AN639: Design of Printed Trace Differential Loop Antennas

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1. Introduction
This application note discusses the general principles involved with designing a printed circuit trace differential loop antenna, suitable for use with sub-GHz RFICs, such as the Si4010/Si4012 from Silicon Labs. This application note also provides a general tutorial on how to design a differential loop antenna, using a combination of design equations and simulation techniques.
Use of loop antennas in small radio devices is often desirable for several reasons. Many modern RFICs use differential circuitry to achieve better performance and provide rejection against common-mode signals; the inherent differential structure of a loop antenna interfaces well to such circuitry. Loop antennas are primarily H-field radiators (compared with E-field radiators, such as monopole antennas) and are somewhat less susceptible to detuning because of hand or body effect. Loop antennas may easily be designed using printed circuit traces, allowing a reduction in Bill of Material (BOM) cost. They may also be designed with a relatively small physical size, allowing integration in a very small form factor.
However, designing a loop antenna (or any antenna) is not the simplest of tasks. It would be convenient to simply select an antenna design from a proven library of existing designs, or to construct an antenna from a template design and then further scale the dimensions to the desired frequency of operation. In practice, however, it is rarely possible for a designer to copy an existing antenna design exactly, without any modification; it is generally necessary to change the antenna layout (at least slightly) to fit within the user’s form factor. Even such minor modifications can result in changes in the performance of the antenna. Thus, new antenna designs generally require simulation and/or bench adjustment, unless they are exact replicas of existing designs.

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Copyright © 2021 by Silicon Laboratories


2. Design Approach
Silicon Labs recommends the following approach to design a loop antenna:
1. Estimation of required antenna dimensions using basic design equations, given the desired link range. 2. Calculation of tuning components required to resonate loop antenna at the desired operating frequency. 3. Simulation of proposed antenna geometry using antenna/EM simulation software. 4. Fabrication of PCB containing printed antenna structure. 5. Bench measurement of antenna resonant frequency and input impedance. 6. Adjustment of discrete tuning capacitors to optimize resonant frequency and input impedance. 7. Modification of physical layout of antenna structure (only if adjustment of discrete components is not sufficient).
Although it is obviously desirable to achieve optimal performance on the initial design, it is generally not possible to calculate or simulate with this degree of accuracy. Most simulations inherently make use of simplifications or approximations in order to speed up simulation time and to reduce simulation memory requirements. These simplifications often introduce small errors in the simulation results that must then be corrected through measurement and adjustment on the bench. While it is sometimes possible to increase the complexity of the simulation model to reduce these errors, the result is generally a drastic increase in required simulation time for only a moderate improvement in simulation accuracy. In short, it is usually quicker to use just enough complexity in the simulation to “get close enough”, and to complete the optimization of the design through bench measurement and adjustment. It is quite normal for an initial fabricated antenna design to match no closer than 5%–10% to the simulation results (e.g., the actual measured resonant frequency differs slightly from the simulated resonant frequency). These differences can generally be corrected through adjustment of discrete tuning components, without the need for another board spin.
2.1. Design Characteristics of Loop Antennas
The following represents a brief list of some of the characteristics of loop antennas; each of these items is discussed in further detail below.
 Differential structure  High input impedance  Narrowband (high-Q)  Resonant frequency is inversely proportional to loop size  More efficient radiator with larger loop size 2.1.1. Differential Structure
The loop antenna is nominally a balanced structure and thus interfaces well to a differential circuit, such as a differential PA output or a differential LNA input. The designer should strive to maintain physical symmetry of the antenna layout in order to obtain optimal performance.
2.1.2. High Input Impedance
The input impedance of a loop antenna at natural resonance is quite high, ranging anywhere from ~10 kΩ to 50 kΩ. This characteristic high input impedance is a result of the loop antenna operating in a parallel-resonant mode at the desired frequency of operation. This impedance value is much higher than the typical impedance of the circuitry to which the antenna is expected to interface (e.g., PA output or LNA input). It is possible to transform the loop antenna impedance to a lower value through the use of discrete reactive components (e.g., capacitors) or through the use of impedance transforming structures (e.g., a tapped loop). However, it may not always be possible to achieve a complex conjugate match (desirable for optimum power transfer), due to other constraints such as peak voltage swing.


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2.1.3. Narrowband (High-Q)
The natural resonance of a loop antenna is quite narrowband, perhaps only 5–10 MHz in bandwidth. The tuning of a loop antenna may be affected by nearby objects (i.e., hand effect or body effect); thus it is recommended that some method of automatically tuning the antenna back to resonance be provided in the RFIC. One advantage of a high-Q antenna is that it provides attenuation of harmonic signal components, allowing the filtering required from discrete circuitry to be relaxed (or even eliminated).
2.1.4. Resonant Frequency
The natural resonant frequency of a loop antenna is inversely proportional to the size of the loop antenna: the larger the antenna, the lower its natural frequency of resonance. However, it is generally possible (through the use of discrete tuning components) to tune a small loop antenna to resonance at frequencies well below its natural resonant frequency. This is usually desirable, as the discrete tuning components also provide a means by which the high native impedance may be transformed to a lower and more useful value.
2.1.5. Radiation Efficiency
The radiation efficiency of a loop antenna generally increases with size. If the designer has a choice in selecting the loop antenna size (assuming both may be tuned to resonance through the use of discrete tuning components), the larger antenna will generally provide better performance.

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3. Loop Antenna Design Equations
A typical rectangular printed loop antenna structure is shown in Figure 1, where:  a1, a2 = dimensions of the sides of the loop antenna (in meters), measured from the center of the traces  t = thickness of the trace conductor (in meters)  w = width of the trace conductor (in meters)

Figure 1. Printed Loop Antenna
From these dimensions the total length (i.e., perimeter) of the loop (in meters) and the area of the loop (in meters2) may be calculated as:
l = 2  a1 + a2 Equation 1.
A = a1  a2 Equation 2.


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Many equations for inductance assume a conductor with a circular cross-section of radius ‘b’ (i.e., a wire). An effective radius ‘b’ of a printed trace conductor may be calculated as:

b = 0.35  t + 0.24  w Equation 3.
The inductance of a square loop (a1 = a2 = a) may be calculated as1:

L = -2-------------o---------a-  lna-- – 0.774

 b

Equation 4.

Here, µo equals the free space permeability and is given by:

o = 4  10–7 = 1.256E-6 H/m
In the event that a rectangular loop antenna is used (a1 ≠ a2), Equation 4 may be still be used with an effective or mean loop dimension:

a = a1  a2 Equation 5. In the event a circular loop antenna is used, the inductance may be calculated as2:

L = o  a 

ln-8---a-- – 2  b

Equation 6.

The radiation resistance of a small loop antenna is given by3:

R = 3204 A-----2-



= 3204 A-----2---f--24

Equation 7.

1C.A. Balanis, “Antenna Theory: Analysis and Design”, John Wiley & Sons, 2005, p. 245. 2Ibid. 3Ibid, p. 238.

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Radiation resistance is a “good” type of loss mechanism, as it is through the mechanism of radiation resistance that power is transferred from the applied conducted signal to the free space wave. However, there are other loss mechanisms present in the antenna as well, including ohmic trace loss (RTRACE), PCB dielectric loss (RPCB), and ESR of discrete tuning components (RESR) due to finite Q-factors. The total series resistance of the loop antenna is the sum of all of these factors:

Equation 8. The radiation efficiency of the loop antenna is given by the ratio of the radiation resistance to the total series resistance:
 = ------------------------------------R----R----A----D------------------------------------- = R-----R----A---D-r RRAD + RTRACE + RPCB + RESR RSER Equation 9.
From these equations, it is clear that the efficiency (gain) of the antenna may be optimized by increasing the radiation resistance while minimizing the other loss factors. The high-frequency resistance of a printed trace (assuming very small skin depth) may be calculated as:


= ---1---- ----f------o-

TRACE 2w 

Equation 10.
Here, sigma represents the conductivity of copper (5.8E7 Siemens/meter).
These equations show that the trace loss RTRACE ~ loop perimeter ~ (loop area)1/2, while RRAD ~ (loop area)2. As loop size is increased, the radiation resistance increases faster than the trace loss and the efficiency of the loop antenna is improved with larger loop area.
Therefore, the antenna designer should (nearly) always strive to maximize the size of the loop antenna within the available board space.
However, the loop antenna size should not be increased so large that it approaches self-resonance on its own, without the need for capacitor tuning elements; the resonant frequency of such an antenna would be quite sensitive to small variations in the design parameters.
The designer may be familiar with the characteristic impedance of more commonly encountered antenna structures such as dipoles (Zo = 73 Ω) and monopoles (Zo = 36.5 Ω). In contrast, the radiation resistance of a typical loop antenna is quite low, on the order of a few ohms (or less).
In practice, the electromagnetic field surrounding a loop antenna exists partially within the dielectric material of the PCB and partially in free space. The velocity factor of a signal within the dielectric material is lower than that of free space, and thus the wavelength is correspondingly smaller. It is well known that the velocity factor in a medium other than free space is inversely related to the square root of the permittivity of the medium, and is given by:

PCB = ---c----r
Equation 11.


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In Equation 11, ‘c’ equals the speed of light in free space (3E8 meters/second). As the electromagnetic field exists simultaneously in both mediums (free space and the board dielectric material), the appropriate value of wavelength (λ) to use in Equation 7 is a value somewhere between the wavelength in free space and the wavelength in the board material. The author is not currently aware of an explicit formula to correctly predict the effective velocity factor (and thus the effective wavelength) of a signal that spans across two different mediums. However, a series of antenna simulations may be performed in which all other loss mechanisms are disabled except that loss due to radiation; from these simulations the radiation resistance RRAD may be extracted and compared to the value predicted by Equation 7. The velocity factor may then be empirically adjusted until the predicted values “curve fit” to the simulated values. From this exercise, an appropriate value of value of velocity factor may be found:
V.F. = 0.82
Equation 12.
That is to say, the appropriate value of wavelength (λ) to be used in Equation 7 is 0.82x that calculated in free space.
The loss due to the dielectric material (RPCB) is also a challenge to estimate. This loss factor is related to the loss tangent () of the dielectric material. The simulations and examples within this application note assume FR-4 board material with loss tangent  = 0.02. Again, the author is not currently aware of an explicit formula to relate the loss resistance RPCB to the loss tangent, operating frequency, and board area. Once again, a series of simulations may be performed in which individual loss mechanisms are sequentially disabled, allowing extraction of the loss resistance due to only the dielectric loss tangent. This simulation exercises results in the following set of curves of dielectric loss resistance RPCB as a function of antenna area and operating frequency. The designer may use these curves to estimate the loss resistance due to the FR-4 board material for a particular antenna design.

Figure 2. Dielectric Loss Resistance RPCB vs. Area vs. Frequency

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4. Loop Antenna Practical Design
Armed with knowledge of these basic design equations and performance curves, it is now possible to design a practical loop antenna. As an example, the area of a loop antenna that fully occupies the available space within a typical remote keyless entry (RKE) keyfob for an automobile may be only 40 mm x 25 mm = 1000 mm2. A trace width of 1.0 mm is assumed with a copper weight of 1 oz.
 F = 434 MHz  a1 = 40 mm  a2 = 25 mm  t = 0.035 mm (1 oz. copper)  w = 1.0 mm
Application of Equation 7 at an operating frequency of F = 434 MHz and applying the empirically-derived velocity factor of V.F. = 0.82 results in:

R = 3204 ----------0----.-0----4---------0----.-0----2---5------2--------- = 0.302 



0.82  3E8/434E6

Equation 13.

Application of Equation 10 results in a calculated value of trace resistance RTRACE of:



 2----------0---.--0---4-----+-----2----------0---.--0---2----5-  2  0.001 

----------4---3----4---E----6----------1---.--2---5---6----E-----–-----6-- = 0.353  5.8E7

Equation 14.
It is evident that the loss due to the trace resistance exceeds the radiation resistance, and thus the radiation efficiency is expected to be relatively low (refer to Equation 9). The radiation efficiency will degrade even further as the remaining loss mechanisms (i.e., dielectric loss and component ESR) are accounted for.
The dielectric loss resistance may be estimated from the curves of Equation 2 as being RPCB = ~0.7 Ω at an operating frequency of 434 MHz and loop area of 1000 mm2.
The inductance of the loop antenna may be calculated from Equation 4 and Equation 5 as:

L = 2--------------o-------------a---1---------a----2-  ln -----a---1---------a----2- – 0.774 = 102.64 nH

 b

Equation 15.

An equivalent lumped-element model of the loop antenna is simply an inductor in series with a resistor. The inductance may be resonated with a capacitor of value:

C = ---------1---------- = 1.31pF 2f2L Equation 16.


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The equivalent series resistance (RESR) of this capacitor is a function of its Q-factor. A reasonable estimate of the Q of surface-mount ceramic capacitors (e.g., Murata GJM1555 series) is Q ≈ 350. The RESR may be calculated as:
R = -X----C-- = --------1---------- = -----------------------------------1------------------------------------ = 0.799  ESR Q 2fCQ 2  434E6  1.31pF  350 Equation 17.
The total series resistance may be calculated as:
RSER = RRAD + RTRACE + RPCB + RESR = 0.302 + 0.353 + 0.7 + 0.799 = 2.154  Equation 18.
The expected antenna efficiency may be estimated by Equation 9 as:
 = ------------------------------------R----R----A----D------------------------------------- = 0----.-3----0---2-- = 0.14 r RRAD + RTRACE + RPCB + RESR 2.154 Equation 19.
The antenna efficiency may be expressed in dBi (dB relative to an isotropic antenna):
GANT_EFF = 10  log r = –8.53 dBi Equation 20.
This quantity represents the radiation efficiency of the antenna when summed over all spatial directions. The antenna gain typically measured in a lab environment or antenna chamber reflects the product of the antenna efficiency and the antenna directivity. The loop antenna may be brought to parallel resonance by placing this capacitor across the input terminals of the differential loop antenna.

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Figure 3. Series and Parallel Lumped Equivalent Models
The series equivalent lumped element model may be transformed into a parallel equivalent model, as shown in Figure 3. Due to the high Q-factor of the network, the parallel values of L and C remain (approximately) the same, while the parallel resistance may be calculated as:

XLSER = 2fL = 2  434E6  102.64nH = 279.89  Equation 21.

R = R 1 + Q 2 = R 1 + X-----L---S---E----R- 2 = 2.1541 + 2----7---9----.-8----9- 2 = 36.37 k




SER  RSER  

  2.154  

Equation 22.

The total series resistance of a loop antenna was observed to be quite low (a few ohms or less); in contrast, the transformed parallel resistance is quite high (several tens of kilohms). Without further impedance matching, this high value of resistance would be observed at the input terminals of the antenna at parallel resonance (i.e., at the frequency where the reactances of LP and CP cancel each other).

As this antenna impedance is much higher than the typical output impedance of a PA circuit or input impedance of an LNA circuit, it is common to transform this high impedance downwards to better match the impedance of the circuit. This impedance transformation may be provided by discrete capacitors, discrete inductors, distributed transmission lines, or a combination thereof; however, the most commonly used method is “capacitive-tapping,” as shown in Figure 4. Capacitors are typically used as discrete matching elements because their Q-factor is usually much higher than inductors, causing them to minimize any additional loss in the match.


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AN639: Design of Printed Trace Differential Loop Antennas