Power waves and scattering parameters in magneto-inductive systems

Difficulties arise in the definition of power flow in transmission-line systems with a complex propagation constant. These were resolved by Kurokawa using quantities known as “power waves,” which contain both voltage and current terms and correctly separate power flow into forward- and backward-traveling components. Similar difficulties must arise for electromagnetic metamaterials since any discrete, periodic structure leads to band-limited propagation, with a complex propagation constant both inside and outside the bands due to loss and cutoff, respectively. Here, discrete power waves are defined for magneto-inductive (MI) systems, metamaterials based on chains of magnetically coupled LC resonators. These waves are shown to satisfy the discrete power conservation equation for MI waves and are used to calculate scattering parameters for multi-port MI devices without the anomalous predictions of conventional methods. The results will allow correct evaluation of internal scattering parameters in MI systems


I. INTRODUCTION
Scattering parameters provide tools to analyze reflection and transmission at discontinuities in electromagnetic systems or media. 1,2 However, it is well known that there may be inconsistencies when port impedances are complex or there is gain or loss. 3,4 These arise from the fact that while amplitudes are linear quantities, powers are not. Conventional definitions may then result in anomalies, such as a reflection coefficient being apparently greater than unity even for a passive load. 5 These difficulties were resolved for transmissionline systems by Kurokawa 6 and others. [7][8][9] For input voltages and currents vi and ii into a set of ports of impedance zi, the new quantities ai = (vi + ziii)/{2 √ Re(zi)} are introduced, where " * " denotes the complex conjugate. Since 1 2 {|a i| 2 − |bi| 2 } = 1 2 Re(iiv * i ), forward and backward power flow may then be separately described using these "power waves" by the terms 1 2 |ai| 2 and 1 2 |bi| 2 . This allowed a consistent definition of scattering parameters, for example, for a two-port device as S 11 = b 1 /a 1 , S 12 = b 1 /a 2 , S 21 = b 2 /a 1 , and S 22 = b 2 /a 2 .
The aim of this paper is to develop a power wave formulation for a periodic system supporting current waves that can be used to obtain scattering parameters at junctions between media in addition to terminations. Important differences from Kurokawa's work are introduced by the discrete nature of the system and lack of voltages at internal elements. In Sec. II, we derive the power conservation relation for MI waves. In Sec. III, we prove that discrete variants of power waves satisfy this relation and show how they may be used to find scattering parameters in MI systems. In Sec. IV,

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scitation.org/journal/adv we illustrate the process with simple examples for N-port MI devices where conventional methods break down. Conclusions are drawn in Sec. V.

II. POWER CONSERVATION IN MAGNETO-INDUCTIVE SYSTEMS
We begin by considering an infinite MI waveguide formed from magnetically coupled, lossy resonant loops, as shown in Fig. 1.
At a low frequency, the elements may be modeled as lumped circuits of inductance L (with associated resistance R) and capacitance C, coupled by mutual inductance M. For nearest-neighbor coupling, the equation relating the currents In in adjacent loops at angular frequency ω is The assumption of forward-going traveling wave solutions in the form In = IF exp(−jnka), where IF is the current amplitude, k is the propagation constant, and a is the element spacing, yields the well-known dispersion equation 12,13 R + jωL + 1/jωC + 2jωM cos(ka) = 0.
This result may be expressed alternatively in the normalized form as Here, ω 0 = 1/ √ LC is the resonant frequency, Q = ω 0 L/R is the quality factor, and κ = 2M/L is the coupling coefficient. Lossless systems are band-limited to a frequency range determined by κ (which may be positive or negative), namely, Within this band, k is real, with ka = π/2 at resonance. Outside it, k abruptly becomes imaginary. It is also well known 23 that the characteristic impedance is In general, Z 0 is complex. However, when k is real, we can obtain Re(Z 0 ) = ωM sin(ka).
In this case, Re(Z 0 ) reduces to ω 0 M at resonance and to zero at the band edges. Lossy systems have a complex propagation constant k = k ′ − jk ′′ and allow propagation out-of-band. Figure 2 shows the frequency dependence of k ′ a and k ′′ a, for example, parameters of κ = 0.6, Q = 10 000. Here, the black dotted lines show the cutoff frequencies, highlighting the band-limited nature of propagation in low-loss systems.   Figure 3 shows the corresponding variation of Z 0 ; the wide variations in both the real and imaginary parts make broad-band impedance matching difficult.
Adding (7) to its own complex conjugate, dividing by 4, and rearranging, we then get Since the first term above is the power P d dissipated in element n, (8) is a power conservation relation, with the second and third terms representing the powers P n+1 and Pn flowing into element n + 1 from element n and into n from n − 1, respectively. Figure 4 shows variations of P d , Pn, and P n+1 with n for a very lossy MI waveguide with parameters κ = 0.6 and Q = 10, assuming that L = 100 nH, f 0 = 1 GHz, and ω/ω 0 = 1. Figure 4(a) shows the results obtained when there is only a forward-going wave present so that In = IFexp(−jnka), with IF = 1 mA. In this case, all three variations decrease exponentially with n as expected. Figure 4(b) shows results when both forward-and backward-going waves are present so that In = IF exp(−jnka) + IB exp(+jnka), with IF = IB = 1 mA. In this case, P d , Pn, and P n+1 are not (as might initially be expected) sums or differences of exponentials; there is an additional oscillation caused by beating. In both cases, we have verified that the power conservation relation (8) is satisfied.

III. DISCRETE POWER WAVES AND SCATTERING PARAMETERS
We now seek to express power flow for the discrete current waves that exist in MI systems. To do so, we introduce a discrete form of power wave. We will need two types. For lines extending to the left, we define type 1 amplitudes A1n and B1n at element n as Note that these expressions contain no voltages. However, analogies with Kurokawa's a and b coefficients follow from the fact that −jωMI n−1 is the voltage induced in element n by element n − 1.
Multiplying out, we get Subtracting these equations from each other and dividing by 2, we then get In a similar way, we can obtain Equations (11) and (12) may then be recognized as Pn and P n+1 , respectively. Substituting into (8) and re-arranging the equation so that all terms are positive, the result is Equation (13) then allows the direction of the separate power flows carried by each power wave to be identified as shown in Fig. 1. Before proceeding, we briefly consider some properties of these waves when k is real and the current consists of the sum of a forwardgoing wave of amplitude IF and a backward wave of amplitude IB so that In can be written as Direct substitution shows that Clearly, A1n is only dependent on IF in this case, while B1n is only dependent on IB. The forward and backward power flows are then These results imply that for real k, there will be no difference between calculations based on power waves and on the moduli squared of the separate current amplitudes. However, differences are to be expected when k is complex. We now consider the second type of discrete power wave for lines extending to the right. For consistency with Kurokawa's direction of current flow as "inward," 6 we define type 2 amplitudes A2n and B2n at element n as Here, jωMI n+1 is the voltage induced in element n by element n + 1.
Carrying out similar manipulations, it is simple to show that This power conservation relation is in a similar form to (13), but the terms are different. The two types of discrete power waves allow sources at different locations, and both are needed to compute the full set of S-parameters. For example, Fig. 5 shows a two-port MI device extending from element m to element n; here, the scattering parameters are S 11 = B1m/A1m, S 12 = B1m/A2n, As expected, these expressions are analogous to Kurokawa's scattering parameters. Simple conclusions may be drawn immediately. For example, if k is purely imaginary, which occurs out-of-band in lossless systems, the conjugate impedance is Z * 0 = −Z 0 . In this case, B1m = A1m and B2n = A2n, so S 11 and S 22 are unity for any k.
Before presenting examples, we show how Kurokawa's scattering parameters can be found for terminated MI circuits; we term this the circuit method. Figure 6(a) shows the arrangement of a two-port device using his notation for voltages v 1 and v 2 and currents i 1 and i 2 at ports 1 and 2.
In this case, the power wave scattering parameters are If v 1 is derived from a source at port 1 with voltage v 01 , v 1 = v 01 − z 1 i 1 . Similarly, in the absence of a source at port 2, v 2 = −z 2 i 2 .
These results may therefore be written as These expressions allow the power wave scattering parameters to be found from a numerical solution of the circuit equations, which can yield i 1 and i 2 for a given v 01 .

IV. EXAMPLES
We illustrate the use of the discrete power waves with examples chosen to demonstrate anomalous predictions of increasing severity using conventional methods, first involving a two-port device with equal and unequal port impedances and then a three-port device.
A. Two-port device, equal port impedances We begin with the case shown in Fig. 6(b), namely, an MI waveguide discontinuity caused by a local variation in the mutual inductance from M to M ′ = μM between two elements. We follow the approach used in Ref. 24 of finding the reflection and transmission coefficients at the junction. The governing equations are as (1) for all elements except these, where (R + jωL + 1/jωC)I −1 + jωM(I −2 + μI 0 ) = 0, (R + jωL + 1/jωC)I 0 + jωM(μI −1 + I 1 ) = 0.
Solutions may be found for incidence from n = −∞ by assuming incident and reflected waves before the discontinuity and a transmitted wave after it so that Away from the discontinuity, these solutions satisfy Eq. (1). Substitution into (22) As expected, Γ = 0 and T = 1 when μ = 1. For lossless systems and in-band operation, these expressions allow calculation of |S 11 | 2 and |S 21 | 2 as |Γ| 2 and |T| 2 ; we refer to this approach as the "modulus square" method. However, for lossy systems and/or out-of-band operation, the results can be anomalous. We illustrate this by comparison with the results obtained by (a) using the circuit method (21) with an equivalent circuit model of the system in Figs. 6(a) and 6(b) substituting (24) into expressions (9) and (17) for discrete MI power waves and then evaluating the scattering parameters using (19). In each case, the ports are effectively located at elements −1 and 0. As a numerical example, we first assume parameters of κ = 0.6, Q = 10 000, and μ = 1.1, defining a weak reflector in a strongly coupled system with unfeasibly low loss. Figure 7 shows the variations with frequency of |S 11 | 2 and |S 21 | 2 on a dB scale obtained using the three methods. The results are qualitatively similar in each case. All three methods agree in-band, where there is high transmission. The circuit and power wave methods agree out-of-band and show high reflection in this range. However, the modulus square method predicts a reflection coefficient greater than unity here.  B. Two-port device, unequal port impedances We now consider the case shown in Fig. 6(c), namely, a discontinuity caused by a global variation in the mutual inductance from M to M ′ = μM between elements −1 and 0 and thereafter, so the system consists of two MI waveguides with different propagation constants k 1 and k 2 connected together. There are now two dispersion equations, R + jωL + 1/jωC + 2jωM cos(k 1 a) = 0 for n < −1, R + jωL + 1/jωC + 2jωμM cos(k 2 a) = 0 for n > 0.
Once again, as expected, Γ = 0 and T = 1 when μ = 1 and k 1 = k 2 . Scattering parameters for ports at elements −1 and 0 may again be calculated by all three methods. For example, we again assume the parameters κ = 0.6 and Q = 100 and take μ = 1.1 so that the right-hand side line has a larger bandwidth than the left-hand side line and |Z 02 | > |Z 01 |. Figure 9 shows the frequency dependence of |S 11 | 2 and |S 21 | 2 calculated by all three methods on a dB scale, with the cutoff frequencies of the two lossless bands marked using black dotted and dotted-dashed lines. Once again, the circuit and power wave methods agree exactly. The modulus square method agrees reasonably inside the smaller band, when k 1 and k 2 are both approximately real. However, it predicts unphysical results when either or both of k 1 and k 2 have significant imaginary parts.
Here, I 0 is the current at the connecting loop, a separate unknown. For incidence from line 1, solutions are assumed as I 1,n = II exp(−jnka) + IR exp(+jnka) for n < 0, I 2n = I T2 exp(−jnka)for n > 0, I 3n = I T3 exp(−jnka)for n > 0. Substituting and making use of the dispersion equation, solutions can be obtained as 24 , , Here, we focus on the case when μ 1 = 1 and μ 2 2 + μ 2 3 = 1 and (31) reduces to Γ = 0, T 2 = μ 2 , and T 3 = μ 3 , implying that the splitter is matched and has a constant splitting ratio. Using the circuit method, equations are solved for the four loops, assuming a voltage source with output impedance Z 0 at port 1 and terminations at ports 2 and 3, as shown in Fig. 10(a). Using discrete power waves, A and B coefficients are found for each line in turn. Figure 11 shows the frequency dependence on a dB scale of |S 11 | 2 and |S 21 | 2 thus obtained for a splitter with parameters κ = 0.6, Q = 100, μ 1 = 1, and μ 2 = μ 3 = 1/ √ 2, so the device is a 3dB splitter. As usual, the black dotted lines show the lossless band edges. The predictions of the modulus square method are entirely incorrect for |S 11 | 2 since this trace is absent. Although the predictions for |S 21 | 2 (and |S 31 | 2 , since this is identical) are qualitatively correct in-band, only the circuit and power wave methods show the expected band-limited performance.
These results highlight the need to obtain realistic estimates of power flow when there is loss and/or when propagation is outof-band. For terminated ports, the circuit method of Eq. (21) gives   FIG. 11. Frequency dependence of |S 11 | 2 and |S 21 | 2 for a 3dB MI splitter with parameters κ = 0.6, Q = 100, μ 1 = 1, and μ 2 = μ 3 = 1/ √ 2 as predicted using three different methods. Black dashed lines show the lossless band edges.

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numerical results that are universally equivalent to those obtained using Kurokawa's method. 6 However, the discrete power wave method also allows scattering parameters to be found when there is no immediately adjacent port, for example, at junctions between semi-infinite media. As we have shown numerically, this allows direct exploitation of the solutions of boundary matching problems. Although these are not presented here due to their length, we have also used known results for Γ and T to construct complete analytic expressions for the S-parameters themselves.

V. CONCLUSIONS
A discrete formulation of power waves has been developed for magneto-inductive waveguides. These waves are analogous to the continuous power waves introduced by Kurokawa for transmissionline systems, satisfy the power conservation relation, and avoid the physical anomalies seen in scattering parameters found using modulus square methods for passive MI systems. They are applicable to a system supporting pure current waves and allow determination of intermediate scattering parameters, for example, at junctions between semi-infinite media.
The importance of the power wave approach has been demonstrated using examples involving multi-port MI devices, which show how analytic solutions can be correctly converted to scattering parameters. Consequently, this approach should assist in designing systems that involve connected MI devices (for example, containing splitting or filtering components as well as simple propagation pathways). Since an analogous power conservation relation may be constructed for MI waves in 2D, assuming mutual inductances Mx and My in the two perpendicular directions, it may be useful for calculating power flow in devices based on metasurfaces. Finally, it is likely to be relevant to other periodic metamaterials, such as electroinductive waveguides, 38 for which discrete power waves may also be constructed.

DATA AVAILABILITY
The data that support the findings of this study are available within the article.