Linearized active circuits:
transfer functions and stability

3. Realistic models for components

We will now provide some examples of component models that are realistic. We start discussing passive components, where the key to being realistic is to have at least some losses. We also show that lossy transmission lines are realistic. Finally, we discuss active components, like diodes and transistors.

3.1 Passive components

To rule out unrealistic behaviors of passive components, it is natural to select lossy multiports that involve perfect short or open circuits at no frequency. This is the object of the following definition which is compatible with the formalism developed in [25]:

A passive multiport is said to be very strictly passive if its complex impedance matrix $Z\!\left(s\right)$ and its admittance $Y\!\left(s\right)=Z\!\left(s\right)^{-1}$ are meromorphic on $\mathbb{C}$ and satisfy, for $\Re(s)\geq0$ :

$(i)$ $Y\!\left(s\right)+Y^{*}\!\left(s\right)\succeq\alpha\,\mathbf{Id}$ for some $\alpha>0$ ,

$(ii)$ $Z\!\left(s\right)+Z^{*}\!\left(s\right)\succeq\beta\,\mathbf{Id}$ for some $\beta>0$ .

A very strictly passive multiport is clearly realistic. Condition $(ii)$ is a frequency domain version of so-called strict input passivity of the system with transfer function $Z$ , while (2.7) expresses its strict output passivity, see [26]. In the literature, a rational function satisfying $(i)$ is called strongly strictly positive real [27]. Note that the strict input passivity of the system with transfer function $Z$ implies the strict output passivity of the system with transfer function $Y$ . It should be noticed that, contrary to the rational case, meromorphic functions may have no limit at infinity.

3.2 Lumped passive components

Most passive components in a design, like resistors, capacitors, inductors and transformers, are modeled as an interconnection of several ideal lumped components. The obtained model is therefore a rational model and will be meromorphic on the whole complex plane.

It should be noted that this definition considers pure inductors and capacitors as unrealistic, since their impedance is unbounded on the imaginary axis. A realistic inductor can be obtained by connecting an ideal inductor in parallel with a large resistance $R$ , and adding a small series resistance $r$ as shown in Figure Figure 3.1. We may think of $R$ , e.g. as being the resistance of the air around the inductor, and of $r$ as the resistance of the conductor constitutive of the element. Similar considerations apply to capacitors.

Figure 3.1 Realistic inductors and capacitors are lossy. A realistic capacitor or inductor model will therefore contain losses modeled as series and parallel resistors.

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