# Application of the Theory of Characteristic Modes to Antenna Analysis and Design

In modern applications, such as smart phones and automotive communication systems, there is an ever growing demand to support multiple wireless communication protocols over a wide frequency range, resulting in the use of several different antennas. Furthermore, as the need for higher data rates steadily grows, more and more applications rely on MIMO operation which requires decoupled antennas. Simultaneously, modern devices have a high degree of integration, in particular with respect to the antennas, to fulfil the requirements of usability and aesthetics. In the end, this means that a lot of (decoupled) antennas have to be placed in a confined space. A promising approach to tackle these conflicting issues is the application of the Theory of Characteristic Modes.

This theory has proved to be a valuable tool in antenna engineering. The complex current distribution of an arbitrarily shaped antenna is decomposed into a set of orthogonal modes. These characteristic modes consist of real surface current densities (characteristic currents) and their corresponding electromagnetic fields (characteristic fields) which are oftentimes much easier to understand than the total current distribution. This allows the antenna engineer to gain deeper insight into the radiation mechanisms of the antenna without placing any driving ports and enables him/her to manipulate the antenna structure and place excitations in order to obtain desired radiation properties.

The n-th characteristic mode of an antenna is described by an eigenvalue \({{\lambda }_{n}}\) and an eigenvector or eigencurrent \({{I}_{n}}\). They are obtained by solving the following generalized eigenvalue equation:

\[{\bf{X}}{{\bf{I}}_n} = {\lambda_n}{\bf{R}}{{\bf{I}}_n}\]

This equation contains the real part \(R\) and the imaginary part \(X\) of a complex impedance matrix \(Z\). Typically, the Method of Moments (MoM) is used to obtain the impedance matrix of the antenna structure.

By examining the eigenvalues and characteristic currents, the antenna engineer can decide which modes are suitable for his/her application. An eigenvalue close to zero generally means that the mode has good radiation performance. Inspecting the characteristic currents yields information on how to excite a desired mode or how to manipulate it, e. g. to shift its resonant frequency.

If an antenna is driven by a port, the total surface current \(I\) of the antenna structure is a weighted linear superposition of its characteristic currents:

\[{\bf{I}} = \sum\limits_n {{\alpha _n}{{\bf{I}}_n} = \sum\limits_n {\frac{{{\bf{I}}_n^H \cdot {\bf{V}}}}{{1 + j{\lambda_n}}}} } {{\bf{I}}_n}\]

The coefficient \({{\alpha }_{n}}\) is called modal weighting coefficient and describes the contribution of the n-th mode to the total current density. It depends on the eigenvalue \({{\lambda }_{n}}\) and the modal excitation coefficient, which is the scalar product of the characteristic current \({{I}_{n}}\) and MoM-excitation-vector \({V}\). This allows the antenna engineer to assess how well a desired mode is excited by the port.

It is now obvious that the Theory of Characteristic Modes has several interesting features which can be leveraged by engineers for present-day and future antenna applications. As described above, it can be used to analyze a given antenna structure and determine how to manipulate it or where to place feed ports in order to achieve a desired performance. Based on this, it is even possible to excite several modes together for multiband operation. Another approach is to selectively excite different modes in order to obtain a multi-port antenna for MIMO applications.

Characteristic Mode Computation

Although the Theory of Characteristic Modes was derived by Garbacz and Harrington in the early 1970s, research interest in the topic really started only during the last ten years. While the basic theory is understood quite well, there are still several issues regarding the characteristic mode computation, like eigenvalue tracking, that need to be addressed. On top of that, there is a lack of reliable commercial software tools for mode computation. That is why there is still need for basic research in the field of characteristic mode computation.

Characteristic Mode Computation with Method of Moments

Since a few years ago there were no suitable software tools available for the characteristic mode computation, an in-house tool called CMC (Characteristic Mode Computation) was developed in MATLAB for the simulation of small antennas consisting of perfectly electrically conducting bodies. Over the years, this tool has evolved into a full characteristic mode computation tool including a graphical user interface with CAD capabilities and capable of calculating the characteristic modes of perfect electric conductors and dielectric bodies.

In addition to the basic research in the field of characteristic mode computation, there are several advanced techniques regarding the computation and manipulation of characteristic modes investigated and utilized at HFT. The most important of these techniques is the eigenvalue tracking which is necessary if the characteristic modes are evaluated at different frequencies. Another useful technique is the source reconstruction which enables the antenna engineer to reconstruct the characteristic modes of an antenna based on its total radiated far field. If the manipulation of characteristic modes is focused, reactive loading may be a promising alternative to changing the antenna’s geometry.

Advanced Techniques for the Computation of Characteristic Modes

Characteristic Mode Based Antenna Design

As already mentioned above, the Theory of Characteristic Modes has several attractive features for antenna design. It is particularly insightful when multiband or MIMO antennas are to be designed. HFT uses the Theory of Characteristic Modes for a wide range of applications like mobile communications and automotive antennas.

If during the design process a set of suitable modes has been identified for a given application, the question remains how to excite these modes. There are various ways of effectively exciting a characteristic mode. The actual implementation depends on the application, i. e. if several modes are to be excited simultaneously (for multiband operation) or if modes are to be excited independent of each other (for MIMO operation).

Excitation of Characteristic Modes

Modern smart phones have to operate in a wide range of different frequency bands. Additionally, in some of these bands multi-antenna operation is desired. The antennas needed have to be accommodated in the confined space of current smart phone designs. A promising approach is to use the smart phone chassis as the actual antenna. The Theory of Characteristic Modes provides the means to analyze the chassis and identify appropriate modes. By utilizing capacitive and inductive coupling elements, a multi-band and/or multi-port antenna system can be implemented.

Small Terminal Antenna Design based on the Theory of Characteristic Modes

Future mobile communication systems like 5G or upcoming versions of WiFi aim at extremely high data rates. Such ultra-high-speed wireless communications with peak data rates of 100 Gbps and beyond, as targeted in the DFG research focus area SPP 1655, require an ultra-wide signal bandwidth in conjunction with suitable antennas and advanced baseband processing techniques to increase spectral efficiency and power efficiency simultaneously. In order to achieve these goals, interdisciplinary research is conducted in the fields of antenna design and baseband processing. We concentrate on the frequency range of 6.0 - 8.5 GHz, where the spectral mask is most relaxed in Europe (-41.3 dBm/MHz EIRP). Hence, a spectral efficiency of at least 40 bps/Hz must be achieved, which is a rather challenging goal with respect to small devices.

There are two main key concepts to achieve this goal: A novel antenna design based on the Theory of Characteristic Modes and an advanced signal processing concept called massive MIMO. With regard to the antenna design, the new feature is that multiple Characteristic Modes are used per antenna element. Each mode is assigned to an individual antenna port. Hence, many antenna ports can be implemented given just a few antenna elements.

In a massive MIMO system, the complexity is preferably concentrated in the base station or the access point antenna array. Using the novel antenna concept explained above, a considerable size reduction is achieved compared to conventional antenna arrays.

Access Point Antenna Design for Multi-Mode Massive MIMO

Correspondingly, the multi-mode antenna concept can be applied to the user terminals. If e. g. a typical smart phone chassis is considered, there is an extraordinarily large number of resonant modes in the frequency range of interest. The use of as many modes as possible is especially crucial with respect to the baseband processing of the huge data rates.

The n-th characteristic mode of an antenna is described by an eigenvalue \({{\lambda }_{n}}\) and an eigenvector or eigencurrent \({{I}_{n}}\). They are obtained by solving the following generalized eigenvalue equation:

\[{\bf{X}}{{\bf{I}}_n} = {\lambda_n}{\bf{R}}{{\bf{I}}_n}\]

This equation contains the real part \(R\) and the imaginary part \(X\) of a complex impedance matrix \(Z\). Typically, the Method of Moments (MoM) is used to obtain the impedance matrix of the antenna structure.

By examining the eigenvalues and characteristic currents, the antenna engineer can decide which modes are suitable for his/her application. An eigenvalue close to zero generally means that the mode has good radiation performance. Inspecting the characteristic currents yields information on how to excite a desired mode or how to manipulate it, e. g. to shift its resonant frequency.

If an antenna is driven by a port, the total surface current \(I\) of the antenna structure is a weighted linear superposition of its characteristic currents:

\[{\bf{I}} = \sum\limits_n {{\alpha _n}{{\bf{I}}_n} = \sum\limits_n {\frac{{{\bf{I}}_n^H \cdot {\bf{V}}}}{{1 + j{\lambda_n}}}} } {{\bf{I}}_n}\]

The coefficient \({{\alpha }_{n}}\) is called modal weighting coefficient and describes the contribution of the n-th mode to the total current density. It depends on the eigenvalue \({{\lambda }_{n}}\) and the modal excitation coefficient, which is the scalar product of the characteristic current \({{I}_{n}}\) and MoM-excitation-vector \({V}\). This allows the antenna engineer to assess how well a desired mode is excited by the port.

It is now obvious that the Theory of Characteristic Modes has several interesting features which can be leveraged by engineers for present-day and future antenna applications. As described above, it can be used to analyze a given antenna structure and determine how to manipulate it or where to place feed ports in order to achieve a desired performance. Based on this, it is even possible to excite several modes together for multiband operation. Another approach is to selectively excite different modes in order to obtain a multi-port antenna for MIMO applications.

Characteristic Mode Computation

Although the Theory of Characteristic Modes was derived by Garbacz and Harrington in the early 1970s, research interest in the topic really started only during the last ten years. While the basic theory is understood quite well, there are still several issues regarding the characteristic mode computation, like eigenvalue tracking, that need to be addressed. On top of that, there is a lack of reliable commercial software tools for mode computation. That is why there is still need for basic research in the field of characteristic mode computation.

Characteristic Mode Computation with Method of Moments

Since a few years ago there were no suitable software tools available for the characteristic mode computation, an in-house tool called CMC (Characteristic Mode Computation) was developed in MATLAB for the simulation of small antennas consisting of perfectly electrically conducting bodies. Over the years, this tool has evolved into a full characteristic mode computation tool including a graphical user interface with CAD capabilities and capable of calculating the characteristic modes of perfect electric conductors and dielectric bodies.

In addition to the basic research in the field of characteristic mode computation, there are several advanced techniques regarding the computation and manipulation of characteristic modes investigated and utilized at HFT. The most important of these techniques is the eigenvalue tracking which is necessary if the characteristic modes are evaluated at different frequencies. Another useful technique is the source reconstruction which enables the antenna engineer to reconstruct the characteristic modes of an antenna based on its total radiated far field. If the manipulation of characteristic modes is focused, reactive loading may be a promising alternative to changing the antenna’s geometry.

Advanced Techniques for the Computation of Characteristic Modes

Characteristic Mode Based Antenna Design

As already mentioned above, the Theory of Characteristic Modes has several attractive features for antenna design. It is particularly insightful when multiband or MIMO antennas are to be designed. HFT uses the Theory of Characteristic Modes for a wide range of applications like mobile communications and automotive antennas.

If during the design process a set of suitable modes has been identified for a given application, the question remains how to excite these modes. There are various ways of effectively exciting a characteristic mode. The actual implementation depends on the application, i. e. if several modes are to be excited simultaneously (for multiband operation) or if modes are to be excited independent of each other (for MIMO operation).

Excitation of Characteristic Modes

Modern smart phones have to operate in a wide range of different frequency bands. Additionally, in some of these bands multi-antenna operation is desired. The antennas needed have to be accommodated in the confined space of current smart phone designs. A promising approach is to use the smart phone chassis as the actual antenna. The Theory of Characteristic Modes provides the means to analyze the chassis and identify appropriate modes. By utilizing capacitive and inductive coupling elements, a multi-band and/or multi-port antenna system can be implemented.

Small Terminal Antenna Design based on the Theory of Characteristic Modes

Future mobile communication systems like 5G or upcoming versions of WiFi aim at extremely high data rates. Such ultra-high-speed wireless communications with peak data rates of 100 Gbps and beyond, as targeted in the DFG research focus area SPP 1655, require an ultra-wide signal bandwidth in conjunction with suitable antennas and advanced baseband processing techniques to increase spectral efficiency and power efficiency simultaneously. In order to achieve these goals, interdisciplinary research is conducted in the fields of antenna design and baseband processing. We concentrate on the frequency range of 6.0 - 8.5 GHz, where the spectral mask is most relaxed in Europe (-41.3 dBm/MHz EIRP). Hence, a spectral efficiency of at least 40 bps/Hz must be achieved, which is a rather challenging goal with respect to small devices.

There are two main key concepts to achieve this goal: A novel antenna design based on the Theory of Characteristic Modes and an advanced signal processing concept called massive MIMO. With regard to the antenna design, the new feature is that multiple Characteristic Modes are used per antenna element. Each mode is assigned to an individual antenna port. Hence, many antenna ports can be implemented given just a few antenna elements.

In a massive MIMO system, the complexity is preferably concentrated in the base station or the access point antenna array. Using the novel antenna concept explained above, a considerable size reduction is achieved compared to conventional antenna arrays.

Access Point Antenna Design for Multi-Mode Massive MIMO

Correspondingly, the multi-mode antenna concept can be applied to the user terminals. If e. g. a typical smart phone chassis is considered, there is an extraordinarily large number of resonant modes in the frequency range of interest. The use of as many modes as possible is especially crucial with respect to the baseband processing of the huge data rates.