Loop antennas are widely used in small wireless products, particularly for UHF bands between 300 and 1000 MHz. They are small in size relative to wavelength, independent from a ground plane, and relatively immune to the influence of nearby objects. They are also easily implemented in printedcircuit form with corresponding low cost. But their low radiation resistance makes them difficult to match and subject to low efficiency. They exhibit high quality factor (Q) and associated narrow bandwidth that can make performance nonuniform over a normal range of matching component tolerances. Fortunately, mutual inductance techniques can help reduce loop-antenna losses and achieve optimal impedance matching.

In many applications, a simple circuit matches the low-resistance component of a loop antenna to the generally higher output impedance of a transmitter or the input impedance of a receiver * Fig. 1(a)*>.

*is an equivalent-circuit representation with the loop impedance shown as loop inductance LL and resistance R1. R1 is the sum of the radiation resistance, antenna losses, and the series resistance of capacitance C1. The antenna efficiency is the ratio of the radiation resistance to R1. In*

**Figure 1(b)***, C1 is shown in series with the loop; L1 in*

**Fig. 1(a)***is the equivalent of the loop inductance LL, reduced by the negative reactance of C1. The equations for L1 and C2 to match R1 to R2 are*

**Fig. 1(b)**The values of R1 and LL must be estimated from the dimensions of the loop, and the impedance to which the loop is matched, and R2 (typically 50 Ohms) must be known.^{1} The value of C1 can then found from

where

f = the frequency of operation.

Loss resistance R1 is the sum of the radiation resistance, loop conductor resistance, component losses, and losses due to surrounding objects. Equation 4 provides the radiation resistance (R_{rad}) in Ohms:

where

A = the area enclosed by the loop

and

= the wavelength with dimension units corresponding to A.

The loop conductor loss (in Ohms) can be approximated by

where

len = the loop perimeter,

w = the conductor width in the same units as len, and

freq = the frequency in MHz.

Equation 6 gives a good approximation for the inductance of a printed rectangular loop (in nH) with sides s1 and s2 and conductor width w, (with units in mm):^{2}

The matching configuration of * Fig. 1* (has the disadvantage that, for a given set of entry parameters, only one value each is found for C1 and C2, and these results may not even be close to standard capacitor values. Matching configurations in

*and*

**Figs. 2(a)***provide an additional degree of freedom so that at least one of the matching component values can be chosen as a standard value, preferably C1 since it is most critical, and the others calculated accordingly (Eqs. 7-13). As before, the value of C1 is found using Eq. 3. The resulting component values are varied by adjusting the chosen Q according to Eq. 7:*

**2(b)**From * Fig. 2 (a)*,

with C2 and C3 in Eqs. 9 and 10, and from * Fig. 2 (b)*,

and C2 and L2 in Eqs. 12 and 13.

Although it is necessary to know the loop inductance and total resistance to calculate matching-network component values, this knowledge may not come routinely. While there are fairly accurate formulas for coil inductance, the actual inductance is often higher than the calculated value due to spurious capacitance across the coil terminals. This capacitance can be due to nearby conducting objects or printed-circuit-board (PCB) traces referenced to ground. An example helps to demonstrate the effect of the spurious capacitance on the effective impedance and resistance of the loop.

Assume a printed square loop measuring 30 x 30 mm with 2-mm trace width and frequency of 434 MHz. From Eqs. 4, 5, and 6, R = 0.274 Ohm and L = 82 nH. Now, assume a spurious capacitance of C = 0.8 pF, and the circuit then has the form of * Fig. 3*. The impedance of this circuit at 434 MHz can be found to be Z = 1 + 436j Ohms. The inductance of the circuit, found from the reactance of 436 Ohms, is 160 nH. The resistance to be matched to the transmitter or receiver output or input impedance is 1 Ohm, plus the resistive loss of the series capacitor of the matching circuit, which will be approximately 0.3 Ohms.

^{3}

The spurious capacitance is difficult to measure directly, but it can be determined by calculation from the resonant frequency of the loop when it is connected in parallel to two different capacitors. Measurements are performed as follows. In the circuit of * Fig. 1(a)*, a short circuit is used in place of C2. A standard value should be chosen for C1 that is close to the value that resonates with the value of LL that is calculated from the loop dimensions; this capacitance is called Ca. The true resonant frequency is measured using a vector network analyzer (VNA) or scalar network analyzer (SNA) configured for one-port measurements. A small round wire loop of approximately 10 mm diameter is attached to the end of a coaxial cable connected to the analyzer test port. When the coil is brought near to the loop, a narrow dip will be observed on the analyzer's display (

*). The trough of the dip is the first resonant frequency, fa. The next step is to replace Ca with a capacitance value that is about 30-percent lower, called Cb, and measuring the resulting second resonant frequency, fb. The spurious capacitance and loop inductance are calculated as follows:*

**Fig. 4**Then, the apparent inductance at the desired frequency f is

For example, assume a 2.2-pF parallel capacitor with observed resonance at 363.8 MHz. The capacitor is changed to 1.5 pF and the resonance is then observed at 415 MHz. From Eqs. 14 and 15, the spurious capacitance is 0.8 pF and the true inductance is 63.8 nH. At an operating frequency of 434 MHz, Eq. 16 shows that the apparent inductance is 102.8 nH. The effective loop resistance, derived from * Fig. 3*, is

In this example, R = 0.274 Ohms and, from Eq. 17, R' = 0.71 Ohms.

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There are two important consequences of the spurious capacitance. Since the effective inductance increases, a smaller serial capacitance is needed to resonate the matching circuit. In the case of a large-area loop, the selfresonant frequency may even be below the operating frequency, making the effective impedance capacitive instead of inductive. Another aspect is that the spurious capacitance may not be stable and, consequently, antenna performance may vary over time and between different production units.

A version of inductive matching in which the feed loop is connected to the radiating loop by wire taps was described previously.^{4} The present approach is to couple the loops electromagnetically, with no physical connection between them. * Figure 5* shows a configuration of two concentric circular loops. The feed loop does not have to be centered on the axis of the radiating loop. The two loops may be in the same plane, but usually their planes are parallel and separated by a distance z. This arrangement has several advantages: the radiating loop is floating and less susceptible to spurious capacitance, even harmonics are suppressed, component loss is reduced, and there is physical flexibility in some applications when the radiating loop is a small distance from the PCB, with no connecting wires.

The two loops effectively form a transformer, coupled through their mutual inductance. * Figure 6* is a simplified equivalent circuit of a transformer,

^{5}with reactive components labeled as impedances. Parameters X1 and X2 are primary and secondary winding impedances, respectively, and XM is the mutual impedance of the transformer. This circuit can be compared to the matching circuit of

*. By making the transformer in such a way that, at the desired operating frequency X1 + XM equals the impedance of L2 in*

**Fig. 2(b)***, X2 + XM equals the impedance of L1, and the mutual impedance XM is the negative of the impedance of C2, the transformer circuit will match R1 to R2. By using Eqs. 11, 12, and 13 to design the matching circuit of*

**Fig. 2(b)***, the implementation can be carried out by a transformer with appropriate values of mutual impedance and primary and secondary winding impedances. The transformer will be in the form of the two electromagnetically coupled loops. As in*

**Fig. 2(b)***, a capacitor is needed in series with the antenna loop to adjust the impedance of the secondary branch of*

**Fig. 2(b)***. Parameter Q in Eqs. 11 and 12 is also adjusted to help make*

**Fig. 6***and*

**Figs. 2(b)***compatible.*

**6**Mutual inductance M must be found from the size and configuration of the antenna loops. The mutual inductance M between two coils can be defined as the flux enclosed by the windings of one of the coils that is created by current flowing in the other coil. Symbolically, M is represented by Eq. 18:

The flux F is related to the magnetic field B by Eq. 19:

which states that the flux is the integral over the dot product of the magnetic field and the area through which the flux flows. From Eqs. 18 and 19, Eq. 20 was derived to allow M to be found when the geometry of the coils, or the loops in our case, are defined mathematically^{6} as

where

_{0} = permeability,

ds, ds' = differential sections on the

two loops, C and C', and

|Rss'| = the distance between those sections on the two curves.

Note that dsds' is a dot product of two vectors, which means that the product is that of their scalar lengths times the cosine of the angle between them. Equation 20 may be more easily understood by referring to * Fig. 7*.

Equation 20 implies that the value of M depends on the shapes of the inner and outer loops, since the integration has taken over their perimeters. However, the mutual inductance depends predominantly on the areas included by the loop perimeters. To use this observation, normalizing factors must be found for the mutual inductance, the areas, and the separation between the planes of the loops. The normalizing factor for the mutual inductance and separation z will be taken to be the square root of the area of the outer loop, and the small loop area will be expressed as relative to the area of the outer loop.

Now, it is possible to calculate the mutual inductance from Eq. 20 and plot relative induction, which is the mutual induction divided by the square root of the area of the larger loop, against the relative areasthe area of the smaller loop divided by that of the larger loop. * Figures 8* and

*plot such curves.*

**9***was plotted for two circular loops whose centers are on the same axis. The five curves are for separations between the parallel planes of the two loops of 0, 0.1, 0.2, 0.3, and 0.4 times the square root of the area of the larger loop. In*

**Figure 8***, the axes of the two loops are skewed. The distance between them are such that the perimeter of the smaller loop almost touches the larger loop when z = 0. Although the curves in*

**Fig. 9.***and 9 were calculated from circular loops, they are usable for rectangular or irregularly shaped perimeter loops, as long as the enclosed areas and selfinductance can be defined.*

**Figs. 8*** Figure 10* shows an example of how mutual inductance curves can be used to design an inductively coupled loop antenna based on the given geometry. The outer loop has sides of 30 mm and an area of 900 mm2, with 2-mm-wide printed conductors. The feed loop is positioned 9 mm from the radiating loop, with its feed point opposite an edge of the large loop. A goal is to estimate the feed loop area needed to match a 50-Ohm transmitter or receiver circuit at 434 MHz.

Using Eqs. 4, 5, and 6, the radiating loop inductance is estimated to be 82 nH and total resistance to be matched, including the resistances of the resonating capacitor and some proximity losses, is 1 Ohm. The feed loop area can then be found through the following steps.

1. Use Eqs. 7, 11, 12, and 13 to find matching values for the components in * Fig. 2(b)* and, consequently, the required mutual inductance. Values of R2 = 50 Ohms and R1 = 1 Ohm are selected, and Q is assigned a value of 14. The mutual coupling capacitance C2 = 29.25 pF. Its equivalent value in terms of inductance at 434 MHz is -4.6 nH, so the required mutual inductance M in

*is 4.6 nH. The required inductance L2 at the impedance step up side, on the left of the matching circuit*

**Fig. 6***, is 31.4 nH.*

**Fig. 2(b)**Continue to page 3

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2. Now it is necessary to estimate the area of the feeding loop. * Figure 11* is a magnified portion of the skewed loop curves of

*. The separation of the loop planes is 9 mm, so the curve in*

**Fig. 9***labeled "z = 0.3xsqrt(Area)" is used. The mutual inductance is normalized by dividing M by the square root of the radiating loop area, so the relative inductance on the ordinate of*

**Fig. 11***is 4.6/30 = 0.153. The corresponding relative area from the z = 9 curve is found to be approximately 0.23. Multiplying by the radiating loop area of 900 mm*

**Fig. 10**^{2}yields an estimate of 207 mm

^{2}for the feed loop area. The feed loop dimensions are assumed to be about 14 x 14 mm.

3. Using Eq. 6 with a conductor width of 2 mm and mean sides of 14 x 14 mm, the self-inductance of the small loop is estimated to be 29.5 nH.

4. The estimates of M and L1 must be tested in the equivalent transformer circuit, * Fig. 6*, to see if the left branch meets the matching requirement of

*. That is, the sum of the selfinduction of the small loop (X1 in*

**Fig. 2(b)***) plus the mutual induction M should equal the calculated value of L2 in*

**Fig. 6***. The inductance of the right branch in*

**Fig. 2(b)***does not have to be considered because it will be adjusted when C is chosen for resonance at the operating frequency. The small loop induction plus M = 29.5 + 4.6 = 34.1 nH. The required inductance found for L2 in*

**Fig. 2(b)***was 31.4 nH, which is close enough for simulation or initial prototype implementation. In the event that the left branch requirement of*

**Fig. 2(b)***is not met, the estimation process must be restarted at step 1, with different value of Q, to change the required mutual inductance and left branch inductance.*

**Fig. 2(b)**A simulation was performed using Sonnet Lite electromagnetic (EM) software from Sonnet Software,^{7} with a small loop geometry as determined in step 2. The layout and dimensions of the loops are shown in * Fig. 10*. The Smith chart in

*shows the simulation result. The return loss was simulated as 14.8 dB at 434 MHz using a resonating capacitance of 1.50 pF. Slight undercoupling is evident from the Smith chart, and the matching can be improved by increasing the area of the small loop, bringing it a little closer to the larger loop, or adding an appropriate capacitance in series with the feed line.*

**Fig. 12**To demonstrate the effectiveness of this approach, a loop antenna matched for 318 MHz was fabricated by means of the mutual inductance coupling technique (* Fig. 13*). A 50- Ohm coaxial cable connects the feed loop to the main RF circuit board with no additional matching components required. Considering the advantages of inductive coupling, the method can be recommended for many compact, short-range devices at UHF.

REFERENCES

1. Alan Bensky, Short-range wireless communication, 2nd Ed., Elsevier, 2004, pp. 49-51.

2. F. L. Dacus, J. Van Niekerk, and S. Bible, "Introducing loop antennas for short-range radios," Microwaves & RF, July 2002.

3. Murata Chip S-Parameter & Impedance Library, Version 3.5.0, Murata Manufacturing Co., Ltd.

4. J. Van Niekerk, F. L. Dacus, and S. Bible, "Matching loop antennas to short-range radios," Microwaves & RF, August 2002.

5. W.W. Lewis and C. F. Goodheart, Basic Electric Circuit Theory, The Ronald Press Company, New York, 1958, p. 223.

6. Wikipedia, definition for inductance.

7. Sonnet Software, North Syracuse, NY.