**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 [9–11].

#### **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 <sup>n</sup>*. The fundamental mode *f* <sup>0</sup>, the transverse magnetic field, is given by:

$$J\_0 = \mathbf{H}\_0 \times \hat{n}\_0 = I\_0 f\_{\;0} \tag{1}$$

*n*^<sup>0</sup> 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**.

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

$$J\_n = H\_n \times \hat{n} = -Y\_{M\_n} E\_n \tag{2}$$

*YMn* is the mode *n* admittance that is equal, respectively, to *γn=jωμ* and *jωε=γ<sup>n</sup>* for TE and TM modes [11]. The minus sign comes from the orientation *n*^0. In Eq. (2), *J<sup>n</sup>* and *E<sup>n</sup>* are the transverse current and fields with a rotation of *π=*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:

$$
\hat{P}\_{\mathfrak{n}} f = f\_n \langle f | f\_n \rangle \tag{3}
$$

*<sup>P</sup>*^*<sup>n</sup>* is also written *<sup>P</sup>*^*<sup>n</sup>* <sup>¼</sup> *<sup>f</sup> <sup>n</sup>* � � � *<sup>f</sup> <sup>n</sup>* � � �: The scalar product is commonly used. It is written as

$$\langle f\_n \vert f \rangle = ^{\text{Definition}} \int\_{\mathcal{S}} f\_n^{\*\text{-}t} f d\mathcal{S} \tag{4}$$

**Figure 1.** *Source model in a homogenous guide.*

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

$$J = -\sum\_{n>0} Y\_{M\_n} \hat{P}\_n \mathbf{E} + \mathbf{J}\_0 \tag{5}$$

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

$$J = -\hat{Y}\_{M\_\pi} \mathbf{E} + \mathbf{J}\_0 \tag{6}$$

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 *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. 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*<sup>1</sup> and *P*<sup>2</sup> 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 *Js*, 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

**Figure 2.** *Excitation of a line by a coaxial.*

**Figure 3.** *Simplified representation of a coaxial excitation.*

choices do not affect the result of the study of a circuit, i.e., the impedance seen from the source.

Assuming that the input *E* of the coaxial is connected to an internal impedance generator *Z*<sup>0</sup> and providing a voltage *V*0, we obtain the relation between *E*<sup>0</sup> and *Js*.

$$E\_0 b = V = V\_0 - Z\_0 I = V\_0 + Z\_0 \mathbf{j}\_r a \tag{7}$$

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 <sup>s</sup>* 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*<sup>1</sup> and *P*2. As before, we write by convention:

$$\begin{aligned} \mathbf{J}\_1 &= \mathbf{H}\_1 \times \hat{n}\_1 \\ \mathbf{J}\_2 &= \mathbf{H}\_2 \times \hat{n}\_2 \end{aligned} \tag{8}$$

*n*^<sup>1</sup> and *n*^<sup>2</sup> are the unit normal outgoing ones taking into account the general conditions of continuity:

$$\mathbf{H}\_1 - \mathbf{H}\_2 = \mathbf{j}\_s \times \hat{n}\_{21} \tag{9}$$

We deduce the relations between *j <sup>s</sup>* and *J*1, *J*2. Vectorically multiplying Eq. (9) by *n*^<sup>21</sup> = *n*^1, it comes:

$$J\_1 + J\_2 = -j\_s \tag{10}$$

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

$$\mathbf{E}\_1 = \mathbf{E}\_2 = \frac{V\_0}{b} - Z\_0 \frac{a}{b} (\mathbf{J}\_1 + \mathbf{J}\_2) \tag{11}$$

#### *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*<sup>0</sup> conducting elements, and an impedance surface *S*<sup>1</sup> (**Figure 5**).

An auxiliary source's analysis consists of simply replacing *S*<sup>1</sup> by a source, calculating the coupling quadripole *Q* between *S*<sup>0</sup> 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 *E*0) to model "localized elements" in planar structures with a method of moments is given in **Figure 6**. **Figure 7** shows the reflection

**Figure 5.** *Circuit comprising main source and an auxiliary source.* *Multiscale Auxiliary Sources for Modeling Microwave Components DOI: http://dx.doi.org/10.5772/intechopen.102795*

**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*<sup>11</sup> *seen from the main excitation source with* l *= 6 mm, ε<sup>r</sup> = 9.8, and* h *= 1.35 mm.*

coefficient *S*<sup>11</sup> seen from the main excitation source versus frequency for different values of the surface impedance *Zs*.

The simulation results corresponding to low values of *Zs* (less than 5 Ω, for example) are identical to those obtained for a microstrip line without surface impedance. In this range of *Zs* 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* ¼ *l=*2 for example, the second resonance frequency of the line disappears, which reappears for a distance *D* ¼ *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*<sup>1</sup> 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*<sup>1</sup> 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*<sup>1</sup> 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*<sup>1</sup> 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:

$$\operatorname{rot}(\mathbf{E}\_1) = \mathbf{0}; \operatorname{div}(\mathbf{j}\_s) = \mathbf{0} \tag{12}$$

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 Ω<sup>1</sup> and the other internal Ω<sup>2</sup> 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*<sup>11</sup> *seen by the main excitation source that varies against frequency with* l *= 6 mm, ε<sup>r</sup> = 9.8, and* h *= 1.35 mm.*

*Multiscale Auxiliary Sources for Modeling Microwave Components DOI: http://dx.doi.org/10.5772/intechopen.102795*

**Figure 9.**

*Decomposition of a circuit into external and internal domains.*

**Figure 10.** *Placement of auxiliary sources.*

In the vicinity of S exist two surfaces *S*1, *S*2; one in Ω<sup>1</sup> and the other in Ω2. The disconnection operation between Ω<sup>1</sup> and Ω<sup>2</sup> is done in two steps (**Figure 10**):


For a planar source, the magnetic wall prohibits, for an electric field source, any component normal to the source (**Figure 11**). Thus *E<sup>z</sup>* ¼ 0 from which: divsð Þ¼ *E* 0 . So, (*<sup>∂</sup>Ex <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *<sup>∂</sup>Ey <sup>∂</sup><sup>y</sup>* ¼ 0) and *E* ¼ �gradð Þ*v* (quasistatic hypothesis).

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, *E*<sup>0</sup> is known, and after electromagnetic calculations, the surface current distribution *J<sup>s</sup>* is obtained.

**Figure 11.** *Auxiliary planar source.*

By posing *E*<sup>0</sup> ¼ *Vf* <sup>0</sup>, we deduce:

$$I = \left< f\_0 | \mathcal{J}\_s \right>\tag{13}$$

where *f* <sup>0</sup> 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:

$$J = \text{If }\_{0}^{\prime}{}^{\prime} \text{ and } V = \left\langle \left. f\_{0}^{\prime} \right| \mathbf{E} \right\rangle \tag{14}$$

In this case, *f* <sup>0</sup> <sup>0</sup> 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 *j <sup>s</sup>* 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 *j <sup>s</sup>* 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 *j <sup>s</sup>* is the tangential electric field E and vice versa. The equivalent schema of an interface with a virtual surface current source *j <sup>s</sup>* and the main field source *E*<sup>0</sup> 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.

The working domain consists of three subdomains: the source domain *DS*, the metallic domain *DM*, and the dielectric domain *DD* (**Figure 13**).

Since the virtual source overlaps the source domain, and the dual quantity *j s* , i.e., the electric field, is annulled in the latter, we deduce the equivalent schema of the interface in *DS* and *DM* (*E*0) is zero on the metal, so in *DM* 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

*Multiscale Auxiliary Sources for Modeling Microwave Components DOI: http://dx.doi.org/10.5772/intechopen.102795*

**Figure 12.** *Representation of an interface with a field source and a perfect conductor.*

**Figure 13.** *Interface made of metal, dielectric, and a source.*

hold a scheme similar to the one in **Figure 1** described by condition (6) (that assuming *J*<sup>0</sup> ¼ 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.
