3.1. Checking the stability

First of all, the stability of the transistor needs to be checked. To do that, two conditions have to be verified, the first condition is that the stability coefficient being greater than one (K > 1), where:

(1)

The second condition is that the module of the determinant of scattering matrix (S) being lower than 1 (|∆S| <1), where:

(2)

All the different S parameters, S₁₁, S₁₂, S₂₁, S₂₂, as well as, the stability coefficient K and the determinant ∆_S are computed using MWO.

To check the stability it is also useful to have the information provided by the scattering matrix obtained from the transistor MESFET (FET2_1 in simulation). The scattering matrix of the transistor and the different graphs are obtained through MWO. The schematic needed to compute both the matrix S and the determinant is the shown in Fig. 1, where the 50 ohms impedance on both input and output corresponds to the requested values of the input and output matching networks. The schematic shown in the Figure 1, is a starting point of the design and it is suggested in some literature as in (^{Pirola, Teppati, & Camarchia, 2007}) and (Camarchia, Teppati, Corbellini, & Pirola, 2007).

Fig. 1: Schematic to check the stability (scattering parameters FET2_1) 

Both, K and |∆S| fulfill with the conditions mentioned above, and their different values in a region of analysis from 5 GHz to 25 GHz are shown in Figure 2.

Fig. 2: K and |∆S| to check the stability 

3.2. Transistor Gain

In order to know the behavior of the active element (transistor FET2), it is also suggested to check the power gains, the MAG Maximum Available Gain (the maximum gain GMax), and |S₂₁|, which is the gain when the device is close on both ports with the normalization impedance (50 Ω). Both gains in dB are shown in the Figure 3.

Fig. 3: GMAX and S₂₁ of Transistor FET2 

In Figure 3, it can be noted that the device behavior is linear (stability condition) in terms of gain. Now, based on the analysis from Fig. 2 (K > 1 and |∆S| <1) and observing the Fig 3., it is noted that when the device is working in a range of frequencies between 5 GHz and 25 GHz, and of course in 14 GHz, it has a behavior “unconditionally stable”. In addition, Fig. 3 shows that the maximum device gain at 14 GHz is Gmax= 9.206 dB, and |S₂₁|= 4.7161 dB, i.e. S₂₁ < Gmax; at end of design it is expected to get Gmax = S₂₁ at 14 GHz. The maximum power transfer occurs when the generator yields the maximum power, i.e. the input power is the generator (input) available power and the power on the load is the output available power. The maximum power transfer implies power impedance matching simultaneously at the input and output (if this is possible). The maximum operational gain is:

(3)

Furthermore, to completely understand the behavior of the device, it is convenient to plot the constant gain circles and the stability circles. According to the gain circles, GOP circle (GOP: operating gain - Gamma L) is respect to output, and GAV circle (GAV: available gain - Gamma G) is respect to input, see Figure 4 (^{Teppati, Ferrero, Pisani, Colantonio & Limiti, 2005}).

Fig. 4: Constant gain circle Γ_G and Constant gain circle Γ_L, f= 14 GHz 

As it is expected, the stability circles (SCIR 1 and SCIR 2) do not appear since the element (transistor) is “unconditionally stable”. Meanwhile in the analysis of the gain circles, it is verified that there is a maximum in the plane ΓG for GAV and a maximum in ΓL for GOP. These maximum values correspond to optimum values ΓG opt. and ΓL opt., It can be noted that MWO is able to directly compute the value of ΓG opt, and ΓL opt.. MWO define these parameters as GM1 (ΓG opt) and GM2 (ΓL opt.). In Table 4, it can be noted the complex values of ΓG opt. and ΓL opt.; these values of GM1 and GM2 expressed in magnitude and angle will be very useful for the design of the matching networks as it will be shown later.

Table 4 GM1 and GM2 at 14 GHz. 

GM1 and GM2 indicate that working with the current active device in a given frequency (e.g. 14 GHz), and considering 50 Ω of input and output impedance, there are only two passive loads, one at the input port (GM1), and the other at the output port (GM2) that ensure the simultaneous match at the two ports of active device (MESFET), hence now the task consists in designing two lossless matching networks.

3.3. Design considerations and simultaneous matching networks

Considering that the design consists of making an amplifier that works with maximum gain, it is necessary to design the corresponding lossless matching networks, remembering that a two-port unconditionally stable system can be simultaneously matched at both ports. In this case, all the power will be transferred from generator (maximum power) to the load ZL, i.e. GOP = GAV = GMAX; in other words, when the device sees GM1 at the input port, and GM2 at the output port, it achieves the maximum gain GMAX (i.e. |S₂₁| = Gmax).

Considering the active device (transistor) and the input and output matching network, the general scheme of the desired amplifier is shown below in Figure 5, (Camarchia, Teppati, Corbellini, & ^{Pirola, 2007}).

Fig. 5: RF Amplifier with simultaneous matching networks 

In order to implement the input and output matching networks, it is necessary to work with conjugate values of GM1 and GM2, so as to eliminate the imaginary parts of GM1 and GM2, as it is show in Table 5 and in Figure 6, i.e. the matching network is a conjugate matching of an active two-port device.

Table 5 GM1 and GM2 conjugate. 

Fig. 6: Specifications GM1* and GM2*, matching networks 

When the matching condition is satisfied, S₁₁ and S₂₂ of the amplifier will be zero and |S₂₁| = GMAX. Now, all the considerations to design the matching networks are available, but before, some parameters in MWO have to be configured, as it is done below.

3.4. Design considerations in MWO

The design must be implemented using microstrip lines, whose parameters (substrate and metal) are described in Table 1 and shown in Figure 7.

Fig. 7: Design parameter microstrip lines 

Remembering that resistivity of the metallization (Rho) is normalized to gold, for copper its value is:

(4)

The TXLINE tool of MWO (Figure 8) will be used with the amplifier data to automatically get the width and the physical length for the transmission line. In particular, for this design we considered a fixed width, while the parameter that could change is the physical length.

Fig. 8: Calculation of width and initial length of the microstrip lines using TXLINE TOOL 

3.5. Matching networks: Input matching network

In the design of the input and output matching networks we considered not only the microstrip lines, but also a serial short circuit stub, i.e. the input and the output are composed by the transmission lines and the stubs (^{Raab, Asbeck, Cripps, Kenington, Popović, Pothecary & Sokal, 2002}).

The desired values of resistance and reactance are obtained with microstrip lines and short-circuit stubs. The aim at the moment is to couple the input and output networks by modifying certain parameter, in this case the physical length of the line and stub. The modification and all necessary changes in the length do not matter because establishing the goals and using optimizer all changes are performed automatically and then these can be tested using Smith chart or other kind of graphs. In general the scheme of the amplifier looks like as the shown in the Figure 9.

Fig. 9: Amplifier diagram included short circuit stubs 

For the design,the width of the transmission line has been considered as a fixed parameter (W= 1585 um) and the variable parameter is the physical length (TL2 and TL4 for the input and the output network, see Fig.9). By means of the optimizer tool of MWO, the best dimensions of the length to satisfy the goals and needs is obtained automatically. The first step is to set the goal for the matching network, in this case the goal is to minimize the S₁₁ (reflection coefficient) in order to transfer the maximum power from the generator toward the load, so, it is set to |S₁₁| < -30 dB and in a range of frequencies between 13.8 GHz and 14.2 GHz (central frequency at 14 GHz), in particular narrow band design has been considered because it is less complex.

Once the goal is established, we have to select the parameters to be optimized (variables). For the input matching network, the physical length of TL2 and TL4 are the parameters to be optimized, then the optimizer is run with the desired values according to the established goal. Furthermore, the size of the elements has been constrained to avoid negative values (Lower=100 um, Upper=9000 um).

Finally, the input matching network (input filter) is obtained using all the elements and dimensions. Its scheme is shown in Figure 10.

Fig. 10: Input matching Network 

Once the input network has been designed, its operation has to be checked. The S₁₁ parameter in the Smith chart and in a rectangular graph give enough details of the behavior of the input network and the fulfillment of the goal (|S₁₁| <-30 dB), see Figure 11.

Fig. 11: Input reflection coefficient in Smith Chart and |S11| dB in rectangular graph (f=14 GHz) 

Analyzing Fig. 11, it can be noted that the design of the input matching network has been completed successfully, since |S11| dB is lower than the fixed goal (|S11| = -32.97 dB) and the reflection coefficient is 0 (real= -0.01664, imaginary=-0.015).

3.6. Matching networks: Output matching network

As before in this case, the goal is |S11| < -30 dB, and the variables to optimize are TL2 and TL4. Following the same procedures as in the input matching networks, the results obtained are shown below, see Figure 12.

Fig. 12: Output matching Network 

Fig. 13: Output reflection coefficient and |S11| dB in rectangular graph (f=14 GHz) 

Analyzing Figure 12 and Figure 13, it can be concluded that the design of the output matching network has been completed successfully, since |S11| dB is lower than the fixed goal (|S11| = -47.838504 dB) and the reflection coefficient is 0 (real= -0.0006447, imaginary=-0.004004).

4.1. Layout of the amplifier

Having designed the amplifier, is very easy to get its layout. MWO gives the possibility to plot 2-D or 3-D layouts, with the option “layout”. Taking in mind that the amplifier contains an active device (transistor), it is necessary to consider a power supply in the design (BIAS), and capacitors in the input and in the output to block the DC component (^{Camarchia, Moreno, Pirola, Quaglia & Wegeland, 2013}), so the final implementation of the amplifier looks like the shown in Fig. 16, (^{Kawai, Ohta & Enokihara, 2010}).

Fig. 16: RF amplifier considering a power supply 

In order to obtain the final layout, the scheme will change a little due to the association of the different elements (transistor, capacitors). The bias capacitor value was calculated based on the consideration about the work frequency, taking into account that the capacitor should act as a short circuit at 14 GHz of frequency, and that:

(5)

The value of the capacitor is 22 pF and the amplifier with all additional elements is the shown in the Figure 17, it is shown a layout in 2-D and 3-D.

Fig. 17: Amplifier layout 2-D and 3-D