**3.2 Mathematical model of the amplifier**

An Amplifier is an electronic device whose role is to increase the power of a signal introduced at its input. To simplify our study we will admit that our amplifier is a quadrupole whose matrix is [S] and connected to a voltage source E with an internal impedance and it is loaded by an impedance (**Figure 4**).

**Figure 4.** *Distribution waves and reflection coefficients at the entry and exit of the quadrupole.*

The matrix [S] of distribution of this linear quadrupole is such that [9]:

$$[b] = [\mathbf{S}][a] \Rightarrow \begin{pmatrix} b\_1 \\ b\_2 \end{pmatrix} = \begin{pmatrix} \mathbf{S}\_{11}\mathbf{S}\_{12} \\ \mathbf{S}\_{21}\mathbf{S}\_{22} \end{pmatrix} \begin{pmatrix} a\_1 \\ a\_2 \end{pmatrix}$$

$$\begin{cases} b\_1 = \mathbf{S}\_{11}a\_1 + \mathbf{S}\_{12}a\_2 \\ b\_2 = \mathbf{S}\_{21}a\_1 + \mathbf{S}\_{22}a\_2 \end{cases} \tag{10}$$

**Z2:** is the output impedance of the quadrupole fed by the Zs impedance source. **a1**: is the incident wave at port 1;

**b1**: is the wave reflected at the access;

**a2**: is the incident wave at port 2;

**b2**: is the wave reflected at port 2.

When making the amplifier, we try to have maximum gain. In other words, it is necessary to perform the adaptation at the input of the transistor and the source simultaneously with the adaptation between the transistor output and the load.