Multiscale Auxiliary Sources for Modeling Microwave Components

Abstract

This chapter presents multiscale auxiliary sources mainly used to solve complex electromagnetic problems, especially those that insert localized elements into circuits. Several equivalence relations (field-circuit) are established to simplify and make more accurate electromagnetic calculations by changing some characteristics of the localized elements known by their field representation as “voltage-current” representation and vice versa. Various examples are illustrated to show the effects of auxiliary sources in planar circuits containing localized elements (dipoles, diodes, transistors) in the millimeter and terahertz bands. An example of a graphene or Gold dipole is demonstrated in this approach. Another typical example of a diode integrated in a radiating structure is also simulated.

1. Introduction

Microwave circuits are now most often made in planar technology on various substrates: Silicon for medium frequencies of mobile telephony, Alumina for hybrid circuits, gallium arsenide for millimeter waves. The active parts have dimensions in the micron range for the lines; we pass to the ten of microns as far as transverse dimensions are concerned; as, for the other dimensions (lines or selves), they can be of the order of a few millimeters. For many years, it appeared that it was inappropriate to simulate such an ensemble with a single software that, by a very tightly meshed, would describe the details of the multiple heterogeneities of a localized element, then, with a looser mesh, would take into account the excitation lines and the box (waveguide).

The first idea to avoid the unnecessary approach of simulating small elements that can be isolated from the surrounding circuit to measure them has resided the introduction of localized elements (“localized elements”) in electromagnetic calculation software and electromagnetic calculations: Finite Elements, finite differences, method of moments [1, 2].

The second idea is more rigorous. It simulates the different parts of the circuit by software adapted to each function (method of moments for the homogeneous parts and finite differences for the strongly heterogeneous elements, for example). In a second step, the values of the fields are equalized to the limits of the domains [3, 4].

In the many studies published on the subject, one can note that the coupling problem is always expressed in a matrix form that Maxwell’s equations take after a spatial or spectral truncation operation. The “compression method” [5] is, in this sense, clear and efficient. It applied to planar circuits modeled by a method of moments. The “basis functions” or test functions are triangular functions (or “roof-top” functions). The problem is then discretized, and the relations between voltage and current in a localized element are then introduced as relations between neighboring roof-tops. We use the edges as impedance ports to introduce localized elements in the finite element method [6, 7]. The approach is similar for the finite difference time domain methods and transmission line method (FDTD and TLM).

Coupling two sub-domains of a circuit, one being the “external” circuit and the other, a “localized” or “interior” part, is an objective that must be achieved independently of the mesh or the chosen numerical method. This objective is achieved in the context of integral methods by applying the “equivalence principle”: An electric field on a given surface is replaced by two opposing magnetic current sources [8]. We can also reason by duality with sources of electric currents. Therefore, the model exists but must be redefined in the sense of a change of scale until it can integrate localized elements. The difficulties appear then because the passage to the limit of the zero dimensions for an element or a source is not possible in general electromagnetism under the penalty of divergences in the series. However, it is easy to admit that the meticulous description of the internal functioning of a transistor is not indispensable to predict its behavior in a circuit. The only thing that counts is its extrinsic electrical characteristics, i.e., those accessible to measurement. Then, we can affirm that the electromagnetic effects in the external domain depend only on certain parameters. No matter how the active element is manufactured, only a few average values of the electromagnetic fields count. Rather than approaching the active element by going to the zero-dimensional limit, which is impossible, we should approach the problem by defining an equivalence between elements by identifying their electrical characteristics.

To clarify these notions and evaluate the limits of validity of the hypotheses, it was judged simpler to detail the calculation on an analytical example after recalling the definition of the different types of sources used in the electromagnetic simulation. In the context of integral methods for studying planar circuits, we place them in the spectral domain (or simply in “transverse resonance”).

2. Different forms of sources in planar circuits

We consider a plane P on which a metal circuit is drawn with surface impedances and active elements. This plane is excited by sources that can be described by giving an electric field or a surface current on a specific surface. We can consider an incident wave on the circuit. These sources will be called “main sources.” The “auxiliary sources” are intended to be used as intermediary in the calculation to pass from the external domain to the internal domain (active element), it is the translation of the equivalence principle: The diffraction of a plane wave by a slot requires, for example, the use of an auxiliary source of magnetic current in the slot to obtain the radiation pattern. If this source is given, it allows for calculating an impedance or a quadripole. Finally, the “virtual sources” are contrary to the auxiliary sources chosen arbitrarily with electromagnetic quantities defined on a conductor or an aperture but conditioned by continuity. It is, for example, the current density on a microstrip line for which we want to calculate the propagation constant of the fundamental mode. This current density that the electric field is zero in its domain of definition. Taking the example of the radiating slot again, from the moment we want to calculate the amplitude of the electric field in the slot, given the amplitude of the incident plane wave, by writing the continuity of the magnetic field; then this electric field is a virtual source. These distributions can be interpreted well from an equivalent scheme [9, 10, 11].

2.1 Principal sources

Three types can be distinguished.

2.1.1 Excitation of a circuit by an incident wave

This source is used for free-space diffraction problems or obstacles in the guide. The latter case is defined as follows: Let S be the straight section of an excitation guide, enclose TE and TM modes whose transverse electric fields are orthonormal functions with two components $f_n$. The fundamental mode $f_0$, the transverse magnetic field, is given by:

$${\mathit{J}}_0={\mathit{H}}_0\times {\widehat{n}}_0={\mathit{I}}_0{f}_0$$E1

${\widehat{n}}_0$ being the normal unit vector to S directed towards the load. In the opposite direction, the higher-order modes on S behave as if the guide is infinite, Figure 1.

Figure 1.

Source model in a homogenous guide.

For these modes, we can write (they are by hypothesis TE or TM modes):

$${\mathit{J}}_n={\mathit{H}}_n\times \widehat{n}=-Y_{M_n}{\mathit{E}}_n$$E2

$Y_{M_n}$ is the mode n admittance that is equal, respectively, to ${\gamma }_n/\mathit{j\omega \mu }$ and $\mathit{j\omega \epsilon }/{\gamma }_n$ for TE and TM modes [11]. The minus sign comes from the orientation ${\widehat{n}}_0$. In Eq. (2), ${\mathit{J}}_n$ and ${\mathit{E}}_n$ are the transverse current and fields with a rotation of $\pi /2$ for the magnetic field of mode order n. Eqs. (1) and (2) can be written compactly by using the operators of projection. For that, we pose by definition, f being a function with two arbitrary components:

$${\widehat{P}}_{n}f=f_n\left.⟨f\right|f_n⟩$$E3

${\widehat{P}}_n$ is also written ${\widehat{P}}_n=\left|f_n\right.⟩\left.⟨f_n\right|$: The scalar product is commonly used. It is written as

$$\left.⟨f_n\right|f⟩{=}^{\text{Definition}}{\int }_S{f}_n^{\ast t}\mathit{fdS}$$E4

Thus, we can write Eq. (2) and Eq. (1) as follows:

$$\mathit{J}=-\sum _{n>0}{Y}_{M_n}{\widehat{P}}_n\mathit{E}+{\mathit{J}}_0$$E5

J and E are the transverse current and fields, respectively, at S, Eq. (5) can also be written as

$$\mathit{J}=-{\widehat{Y}}_{M_n}\mathit{E}+{\mathit{J}}_0$$E6

This formula can be represented by a scheme analogous to a dipole source in the circuit theory. The two borders represent the surface S, E is a function of two variables with two components, and J behaves as a line current oriented in the direction ${\widehat{n}}_0$.

2.1.2 Excitation of a circuit by a very fine cable

Figure 2 shows the excitation of a metal line by a coaxial crossing the circuit box.

Figure 2.

Excitation of a line by a coaxial.

The dimension AB is tiny in wavelength, so we can assume that its surroundings satisfy the quasi-electrostatic hypothesis. In principle, a rigorous study requires calculating the circuit with a current source in the aperture C. However, since C is assumed to be small, we can directly evaluate the boundary conditions between the planes $P_1$ and $P_2$ enclose the excitation. This approximation has the advantage of allowing planar integral modeling without having to resort to a three-dimensional approach in the proximity of AB. A second hypothesis concerns the field fluctuations around the aperture and the excitation line influence the diffraction of the waves from the coaxial cable. This influence will be neglected as long as the average value of the corresponding fields on the source extent is zero. This assumption must be justified and clarified. If it is not valid, then parasitic elements will have to be introduced in the equivalent scheme.

Any field fluctuation is eliminated by arbitrarily limiting the source to the microstrip line’s limits. The result between A and B is a constant electric field and a constant surface current density ${\mathit{J}}_s$, it corresponds to the conduction current in the wire distributed over the whole width of the source. In this operation, as the dimensions of sources a and b (Figure 3) are arbitrary, it is important to ensure that these choices do not affect the result of the study of a circuit, i.e., the impedance seen from the source.

Figure 3.

Simplified representation of a coaxial excitation.

Assuming that the input E of the coaxial is connected to an internal impedance generator $Z_0$ and providing a voltage $V_0$, we obtain the relation between ${\mathit{E}}_0$ and ${\mathit{J}}_s$.

$${\mathit{E}}_{0}b=V=V_0-Z_{0}I=V_0+Z_0{\mathit{j}}_{s}a$$E7

From Eq. (7), we deduce the electric field source $Z_{0}a/b$. In the second member of Eq. (7), we notice a plus sign at the orientation change $j_s$ after its simplified representation (Figure 3). The scheme in Figure 3 translates the relation (7) of electric and magnetic fields at the level of the planes $P_1$ and $P_2$. As before, we write by convention:

$$\begin{array}{l}{\mathit{J}}_1={\mathit{H}}_1\times {\widehat{n}}_1\\ {\mathit{J}}_2={\mathit{H}}_2\times {\widehat{n}}_2\end{array}$$E8

${\widehat{n}}_1$ and ${\widehat{n}}_2$ are the unit normal outgoing ones taking into account the general conditions of continuity:

$${\mathit{H}}_1-{\mathit{H}}_2={\mathit{j}}_s\times {\widehat{n}}_{21}$$E9

We deduce the relations between ${\mathit{j}}_s$ and ${\mathit{J}}_1$, ${\mathit{J}}_2$. Vectorically multiplying Eq. (9) by ${\widehat{n}}_{21}$ = ${\widehat{n}}_1$, it comes:

$${\mathit{J}}_1+{\mathit{J}}_2=-{\mathit{j}}_s$$E10

The transition relation from Eq. (7), considering Eq. (10), is then written as:

$${\mathit{E}}_1={\mathit{E}}_2=\frac{V_0}{b}-Z_0\frac{a}{b}\left({\mathit{J}}_1+{\mathit{J}}_2\right)$$E11

2.1.3 Unilateral excitation of circuit

Previously, excitation by a wire produced a negligible effect. It is interesting to consider the opposite case to compare the results obtained by the two approaches. It is now assumed that a metal strip entirely masks the aperture AB (Figure 4) so that the metal strip integrally reflects the incident waves over the source.

Figure 4.

Excitation of a circuit by a metal tongue.

In the upper part, at the source level, this one being masked, only a short-circuit appears. On the other hand, the schematic of the source with its internal impedance appears in the lower part (Figure 4).

Note: The unilateral or bilateral planar source is not identical to the source on the fundamental mode. In fact, everything happens as if a “coupling quadrupole” existed between the source and the fundamental mode line [12]. The determination of this coupling has been the subject of numerous studies [13].

2.2 Auxiliary sources

These sources are used to study two distinct parts of a circuit separately, especially when their respective dimensions are very different from each other.

In the planar circuit approach, it is assumed that the thickness of the “small circuit” is infinitely small.

Therefore, it is equivalent to a possibly nonlinear surface impedance or, as for the main sources, to a field or current source with internal impedance. The simplest circuit that can be studied by auxiliary sources has the main source, $S_0$ conducting elements, and an impedance surface $S_1$ (Figure 5).

Figure 5.

Circuit comprising main source and an auxiliary source.

An auxiliary source’s analysis consists of simply replacing $S_1$ by a source, calculating the coupling quadripole Q between $S_0$ and $S_1$, then closing the quadrupole by the impedance present in the circuit. We deduce the impedance seen from the main source $S_0.$

An illustration of the use of a surface impedance or an auxiliary source (defined by a constant imposed electric field ${\mathit{E}}_0$) to model “localized elements” in planar structures with a method of moments is given in Figure 6. Figure 7 shows the reflection coefficient $S_{11}$ seen from the main excitation source versus frequency for different values of the surface impedance $Z_s$.

Figure 6.

Microstrip line: With a localized surface impedance (a) or with a localized auxiliary source (b).

Figure 7.

Effects of localized surface impedance value on the reflection coefficient $S_{11}$ seen from the main excitation source with l = 6 mm, ${\epsilon }_r$ = 9.8, and h = 1.35 mm.

The simulation results corresponding to low values of $Z_s$ (less than 5 Ω, for example) are identical to those obtained for a microstrip line without surface impedance. In this range of $Z_s$ variation, the two approaches by localized surface impedance or auxiliary source are equivalent. As the value of the surface impedance increases, the reflection coefficient increases, inducing a lower transmission between the accesses Eqs. (1) and (2) defined in Figure 6.

Note:

The influence of the auxiliary source position in relation to the main excitation source as a function of frequency shows that by placing the auxiliary source at a distance, $D=l/2$ for example, the second resonance frequency of the line disappears, which reappears for a distance $D=l/8$, the width of the auxiliary source is taken equal to the tenth of the line length l.

Thus, except for the differences relative to the previous remark related to the position of the auxiliary source with respect to the main excitation source, the two approaches by localized surface impedance and auxiliary source give identical results as long as the dimensions of the auxiliary source (δ) remain small with respect to the operating wavelength.

Relative to l, these dimensions become of the order of the operating wavelength for very high frequencies. For the auxiliary source dimensions of the order of the operating wavelength, the propagation is no longer negligible in the domain of the auxiliary source. The results obtained with the two approaches are different. In this case, imposing an electric field or a magnetic field does not necessarily give the same results.

If the surface $S_1$ is large enough, the description of the fields and currents surface at its level must require a set of basis functions, the output $E_1,J_1$ of Q is then a set of ports (one per basis function), Q is a multipole that must be calculated. The interest in introducing auxiliary sources is not obvious in this case. On the other hand, if $S_1$ is sufficiently small, then a single function is sufficient. There is a bottleneck at its level, and thus only one electromagnetic calculation (the elements of the Q matrix) for a set of load impedances in $S_1.$.

There is a case where the impedance of $S_1$ is defined in terms of current and voltage (localized elements). Under this assumption, as for the main sources, quasistatic limit is verified, and distribution of fields and currents for this basis function must satisfy the uniqueness of the definition of voltages and currents:

$$\mathit{rot}\left({\mathit{E}}_1\right)=0;\mathit{div}\left({\mathit{j}}_s\right)=0$$E12

The choice of the auxiliary source is then very small; the knowledge of the limits and the connectors is enough to define it. Returning to the concept of coupling between parts of a circuit, one external ${\mathrm{\Omega }}_1$ and the other internal ${\mathrm{\Omega }}_2$ or localized, the description of an auxiliary source then becomes natural. The two parts of the circuit are connected; two or more metallic parts exist at their borders Figures 8 and 9).

Figure 8.

Position’s effects of the auxiliary source described by the reflection coefficient $S_{11}$ seen by the main excitation source that varies against frequency with l = 6 mm, ${\epsilon }_r$ = 9.8, and h = 1.35 mm.

Figure 9.

Decomposition of a circuit into external and internal domains.

In the vicinity of S exist two surfaces $S_1$, $S_2$; one in ${\mathrm{\Omega }}_1$ and the other in ${\mathrm{\Omega }}_2$. The disconnection operation between ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ is done in two steps (Figure 10):

• Setting up a magnetic wall in place of S (this is an equivalent schematic of an open circuit).

• Introduction of a source over $S_1$ for the study of ${\mathrm{\Omega }}_1$ or over $S_2$ for the study of ${\mathrm{\Omega }}_2$.

Figure 10.

Placement of auxiliary sources.

For a planar source, the magnetic wall prohibits, for an electric field source, any component normal to the source (Figure 11). Thus ${\mathit{E}}_z=0$ from which: $\mathrm{di}{\mathrm{v}}_{\mathrm{s}}\left(E\right)=0$ . So, ($\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}=0$) and $E=-grad\left(v\right)$ (quasistatic hypothesis).

Figure 11.

Auxiliary planar source.

These two conditions allow us to derive a general formulation for auxiliary sources of any shape [11]. In the case of a rectangular source of dimension a, b, we pose:

• For a field source, ${\mathit{E}}_0$ is known, and after electromagnetic calculations, the surface current distribution ${\mathit{J}}_s$ is obtained.

By posing ${\mathit{E}}_0={\mathit{Vf}}_0$, we deduce:

$$I=⟨f_0\left|{\mathit{J}}_s\right.⟩$$E13

where $f_0$ is a function with two components ($1/b,0$). The scalar product is defined as the integral over the whole surface of the source; we obtain the total intensity going through the source by Eq. (13), which gives:

$$\mathit{J}={{\mathit{\text{If}}}_0}^{\text{'}}\text{and}V={⟨f_0}^{\text{'}}\left|\mathit{E}\right.⟩$$E14

In this case, ${f_0}^{\text{'}}$ is equal to $1/a$. This procedure is, of course, valid for the main sources.

Note: An auxiliary source can be considered a mode source for the main sources. This is the case, for example, if we consider a circuit inserted on a ground plane with a periodic motif. The latter can be studied independently employing modal excitations, which coincide with the modes of the case in transverse resonance. The impedance seen by each mode will be introduced in the operators necessary to study the circuit placed above the ground plane. The periodic motif is equivalent to excitation by the fundamental mode of such a guide under a variable incidence. We find in this process the essential quality of the auxiliary source the part not concerned (here the circuit above the periodic motif plane) can change; the impedance seen by each mode in transverse resonance will be the same. Global modeling will not be necessary when only one part of the circuit is variable.

2.3 Auxiliary sources

Virtual sources are well known to users of the method of moments. For example, we give them the current density ${\mathit{j}}_s$ on a perfect conductor; we deduce the tangential electric field, and, writing that the latter is null on the conductor, we find an equation that allows us to determine the unknown current. If we consider ${\mathit{j}}_s$ as a source, we can see that it does not deliver any power, neither real nor complex. Hence, the term virtual, as opposed to real, is attributed to this type of source. This property can be expressed more operationally. The definition of a virtual current source is:

A source defined in a domain D is virtual if its dual magnitude is nullified in D.

The dual quantity of ${\mathit{j}}_s$ is the tangential electric field E and vice versa. The equivalent schema of an interface with a virtual surface current source ${\mathit{j}}_s$ and the main field source ${\mathit{E}}_0$ is presented as follows (Figure 12): As the value of the virtual source at each point is not known a priori, it is represented as an adjustable current source with an inclined arrow.

Figure 12.

Representation of an interface with a field source and a perfect conductor.

The working domain consists of three subdomains: the source domain $D_S$, the metallic domain $D_M$, and the dielectric domain $D_D$ (Figure 13).

Figure 13.

Interface made of metal, dielectric, and a source.

Since the virtual source overlaps the source domain, and the dual quantity ${\mathit{j}}_s$, i.e., the electric field, is annulled in the latter, we deduce the equivalent schema of the interface in $D_S$ and $D_M$ (${\mathit{E}}_0$) is zero on the metal, so in $D_M$ only the short-circuit appears).

Concerning the dielectric, being outside the virtual source, the latter is zero. The equivalent schema shows an open circuit. At each point of D, the boundary conditions are correctly expressed by the schema of Figure 12. It is thus sufficient to describe the behavior of the electromagnetic field in the vicinity of D. By expressing the relations between the waves in the upper and lower parts of D using the transverse modes; we hold a scheme similar to the one in Figure 1 described by condition (6) (that assuming ${\mathit{J}}_0=0$). The following paragraph will discuss this process and the resolution of a circuit problem. It will highlight the interest and limitations of auxiliary sources. This will also lead to a rational definition of what we will understand by localized elements in the last part.

3. Use of auxiliary sources in planar circuits

The auxiliary sources allow the passage from a large scale at which the calculation is purely electromagnetic, to a smaller scale that corresponds to the connection of purely electromagnetic, to a smaller one which corresponds to the connection of circuit elements, most often characterized elsewhere.

The main and auxiliary sources are of such dimensions that the definition of voltage and current are intensity are unambiguously defined, as mentioned in the previous paragraph. Taking the general scheme of a planar circuit comprising one or more main sources and auxiliary sources, we can represent the different interactions in the following way (Figure 14): Q is the multipole of electromagnetic interactions. At the junction level, surfaces between sources and the surrounding medium, $S$ and $S^{\text{'}}$, it must be possible to define at the same time an electric field, a voltage, a current, and a magnetic field; this is why the condition of Eq. (12) has been imposed.

Figure 14.

Interactions between main and auxiliary sources.

However, this is not without problems because a global calculation (without auxiliary sources) gives a form of the field which does not verify this condition. This is shown by the simple example treated in the following.

This raises the problem of the validity of introducing the auxiliary sources; the analysis of this example gives an order of magnitude of the maximum dimensions to take for these last ones (Figure 15).

Figure 15.

Microstrip line loaded by a surface impedance with sidewalls magnetic and top and bottom electric.

4. Modeling localized impedance in microstrip circuit

Let us consider a microstrip line loaded by an impedance represented by a surface impedance of value $Z_s$. The line is assumed to be excited by the TEM mode. The edge effects are neglected and, to simplify the numerical approach, it is assumed that the line is simply bordered by magnetic walls (Figure 16). A magnetic wall P also limits the functional area.

Figure 16.

Relationship between measured impedance input impedance of a device and intrinsic impedance.

This problem can be addressed in two ways. Either the circuit is modeled globally by taking a virtual source (test function) instead of $Z_s$ for $0<y<c$, or we take an auxiliary source for $0<y<c$. We calculate the quadrupole Q between this source and the main excitation, then we close this source by impedance $Z_s$, following the circuit shown in Figure 14.

From the calculation developed in the appendix, the following conclusions can be drawn: first, if we want the same result between a direct approach and an indirect approach through auxiliary sources, it is necessary to take as many auxiliary sources as there are test functions in the first approach; it is necessary to take as many sources. This means that if one seeks precision of the results by taking into account, for example, the effects of edges to define the shape of the electric field of a single auxiliary source, other the difficulty that this way of making will present to evaluate the impedance seen of the source (as it was specified in II-2), this selected shape may be inadequate for a different impedance. If the precision is judged insufficient with the only auxiliary source, it will be necessary to introduce others and to study the action of these on the localized element, if possible. The second remark concerns the problem that arises as soon as the dimensions of the subdomain corresponding to a localized element tend to zero. In the example given in an appendix, a capacitance that tends to infinity appears parallel to an impedance whose dimension tends to zero; the capacitive nature comes from the TM modes alone present in the structure studied. For a more realistic impedance placed at the end of a microstrip line, we would find an inductance in series with the impedance tending to infinity with the width of the latter, in addition to the capacitance in parallel.

These remarks raise the difficulty of defining the behavior of a circuit of very small dimensions placed in a box. It is intuitively clear that the details of the internal functioning of this circuit are not affected by the large external circuit; only its electrical characteristics matter, the geometrical parameters do not influence in themselves. Hence, the idea of locating the impedances by admitting that their dimensions are zero. As this approach is not possible, because of the divergences of this discrepancy, we must return to the process of measuring an impedance and see how it can help to define its behavior when it is inserted in any circuit, because the circuit, because the only accessible parameters are those measured, or modeled from a fine description of the element.

5. Equivalent localized elements

To introduce the equivalent impedances, the simplest way is to go back to the previous example, taking into account the fact that the impedance coming from the surface impedance, of value:

$$Z_0=Z_s\frac{c}{a}$$E15

which can be called “nude impedance,” is not directly accessible.

This impedance is introduced into a measuring device that can be taken similar to the previous one with parameters $a_0$, $b_0$, $c_0$ instead of a, b, c of the device in which impedance is inserted for its practical use. However, in the electromagnetic calculation, the auxiliary source is charged by $Z_0$, so it is important to know the latter. To do this, a process of “de-embedding” is used.

Knowing the input impedance of the measuring device, we deduce $Z_0$ by an electromagnetic calculation. Using the results obtained in Eq. (15), we can write the measured impedance $Z_M$ as:

$$Z_M=\frac{1}{\mathit{j\omega }{C}_0\left(a_0,b_0,c_0\right)+\frac{1}{Z_0}}$$E16

Hence, the expression of $Z_0$:

$$Z_0=\frac{1}{-\mathit{j\omega }{C}_0+\frac{1}{Z_M}}$$E17

Placing this expression in a used device, homogenizing the load with a surface impedance,

$$Z_s=Z_0\frac{a}{c}$$E18

We deduce the input impedance $Z_E$

$$Z_E=\frac{1}{\mathit{j\omega C}\left(a,b,c\right)+\frac{1}{Z_M}}=\frac{1}{\mathit{j\omega }\left(C-C_0\right)+\frac{1}{Z_M}}$$E19

There is a relationship between the intrinsic impedance $Z_0$ and the input impedance (or the measured impedance), a homographic relation represented by a coupling quadrupole $Q_E$ or $Q_M$ (Figure 13).

$Z_0$, in the case considered here, is made with a surface impedance. As shown in Eq. (17), several values of $Z_s$, a, c, can give the same impedance $Z_0$. Due to the small size of the localized elements, several different devices can have identical electromagnetic behavior, which is reflected here by the same impedance value $Z_0$. Therefore, the measured impedances will be identical for the same measuring device. By definition, two impedances are “equivalent” when placed in the same device; their measured values are the same. Taking two surface impedances of dimensions a, c and $a\text{'}$, $c\text{'}$ (Figure 17), placed in a device similar to that of Figure 12, we see that the surface impedances $Z_s$ and $Z_s\text{'}$chosen so that the impedances are equivalent and roughly proportional to a and$a\text{'}$, according to formula of Eq. (17).

Figure 17.

Equivalent admittances.

However, the calculation corresponding to Figure 17b shows that the TE modes must be taken into account, which leads to the appearance of an inductance; the need to obtain an identical measured impedance in both cases, therefore, requires the surface impedance $Z_s\text{'}$ to have a capacitive part to compensate the effect of this inductance.

A common case concerns active or passive dipoles [14, 15, 16, 17, 18]. The measuring device is not specified in the manufacturer’s data, but an equivalent scheme is usually proposed. This scheme integrates the environment of the dipole in the measurement process. In particular, the capacitance between the terminals is taken into account and should be removed to reach the intrinsic (or nude) impedance (Figure 18a): If this precaution is not taken, the capacitance C will be integrated into the localized impedance or in the equivalent medium [19] but, in the calculation of the global device, there will be the presence of external field lines (Figure 18a) in the electromagnetic resolution. These field lines contribute to capacity C, and they have already been taken into account to establish the equivalent scheme.

Figure 18.

Dipole and equivalent scheme for two choices of surfaces limits of the active element (magnetic walls: m.m.).

In summary, a part of C is counted twice. To avoid this, it is necessary to enclose the “box” in the magnetic wall (surface $S\text{'}$ in Figure 11) the field lines surrounding the box. Thus, the impedance of the equivalent scheme becomes an intrinsic impedance (Figure 18b).

The case of transistors is the case of quadrupoles. Assuming that the dimensions are small enough for the equivalence hypothesis between localized elements to be verified, the shape of the transistor does not have to be specified in detail (exact dimensions of the electrodes, interdigitated or in-line nature, etc.); they are all equivalent to each other by hypothesis. We can choose a simple one, integrated with a classical coplanar measuring device assumed known. This device is shown in Figure 19a. Modeling allows deducing the coupling matrix between two auxiliary sources $S_1$, $S_2$. Let $Z_0$ be the impedance matrix of this intrinsic transistor. This matrix will be used in the final device to model the transistor in an arbitrary environment symbolized here by a quadrupole Q connected enclosure to the sources $S_1$, and $S_2$ themselves closed on $Z_0$.

Figure 19.

Transistor in a measuring device (a) and inserted in a circuit (b).

The process is led in three stages:

• Choice of a geometrical shape for the sources and accesses of the G, S, D transistor (Figure 19a)

• From the known $Z_M$, we deduce the intrinsic coupling matrix $Z_0$ between S₁, S₂.

• We introduce the sources S₁ and S₂ in the circuit to be studied without modifying their dimensions and relative positions. An electromagnetic calculation allows determining the multipole $Q_E$ and the input impedance$Z_E$.

Therefore, in introducing active elements in a circuit, there is the necessity to first reach an intrinsic element by an operation of “deshabillage.”

This element is then connected to one or more auxiliary sources, linking this element and the circuit studied. This operation is simplified in the case of low frequencies. In the first step, the inductance of the supply wires (Figure 19a) and capacitances due to connections can be neglected.

Let $a_1$, $b_1$, $a_2$, $b_2$ be the dimensions of $S_1$, $S_2$. We have successively:

$$V_{\mathit{GS}}=E_1{b}_1\text{and}{V}_{\mathit{DS}}=E_1{b}_1+E_1{b}_1$$

$$I_{\mathit{DS}}=J_2{a}_2\text{and}{I}_{\mathit{GS}}+I_{\mathit{DS}}=J_1{a}_1.$$E20

We deduce the relation between $Z_M$ and $Z_0$:

$$\left|\begin{array}{l}{E}_1\\ E_2\end{array}\right|=Z_0\left|\begin{array}{l}{J}_1\\ J_2\end{array}\right|,\left|\begin{array}{l}{V}_{\mathit{GS}}\\ V_{\mathit{DS}}\end{array}\right|=Z_M\left|\begin{array}{l}{I}_{\mathit{GS}}\\ I_{\mathit{DS}}\end{array}\right|$$E21

$$Z_0=\left|\begin{array}{cc}\frac{1}{b_1}& 0\\ -\frac{1}{b_2}& \frac{1}{b_2}\end{array}\right|Z_0\left|\begin{array}{cc}{a}_1& -a_2\\ 0& a_2\end{array}\right|$$E22

Thus, access to $Z_0$ is simplified and does not require any electromagnetic calculation. This will be the case when the capacities at the terminals of $S_1$ and $S_2$ are negligible. It is enough for this that the sources $S_1$ and $S_2$ are of dimensions $a_1$, $a_2$, sufficiently small.

Therefore, the localized element is an equivalence class, and in the use of localized elements, one arbitrarily chooses one of the elements for electromagnetic modeling. The localized elements are thus accessible by an operation that gives rise to equivalence. This equivalence relation makes it possible to avoid the introduction of zero dimensions that are not possible in electromagnetic calculations.

6. Evaluating surface impedance models for terahertz applications

The design of the terahertz antenna demands particular attention on the choice of materials to fulfill the microfabrication process.

As illustrated in Figure 20, a resonant antenna element for the terahertz application proposed here consists of three multilayers, from top to bottom: a gold or graphene dipole, polydimethylsiloxane (PDMS) substrate, and a ground plane. Gold and graphene are good conductive and are not susceptible to oxidation in air, while PDMS has comparatively low losses in the terahertz range.

Figure 20.

Dipole-like antenna made of two graphenes (or gold) strips on a substrate and integrated with a located gap source in the center (example of the photonic mixer).

Various metals are selected for the top and bottom layer metallizations due to their selectivity for patterning, as they respond to various etchants. If the same metal is employed, the leakage of the etchant through the PDMS will damage the ground layer during the modeling of the upper metallization.

In the simulations stage, the material properties of the upper metals are obtained from a Drude model to evaluate the surface impedance $Z_{\mathit{SR}}$ [20]. More details are given in [21] knowing that $Z_{\mathit{SR}}$ is expressed by:

$$Z_{\mathit{SR}}=\sqrt{\frac{j{\mathit{\omega \mu }}_0{\mu }_r}{{\sigma }_r+j{\mathit{\omega \epsilon }}_0}}\text{With}{\sigma }_r=\frac{{\sigma }_0}{1+\mathit{j\omega \tau }}$$E23

where angular frequency $\omega =2\mathit{\pi f}$, and f = frequency of the driving electric field, ${\mu }_0$ = permeability of free space, ${\mu }_r$ = relative permeability, ${\epsilon }_0$ = permittivity of free, ${\sigma }_0$ = intrinsic bulk conductivity at DC, and $\tau$ = phenomenological scattering relaxation time for the free electrons (i.e., mean time between collisions).

Except for the graphene model, as proposed in [22, 23], a single graphene layer can be modeled by a 2D surface conductivity. Other models (with a classical, semi-classical, and quantum mechanical treatment) are proposed to evaluate the surface impedance, as provided in [24]. We utilize the frequency-dependent conductivity of a monolayer of graphene based on Kubo’s formula [22, 23].

$$\sigma \left(\omega ,{\mu }_c,\mathrm{\Gamma },T\right)\approx -j\frac{q_e{k}_{B}T}{\pi {\mathrm{\hslash }}^2\left(\omega -2j\mathrm{\Gamma }\right)}\left(\frac{{\mu }_c}{k_B\mathrm{\Gamma }}+2ln\left(e^{-\frac{{\mu }_c}{k_B\mathrm{\Gamma }}}+1\right)\right)$$E24

Where $\omega$ is the angular frequency; ${\mu }_c$ is the chemical potential; $\mathrm{\Gamma }=1/2\tau$; $\tau$ is the transport relaxation time; T is the temperature; $k_B$ is the Boltzmann constant, and $\mathrm{\hslash }$ is the reduced Planck constant. It is important to note that only the intra-band term is considered in relation to Eq. (24), which gives good and exact results for frequencies limited to a few THz.

The implementation of surface impedance in the formulation of a simplified equivalent circuit is described in Eq. (34) and is detailed in the given articles [23, 25]. The terahertz antenna is studied based on graphene nanoribbon using the MoM-GEC formulation [23, 25]. For this purpose, we propose Figure 21, which depicts the radiated electric field for several z planes (near field). For a minimal value of Z, the electric field distribution is perturbed by the evanescence modes in the vicinity of the discontinuity surface, which gives information about the antenna structure. For a distance so far, it gives the fundamental mode of the waveguide.

Figure 21.

Normalized radiated electric field of the GNR antenna for different z-planes (planar monopole antenna).

7. Microwave diode modeling: from the circuit concept (U-I) to the field concept (E-H)

The diode structure can be described as an RCL boundary with two different impedances: field and localized [26, 27]. The definition of the lumped and field impedances are as follows:

$$Z_{\text{Lumped}}=\frac{U}{I},Z_{\text{Field}}=\frac{E_t}{H_t}$$E25

The circuit quantities for U and I may be expressed by the field components as shown below:

$$U=\underset{0}{\overset{l}{\int }}{\mathit{E}}_t\mathit{dl}\text{and}\oint {\mathit{H}}_t\mathit{dc}={\mathit{H}}_{t}w=I$$E26

By replacing the field quantities in the definition of the impedance, we obtain:

$$Z_{\text{Lumped}}=\frac{U}{I}=\frac{E\times l}{H\times w}=Z_{\text{field}}\frac{l}{w}\Rightarrow Z_{\text{field}}=\frac{w}{l}{Z}_{\text{Lumped}}$$E27

Most commercial software do not have a specialized program for the simulation of RLC components. Thus, the simulation is carried out by taking a set of boundaries in the desired impedance values. A component of an arbitrary surface is matched with an equivalent rectangular plate of the identical surface. The width of the equivalent rectangular surface boundary is:

$$w=\frac{S}{l}$$E28

where S is the sheet area. The relationship between the lumped and field impedances is

$$Z_{\text{lumped}}=Z_{\text{field}}\frac{l^2}{w}$$E29

The field values are given by $R_{\text{field}}=\mathit{Rl}/w$, $L_{\text{field}}=\mathit{Ll}/w$ and $C_{\text{field}}=\mathit{Cw}/l$ (Figure 22).

Figure 22.

Transform an arbitrarily shaped surface into an equivalent rectangular-shaped surface.

Generally, in the microwave domain, the particular case of PIN diodes is modeled by an equivalent electric circuit according to its polarization state (ON or OFF). The equivalent models of the PIN diode are shown in Figure 23. It is assumed that the frequencies considered are much higher than the transition frequency (in the direct regime) and the relaxation frequency (in the inverse regime).

Figure 23.

Equivalent circuit of the PIN diode: (a) direct regime with $f>f_T$ (b) inverse regime with (V = 0), (c) inverse regime $V>V_{\mathit{PT}}$ .

Let Z the impedance of the diode in the concept (U-I). It is defined as follows:

$$U=\mathit{ZI}\text{Such}\mathrm{as}Z=\left\{\begin{array}{c}{R}_s+R_{\mathit{HF}}+\mathit{jL\omega };\text{direc}\mathrm{t}\text{regime}\\ R_s+\frac{-j}{{C_{\mathit{min}}}_{\mathit{PT}}};\text{inverse}\text{regime};\left(V>V_{\mathit{PT}}\right)\end{array}\right.$$E30

In an attempt to transpose the concept (U-I) to the concept (E-J), the PIN diode was modeled by a surface impedance $Z_{\text{diode}}$ of width $W_D$ and height $d_D$. Let E and J be the field and current relative to $Z_{\text{diode}}$. The relationship between E and J is written as follows:

$$E=Z_{\text{diode}}J$$E31

Considering the TEM mode, the relationship from concept (U-I) to concept (E-J) is:

$$\left\{\begin{array}{l}\mathit{E}=\frac{U}{d_D}\\ \mathit{J}=\frac{I}{w_D}\end{array}\right.$$E32

Thus, we have: $\left\{\begin{array}{l}U=\mathit{ZI}\\ E=Z_{\text{diode}}J\end{array}\right.\Rightarrow \left\{\begin{array}{l}U=\mathit{ZI}\\ \frac{U}{d_D}=Z_{\text{diode}}\frac{I}{w_D}\end{array}\right.\Rightarrow Z_{\text{diode}}=\frac{U}{d_D}\frac{w_D}{I}$.

The relation between $Z_{\text{diode}}$and Z is written:

$$Z_{\mathit{\text{diode}}}=\frac{w_D}{d_D}Z=\left\{\begin{array}{l}\frac{w_D}{d_D}{R}_s+R_{\mathit{HF}}+\mathit{jL\omega };\mathit{\text{directregime}}\\ \frac{w_D}{d_D}{R}_s+\frac{-j}{C_{\min }\omega };\mathit{\text{inverseregime}}\left(V>V_{\mathit{PT}}\right)\end{array}\right.$$E33

Several examples of diode modeling by an integral method based on the MoM and multiscale methods are presented in the thesis of Sonia Mili [27]. In this context, we propose a simple descriptive structure of an elementary motif in the presence of a diode and its equivalent circuit (CEG), as given in Figure 24.

Figure 24.

An elementary motif in the presence of a diode and its equivalent circuit (CEG).

Figure 25 ures represent the distribution of the surface current and the field diffracted by the elementary motif when it is excited by the TEM mode. The current and the field check the boundary conditions on the metal and the dielectric well. The diode is also detected and characterized by a non-zero field and current compared to the metal platelets. The interface between the diode and each of the metal platelets is an important discontinuity translated at the level of the diffracted by an abrupt variation of important amplitude.

Figure 25.

Distribution of (a) the surface current and (b) the diffracted field on the elementary motif of active array with PIN diodes, f = 2,45 GHz, a = 10,2 mm, b = 22,9 mm, w = 0; 5 mm, diode ON.

We also note the existence of the Gibbs effect due to the large values of the field located at the interfaces between the PIN diodes and the surrounding metal platelets. This Gibbs effect reflects the difficulty of approximating such discontinuities by a finite series of continuous modes. Some standard and commercial RF PIN diodes utilized on metasurface designs and programmable antenna arrays are listed in [26].

8. Conclusions

Problems related to scaling in circuits can be solved by introducing auxiliary sources. The coherence of the approach has led to relatively simple models of auxiliary sources. These sources have already been used in the literature, particularly in diffraction, and have been applied here to study the insertion of localized elements in circuits. A simple example has highlighted the limits of the validity of using these sources and has defined a protocol for describing the influence of localized elements in electromagnetism. We have been led to introduce an equivalence relation that allows us to get away from the element’s geometry. This last hypothesis, if verified, justifies the intuitive approach, which consists in considering that one must be able to realize an exact electromagnetic calculation by assuming certain parts of the circuit (localized elements) by their “current–voltage” characteristics.

Notes/thanks/other declarations

The author thanks Mr. Henri Baudrand of the N7 Toulouse for his help.

The equivalent scheme of Figure 12 includes an excitation part using the formula of Eq. (6) (the upper modes are assumed evanescent) and an interface part, which is, in this case, a dipole. The global scheme is thus, after using the one of Figure 1, the following is:

By posing $Y_s={Z_s}^{-1}$, we obtain the relations (Figures 26 and 27):

Figure 26.

Equivalent scheme of the closure at z = 0.

Figure 27.

Equivalent scheme of the structure of Figure 12.

$$\left|\begin{array}{l}{\mathit{E}}_0\\ \mathit{J}\end{array}\right|=\left|\begin{array}{cc}0& 1\\ 1& {\widehat{Y}}_M+{\widehat{Y}}_S\end{array}\right|\left|\begin{array}{l}{\mathit{J}}_0\\ \mathit{E}\end{array}\right|$$E34

We can notice that ${\widehat{Y}}_S$ is a diagonal operator because the surface impedance is uniformly defined on the whole surface. Its annulment on the metal subdomain is indeed by the short circuit shown in Figure 1.

We define the scalar product as an integral over y between 0 and b of the product of functions:

$$\left.⟨f\right|g⟩=\underset{0}{\overset{b}{\int }}{f}_y{g}_y\mathit{dy}\left(f_x=g_x=0\right)$$E35

In this problem, the E and J fields have no components at x and are independent of x (Invariance by translation along O_x of the structure) so, the integral on the surface can be simplified into a y-integral. The problem is two-dimensional, leading to many simplifications: simple series in the developments, only one mode present, and a scalar formulation. We pose, in a slightly different way than in Eq. (14):

$${\mathit{J}}_0=\frac{1}{a}{{\mathit{\text{If}}}_0}^{\text{'}}\text{and}\left.⟨{f_0}^{\text{'}}\right|\mathit{E}⟩$$E36

${f_0}^{\text{'}}=1$ in this case, with the definition of the scalar product of Eq. (35). The unknown field in the aperture is expressed on a basis defined from 0 to c. We pose successively:

$$\mathit{E}\left(y\right)=\sum _{p=0}^{\infty }{x}_p{\mathit{g}}_p\left(y\right)$$E37

with${f_0}^{\text{'}}=1$, ${\mathit{g}}_0=\frac{1}{\sqrt{c}}$, ${\mathit{g}}_p\ne \sqrt{\frac{2}{c}}cos\left(\frac{\mathit{p\pi y}}{c}\right)$. The unknowns are ${\mathit{g}}_p$; by applying the Galerkin method, the first Equation is scalarly multiplied by ${f_0}^{\text{'}}$ and the second one successively by ${\mathit{g}}_0$, ${\mathit{g}}_1$, ${\mathit{g}}_2$ … ${\mathit{g}}_p$, p is the order of truncation of the test functions. This gives us:

$$V=\sum _{p=0}^P\left.⟨{f^{\text{'}}}_0\right|{\mathit{g}}_p⟩x_p$$E38

The other equations have a zero first member because J is zero in the virtual source domain and thus:

$$\left.⟨{\mathit{g}}_p\right|{f_0}^{\text{'}}⟩=0=-⟨{f_0}^{\text{'}}\left|{\mathit{g}}_p\right.⟩\frac{I}{a}+\sum _{q=0}^P\left.⟨{\mathit{g}}_p\right|\left({\widehat{Y}}_M+{\widehat{Y}}_s\right)\left|{\mathit{g}}_q\right.⟩x_q$$E39

Eliminating the unknown vector of $x_p$ components between Eq. (38) and Eq. (39), we find:

$$\frac{V}{I}=\frac{1}{a}{A}^t{\left[Y\right]}^{-1}A$$E40

with, $A=\left[\left.⟨{f_0}^{\text{'}}\right|{\mathit{g}}_1⟩,\left.⟨{f_0}^{\text{'}}\right|{\mathit{g}}_2⟩,\dots ,\left.⟨{f_0}^{\text{'}}\right|{\mathit{g}}_p⟩\right]$.

${\left[Y\right]}^{-1}$ is the inverse of the matrix and $Y$ is built on the operator ${\widehat{Y}}_M+{\widehat{Y}}_s$. The general term of $Y$ is written, in this definition:

$${\left[Y\right]}_{\mathit{pq}}=\left.⟨{\mathit{g}}_p\right|\left({\widehat{Y}}_M+{\widehat{Y}}_s\right)\left|{\mathit{g}}_q\right.⟩$$E41

with as in Eq. (6):

$${\widehat{Y}}_M=\sum _{n>0}{Y}_{\mathit{Mn}}{\widehat{P}}_n=\sum _{n>0}\left.⟨f_n\right|\left.Y_{\mathit{Mn}}\right|f_n⟩$$E42

We can find Eq. (40) with the help of auxiliary sources by representing the problem in Figure 12 differently: at the level of P, we consider a set of sources for each mode ${\mathit{g}}_p$ defined for $y<c$. The multipole Q represents the discontinuity between the guide of height b and that of height c; the sources ${\mathit{g}}_p$ will then be closed on the impedance $Z_s$. A classical modal connection studies the multipole Q. The conditions at z = 0 are written:

$$\left\{\begin{array}{l}\mathit{E}={\mathit{E}}^{\text{'}}\mathit{\text{with}}0<y<b\\ \mathit{J}+{\mathit{J}}^{\text{'}}=0\mathit{\text{with}}0<y<c\end{array}\right.$$E43

We deduce that, by scalar multiplication by $f_n$ of the first Equation and by ${\mathit{g}}_p$ of the second, the generalized transformer relations:

$$\left\{\begin{array}{l}{\mathit{E}}_n=\sum _{q=0}^P\left.⟨f_n\right|{\mathit{g}}_q⟩{{\mathit{E}}^{\text{'}}}_q\\ {{\mathit{J}}^{\text{'}}}_p=-\sum _{m=0}^{\infty }⟨{\mathit{g}}_p\left|f_m\right.⟩{\mathit{J}}_m\end{array}\right.$$E44

Using the closure relation on the mode admittances of evanescent modes $f_m$, (m > 0), the second relation of Eq. (44) combined with the first gives:

$$\left\{\begin{array}{l}{{\mathit{J}}^{\text{'}}}_p=\left.⟨{\mathit{g}}_p\right|f_0⟩{\mathit{J}}_0+\sum _{m>0}^{\infty }\left.⟨g_p\right|f_m⟩Y_{\mathit{Mn}}\left.⟨f_m\right|{\mathit{g}}_p⟩{{\mathit{E}}_q}^P\\ {\mathit{E}}_0=\sum _{q=0}^P\left.⟨f_0\right|{\mathit{g}}_q⟩{{\mathit{E}}^{\text{'}}}_q\end{array}\right.$$E45

We notice the similarity between Eq. (44) and Eq. (38), Eq. (41). The difference comes from the definition of I using a function ${f_0}^{\text{'}}$Eq. (38), whereas here, we use the fundamental mode of the guide.

Now posing the closure relation:

$${\mathit{J}}^{\text{'}}=-Y_s{\mathit{E}}_p^{\text{'}}$$E46

We find the expression of the impedance seen from the source:

$${\mathit{E}}_0=B^t\left[{\widehat{Y}}_M+{\widehat{Y}}_S\right]B{\mathit{J}}_0$$E47

With

$$B^t=\left|\left.⟨f_0\right|{\mathit{g}}_1⟩,\left.⟨f_0\right|{\mathit{g}}_2⟩,..\dots \dots ,\left.⟨f_0\right|{\mathit{g}}_p⟩\right|$$E48

Considering that $f_0=\frac{1}{\sqrt{\mathit{ab}}}$, (normed fundamental mode), that the scalar product of Eq. (48) concerns the integral in the whole right section, therefore is equal to that Eq. (35) multiplied by a. Finally, $V={\mathit{E}}_{0}b$ and $I={\mathit{J}}_{0}a$, we find exactly the expression of Eq. (40). In terms of auxiliary sources, we can say that there is an identity of treatment between the direct approach and the passage through the auxiliary sources provided that these are sufficiently numerous, each having the field distribution corresponding to the functions $g_p$. The formula of Eq. (40) is simple in the case of a single test function $g_0$. We can establish the expression of the input admittance and examine to what extent the introduction of a second test function modifies the result.

By taking:

$${\mathit{g}}_0=\frac{1}{\sqrt{c}};{\mathit{g}}_p\ne \sqrt{\frac{2}{c}}cos\left(\frac{\mathit{p\pi y}}{c}\right)$$E49

$$Y_{\mathit{Mn}}=\frac{j{\mathit{\omega \epsilon }}_0}{{\gamma }_n};{\gamma }_n=\sqrt{\frac{n^2{\pi }^2}{b^2}-k_0^2}$$E50

The excited evanescent modes are of type ${\mathit{TM}}_{0n}$. Hence, the expression of Eq. (44) is as follows.

$$Y=\frac{I}{V}=a\frac{\sum _{n>0}\frac{\mathit{j\omega \epsilon }}{{\gamma }_n}{\left|\left.⟨{\mathit{g}}_0\right|f_n⟩\right|}^2}{{\left|\left.⟨{\mathit{g}}_0\right|f_a^{\text{'}}⟩\right|}^2}+Y_s\frac{a}{c}$$E51

takes the form after development:

$$Y=2\mathit{j\omega \epsilon }\frac{a}{b}\sum \frac{1}{\sqrt{\frac{n^2{\pi }^2}{b^2}-k_0^2}}{\left(\frac{sin\left(\frac{\mathit{nc}}{b}\right)}{\frac{\mathit{nc}}{b}}\right)}^2+Y_s\frac{a}{c}$$E52

Assuming that the impedance is localized, b representing a quantity of the order of dimension of the box (lower however than the half-wavelength in the vacuum), we can admit that b is very large in front of c. By posing:

$$x=n\frac{c}{b};\mathit{dx}=\frac{c}{b}$$E53

x is a practically continuous variable, and the series that appears in Eq. (52) turns into an integral. Using the approximations in Eq. (53), we have:

$$Y=Y_s\frac{a}{c}+2j\sqrt{\frac{\epsilon }{\mu }}\underset{\frac{c}{b}}{\overset{\infty }{\int }}\frac{{\mathit{ak}}_0}{\sqrt{{\pi }^2{x}^2-K_0^2{c}^2}}\frac{{sin}^2\left(x\right)}{x^2}\mathit{dx}$$E54

When c tends to zero, we see that the integral of Eq. (54) diverges as $log\left(K_{0}c\right)$, while the purely surface admittance part can be kept constant by making $Y_s$ tend to zero proportionally to c; the capacitive part placed in parallel tends towards infinity, which forbids to consider an impedance of zero dimension.

If we desire to consider the edge effects, it is necessary to introduce other test functions $g_p$. This gives an idea of the precision of the auxiliary source concept. For example, considering two test functions, we find for the input admittance [according to Eq. (40)]:

$$Y^{\text{'}}=a\frac{Y_{11}\left(1-\frac{Y_{11}^2}{Y_{11}Y}\right)}{{\left|\left.⟨g\right|f_0^{\text{'}}⟩\right|}^2}+Y_s\frac{a}{c}$$E55

The relative degree of accuracy of the capacitive part is given by the term:

$$\frac{\partial C}{C}=\frac{-Y_{12}^2}{Y_{11}{Y}_{22}}\text{With}\left\{\begin{array}{l}{Y}_{11}=\sum _{n>0}\frac{\mathit{j\omega \epsilon }}{{\gamma }_n}{\left|\left.⟨{\mathit{g}}_0\right|f_n⟩\right|}^2\\ Y_{22}=\sum _{n>0}\frac{\mathit{j\omega \epsilon }}{{\gamma }_n}{\left|\left.⟨{\mathit{g}}_1\right|f_n⟩\right|}^2\\ Y_{11}=\sum _{n>0}\frac{\mathit{j\omega \epsilon }}{{\gamma }_n}\left.⟨{\mathit{g}}_0\right|f_n⟩\left.⟨f_n\right|{\mathit{g}}_0⟩\end{array}\right.$$E56

From these expressions, we can see that the accuracy depends essentially on the ratio $c/b$, as a first approximation, the relative error is proportional to ${\left(c/b\right)}^2$; we can therefore admit that the auxiliary sources are a good approximation for dimensions between one-tenth and one-hundredth of the dimensions of the case.