**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:

$$Z\_0 = Z\_s \frac{c}{a} \tag{15}$$

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*<sup>0</sup> 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*<sup>0</sup> by an electromagnetic calculation. Using the results obtained in Eq. (15), we can write the measured impedance *ZM* as:

$$Z\_M = \frac{1}{j\alpha \mathcal{C}\_0(a\_0, b\_0, c\_0) + \frac{1}{Z\_0}}\tag{16}$$

Hence, the expression of *Z*0:

$$Z\_0 = \frac{1}{-j\alpha C\_0 + \frac{1}{Z\_M}}\tag{17}$$

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

$$Z\_s = Z\_0 \frac{d}{c} \tag{18}$$

We deduce the input impedance *ZE*

$$Z\_E = \frac{1}{j\alpha C(a, b, c) + \frac{1}{Z\_M}} = \frac{1}{j\alpha (C - C\_0) + \frac{1}{Z\_M}} \tag{19}$$

There is a relationship between the intrinsic impedance *Z*<sup>0</sup> and the input impedance (or the measured impedance), a homographic relation represented by a coupling quadrupole *QE* or *QM* (**Figure 13**).

*Z*0, in the case considered here, is made with a surface impedance. As shown in Eq. (17), several values of *Zs*, *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*<sup>0</sup> , *c*<sup>0</sup> (**Figure 17**), placed in a device similar to that of **Figure 12**, we see that the surface

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

**Figure 17.** *Equivalent admittances.*

impedances *Zs* and *Zs* 0 chosen so that the impedances are equivalent and roughly proportional to *a* and*a*<sup>0</sup> , according to formula of Eq. (17).

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 *Zs* <sup>0</sup> to have a capacitive part to compensate the effect of this inductance.

A common case concerns active or passive dipoles [14–18]. 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.

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*<sup>0</sup> 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*<sup>0</sup> 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*<sup>2</sup> themselves closed on *Z*0.

The process is led in three stages:


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*<sup>2</sup> be the dimensions of *S*1, *S*2. We have successively:

$$V\_{GS} = E\_1 b\_1 \text{ and } V\_{DS} = E\_1 b\_1 + E\_1 b\_1$$

**Figure 19.** *Transistor in a measuring device (a) and inserted in a circuit (b).* *Multiscale Auxiliary Sources for Modeling Microwave Components DOI: http://dx.doi.org/10.5772/intechopen.102795*

$$I\_{\rm DS} = I\_2 a\_2 \text{ and } I\_{\rm GS} + I\_{\rm DS} = I\_1 a\_1. \tag{20}$$

We deduce the relation between *ZM* and *Z*0:

$$
\begin{vmatrix} E\_1 \\ E\_2 \end{vmatrix} = Z\_0 \begin{vmatrix} J\_1 \\ J\_2 \end{vmatrix}, \begin{vmatrix} V\_{GS} \\ V\_{DS} \end{vmatrix} = Z\_M \begin{vmatrix} I\_{GS} \\ I\_{DS} \end{vmatrix} \tag{21}
$$

$$Z\_0 = \begin{vmatrix} \mathbf{1} & \mathbf{0} \\ \hline b\_1 & \mathbf{0} \\ -\mathbf{1} & \mathbf{1} \\ -\frac{\mathbf{1}}{b\_2} & \frac{\mathbf{1}}{b\_2} \end{vmatrix} Z\_0 \begin{vmatrix} a\_1 & -a\_2 \\ \mathbf{0} & a\_2 \end{vmatrix} \tag{22}$$

Thus, access to *Z*<sup>0</sup> is simplified and does not require any electromagnetic calculation. This will be the case when the capacities at the terminals of *S*<sup>1</sup> and *S*<sup>2</sup> are negligible. It is enough for this that the sources *S*<sup>1</sup> and *S*<sup>2</sup> 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.
