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The characteristic mode theory (CMT) can provide physically intuitive guidance for the analysis and design of antenna structures. In CMT applications, the antenna current distribution is decomposed into the superposition of multiple characteristic modes, and the proportion of each current mode is characterized by the modal weighting coefficient (MWC). However, different characteristic currents themselves have different radiation efficiencies reflected by the eigenvalues. Therefore, from the perspective of the contribution to the radiation field, the modal proportion should be more accurately determined by the combination of the modal weighting coefficient and the mode current itself. Since the discrete mode currents calculated using the electromagnetic numerical method are distributed on the whole conductor surface, we can actually use the radiation field to quantify the modal proportion or estimate it using the far field in the maximum radiation direction. The numerical examples provided in the paper demonstrate that this modal proportion can effectively evaluate antenna performance.

The characteristic mode theory (CMT) was first formulated by Garbacz in 1968 [

In fact, what makes characteristic modes really attractive for antenna design is the physical insight into antenna radiation. There is an eigenvalue or characteristic angle associated to each characteristic mode that can provide information about the mode resonance and radiating characteristics. Additionally, since characteristic modes only depend on the shape and size of the conducting object, antenna design can be performed in a controlled way.

However, CMT has once practically fallen into disuse. Later, in order to design and layout platform antennas [

When using CMT to design or analyze the antenna structure, we should first give the distributions of characteristic currents on the conductor surface. The selection of the feed point and the feed structure then has to be performed to excite the corresponding characteristic modes or combination thereof to meet the desired radiation field. For example, multiple CCEs (capacitive coupling elements) or ICEs (inductive coupling elements) are used to stimulate multiple modes of the mobile terminal to implement the MIMO antenna function [

That is to say, to design an antenna is to stimulate the fundamental mode or the first few characteristic modes of the antenna structure and suppress the other modes as much as possible. Usually, the proportion level of each mode is characterized by the modal weighting coefficient (MWC) [

Nevertheless, when the electromagnetic numerical calculation is used to determine the modal proportion, each characteristic current is discretely distributed on structure meshes, which makes it difficult to determine MP using mode currents. On the other hand, the far field of each mode is contributed by the overall current of the corresponding mode. Thus, it is proposed using the radiation field of the corresponding mode to quantify the modal proportion. We can also estimate the modal proportion using only field values in a certain spatial direction to verify that the design antenna is theoretically in compliance with the desired requirements.

According to CMT, the actual current

On the ideal conductor surface, there is the relationship equation between the source and the field:

Then, the dot product of equation (

The characteristic modes can have the following orthogonality [

Substituting equation (

It is seen that the modal weight coefficient

In equation

The inner product

If the feeding position is not appropriate, even the resonant eigenmode (

Along with the modal excitation coefficient, the modal significance measures the contribution of each mode to the overall electromagnetic response to some extent. It should be noted that the MWC shown in equation (

In the numerical calculation of the electromagnetic field, the solved area must be discretized using the grids. At this time, the inner product of the vector in equation

After the current distribution of the first few characteristic modes with smaller eigenvalues is analyzed, the mode selection and combination are carried out with the desired antenna radiation field requirements. In that case, MWC is an important guiding parameter for the antenna design to determine the appropriate feed point and feed structure. Here, we illustrate the application of MWC with the radiation of a

The eigenvalues of the first six modes for the rectangular metal plate within the frequency range from 1.0 GHz to 4.0 GHz are shown in Figure

Eigenvalues of the first six modes for the

The current distributions of the first six modes on the rectangular metal plate at 2.5 GHz are shown in Figure

Characteristic currents for the first six modes of the rectangular metal plate at 2.5 GHz: (a) mode 1, (b) mode 2, (c) mode 3, (d) mode 4, (e) mode 5, and (f) mode 6.

The desired characteristic modes on a single conductor plate can be excited by magnetic field coupling or electric field coupling [

The mode current distribution is consistent with that of the near-field magnetic field. Thus, the ICE should be placed at the maximum mode current or mode magnetic field position for good coupling. On the other hand, the CCE should be placed at the minimum point of the mode current or at the maximum position of the mode near-field electric field. To this end, Figure

Normalized near-field map of the first two modes at 2.5 GHz for the rectangular metal plate: (a) electric field of mode 1, (b) electric field of mode 2, (c) magnetic field of mode 1, and (d) magnetic field of mode 2.

According to the mode distributions shown in Figures

The rectangular metal plate excited using CCE and the corresponding

The CCE size and position have a great influence on the coupling effect and port matching. The antenna return loss in this case is shown in Figure

In order to determine whether mode 1 is effectively excited, the weighting coefficients of the first six modes are analyzed. Figure

From Figure

As can be seen from Figure

The rectangular metal plate excited using ICE and the corresponding

In order to match the excitation port, a capacitor of

Figure

It should be noted that the solution of the eigenvalue equation is related to the frequency, so the amplitude of characteristic current at different frequencies is not the same. In fact, the oscillation of the mode current magnitude will become more and more obvious with the increase of frequency, which is also reflected in Figure

The MWC analysis has utilized the orthogonality of the characteristic current wave mode, and each mode radiation power is first normalized to the unit value. On the other hand, the reactive power is proportional to the eigenvalue magnitude. Thus, the amplitude of a characteristic current is related to the radiation efficiency of the corresponding mode, i.e., its eigenvalue magnitude. Higher-order modes require large mode currents to reach the same unit radiation power level as lower-order modes.

That is, the amplitudes of the expanding respective characteristic currents on the open conductor are not equal. On this basis, the modes are weighted in equation (

If the modal weighting coefficient is calculated using the mode current normalized by input power rather than radiation power, the actual proportion of the corresponding mode can be correctly determined by MWC squared [

According to equation (

In general, only a few lower order modes play a major role in the radiation field. The number of modes can be selected according to the modal significance being greater than

Each characteristic mode current of the antenna structure produces a corresponding radiation pattern of the far field. The proportion of the far field to the total radiation field in this mode is the same as the proportion of the mode current in the total current. Therefore, the corresponding modal proportion of the antenna can be verified by the proportion of the radiated field. The total electric field

Similarly, the actual radiation field is mainly determined by the first few modes, and only the first 6 modes are also analyzed for the antennas shown in Figures

However, it should be noted that the radiation field of each mode has directionality. The calculation of the modal proportion should use the integrated value of the corresponding mode energy flow density on the enclosing surface surrounding the antenna. If so, the calculation process is also complicated. However, since the antenna total radiation field is mainly determined by the first one or two modes [

For a specific calculation, the maximum radiation direction of the main mode or total far-field electric field is firstly obtained by the full-wave method. Then, a cut plane containing the maximum field is selected, the radiation pattern of each mode on this plane is given, and the field value of each mode in the maximum radiation direction is read. Combined with the modal weighting coefficient, the proportion of the corresponding mode is estimated using equation (

For the antenna in Figure

Radiation patterns of the total field and the first six modes for the CCE-fed antenna on the

In Figure

MWCs and far field values (

Mode | MWC from Figure |
MWC proportion | EFV from Figure |
Modal proportion | |
---|---|---|---|---|---|

1 | 0.120 | 66.3% | 6.849 | 0.822 | 83.5% |

2 | 0 | 0 | 7.715 | 0 | 0 |

3 | 0 | 0 | 2.518 | 0 | 0 |

4 | 0.058 | 32% | 2.589 | 0.150 | 15.2% |

5 | 0 | 0 | 1.814 | 0 | 0 |

6 | 0.003 | 1.7% | 4.448 | 0.013 | 1.3% |

MWC, modal weighting coefficient; EFV, electric field value.

At this point, the proportion of mode 1 is

Under the condition that the excitation source voltage is 1 V, the total electric field value (0.985 mV) calculated by the first six modes in the maximum radiation direction is substantially the same as the value (0.967 mV) of Figure

For the antenna based on the ICE feed in Figure

Radiation patterns of the total field and the first six modes for the ICE-fed antenna on the

The field values for each mode in the maximum radiation direction (

MWCs and far field values (

Mode | MWC from Figure |
MWC proportion | EFV from Figure |
Modal proportion | |
---|---|---|---|---|---|

1 | 0.117 | 53.7% | 6.564 | 0.768 | 66.3% |

2 | 0.015 | 6.9% | 0.903 | 0.014 | 1.2% |

3 | 0.055 | 25.5% | 1.881 | 0.103 | 8.9% |

4 | 0.002 | 0.9% | 1.563 | 0.003 | 0.3% |

5 | 0.016 | 7.3% | 10.014 | 0.160 | 13.8% |

6 | 0.013 | 6.0% | 8.431 | 0.011 | 9.5% |

MWC, modal weighting coefficient; EFV, electric field value.

At this time, mode 1 accounts for about

We can also observe that the proportion of mode 1 in the CCE-fed antenna is larger than that in the ICE-fed antenna. This coincides with the conclusion that the radiation efficiency of the capacitive coupling antenna is greater than that of the inductive coupling antenna. It indicates that the modal proportion can also reflect the antenna radiation efficiency to some extent.

In summary, in order to determine the contribution of a certain mode to total radiation, using the modal proportion calculated by the far-field electric field is more reasonable than using MWC. However, there are two main error sources for the modal proportion calculation using far fields. One is that the antenna radiation fields are approximated by the electric field in the maximum direction, and the other is introduced by the numerical calculation. The final error can be estimated by the difference between the total field calculated by each mode and that directly obtained by the full-wave method. Thus, the relative error for the above CCE-fed antenna is about 1.9% and that for the ICE-fed structure is approximately 15.7%, which is relatively large. The reason is that the antenna structure using the ICE excitation has a poor ability to suppress high-order modes. From this point of view, it cannot be said that it is easier to use ICE than CCE to achieve a high purity mode.

The application of the characteristic mode theory in antenna structures is to decompose and synthesize surface currents or radiation fields. As the radiation field is decomposed into the superposition of orthogonal modes, the antenna performance is easy to optimize and realize from the physical principle. Each characteristic mode contribution to the total antenna performance cannot be determined only by the modal weighting coefficient, because different characteristic currents have different radiation efficiencies reflected by the eigenvalues. The modal proportion should be calculated by the combination of the modal weighting coefficient and mode current. However, the extraction of discrete mode currents is cumbersome in electromagnetic numerical methods. Therefore, we use the far field in the maximum radiation direction to quantify the modal proportion. This modal proportion can effectively evaluate the antenna performance.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was supported by the Open Fund of the State Key Laboratory on Integrated Optoelectronics (IOSKL2017KF02), China.