The problems in the quarter-wavelength model and impedance matching theory in analysising microwave absorption material


 It is shown here that many concepts in current mainstream microwave absorption theory are used inappropriately. Reflection loss RL has been used to characterize microwave absorption from material instead of film and the results have been rationalized by impedance matching theory. The quarter-wavelength model states that the reflection of microwaves with wavelength l from a film is minimized if the thickness of the film is m l /4 where m is an odd integer. But we show here that the model is wrong because the phase effects from interfaces have been overlooked. RL is an innate property only for metal-backed film. Impedance matching theory is developed from transmission-line theory for scattering parameter s 11 but cannot be generalized to RL.


Introduction
In current microwave absorption theory, reflection loss RL has been used to characterize absorption from material. 1,2,3,4,5,6 The results obtained have been rationalized by using the impedance matching theory (IM). 1 -4, 7, 8 In order to search for the minimum value of RL, the quarter-wavelength model (QWM) has been used. 1,5,6,8,9 Based on IM, ∆ parameter 10,11,12,13,14,15 have been devised and widely used, but this parameter cannot be a validation of IM since it is developed from the wrong theory thus is problematical itself.
RL is defined as the reflection coefficient or the scattering parameter s 11 for metal-backed film (MBF) shown by Fig. 1. It has been pointed out previously 16 that when RL is used to characterize material, this leads to erroneous conclusions, such as that less microwaves would be absorbed if the microwaves traveled a longer distance in a material. To get around this difficulty but retaining RL to characterize material, the theory of IM has been introduced but this theory is inadequate. IM has been developed in transmission-line theory based on the fact that s 11 (dB) for film without metal-back (NMBF) achieves its minima when the absolute value of the maximum or minimum of the input impedance |Z in | of the film is close to the characteristic impedance Z 0 of the open space, or when |Z in -Z 0 | is close to zero. It is demonstrated in this work that IM cannot be applied to RL(dB) or be generalized to MBF. QWM has also been used to search for the minima of RL. The model states that reflection of microwaves from a film is minimized if the thickness of the film is mλ/4 where m is an odd integer. It is argued that the incident i and backward b beams, 15,17,18,19,20,21,22 or the beams r and t 23,24 in Fig. 1 are out of phase by π if the thickness of the film is mλ/4. However, this claim takes no account of the phase effects from the interfaces. "Strict proof" of the model exists 9 but there are serious problems with the proof.
The methods used herein are simple and straightforward. By introducing the phase effects of interfaces in a film, the problems with the QWM have been identified and the results then verified rigorously from wellknown formulae and confirmed from experimental data. Along with the discussions, all the above issues concerning IM and RL have been addressed.
2. The problems with the quarter-wavelength model

wave addition and phase effects from interfaces
There are two key points in this discussion. Firstly, the phase difference required to cancellate beams r and t in Fig. 1 and secondly the phase effects of the interfaces at x 1 and x 1 + d which need to be applied with wave addition to obtain the minima of RL(dB) for MBF and s 11 (dB) for NMBF. ψ 1 is the phase of Z M which is usually very small for microwave absorbing material. The reflection coefficient Fig. 1 is given by Since It will be noted that there is a phase difference of π represented by Eq. 5 and 6. Phase differences for the incident and the reflected beams at interfaces have already been included in Eq. 3 and 4. Equation 7 is applicable for interfaces in the film equivalent to that shown in Fig 1 but with the metal-back removed.
For MBF, only The phase difference of π + ϕ, or approximately π since ψ 1 is small, also applies to reflected beams between films with Z M > Z 0 and Z M < Z 0 .
The small phase effects from Z M shown in Eqs. 8 and 9 also need to be considered for transmitted beams from interfaces at x 1and x 1 + .The transmission coefficients γ M at interfaces for NMBF are given in Eqs. 8 and 9.
φ is the phase for (Z M + Z 0 ). Beams f M , b M , and t in the film shown by Fig. 1 Minimum: |s 11 | = 0 when d = 0 Maximum: Minima:

Verification
The results of Table 1 have been obtained using wave addition theory which involves the phase effects of the interfaces. It can be seen with real numbers from Eq. 2 that Z M > Z 0 when µ r > ε r and Z M < Z 0 when µ r < ε r .
Omitting the phase in Z M simplifies wave addition theory though it will be problematical for microwave absorption. In this section, the results shown in Table 1 are verified theoretically from concreate calculations based on experiments or well accepted formulae. The precise phase effects are already included in the expressions and the data though these are omitted in the qualitative discussion for simplicity where "approximately" or "near" are often used, which will be stand by "~". We now consider each row in Table 1 in turn.      Fig. 2 The verification of row 1. Values of ε r and µ r are measured using the composite Cu@ZnFe 2 O 4 at the frequency indicated, except in the bold black curve in (b) which represents |Z in (x 1 )/Z 0 | with ε r ″ = µ r ″ = 0 for the example in row 2. s 11 (dB) is calculated from Eq. 10 and Z in (x 1 )/Z 0 from Eq. 13, using the measured ε r and µ r .
Whether Z M > Z 0 (a) or Z M < Z 0 (b), the minima of |s 11 | occur at ~nλ/2 which is contrary to QWM. The same conclusion can be obtained using the data measured at every frequency in the supplementary data file and can be generalized to published experimental data.
where Re(x) and Im(x) are the real and imaginary parts of x, respectively. From Eqs. 10 and 11, the minima of s 11 (dB) occur near exp(-jα j d) = 1 where d is ~nλ/2 when beams r and t in Fig s 11 is usually expressed by two equations as (13). Fig. 2 is obtained from Eq. 13 and can be understood along with Eqs. 2 and 3. From The amplitudes of the minima of |Z in (x 1 )/Z 0 | decay as d increases which signifies that the minimum |Z in (x 1 )| deviates more from |Z 0 |. Thus, s 11 (dB) = -∞ can never be obtained when d ≠ 0 even though the phase requirement for the minimum of s 11 (dB) has been met at the minima of |Z in (x 1 )|.
As shown by Fig. 2b, a maximum of |Z in (x 1 )/Z 0 | occur at d = 12.36 mm = 0.53λ and conforms to condition that d = ~nλ/2 where exp(-jα j d) = 1, and its minima at 6.66 and 18.14 mm (0.29 and 0.78λ) which are close to The concept of IM as developed in transmission-line theory specifies that s 11 (dB) achieves its minima when the maxima or minima of |Z in (x 1 )| approaches |Z 0 | or the minima of |Z in (x 1 ) -Z 0 | approaches zero. As shown by Fig. 2, this concept fits quite well for the minima values of s 11 (dB). However, this IM theory can only be applied to s 11 . It should be noted that at these maxima or minima of |Z in (x 1 )|, not only |Z in (x 1 )| does approach |Z 0 |, but also these maxima or minima also occur at positions which meet the phase requirement of the minima of s 11 (dB), i.e., beam r and t are out of phase exactly by effectively π, i.e. mπ. It should be pointed out that there are positions where |Z in (x 1 )/Z 0 | = 1 or |Z in (x 1 )| = |Z 0 | such as those when the value of d = ~1.66 mm in Fig. 2b.
However, this position does not correspond to minima of s 11 (dB) because the phase requirement has not been met. Unfortunately, IM is inappropriately generalized from s 11 (dB) to RL(dB). It will cause more serious problems for RL(dB) if only emphasizing the condition that |Z in (x 1 )| approaches |Z 0 | but ignoring the condition for phase difference. The subject will be discussed later.
These contradictions to the QWM should be readily apparent from experimental data but have remained unrecognized over the years. We have provided a file obtained from the measurement of Cu@ZnFe 2 O 4 provided in the Supplementary Material. Each entry in the file contains a pair of ε r and µ r values measured at a particular frequency. The above results apply to each entry.

Row 2:
When both ε r ″ and µ r ″ are zero, the film does not dissipate microwaves. Thus, the oscillations of |s 11 | and |Z in (x 1 )| are similar to those represented by row 1 but without decay, as shown by the bold line in Fig.   2b for |Z in (x 1 )| where both ε r ″ and µ r ″ have been set to zero. The bold line is indeed oscillating without decay.
The results for s 11 (dB) are similar and can be readily inferred from Fig. 2. |s 11 | and |Z in (x 1 )| oscillate asymmetrically around |R M | and |Z M |, respectively. From Eqs. 3, 8, 9, and 13, it is clear that |Z in (x 1 )| oscillates between |Z 0 | and |Z M | 2 /|Z 0 |. With Z M > Z 0 , |Z in (x 1 )| achieves its minima when |Z in (x 1 )| = |Z 0 | and its maxima when These maximum and minimum values of |Z in (x 1 )| are unaffected by the value of d because ε r and µ r are real numbers and α P in Eq. 11 is zero. IM applies at |Z in (x 1 )| = |Z 0 | here but the film does not dissipate microwave since ε r ″ = µ r ″ = 0 while IM has been developed in field material for characterize microwave absorption. The numerical verification is included as Fig. S2 in the Supplementary Materials. When Z M = Z 0 , both the maxima and the minima of |Z in (x 1 )| become |Z 0 | which signify that |Z in (x 1 )| is a constant, s 11 (dB) = -∞ at any value of d and the case reduced to that represented by row 3.
It should be noted that when s 11 (dB) = -∞, this condition does not characterize the microwave dissipation power of the film with ε r ″ = µ r ″ = 0. When d = nλ/2 or exp(j2πd/λ) = ±1, the energy of the microwaves has not leaked back to the open space from interface at x 1 , but it is so leaked from interface at x 1 + d and |s 21 | = 1 according to Eq. 14. The result here illustrates that s 11 (dB) is a parameter faithfully characterizing film rather than material.
The expression of |Z in (x 1 )| can be obtained from Eq. 13 as when ε r " = µ r " = 0 since both the phase of R M (x 1 -) and the value of α P d are zero (the bold line in Fig. 2b). The value of s 11 (dB) is -∞ exactly at d = nλ/2 when both α P and ϕ are zero from Eq. 16 which is obtained from Eq. 10.
However, when ε r ″ or µ r ″ is not zero, the phase of R M (x 1 -) and the value of α P d will shift the minima and deviates from zero. These changes in |Z in (x 1 )| will also affect the position and the value of the minimum of s 11 (dB).
Using wave addition theory, beams r and t in Fig. 1 with the metal-back removed are out of phase by π at d = nλ/2 when ε r " = µ r " = 0. When ε r " ≠ µ r " , the phase of R M (x 1 -) will affect the phase of beam r as described by Eq. 3. Thus, the values of d must be adjusted from nλ/2 so that beam r is still out of phase by π to provide the minima of s 11 (dB). The phase effects from interfaces expressed by Eqs. 7 -9 also affect the phase of beam t which in turn affects the adjustment of d. The different amplitudes of each beam in t also make a contribution.
All these effects have been included in Eq. 15 for |Z in (x 1 )| and Eq. 16 for |s 11 (x 1 -)|. From Eq. 10, s 11 (dB) = -∞ when d = 0 which is true for all cases in rows 1-4. Also it should be noted that From Eqs. 8, 9, and 13, |Z in deviates more from |Z 0 | in both cases and from Eq. 12, s 11 (dB) increases and approaches the R M (dB) with the value of -9.54. This result signifies that the further the microwaves travel in the material, then the more intense will be the backward beam b that emerges from the film. (Fig. 3) Thus, s 11 is a parameter faithfully characterizing film 26 rather than material. 16 This result of |s 11 | for NMBF is equivalent to that of R M (x 1 -) which is the s 11 for interface. From Eq. 3, the intensity of the reflected beam r from the interface at x 1 will be weak when Z M is nearly equal to Z 0 , but will become stronger when Z M differs more from Z 0 .
The voltages before and after a node between a wire and a resistor are equal. But the resistances of the wire and the resistor are not equal. Reflecting current at a node is used in transmission line theory to solve the problem that the currents at both sides of the node are not equal with the Ohm's law. 16 The analyses for microwave and conventional circuits are essentially the same except the former involves high frequency. A node in circuit is equivalent to an interface. When the resistance of a resistor in circuit is zero, the current is infinity which signifies that all current goes through the node without reflection; Similarly, when Z M = Z 0 at an interface, all the incident microwaves penetrate the interface without reflection, thus from Eq. 3 R M = 0. As the resistance increases, more current is reflected at the node and less passes through, so that the current of the circuit is reduced; as Z M increases from Z 0 , reflection at interface is increased and less microwave transmitted, thus, R M is increased. When the resistance increases to infinity, the current is zero so    |Z RL are also expressed by two equations familiar to material scientists. Z 0 | deviates more from zero. Thus, the fact that the minima of both RL(dB) and |Z in (x 1 )| occur at d = ~nλ/2 is not because of IM but because beams r and t in Fig. 1 are out of phase by π. There are many reports published with data similar to those shown in Fig. 4 but IM theory has never been questioned. 27,28,29 It worth noting that both |Z in (x 1 ) -1| and |Z in (x 1 ) + 1| are greater than |Z in (x 1 )| at the minimum positions of |Z in (x 1 )| in Fig. 4a because Z in (x 1 ) is near 0. The fact shows that IM theory is problematical since in the theory only the effect of |Z in (x 1 ) -1| in Eq. 17 is taken into account.
In The results from Fig. 4a apply to every entry of data when the frequency is higher than 8.14 GHz and the results from Fig. 4b apply to every entry of data when the frequency is lower than 7.97 GHz in the file provided in the Supplementary Material. The result is natural since Z M > Z 0 at high frequency and Z M < Z 0 at low frequency for magnetic material. 30 Because the data acquisition step is ∆ν = 170 MHz, the exact frequency at which Z M -Z 0 changes sign cannot be established.
Indeed, using the IM concept developed from transmission line theory illustrated in Fig. 2, the minima of The IM theory cannot be generalized from s 11 (dB) to RL(dB). In Fig. 4a, the value RL(dB) becomes its minima when |Z in (x 1 )| approaches zero where |Z in (x 1 )| deviates the most from |Z 0 |. In Fig. 4b, RL(dB) becomes its minima when |Z in (x 1 )| approaches its maximum values. In other word, contrary to IM, RL(dB) achieves its minima when |Z in (x 1 )| deviates the most from |Z 0 | and |Z in (x 1 ) -Z 0 | deviates the most from 0. The result signifies that Z in (x 1 ) and Z 0 are in phase with each other is not the same as that beams r and t are out of phase with each other. Thus, to minimize RL(dB), it is not sufficient to just consider the numerator |Z in (x 1 ) -Z 0 | in Eq. 17 and ignore the effect of the denominator |Z in (x 1 ) + Z 0 | on |RL|.    |Zin( When both ε r and µ r are real, from Eq. 1 we obtain can also be obtained from Eq. 19 when both αP and ψ 1 are zero. Since Z in (x 1 ) is an imaginary number by Eq. 23 and Z 0 is a real number, |Z in (x 1 ) -Z 0 | = |Z in (x 1 ) + Z 0 | as shown by Fig. 5a. Thus, |RL| = 1 from Eq. 17. The result shows that the effect of |Z in (x 1 ) + Z 0 | on |RL| cannot be neglected which invalidates the argument from IM.

Row 8: From IM theory, all the incident microwaves penetrate into the film when
2). Thus, it is suggested by IM that the minima of RL(dB) could be achieved when ε r = µ r 19, 24, 31 .
However, such a condition leads to a case violating the QWM and the minima of RL(dB) cannot be achieved.
Z M = Z 0 when ε r = µ r (Eq. 2), and the interface at x 1 disappears so that there is only a forward beam and a backward beam from the interface at x 1 + d in Fig. 1. This makes RL(dB) correlate inversely with d, i.e., the thicker the film, the smaller the value of RL(dB). Since R M (x 1 -) = 0 from Eq. 3, we obtain Eq. 24 from Eq. 1 or 21.  Fig. 5b can be drawn from Eq. 1 or 24 while Z in (x 1 ) is drawn from Eq. 18 which shows that both QWM and IM theory cannot be correctly applied to |RL| since the value of |Z in (x 1 -) -1| is not directly related to that of RL(x 1 -).

RL(dB) in
The proof of QWM published by Zhang et al 9 is based on the condition that ψ 1 in Eq. 2 is zero but this can only be achieved when ε r = µ r or when both ε r and µ r are real numbers. As shown in rows 7 and 8, no minimum of RL(dB) can be found for either condition, the proof is flawed.  (Fig. 6a). On the other hand, if Z M < Z 0 , |RL| decreases when Z in (x 1 ) increases toward Z 0 (Fig. 6b). If Z M = Z 0 , the result reduces to that represented by row 8 (Fig. 5b) except that Z in (x 1 ) is no longer periodic (Fig. 6c).  (b) ε r = -0.8j, µ r = 0.2j ν = 12.73 GHz

RL(dB) increases
On the right side of Eq. 25, the first term R M (x 1 -) is the contribution from the interface at x 1 , while the second term is the contribution from the interface at x 1 + d.
A derivation of Eq 25 is provided as S8 -S11 in the Supplementary Materials.

Conclusions
By considering the phase effects from interfaces in a film, the flaws in QWM have been found and a complete solution has been obtained for numerous different conditions which are summarized in Table 1. The results have been verified from experimental data and from well-known formulae. Along with the analysis, it is proved that the concept of IM developed from transmission line theory for s 11 (dB) cannot be applied to RL(dB) with MBF. Although the minima of s 11 (dB) are achieved when |Z in (x 1 )| is the closest to |Z 0 | or |Z in (x 1 ) -Z 0 | is the closest to zero as shown by Figs. 2 and S2, it has proved that the more relevant reason is not IM but because beams r and t in Fig. 1  It is shown by Figs. 3, 6a and 6b that IM is valid only when Z in (x 1 ) is a real function since the periodicity of |Z in (x 1 )| has been losen when Z in (x 1 ) is a real function. s 11 (dB) and RL(dB) have been shown to be parameters which faithfully characterize device rather than material. As shown by Figs. 2, 3, 4, 6a, s 11 (dB) and RL(dB) can be larger when the microwaves travel longer in a material.
Our work shows that there is a general practice of using established theory believing that it must be correct because it has been used for many years by many researchers. However, such theories need to be constantly checked, particularly when they are inconsistent with published experimental data and fundamental physical principles

Supplementary Material
The procedures of wave addition for obtaining the results of Table 1  The quarter-wavelength model is widely used as a general model to find the minima of reflection loss RL(dB) of microwaves. 1,2,3,4,5,6,7,8 In the model it is claimed that the reflection coefficient of a film is minimized when the thickness d of the film is mλ/4 where m is an odd integer. The model is based on the reasoning that beams r and t in Fig. 1 are out of phase by π when d = mλ/4. 9, 10 What is wrong with the model is that the phase effects of interfaces have been overlooked.
We  From Eq. 3 or S1 it can be seen that at position x 1 -, the phase difference between beams r and i is ϕ.
The phase contribution from R M (x 1 ) is usually small for microwave absorption materials. So, beam r is approximately in phase with beam i.
Also, at x 1beam t1 is out of phase by mπ or effectively π with beam r if d is ~nλ/2 since: The phase difference between beam f1 at x 1 + and beam i at x 1is ψ 1 -φ as shown by Eq. 8 or S2 which defines the transmission coefficient γ M (x 1 + ) of interface at x 1 . The phase contribution from γ M (x 1 ) is small. So, beam f1 at x 1 + is approximately in phase with beam i at x 1 -.
At x 1 + dbeam b1 is out of phase by π + ϕ, or ~π with beam f1 shown by Eq. 7 or S3 which defines the reflection coefficient R M (x 1 + d -) of interface at x 1 + d for film without metal-back. b1 1 1 f1 1 The phase difference between beam t1 at x 1and beam b1 at x 1 + is -φ or ~0 shown by Eq. 9 or S4 which provides the transmission coefficient r(x 1 -) of interface at x 1 .
Only an optical distance of ~nλ/2 can result in a phase difference of exactly mπ between beams r and t1. A phase difference of mπ is effectively π since m is an odd number. If d = mλ/4 as specified in the quarter-wavelength model, then beam t1 would be approximately in phase with beam r at x 1 -, which 4 is near the maximum of s 11 (dB).
It can be proved similarly that at x 1each beam of t2, t3, etc. is out of phase by ~mπ with beam r if d is ~nλ/2 since: At x 1 + beam f2 is out of phase by ~π with beam b1 by Eq. 4, 7 or S5 for the reflection coefficient With similar analysis, it is easy to prove that each beam of t2, t3, etc. is out of phase by ~mπ with beam r and thus beam t is exactly out of phase with beam r by effectively π when d is ~nλ/2. Therefore, the minimum of s 11 (dB) is achieved at d = ~nλ/2 but not at d = ~mλ/4 as specified in the quarterwavelength model.

For film with Z M ′ < Z 0
When Z M ′ < Z 0 beam r and i in Fig. S1a are out of phase by ~π, and beams b1 and f1 are approximately in phase with each other. Following the same argument, beams t and r will be out of phase exactly by π if d is ~nλ/2.
The results for ZM ′ > Z0 and ZM ′ < Z0 for film without metal-back have been presented by rows 1 and 2 in Table 1. All the data measured at different frequencies for Cu@ZnFe 2 O 4 in the data file provided give consistent results to those shown in row 1 in Table 1. Examples have been given in Fig.   2 in the main text.  Table 1 provides the result from wave addition for metal-backed film with Z M ′ > Z 0 , in which RL(dB) achieves its minima at ~nλ/2 where beam r and t in Fig. S1b are out of phase by exactly mπ, and its maxima at ~mλ/4 where beam r and t in Fig. S1b are exactly in phase with each other. The result is contrary to that predicted by the quarter-wavelength model as shown below.
At position x 1 -, beam r is approximately in phase with beam i. Also, at x 1beam t1 is out of phase by ~π with beam r if d is ~nλ/2 since: Beam f1 at x 1 + is approximately in phase with beam i at x 1 -. At x 1 + dbeam b1 is out of phase by π with beam f1 as shown by Eq. S6 which defines the reflection coefficient R M (x 1 + d -) of interface at x 1 + d for metal-backed film.
In Eq. S6 the characteristic impedance Z metal for metal is zero.
Beam t1 at x 1is approximately in phase with beam b1 at x 1 + .
The optical distance of nλ/2 will result in a phase difference of mπ or an effectively π between beams f1 and b1 at x -. If d = mλ/4 as specified in the quarter-wavelength model, then beam t1 would be approximately in phase with beam r at x 1 -. Only if d is ~nλ/2 can beam t1 be out of phase by effectively π with beam r at x 1 -.
It can be proved similarly that at x 1each beam of t2, t3, etc. is out of phase by effectively π with beam r if d is ~nλ/2 since: At x 1 + beam f2 is out of phase by ~π with beam b1. Similarly, it is easy to prove that each beam of t2, t3, etc. is out of phase by mπ with beam r and thus beam t is out of phase with beam r by effectively π when d = ~nλ/2. Therefore, the minimum of RL is achieved at d = ~nλ/2 but not at d = mλ/4 as specified in the quarter-wavelength model.
At high frequency for magnetic materials, Z M ′ > Z 0 . 11 The data measure for Cu@ZnFe 2 O 4 at frequencies higher than 8.14 GHz are consistent with this result which is represented by row 5.

For film with Z M ' < Z 0
When Z M ′ < Z 0 , beam r and i in Fig, S1b are out of phase by ~π, and beams b1 and f1 are out of phase by π with each other. Following the same argument, beams t and r are out of phase by exactly π if d is ~mλ/4. This is the only result conforms to the quarter-wavelength model.
At low frequency, Z M ′ < Z 0 . 11 The data for Cu@ZnFe 2 O 4 at frequencies lower than 7.97 GHz conform to this result which is represented by row 6. All the data measured at different frequencies for Cu@ZnFe 2 O 4 give consistent results as that shown by row 1.

3 The positions of minimum reflection loss and phase effect
For the quarter-wavelength mode, it is usually claimed that minima of RL occur when the phase difference between the incident and reflected beams is π. 12 , 13 , 14 , 15 , 16 , 17 , 18 In fact, return loss L R for metal-backed film is the negative of reflection loss RL in units of dB and is defined as where P i and P b are the powers of beam i and b in Fig. 1