### Section 2: First Attempt at Improving the Quarter-Wave Transformer

In the previous lesson, we simulated a quarter-wave transformer. The quarter-wave transformer minimized the return loss at particular frequencies; in particular, the circuit had good performance over a 40% bandwidth (defined as better than 20dB return loss). We would like to improve this bandwidth.

Consider the equation for return loss:

We can make the return loss smaller if the transition is smaller. The circuit in Figure 4.1 breaks the 90º line length of the transmission line “T1” into two 45º transmission lines, “T2” and “T3”. Therefore, the total length is the same, but rather than one 70.7Ώ line we will gradually increase the impedance. Therefore, we will go from the 50Ώ source impedance to a 65Ώ transmission line, to an 85Ώ transmission line, to a 100Ώ transmission line.

In order to make the circuit in Figure 4.1, you will need the variables and calculation blocks which can be found the the “mason” library, along with the port, flag, and frequency blocks from before.

Simulating this circuit, we find that the performance of the two transmission line circuit does have a wider bandwidth... but that bandwidth is not centered at our design frequency. In particular, the circuit has an 80% bandwidth but that is centered at 2GHz, twice our design frequency. We note that the individual transmission lines of “T2” and “T3” are each half the electrical length, which means that they are effective at twice the frequency of “T1”.

**An important (and useful) tip: if
you replace the “Ref” of the optimization block with “_NoOpt1”
or the statistical block with “_NoStat1” (or even a variable
block with “_NoVar1”, if you're getting tricksy), you turn off
that block. Useful for toggling between optimization runs and single
runs.**

**Main
Point**

Adding more matching elements does improve the bandwidth of the matching performance, however it appears that we might have to increase the physical size of the circuit to cover the band of interest.

Questions:

4.1.1 Can you improve the values by hand?

4.1.2 If
the line is 90° at 1GHz, how many degrees is the line at 3GHz? What
relationship can we see? Why do we see the second null for the S22
at 3GHz? Where would we predict the third null?