International Journal of Antennas and Propagation

Volume 2018, Article ID 5313590, 17 pages

https://doi.org/10.1155/2018/5313590

## Electromagnetic-Power-Based Modal Classification, Modal Expansion, and Modal Decomposition for Perfect Electric Conductors

School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

Correspondence should be addressed to Renzun Lian; moc.361.piv@nailzr

Received 24 March 2018; Revised 1 July 2018; Accepted 17 July 2018; Published 25 November 2018

Academic Editor: Miguel Ferrando Bataller

Copyright © 2018 Renzun Lian and Jin Pan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Traditionally, all working modes of a perfect electric conductor are classified into capacitive modes, resonant modes, and inductive modes, and the resonant modes are further classified into internal resonant modes and external resonant modes. In this paper, the capacitive modes are further classified into intrinsically capacitive modes and nonintrinsically capacitive modes; the resonant modes are alternatively classified into intrinsically resonant modes, which are further classified into nonradiative intrinsically resonant modes and radiative intrinsically resonant modes, and nonintrinsically resonant modes; the inductive modes are further classified into intrinsically inductive modes and nonintrinsically inductive modes. Based on the modal expansion corresponding to these new modal classifications, an alternative modal decomposition method is proposed. In addition, it is also proved that all intrinsically resonant modes and all nonradiative intrinsically resonant modes constitute linear spaces, respectively, but other kinds of resonant modes cannot constitute linear spaces; by including the mode 0 into the intrinsically capacitive mode set and the intrinsically inductive mode set, these two modal sets become linear spaces, respectively, but other kinds of capacitive modes and inductive modes cannot constitute linear spaces.

#### 1. Introduction

Resonance is an important concept in electromagnetics. Based on whether the resonant modes radiate electromagnetic (EM) energy, they are classified into internal resonant modes and external resonant modes, and these two kinds of resonant modes are widely applied in EM cavities [1–5] and EM antennas [6–10], respectively.

The most commonly used mathematical method for researching internal resonant modes is the eigenmode theory (EMT) [1, 2, 11], and the EMT can construct the basis of the internal resonance space (which is constituted by all internal resonant modes [12]), and the basis is called eigenmodes. The most commonly used mathematical methods for researching external resonant modes are the singularity expansion method (SEM) [13–20] and characteristic mode theory (CMT) [21–30], and the modes constructed by SEM and CMT are, respectively, called natural modes and characteristic modes (CMs). Based on the results given in [29], it is easy to conclude that all the natural modes are resonant. Recently, [30] generalizes the traditional CMT to internal resonance problem, and it also proves that all nonradiative modes are resonant; all nonradiative modes constitute a linear space called nonradiation space, which is the same as the internal resonance space; all nonradiative CMs constitute the basis of the nonradiation space and internal resonance space, and then they are equivalent to the eigenmodes from the aspect of spanning whole space. Based on the above observations, the fundamental modes which are resonant can be classified into four categories internal resonant eigenmodes, external resonant natural modes, radiative resonant CMs, and nonradiative resonant CMs, and the relationships and differences among these fundamental resonant modes are analyzed in papers [27, 28, 30].

This paper alternatively classifies all resonant modes into three categories: nonradiative intrinsically resonant modes, radiative intrinsically resonant modes, and nonintrinsically resonant modes, and discusses the relationships and differences among them. Following this alternative modal classification for resonant modes, this paper further classifies all capacitive modes into two categories intrinsically capacitive modes and nonintrinsically capacitive modes and further classifies all inductive modes into two categories intrinsically inductive modes and nonintrinsically inductive modes. By employing the modal expansions corresponding to these new modal classifications, an alternative modal decomposition method is proposed in this paper, and at the same time, some further conclusions are obtained. In addition, some typical examples are also provided to verify the conclusions obtained in this paper.

#### 2. Modal Classification

When the EM field is incident on a perfect electric conductor (PEC), an electric current will be induced on the PEC. All possible working modes constitute a linear space called *modal space* [12, 21, 22, 30]. If is expanded in terms of independent and complete basis functions, there exists a one-to-one correspondence between and its expansion vector [12, 22, 30], and the linear space constituted by all possible is called *expansion vector space* (where is the vector constituted by all expansion coefficients). The following parts of this paper are discussed in the expansion vector space and frequency domain.

In the expansion vector space, the complex power done by on has the matrix form , and then the radiated power and imaginary power can be expressed in their matrix forms as and [29, 30]. Here, the superscript “” represents the transpose conjugate of a matrix or vector, and the method to obtain the matrix (which is identical to the impedance matrix derived from discretizing the electric field integral equation by employing Galerkin’s method, except a coefficient 1/2) can be found in [22, 25, 29, 30]; and [29, 30].

##### 2.1. Traditional Modal Classification

The matrix is positively semidefinite [30], so for any , and the modes corresponding to and are called *nonradiative modes* and *radiative modes*, respectively. In addition, the semidefiniteness of the matrix implies that if and only if [31], i.e.,

Thus, all nonradiative modes constitute a linear space (i.e., the null space of [31]) called *nonradiation space* (which is identical to the *internal resonance space* [30]), and any satisfies the following orthogonality:
for any working mode (because is Hermitian [30]).

The matrix is indefinite [22, 30], so can be negative, zero, or positive, and the modes corresponding to , , and are called *capacitive modes*, *resonant modes*, and *inductive modes*, respectively [11, 22, 29, 30]. According to whether the resonant modes radiate EM energy, the resonant modes are further classified into *internal resonant modes* (which do not radiate, so this paper calls them *nonradiative resonant modes*) and *external resonant modes* (which radiate, so this paper calls them *radiative resonant modes*) [12, 27, 30]. As demonstrated in [30] and Section 5 of this paper, the nonradiative modes must be resonant, so all capacitive and inductive modes must be radiative, and then this paper calls them *radiative capacitive* and *inductive modes*, respectively.

##### 2.2. New Modal Classification

Besides traditionally classifying all modes into radiative capacitive modes, resonant modes (including nonradiative resonant modes and radiative resonant modes), and radiative inductive modes, an alternative classification for the resonant modes is proposed in this section.

The matrix is indefinite, so does not imply that [31], though always implies that . This is equivalent to
i.e., the condition is stronger than the condition to guarantee resonance. Based on this, can be particularly called *intrinsic resonance condition*, if is viewed as *resonance condition*.

Correspondingly, the modes satisfying are called *intrinsically resonant modes*, and the resonant modes not satisfying are called *nonintrinsically resonant modes*. Obviously, all intrinsically resonant modes constitute a linear space, i.e., the null space of , and this space is called *intrinsic resonance space*. Similar to (2), any intrinsically resonant mode satisfies (4) for any :

When the intrinsically resonant mode satisfies the condition , it is called *nonradiative intrinsically resonant mode* and correspondingly denoted as . When the intrinsically resonant mode satisfies the condition , it is called *radiative intrinsically resonant mode* and correspondingly denoted as . As demonstrated in [30], , if . This implies that the intrinsic resonance space contains the whole nonradiation space. Then, the set constituted by all must be a linear space, and this space is just the nonradiation space; all nonintrinsically resonant modes are radiative, and they are particularly denoted as ; for any mode , satisfies orthogonality:

In summary, by introducing the concepts of intrinsic resonance and nonintrinsic resonance, this section alternatively classifies all resonant modes into nonradiative intrinsically resonant modes , radiative intrinsically resonant modes , and radiative nonintrinsically resonant modes . Because the nonradiative intrinsically resonant modes are just the traditional internal resonant modes, the introduction of the radiative intrinsically resonant modes and the radiative nonintrinsically resonant modes is essentially a subdivision for the traditional external resonant modes.

In addition, a similar subdivision for the capacitive modes and inductive modes will be provided in Section 4.

##### 2.3. Classification for Characteristic Modes

Because both the above traditional modal classification and new modal classification are suitable for the whole modal space, they are also valid for the CM set . Here, the symbol “” is used to represent the expansion vector of CM in order to be distinguished from the expansion vector of the general mode .

###### 2.3.1. Traditional Classification for CMs

Traditionally, the CM set is divided into four subsets [22, 30]: radiative capacitive CM set , nonradiative resonant CM set , radiative resonant CM set , and radiative inductive CM set . For the convenience of the following parts of this section, the nonradiative and radiative resonant CMs are collectively referred to as resonant CMs, and the union of sets and is correspondingly denoted as , i.e., .

###### 2.3.2. An Alternative Classification for Resonant CMs

As demonstrated in [22, 30], all satisfy the characteristic equation . In fact, this equation is just the intrinsic resonance condition introduced in Section 2.2, so all are intrinsically resonant, and then they are particularly denoted as . Correspondingly, and are particularly denoted as and , respectively.

All are independent of each other [22–30], and the rank of the set equals to the rank of the null space of , so they constitute the basis of the intrinsic resonance space [31], i.e., any intrinsically resonant mode can be uniquely expanded in terms of . In addition, constitute the basis of the nonradiation space [30], i.e., any nonradiative mode can be uniquely expanded in terms of .

#### 3. Modal Expansion

In this section, a further discussion on the CM-based modal expansions for various modes is provided, based on the new modal classification proposed in Section 2.

##### 3.1. Modal Expansion for General Modes

Based on the independence property and completeness of the CM set [22–30], any mode can be uniquely expanded in terms of some radiative capacitive CMs , some nonradiative resonant CMs , some radiative resonant CMs , and some radiative inductive CMs as where the reason to use “” instead of “” will be explained in Section 4. Based on the expansion (6), some valuable conclusions shown in Figure 1 can be derived, and they are proved as follows: (i)The proof for “” is obvious(ii)The proof for “”: it is obvious that , so mode 0 is intrinsically resonant. Thus, if , then is intrinsically resonant(iii)The proofs for “” and “”: it is obvious that the term is intrinsically resonant. Thus, the mode is intrinsically resonant, if and only if the term is intrinsically resonant, based on the intrinsic resonance condition introduced in Section 2.2(iv)The proof for “” is obvious, because of (3)(v)The proofs for “” and “”: because the term is intrinsically resonant, the imaginary power of mode equals to the imaginary power of the term due to the orthogonality (4). Thus, both the “” and “” hold(vi)The proofs for “” and “”: if is intrinsically resonant, then the mode is intrinsically resonant due to “”. This implies that can be expanded in terms of as concluded in Section 2.3. Because of the uniqueness of the CM-based modal expansion for a certain , the coefficients and in (6) must be zero, and then both the terms and must be zero. Based on this and the “” and “”, it is easy to conclude that both “” and “” hold.