero or finite R

_{RAD}. With the assumption of constant power loss, we can make use of S

_{11}
magnitude measurements with the cap on and off as follows:

Cap on. The radiation resistance is zero, and the antenna reflection coefficient is measured and referred to
as S

_{11WC. } Then

(3)

Cap off. The radiation resistance is that of the antenna radiating into free space, and the antenna reflection
coefficient is measured and referred to as S

_{11FS.} Then

(4)

We need only measure the magnitudes (and not the signs) of S

_{11WC} and S

_{11FS}. The antenna
efficiency becomes

h =
(5)

h =
(6)

The efficiency can therefore be found directly from the reflection coefficient magnitude measurements, without
any need to actually determine R

_{RAD }and R

_{LOSS}. It should still be noted that the
measurements must be made at resonance, because the loss model is based on vector S

_{11} values that are
all-real, even though only their magnitudes are needed in the above equations.

Eq. (6) appears in [3] in a survey of previous techniques. This method was likely developed to make it
possible to obtain an antenna efficiency measurement using only reflectometer-type reflection-coefficient
measurements, where only the magnitude of S

_{11} is measured. In this case the method further depends on
the close proximity of the minimum of |S

_{11}| to its all-real point.

For our example of a 1/20-wavelength monopole, suppose the measurements are

S_{11WC} = -0.966
S_{11FS} = -0.823

Then the efficiency calculated from eq. (6) is

h = 79.3%

This antenna causes a performance penalty to the radio of

10log(79.3%) = -1.0 dB

Such an efficiency is not atypical, and some antennas have been measured to be even less than 50% efficient,
which corresponds to a power loss of more than 3dB. This power loss is a direct degradation of the receiver
sensitivity and the transmitter output power, relative to a cable-connection test. Such a transmitter power loss
would greatly degrade the handset’s battery life, and, in cellular systems that tend to be uplink-limited, it
would affect the ability of the handset to obtain service in marginal areas. In a 1/8-duty-cycle GSM system
operating at full transmit power of 2W with a 50% efficient antenna, the resistive heating of the antenna would
amount to 1/8W!

**Efficiency of Moderate-Length Antennas: The
Constant-Loss-Resistor Method**
As the antenna becomes longer, its radiation resistance increases, and the assumption of constant power loss
with and without the cap breaks down. In this case a method of efficiency measurement that directly makes use of
the quantities R

_{RAD} and R

_{LOSS} is preferred. Fortunately, modern vector network analyzers can
provide a direct display of the impedance of a measured device when performing a reflection coefficient
measurement. So we make use of the resistance ratio in (1) rather than the power ratio:

h =
(7)

Here the key assumption is that R

_{LOSS} itself, rather than the power lost in it, remains constant
with the cap in place or removed. If desired, we can still express the efficiency in terms of the reflection
coefficients:

Cap on. The radiation resistance is zero, and the antenna reflection coefficient is S

_{11WC.
} Then

(8)

Cap off. The radiation resistance is that of the antenna radiating into free space, and the antenna reflection
coefficient is S

_{11FS.} Then

(9)

(7) and (8) are transposed to become

(10)

(11)

And the efficiency is

h =
(12)

h =
(13)

Eq. (13) is actually more accurate then eq. (6) regardless of the absolute value of R

_{RAD}
(and thereby of the antenna length). Its one disadvantage lies in the fact that the signs of the (all-real) S

_{11}
measurements must be retained and accounted for in the calculation, making for somewhat less convenience. But it
exactly reproduces the resistance ratio at any level of antenna efficiency, as opposed to eq. (6) which becomes
less accurate at lower efficiencies.

As an example, consider a low-efficiency monopole, where the S

_{11} measurements are:

S_{11WC} = -0.626
S_{11FS} = -0.325

Then the efficiency calculated from eq. (13) is

h = 54.9% = -2.6dB

while that calculated from eq. (6) is

h = 32.0% = -4.9dB

The value of efficiency calculated by the constant-power-loss method is unnecessarily pessimistic. This
discrepancy between the methods occurs with longer antennas that inherently exhibit a large R

_{RAD}.

Next we plot the calculated efficiency vs. a swept value of R

_{LOSS}, in order to further compare the
two methods of calculation. Figure 4 is plotted for an R

_{RAD} of 4

W
(our short-antenna case) while Figure 5 is for an R

_{RAD} of 14

W
(a longer antenna). In the 4

W case the curves agree down to about 75%,
while in the 14

W case they quickly diverge. In either case the
constant-loss-resistor method is more accurate, as it agrees with the resistance ratio. This illustrates that the
constant-power-loss method is accurate only for small radiation resistances and high efficiency.

Figure 4 Efficiency vs. R_{LOSS} for Antenna with 4W
Radiation Resistance

Figure 5 Efficiency vs. R_{LOSS} for Antenna with 14W
Radiation Resistance

**Making the Measurements: Practical Considerations**

In the above derivations it was assumed that the radiation and loss resistances are not accompanied by any
reactive impedances; therefore the S

_{11} measurements need to be made at the antenna’s actual resonance,
as defined by the point where S

_{11} is all-real. For an ideal lossless antenna and perfectly-reflecting
Wheeler cap, the S

_{11} measurements would be -1 with the cap on and zero with the cap off, and we need to
find the points of all-real impedance that most closely approach these. They may or may not be precisely the

same as where the magnitude |S

_{11}| is minimized, as illustrated in Figure 6, and so it is advisable
to view the measurements on a Smith chart display rather than a log-magnitude grid. These points should not be too
far apart.

Figure 6 Smith Chart Display of Free-Space S_{11}

It is especially true that the free space measurement should be done at the actual resonance instead of the
antenna’s nominal operating frequency F

_{0}. This is because the antenna is normally loaded down when
installed on the handset, and it is expected that there should be a small shift when it is removed and placed on a
different ground plane. A rule of thumb that is normally used for the maximum limit of this shift is

±
10% of F

_{0}. De-tuning beyond this could impact the accuracy of the measurement, as it means the
actual-usage environment differs too much from the measuring setup.

Figure 7 Allowable De-tuning for Measurement

Conclusion

The Wheeler cap provides a convenient and reasonably accurate method of determining antenna efficiency. For
practical use it consists of a cap over a ground plane, usually of a simpler shape than the ideal half-sphere,
such as a cylinder, keeping the
l/2p radius. The efficiency is best determined by measurement of
the antenna resistance with the cap in place and removed, each taken at the resonance defined as the all-real
impedance point.

Appendix

The derivation of equation (1), which also appears in ref. [2] page 48, is duplicated here. In the ideal case,
where there are no disturbances to the antenna, and a perfect match, the input resistance represents the
dissipation loss of the antenna. This resistance represents the sum of radiation resistance and ohmic resistance.

R_{in} = R _{RAD} + R_{LOSS
}(a1)

_{ }

Given that the peak current flowing in to the antenna is I_{in} then the average power dissipated in
an antenna is

Pin = ½ R_{in} | I_{in} |^{2
}(a2)

^{ }

Inserting (a1) into (a2) yields,

Pin = ½ ( R _{RAD} + R_{LOSS})
| I_{in} |^{2} = ½ R _{
RAD} | I_{in} |^{2 }+ ½ R_{LOSS}
| I_{in} |^{2}

Or equivalently,

P_{RAD} = ½ R
_{RAD} | I_{in} |^{2}
(a3)

P_{LOSS} = ½ R_{LOSS}
| I_{in} |^{2}
(a4)

Using the efficiency equation and inserting (a3) and (a4) yields,

h
=
=
=

After canceling similar terms, it yields,

h
=

References:

1) H. A. Wheeler, “The radiansphere around a small antenna,” Proc. Of the IRE, vol.47, pp.1325-1331, Aug.
1959.

2) W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, Wiley, New York, 1981.

3)
R. H. Johnston, L. P. Ager, and J. G. McRory, “A new small antenna efficiency measurement method,”

IEEE
1996 Antennas and Propagation Society International Symposium, vol.1, pp. 176-179

Authors:

Darioush Agahi, P.E. is director of GSM RF systems engineering at Conexant Systems Inc. in Newport Beach CA.
He has 17 years of industry experience of which the last 4 years he has been with Conexant and 9 years with
Motorola’s (GSM) cellular subscriber division. Darioush received his BS in electronics (1981) and MS in Medical
Engineering from the George Washington University in Washington DC (1983). Also he received an MSEE from Illinois
Institute of Technology in Chicago Illinois (1993) and an MBA from National University in 1997. Darioush holds ten
US patents and several more pending, he is a Professional Engineer (P.E.) registered in the state of Wisconsin.

William Domino is Principal Engineer, GSM RF Systems, at Conexant Systems Inc. in Newport Beach, CA,
where he has been employed since 1992 in the area of digital-radio system architecture development. He received
the BSEE degree from the University of Southern California in 1979 and the Master of Engineering from the
California State Polytechnic University, Pomona, in 1985. His interests currently include receiver and transmitter
system design for various cellular standards, as well as filter design. In these areas he has one patent issued
and ten patents pending.