**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:

$$Z\_{\text{Lumped}} = \frac{U}{I}, Z\_{\text{Field}} = \frac{E\_t}{H\_t} \tag{25}$$

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

**Figure 21.** *Normalized radiated electric field of the GNR antenna for different z-planes (planar monopole antenna).*

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

$$\mathbf{U} = \int\_{l} \mathbf{E}\_{l} dl \text{ and} \oint \mathbf{H}\_{l} dc = \mathbf{H}\_{l} w = I \tag{26}$$

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

$$Z\_{\text{Lumped}} = \frac{U}{I} = \frac{E \times l}{H \times w} = Z\_{\text{field}} \frac{l}{w} \Rightarrow Z\_{\text{field}} = \frac{w}{l} Z\_{\text{Lumped}} \tag{27}$$

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:

$$w = \frac{\mathcal{S}}{l} \tag{28}$$

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

$$Z\_{\text{lumped}} = Z\_{\text{field}} \frac{l^2}{w} \tag{29}$$

The field values are given by *R*field ¼ *Rl=w*, *L*field ¼ *Ll=w* and *C*field ¼ *Cw=l* (**Figure 22**).

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

#### **Figure 23.**

*Equivalent circuit of the PIN diode: (a) direct regime with f* > *f <sup>T</sup> (b) inverse regime with (*V *= 0), (c) inverse regime V* >*VPT .*

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

$$U = ZI \text{ such as } Z = \begin{cases} R\_t + R\_{HF} + jLw; \text{direct regime} \\ R\_t + \frac{-j}{C\_{minPT}}; \text{ inverse regime}; \ (V > V\_{PT}) \end{cases} \tag{30}$$

In an attempt to transpose the concept (*U*-*I*) to the concept (*E*-*J*), the PIN diode was modeled by a surface impedance *Z*diode of width *WD* and height *dD*. Let *E* and *J* be the field and current relative to *Z*diode. The relationship between *E* and *J* is written as follows:

$$E = Z\_{\text{dide}} J \tag{31}$$

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

$$\begin{cases} \mathbf{E} = \frac{U}{d\_D} \\ J = \frac{I}{w\_D} \end{cases} \tag{32}$$

Thus, we have: *U* ¼ *ZI E* ¼ *Z*diode*J* � ) *U* ¼ *ZI U dD* ¼ *Z*diode *I wD* 8 < : ) *<sup>Z</sup>*diode <sup>¼</sup> *<sup>U</sup> dD wD I* .

The relation between *Z*diodeand *Z* is written:

$$Z\_{\text{diode}} = \frac{w\_D}{d\_D} Z = \begin{cases} \frac{w\_D}{d\_D} R\_s + R\_{\text{HF}} + jLov; \text{direction} \\\\ \frac{w\_D}{d\_D} R\_s + \frac{-j}{C\_{\text{min}}o}; \text{inverseregime} (V > V\_{PT}) \end{cases} \tag{33}$$

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 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

**Figure 24.** *An elementary motif in the presence of a diode and its equivalent circuit (CEG).*

**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.*

an important discontinuity translated at the level of the diffracted by an abrupt variation of important amplitude.

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].
