Michio Miyakawa

Nozomu Ishii

Yuji Tsuboike

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Uncertainty Evaluation of Wheeler and Reflection 
Methods for Radiation Efficiency Using Monte 
Carlo Simulation 
 
#Yuji Tsuboike 1, Nozomu Ishii 1, Michio Miyakawa 1 
1 Graduate School of Science and Technology, Niigata University 
1 8050, Ikarashi 2-nocho, Nishi-ku, Niigata, 9050-2181, JAPAN 
E-mail: nishii@eng.niigata-u.ac.jp 
 
 
 
1. Introduction 
 
 The radiation efficiency is one of the most important parameters for the antenna built in the 
mobile communication devices. The Wheeler and reflection methods are known as simple and low-
cost method of measuring the radiation efficiency for the small antenna [1],[2]. In these methods, 
the reflection coefficients in free space and cavity are required. However, the magnitude of the 
reflection coefficient covered with the cavity is nearly equal to unity so that the uncertainty of the 
resulting efficiency can deteriorate. Therefore, in order to confirm the reliability of the measurement, 
the uncertainty of the measurement should be examined.  We derived and evaluated the uncertainty 
expression of the measurement for the Wheeler and reflection methods. In this paper, to validate 
these expressions, we will evaluate the uncertainty by use of a different method, i.e., Monte Carlo 
simulation method.  
 
2. Two Ways of Evaluating Uncertainty in Wheeler/Reflection Methods 
2.1 Formula of Uncertainty of Wheeler/Reflection Methods 
 In the Wheeler method [1], the radiation efficiency of the antenna is given by 
                                                                   
|Γ | Γ
1 Γ
,                                                                    1  
where, |Γ | and |Γ | donote the magnitude of the reflection coefficients at the input port of the 
antenna in free space and the cavity as shown in Fig. 1.  The uncertainty for  is given by 
                                            
 Γ
Γ
|Γ |
|Γ |
,                                            2   
where  denotes a combined standard uncertainty for , and  denotes a standard 
uncertainty for , and ,  are given in [3]. 
 In the refection method [2], the radiation efficiency of the antenna is given by 
                                                         f
1
1 Γ
| |
 ,                                                        3   
where  and  are center and radius of a circle formed by the points Γ , Γ , 1,2, ,  
on the Smith chart. Γ ,  are the reflection coefficient at the input port of the antenna inserted into 
the rectangular waveguide with two sliding short as shown in Fig. 2. The uncertainty for f  is 
given by 
                           
2 Γ | | /
1 |Γ |
Γ
2| |
1 |Γ |
| |                   
                                           
1  | | /
1 |Γ |
,                                                                      4  
where | |  and  are given in [4]. 
The 2009 International Symposium on Antennas and Propagation (ISAP 2009)
October 20-23, 2009, Bangkok, THAILAND
- 265 -
 
 
 
 
 
 
 
 
Figure 1: Rectangular Shielding Used in         Figure 2: Rectangular Waveguide with Two Sliding 
Wheeler Method                                             Shorts Used in Reflection Method 
 
2.2 Monte Carlo Simulation for Efficiency Based on Wheeler/Reflection Methods 
By generating the normal random numbers for the magnitude and phase of measured 
reflection coefficients, we can estimate the uncertainty of the radiation efficiency. The average and 
standard deviation of the normal random numbers were selected as the corresponding measured 
value and corresponding standard uncertainty which can be given by the spreadsheet provided by 
the manufacturer of a vector network analyzer. Lots of resulting efficiency can lead to an alternative 
evaluation of the standard uncertainty for the radiation efficiency by calculating their estimated 
standard derivation. 
 
3. Examples of Uncertainty Evaluation 
3.1 Uncertainty in Wheeler Method 
In this paper, a monopole antenna with a length of 40mm on the ground plane was used as 
shown in Fig. 1. Reflection coefficients were measured in a grounded half space and a metallic 
shielding as a function of the frequency by use of a vector network analyzer, Agilent 8720ES, with 
Calc/Meas power of -5dBm, IFBW of 10Hz and Average Factor of 1. The rectangular shielding has 
a dimension of 150mm 150mm 75mm  as shown in Fig. 1. The antenna under test was 
inserted at the center on the undersurface of the cavity. 
In the Wheeler method, Fig. 3 shows evaluated efficiency and corresponding standard 
uncertainty by use of the equations (1), (2) and the Monte Carlo simulation. In the Monte Carlo 
simulation, we generated 100 normal random numbers for the measured reflection coefficients. The 
cavity has resonant frequencies of 1.00GHz, 1.25GHz and 2.22GHz for TE101, TE102 and TE103 
modes. Note that the estimated efficiency has dips in the neighbourhood of these frequencies. 
Except the above case, the standard uncertainty evaluated by the equation (2) has a good agreement 
with the one obtained by the Monte Carlo simulation over 1GHz. This fact confirms the validity of 
the equation (2) which can evaluate the uncertainty for the radiation efficiency in the Wheeler 
method that we derived in [3]. However, below 1GHz, the curves of efficiency and its standard 
uncertainty obtained by the equation (1), (2) are quite different from those obtained by the Monte 
Carlo simulation. This is because |Γ | is often larger than |Γ | in the Monte Carlo simulation as 
shown in Fig. 4 so that the efficiency calculated by the equation (1) can be often negative. For 
reference, over 1GHz, there is no case of |Γ | |Γ |, as shown in Fig. 5. 
3.2 Uncertainty in Reflection Method 
In the reflection method, the 40mm monopole antenna was inserted along the centerline on 
the board wall of the rectangular waveguide, whose cross-section has a dimension of 150mm
75mm as shown in Fig. 2. Reflection coefficients were measured in a grounded half space and a 
cavity formed by the waveguide with two sliding shorts. Setting parameters of the vector network 
analyzer were same as those in the above case of the Wheeler method. To avoid the dips of the 
efficiency, we excluded the measured data with |Γ , | 0.9 in finding a circle. The number of the 
combination of ,  is 64, that is,  and  ranged from 60mm to 130mm at intervals of 10mm, 
where  and  denote the distance from the center of the antenna to the left and right sliding shorts. 
 Fig. 6 shows evaluated efficiency and corresponding standard uncertainty by use of the 
equations (3), (4) and the Monte Carlo simulation. In the Monte Carlo simulation, we generated 100 
normal random numbers for the measured reflection coefficients. Over 1.25GHz, the standard 
lL lR
150mm
75mm
150mm
150mm
75mm
The 2009 International Symposium on Antennas and Propagation (ISAP 2009)
October 20-23, 2009, Bangkok, THAILAND
- 266 -
 
 
 
 
 
 
 
 
 
 
 
(a) Equations (1) and (2)                       (b) Monte Carlo Simulation (N=100) 
Figure 3: Wheeler Efficiency and its Uncertainty 
 
 
 
 
 
 
 
 
 
 
 
 
(a) Free Space                                                  (b) Rectangular Shielding 
Figure 4: Frequency distribution of reflection coefficients at 0.8GHz in the Monte Carlo simulation 
of Wheeler efficiency. 
  
 
 
 
 
 
 
 
 
 
 
(a) Free Space                                                  (b) Rectangular Shielding 
Figure 5: Frequency distribution of reflection coefficients at 1.7GHz in the Monte Carlo simulation 
of Wheeler efficiency. 
 
uncertainty obtained by the equation (4) has a fairly good agreement with the one obtained by the 
Monte Carlo simulation. This fact confirms the validity of the equation (4) which can evaluate the 
uncertainty for the radiation efficiency in the reflection method that we derived in [4]. However, 
below 1.25GHz, the curves of efficiency and its uncertainty obtained by the equations (1), (2) are 
quite different from those obtained by the Monte Carlo simulation. This is because the circle 
required for finding the efficiency cannot be exactly drawn in the Smith chart below 1.25GHz, 
where the guide wavelength is too long to change Γ ,  or the guide wavelength is infinite so that 
Γ ,  could not vary for moving the sliding short as shown in Figs. 7 (a), (b). For reference, over 
1.25GHz, Γ ,  can vary along the circle as the sliding short is moved as shown in Fig. 7(c). 
0
10
20
30
40
0.14 0.16 0.18 0.20 0.22 0.24 0.26 
C
ou
nt
s
|Γf |
0.75 1.00 1.25 1.50 1.75 2.00 2.250
0.2
0.4
0.6
0.8
1.0
0
20
40
60
80
100
Frequency[GHz]
R
ad
ia
tio
n 
Ef
fic
ie
nc
y
 ηΓ
 uc(η
Γ)
U
ncertainty[%
]
0.75 1.00 1.25 1.50 1.75 2.00 2.250
0.2
0.4
0.6
0.8
1.0
0
20
40
60
80
100
Frequency[GHz]
R
ad
ia
tio
n 
Ef
fic
ie
nc
y
U
ncertainty[%
] ηΓmc
 uc( η
Γ
mc)
0
10
20
30
40
0.97 0.98 0.99 1.00 1.01 1.02 1.03 
C
ou
nt
s
|Γf |
0
10
20
30
40
0.97 0.98 0.99 1.00 1.01 1.02 1.03 
C
ou
nt
s
|Γs |
0
10
20
30
40
0.95 0.96 0.97 0.98 0.99 1.00 1.01 
C
ou
nt
s
|Γs |
The 2009 International Symposium on Antennas and Propagation (ISAP 2009)
October 20-23, 2009, Bangkok, THAILAND
- 267 -
 
 
 
 
 
 
 
 
 
 
 
(a) Equations (3) and (4)                                (b) Monte Carlo Simulation (N=100) 
Figure 6: Efficiency Based on Reflection Method and its Uncertainty 
 
  
 
 
 
 
 
 
 
 
 
(a) 0.8GHz                                (b) 1.1GHz                                    (c) 1.7GHz 
Figure 7: Γ ,  in the Smith chart for Monte Carlo Simulation 
 
 
 
5. Conclusion 
 
 By use of the Monte Carlo simulation, we validate the expressions of the uncertainty for the 
Wheeler and reflection methods that we derived. And we find that the efficiency and its uncertainty 
cannot be estimated by the Monte Carlo simulation in the neighborhood of the resonant frequencies 
and below the first resonant frequency of the cavity. Therefore, the use of the expressions for the 
uncertainty is still powerful way to avoid this difficulty. 
 
References 
 
[1] E. H. Newman, P. Bohley, C. H. Walter, “Two methods for the measurement of antenna 
efficiency,” IEEE Trans. Antennas & Propagat., vol.AP-23, no.4, pp.457-461, 1975. 
[2] R. H. Johnston and J. G. McRoy, “An improved small antenna radiation efficiency measurement 
method,” IEEE Antennas Propagat. Mag., vol. 40,no. 5, pp.40-48, 1998.  
[3] N. Ishii, Y. Kobayashi, Y. Katagiri, M. Miyakawa, “Systematic uncertainty of radiation 
efficiency based on some formulas for Wheeler method,” Proc. 2008 IEEE International 
Workshop on Antenna Technology: Small Antennas and Novel Metamaterials, Chiba, Japan, 
pp.290-293, 2008. 
[4] N. Ishii, Y. Katagiri, M. Miyakawa, “Systematic uncertainty of efficiency by reflection method," 
Proc. 2008 IEEE AP-S International Symposium, 439.8, San Diego, CA, USA, July 2008 
 Γf
 Γwg,i  Γf Γwg,i
0.75 1.00 1.25 1.50 1.75 2.00 2.250
0.2
0.4
0.6
0.8
1.0
0
20
40
60
80
100
Frequency[GHz]
R
ad
ia
tio
n 
Ef
fic
ie
nc
y
U
ncertainty[%
] ηrefl
 uc(η
refl)
0.75 1.00 1.25 1.50 1.75 2.00 2.250
0.2
0.4
0.6
0.8
1.0
0
20
40
60
80
100
Frequency[GHz]
R
ad
ia
tio
n 
Ef
fic
ie
nc
y
U
ncertainty[%
] ηreflmc
 uc(η
refl
mc)
 Γf
 Γwg,i
The 2009 International Symposium on Antennas and Propagation (ISAP 2009)
October 20-23, 2009, Bangkok, THAILAND
- 268 -


Uncertainty Evaluation of Wheeler and Reflection Methods for Radiation Efficiency Using Monte Carlo Simulation

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Uncertainty Evaluation of Wheeler and Reflection Methods for Radiation Efficiency Using Monte Carlo Simulation

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