### ETC ALL4XXAND9XX

```APPLICATION NOTE
Small loop antennas
1.
nAN400-03
General
For RF designers developing low-power radio devices for short-range applications,
antenna design has become an important issue for the total radio system design.
Taking the demand for small size and low cost into account in the development of
such radio modules, a small-tuned loop antenna on the same printed circuit board as
the radio module is a good solution.
An overview of the basics for electrically small loop antennas is presented. The
overview is mainly based on reference [1]. An effective shunt-matching technique for
loop antennas, the T-match, is also discussed.
Four different loop antennas for 433 MHz have been fabricated, and impedance and
gain measurements have been made on these antennas in an antenna laboratory.
2.
Loop antenna basics
Electrically small loop antennas are antennas where the circumference is less than
about one-tenth of a wavelength [1]. The field pattern of such loop antennas is similar
to that of an infinitesimal dipole with a null perpendicular to the plane of the loop and
with its maximum along the plane of the loop.
This chapter describes the geometry and the electrical equivalent circuit for a
rectangular loop antenna. Physical dimensions for the antenna is used to calculate the
components in the antenna equivalent circuit and the antenna efficiency.
The T-matching method is presented in order to match the impedance of the antenna
Formulas for range calculation is also presented in order to make the designer of radio
modules for short range applications able to calculate either the range for a device or
the power needed for a specified range.
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2.1. Loop antenna physical parameters
Figure 1 shows the geometry of the rectangular loop antenna.
a2
C1
C2
Loop conductor
Rq
a1
b1
d
b2
s
ZANT
P RF
Figure 1. Geometry of rectangular loop antenna
The loop antenna physical parameters used in the calculations are
a1
a2
b1
b2
=
=
=
=
loop antenna width [m]
loop antenna length [m]
thickness of loop conductor [m]
width of loop conductor [m]
For loop antenna fabricated on a printed circuit board (PCB), the thickness of the loop
conductor b1 means the thickness of the copper layer on top of the substrate.
During calculation of the antenna electrical parameters, the rectangular loop has to be
modelled as an equivalent quadratic loop, and the planar loop conductor has to be
modelled as a wire conductor with equivalent circular radius.
From the parameters above the equivalent quadratic sides of the loop are given by
[m]
a = a1 a 2
The calculated equivalent quadratic sides are used in the formulas below for the loop
area A, and the inductances LA and LI.
The loop area is given by
[m ]
A = a2
2
The equivalent electrical circular radius of the loop conductor is given by
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b = 0.35 ⋅ b1 + 0.24 ⋅ b2
[m]
In electrostatic, the equivalent radius represents the radius of a circular wire whose
capacitance is equal to that of the noncircular geometry, see [1] Table 9.3 pp. 456.
2.2. Loop antenna electrical equivalent circuit
To be able to estimate the capacitor CP used to resonate the antenna, the input
impedance of the loop antenna has to be determined. To estimate the antenna
efficiency, radiation resistance, loss resistance of the loop conductor and other ohmic
losses has to be determined.
According to [1] the equivalent circuit for the input impedance of a small loop when
the loop is used as a transmitting antenna is shown in Figure 2.
RL
XI
ZG
RR
+
VG
CP
_
Z'IN
XA
ZIN
RX
Figure 2. Loop antenna equivalent circuit (transmit mode).
The loop antenna input impedance ZIN is given by:
Z IN = (RR + RL + R X ) + j 2π f 0 ( L A + LI )
[Ω]
where
RL = loss resistance of loop conductor []
Rx = additional ohmic losses (ESR in capacitor CP etc.) []
LA = inductance of loop antenna [H]
LI = inductance of loop conductor [H]
The radiation resistance is given by
 A2 
RR ≈ 31171 ⋅  4 
λ 
[Ω]
where
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Small loop antennas
λ=
[m]
c
f0
where
c is the speed of light equal to 3⋅108 m/s
f0 is the resonance frequency in Hz.
The loss resistance of the loop inductor is given by
RL =
a + a2
l
RS = 1
P
b1 + b2
π f0 µ0
σ
[Ω]
where
l = length of the metal loop conductor
P = perimeter of the cross section of the loop conductor
RS = conductor surface resistance
µ0 =
⋅10-7 H/m
σ = conductivity of the conductor equal to 5.8⋅107 S/m for copper.
The additional ohmic losses that is introduced mainly because of ESR (Equivalent
Series Resistance) of the capacitor CP is given by
RX =
2π f 0 (L A + LI )
− RR − RL
Q
[Ω]
As can be seen from the above expression, the maximum possible quality factor Q of
a loop antenna is mainly determined by the ESR (i.e. the quality factor) of the
capacitor CP. A resistor RQ in parallel with CP can be used to control the Q-value of
the antenna. The insertion of this parallel resistor will reduce the antenna input
impedance.
In Figure 2 the capacitor CP is used in parallel to ZIN to resonate the antenna, that is to
cancel out the imaginary part of the input impedance ZIN at the operating frequency.
CP can also be used to represent distributed stray capacitances. It can be shown that
the parallel capacitor CP at resonance is given by
CP =
L A + LI
( RR + RL + R X ) 2 + [2πf ( L A + LI )]
2
[F ]
'
Under resonance the input impedance Z IN
can be shown to be equal to
Z
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'
IN
= RR + RL + R X
2
[
2πf (L A + LI )]
+
[Ω]
RR + RL + R X
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Small loop antennas
The inductive reactance XA of the loop is computed using the inductance LA of, [1]
  8a  
L A = µ 0 a ln   − 2
  b 
[H ]
Square loop with sides a and wire radius b:
LA = 2µ 0
a
π
 a

ln  b  − 0.774
  

[H ]
The reactance XI of the loop conductor can be computed using the inductance LI of the
loop. For a single turn this can be approximated by [2]
LI = µ 0
A
2a
[H ]
where A is the area of the loop.
The antenna efficiency can then be expressed as
η=
RR
RR + RL + RESR
alternatively
η=
QRR
2π f 0 (LA + LI )
2.3. Impedance matching
Under resonance the resistive input impedance of the loop is high, and has to be
transformed down to a lower value to match the transmitter output
impedance/receiver input impedance. An effective shunt-matching technique is the Tmatch connection as shown in Figure 1. This method of matching is based on the fact
that the impedance between any two points equidistant from the center along a
resonant antenna is resistive, and has a value that depends on the spacing between the
two points (feed length). It is therefore possible to choose a pair of points between
which the impedance will have the right value to match the transmitter output
impedance/receiver input impedance. By reducing the distance between the
connection points the impedance is reduced. In practice, the transmitter/receiver
cannot be connected directly at these points because the distance between them is
much greater than the pin spacing of an integrated circuit. The T-match arrangement
in Figure 1 overcomes this difficulty by using a second conductor paralleling the
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antenna to form a matching section to which the transmitter/receiver may be
connected.
A trial and error procedure is used to vary the feed length to make the total input
impedance of the loop antenna equal to the transmitter output impedance/receiver
input impedance. The estimated capacitor CP must be tuned for maximum radiated
power from the antenna for every position of the connection points.
2.4. Antenna impedance and Q-value with chip capacitors in the loop
The antenna impedance is dependent of both feed length and Q-value (read parallel
resistor RQ). The Q-value is independent of the impedance of the antenna, which
means that one chooses a Q-value and then chooses the feed length.
in mass production, the Q-value of the antenna has to match the capacitors that will be
used to tune the antenna to the right resonance frequency. The Q-value of the loop
should be chosen according to
Q=
1
tol
1+
−1
100
The tol variable is the tolerance of the capacitors in %. The equation is based on the
assumption that the variation in radiated power due to capacitor variations should be
lower than 3dB. If the tolerance is 4% the Q-value will be 50. If a higher Q-value is
chosen, each antenna has to be tuned to keep the variation in radiated power lower
than 3dB.
2.5. Range calculations
RF systems operation in the UHF band are not restricted to the line-of-sight coverage
of optical systems (IR systems) due to diffraction and reflection of radio waves at
edges and conductive surfaces as well as their capability to penetrate dielectric
materials.
Range calculation parameters are
•
•
•
•
•
•
transmitter output power, PRF [dBm]
transmitter and receiver antenna efficiency, η
antenna separation, R [m]
free-space loss, LP [dB]
additional propagation losses other than free-space loss, LX [dB]
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Small loop antennas
The free space propagation model is used to predict received signal strength when the
transmitter and receiver have a clear, unobstructed line-of-sight path between them.
The free-space loss factor LP is given by
 λ 

LP = 
 4π R 
2
 λ 

LP [dB ] = 20 ⋅ log
π
R
4


The free-space loss factor takes into account the losses due to the spherical spreading
of the energy by the antenna. The equation shows that the received power will fall off
as the square of the transmitter-receiver separation distance. This implies that the
received power decays with distance R at a rate of 20 dB/decade, (i.e. 6dB extra loss
for doubling of the distance).
Assuming reflection and polarisation-matched antennas, aligned for maximum
directional radiation and reception, it can be shown that the communication range
with given output power PRF, sensitivity S and equal TX/RX antennas is given by
R=
λ
S
4π
2
η PRF
[m]
This equation is based on the assumption that the two antennas are separated by a
distance R > 2 D 2 / λ . D is the largest dimension of either antenna. Wave guidance
occurring along conductive surfaces may increase the operation range as well.
The free-space path analysis applies to line-of-sight propagation, which means you
have to correct for various other propagation losses LX such as signal reflection,
diffraction, scattering and polarisation losses. When these losses are included, the
communication range is given by
R=
λ
S
4π
L X η 2 PRF
[m]
Given the required range R, assumed losses LX, sensitivity S and equal TX/RX
antennas, the necessary output power PRF is given by
PRF =
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S
L X LPη 2
-
[W ]
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Small loop antennas
2.6. Range calculation example
Given a rectangular loop antenna with dimensions a1 = 30mm and a2 = 50mm
fabricated on a standard FR-4 substrate. The thickness of the copper layer on top of
the substrate is b1 = 35µm and the width of loop conductor b2 = 1mm. The antenna
quality factor is limited to Q = 50 by a resistor in parallel with the capacitor CP which
is used to resonate the loop. The antenna operates at f0 = 433.936MHz.
1. Estimate the capacitor CP at resonance and the antenna efficiency.
2. Calculate the free-space communication range assuming equal loop antennas as
given above. The transmitter output power is 10dBm (10mW), and the receiver
sensitivity is –103dBm (0.05012 pW).
Solution:
1. To estimate the capacitor CP, the parameters LA, LI, RR, RL and RX has to be
calculated. To calculate LA and LI, the equivalent quadratic sides, a, and the
equivalent electrical circular cylinder radius of the loop conductor b, must be
calculated first.
a = a1 a 2 = 0.03 ⋅ 0.05 = 0.03873 m
b = 0.35 ⋅ b1 + 0.24 ⋅ b2 = 0.35 ⋅ 0.000035 + 0.24 ⋅ 0.001 = 0.00025 m
LA = 2µ 0
a
π
 a


− 7 0.03873   0.03873 
ln  b  − 0.774 = 2 ⋅ 4π ⋅10 ⋅ π
ln  0.00025  − 0.774

  

 

L A = 132.27 nH
LI =
1
1
µ 0 a = ⋅ 4π ⋅ 10 −7 ⋅ 0.03873 = 24.33 nH
2
2
 (0.03873 ⋅ 0.03873)2 
 A2 
 = 0.30710 Ω
RR ≈ 31171 ⋅  4  = 31171 ⋅ 
4

λ
(
)
0
.
6913




π f0 µ0
0.03 + 0.05
π ⋅ 433.936 ⋅ 10 6 ⋅ 4π ⋅ 10 −7
=
= 0.42008 Ω
σ
0.000035 + 0.001
5.8 ⋅ 10 7
RL =
a1 + a 2
b1 + b2
RX =
2π f 0 (L A + L I )
− RR − RL
Q
RX =
2π ⋅ 433 , 936 ⋅ 10 6 132 . 27 ⋅ 10 − 9 + 24 . 33 ⋅ 10 − 9
− 0 . 30710 − 0 . 42008
50
(
)
R X = 7.81222 Ω
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Then the capacitor CP can be calculated:
CP =
CP =
L A + LI
( RR + RL + R X ) 2 + [2πf ( L A + LI )]
2
132.27 ⋅ 10 −9 + 24.33 ⋅ 10 −9
(0.30710 + 0.42008 + 7.81222 )2 + [2π ⋅ 433.936 ⋅ 10 6 (132.27 ⋅ 10 −9 + 24.33 ⋅ 10 −9 )]2
C P = 0.86 pF
The antenna efficiency is
η=
RR
0.30710
=
= 0.03596 (−14.4 dB)
RR + RL + RESR 0.30710 + 0.42008 + 7.81222
2. The communication range R is
R=
λ
S
4π
2
η PRF
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=
0.6913
0.05012 ⋅10 −12
4π
0.03596 2 ⋅ 10 ⋅ 10 −3
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= 884 m
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3.
Loop antenna measurements
Four different tuned loop antennas have been tested in an antenna laboratory. The
loop antennas are made on a standard 1.6mm FR4 printed circuit board. The tested
loop antenna sizes are:
1.
2.
3.
4.
50x30mm
35x20mm
25x15mm
18x10mm
Each antenna is tuned to a resonance frequency of 433,936MHz with a fixed chip
capacitor (5%) in series with a variable capacitor.
All loop antennas are tuned to approximately 400Ω with a T-match. The resistor RQ
for controlling the Q-value is not used in these measurements. The chip capacitor in
series with the variable capacitor determines the maximum possible Q-value of the
loop antennas. The measured antennas has Q-values of Q = 50±10%.
To be able to compare the measurement results of the loop antennas to some known
antenna response, measurements where made on a λ/4 dipole antenna mounted on a
40x40cm ground plane.
We have used a standard log periodic antenna in the antenna laboratory as the
transmitter (TX) antenna for all measurements. The λ/4 dipole antenna and the loop
antennas under test have been used as receiver (RX) antennas. By doing the
measurements this way, we measure the difference between the antennas, not the
actual gain for each antenna.
3.1. Antenna transmission
For all measurements, the log periodic dipole is used as the transmitter antenna. The
measurement presents S21 for the complete transmission. Both the TX and RX
antennas are part of the transmission budget. Due to this we can not extract the
absolute gain of the measured λ/4 dipole and loop antennas.
Figure 3 shows the measured S21 for all the loop antennas and the λ/4 dipole antenna.
S21 a = S21 for λ/4 dipole reference antenna
S21 b = S21 for 50x30mm loop antenna
S21 c = S21 for 35x20mm loop antenna
S21 d = S21 for 25x15mm loop antenna
S21 e = S21 for 18x10mm loop antenna
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Figure 3. Plot of S21 for all loop antennas and λ/4 dipole reference antenna
Each antenna measurement is discussed in the following chapters.
3.1.1. λ /4 dipole reference antenna
The λ/4 dipole antenna is a whip made of copper and has a length of 16.1cm. The
antenna was mounted on a 40x40cm ground plane. From Figure 3 we see that the λ/4
dipole antenna has a measured value of –46.5dB at 434MHz.
3.1.2. 50x30mm loop antenna
This antenna is tuned to 400Ω, and has a quality factor of Q = 48. Figure 3 shows a
measured peak value of –52,5dB at 434MHz for this antenna. Compared to the λ/4
dipole antenna, the 50x30mm loop antenna has 6dB lower gain.
3.1.3. 35x20mm loop antenna
This antenna is tuned to 386Ω, and has a quality factor of Q = 54. Figure 3 shows a
measured peak value of –57.5dB at 434MHz for this antenna. Compared to the
50x30mm loop antenna, the 35x20mm loop antenna has 5dB lower gain. The
calculated difference in efficiency is 4.1dB. Gerber files for layout is available, see [3,
4, 5].
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3.1.4. 25x15mm loop antenna
This antenna is tuned to 414Ω, and has a quality factor of Q = 48. Figure 3 shows a
measured peak value of –61.5dB at 434MHz for this antenna. Compared to the
50x30mm loop antenna, the 25x15mm loop antenna has 9dB lower gain. The
calculated difference in efficiency is 8.4dB. Gerber files for layout is available, see [3,
4, 5].
3.1.5. 18x10mm loop antenna
This antenna is tuned to 400Ω, and has a quality factor of Q = 48. Figure 3 shows a
measured peak value of –65.5dB at 434MHz for this antenna. Compared to the
50x30mm loop antenna, the 18x10mm loop antenna has 13dB lower gain. The
calculated difference in efficiency is 12.8dB. Gerber files for layout is available, see
[3, 4, 5].
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Small loop antennas
4.
References
1. C. A. Balanis, “Antenna Theory, Analysis and Design”, second edition, John Wiley
& Sons, Inc., 1997.
2. J. D. Kraus, “Electromagnetics”, 4th ed., McGraw-Hill Book Co., New York, 1992.
3. Application note nAN400-04, “nRF0433 RF and antenna layout”,
Nordic VLSI ASA.
4. Application note nAN400-05, “nRF401 RF and antenna layout”,
Nordic VLSI ASA.
5. Application note nAN400-06, “nRF402 RF and antenna layout”,
Nordic VLSI ASA.
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LIABILITY DISCLAIMER
Nordic VLSI ASA reserves the right to make changes without further notice to the
product to improve reliability, function or design. Nordic VLSI does not assume any
liability arising out of the application or use of any product or circuits described
herein.
LIFE SUPPORT APPLICATIONS
These products are not designed for use in life support appliances, devices, or systems
where malfunction of these products can reasonably be expected to result in personal
injury. Nordic VLSI ASA customers using or selling these products for use in such
applications do so at their own risk and agree to fully indemnify Nordic VLSI ASA
for any damages resulting from such improper use or sale.
Application Note. Revision Date : 29.02.2000.
Application Note order code : 290200-nAN400-03
written permission of the copyright holder.
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