# Mutual Inductance Matching for Small Loop Antennas

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2 (2) In order to use these equations, the values of R1 and LL must be estimated from the dimensions of the loop, and the impedance to which the loop is matched, R2 (typically 50 ohms) must be known [1]. C1 is found from (3) The matching configuration of Figure 1 has the disadvantage that for a given set of entry parameters, only one value each is found for C1 and C2. These results may not even be close to standard capacitor values. Matching configurations Figures 2(a) and 2(b) provide an additional degree of freedom so that at least one of the matching component values can be chosen to be a standard value, preferably C1 since it is most critical, and the others calculated accordingly. The equations are shown in (7) to (13). As before, C1 is found using (3). The resulting component values are varied by adjusting the chosen Q according to equation (7). (7) where f is the frequency of operation. The loss resistance R1 is the sum of the radiation resistance, loop conductor resistance, component losses and losses due to surrounding objects. The radiation resistance in ohms is (4) Figure 2(a): (8) where A is the area enclosed by the loop and λ is the wavelength with dimension units corresponding to A. Loop conductor loss in ohms can be approximated by (9) (10) (5) where len is the loop perimeter, w is conductor width in the same units as len, and freq is the frequency in MHz. Equation (6) below gives a good approximation for the inductance of a printed rectangular loop in nh with sides s1 and s2 and conductor width w, lengths in mm[2]. (6) Figure 2(b): (11) (12) (13) 2

3 Problem of Spurious Capacitance We have seen that we have to know the loop inductance and total resistance in order to calculate the matching network component values. While there are fairly accurate formulas for coil inductance, it is often found that the actual inductance is higher than that calculated. The reason is spurious capacitance across the coil terminals. This capacitance can be due to nearby conducting objects or PC board traces that are referenced to ground. An example demonstrates the effect of the spurious capacitance on the effective impedance and resistance of the loop. Assume a printed square loop, 30mmx30mm with 2mm trace width. Operating frequency is 434 MHz. From formulas (4), (5) and (6) we find R=.274 ohm and L=82 nh. Now we assume there is spurious capacitance of C=0.8 pf and the circuit then looks like Figure 3. We can find the impedance of this circuit at f=434mhz, which is 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.3 ohms [3]. It is very difficult to measure the spurious capacitance 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. The measurements are done as follows. In the circuit of Figure 1(a), put a short circuit in place of C2. Use a standard value for C1 that is close to the value that resonates with the value of LL that is calculated from the loop dimensions. This is Ca. The true resonant frequency is measured using a vector or scalar network analyzer configured for one-port measurements. Connect a small round wire loop of approximately 10mm diameter to the end of a coaxial cable that is connected to the analyzer port. When the coil is brought near to the loop, a narrow dip is observed on the analyzer display (Figure 4). The trough of the dip is the first resonant frequency, fa. Now replace Ca with a value, Cb, that is approximately 30% lower and measure a second resonant frequency, fb. Spurious capacitance and loop inductance are calculated as follows: (14) (15) Then, the apparent inductance at the desired frequency f is (16) For example, assume you use a parallel capacitor of 2.2pF and observe resonance at 363.8MHz. Then you change the capacitor to 1.5pF and see resonance at MHz. From equations (14) and (15), spurious capacitance is.8pf and the true inductance is 63.8nH. At the operating frequency of 434 MHz, (16) shows that the apparent inductance is 102.8nH. The effective loop resistance, derived from Figure 3, is (17) In this example, R=.274 ohms and from (17), R' =.71 ohms. There are two important consequences of the spurious capacitance. Since the effective inductance increases, a smaller serial capacitance is needed for resonating the matching circuit. In the case of a large area loop, the self resonant frequency may even be below the operating frequency, making the effective impedance capacitive instead of 3

4 inductive. Another aspect is that the spurious capacitance may not be stable, and consequently the antenna performance will vary over time and between different production units. Inductive matching A version of inductive matching in which the feed loop is connected to the radiating loop by wire taps was described previously in Microwaves and RF [4]. Our approach is to couple the loops electromagnetically, with no direct connection between them. A configuration of two concentric circular loops is illustrated in Figure 5. 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 but separated by a distance z as shown in the figure. 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 circuit board, 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]. Reactive components are labeled as impedances. X1 and X2 are primary and secondary winding impedances and XM is the mutual impedance of the transformer. This circuit can be compared to the matching circuit of Figure 2(b). By making the transformer such that, at the desired operating frequency, X1+XM equals the impedance of L2 in 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. Thus, by using equations (11), (12), and (13) to design the matching circuit of Figure 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 Figure 2(b), a capacitor is needed in series with the antenna loop to adjust the impedance of the secondary branch of Figure 6. The parameter Q in (11) and (12) is also adjusted to help make Figures 2(b) and 6 compatible. Now we turn to the problem of finding the mutual inductance M 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 which is created by current flowing in the other coil. Symbolically: (18) The flux φ is related to the magnetic field B by the formula (19) which states, in words, that the flux is the integral over the dot product of the magnetic field and the area through which the flux flows. Starting from (18) and (19) the following equation was derived from which the M can be found when the geometry of the coils, or the loops in our case, are defined mathematically [6] (20) where µ 0 =permeability, ds, ds' are differential sections on the two loops, C and C', and R ss' is the distance between those sections on the two curves. Note that ds ds' 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. (20) may be more easily understood by referring to Figure 7. 4

5 Equation (20) implies that the value of M depends on the shapes of the inner and outer loops, since the integration is taken over their perimeters. However, the mutual inductance depends predominantly on the areas included by the loop perimeters. In order to use this observation, we must decide on normalizing factors 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 we can calculate the mutual inductance from (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 areas the area of the smaller loop divided by that of the larger loop. Two such curves are plotted in Figures 8 and 9. Figure 8, was plotted for two circular loops whose centers are on the same axis. The five curves shown are for separations between the parallel planes of the two loops of 0,.1,.2,.3, and.4 times the square root of the area of the larger loop. In Figure 9, 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 Figures 8 and 9 were calculated from circular loops, they are usable for rectangular or irregular shaped perimeter loops, as long as the enclosed areas and self inductances can be defined. Example An example of the use of mutual inductance curves to design an inductively coupled loop antenna is based on the geometry shown in Figure 10. The outer loop has sides of 30 mm and an area of 900 mm 2. The printed conductors are 2 mm wide. The feed loop is positioned 9 mm from the radiating loop, with its feed point opposite an edge of the large loop. We want to estimate the feed loop area that is needed to match a 50 ohm transmitter or receiver circuit. Operating frequency is 434 MHz. Using (4), (5) and (6), we estimate the radiating loop inductance to be 82 nh and total resistance to be matched, including the resistances of the resonating capacitor and some proximity losses, at 1 ohms. We proceed to find the feed loop area through the following steps. 1) Use (7), (11), (12) and (13) to find matching component values for Figure 2(b), and consequently the required mutual inductance. R2 = 50 ohms, R1 = 1 ohm, and we decide to start with Q = 14. The mutual coupling capacitance C2 = pf. Its equivalent value in terms of inductance at 434 MHz is nh, so the required mutual inductance M in Figure 6 is 4.6 nh. The required inductance L2 at the impedance step up side, on the left of the matching circuit Figure 2(b), is 31.4 nh. 2) Now we need to estimate the area of the feeding loop. Figure 11 is a magnified portion of the skewed loop curves of Figure 9. The separation of the loop planes is 9 mm, so we use the curve in Figure 11 labeled "z=.3xsqrt(area)" (which equals.3 x sqrt(900)=9). 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 Figure 10 is 4.6/30 = The corresponding relative area from the z = 9 curve is found to be approximately Multiplying by the radiating loop area of 900 mm 2 the estimate of the feed loop area is 207 mm. We take the feed loop dimensions to be 14 mm x 14 mm. 3) Using equation (6) with a conductor width of 2 mm and mean sides of 14 mm x 14 mm we estimate the self inductance of the small loop to be 29.5 nh. 4) The estimates of M and L1 must be tested in the equivalent transformer circuit, Figure 6, to see if the left branch meets the matching requirement of Figure 2(b). That is, the sum of the self induction of the small loop (corresponding to X1 in Figure 6) plus the 5

7 Figures Figure 1. A simple matching network for a loop antenna is shown in (a), whereas (b) shows the antenna as equivalent inductor and resistor circuit elements. (a) (b) Figure 2. Alternate coupling circuits (a) and (b) give additional freedom to chose component values. (a) (b) 7

8 Figure 3. Spurious capacitance C affects the apparent inductance and resistance of the loop. Figure 4. A network analyzer and small loop can be used to find resonant frequency, without a direct connection to the circuit. 8

9 Figure 5. Two concentric single turn loops. The small (feed) loop connects to the device RF circuit and the large loop is the antenna. y x z z Figure 6. This equivalent circuit of a transformer presents the reactive components as inductive reactances. X1 XM XM X2 R2 -XM R1 9

10 Figure 7. ds and ds are infintesmal sections on circular loops C and C whose directions are shown by the arrows. Rss is the distance between them. ds' Rss' C' ds C Figure 8. The figure shows normalized curves of mutual induction vs. relative loop area, with the normalized distance between loop planes as a parameter. The normalizing factor for inductance and loop separation is the square root of the area of the antenna loop. The centers of the two loops are on the same axis. 10

11 Figure 9. This figure is similar to that of Figure 8 except that the feed loop axis is skewed such that its perimeter at one point is opposite that of the antenna loop. Figure 10. An inductively coupled and matched loop antenna configured as shown was simulated on Sonnet Lite. 11

12 Figure 11. This magnified portion of the skewed loops curve in Figure 9 is convenient for finding the required small loop area. Figure 12. The Smith chart from a Sonnet Lite simulation indicates some under coupling. 12

13 Figure 13. Loop antenna for 318 MHz with mutual inductance coupling. 13

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